Simple, elegant models developed in the 1960s with 2 important assumptions:

Only systematic risk matters, idiosyncratic risk is not priced (Sharpe 1964)

Systematc risk = co-variance with market factor(s) (i.e. β)

Idiosyncratic risk can always be diversified away

Efficient Market Hypothesis (EMH) (Samuelson 1965)

All new information is reflected in prices immediately, entirely and correctly

No arbitrage opportunities, no value to private information

Both assumptions have been questioned early on (Lintner 1965b; Simon 1955). Debate long remained academic only since mainstream models performed well for investors in 20th century

Several related linear factor models

Capital Asset Pricing Model (Sharpe 1964 & Lintner 1965a)

Intertemporal CAPM (Merton 1973)

Arbitrage Pricing Theory (Ross 1976)

3-Factor model (Fama & French 1992)

5-Factor model (Fama & French 2015)

Literature strand 1 – summary Mainstream Asset Valuation

Idiosyncratic Risk and the Cross Section of Stock

Returns

A thesis submitted for the degree of Doctor of Philosophy

by

Stanislav Bozhkov

Brunel Business School

Brunel University

November 2017

Abstract

A key prediction of the Capital Asset Pricing Model (CAPM) is that idiosyncratic risk

is not priced by investors because in the absence of frictions it can be fully diversified away.

In the presence of constraints on diversification, refinements of the CAPM conclude that the

part of idiosyncratic risk that is not diversified should be priced. Recent empirical studies

yielded mixed evidence with some studies finding positive correlation between idiosyncratic

risk and stock returns, while other studies reported none or even negative correlation. In this

thesis we revisit the problem whether idiosyncratic risk is priced by the stock market and what

the probable causes for the mixed evidence produced by other studies, using monthly data for

the US market covering the period from 1980 until 2013.

We find that one-period volatility forecasts are not significantly correlated with stock

returns. On the other hand, the mean-reverting unconditional volatility is a robust predictor of

returns. Consistent with economic theory, the size of the premium depends on the degree of

‘knowledge’ of the security among market participants. In particular, the premium for

Nasdaq-traded stocks is higher than that for NYSE and Amex stocks. We also find stronger

correlation between idiosyncratic risk and returns during recessions, which may suggest

interaction of risk premium with decreased risk tolerance or other investment considerations

like flight to safety or liquidity requirements. The difference between the correlations between

the idiosyncratic volatility estimators used by other studies and the true risk metric – the

mean-reverting volatility – is the likely cause for the mixed evidence produced by other

studies. Our results are robust with respect to liquidity, momentum, return reversals,

unadjusted price, liquidity, credit quality, omitted factors, and hold at daily frequency.

Acknowledgement

This thesis would not have been possible without the support of many people, to whom I am

greatly indebted.

I wish to thank my supervisor Prof Habin Lee for the endless patience and Prof Sumon

Bhaumik for his encouragement to embark on this study and to see it through despite all the

odds.

Many thanks are due to Ms Vasilka Stamatova for her encouragement, support, optimism, and

for proof-reading of this thesis (all errors that remain are solely mine).

Apologies are due to my immediate family for the edgy moods that occasionally accompanied

my studies.

I am also grateful to my workmates who covered for my absences, often at very short notice.

Table of Contents

Abstract .......................................................................................................................................... 2

Acknowledgement .......................................................................................................................... 3

1. Introduction ................................................................................................................................ 9

1.1. Background to the work .................................................................................................................. 9

1.2. Research motivation, aims and objectives .................................................................................... 14

1.3. Structure of the thesis ................................................................................................................... 17

2. Related studies .......................................................................................................................... 19

2.1. Introduction ................................................................................................................................... 19

2.2. Underlying economic theories ....................................................................................................... 21

2.3. Empirical findings concerning idiosyncratic volatility and the cross-section of stock returns ...... 42

2.4. Conclusions .................................................................................................................................... 57

3. Research Methodology and Data Sources ................................................................................... 59

3.1. Introduction ................................................................................................................................... 59

3.2. Methodological notes ................................................................................................................... 60

3.2.1. Research philosophy: positivism and its limitations .............................................................. 60

3.2.2. Deductive and inductive research ......................................................................................... 63

3.2.3. Quantitative and qualitative research ................................................................................... 72

3.2.4. Statistical methods ................................................................................................................ 75

3.2.5. Regression models ................................................................................................................. 79

3.2.5.1. Correlation and causation ............................................................................................................. 79

3.2.5.2. Factor model estimation ............................................................................................................... 80

3.2.5.3. Selection of volatility models ........................................................................................................ 84

3.2.5.4. Comparison of volatility forecasts ................................................................................................ 93

3.2.5.5. Assessing the correlation between expected idiosyncratic volatility and returns ....................... 98

3.2.5.6. Securities as assets ..................................................................................................................... 101

3.3. Control variables ......................................................................................................................... 102

3.3.1. Splitting total return into systematic and idiosyncratic returns .......................................... 102

3.3.2. Control variables in the cross-section regressions .............................................................. 107

3.4. Data sources, transformations, and summary statistics ............................................................. 108

3.4.1. Data sources ........................................................................................................................ 108

3.4.2. Calculated covariates ........................................................................................................... 111

3.5. Classification of volatility regimes............................................................................................... 122

3.6. Conclusions .................................................................................................................................. 128

4. Idiosyncratic Risk and the Cross-Section of Stock Returns: Empirical Findings ............................ 130

4.1. Introduction ................................................................................................................................. 130

4.2. Comparison of volatility forecasts ............................................................................................... 131

4.3. Idiosyncratic volatility and the cross-section of stock returns: results from Fama–Macbeth

cross-sectional regressions ................................................................................................................. 144

4.4. Mean-reverting level of volatility ................................................................................................ 157

4.5. Further tests of robustness .......................................................................................................... 181

4.5.1. Was there an omitted factor? ............................................................................................. 181

4.5.2. Evidence from daily data ..................................................................................................... 185

4.5.3. Portfolios as assets .............................................................................................................. 191

4.5.4. Interaction effects ............................................................................................................... 198

4.6. Summary ..................................................................................................................................... 216

5. Discussion ............................................................................................................................... 217

5.1. Introduction ................................................................................................................................. 217

5.2. Forecasts quality and goodness of fit.......................................................................................... 217

5.3. Mean-reverting level ................................................................................................................... 220

5.4. Comparison with other studies ................................................................................................... 222

5.5. Liquidity premium and other premia .......................................................................................... 227

5.6. Is it tradable? .............................................................................................................................. 232

6. Conclusions and directions for future research ......................................................................... 236

6.1. Conclusions .................................................................................................................................. 236

6.2. Limitations and directions for future research ............................................................................ 238

List of tables

Table 1: Summary of selected theoretical models and their predicted correlation between

idiosyncratic risk and stock returns .....................................................................................40

Table 2: Summary of selected empirical studies of the link between idiosyncratic volatility and

stock returns ........................................................................................................................56

Table 3: Deductive research workflow .......................................................................................71

Table 4: Average sector weights, Pearson correlations of sectors with the market, and

cross-sector correlations ....................................................................................................110

Table 5: Descriptive statistics, 1/1980–3/2013 .........................................................................113

Table 7: Mean volatilities by portfolios sorted by beta and capitalisation, 1980/7–2013/3 .....120

Table 8: Parameter estimates for Hidden Markov Model with two states and Normal

distribution of volatilities in each state .............................................................................124

Table 9: Parameter estimates for Hidden Markov Model with three states and Normal

distribution of volatilities in each state .............................................................................125

Table 10: Periods with high market volatility ...........................................................................126

Table 11: Average cross-sectional Spearman rank correlations of idiosyncratic volatilities....133

Table 12: Dickey–Fuller tests for ex ante and ex post volatility estimates ...............................136

Table 13: Predictive accuracy from Mincer-Zarnowitz regressions .........................................140

Table 15: Fama–Macbeth cross-sectional regressions with historical and ARMA forecasts ...154

Table 16: Fama–Macbeth cross-sectional regressions with mean-reverting volatility .............160

Table 17: Descriptive statistics, 1/1980 - 3/2013 ......................................................................163

Table 18: Fama-Macbeth cross-sectional regressions with mean-reverting volatility – return

persistence .........................................................................................................................178

Table 19: Summary statistics of the first common factor of idiosyncratic returns and loadings

on that factor, 07/1982-03/2013 ........................................................................................183

Table 20: Fama–Macbeth cross-sectional regressions with loading on the principal factor

affecting idiosyncratic returns, 07/1982 – 03/2013 ...........................................................184

Table 21: Summary statistics of expected volatilities from daily data .....................................187

Table 24: Portfolio Alphas Relative to Fama–French–Carhart Model .....................................196

Table 26: Fama–Macbeth regressions with mean-reverting level of volatility – interactions with

Roll's bid-ask spread, 7/1980-03/2013 ..............................................................................206

Table 27: Fama–Macbeth regressions with mean-reverting level of volatility – interaction with

unadjusted prices, 07/1980-03/2013 .................................................................................209

Table 28: Fama–Macbeth regressions with mean-reverting level of volatilities – interaction

with traded volume in the last 36 months, 07/1980-03/2013 ............................................212

List of figures

Figure 1: Structure of Chapter 2 ..................................................................................................19

Figure 2: Efficient frontier with 30 stocks as of December 31, 2013, and a subset of 7 stocks .32

Figure 3: Volatility of DataStream market return index (all sectors) .......................................123

Figure 4: Moving average and Exponentially-weighted moving average as filters ..................224

9

1. Introduction

1.1. Background to the work

Modern portfolio theory was developed in the middle of 20th century and over time

became one of the thriving fields of economic research. In the span of less than ten years

between the works of Markowitz and Lintner, it offered a new formulation of the

decision-making problem in portfolio optimisation, which served as a basis for many

subsequent theoretical enquiries and practical applications.1

Markowitz (1952) proposed that rational investors in stocks should prefer higher

expected return and lower variability of returns. This proposition was formalised in the

concept of the efficient frontier, which was defined as the set portfolios offering lowest

variance of returns for a given level of expected return.

The next significant step to solving the portfolio optimisation problem was made by

Tobin (1956). He assumed that there existed one risk-free asset, usually referred to as cash.

He demonstrated that investors could improve their expected return if they invested in a mix

between the risk-free asset and some diversified portfolio on the efficient frontier. The returns

obtainable through such a mix between risky and risk-free assets formed a straight line in the

expected return/portfolio variance space, passing through the risk-free rate point and the

selected efficient portfolio, and the steeper the line, the better the risk-return trade-off. Hence,

the optimal solution was choosing a portfolio in which the line connecting the portfolio and

the risk-free rate was tangent to the efficient frontier. In that setting all investors invested in a

fraction of that super-efficient portfolio, irrespective of their risk appetite. The composition of

that super-efficient portfolio still had to be estimated through the full Markowitz optimisation.

Sharpe (1964) and Lintner (1965b) developed the mean-variance reasoning further by

formulating the Capital Asset Pricing Model, or “CAPM”. The CAPM demonstrated that

1 Harry Markowitz shared the 1990 Sveriges Riksbank Prize in Economic Sciences in

Memory of Alfred Nobel with William Sharpe and Merton Miller. He was praised in the

award press release for “developing a rigorously formulated, operational theory for portfolio

selection under uncertainty - a theory which evolved into a foundation for further research in

financial economics”. Nobelprize.org, ‘“The Prize in Economics 1990 - Press Release”’,

Nobel Media AB 2014, 1990

<http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1990/press.html>

[accessed 19 July 2016].

10

asset risk could be decomposed into two components – systematic risk and idiosyncratic risk.

In their approach systematic risk referred to changes in asset prices that were due to common,

market-wide changes that affected all securities, albeit to different extent, while idiosyncratic

risk was the risk of change of individual asset prices due to reasons specific for the asset and

uncorrelated with the overall market movements. Investors could diversify away all

idiosyncratic risks by holding the market portfolio, defined as the value-weighted index of all

financial instruments. Therefore, the CAPM predicted that asset returns in excess of the

risk-free rate should be proportionate to the market excess return, and the coefficient of

proportionality (referred to as asset’s beta) should be determined by the covariance of the

asset’s excess returns with the market excess returns. In that context the idiosyncratic risk was

the residual change of the values of individual assets that was uncorrelated with changes of

the excess returns on the market portfolio, and that was assumed to be driven by random

events concerning the individual issuer that did not have a broad market impact. The

interpretation of the core CAPM proposition was then straightforward – only systemic risk

was priced, idiosyncratic risk was not.

The conclusions of Sharpe and Lintner were surprising, yet intuitive and deceptively

testable. However, the predicted optimal investor behaviour – holding a diversified portfolio

of assets – was in stark contrast with surveys of the composition of individual portfolios. For

example, Blume and Friend (1975) found that contrary to the predictions of CAPM, investors

held rather concentrated portfolios. They examined a sample of tax returns from the 1971 tax

year, and found as many as 34.1% of the tax returns listed only one dividend-paying stock2,

and 50.9% listed up to two dividend-paying shares.

Such apparently suboptimal behaviour could be at least explained by the restrictive

assumptions of the CAPM, and economists were quick to explore the implications of the

relaxation of some of these assumptions on equilibrium prices and returns. In particular, the

CAPM assumed that there were no market frictions like transaction costs and asset

indivisibility. Levy (1978) and Merton (1987) developed theoretical extensions of the CAPM

that assumed that market imperfections prevented investors from investing in the entire

available investment universe, and concluded that optimal investors would be seeking reward

for the undiversified idiosyncratic risk.

Empirical studies to date, however, produced no conclusive evidence on the pricing of

2 For tax purposes households were required to disclose only the dividend-paying

stockholdings; information on the ownership of non-dividend-paying shares was not available

in that sample.

11

idiosyncratic risk. The first tests of the CAPM performed by Lintner (1965a) confirmed that

beta was a significant predictor of average asset returns, and the slope was positive, consistent

with the theory. However, the coefficient for idiosyncratic risk turned out to be positive and

statistically significant, contrary to the prediction of CAPM. Miller and Scholes (1972) also

confirmed the findings of Lintner over an extended period and noted that idiosyncratic

variance alone had higher correlation with average returns compared to beta alone. They also

pointed out that the specification of the cross-sectional regression of realised returns on betas

required the use of the true betas but in fact used the estimated ones, introducing

error-in-variable problem.

The first tests of CAPM that addressed that error-in-variable problem where

performed by Black et al. (1972) and by Fama and MacBeth (1973). Black et al. (1972)

pooled securities into portfolios of similar securities in order to overcome the problem of the

correlations between the pricing errors of individual securities. Fama and MacBeth (1973)

further refined the method for testing the CAPM, and found that the idiosyncratic volatility

was not a significant predictor of the cross-section of returns, which was consistent with the

predictions of the CAPM and suggested that the previous positive results were the likely

result of limitations of the testing methodology.

Since 2000, interest in the empirical investigation of significance of idiosyncratic risk

in explaining the cross-section of stock returns rebounded. Such an interest was spurred by a

variety of motivations. For example, the significance of beta as the sole, or at least a

significant factor in explaining the cross-section of returns, has declined and some studies

found it insignificant, while other market factors3 or characteristics of the issuer4 were added

to empirical asset pricing models. The growing interest in behavioural asset pricing also

contributed to the increasing interest in re-examining the assumptions of classical finance

theory. Finally, the surge of studies in a related field – the correlation between idiosyncratic

risk and aggregate market returns – also spilled over to the renewed interest in its contribution

to explaining the cross-section of returns. Whichever the cause, in the past fifteen years a

number of interesting studies were published that reported contradictory results concerning

the significance of idiosyncratic risk as a predictor of expected returns.5

Malkiel and Xu (2004) found that CAPM beta was an important factor in explaining

3 e.g. the small-minus-big factor and the high- minus-low factors in Fama and French (1993) 4 e.g. size, price/earning ratio, dividend yield, past returns – see Daniel and Titman (1997,

1998) 5 Ang et al. (2006); Bali and Cakici (2008); Fu (2009); Huang, Liu, Rhee and Zhang (2012)

12

cross-sectional differences of returns but that its effect declined over time, but also confirmed

that idiosyncratic risk was also a significant factor – both statistically and economically.

Spiegel and Wang (2005) noted that there was a significant negative correlation between

liquidity and idiosyncratic volatility, and that expected stock returns were positively

correlated to idiosyncratic risk and negatively correlated to liquidity. They found that

idiosyncratic risk was consistently positively correlated to expected stock returns, but the

impact of one standard deviation change in idiosyncratic risk was on average between 2.5 and

8 times stronger than the impact of a corresponding one standard deviation increase in

liquidity.

The debates were stirred by the surprising results of Ang et al. (2006) who reported

that portfolios ranked on idiosyncratic volatility exhibited a consistently negative correlation

to expected returns after controlling for various factors, across sub-samples, and for various

portfolio formation strategies. Their results were puzzling because they contradicted both the

CAPM (idiosyncratic volatility should not be priced at all), and the Levy and Merton models

(idiosyncratic risk should earn positive risk premium). Ang et al. (2009) reviewed evidence

from the G7 countries and other developed markets and found that the spread between the

first and fifth quintiles of portfolios sorted on idiosyncratic risk was again negative, standing

at −1.31 per cent per month after controlling for world market, and size and value factors.

Bali and Cakici (2008) also examined how idiosyncratic risk was priced using

NYSE/AMEX/NADAQ data over the period July 1963 – December 2004 and found that the

results of Ang et al. (2006) were not robust with respect to weighting scheme, time frequency,

portfolio formation, and screening for size, price, and liquidity.

In a significant contribution to the debate, Fu (2009) observed that the theoretically

correct variable to explain expected returns was the expected idiosyncratic risk in the current

period, rather than the actual idiosyncratic risk in the preceding period, which was used by the

studies of Ang et al. (2006) and Ang et al. (2009). He proposed to use the Exponential

Generalised Autoregressive Conditional Heteroscedasticity (EGARCH) model in order to

forecast next-period expected idiosyncratic volatility while allowing an asymmetric response

to shocks. He used Fama and MacBeth (1973) with individual securities as assets and found

that cross-sectional returns were statistically and economically significant and positively

correlated with idiosyncratic risk. He found that reversals of returns were a significant factor

for the puzzling results obtained by Ang et al., (2006). Brockman et al. (2009) reached similar

conclusions using an international data set and employing Fu’s methodology. However, Guo

et al. (2014) who argued that the way it was implemented involved a look-ahead bias that was

13

aggravated by the particular type of GARCH model used in the estimation (EGARCH(𝑝,𝑞)

with 𝑝, 𝑞 = 1…3). When they controlled for that bias they found that idiosyncratic risk was

no longer significantly correlated with returns. In a similar vein, Huang, Liu, Rhee and Zhang

(2012) argued that the omission of last-period return resulted in omitted variable bias, which

could impact the significance of the idiosyncratic risk proxies.

Thus, the studies in the last fifteen years produced mixed evidence on the correlation

between idiosyncratic risk and market returns. That topic, however, is of great importance

from both a practical and a theoretical perspective. Finding the root cause for the mixed

evidence produced by existing studies would allow improved portfolio construction and

would optimise the risk-return trade-off for both individual and professional investors. For

example, if idiosyncratic risk is correlated with returns, a failure to recognise it as a

characteristic in portfolio construction could result in apparent over-performance of portfolios

loaded on securities with higher idiosyncratic risk over portfolios with lower idiosyncratic

risk; thus, hedge portfolios loaded on idiosyncratic risk could mislead investors into believing

superior forecasting performance of their asset managers (high alpha) when in fact they are

bearing higher idiosyncratic risk. On the other hand, if idiosyncratic risk is not priced by the

market, then investment funds marketing portfolios constructed to exploit exposure to

idiosyncratic risk could be overcharging investors and reducing their risk-adjusted

performance after trading costs. Therefore, irrespective of the direction of the correlation,

understanding its sign, magnitude of underlying cause should benefit investment managers.

The issue is also important from a theoretical perspective. Academic research has

identified a number of stock market anomalies, as well as factors that seem to explain stock

returns, but which lack a solid theoretical understanding. Understanding of the true role of

idiosyncratic risk could allow improved return attribution and ultimately – improved

identification of the underlying economic factors. For example, smaller stocks are known to

have higher volatility; if idiosyncratic risk is priced by the market, then some of the

documented variation of the small-minus-big factor could be due to changes of idiosyncratic

volatility of small companies due to aggregate economic shock. Therefore, resolving the

idiosyncratic volatility puzzle should benefit both practitioners and theorists and would

further the cause of wealth management.

14

1.2. Research motivation, aims and objectives

In our view there are two important and closely interrelated aspects of this debate that

deserve careful attention, and which are in the centre of this study: firstly, an analysis of the

differences and similarities of the employed measures of idiosyncratic risk; secondly, analysis

of the correlation between idiosyncratic risk and the cross-section of returns.

The first aspect concerns the use of different definitions of idiosyncratic volatility

across studies. Thus, Malkiel and Xu (2004) measure idiosyncratic volatility as the standard

deviation of residuals from the one-factor CAPM model or the three-factor Fama–French

model. Ang et al. (2006) and Ang et al. (2009) use a similar approach but instead of applying

it on a rolling window of monthly data, they use daily returns from the previous month in

order to split volatility into systematic and idiosyncratic components and to calculate the

mean daily variance in the respective month. Bali and Cakici (2008) employ two measures of

idiosyncratic risk: the one used by Ang et al. (2006), and another version calculated for

monthly data over the preceding 24 to 60 months, as available. Cao (2010) and Cao and

Xu (2010) measure idiosyncratic volatility in terms of the exponentially-weighted moving

average of the residuals from OLS regression (24 to 60 months) with weights of 0. 9𝑘. Fu

(2009), emphasising the importance of using forward-looking models in order to obtain ex

ante risk measures, employs EGARCH(p,q) to produce forecasts from monthly data; a similar

approach is followed by Spiegel and Wang (2005). Finally, Huang, Liu, Rhee and Zhang

(2012) use forecasts from an ARIMA model fitted on realised monthly volatilities.

There is limited information on how these measures compare with one another in

terms of forecasting the unobservable true volatility. Some of the studies that address the issue

employ loss functions to compare idiosyncratic volatility estimates. However, loss functions

could be of limited use when one compares estimates based on different frequencies (monthly

vs daily). Bali and Cakici (2008), on the other hand, use Mincer and Zarnowitz (1969)

regressions to compare predictive performance of monthly-based vs daily-based forecasts.

Spiegel and Wang (2005) also perform a comparison of prediction accuracy but use only

monthly data and loss-function approach to the test. Nonetheless, the choice of measure of

true volatility may pre-determine the ‘winner’ in those forecasts.

Therefore, the first gap that this study aims to address is to compare the types of

idiosyncratic volatility proxies employed by previous studies and to analyse how conclusions

depend on the quality of source data.

15

Addressing the first gap would allow us to examine empirically whether idiosyncratic

variance explains the cross-section of returns. Indeed, one should expect that if idiosyncratic

risk explains the cross-section of returns, then the better forecasts would result in more robust

prediction of returns. Furthermore, we can then analyse the role of other characteristics, e.g.

the mean-reverting level of idiosyncratic volatility, in explaining the cross-section of returns.

Such an analysis is warranted by the finding of Fu (2009) that idiosyncratic volatility is

stationary for 90 per cent of all securities.

Another gap for our empirical study is to address some of the methodological

challenges to the inspiring contribution of Fu (2009) – the alleged presence of look-ahead bias

and the omitted-variable bias. Indeed, Huang, Liu, Rhee and Zhang (2012) report that

introducing the lagged return as a control variable, which is omitted in other studies, including

that of Fu (2009), renders idiosyncratic volatility statistically insignificant. Another criticism

came from Guo et al. (2014) who argue that there might be look-ahead bias inherent in the

studies employing EGARCH models, which explains the predictive performance of those

forecasts.

In this study we aim to address those research gaps by addressing two interrelated

research questions: firstly, to compare alternative volatility forecasts in terms of their

accuracy, in order to understand if such differences explain the mixed empirical evidence in

existing literature; secondly, to re-examine the relevance of under-diversification for asset

prices by testing whether idiosyncratic risk is priced by investors in stock markets.

Therefore, the objectives of this research are as follows:

1. To compare the predictive performance of the main classes of estimators of monthly

idiosyncratic volatility using Mincer-Zarnowitz regressions;

2. To test statistically whether the different estimates of idiosyncratic volatility explain

the cross-section of monthly stock returns using the Fama-Macbeth methodology;

3. To test statistically whether other characteristics of the idiosyncratic volatility

process, especially the mean-reverting level of volatility, explain the cross-section of stock

returns using the Fama-Macbeth methodology.

The statistical tests use public domain secondary data on securities traded on stock

markets in the United States of America over the period from January 1972 until March 2013.

The source data is from the Thomson Reuters Datastream service. Most existing studies

employ the dataset of the Center for Research in Security Prices (CERP), and in that respect

the validation of the results from previous studies, in particular that of Fu (2009), against the

Thomson Reuters dataset can be also construed as a test of whether previous results are

16

exposed to the data snooping problem.

The scope of this study is limited only to US stocks listed at the largest American

stock exchanges during the period of study: the New York Stock Exchange (NYSE), the

American Stock Exchange (AMEX), and the NASDAQ Stock Market (NASDAQ). The

choice of market (the US) is aimed to facilitate comparisons with other existing studies in the

field, while international (non-US) evidence is deferred for further study.

The underlying economic theory – the CAPM and its extensions by Levy (1978) and

Merton (1987) – is not bound to a specific time frequency, and the results in principle should

hold at all frequencies, e.g. daily, weekly, monthly, quarterly, or annual. Existing evidence on

the significance of idiosyncratic risk is somewhat stronger with respect to daily frequency.

However, there might be market microstructure effects that affect those findings. This is

especially true for smaller stocks that have a narrower investor base and should earn higher

return in equilibrium, according to the underlying economic model. Therefore, in this study

we focus on the monthly frequency and leave the further cross-frequency validation for

further studies.

In this dissertation we contribute to the debate in three ways. Firstly, we analyse how

the different forecasts of the idiosyncratic variance perform in predicting next-period

variance. The evidence available in that respect is scarce, and either compares only estimates

from monthly data with one another, or uses proxies of true idiosyncratic variance that might

be suspected to pre-determine the result of the comparison. Secondly, we re-examine the

existing evidence concerning the link between idiosyncratic variances and the cross-section of

stock return and we address the recent methodological objections concerning studies

documenting the existence of a positive correlation. In that way we are able to bridge the gap

between the quality of forecasts and the resulting explanations of the cross-section. We also

address the criticisms (omitted variable and look-ahead biases) of the (E)GARCH

methodology employed in some of the studies that find positive correlation between

idiosyncratic risk and the cross-section of returns. Thirdly, we identify the mean-reverting

level of volatility as a key variable in explaining the cross-section of returns. The outcome of

our tests puts into perspective and reconciles the existing evidence by identifying the

components of idiosyncratic volatility that explain the cross-section of returns.

Our results shall be useful from both a theoretical and a practical perspective. From

the theoretical perspective evidence on the significance of idiosyncratic volatility would

provide support to the Merton model. Furthermore, it would shed light on how significant that

premium is, and indirectly – on whether under-diversification is widespread enough to affect

17

observed prices significantly. From a practical perspective the results could be used to

construct investor portfolios and to assess their performance after taking into account their

exposure to idiosyncratic risk.

1.3. Structure of the thesis

In Chapter 2 we review the findings of related studies. These studies fall into three

broad categories, and we dedicate a section to each of those. In Section 2.2 we review the

milestones of the modern portfolio theory until the formulation of the Capital Asset Pricing

Model (CAPM). The CAPM predicts that idiosyncratic risk can be diversified away by

investors completely, and therefore it is not priced. In particular, we see that there are at least

two lines of extension, from which possible deviations from the predicted CAPM equilibrium

could emerge. Firstly, this is the axiomatic basis of the Expected Utility theory. Criticisms

from that direction are the basis of the behavioural asset pricing theory; such behavioural

explanations resulting in aversion to idiosyncratic risk could be due to behavioural biases or

regulatory requirements concerning the disclosure of significant individual loss-making

positions. However, in this study we focus on concerns about the limits on portfolio

diversification due to transaction costs, asset imperfect divisibility, or other motives that result

in investors holding undiversified portfolios. In those cases the CAPM extensions of Levy

(1978), Merton (1987) and Malkiel and Xu (2004) predict that idiosyncratic risk could be

priced in equilibrium, if under-diversification is pervasive. In Section 2.3 we review the

principal empirical findings concerning the idiosyncratic risk and the cross-section of returns.

Chapter 3 explains the principal methodological tools employed in this study. The

reviews are fairly brief because we employ existing methods, so the chapter is more

concerned with explaining the use of those methods in this study. In particular, Section 3.2

motivates the positivist philosophy that underpins the testing methodology of this study.

Section 3.3 then explains how we split the observed excess return into a systematic

component and an unexpected idiosyncratic return. The excess return is the variable that we

aim to explain. It comprises of two components – systematic and idiosyncratic returns, and

the latter are used to estimate idiosyncratic volatilities. Section 3.4 then explains how the

idiosyncratic returns can be used to estimate idiosyncratic volatilities. In particular, we

employ four such alternative estimators, and we dedicate a subsection to each of those.

18

Section 3.5 then explains our approach to comparing the predictive performance of these

alternative volatility estimators. This comparison allows us to check how well each of the four

estimators predicts returns, and which should shed light on whether it is indeed idiosyncratic

risk that explains returns, and not some other factor or characteristic which correlates with

some of the measure. Thus, if the superior estimators of volatility are significant predictors of

the cross-section, then we could conclude that the noise in the inferior estimators is the reason

for the mixed empirical performance. As it happens, it turns out that the opposite is the case,

which suggests that there is another characteristic, that is more important than next-period

expected volatility.

Section 3.6 lays out the Fama–Macbeth methodology that is used throughout Section 4

to perform the empirical tests. Section 3.7 then explains what data sources were employed in

the empirical analysis, what transformations of those data were implemented, as well as how

we classify the volatility states of the market, which is used in some of the robustness tests in

Chapter 4.

Chapter 4 presents our empirical findings concerning idiosyncratic risk and stock

returns. The structure of the chapter mirrors much of the structure of Chapter 3. In particular,

Section 4.2 compares the volatility forecasts using the methodology of Section 3.5. Section

4.3 presents the results of the empirical tests concerning the correlation between idiosyncratic

risk and returns using the Fama–Macbeth methodology described in Section 3.6. Section 4.4

presents further tests using the mean-reverting level of volatility as an explanatory variable

instead of next-period volatility. The section also provides a first set of confirmatory

robustness checks. These checks were further extended in Section 4.5, using additional tests

that were not described in Chapter 3; the relevant methods are explained in each of the

respective sub-sections.

Chapter 5 discusses the results from our analysis, providing further motivations for the

significance of the mean-reverting level of volatility, as well as the difficulties in interpreting

the idiosyncratic volatility premium due to its inseparability from other characteristics,

especially liquidity. The chapter also highlights the risk in practical implementation of trading

strategies aiming to exploit the higher returns on high-volatility stocks. Chapter 6 concludes

the thesis.

19

2. Related studies

2.1. Introduction

In this chapter we shall explore the theoretical foundations underpinning our study and

the available empirical evidence. The body of related literature is quite vast, and this is related

to the fact that the proposition that idiosyncratic risk is irrelevant for equilibrium prices is a

core proposition of the classical finance theory. Therefore, in order to keep this chapter

reasonably short and focused, we need to be very selective in our choice of related studies.

Nevertheless, we also aimed to provide the readers with just enough context to enable them to

relate our results not only with the results of classical finance theory, but also with the

associated field of behavioural finance. Therefore, in Section 2.2 we shall outline the

historical development of the Capital Asset Pricing Model and its refinements aimed to relax

some of its assumptions. For convenience, we outline the structure of this chapter in Figure 1.

Figure 1: Structure of Chapter 2

Source: the author

The review of the CAPM serves two purposes: on the one hand, it outlines the key

Secion 2.1: Introduction

• Chapter outline

Section 2.2: Underlying economic models

• Structure and predictions of the CAPM

• Assumptions of the CAPM

• Approaches to relaxing the assumptions of the CAPM

• Relaxations that assume limits to portfolio diversification: Levy (1978), Merton (1987), Malkiel & Xu (2004)

Section 2.3: Empirical evidence

• Evidence concerning number of assets held in investor portfolios

• Evidence concerning correlation between returns and idiosyncratic risk

Section 2.4: Conclusions

• Chapter wrap up

20

model that predicts that idiosyncratic risk should not be priced by investors. On the other

hand, the approach highlights the key assumptions underpinning that result. The key

assumptions of the CAPM that are relevant for our topic are that investors’ preferences can be

represented by a quadratic utility function, and that there are no frictions. The simple

statement of the first assumption partially conceals the true underlying assumptions, and we

aim to highlight those by placing them in the appropriate historical context. In particular, we

first need to assume that investor’s preferences can be expressed in terms of a utility function,

and then that function can be reasonably represented as a quadratic one. Furthermore, there

should be no frictions like transaction costs or asset indivisibility.6 Relaxation of any of these

assumptions could be relevant for our study. For example, relaxation of the expected utility

theory axioms could give rise to a behavioural explanation of idiosyncratic risk.7 Absence of

quadratic approximation to the utility function could imply significance of the higher

moments of the utility function.8 Finally, in the presence of frictions that prevent full

diversification, investors may require a risk premium for idiosyncratic risk.9 Therefore, we

believe it is important to highlight how the predictions of the CAPM depend on those

assumptions.

There are various relaxations that may give rise to deviations from the CAPM

predictions. These can be broadly classified into behavioural extensions, which embed

specific investor behaviours into the models in order to accomplish more realistic models of

asset pricing. However, as we shall explain in somewhat greater detail in Chapter 3, these

assumptions could be too ad hoc, which motivates our preference for the approach of classical

finance to our problem. Therefore, we shall not explore behavioural asset pricing theories

here; instead, in Section 2.2 we explore how the introduction of market frictions to the CAPM

affects the predictions of that model. We review the models of Levy (1978), Merton (1987),

and Malkiel and Xu (2004) in order to elucidate their assumptions and highlight their key

predictions concerning the sign and the magnitude of such idiosyncratic risk premium.

Then, in Section 2.3 we review the available empirical evidence on the sign and

significance of the link between idiosyncratic variance and stock returns. Such evidence

broadly comprises two types of studies. Some studies observe investor behaviours (portfolio

6 See DeGenarro and Robotti (2007) for a discussion of financial market frictions. 7 For example, preference for own skewness, as proposed by Barberis and Huang (2008). 8 For example, Kraus et al. (1976) use a higher-order Taylor approximation of investor’s

utility function and conclude that there should be preference for co-skewness with the market. 9 Levy (1978); Merton (1987); Malkiel and Xu (2004)

21

compositions) directly, while most of the reviewed studies explore factors and characteristics

that explain stock returns. Our review of those results has two goals: on the one hand, to

understand the empirical support for the two competing theories (the CAPM and its

extensions), and on the other hand to identify control variables that should be included in our

study in order to avoid inasmuch as possible omitted variable bias. Finally, Section 2.4 sums

up the arguments presented in this chapter.

2.2. Underlying economic theories

“Investing should be more like watching paint dry or

watching grass grow. If you want excitement, take

$800 and go to Las Vegas.”

Paul Samuelson

“Wide diversification is only required when

investors do not understand what they are doing.”

Warren Buffett

In 1713, Nicolas Bernoulli, brother of the famous mathematician Daniel Bernoulli,

posed a problem in a letter to Pierre Raymond de Montmort. The problem later became

known as the St Petersburg paradox and is stated as follows. Suppose that we are offered to

bid in a lottery, the pay-off of which is determined by tossing a fair coin. The pay-off for the

lottery starts at 1 ducat and is doubled every time heads appear. The first time tails appear, the

lottery ends with the gambler winning the accumulated pay-off. So, if tails appear at the first

toss, the gambler receives one ducat; if heads appear, the pay-off is doubled to two ducats. If

at the second toss heads appear once more, the pay-off is doubled to four ducats and so forth.

So, if 𝑘 is the number of consecutive heads that appear, the pay-off of the bet equals 2𝑘

ducats. The expected value of the bet would then be ∑∞𝑘=1 ( 1

2 ) 𝑘

2𝑘 = ∑∞𝑘=1 1 = ∞. Hence the

expected pay-off from playing this game is infinite. In contrast to the infinite expected

pay-off, people are actually willing to pay a very modest entrance fee to bet on this game.

This discrepancy between the finite (and in fact rather low) entrance fee people are willing to

pay to play the game, and its infinite expected pay-off became known as the St Petersburg

paradox.

22

Daniel Bernouli published his solution to the puzzle in 1738 in the Commentaries of

the Imperial Academy of Science of Saint Petersburg, where he argued that the price people

are willing to pay to enter that lottery is determined by their evaluation of the likelihood of

various outcomes and the utility those outcomes yield to them: “The determination of the

value of an item must not be based on the price, but rather on the utility it yields. There is no

doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man

though both gain the same amount” (quoted by Stewart, 2010: 26).

Bernoulli did not develop his thoughts on the St Petersburg paradox into a consistent

theory. This was done by von Neumann and Morgenstern (1944) who developed the first

coherent axiomatic theory of preferences under uncertainty. In their approach, economic

agents chose between lotteries, where lottery 𝑢 is defined as a list of possible states of the

world, each occurring with a non-negative probability 𝑝𝑘 ≥ 0 and yielding a win of 𝑢𝑘 at

each state of the world; clearly ∑𝑁𝑘=1 𝑝𝑘 = 1, 𝑝𝑘 ≥ 0, ∀𝑘. They demonstrate that if there

exists a preference relation that is complete, transitive, continuous, and independent, then

there exists a utility function that represents these preferences. Thus, decision-making under

uncertainty is reduced to the optimisation problem of maximising the expected utility subject

to resource constraints. This result is known as the Expected Utility Theorem and bridges

preference ordering to numerical utility values.

It is usually assumed that utility functions in money are continuous and monotonously

increasing; the rationale is that more wealth is preferred to less wealth. The attitude towards

risk is summarised by the sign of the second derivative of the utility function. Conventionally,

a decision-maker is defined to be (strictly) risk-averse if receiving the expected value of a

lottery is (strictly) preferred to partaking in the uncertain lottery. If he is indifferent between

receiving the expected value of a lottery and the lottery itself, he is defined to be risk-neutral.

It is known that an investor is strictly risk averse if and only if the utility function that

represents their preferences is strictly concave and risk-neutral if the utility function is

linear.10

There is an infinite number of utility functions that would satisfy the expected utility

framework. Convenience and tractability, however, dictate the use of just a few utility

functions that allow analytical tractability (ideally, closed-form solutions) of the implications

of choosing a particular utility form. The three most commonly used utility functions are11:

10 See proposition 6.C.1 in Mas-Colell, Whinston and Green, 1995: 187 11 see Levy, 2012

23

1. Constant Absolute Risk Aversion (CARA) exponential utility function: 𝑢(𝑤) =

−𝑒−𝛼𝑤;

2. Constant Relative Risk Aversion (CRRA) utility 𝑢(𝑤) = 𝑤1−𝛼

1−𝛼 for 𝛼 ≠ 1, with

limiting case the logarithmic utility 𝑢(𝑤) = ln𝑤 ;

3. Quadratic utility function 𝑢(𝑤) = 𝑤 − 𝛼

2 𝑤2.

The CARA utility function owes its name to the fact that it is the only function for

which the Arrow-Pratt risk aversion function is constant.12 Normally one expects the absolute

risk aversion 𝑟𝐴(𝑤) to be a decreasing function of wealth. This translates into

decision-maker becoming more risk-tolerant, requiring smaller risk premium, with increase of

wealth.

CRRA, on the other hand, represents the class of utility functions, for which relative

risk aversion is constant.13

The third type of utility function listed above, the quadratic utility function, is

straightforward to work with, and is the one that links the mean-variance optimisation

framework with the expected utility theory. The downside of the quadratic utility is that it

exhibits increasing absolute risk aversion, thus implying that decision makers should require a

higher risk premium for identical lotteries with the increase of their wealth, which is a

counter-intuitive prediction as wealthier investors should be more able to take risk and thus

require a lower risk premium.14

12 The Arrow-Pratt absolute risk aversion function (𝑟𝐴) is given by 𝑟𝐴 = − 𝑢′′

𝑢′ = −(ln(𝑢′))

′ .

The intuition behind that ratio is that it describes the curvature of the utility function around

point 𝑤, while the metric remains invariant under linear transformations. Through Taylor expansion of the utility function it can be shown that the value of the risk premium that a

utility maximiser would expect for taking part in a small lottery is given by 𝜋 ≈ − 1

2 𝜎2

𝑢′′(𝑤)

𝑢′(𝑤)

(Arrow, 1984: 151).

13 The relative risk aversion function (𝑟𝑅 ) is defined as 𝑟𝑅 = −𝑤 𝑢′′

𝑢′ . It measures the

curvature of the utility function with respect to percentage changes of its argument, while the

absolute risk aversion function quantifies that curvature with respect to absolute (dollar)

changes of wealth. To see this consider the function �̃�(𝑡) = 𝑢(𝑡𝑥) which links the utility of the initial wealth (at 𝑡 = 1) to the utility of wealth increased by (𝑡 − 1)% to become 𝑡𝑥. For fixed 𝑥, �̃�′ = 𝑥𝑢′(𝑡𝑥) and �̃�′′ = 𝑥2𝑢′′(𝑡𝑥), so that for 𝑡 = 1 (the initial value of wealth) the absolute risk aversion becomes �̃�′′(1)/�̃�′(1) = 𝑥𝑢′′(𝑥)/𝑢′(𝑥) (Arrow, 1984: 152). The requirement of decreasing absolute risk aversion of plausible utility functions is sometimes

complemented by a requirement for non-increasing relative risk aversion.

14 Indeed, for the quadratic utility function, 𝑟𝐴 = − 𝑢′′(𝑤)

𝑢′(𝑤) = −

𝛼

1−α𝑤 and 𝑟𝐴

′ = 𝛼2

(1−𝛼𝑤)2 > 0.

24

In an important contribution to the modern portfolio theory, Markowitz (1952)

introduced the concept of mean-variance efficiency. He considers an investment universe of

𝑛 assets. The returns on those assets are random variables; for convenience, returns here are

defined as the ratio of the uncertain future price of the risky asset to the known (certain)

current price of the asset. Let 𝜇𝑖 denote the expected return on asset 𝑖 and let 𝜎𝑖𝑗 denote

the covariance between the returns on assets 𝑖 and 𝑗 . These portfolio returns can be

summarised as a column vector 𝜇 = (𝜇1, 𝜇2, … , 𝜇𝑛) ′, and a square covariance matrix 𝑉 =

[𝜎𝑖𝑗]. It is also assumed that there are no redundant assets, i.e. assets the returns of which were

a linear combination of the returns from other assets under any state of the world; i.e. the

model excludes the possibility of rows (columns) in 𝑉 that are linear combinations of other

rows (columns). Also, the model assumes that there is no risk-free asset, i.e. there is no row

(column) of zeros in 𝑉. By construction, 𝑉 is square and symmetric, and the model assumes

that it is also positive definite.15

Using the notation above, Markowitz defines variance-efficient portfolios as those

having minimum variance of returns among those attaining certain minimum return, i.e.

Min 𝑥 { 1

2 𝑉𝑥 | 𝜇′𝑥 ≥ 𝜇𝑝, 1′𝑥 = 1},

where 𝑥 is the vector of portfolio weights (allocations) to each of the 𝑛 available assets, and

1 is a vector of ones. Thus, an efficient portfolio allocation 𝑥 minimises portfolio variance

for some fixed expected return 𝜇𝑝 subject to investment of the entire wealth, i.e. 1′𝑥 =

∑𝑛𝑖=1 𝑥𝑖 = 1. A dual formulation is also possible, where investors chose weights 𝑥 that

maximise the expected return subject to the constraint that the variance of portfolio returns

does not exceed the desired fixed threshold, i.e.

max 𝑥 {𝑥 |

1

2 𝑥′𝑉𝑥 ≤ 𝜎𝑝

2, 1′𝑥 = 1}.

Solving the variance-efficient portfolio allocation is thus reduced to solving a standard

problem of quadratic programming. The Lagrangian for the problem is:16

ℒ = 1

2 𝑥′𝑉𝑥 − 𝜆1(𝜇′𝑥 − 𝜇𝑝) − 𝜆2(1′𝑥 − 1),

15 A matrix 𝑉 is positive semi-definite if for every non-zero real column vector 𝜇, 𝜇′𝑉𝜇 ≥ 0. If 𝜇′𝑉𝜇 > 0, the matrix is positive definite. By construction, covariance matrices are positive semi-definite. Portfolio optimisation, however, requires the assumption of positive

definitiveness, which rules out perfectly correlated efficient portfolios. A positive-definite

covariance matrix can be inverted, and that would allow unique determination of equilibrium

prices. 16 The derivation outlined here follows the modern presentation of the mean-variance

optimization problem rather than the original solution strategy of Markowitz.

25

and the corresponding first-order conditions (FOC) are

𝑉𝑥 = 𝜆1𝜇 + 𝜆21,

together with the two original budget constraints. Hence the optimum weights are

𝑥 = 𝜆1𝑉 −1𝜇 + 𝜆2𝑉

−11,

which could be substituted in in the two budget constraints, resulting in a system of two

equations in two unknowns (the Lagrange multipliers); as long as 𝑉 is positive definite, the

system can be solved and so an optimal 𝑥 exists. The system for the Lagrange multipliers

has the form

𝜇𝑝 = 𝜆1𝑥′𝑉 −1𝑥 + 𝜆2𝑥′𝑉

−11,

1 = 𝜆11′𝑉 −1𝑥 + 𝜆21′𝑉

−11.

Since 𝑉 is symmetric by construction, 𝑥′𝑉−11 = 1′𝑉−1𝑥 and the system can be written as

( 𝐴 𝐵 𝐵′ 𝐶

) ( 𝜆1 𝜆2 ) = (

𝜇𝑝 1 ),

where 𝐴 = 𝑥′𝑉−1𝑥, 𝐵 = 𝑥′𝑉−11, and 𝐶 = 1′𝑉−11, so

( 𝜆1 𝜆2 ) =

1

𝐴𝐶−𝐵2 ( 𝐵 −𝐶 −𝐶 𝐴

) ( 𝜇𝑝 1 ).

Substituting the solution for the Lagrange multipliers in the variance of the variance-efficient

portfolio weights produces the formula for the efficient frontier:

𝜎𝑝 2 =

𝐴−2𝐵𝜇𝑝+𝐶𝜇𝑝 2

𝐴𝐶−𝐵2 ,

which shows that the efficient portfolio is a parabola in the (𝜇𝑝, 𝜎𝑝 2) space and hyperbola in

the (𝜇𝑝, 𝜎𝑝) (standard deviation) space. 17

The mean-variance formulation of optimal investment could be interpreted as utility

maximization with a quadratic utility function. That approach assumes that investors have

quadratic utility function, i.e. 𝑢(𝑤) = 𝑤 − 𝛼

2 𝑤2. A rational investor seeks to maximise the

expected utility, so

𝔼𝑢(𝜇) = 𝔼(𝜇 − 𝛼

2 𝜇2)

= (𝔼𝜇 − 𝛼

2 𝔼(𝜇2))

= (𝔼𝜇 − 𝛼

2 (𝔼𝜇)2 −

𝛼

2 𝑣𝑎𝑟(𝜇)),

17 Another useful result for variance-efficient portfolios is the two-fund spanning of the

efficient frontier. It can be demonstrated that every variance-efficient portfolio can be

constructed as a linear combination of two other efficient portfolios (‘mutual funds’) and thus

the linear combination of any two efficient portfolios is also an efficient portfolio.

26

where 𝑣𝑎𝑟(𝜇) denotes the variance of expected returns. This shows that for an investor with

quadratic preferences, for fixed variance of returns, the expected utility is maximised when

the expected return is maximised.18 Alternatively, for a fixed level of expected return, the

expected utility is maximised when the variance of the portfolio is minimised. Markowitz

justifies the mean-variance optimisation framework as assuming a quadratic utility function

(Markowitz, 1959, Chapter 13). He does not argue that investors’ preferences are indeed

quadratic but rather that the quadratic function could be a good local approximation to various

plausible utility functions. He specifically considers logarithmic utility ln(1 + 𝑅), as well as

quadratic approximation to (1 + 𝑅)1/2 and (1 + 𝑅)1/3.

Tobin (1956) and Tobin (1958) extend the portfolio model by adding a risk-free asset

to the investment set, assuming that asset return distribution is a two-parameter distribution of

the location-scale family. Under those assumptions Tobin shows that the problem of asset

allocation is separate from the decision of risk tolerance. In particular, by definition the

risk-free asset has no covariance with any of the risky assets, and since the covariance matrix

for the risky assets is assumed to be positive definite, a risk-free asset could not be created

through a combination (portfolio) of risky assets. Thus a portfolio comprising of the risk-free

asset is variance-efficient among the portfolios with zero variance so it must belong to the

efficient frontier. On the other hand, there must be another (risky) portfolio, also on the

efficient frontier. Since all investors would be splitting their wealth between the risky asset

and the risky fund, there would exist a risky portfolio that is efficient and all investors would

be holding a share of it, hence by market clearance it must be simply the market portfolio

comprising all risky assets in the economy. Just as in the case of the variance-efficient

portfolio, the combination of the two efficient portfolios – the risk-free one and the market

portfolio – spans the efficient frontier (two-fund spanning), and because the risk-free asset has

no covariance with any of the risky assets, the efficient frontier transforms into a straight line

through the market portfolio and the risk-free portfolio. In particular, the Sharpe ratio on any

efficient portfolio 𝑐 would equal the ratio for the market portfolio 𝑚, a result that became

known as the capital asset market line, i.e. (𝔼𝜇𝑐 − 𝑟𝑓)/𝜎𝑐 = (𝔼𝜇𝑝 − 𝑟𝑓)/𝜎𝑝, where 𝑟𝑓 is the

known (non-random) risk-free rate.

The work of Sharpe (1964) and the subsequent contribution by Lintner (1965b)

formulate the Capital Asset Pricing Model (CAPM). Sharpe takes off from the observation of

18 Quadratic utility preferences make sense only in the increasing section of the function,

hence higher 𝔼𝜇 guarantees higher 𝔼𝜇 − 𝛼

2 (𝔼𝜇)2.

27

Markowitz (1959) that share prices tend to co-vary with the market, and demonstrated that

what really matters for pricing is the systematic risk. The presentation of CAPM, as we know

it presently, is formulated by Lintner (1965b), who shows that an investor’s excess expected

rate of return is related linearly to the risk of his total investment as measured by the standard

deviation of his return.

An informal derivation of CAPM could follow the same variance-minimisation steps

as above, but subject to a slightly modified budget constraint that adds a risk-free asset to the

investment universe that yields a risk-free rate of return of 𝑟𝑓:

min 𝑥 { 1

2 𝑥′𝑉𝑥 | (𝜇 − 𝑟𝑓1)

′𝑥 ≥ 𝜇𝑝 − 𝑟𝑓},

where (𝜇 − 𝑟𝑓1) is the vector of excess returns. As before, one can form the Lagrangian and

differentiate it with respect to portfolio weights 𝑥 to obtain the first-order condition (FOC):

𝑉𝑥 = 𝜆(𝜇 − 𝑟𝑓1).

Let 𝜎𝑖𝑒 be the covariance between some asset 𝑖 (this could also be a portfolio of assets), and

some efficient portfolio 𝑒, i.e. a portfolio that satisfies the FOC above. Then by definition

𝜎𝑖𝑒 = 𝑥𝑖 ′𝑉𝑥𝑒, and upon substitution of the FOC, 𝜎𝑖𝑒 = 𝜆𝑥𝑖

′(𝜇𝑒 − 𝑟𝑓1). This relation should

hold for any efficient portfolio, and since the market portfolio 𝑚 is an efficient one19, it

follows that 𝜎𝑚 2 = 𝜎𝑚𝑚 = 𝜆(𝜇𝑚 − 𝑟𝑓). This gives the solution for the Lagrange multiplier 𝜆

and one can then eliminate it from the FOC formula above, giving the following link between

the expected return of any asset 𝑖 in relation to its risk as captured by 𝜎𝑖 and its covariance

with the expected return on the market portfolio:

𝜇𝑖 − 𝑟𝑓 = 𝛽𝑖(𝜇𝑚 − 𝑟𝑓), (1)

where 𝛽𝑖 = 𝜎𝑖𝑚

𝜎𝑚 2 . Therefore, in equilibrium the excess return on each asset would be

19 The market portfolio would be an efficient portfolio if the portfolios of the individual

investors are on the efficient frontier. This follows because a combination of portfolios on the

efficient frontier is also on the efficient frontier, and the market portfolio is the sum of

individual investors’ portfolios, which are assumed to be on the efficient frontier. In this

informal outline we do not reproduce the proof of market clearance, but the issue of what

happens when the portfolios of the individual investors are not on the efficient frontier is

studied by Levy (1978), which we discuss in the following section. However, one should note

that this general equilibrium solution is what sets out CAPM from all other models of asset

returns: whereas other models simply presume the existence of some observed or unobserved

factors driving returns and identify the arbitrage-free values of assets implied by exposures to

those factors, the CAPM actually predicts what the sole factor of asset return is (the excess

return on the market portfolio). Hence the special place of CAPM among the asset pricing

models.

28

proportionate to the excess return of the market portfolio, and the coefficient of

proportionality 𝛽𝑖 equals the ratio of the covariance between the asset 𝑖 and the market

portfolio, divided by the variance of the market portfolio.

The predictions of CAPM are quite unequivocal in mathematical terms, yet the

model’s prediction might seem counter-intuitive. The CAPM denies the existence of

abnormal returns. No matter how experienced the investment manager is, or what his

cherished model for forecasting future returns is – technical, statistical, or a fundamental

model – CAPM predicts that the expected return on his portfolio would depend solely on the

betas of the assets in his portfolio and the share of investment in the risk-free asset.

Concerning the composition of portfolios of rational investors, CAPM makes two

claims: firstly, all investors shall invest in identical portfolios of risky assets, with each

portfolio having the same weights as the market portfolio. Differences in risk-tolerance of the

different investors would be accommodated through changes in the share of investment in the

risk-free asset. Risk-shy investors would invest heavily in risk-free assets, while investors

seeking higher return would invest a larger share of their wealth in risky assets, and those

seeking even higher returns could leverage their position by borrowing at the risk-free rate

and investing in the market portfolio.

The important implication of CAPM that in equilibrium only the covariance of the

stock with the market portfolio (systematic risk) is priced, while the idiosyncratic risk of the

individual securities is not priced, is dependent on a number of explicit or implicit

assumptions that are made in the derivation of the model: (i) investor preferences satisfy the

axioms of von Neumann and Morgenstern and thus can be represented in terms of a utility

function; (ii) investors have homogeneous beliefs; (iii) the utility function can be well

approximated by a quadratic utility function; (iv) investors are price-takers; (v) the markets

are frictionless and investor decisions are not affected by other concerns like taxes,

transaction costs, asset indivisibility, asset liquidity; (vi) investors can borrow and lend in

unlimited amounts at the market risk-free rate; (vii) all relevant information is available to all

investors; (viii) markets are complete. This is indeed a long list of assumptions, and invited

many subsequent studies. Some of these examine whether investor portfolios are sufficiently

diversified, as the CAPM predicted that rational investors should seek a broad diversification

of their asset holdings. Other studies seek to relax some of the assumptions and examine the

impact of those relaxations on the market equilibrium. In the following paragraphs we briefly

review the most relevant results for our study, while in a subsequent section we shall examine

whether the CAPM performs well empirically despite the long list of restrictive assumptions.

29

The early presentations of the CAPM considered quadratic utility as a local

approximation of some well-behaved utility function. That assumption is explored in more

detail by Kraus et al. (1976), who employ a third-order approximation of the investor utility

function, i.e.:

𝑢(𝑤) = 𝑢(�̅�) + 𝑢′′(�̅�)

2! 𝜎(𝑤) +

𝑢′′′(�̅�)

3! 𝑚3(𝑤) + 𝑂(𝑤

4),

where 𝑚3(𝑤) is the third moment of the random return process. Somewhat similar

conclusions could be reached based on the prospect theory of decision-making under

uncertainty, developed by Kahneman and Tversky (1979), which assigns an important role on

the perceptions of potential gains and losses, rather than on expected return and standard

deviation, and which could give rise to preference for lottery-like investments, characterised

by higher skewness or kurtosis, and possibly – also higher idiosyncratic risk. Thus, Barberis

and Huang (2008) build on the cumulative prospect theory and argue that the demand for

lottery-like stocks (i.e. ones with significant positive skewness that have large growth

potential) gives rise to the own skewness of the stocks being priced and stocks with high

positive skew being overpriced.20

The mean-variance framework underpinning the CAPM assumes that investors’

preferences are fully captured in terms of their expected return and the variance of returns, so

that the effects of higher-order terms of the Taylor expansion of the utility function, capturing

skewness, kurtosis and higher moments could be ignored. This raises the problem in what

cases could such an assumption be justified. This problem could be approached by asking

what should be the joint distribution of stock returns that would allow investor preferences to

be expressed only in terms of expected return and variance, irrespective of her utility function.

The prime example of this situation is the case of normal returns because the Gaussian

distribution is stable under addition21 and the distribution is entirely defined by its mean and

standard deviation. In such a situation, the expected utility of an investor would be a function

of the mean and standard deviation of the return distribution, irrespective of the functional

20 Special care should be taken to distinguish the skewness preference in Barberis and Huang

(2008) from that of Kraus et al. (1976), who predict preference for positive (co-)skewness

based on Taylor approximation of the utility function of investors. The difference between the

predictions of the two models is that Barberis and Huang (2008) predict significance of own

skewness of the stocks, i.e. 𝛾(𝑥) = 𝔼(𝑥−𝔼𝑥)3

𝜎𝑥 3 , whereas Kraus et al. (1976) predict significance

of the co-skewness with the market, i.e. 𝛾(𝑥,𝑚) = 𝔼(𝑚−𝔼𝑚)2𝔼(𝑥−𝔼𝑥)

𝜎𝑚 3 .

21 The sum of two normal variables is also normally distributed

30

form of the utility function. As long as the investor is risk-averse, for a fixed expected return

she would choose the portfolio with the lowest variance; hence, if asset returns are jointly

normally distributed, the optimal portfolio for the utility-maximising risk-averse investor

would necessarily be a variance-efficient one.

Chamberlain (1983) and Owen and Rabinovitch (1983) examine formally the

circumstances in which investor’s preferences could be formulated in terms of the mean and

variance of portfolio returns alone. Chamberlain (1983) starts from the observation that if two

portfolios yield the same expected utility under arbitrary utility function, then they should

have the same distribution of returns and have coinciding mean and variance. Hence he

establishes the conditions for the joint distribution of asset returns which would ensure that if

the mean and variance of two portfolios coincide, then the portfolio return distributions are

the same, i.e. the two portfolios would yield identical expected utility, irrespective of the

utility function. Chamberlain demonstrates that an investor’s utility function can be

formulated in terms of mean and variance of returns if and only if returns are jointly

elliptically distributed. Elliptic distributions are generalisation of the multivariate normal

distribution. Formally they are defined in terms of the form of the distribution characteristic

function. A multivariate distribution is said to be elliptical if its characteristic function is of

the form 𝑒𝑖𝑡 ′𝜇 Ψ(𝑡′Σ𝑡), where Σ is a positive semi-definite matrix, and 𝜇 is a vector of

parameters. In the case of multivariate normal distribution, Ψ(𝑢) = 𝑒−𝑢/2, and 𝜇 and Σ are

the vector of means and the covariance matrix respectively. Linear combinations of elliptical

distributions are elliptical too; the marginal distributions of a multivariate elliptical

distribution are also elliptical. The class of elliptical distributions includes various symmetric

distributions; in particular, it includes the normal distribution, Student’s t, the logistic

distribution, Laplace distribution, and symmetric stable distributions. While these

distributions allow for a very broad set of return distributions that are characterised with

varying tail fatness, all elliptical distributions have the limitation of being symmetric. The

evidence concerning the joint distribution of returns is inconclusive. Levy (2012) surveys

existing research and concludes that logistic distribution could not be rejected (as measured

by the Kolmogorov-Smirnov test statistic) for daily, weekly and monthly returns for Dow

Jones constituent stocks; other distributions that are reported as not rejected for certain stocks

were beta, normal and log-normal distributions for monthly data, and Levy-stable

distributions for the daily returns of some companies (Levy, 2012, Chapter 8). He also reports

that at portfolio level and horizons up to nine months, logistic distribution cannot be rejected

31

at 10% confidence level, while on a 10- to 12-month horizon suitable distributions are beta,

logistic, and Weibull. The evidence of Levy provides solid ground to consider the

mean-variance framework as the baseline case of modern finance theory. It is clear that such

evidence depends on the set of investigated distributions, the list of considered assets, as well

as the employed goodness of fit statistic, However, there are other studies, e.g. Chicheportiche

and Bouchaud (2012), that reject the joint elliptical distribution of returns.

The CAPM assumes a frictionless market where assets are perfectly divisible, there

are no transaction costs for selling, buying of holding securities, and no liquidity differences.

These are strong assumptions, and both researchers and practitioners alike have sought to

relax those assumptions and to shed light on the structure of the resulting market equilibrium.

Thus, Levy (1978) poses an identical variance-minimisation problem as that solved by

Markowitz and CAPM. However, he assumes that for some reason each investor invests in

some subset of the investment universe. The number of securities for the 𝑘-th investor was

𝑛𝑘, while the total number of shares in the market was 𝑛, 𝑛𝑘 ≤ 𝑛. Levy assumes that each

investor can invest in some 𝑛𝑘 pre-determined securities and the risk-free asset. 22 This

means that in mathematical terms, the optimisation is identical to CAPM, except that the

investment decision for investor 𝑘 involves not the market efficiency frontier (the efficiency

frontier that was feasible when each of the all 𝑛 assets were used in portfolio), but the

efficiency frontier spanned by using the 𝑛𝑘 assets, in which investor 𝑘 could invest. Thus

the efficiency frontier attainable by investor 𝑘 is in the interior of the market efficiency

frontier, and the capital allocation line for investor 𝑘 is below the capital allocation line for

the unconstrained investors (those who could invest in all assets).

22 Selecting an optimal portfolio subject to cardinality constraint, i.e. selecting 𝑛𝑘 shares out of 𝑛 available shares, is a hard problem of combinatorial optimisation.

32

Figure 2: Efficient frontier with 30 stocks as of December 31, 2013, and a subset of 7

stocks

Source: author’s calculations

This point is illustrated on Error! Reference source not found. above. There we use 3

0 large stocks (approximately those of Dow Jones Industrial Average) to calculate the

efficiency frontier as at the end of 2013. The risk-free portfolio almost coincides with the

origin, as the risk-free rate of return (in that case: one-month constant maturity US treasury

bills) is just 0.02% p.a. The higher curve on that plot shows the efficiency frontier that could

be reached by investing in all of the 30 assets. For the purposes of the example, we consider

that these 30 DJIA stocks are all the available securities in the market. Then the tangency

portfolio is given by the point 𝑀. According to CAPM, each investor in this economy would

hold some combination of the risk-free asset 𝑅 and the market portfolio 𝑀. Thus, the

optimal allocations for investors (the capital market line) would be the line 𝑅𝑀. Should some

investors be willing to earn higher return than 𝑀, they should leverage their position, i.e.

borrow money at the risk-free rate 𝑟𝑓 and invest in the market portfolio 𝑀. Thus, 𝑅𝑀 is the

capital market line to the unconstrained investors.

Then we consider an investor who prefers another investment strategy: the ABC

strategy. This strategy reasons as follows: holding all 30 stocks is impractical because of costs

for maintaining and rebalancing the portfolio. Instead, that investor choses to invest only in

stocks with tickers starting with letters A, B, or C. This could be a practical strategy in its

idiosyncratic way because one could find the price quotes in the newspaper easily – they

33

would be on the first page of market data. So in this particular case our investor is limited to

investing only in the following stocks: AA (Alcoa Inc), AXP (American Express), BA

(Boeing), BAC (Bank of America), CAT (Caterpillar), CSCO (Cisco Systems), and CVX

(Chevron). This is not too bad a strategy as the stocks are from quite diverse sectors of the

economy. Also, the expected return upon investment only in the ABC-stocks (portfolio 𝐾)

even slightly outperforms portfolio M in terms of expected return. Unfortunately, the risk

taken by the ABC investors is 2.6 times higher compared to the market risk as measured by

the standard deviations of returns for portfolio 𝑀.

Solving the same problem as the CAPM in the previous section, the ABC investor

would be in equilibrium if

𝜇𝑖 − 𝑟𝑓 = (𝜇𝑘 − 𝑟𝑓) 𝜎𝑖𝑘

𝜎𝑘 2 .

Levy points out that this equilibrium condition is very different from that implied by CAPM:

the market price of risk (= (𝜇𝑘 − 𝑟𝑓)/𝜎𝑘 2, i.e. the slope of the capital market line) is measured

relative to the co-variation of asset 𝑖 with the constrained portfolio 𝐾 instead with the

unconstrained portfolio 𝑀. The covariance of asset 𝑖 is also measured relative to portfolio

𝐾 rather than the market portfolio 𝑀. Thus, investors in the same security 𝑖 face different

prices of risk and covariance with the constrained portfolio that depended on what other assets

the investor has selected for his portfolio. For example, the securities of Alcoa are included in

both portfolio 𝐾 and portfolio 𝑀, yet for portfolio 𝐾 the Sharpe ratio (the price of risk) was

0.4 compared to 1.0 for portfolio 𝑀. This plurality of individual variance-efficient frontiers

obscures its implications for the equilibrium prices. However, Levy (1978) demonstrates that

if investors hold very undiversified portfolios, then expected returns in equilibrium would

depend on idiosyncratic risk. At the same time, betas with the whole market could be

irrelevant because they would include many covariance terms with other securities, which

investors actually may not hold in their portfolios. One way to look at this could be to note

that the covariance of returns of a portfolio with 𝑛 assets and returns 𝜇𝑛 would by

definition equal the sum of 𝑛 individual variance terms plus (𝑛2 − 𝑛) covariances between

individual assets. When 𝑛 is large, the variance of the market portfolio would depend on the

covariance terms and the impact of individual variances on portfolio variance would be

smaller; with a smaller number of assets in the portfolio, the impact of individual variances on

portfolio variance could not be neglected. Upon aggregation of the individual equilibrium

conditions, Levy demonstrates that in his model the market price of risk from the CAPM

34

model (𝛾)23 is not applicable. Instead, he finds that the market price in the constrained model

(𝛾1) equals 𝛾1 = 𝛾 𝜎𝑚 2

∑𝑘 𝑡𝑘𝜎𝑘 2, where the sum in the denominator runs over the optimal portfolios

for each investor 𝑘, and 𝑡𝑘 is the share of the risky portfolio of investor 𝑘 in the total

market. This leads him to conclude that “the classic CAPM may be the approximate

equilibrium model for stocks of firms which are held by many investors (for example,

AT&T), but not for small firms whose stocks are held by a relatively small group of

investors.” (Levy, 1978: 650).

In terms of its implications for empirical testing, the model developed by Levy (1978)

highlights the difficulties involved in such tests. In particular, allocation decisions are based

on betas with some unobservable portfolios that are different from the market portfolio.

Therefore, beta with market portfolio needed not be a good measure of the risk faced by the

under-diversified investors. In fact he shows that the betas for constrained investors would be

higher than those in the unconstrained CAPM world, and hence the average beta across all

investors would also be strictly higher than the CAPM beta. In the absence of a good proxy of

true betas, Levy (1978) reasons that idiosyncratic variances could be expected to be superior

predictors of the price behaviour than the traditional beta. (p. 654)

Merton (1987) develops a similar model but due to the somewhat different structure of

the model, he is able to draw more specific conclusions about the size of the premium

attributable to idiosyncratic risk. His model is again one of a two-period economy where some

of the investors ‘know’24 only a proper subset25 of the universe of available securities. In the

complete information case where all investors know all securities, the model reduces to the

CAPM. In Merton’s model some investors may know all securities, but at least some of the

investors are assumed to ‘know’ only some of the securities. The ‘knowledge’ of a security in

that model is perfect – if an investor was ‘informed’ of a security, he or she knows the return

generation process for that security and agrees with all other investors who know that security

on the parameter values for that security (conditional homogeneous beliefs). The key

behavioural assumption in the model is that investor would include a certain security 𝑘 in his

23 𝛾, also called Sharpe ratio, equals to excess return on the market portfolio divided by the standard deviation of the market portfolio, i.e. 𝛾 = (𝜇𝑚 − 𝑟𝑓)/𝜎𝑚. 24 ‘Knowing’ here has essentially the same meaning as in the work of Levy (1978), i.e. if an

investor ‘knows’ a security, it means that he has somehow pre-selected that security for his

portfolio. The model makes no assumptions as to how investors came to know or chose to

know some security. 25 A subset is said to be proper if there are elements of the superset that are not in the subset.

35

portfolio optimisation only if the investor ‘knows’ that security. Merton (1987) motivates that

assumption as follows (op. cit., p. 488):

“The prime motivation for this assumption is the plain fact that the

portfolios held by actual investors (both individual and institutional) contain only

a small fraction of the thousands of traded securities available. There are, of

course, a number of other factors (e.g., market segmentation and institutional

restrictions including limitations on short sales, taxes, transactions costs,

liquidity, imperfect divisibility of securities) in addition to incomplete

information that in varying degrees, could contribute to this observed behavior.

Because this behavior can be derived from a variety of underlying structural

assumptions, the formally-derived equilibrium-pricing results are the theoretical

analog to reduced-form equations.”

The model assumes a two-period economy comprising 𝑛 firms. The shares of those

firms are traded at the beginning of the period. The cash flows generated from investing in

those firms consist of three components – an expected return on the investment, an

economy-wide shock, the exposure (loading) to which is company-specific, and idiosyncratic

shock for each company. The economy-wide shock in this model specifies the correlation

structure between the returns of the different companies.26 If the starting investment for

company 𝑘 is 𝐼𝑘, then the end-of-period cash flows (�̃�𝑘) are given by

�̃�𝑘 = 𝐼𝑘[𝜇𝑘 + 𝑎𝑘�̃� + 𝑠𝑘𝜀�̃�], (2)

where 𝜇𝑘 , 𝑎𝑘 and 𝑠𝑘 are fixed company-specific parameters describing the production

technology of company 𝑘. �̃� is assumed to be a random variable, describing the state of the

world at the terminal period. It is a common factor reflecting the uncertainty in the model and

is the common factor underlying the cash flows of all companies 𝑘 = 1…𝑛. It is assumed

that it has zero mean and unit variance, i.e. 𝔼(𝑌) = 0 and 𝔼(𝑌2) = 1 (in order to avoid

adding indices 0 and 1 to starting and end-of-period values we follow the notation of

Merton (1987) and use tilde to mark realised, end-of-period values, and bar to mark the

expected values). 𝜀�̃� represents the idiosyncratic risk in the model, i.e. shocks to the

26 Let two random variables 𝑋 and 𝑌 depend on a common random variable 𝑍 and independent shocks 𝜀𝑥 and 𝜀𝑦 , i.e. 𝑋 = 𝛽𝑥𝑍 + 𝜀𝑥 , and 𝑌 = 𝛽𝑦𝑍 + 𝜀𝑦 . Then, (𝑋 − 𝛽𝑥𝑍)

and (𝑌 − 𝛽𝑦𝑍) are independent, i.e. 𝔼(𝑋 − 𝛽𝑥𝑍)(𝑌 − 𝛽𝑦𝑍) = 0 , so that 𝑐𝑜𝑣(𝑋, 𝑌) =

𝛽𝑥𝛽𝑦𝜎𝑧 2.

36

end-of-period cash flows that are specific to firm 𝑘 and do not depend on either the

idiosyncratic shocks to the rest of to firms, or on the state of the world ( �̃� ), i.e.

𝔼(𝜀�̃�|𝜀1̃, 𝜀2̃, … , 𝜀�̃�−1, 𝜀�̃�+1, … , 𝜀�̃�, 𝑌) = 0, ∀𝑘.

Let 𝑉𝑘 denote the equilibrium value of company 𝑘 at the initial point, and let �̃�𝑘 be

the realised return per dollar from investing in company 𝑘. Since this is a two-period model,

all of the cash flows of company 𝑘 are distributed to shareholders at the end of the period.

Hence, �̃�𝑘 ≡ �̃�𝑘/𝑉𝑘 = �̅�𝑘 + 𝑏𝑘�̃� + 𝜎𝑘𝜀�̃� , where �̅�𝑘 = E(�̃�𝑘) = 𝐼𝑘𝜇𝑘/𝑉𝑘 , 𝑏𝑘 = 𝑎𝑘𝐼𝑘/𝑉𝑘 ,

and 𝜎𝑘 = 𝑠𝑘𝐼𝑘/𝑉𝑘. Besides the 𝑛 shares, the economy also has a risk-free asset, the return

on which is denoted by 𝑅 (note that 𝑅 denotes one plus the risk-free rate 𝑟𝑓, 𝑅 ≡ 1 + 𝑟),

and one other composite asset that combines the risk-free asset with a forward contract on the

state of the world 𝑌,27 the return on which is given by �̃�𝑛+1 = �̅�𝑛+1 + �̃� (the index 𝑛 + 1

shall be used to denote that composite security). The risk-free and the composite securities are

assumed to be interior to the efficient frontier so that investment in these securities is zero.

Investors are assumed to be risk-averse. Each investor 𝑗 is endowed with an initial

endowment 𝑊𝑗 that he invests in a portfolio composed of some or all of the (𝑛 + 2)

securities traded in the market at the initial date. The return on that portfolio is �̃�𝑗, so the

wealth at the terminal period is �̃�𝑗 = �̃�𝑗𝑊𝑗. The investor preferences are assumed to be

quadratic and specified as 𝑈𝑗 = 𝔼(�̃� 𝑗) −

𝛿𝑗

2𝑊𝑗 𝑉𝑎𝑟(�̃�𝑗). For the latter parts of the derivation

it is also assumed that investors have identical risk preferences (i.e., 𝛿𝑗 = 𝛿, ∀𝑗) and initial

endowments (𝑊𝑗 = 𝑊, ∀𝑗).

In this setting Merton demonstrates that the resulting equilibrium value �̃�𝑘 of firm 𝑘

equals the equilibrium value 𝑉𝑘 ∗ when all investors are informed of security 𝑘, discounted by

a factor 1/[1 + 𝜆𝑘/𝑅], where 𝑅 = 1 + 𝑟𝑓 is the risk-free return, and 𝜆𝑘 = (1 − 𝑞𝑘)Δ𝑘

measures the information diffusion about security (company) 𝑘, with 𝑞𝑘 being the share of

investors ‘knowing’ security 𝑘 and Δ𝑘 being the shadow cost (Lagrange multiplier) of

‘knowing’ security 𝑘. If all investors are informed of security 𝑘, then 𝑞𝑘 = 1, ∀𝑘, and so

𝑉𝑘 = 𝑉𝑘 ∗. However, when there are investors who were not informed of that company, then

𝑞𝑘 < 1 and the value of the company would be below the equilibrium with complete

knowledge (the CAPM equilibrium). From there Merton deduces that the spread between the

expected return on security 𝑘 under incomplete (�̅�𝑘) and complete (�̅�𝑘 ∗) information would

be proportionate to its shadow cost:

27 Hence the market is complete.

37

�̅�𝑘 − �̅�𝑘 ∗ = 𝜆𝑘(�̅�𝑘/𝑅).

Aggregating across securities he arrives at the following Security Market Line (SML)

equation under incomplete information

�̅�𝑘 − 𝑅 = (𝜆𝑘 − 𝛽𝑘𝜆𝑀)⏟ 𝛼𝑘

+ 𝛽𝑘(�̅�𝑀 − 𝑅), (3)

where 𝑅𝑀 equals (one plus) the return on the market and 𝜆𝑀 is the weighted average

shadow cost of incomplete information over all securities. Again, in case of complete

information where 𝜆𝑘 = 0 for all 𝑘 (and so 𝜆𝑀 = 0), the SML reduces to the well-known

CAPM case. Hence the prediction of the model is that the less known securities should earn a

spread over securities which were well known to the market.

The model comparative statistics (op. cit., p. 494-499) examines the cross-sectional

differences implied by the model. In particular, if 𝜓(𝑌) denotes the elasticity of the expected

excess return for security 𝑘 (|�̅�𝑘 − 𝑅|) with respect to some parameter 𝑌 of the model, i.e.

𝜓(𝑌) = 𝑑log|�̅�𝑘 − 𝑅|/𝑑log(𝑌) , Merton (1987) deduces that 𝜓(𝑏𝑘) > 0 , 𝜓(𝑥𝑘) > 0 ,

𝜓(𝜎𝑘 2) > 0, and 𝜓(𝑞𝑘) < 0 , where 𝑏𝑘 is the loading of returns on security 𝑘 on the

common factor 𝑌, 𝑥𝑘 is the share of company 𝑘 in the total market, 𝜎𝑘 2 is the idiosyncratic

variance of company 𝑘, and 𝑞𝑘 is the share of investors ‘knowing’ security 𝑘.

Thus the model predicts that the expected excess returns are higher for companies of

larger size (higher 𝑥𝑘) and higher exposure to the common factor (higher 𝑏𝑘). Since 𝛽𝑘 is

an increasing function of 𝑏𝑘, 28, the latter implies that companies with higher beta should also

earn higher expected return. On the other hand, more obscure companies (low 𝑞𝑘) should

earn higher expected returns over the well-known ones. For our study, an important prediction

of the model is that the positive elasticity of expected excess returns with respect of

idiosyncratic variance (𝜎𝑘 2) means that idiosyncratic risk is priced and investors in companies

with higher idiosyncratic risk earn a premium for assuming idiosyncratic risk. This relation is

not without qualifications: the amount of the premium is not constant but dependent on how

widely followed the security is. In equation (3) above the excess return depends on the

magnitude of the shadow cost of knowing security 𝑘 (𝜆𝑘) net of the average shadow cost of

knowing the securities on the market (𝜆𝑀). Merton (1987) shows that 𝜆𝑘 = (1 − 𝑞𝑘)Δ𝑘,

where Δ𝑘 = �̅�𝑘 − 𝑅 − 𝑏𝑘(�̅�𝑛+1 − 𝑅) is the excess return on asset 𝑘 relative to the

predicted return from exposure on the market factor as captured in the forward asset

dependent on the state of 𝑌. For a given delta, the amount of the premium on holding security

28 𝛽𝑘 = (𝑏𝑘𝑏 + 𝑥𝑘𝜎𝑘 2)/𝑣𝑎𝑟(�̃�𝑚), eq. 22 on p. 493 in Merton (1987)

38

𝑘 depends on the share of investors aware of the asset. In particular, we should not expect to

find a constant premium for idiosyncratic risk in the cross-sectional regressions. If we restrict

ourselves to widely followed securities, we should expect 𝛼𝑘 close to zero. Thus, depending

on the types of shares that we have in the test sample, failure to reject the null hypothesis that

𝐻0: 𝛼𝑘 = 0 could in fact be a finding consistent with Merton’s rather than a contradicting

one.

Empirical tests of the proposition that idiosyncratic risk should be priced are hindered

by the complex interplay between idiosyncratic risk and the rest of the variables in the

estimation of equilibrium returns. Much of the empirical literature focuses on single point

estimates of the idiosyncratic risk premium. Furthermore, the aggregation in portfolios using

equal capitalisation of NYSE breakpoints could reduce the share of smaller, less known

securities and could apparently reject the predictions of Levy (1978) and Merton (1987),

whereas in fact the results could be interpreted as non-contradicting the two extensions of

CAPM. In view of that, it seems particularly important to be cautious about discarding stocks

from the sample, or aggregating in portfolios biased towards larger and presumably better

known stocks, for which the idiosyncratic risk premium is predicted to be small or

non-existent.

More recently, Malkiel and Xu (2004) revisited the problem of market equilibrium

when investors are constrained from holding all securities. In their model the covariance

matrix includes three (groups of) assets, hence the matrix has the form

𝑉 = (

𝜎𝑎 2 𝜎𝑎𝑏 𝜎𝑎𝑐

𝜎𝑎𝑏 𝜎𝑏 2 𝜎𝑏𝑐

𝜎𝑎𝑐 𝜎𝑏𝑐 𝜎𝑐 2

).

There are three investors in the economy. The second investor could invest in all three assets

(𝑎, 𝑏, and 𝑐), while the first and the third investor are constrained from investing in assets 𝑎

and 𝑐, respectively. Thus the variance matrices perceived by the constrained investors, 𝑉𝑏𝑐

and 𝑉𝑎𝑏, are obtained from the original matrix 𝑉 where the row and column corresponding

to the prohibited assets are replaced by zeros. The covariance function perceived by the

constrained investor then has the form

𝑉∗ = ( 𝑛1

𝑛1+𝑛3 ( 0 0 0 𝑉𝑏𝑐

−1) + 𝑛3

𝑛1+𝑛3 (𝑉𝑎𝑏

−1 0 0 0

))

−1

,

where 𝑛1 and 𝑛3 were the number of constrained investors in the first and the third group,

respectively. All investors maximise quadratic utility function with risk-tolerance parameter

𝜏, i.e. 𝑢(𝑊) = 𝔼(𝑊) − 1

2𝜏 𝑣𝑎𝑟(𝑊). In that setting Malkiel and Xu (2004) demonstrate that

39

instead of the usual CAPM equilibrium ((𝜇 − 𝑟𝑓1) = 1

𝑛𝜏 𝑉𝑆), where 𝑆 is the supply vector for

each security, the equilibrium relationship would instead has the form

(𝜇 − 𝑟𝑓1) = 1

𝑛𝜏 𝑉𝑆 +

𝜂1.3

𝑛𝜏 𝑉𝜔,

where 𝜂1.3 = (𝑛1 + 𝑛3)/𝑛 is the share of constrained investors, 𝑛 = 𝑛1 + 𝑛2 + 𝑛3 is the

total number of investors, and 𝜔 = [(𝐈 − 𝑉∗ −1𝑉)−1 − 𝜂1.3𝐈]

−1𝑆 is a supply adjustment. In

particular, Xu and Malkiel (ibid., eq. 13) show that under the assumption of uncorrelated

idiosyncratic innovations, the idiosyncratic risk premium for security 𝑖 has the form

(𝑘𝛾𝜔∗,𝑖𝜎𝑖 2), where 𝑘 is a constant, 𝛾 is the Sharpe ratio (market price of risk) that translates

the idiosyncratic variance into risk premium,29 𝜎𝑖 2 is the usual measure of idiosyncratic

variance, and 𝜔∗,𝑖 was the 𝑖-th element of 𝜔 vector. The presence of the product (𝜔∗,𝑖𝜎𝑖 2)

in the formula for the premium, rather than just the idiosyncratic variance, suggests that the

risk premium for assuming idiosyncratic risk depends on the ‘supply adjustment’ for the

particular security (they term this undiversified idiosyncratic risk), rather than the simple

(total) idiosyncratic risk. As with the models of Levy and Merton, a principal problem in

testing the model is estimating the price adjustment due to the unobservability of the reference

portfolios against which investors measure risk and return. In addition to the cross-sectional

tests of idiosyncratic risk, Malkiel and Xu (2004) propose to perform also time-series tests

and to employ a work-around for testing the significance of idiosyncratic risk: to estimate the

return on an idiosyncratic risk hedging portfolio. To that end they sort stocks into three size

buckets and two volatility buckets in the spirit of Fama and French (1993) and Fama and

French (1996) and find evidence for significant positive return on idiosyncratic risk portfolios.

A subtle implicit assumption of the CAPM and its extensions is the composition and

observability of the market portfolio. That assumption was identified by Roll (1977), who

pointed out that in any sample of assets there exist infinitely many mean-variance efficient

portfolios, and the relation between returns and the betas with any of these portfolios would

be linear, irrespective whether or not the market portfolio is mean-variance efficient.

Moreover, the market portfolio must include not only the assets traded at stock markets, but

also all other assets available to investors, including real estate, precious metals, commodities,

and human capital. Building on that critique, Eiling (2013) suggested an extension of the

CAPM pricing model to a multifactor pricing model that explicitly distinguished tradable and

29 Normally the denominator of the Sharpe ratio is the standard deviation of the market

portfolio; in this case, however, Xu and Malkiel defined the ratio as having the variance of the

market portfolio in the denominator, i.e. 𝛾 = (𝜇𝑚 − 𝑟𝑓)/𝜎𝑚 2 .

40

K non-tradable assets:

𝔼𝑟𝑡𝑟,𝑖 = 𝛽𝑚𝑘𝑡,𝑖 (𝔼𝑟𝑚𝑘𝑡 − �̅�∑𝐶𝑜𝑣(𝑟𝑚𝑘𝑡, 𝑅𝑛𝑡,𝑘)𝑞𝑛𝑡,𝑘

𝐾

𝑘=1

) +∑𝛽𝑛𝑡,𝑘,𝑖(𝑉𝑎𝑟(𝑅𝑛𝑡,𝑘)𝑞𝑛𝑡,𝑘)

𝐾

𝑘=1

,

where 𝔼𝑟𝑡𝑟,𝑖 is the expected return on tradable asset i, �̅� is the market aggregate risk

aversion coefficient, 𝑞𝑛𝑡 is the K × 1 vector of aggregate wealth due to the nontradable

assets divided by the total value of tradable assets, and 𝛽𝑚𝑘𝑡,𝑖 and 𝛽𝑛𝑡,𝑘,𝑖 are betas with the

traded market and each of the K non-traded assets. This shows that if idiosyncratic risk

correlates with some non-traded asset, then idiosyncratic risk could be priced by the market,

but the reason for that premium is not idiosyncratic risk per se, but rather the correlation with

the non-traded asset. In particular, Eiling (2013) finds that the non-tradable assets model with

(orthogonalized) industry-specific human capital can reduce the premium for idiosyncratic

risk by 36%.

In Table 1 we have summarised the principal theoretical models as they relate to this

study. The predictions of the CAPM should be considered as a baseline case, upon which the

remaining models extend. We have not covered the Arbitrage Pricing Theory developed by

Ross (1976) because the theory as such does not identify the factors that drive the stock

returns and thus would be of limited use for the present debate. Nonetheless, the structure of

that model requires that the explanatory factors should not be diversifiable, and thus the

predictions of that theory would overlap with those of the CAPM extensions, viz in absence

of limits to diversification idiosyncratic risk would be diversified away.

Table 1: Summary of selected theoretical models and their predicted correlation

between idiosyncratic risk and stock returns

Study Predicted

correlation

Comment

Sharpe (1964);

Lintner

(1965b)

None The studies formulate the Capital Asset Pricing Model (CAPM)

that predicts that in equilibrium idiosyncratic risk will be

diversified away and stocks with higher idiosyncratic risk will not

earn higher returns.

41

Study Predicted

correlation

Comment

Kahneman and

Tversky (1979)

Unspecified The authors develop the prospect theory of decision-making, which

places higher value on extreme gains or losses rather than on

expected lottery outcomes. Such a setting could allow higher

returns on skewed outcomes, e.g. higher prices (lower returns) on

securities with small probability of high gains, as in Barberis and

Huang (2008).

Chamberlain

(1983); Owen

and

Rabinovitch

(1983)

Unspecified The studies suggest that if returns are jointly elliptically distributed

then investor preferences could be expressed as a function solely of

expected return and variance, lending indirect support on the

mean-variance optimisation approach of Markowitz (1952).

Levy (1978) Positive The model predicts that if transaction costs, taxes, or other market

frictions prevent the full diversification of idiosyncratic risk, then

in equilibrium the market should price idiosyncratic risk and stocks

with higher idiosyncratic risk should earn, ceteris paribus, higher

returns.

Merton (1987) Positive The model extends the results of Levy (1978) and derives

comparative statics of the model, predicting the direction of

correlation between idiosyncratic premium and security parameters

(share of investors, size, beta with market, and share of investors

following the security).

Malkiel and

Xu (2004)

Positive The study extends the results obtained by Levy (1978) and Merton

(1987) and predicts that the premium on idiosyncratic risk would

depend on the total idiosyncratic risk scaled with a supply

adjustment.

Eiling (2013) Unspecified /

spurious

The author extends the pricing model of the CAPM with

non-tradable assets, as suggested by Roll (1977) and finds that

about a third of the premium to idiosyncratic risk is in fact due to

exposure to industry-specific human capital.

Source: the author

42

In this section we explored the principal theoretical models that explore the pricing of

idiosyncratic risk in frictionless and frictional markets. The development followed the

historical evolution of the debate in order to reveal in context the implicit and explicit

assumptions that underlie the CAPM. This enabled us to see the range of situations that could

invalidate the key prediction of the model, namely – the irrelevance of idiosyncratic risk for

the pricing of individual assets, namely: alternative decision-making rules than those assumed

by von Neumann and Morgenstern (1944), e.g. the behavioural finance theories; inadequacy

of the quadratic approximation to investor’s utility function that could ascribe some role to

higher moments like skewness and kurtosis; asset returns that are not jointly elliptically

distributed so that investor utility function may not be specified in terms of expected return

and variance alone. Special emphasis was placed on three studies that relate most directly

with the topic of our enquiry – Levy (1978), Merton (1987), and Malkiel and Xu (2004). The

three models offered alternative formulations of the portfolio-construction problem when

investors are constrained from holding the market portfolio and from diversifying entirely

their idiosyncratic risk. Those models provide some insight into how investor constraints and

asset characteristics jointly determine the equilibrium prices and returns of the risky assets.

Admittedly, unlike the CAPM which allowed analytical solutions, the idiosyncratic risk

premia of those models depend on characteristics that we are unable to observe directly, like

the limits on number of securities, the investment universe used by each investor in his

portfolio formation, and ultimately – the true distribution of idiosyncratic shocks.

Nevertheless, the models yield insight into the factors that could affect the correlation of the

idiosyncratic risk premium with other asset characteristics. In the following section we shall

review the findings of some key empirical studies on the correlation between idiosyncratic

risk and stock returns.

2.3. Empirical findings concerning idiosyncratic volatility

and the cross-section of stock returns

Empirical evidence on the composition of portfolios of investors diverges from the

predictions of CAPM. Blume and Friend (1975) are among the first to document that contrary

to the predictions of CAPM, investors hold rather concentrated portfolios. They examine a

sample of 17,056 tax returns from the 1971 tax year, as well as the distribution of assets in the

43

Federal Reserve Board’s 1962 Survey of the Financial Characteristics of Consumers, and find

that many portfolios are highly undiversified. They report that as much as 34.1% of the tax

returns list only one dividend-paying stock30, and 50.9% list up to two dividend-paying

shares. The results of the examination of the 1962 survey are similar: 50% of households

holding at least 63% in one asset, excluding holdings in mutual funds; if investment in mutual

funds is added as well, the concentration increases to 90% held in one asset.

Kelly (1995) studies portfolio diversification of a sample of 3665 households (the

‘regular sample’) from 1983 Survey of Consumer Finances, and another sample of 438

households from the top two percentiles of the income distribution. Of the 635 stockholders in

the regular sample he finds that only 35 (i.e., 5.5%) hold 10 or more shares, and only 11 of

these – 20 or more shares. In a regression analysis of the likelihood of holding more than 10

shares on a set of explanatory variables31 he finds that only the portfolio value and the

number of trades are significant in all specifications, and the share of stocks in a household’s

financial assets is statistically significant only in some specifications.

In the high-income sample, Kelly (1995) finds that the median number of shares held

(excluding shares in companies where household members works) is 10. With the increase of

portfolio values, diversification improves and for the highest bucket in that study ($1.3

million to $52.5 million) the share of portfolios with over 20 stocks increases to 61%. Many

more variables are found to be significant in the explanatory regressions, however the

explanatory power of those regressions (as measured by the pseudo-𝑅2) is significantly lower.

Goetzmann and Kumar (2001) examine 79,995 equity investment accounts in a large

(unnamed) discount brokerage house held by individual investors during a six-year period

from 1991 to 1996. In order to exclude dormant accounts, the study focuses on 41,039

accounts that have at least five trades over the surveyed period. The aggregate value of the

examined portfolios exceeds $2.5 billion. The median portfolio in the database is $13,869,

and the average portfolio is worth $35,629. The authors report that the average number of

stocks in a portfolio is just 4, and the median number is 3. More than a quarter of the

portfolios contain just 1 share, and less than 5% are reported to include 10 or more shares.

30 For tax purposes households were required to disclose only the dividend-paying

stockholdings; information on the ownership of non-dividend-paying shares was not available

in that sample. 31 Portfolio value, number of trades, share of portfolio, age, college education, management

job, self-employed, advice from broker, attitude to risk, term life insurance, whole life

insurance, investment in mutual funds, defined contribution pension plan, IRA (as % of

household financial assets), trust fund (as % of household financial assets)

44

The authors discuss whether those portfolios might be used for gambling on the stock market,

but dismiss that as unlikely in view of the material balances on the accounts in absolute and in

relative terms, compared to the annual household income. The authors find that the degree of

diversification increases with the increase of age, possibly reflecting increasing risk aversion.

They also find that the number of shares held increases with the value of the portfolio.

However, investors are not found to use correlations when structuring their portfolios, which

is found to result in suboptimal allocations. They report that just 15% of portfolio values are

invested in mutual funds and do not find evidence that less diversified investors compensate

the under-diversification by investing more in mutual funds. They conclude that the

insufficient diversification observed in their sample could result in pricing of idiosyncratic

risk, as suggested by the works of Goyal and Santa-Clara (2003) and Malkiel and Xu (2004).

Such findings are in stark contrast to the recommendations of various empirical

analyses of required levels of diversification. Statman (1987) examines the diversification

benefits of increasing the number of shares, assuming randomly constructed portfolios, and

finds that a minimum of 30 stocks for borrowing investors and 40 stocks for lending investors

are required for sufficient diversification.32 Other studies, however, suggest other thresholds.

For example, an earlier study by Elton and Gruber (1977) indicates that the benefits of

diversification are largely exhausted with a portfolio of just 10 to 15 securities. They observe

that increasing the number of securities from 1 to 10 eliminates about 50% of variance, and

increasing the number of assets to 20 improves that by just 5 percentage points.

Levy (1978) points out that the evidence on the composition of individual portfolios

need not imply a rejection of CAPM itself, and that the model could be still valid on

normative grounds as producing results that are consistent with the observed equilibrium

expected returns. However, since the empirical evidence in favour of CAPM is mixed too, he

examines the equilibrium expected returns when the number of assets, in which the investor

could invest, was limited. Such a cap on the number of assets is consistent with the empirical

evidence on the number of assets in portfolios, but could also reflect transaction costs or

imperfect divisibility of assets.

Following the introduction of CAPM, a number of papers investigated whether the

model held in practice, i.e. whether the expected market excess return was the sole factor

32 The numbers for borrowing and lending investors were different because the borrowing

investors were assumed to borrow at the risk-free rate plus a marking of 2 percentage points

while the lending investors were assumed to lend at the risk-free rate.

45

predicting asset returns. The first tests of the model were performed by Lintner (1965a)33,

who introduced a two-step procedure for testing the CAPM, which was further refined by

subsequent studies. The first step uses cross-sectional regression of realised returns on an

intercept and market returns: 𝑅𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖𝑅𝑚,𝑡 + 𝑒𝑖,𝑡. At the second step, returns 𝑅𝑖,𝑡 are

averaged to obtain average asset returns �̅�𝑖, �̅�𝑖 = 1

𝑇 ∑𝑡 𝑅𝑖,𝑡, and average returns are regressed

on betas: �̅�𝑖 = 𝑎1 + 𝑎2𝑏𝑖 + 𝑒𝑖. Taking into account that CAPM is formulated in terms of

excess returns, the expected values of 𝑎1 and 𝑎2 if CAPM holds are 𝑎1 = 𝑟𝑓, 𝑎2 = 𝑅𝑚 −

𝑟𝑓 . Lintner runs that two-step procedure using 301 stocks and returns for the period

1954-1963. In order to test if idiosyncratic risk is a priced by investors, at the second step

Lintner (1965a) includes not only intercept and asset betas, but also the variance of the error

term from the first-step regressions, 𝑆𝑒𝑖 2 ; if CAPM holds, one expects the variance term to be

redundant and its coefficient to be insignificant. Lintner (1965a) obtains the following

estimates:

�̅�𝑖 = 0.108 + 0.063𝑏 + 0.237𝑆𝑒𝑖 2

(0.009) (0.035)

𝑡 = 6.9 𝑡 = 7.8

.

These results provide partial support for CAPM. Indeed, they confirm that beta is a significant

predictor of average asset returns, and the coefficient is positive, consistent with the theory.

The slope of the regression equation (0.063) is significantly below the spread between market

returns and the risk-free rate, which over the examined period is about 0.165. The intercept

(0.108) is significantly higher than the risk-free rate. Finally, the coefficient of idiosyncratic

risk turns out positive and statistically significant, contrary to the prediction of CAPM.

Miller and Scholes (1972) confirm the findings of Lintner (1965a) over an extended

period. Furthermore, comparing the coefficient of multiple correlation they note that

idiosyncratic variance alone has higher correlation with average returns compared to beta

alone, with corresponding coefficients of multiple correlation of 0.28 and 0.19 ,

respectively. When both beta and idiosyncratic variance are used in the second-stage

regression, the coefficients of the two are positive and significant, and the coefficient of

multiple correlation stands at 0.33. The estimated model, however, shows the same deficits

as in Lintner’s study – the value of the intercept is significantly higher than the risk-free rate,

and the coefficient of beta is significantly below the spread between the market return and the

risk-free rate:

33 Reported on p. 192-95 in Levy (2012)

46

�̅�𝑖 = 0.127 + 0.042𝑏 + 0.310𝑆𝑒𝑖 2

(0.006) (0.026)

𝑡 = 7.40 𝑡 = 11.76

.

More importantly, Miller and Scholes (1972) point out that the specification of the

second-stage regression requires the use of the true betas but in fact uses the estimates from

the first-stage regression, and therefore the betas are measured with some error and are in fact

random. They show that this issue results in downward bias of the coefficient of beta. The

first tests of CAPM that addresses this error-in-variable problem are performed by Black et al.

(1972) and Fama and MacBeth (1973). Black et al. (1972) argue that correlations between the

pricing errors of individual securities prevent the existing methodologies from testing the

market model. To address these correlations, they propose to pool securities into portfolios of

similar securities. Then the pricing errors for the individual assets would cancel one another,

while any non-independence effect would be absorbed into the intercept; furthermore, the

error in the estimation of betas should also diminish as a result of the aggregation, and the

standard regression techniques and tests could be employed. In order to test the CAPM one

needs observations for returns for various values of beta, hence Black et al. (1972) propose to

form these portfolios based on values of beta, thus ensuring that the variance of beta will be

the highest possible. They observe that creating portfolios using estimates of beta obtained

from the tested sample would introduce sample selection bias; indeed, the portfolio containing

the securities with highest betas would likely also be exposed to the highest positive pricing

errors; similarly, the portfolio with the lowest betas would also be exposed to the securities

with the lowest negative pricing errors. Therefore, they propose to use estimated beta from a

previous time period as an instrument variable to construct portfolios, and then test portfolio

returns using their subsequent returns, which were not used to estimate portfolio betas. They

use monthly returns from the five years preceding the examined period to estimate betas that

are then used for portfolio formation for the following one year. For example, for all securities

that were traded as of January 1, 1931, and were traded for at least two of the preceding five

years, they estimate security beta using the available history from January 1926 until

December 1930. Then they form 10 portfolios based on the deciles of beta, and track the

monthly returns for each portfolio for each of the next 12 months of 1931. Then the

procedure is repeated with all securities traded as of January 1, 1932, with new betas

estimated using data from 1927–1931, and formation of 10 new portfolios. In that way they

obtain 35 years of monthly returns on ten portfolios from the 1,952 securities available to

47

them. For the second stage – the cross-sectional regression – they average returns over time

periods and test the model with average returns on averaged betas.

Black et al. (1972) estimate the coefficients of the market model for each of the 10

decile portfolios for each of the 35 years, obtaining coefficients separately for each portfolio.

They find that portfolio average return is indeed commensurate with portfolio beta. While

most of the intercepts are insignificant, the authors warn that due to non-stationarity of the

estimated coefficients, as well as some limitations of aggregations, the values of 𝑡-statistics

might materially understate the significance of the results. Using the data for all portfolios

they obtain the following pricing relationship:

�̅�𝑖 = 0.00359 + 0.01080𝑏

(0.00055) (0.00052)

𝑡 = 6.52 𝑡 = 6.53

.

The equation shows that market excess return is confirmed as a priced factor. The intercept

(𝛾0) in the above cross-sectional equation is significantly positive and different from zero,

which contradicts CAPM. Black et al. (1972) hypothesise that this suggests the existence of a

second factor. The time series regressions suggest that the intercept from time-series

regressions decreases with beta, as noted by Black et al. (1972), and these results also hold,

albeit in a weaker form, for the four sub-periods, except for the sub-period January, 1931 –

September, 1939, when the relationship is inverted.

Fama and MacBeth (1973) further refined the method for testing CAPM, and their

procedure became one of the most widely used methods for testing factor pricing models. The

first step in their procedure is similar to the method used by Black et al. (1972). Fama and

MacBeth (1973) firstly use seven years of data to form portfolios. Then for the next five years

they re-estimate the time series regressions in order to re-evaluate the betas of individual

securities. Finally, the last four years are used to fit the cross-sectional regressions. The end of

the period formation and initial estimation windows are then recalculated, so that the next

testing four-year period follows the preceding one without leaving gaps or intersecting it. In

the second step they run a cross-sectional regression of returns on intercept, beta, squared

beta, and idiosyncratic standard deviation (the standard deviation of the error term of the

time-series regression). The last two terms are intended to test for non-linearity in beta and for

the significance of idiosyncratic volatility as a pricing factor. Unlike Black et al. (1972), Fama

and MacBeth (1973) estimate that regression separately for each month rather than averaging

returns and betas in the testing period:

𝑅𝑖,𝑡 = 𝛾0,𝑡 + 𝛾1,𝑡𝛽𝑖,𝑡−1 + 𝛾2,𝑡𝛽𝑖,𝑡−1 2 + 𝛾3,𝑡𝑆𝑖,𝑡−1.

48

The significance of the hypothesised factor loadings (𝛾𝑖, 𝑖 = 1,2,3) is tested using the standard

𝑡-statistic:

𝑡(�̅�𝑖) = �̅̂�𝑖

𝑠(�̂�𝑖)/√𝑛 ,

where 𝑛 is the number of cross-sectional regressions, 𝑠(𝛾𝑖) is the standard error of the

regression coefficient, and �̅�𝑖 is the mean value of the respective factors loading, averaged

across all monthly cross-sectional regressions. Fama and MacBeth (1973) find that the factors

for idiosyncratic volatility and the non-linearity term are not significantly different from zero.

On the other hand, as pointed out by Levy (2012), the Fama and MacBeth (1973)

methodology “employs portfolios rather than individual assets; therefore, it has the advantage

of minimising the measurement errors in beta and the disadvantage of not testing asset pricing

of individual assets. Thus, in the case of supporting the CAPM, one cannot generalise it to

individual risky assets.” (p. 200)

Roll (1977) questions the previous tests of CAPM pointing out that the market

portfolio to which CAPM refers is the portfolio of all assets in the economy, including human

capital, real estate, privately held businesses, overseas assets. Clearly that portfolio is not

observable by econometricians, and its replacement by equity index introduces an

error-in-variable problem, which could account for observed anomalies detected by some of

the empirical tests. Furthermore, within the limited number of assets traded at stock markets

there is always a mean-variance efficient portfolio, and finding that a portfolio is or is not

mean-variance efficient has no bearings on the validity of CAPM.

The recent resurgence of the interest in idiosyncratic risk is triggered by Campbell et

al. (2000) who decompose individual stock returns into market, industry and idiosyncratic

components under the assumption of constant betas equal to 1. Using data from the Center

for Research on Security Prices (the CRSP data set) spanning the period from July 1962 to

December 1997 they calculate the monthly volatility series constructed from daily data. They

reject the unit root hypothesis at 5% confidence level for average market, industry, and

idiosyncratic volatilities. They also find a positive and statistically significant trend in average

idiosyncratic risk. Similar results are obtained using various frequencies (daily, weekly,

monthly) and according to the authors are not attributable to outliers. The implications of the

results of Campbell et al. (2000) are intriguing. They document that idiosyncratic risk

increased from 65% of total risk to 72% of total risk over the period from 1962 to 1997. This

increase suggests that correlations between individual assets should be declining, a

phenomenon also documented by their study, implying increasing benefits from

49

diversification. Furthermore, the authors document cyclicality of all volatility components,

with market, industry, and idiosyncratic volatilities all increasing during economic downturn.

Following the publication of Campbell et al. (2000), a number of authors published

studies concerning idiosyncratic risk, raising the questions whether idiosyncratic risk is

increasing over time, whether average idiosyncratic risk predicts market return, and whether

idiosyncratic risk predicts the cross-section of returns, which is the question also pursued by

this study.

In an important contribution to the debate Malkiel and Xu (2004) examine if

idiosyncratic risk is a significant predictor of expected returns. In their study they find that in

portfolio context (using the Fama–Macbeth methodology), CAPM beta is an important factor

in explaining cross-sectional differences of returns, but its effect declines over time. They find

that idiosyncratic risk is significant, irrespective of whether it is measured as volatility of

residuals from CAPM regressions, or Fama–French three-factor regressions. They

furthermore report that the size factor is dominated by the idiosyncratic risk as an explanatory

variable, and while the two of them are significant predictors when used individually, when

used jointly in multiple regression only the idiosyncratic risk variable remains significant (at

94% level). Running cross-sectional regressions with individual securities, they again find

that idiosyncratic volatility is a significant predictor that outweighs the size factor.

Spiegel and Wang (2005) note that there is a significant negative correlation between

liquidity and idiosyncratic volatility, and that expected stock returns are positively correlated

with idiosyncratic risk and negatively correlated with liquidity. They estimate idiosyncratic

risk using an Exponential Generalised Autoregressive Conditional Heteroscedasticity model

(EGARCH) with the three Fama–French factors as predictors of excess returns using monthly

data and employing two types of liquidity measures – “cost based” measures (Gibbs, Gamma,

Amihud, and Amivest) and “reflective” measures like traded volume or number of investors

holding a security. They find that when one controls for idiosyncratic risk, only the

dollar-traded volume has some predictive power over the cross-section of stock returns, while

idiosyncratic risk is consistently positively correlated to expected stock returns. Their results

suggest that the impact of a one standard deviation change in idiosyncratic risk is on average

between 2.5 and 8 times stronger than the impact of a corresponding one standard deviation

difference in liquidity.

In an important contribution to the field, Ang et al. (2006) investigate whether

idiosyncratic volatility explains the cross-section of market returns. They fit the Fama–French

50

three-factor model 34 to the daily excess returns on individual stocks and calculate

idiosyncratic volatility as standard deviation of the error term over the preceding month. They

use a ‘1/0/1’ portfolio formation strategy35 in order to examine whether portfolios formed on

total volatility and idiosyncratic volatility have significantly different yields. They find that

stocks ranked on idiosyncratic volatility exhibit a consistently negative correlation with

expected returns after controlling for various factors (size, book-to-market, leverage, liquidity,

volume, turnover, bid-ask spreads, co-skewness, dispersion of analyst forecasts), across

sub-samples (NYSE stocks only, NBER recessions and expansions, high and low volatility

episodes, and sub-periods), and for various ‘L/M/N’ strategies. Their findings are puzzling as

they contradict both CAPM (idiosyncratic volatility should not be priced at all) and Levy

(1978) and Merton (1987) (idiosyncratic risk should earn positive risk premium).

In a follow-up contribution, Ang et al. (2009) provide further evidence from the G7

countries and other developed markets that the spread between the first and fifth quintiles of

portfolio sorts based on idiosyncratic volatility is again negative, standing at −1.31 per cent

per month after controlling for world market, and size and value factors.

The findings of the two studies of Ang et al. (2006, 2009) caused significant

controversy and prompted many authors to re-examine their findings. Bali and Cakici (2008)

investigate how idiosyncratic risk is priced using NYSE/AMEX/NADAQ data over the period

July 1963 – December 2004 and find that the results of Ang et al. (2006) are not robust with

respect to weighting scheme, time frequency, portfolio formation, and screening for size,

price, and liquidity. Using equally-weighted instead of value-weighted portfolio returns, they

find that the spread between the highest-risk portfolio and the lowest-risk portfolio averaged

insignificant positive +0.02 percentage points. Similar small positive spread (+0.08) is also

obtained using reciprocal idiosyncratic volatilities as weights for averaging. Furthermore, they

point out that the capitalisation of the high-risk and low-risk quintile portfolios used by Ang et

al. (2006) are quite different: the high-risk portfolio contains 20% of all stocks but accounts

on average for just 2% of market capitalisation; in contrast, the low-risk portfolio

comprising of 20% of the stocks with the lowest idiosyncratic risk accounts for 54% of

market capitalisation. Using the NYSE quintile breakpoints results in more balanced portfolio

34 The model is briefly discussed in the next chapter. 35 The ‘L/M/N’ notation means that at moment 𝑡 the idiosyncratic volatilities were estimated using ‘L’ months of daily data from month (𝑡 − L − M) to month (𝑡 − M); then at time 𝑡 portfolios are formed based on the quantiles of the distribution of idiosyncratic volatility, and

these portfolios were held for ‘N’ months.

51

capitalisations; with those NYSE breakpoints they find that all weighting schemes (value,

equal, inverse of idiosyncratic risk) yield statistically insignificant spreads between high-risk

and low-risk portfolios. The spread is also insignificant when the breakpoints are selected

based on market capitalisations so that each portfolio has approximately equal market value.

Bali and Cakici (2008) also observe that the results appear sensitive to calculation

frequencies. For all breakpoints and weighting schemes, they find no significantly positive or

negative average return spread for the NYSE/AMEX/NASDAQ stocks. Finally, they also

argue that the results of Ang et al. (2006) are not robust with respect to price, size and

liquidity factors. In the three sub-samples of large/liquid stocks, large/high priced stocks,

liquid/high priced stocks they find no evidence for a significant link between idiosyncratic

volatility and the cross section of expected returns. However, it should be noted that in Bali

and Cakici (2008) idiosyncratic risk tends to lose its predictive significance when re-assigning

stocks with high capitalisation, and with presumably higher visibility, to high-volatility

portfolios. Such re-assignment should reduce the spread of yields between the portfolios,

consistent with the comparative statics of Merton (1987), and hence make more difficult the

rejection of the null hypothesis of no difference in smaller samples.

In an important contribution, Fu (2009) commented that the theoretically correct

variable to explain expected returns is the expected idiosyncratic risk rather than the past

idiosyncratic risk. He points out that in his sample the first-order autocorrelation of

idiosyncratic volatilities is just 0.33 , which according to him renders previous-month

idiosyncratic volatility an unsatisfactory proxy for current-month idiosyncratic volatility. The

same conclusion is reached when Fu (2009) tests the series of realised idiosyncratic

volatilities for unit roots. The null hypothesis of unit root is rejected for 90% of the series in

his sample36, which implies that previous-month volatility is not an unbiased predictor of

current-month volatility. In order to overcome the limitations of the approach of Ang et al.

(2006), he proposes to use the Exponential Generalised Autoregressive Conditional

Heteroscedasticity (EGARCH) model to forecast next-period expected idiosyncratic volatility.

He addresses concerns raised by other authors that the pooling of securities in quantile

36 Fu (2009) uses Dickey-Fuller test (see Dickey and Fuller (1979)) with drift using levels

and logs, i.e.

𝐼𝑉𝑂𝐿𝑡+1 − 𝐼𝑉𝑂𝐿𝑡 = 𝛾0 + 𝛾1𝐼𝑉𝑂𝐿𝑡 + 𝜂𝑡 , ln𝐼𝑉𝑂𝐿𝑡+1 − ln𝐼𝑉𝑂𝐿𝑡 = 𝛾0 + 𝛾1ln𝐼𝑉𝑂𝐿𝑡 + 𝜂𝑡 . Fu required at least 30 volatilities to perform the test. The null hypothesis that 𝛾1 = 1 was rejected at 1% confidence level for 89.97% of the tested securities in levels (𝐼𝑉𝑂𝐿𝑡) and for 87.81% of the tested log-volatilities (ln𝐼𝑉𝑂𝐿𝑡).

52

portfolios is one of the reasons of the puzzling results of the studies of Ang et al. (2006, 2009)

by using Fama–French regressions with individual securities as assets. He finds that

cross-sectional returns are significantly (in both statistical and economic sense) positively

correlated with idiosyncratic returns, with stocks with idiosyncratic volatility of one standard

deviation higher than the average, earning a risk premium of approximately 1 percentage

point, and a zero-investment portfolio that is long in the 10% of the stocks with the highest

idiosyncratic volatility and short in the 10% with the lowest idiosyncratic volatility earns on

average a premium of 1.75 percentage points per month. He argues that reversals of

idiosyncratic volatilities are a significant contributing factor for the puzzling results obtained

by Ang et al. (2006). The findings of Fu (2009) are qualitatively similar to the results of Bali

et al. (2011) who report that once they control for the maximum positive return in the

previous month, the anomalous regression slopes reported by Ang et al. (2006) are reversed.

One limitation of the study of Fu (2009) that is identified by Huang, Liu, Rhee and

Zhang (2012) is the omission of lagged return as an explanatory variable in the cross-sectional

regression, which could result in omitted-variable bias. Furthermore, the study design did not

allow for explicit analysis of how idiosyncratic risk premium changes with the profile of

investors holding the security. In a follow-up study, Fu and Schutte (2010) revisit these

issues. Idiosyncratic risk is confirmed as a significant factor of the cross-section even in the

presence of the lagged return. Consistent with the underlying economic theory, the slope for

idiosyncratic risk in the cross-sectional regressions is found to be higher in samples involving

securities with higher individual ownership and small size of orders (an indication of

individual investing). On the other hand, for stocks with significant institutional ownership

there is no robust link between idiosyncratic risk and returns.

Brockman et al. (2009) extend the study of Fu (2009) by examining a set of 44

international markets including 58,000 shares and spanning 27 years. Using Fama–

Macbeth cross-sectional regressions and individual securities as assets, they find that

idiosyncratic risk (measured in terms of a monthly EGARCH model fitted on the full series

rather than on expanding windows), correlates positively with expected returns. Furthermore,

they estimate the link between the idiosyncratic risk premium and various explanatory

variables like turnover, breadth of analysts’ forecasts, trading costs, errors of earnings

forecasts.

The use EGARCH models to predict volatilities in the methodology of Fu (2009) is

questioned by Guo et al. (2014); they argue that Fu uses month-𝑡 data to estimate the

parameters of EGARCH. This in their view introduces a look-ahead bias, which may have

53

impact on the observations with higher return (as the likelihood of significant deviation is

small, it implies material change in the parameters of the EGARCH model). Once this

look-ahead bias is controlled, Guo et al. (2014) find no significant relation between

idiosyncratic risk and the cross-section of returns.37 The authors also point out that their

criticism also extends to the use of EGARCH with full sample.

Brooks et al. (2011) examine how the benefit of diversification increases with the

number of securities in an investor’s portfolio. They construct portfolios with a number of

securities ranging from 15 to 70 with an increment of 5 and run Fama–Macbeth

cross-sectional regressions (with only beta and idiosyncratic risk as dependent variables).

They find that idiosyncratic risk is priced in portfolios of up to 50 securities; after that

threshold idiosyncratic risk becomes insignificant. However, one should note that these

cross-sectional tests seem to confirm that the idiosyncratic risk premium is concentrated in a

relatively small number of smaller securities with limited investor base and that construction

of portfolios with a large number of securities blurs the risk-return relationship along the lines

suggested by Ang et al. (2010). A better understanding of the implications of

under-diversification on equilibrium prices might be achievable via simulations. Relative to

the three-factor model of Fama and French (1993), the study finds that idiosyncratic risk

pricing is detectable only for portfolios containing less than 20 shares. Overall, these results

may suggest that using a constant number of portfolios in studies of the cross-section would

result in an increasing number of assets in each cell and might result in loss of explanatory

power of some variables over time.

Eiling (2013) explores one possible explanation for the significance of idiosyncratic

volatility. She builds on the observation of Roll (1977) that the true market portfolio in

CAPM includes all available assets, and in particular – human capital. She develops a model

that suggests that systematic risk related to industry-specific human capital is omitted as a

source of risk and ends up in the idiosyncratic residuals. She suggests that this mechanism

might explain the predictive performance of idiosyncratic risk in the cross-section of returns.

She calculates the monthly human capital returns in terms of per-worker labour income

growth using two-month averages. She finds that industry-specific human capital explains

between 10% and 36% of the idiosyncratic risk premium.

Cao (2010) and Cao and Xu (2010) argue that a distinction should be made between

37 We shall deal with this criticism and how our approach avoids this look-ahead bias in the

methodology section.

54

long-term and short-term risk. They suggest that it is the long-term idiosyncratic risk that

actually explains the cross-section of returns. However, the interpretation of their results is

obscured by the use of a Hodrick–Prescot filter to extract the cyclical component of volatility

using parameters applicable to quarterly macroeconomic series. Furthermore, the long-term

volatilities from the Hodrick-Prescot filter are obtained using the full series and are thus

exposed to look-ahead bias (cf. the critique of Huang et al on the approach of Fu above), with

different horizons of application of the filter resulting in a different split of the overlapping

volatilities into long-term and short-term components.38 More importantly, the outcome of

the application of an HP filter may be very closely related to the residuals from the simple

OLS regression. Indeed, observe that squared idiosyncratic returns are an estimator of

idiosyncratic volatility, and therefore the variance of the residuals from the OLS regression

could be interpreted as a moving average filter of the trailing series of volatilities. In the

absence of a strong trend in volatilities39, the output of the HP filter would likely be close to

the moving average, and thus while the arguments of Cao (2010) and Cao and Xu (2010) are

valuable, their evidence cannot be seen as conclusive.

A related study by Ruan et al. (2010) explores the idea that the contradictory empirical

results are due to noise in the estimated volatilities by applying a dual predictor approach to

estimating the unobservable aggregate idiosyncratic risk from the volatilities of

equally-weighted and value-weighted portfolios of different sizes. In that setting the authors

find that aggregate idiosyncratic risk is a significant predictor of market excess returns.40

Huang, Liu, Rhee and Zhang (2012) base their forecasts on an ARIMA model fitted

using realised monthly volatilities calculated by the method employed by Ang et al. (2006).

They test the hypothesis with four estimators of idiosyncratic volatility and find that

38 To clarify this point, consider a security X over the periods 2000-2008 and 2000-2016. If

we use the HP filter over the longer period (2000-2016), the resulting split of volatilities into

short-term and long-term components over the sub-period 2000-2008 would be different from

what would be obtained using data only from the period 2000-2008. Furthermore, the values

from the sub-period 2009-2016 would have affected the split in the first sub-period,

2000-2008, potentially introducing look-ahead bias. Even when the method is applied on

expanding window designs, the current split into long-term trend and noise or cyclical

component depends on the future. 39 If there was a significant trend, this would have manifested in a difference in the

Augmented Dickey Fuller tests with and without trend, but our analysis did not indicate such

a problem. 40 Note that this study explores a related but still distinctly different problem from ours, i.e.

whether aggregate volatility predicts aggregate returns, which was pioneered by Goyal and

Santa-Clara (2003)

55

introducing the lagged return (𝑅𝑖,𝑡−1) as a control variable, omitted in other studies, renders

idiosyncratic volatility (lagged or predicted by the ARIMA model) statistically insignificant.

In contrast, they report a significant positive correlation between the predicted volatility from

the EGARCH model and the expected return.

Li et al. (2014) investigate whether there are significant trading gains from holding a

portfolio that is long in low-risk securities and short in high-risk securities. They find that the

composition of the high-risk portfolio changes quickly and the associated trading costs, as

well as the small capitalisation and low liquidity of the high-risk stocks, wipes out most, if not

all, gains from trading strategies aimed at exploiting the low-volatility anomaly.41 Their

portfolios are formed based on the estimator of Ang et al. (2006), and the criticism that there

are significant return reversals of high-volatility stocks cast doubt on the empirical

significance of their results.

In a recent contribution, Fan et al. (2015) explore whether idiosyncratic volatilities are

related to a range of asset pricing anomalies, including asset growth42, book-to-market

value43, investment-to-assets44, short-term return momentum45, new stock issues46, size47, and

total accruals effects48. Using five-by-five quintile portfolios constructed on idiosyncratic

volatility and each of the anomaly dimensions, the study finds a significant link between

idiosyncratic risk and the analysed stock anomalies, and that the impact of idiosyncratic risk

on stock anomalies in developed countries is significantly weaker than its impact in

developing markets.

41 In their view it was the securities with low volatility that earn abnormally low return, hence

the reference to low-volatility anomaly. 42 The anomaly concerns the finding that companies experiencing higher asset growth earned

lower returns; see Cooper et al. (2008) 43 The book-to-market value anomaly concerns the finding that companies with high

book/market ratio earn higher returns than companies with low book/market ratio; see Fama

and French (1992), 1993), 2007) 44 The anomaly concerns the finding that companies with a high investment-to-assets ratio

earn lower returns than ones with a low investment-to-asset ratio; see Lyandres et al. (2008) 45 The anomaly concerns the finding that securities with higher cumulative past returns (e.g.

over six months) were likely to continue to perform better; see Jegadeesh and Titman (1993) 46 Concerns the finding that companies with newly issued stocks underperformed otherwise

comparable securities; see Fama and French (2008) 47 Concerns the finding that companies with smaller market capitalization earned higher

returns than those with larger market capitalization; see Fama and French (1992), 1993),

2007) 48 Concerns the finding that cash flows and accruals are not fully reflected in prices until they

impact prices, and thus companies with high total accruals earning lower returns than

companies with lower total accruals; see Sloan (1996)

56

Table 2: Summary of selected empirical studies of the link between idiosyncratic

volatility and stock returns

Study Correlation Comment

Fama and

MacBeth

(1973)

None The study introduces the cross-sectional methodology for asset

pricing tests. Using portfolios as assets, no significant correlation

between idiosyncratic volatility and expected returns is found

Malkiel and

Xu (2004)

Positive The authors propose an extension of the Merton model and using

monthly data on individual US equities and a sample of Japanese

equities find positive correlation between undiversified

idiosyncratic risk and expected returns. Idiosyncratic risk is

reported to be more reliable a predictor than beta or size

Ang et al.

(2006, 2009)

Negative The study uses portfolios formed on daily idiosyncratic volatility in

the preceding month. The authors find that for US & G7 countries,

high-volatility securities have abnormally low expected returns

compared to low-volatility equities. They report strong

co-movement of negative returns for high-volatility stocks.

Fu (2009); Fu

and Schutte

(2010)

Positive The studies use the EGARCH model to estimate expected returns.

The papers use the methodology of Fama and French (1992) with

monthly data and individual securities as assets and find robust

positive correlation between expected idiosyncratic volatility and

realised returns. The study of Fu also examines the stationarity of

the volatility series and finds that the unit root null hypothesis is

rejected for 90% of all securities. The study attributes the negative

spread obtained by the studies of Ang et al. (2006) to return

reversals.

Spiegel and

Wang (2005)

Positive The paper uses EGARCH model to forecast volatility and

documents negative correlation between liquidity and idiosyncratic

volatility. However, the impact of idiosyncratic volatility is 2.5 to 8

times stronger than the impact of the liquidity characteristic.

57

Study Correlation Comment

Bali and

Cakici (2008)

Mixed The study finds that data frequency, breakpoints, weighting

schemes, and data filters all affect the significance and sign of

correlation and concludes that the evidence is mixed and not

convincing.

Cao (2010);

Cao and Xu

(2010)

Positive with

the long-term

component.

The dissertation and the working paper propose that long-term

volatility as estimated from a digital filter explains the

cross-section of returns, while the link between short-term

volatility and returns is negative. The estimated split is, however,

not forward-looking and is exposed by construction to look-ahead

bias.

Li et al. (2014) Economically

insignificant

The authors find that the gains obtainable from holding low-risk

minus high-risk portfolios are small and likely to be wiped out by

frequent rebalancing and trading costs for small, low-liquidity

securities.

Source: the author

2.4. Conclusions

In this chapter we covered two complementary aspects of our study. Firstly, we

surveyed the theoretical models concerning our research problem. We found that these fall

into two related strands: firstly, the Capital Asset Pricing Model (CAPM), which predicted

that in a frictionless market investors solving the mean-variance portfolio optimisation

problem would invest in the market portfolio, and that individual equilibrium returns would

be determined solely by the beta of the asset with the market, and not by idiosyncratic risk,

which investors can diversify completely. Secondly, when transaction costs, asset

indivisibility or other frictions prevent investors from completely diversifying away

idiosyncratic risk, idiosyncratic risk should be priced by the markets and the lower the number

of assets in investors’ portfolios, the higher the equilibrium return. These models are

refinements of the CAPM resulting from relaxation of its assumption that markets are

frictionless.

The theoretical models alone are unable to resolve the problem whether idiosyncratic

58

risk is priced by markets. There are a couple of reasons for that: firstly, there is a trade-off

between model tractability and realistic assumptions, e.g. assuming a two-period economy

with deterministic volatility that is known to all investors; secondly, the premium for

idiosyncratic risk is a non-linear function of the limits on diversification imposed by market

frictions; thirdly, idiosyncratic risk is unobservable. Thus, we see that while the theoretical

models provide a solid framework for analysis of equilibrium prices and returns, the problem

of the correlation between idiosyncratic risk and returns is ultimately an empirical one.

The survey of the empirical evidence concerning the predictability of the cross-section

of stock returns by idiosyncratic risk reveals that although significant contributions have been

made towards understanding the patterns of market returns. Nonetheless, the existing

empirical evidence is mixed, with some studies finding no robust correlation between

idiosyncratic risk and returns, other reporting economically and statistically significant

positive correlation, yet other studies finding a negative correlation. Therefore, some

important questions concerning the link between idiosyncratic risk and returns remain

unanswered.

The first of these questions is how the various idiosyncratic volatility estimators

compare to one another. Indeed, few studies employ comparable estimators of idiosyncratic

volatility, which invites the question how these different measures correlate with each other

and with the unobservable true volatility. The answer to that question would elucidate

whether the explanatory power of the different estimators lies in their correlation with the

expected volatility or with other characteristics of the volatility dynamics. The second

question is how the recent critiques on previous studies on methodological grounds (e.g.

omitted-variable and look-ahead biases) impact the findings of (in)significance of

idiosyncratic volatility in explaining the cross-section of returns.

In the following chapter we shall formulate our methodology for answering those

empirical questions and we shall elaborate on how it relates to existing studies.

59

3. Research Methodology and Data Sources

3.1. Introduction

“I often say that when you can measure

what you are speaking about, and express it

in numbers, you know something about it;

but when you cannot measure it, when you

cannot express it in numbers, your

knowledge is of a meagre and

unsatisfactory kind; it may be the beginning

of knowledge, but you have scarcely in

your thoughts advanced to the state of

Science, whatever the matter may be.”

Lord Kelvin, 1883

The preceding chapter suggested that both nil and positive correlation between returns

and idiosyncratic risk would be compatible with underlying economic models. However,

empirical evidence thus far did not provide unequivocal guidance on the direction or strength

of that correlation. We saw that while existing studies employed the same data set (the series

of the Center for Research in Security Prices, augmented with Compustat data), the studies

diverged greatly in terms of how to measure idiosyncratic risk and how to test thepropositions

of the underlying theoretical models. In this chapter we develop the methodological aspects of

our empirical examination. We start with Section 3.2 that discusses and motivates the choice

of our research methodology, which can be described as positivist, deductive statistical study.

In that section we also highlight the strength and sacrifices that this choice imposed upon our

study. Section 3.3 then discusses the choice of factor model and its estimation, as well as the

selection of control variables for the cross-sectional regressions. Section 3.4. elaborates on the

sources of data, as well as the calculation procedures for the calculated covariates. Section

3.5. outlines the Markov chain methodology for classification of volatility regimes, and

provides an overview of the volatility episodes in the surveyed period. Section 3.6 concludes

the chapter.

60

3.2. Methodological notes

3.2.1. Research philosophy: positivism and its limitations

The rise of positivism in economic research dates from at least the early 20th century.

Inspired by the success of natural sciences, positivist ideas became increasingly adopted in

economics, which eventually lead to the separation of economics from its precursor – the

political economy. Economic research became increasingly concerned with the search for

objective laws governing the economy, which were not particular to a given economic or

political system, but prevailed universally in every economic system, irrespective of space,

time, and politics.

Asset pricing theory, as a branch of finance theory, followed a similar pattern of

development. Ryan et al. (2002) note that most of finance studies adopt a capital market

perspective, which is reflected in the treatment of transactions and agents and investments and

investors, while relatively few studies adopted a managerial or intercompany perspective

(p. 51-52). They highlight three core propositions of finance: firstly, individual economic

agents are formally rational; secondly, financial markets are perfectly competitive; and

thirdly, information is freely available to agents. (ibid, p. 51)

The positivist approach to analysis of the social phenomena, however, is open to

certain criticisms pointing out that social reality is constructed by humans and cannot be

modelled with the deterministic certainty characteristic of most of physics. Salmon et al.

(1999) highlight three alternative approaches to positivism in the context of social research:

“Contemporary critics of the view that studies of human behavior are or can be scientific fall

into three categories. The first group, called interpretivists, claim that explanations of human

behavior are structured entirely differently from explanations of the behavior of physical

objects since human behavior, they say, consists of actions done for reasons rather than events

resulting from causes. The second group, called here nomological skeptics, do not deny that

human behavior is subject to causal laws, but doubt that it will ever be possible to find laws of

human behavior that are similar in power and scope to those in physical science. The third

group, called critical theorists, claim that it is inappropriate even to try to explain human

behavior in terms of the laws of cause and effect because to do so denies the value of human

autonomy (free will). In addition, they say, any attempt to construct a social science on the

model of physical sciences promotes unethical manipulation of humans and discourages any

attempts to improve the conditions of social life.” (p. 407-408)

61

The positivist approach to economics (and finance) is not free of criticisms. For

example, Caldwell (1980) outlines three criticisms to positivist economics. His first critique

concerns the confirmability of such propositions. He points out that a rigorous test of a

positive deductive argument requires clear specification of all assumptions and auxiliary

hypotheses, but that is difficult to achieve, especially in social sciences, as there might be

many control variables and assumptions. Secondly, Caldwell (1980) points out the distinction

between theoretical terms in the positivist axiomatic-deductive models and the real world.

Theoretical models work with theoretical terms, which may not exist in the real world, or may

be unobservable. In our case the CAPM and Merton (1987) models work with abstract,

theoretical concepts like the return-generating process, idiosyncratic risk, conditional

homogeneous beliefs, two-period economy, or complete markets. Some of these are

convenient assumptions that allow us to solve the model and produce testable predictions,

while others are theoretical constructs with no direct match in the real world. Therefore, even

in the core term of this study – idiosyncratic risk – we need to find a way to operationalize the

term and find some way to measure it. However, there is no unique or correct way to do that –

all studies in the field employ equally valid measures of idiosyncratic risk, yet they reach

starkly different conclusions. This requires understanding how these alternatives relate to one

another and to the underlying model, in order to interpret the results. Nevertheless, such

interpretation is necessarily subjective and could be rejected by other researches, leading to

different conclusions regarding the validity of the underlying positivist model. Finally,

Caldwell (1980) notes that the role of scientific explanations may not fit nicely into the two

models – the deductive-nomological or the inductive-probabilistic explanations.49

We recognise the validity of the criticisms of positivism reviewed by Salmon et al.

(1999) and Caldwell (1980), and we concur that economic ‘laws’ are qualitatively different

from those of classical mechanics. Nevertheless, it should also be recognised that a number of

results in economics can be found to be statistically valid, and although the next state of the

social system cannot be predicted in the mechanical sense, some states are more likely than

other. In a context akin to ours – a study of evidence in favour of technical analysis – Aronson

(2007) points out that scientifically valid predictions should pass the discernable-difference

test, so that they are either true or false, and the scientist should be able to discern between the

predicted outcomes. Such a test requires two ingredients: the ability to reproduce objectively

the assertion and the ability to distinguish between the ex-post events. In particular, theories

49 For a more recent review of the explanatory schemes, see Woodward (2014)

62

that rely on subjective interpretation of information (e.g. stock prices) are not reproducible

and may be consistent with conflicting outcomes; such theories might be true but would not

be scientifically testable.

In some studies such a situation may be difficult to avoid, but in the present study the

predicted outcomes (prices and returns) are readily observable, which might suggest that

positivist philosophy might be a more useful context for our exploration. However, adoption

of the positivist approach would also require that not only the outcome (returns) be objective,

but also the risks that affect returns. One approach could be interpretivist and seek to elucidate

how investors perceive risk and how they assess the risk of their holdings. Such assessment is

likely to be a multicriterial one and could include how prices change in normal conditions

(e.g. stock volatility), under stress (e.g. tail losses), soft facts about the company and its

management, market strategy of the management team, transparency and reliability of

reporting, and many more. Such an approach could uncover how investors assess risk much

more reliably and realistically than simple standard deviation of returns. On the other hand, it

would be difficult to extend those findings on the link between risk and returns, because that

link is a statistical one and its validation would require a large sample of both high-risk and

low-risk securities.

Summing up, we recognise that in the present context the positivist analysis has its

limitations. These relate to its objective to ensure that the uncovered principals are universally

valid across issuers and in time. Therefore, in our study we aim to ensure maximal size of our

studies sample. Furthermore, we need to ensure that all our conclusions are reproducible, so

we cannot rely on soft facts that might have different interpretation by other researchers.

Despite these limitations, we shall nevertheless employ the positivist approach in order to

pursue the external validity of our conclusions and to facilitate comparison with the existing

body of literature surveyed in Chapter 2. This necessitates that we limit the basis of risk

assessment in this study to objectively reproducible measures of risk that can be estimated for

significant part of the market, and hence is based only on public-domain information that was

available to all market participants (e.g. stock prices or publicly available financial reports).

Moreover, we will focus on single measure of risk – the idiosyncratic volatility - and we shall

not explore how investors ‘weigh’ the different measures in order to reach their overall risk

assessment of the available securities.

63

3.2.2. Deductive and inductive research

The scientific arguments are broadly classified into two streams, depending on their

approach: inductive and deductive ones. Inductive reasoning starts with analysis of individual

instances and seeks to generalise the observations from individual instances into fundamental

principles or regularities. Deductive reasoning, on the other hand, starts with some

fundamental law and seeks to operationalise or confirm it for some specific instances.

Inductive reasoning starts with the observations of the world and notes patterns in it.

These observations are then combined to build a theory of the explored phenomenon. This

theory can be confirmed and refined by exploring its implications in other settings and

examining whether the predicted outcomes are consistent with the actual data.

Deductive reasoning builds knowledge in the opposite way. It builds a theory of the

world that produces some falsifiable predictions. Then the researcher performs tests in order

to test if the general theory can be rejected. An example of that approach could be the

development of the Expected Utility Theory, which started with a set of axioms concerning

the behaviour of rational decision-makers under uncertainty, and produced a set of predictions

that could be tested, as well as deduced implications from those axioms – the possibility to

represent preferences in terms of utility functions.50

Both inductive and deductive reasoning have their strengths and weaknesses. The

principal characteristics of the inductive approach, as summarised by Salmon et al. (1999), are

that it is ampliative51, but not necessarily truth-preserving52 and erosion-proof53, and its

arguments come with varying degrees of strength (p. 11). On the other hand, deductive

reasoning has the advantages of being truth-preserving and erosion-proof, but non-ampliative

and absolute in the sense that the deduction is either completely valid, or completely invalid.

Aronson (2007) notes that the most common type of inductive reasoning is induction

by enumeration, which enumerates the evidence from some set of data and generalises it to

some principle. For example, we may observe that in a sample of securities, those with higher

idiosyncratic risk earned on average higher return, and therefore we could conclude that

securities with higher idiosyncratic risk earn higher returns. Such generalisations should be

made with care in order to ensure that they are not based on a non-representative sample (e.g.

50 See Chapter 3, pp 57–62 in Ryan et al. (2002) for an exposition of the methodological

tradition underpinning the modern theories of asset pricing. 51 i.e., the conclusions of the inductive reasoning go beyond the content of its premises 52 i.e., correct premises may lead us to wrong conclusions 53 i.e., new premises may completely undermine a valid inductive conclusion

64

encompassing only periods of economic growth, or one including only large companies).

Furthermore, the analysis should also take into account other characteristics and

considerations that may have affected the observations, e.g. liquidity. Aronson (2007) points

out that the most common error in inductive research is the hasty generalisation due to a small

sample or low quality of evidence.

A problem with inductive financial market research is the data mining bias (or data

snooping). Data mining is the name of a group of techniques that aim to uncover patterns in

large data sets. Those methods have extensive applications in machine learning and artificial

intelligence studies. However, such algorithms may also “overfit” the data, e.g. by using a too

complex neural network; the outcome would be that the in-sample performance could be

good, but that performance would not be sustained out of sample.

Due to the standardisation of stock market contracts and the available public

information on prices, it has grown to be probably the most scrutinised market. Nevertheless,

complete information is available for only a limited number of years and stocks. As a result,

Black (1993) warned that the growing research in finance using the common data set (the data

of the Center for Research in Security Price and the Compustat data on financial ratios) is

likely to uncover patterns in the data where none exist. The ‘data mining bias’ therefore refers

to the risk of finding spurious patters in data that do not generalise out of sample. The

sample-splitting approach used in the field of machine learning could be applied in finance as

well, in order to mitigate that problem. However, McLean and Pontiff (2016) find that the

abnormal returns to many of the stock market “anomalies” discovered by academic research

shrink rapidly after publication, although they do not disappear entirely. This suggests that

either the anomalies were due to data snooping54, or patterns lose their predictive powers as

investors attempt to exploit the anomaly.

The latter explanation of the shrinking of market anomalies after publications is

particularly important in terms of choosing between inductive and deductive research. If

investors change their investment behaviour over time, then the results of many techniques

used in inductive research might not generalise well in time.

Deductive research starts from a theory explaining some phenomenon and

operationalises it to specific instances in order to explain a certain phenomenon or validate the

54 Researchers in finance rarely employ splits of the full sample into training, test, and

validation subsets. Instead, they control for spurious results through robustness check using

subsamples of the original sample in various dimensions or through reporting data for

sub-periods.

65

theory. Deductive research is less concerned with social context compared to inductive studies

and does not create new knowledge (non-ampliative), but its results are, in principle,

erosion-proof and therefore it generalises well. It is the approach that we select for this study,

which is motivated by the fact that it generalises and it is less exposed to data mining

problem. Inductive research is more sensitive to social context and thus able to accommodate

soft facts like attitudes and perceptions, but in view of the evidence of McLean and Pontiff

(2016) it would be difficult to generalise the findings both in the past and in the future.

Likewise, the small sizes implied by some research methods (e.g. interviews and

questionnaires) mean that the conclusions may not generalize over the entire markets over the

studied decades, and therefore the study would not be able to convincingly resolve the

puzzling conflicting evidence on the correlation between idiosyncratic risk and returns. On the

other hand, inductive use of techniques like regressions or neural networks could result in data

mining, which may uncover patterns in the past, but those patterns could perish in the future,

as documented by McLean and Pontiff (2016), unless there is some fundamental reason

underpinning the pattern.

Another benefit of deductive research in that setting is the ability to make

generalisations about the future developments on the market. For example, in recent years we

observe various trends in the market that could affect the magnitude of the risk premium

accruing to investors. The advances in information technology result in gradual reduction of

transaction costs, which should encourage broader diversification and hence reduce

undiversified idiosyncratic risk. On the back of automated trade execution, the recent years

see increasing use of trading algorithms that shorten investment horizons and could be more

able to trade on volatility changes. Another recent trend closely intertwined with the former

two (reduction of trading costs and algorithmic trading) is robo-advising, which could

increase the share of investment universe followed by smaller investors and promote their

diversification, thus reducing idiosyncratic risk premium. Thus, the guidance of the

theoretical model allows not only an exploration of the status quo, but also anticipates how

technological and institutional advances on the marketplace could impact idiosyncratic risk

premia.

We demonstrate the deductive research workflow using the present study as an

example in Table 3. The table shows the key decisions that need to be made at each phase of

the research: choosing a theory that could explain the link between idiosyncratic risk and

returns; formulating the research hypotheses; operationalizing the theory; performing the

tests; drawing conclusions and refining the theory.

66

Following the context phase, which in our case was concerned mostly with literature

research and fact-finding, the first important decision in the deductive research workflow was

choosing a theoretical model. As a baseline case we considered the Capital Asset Pricing

Model (CAPM), which made a positive prediction of what determines stock returns. The

CAPM extensions of Levy (1978) and Merton (1987) incorporate the assumption of

incomplete diversification, while incorporating CAPM as a particular case. Thus, they appear

as a natural choice for a theoretical model. Nevertheless, it should be recognised that there

were other possibilities. Thus, Iwasawa and Uchiyama (2013) explore a behavioural finance

explanation of the volatility anomaly in the Japanese market. Their behavioural exploration is

based on the noise trader approach to finance in De Long et al. (1990), which proposes that a

deviation of market prices from fundamental values may persist in markets where there are

traders who trade without reference to the fundamental value of the traded assets (‘noise

traders’) and the capacity or willingness of the arbitrageurs to take sufficiently large risky bets

to close the gap between current price and fundamental value is limited.55 In that setting

Iwasawa and Uchiyama (2013) note that “[a]pplying this logic to interpret the “volatility

anomaly,” we can view that investors who like to buy stocks with high beta and/or high

idiosyncratic volatility, even if they are overvalued, cause overpricing of these stocks on

average, and low long-run average return of them.” (p. 465) That market prices are more

volatile than price fundamentals is a known fact, and the noise trader approach could

accommodate such stylized facts. However, such an explanation is too ad hoc and

non-specific, in the sense that a hypothetical proposition that investors like big companies or

ones with stable cash flows and are buying them even when they are overvalued could be

equally consistent with the noise trader approach. Therefore, we recognise that behavioural

finance is motivated by actual behaviours and imperfections observed on the markets,

although it is still not sufficiently mature to match the rigour of the traditional approach. The

debate between the proponents of the two approaches goes well outside the scope of this

discussion56 and we are willing to agree that in cases where a theoretical model is not

55 E.g. because trading involves assuming risk and the market price could go against the

traders, causing significant loss to the arbitrageurs. 56 The following quote from Subrahmanyam (2007) offers a fair summary of the points of the

two sides: “Traditional finance academics often offer a few common objections to

behavioural finance. First, it is often said that theoretical behavioural models are somewhat ad

hoc and designed to explain specific stylised facts. The response is that behavioural models

are based on how people actually behave based on extensive experimental evidence, and

explain evidence better than traditional ones. Another common objection is that the empirical

67

consistent with empirical evidence, a new or refined model should be developed. However,

we also believe that valid scientific models should produce testable non-trivial (non-obvious)

predictions and in this instance the present author is more convinced by the arguments of

classical finance. The concept of noise-trading may be consistent with some behaviours

observed in the markets, but the limits to arbitrage do not seem consistent with the evidence

of McLean and Pontiff (2016) of a significant decrease of anomalous gains after publication.

Another relevant prediction could be based on the theory of Barberis and Huang (2008) who

build on the cumulative prospect theory and argue that in the presence of demand for

lottery-like stocks (i.e. ones with significant positive skewness that have large growth

potential) gives rise to the own skewness of the stocks being priced and stocks with high

positive skew being overpriced. Nevertheless, from the perspective of traditional finance it is

not immediately clear what is the non-trivial insight from that theory, in the sense that if a

model assumes preference for a certain type of assets, it will predict that that preference

would be priced. Therefore, evaluating the arguments of both sides we prefer to base our

deductive research on classical finance model, while recognising that behavioural models

could also offer an alternative view on the matter.

The next step in the deductive workflow was the formulation of a research hypothesis.

It was treated in the previous chapter and we shall not reproduce it here, but in summary we

shall only note that since the theoretical model predicts that idiosyncratic risk is priced in the

presence of under-diversification, this was naturally our principal research hypothesis. Its

validation and reconciliation with the conflicting evidence of previous studies requires a

comparison of volatility forecasts in order to find that the inconclusive results are due to

discrepancies in predictive accuracies, which thus became a secondary hypothesis of this

study.

work is plagued by data-mining (that is, if researchers set out to find deviations from rational

pricing by running numerous regressions, ultimately they will be successful). However, much

empirical work has confirmed the evidence out-of-sample, both in terms of time-periods as

well as cross-sectionally across different countries. Finally, it is often claimed that

behavioural finance presents no unified theory unlike expected utility maximisation using

rational beliefs. This critique may well be true at this point, but traditional risk-based theories

do not appear to be strongly supported by the data. Thus, it appears that there is a strong case

to build upon some theories that are consistent with evidence, than theories based on rational

economics whose empirical support appears quite limited. Indeed, a ‘normative’ theory based

on rational utility maximizers cannot be construed as a superior alternative to behavioural

approaches merely because it discusses how people should behave. If people do not behave in

this way, this approach has limitations in helping us understanding financial phenomena.”

(p. 13)

68

As a next step the tests have to be operationalized. Indeed, the model of Merton (1987)

assumes a two-period setting with deterministic and known volatilities and homogeneous

beliefs on the parameters of the model. In this way the model concentrated on predicting

equilibrium prices (and therefore, in the two-period setting, returns) and abstracted the

parameters that were easier to estimate like variances. The model also did not make any

suggestions about what the investment horizon for the decision-makers is. Therefore, the

operationalization of the theoretical model requires making decisions on how to apply the

model to the empirical setting. The first choice is the horizon of the tests. The model of

Merton (1987) is set up as a two-period economy where at time 𝑡 = 0 investors make

allocation decisions, and at the end of the period, time 𝑡 = 𝑇, investors collect the uncertain

dividend (positive or negative) on their investments. Thus, the predictions of the model should

be valid at any frequency – daily, weekly, monthly, quarterly, yearly, or any other frequency.

In practice daily, weekly and monthly frequencies tend to be the most common ones. Tests

involving all time frequencies would increase significantly the scope of this research,

especially given that different frequencies are associated with different amounts of data and

face different challenges. For example, five years of data at a monthly frequency imply 60

realisations, which limits the types of models that can be estimated with reasonable accuracy.

On the other hand, five years of daily data translate into over 1250 realisations, which allows

estimation of much more complex models. High frequencies (daily and intra-daily), however,

are exposed to market microstructure effects that may affect tests trying to match daily returns

to risk. Market microstructure noise refers to short-term deviations between reported prices

and the actual fundamental or equilibrium prices due to the way stock markets operate, e.g.

bid-ask spreads, depth of the order book, rounding errors, screen fighting or other

considerations. Whatever the cause, that noise obscures the ‘true’ market prices and returns;

for example, a large buy bid may not be executed in one transaction but may take a couple of

trades with increasing prices, which could result in reporting higher daily return, whereas in

fact the prices subsequently revert to the equilibrium value. Therefore, the estimation

methodologies suitable for monthly data may not be well suited for drawing conclusions with

daily data. In our problem most of the controversial findings are based on tests using monthly

data, and therefore we decide to concentrate on that frequency as well, and at that frequency

the support for the prediction of the Merton model is weaker.

Another decision in the operationalization of the model concerns the selection of

measures of idiosyncratic risk. The models of Levy (1978) and Merton (1987) associate

idiosyncratic risk more or less directly with the distribution of the company-specific shocks,

69

and more specifically, with the standard deviation of that distribution. The model of Levy

(1978) considers investors’ individual portfolio optimisation problems in the mean-variance

efficient optimisation framework where idiosyncratic risk is naturally associated with the

standard deviation of idiosyncratic shocks in the linear factor model. In the model of Merton

(1987) idiosyncratic risk is introduced as a random shock on company cash flows in an

economy characterised with quadratic utility function, so that again the standard deviation of

the shock distribution is the natural measure of idiosyncratic risk. Nevertheless, we should

also recognise that those models are an abstraction, and individual risks may not be linear,

while utility functions need not be well approximated by quadratic utility function.

Concerning the first objection we shall note here that in practice exposures to risks are not

known to investors accurately, but are learned from many factors. This process of learning

and mixing market data with soft information and individual perceptions to reach an

assessment of the riskiness of an investment may suggest that qualitative factors may be

important in that respect, a consideration that we discuss in somewhat greater detail in the

following subsection.

Concerning the second objection – that investor utility may not be approximated well

by quadratic utility, we note that such considerations may give rise to significance of other

idiosyncratic risk characteristics like skewness, as suggested by Barberis and Huang (2008),

or tail risk, as reported by Huang, Liu, Rhee and Wu (2012). The options to pursue these

directions were limited by the choice of the monthly frequency for the tests. We have

nevertheless attempted to estimate tail risk from the parameters of the distribution of

idiosyncratic innovations from GARCH models, as well as sample own skewness,

co-skewness, and co-kurtosis, but we found that in view of the short series these analyses

were not conclusive and are not reported here. Therefore, we concentrate on idiosyncratic

volatility (standard deviation) as a principal idiosyncratic risk measure in this study, while

recognising that other qualitative and quantitative risk characteristics could be added in the

analysis to improve accuracy.

We will defer the discussion of the tests employed in this study to the subsequent

subsection on statistical tests. At this juncture we can only note that the step involved

selecting methods to compare predictive accuracy from different models, e.g. though

loss-functions, Mincer and Zarnowitz (1969) regressions, or other tests. Similarly, the choice

of test of the link between idiosyncratic risk and returns allowed the use of portfolio spreads,

Fama and MacBeth (1973) regressions, Generalised Least Squares (GLS), or Generalised

Method of Moments (GMM). We will motivate our choices in the subsection dedicated to that

70

topic.

Finally, in view of the evidence contradicting the theoretical model we were invited to

either revisit our operationalization of the model, or reject the model altogether. We found

that the introduction of the mean-reverting level of volatility was in fact a plausible

rationalisation in the context of dynamic, stationary volatilities and it resulted in overall

confirmation of the theoretical models.

Wrapping up the discussion, we find that the research problem can be approached by

an array of inductive and deductive research strategies, each having its advantages and

limitations. Our first reason to prefer deductive over inductive research is the material risk of

data mining, which could result in a discovery of spurious correlations that do not generalise

outside of our sample. The second driver for our preference for the deductive approach is the

goal to reach conclusions the generalise outside of the studied sample, e.g. in other time

periods or other marketplaces, institutional arrangements, market segments, as well as to

anticipate the effects of market infrastructure changes that could result in lower transaction

costs (technological advances), shorter planning horizons (e.g. algorithmic trading), or

improved transparency (e.g. robo-advisors). We see these as attractive aspects of deductive

research, and therefore we choose for our research a deductive approach, though we do adapt

our operationalisation of the theoretical models if required by the available evidence.

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Table 3: Deductive research workflow

Stages in the

deduction

process

Actions taken Example: application to the present research

Context Read and consider Review of the factors affecting stock returns

revealed contradictory evidence, e.g. Ang et al.

(2006); Bali and Cakici (2008); Fu (2009), as well

as the evidence of wide under-diversification of

investor portfolios.

Theory Select a theory or set

of theories most

appropriate to the

subject under

investigation.

Review of the theories purporting to predict the

phenomenon, e.g. CAPM, the models of Levy

(1978) and Merton (1987), and select between

those and alternative explanations, e.g.

behavioural models.

Hypothesis Produce a hypothesis

(a testable proposition

about the relationship

between two or more

concepts).

We hypothesise that there is sufficient

under-diversification in the market to produce

positive correlation between idiosyncratic risk and

stock returns. We hypothesise that the conflicting

evidence in previous research is due to

discrepancies in the operationalization of the tests.

Operationalize Specify what the

researcher must do to

measure a concept.

The theoretical model is an abstraction (two

period, homogeneous beliefs) and can be

employed in various horizons (daily, monthly,

etc), different measures of idiosyncratic risk (e.g.

volatilities, tail risk, dispersion of analysts’

forecasts, etc).

Testing by

corroboration or

attempted

falsification

Compare observable

data with the theory. If

corroborated, the

theory is assumed to

have been established.

Test the differences in forecast accuracy through

Mincer and Zarnowitz (1969) regressions.

Test the correlation between volatilities and

returns using different tests (portfolios vs

individual securities, portfolio spreads vs

Fama-Macbeth vs Generalized Method of

Moments).

Examine

outcomes

Accept or reject the

hypothesis from the

outcomes.

Analyse the results from the tests above.

Modify theory

(if necessary)

Modify theory if the

hypothesis is rejected.

Adapt the operationalization of the theory, e.g.

exclude idiosyncratic tail risk as insignificant and

introduce the mean-reverting level.

Source: adapted by the author from Table 2.1, p. 17 in Gray (2014)

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3.2.3. Quantitative and qualitative research

Research approaches can be broadly classified into quantitative and qualitative,

depending on the type of interpreted information. Examples of problems of qualitative nature

are the analysis of meanings expressed in words, images or sounds, analysis of

non-standardised data requiring classification into categories, or analyses conducted through

the use of conceptualisation; see Table 13.1, p. 482 in Saunders et al. (2009). Quantitative

research, on the other hand, applies to cases where meaning is extracted from numerical,

standardised data and analysis is conducted through the use of diagrams and statistical

methods.

It is difficult to give a precise definition of what qualitative research means because

the scope of research approaches is quite wide. Instead, Yin (2011) identifies five key traits of

qualitative research:

“1. Studying the meaning of people’s lives, under real-world conditions;

2. Representing the views and perspectives of the people [...] in a study;

3. Covering the contextual conditions within which people live;

4. Contributing insights into existing or emerging concepts that may help to explain

human social behavior; and

5. Striving to use multiple sources of evidence rather than relying on a single source

alone.” (p. 7-8)

Qualitative research methods are often beneficially used in studies that interpret social

phenomena in order to reveal their causes and effects. Such interpretative studies recognise

that unlike the natural world, the social world is constructed by humans and has valid

subjective description. Qualitative studies often use inductive57 reasoning in order to uncover

the causes and effects of the studied phenomena using tools like case studies, interviews,

focus group discussions, in-depth interviews, observations and field trips. In contrast,

quantitative studies interpret numerical, standardised data; quantitative methods are preferred

for confirmation of positive theories and deductive arguments that extract meaning from

57 The use of the deductive approach in qualitative research is subject to some debate. Some

authors support its application, but there are also arguments against that. For example,

Bryman (1988:81) (quoted in Saunders et al. (2009), section 13.4) argues as follows: “The

prior specification of a theory tends to be disfavoured because of the possibility of introducing

a premature closure on the issues to be investigated, as well as the possibility of the

theoretical constructs departing excessively from the views of participants in a social setting.”

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numbers through measurement, statistical analyses and test of hypotheses.

An empirical analysis of the link between idiosyncratic risk and returns could be

tackled with both qualitative and quantitative methods. For example, in-depth interviews with

decision-makers could be conducted in order to understand how they form their portfolios in

terms of balancing the various criteria – some of which qualitative – and making allocation

decisions. Similarly, how decision-makers perceive of risks could also be fruitfully examined

through such qualitative methods. In business research one often deals with notions like

corporate image or social responsibility, which may not have a standardised quantitative

measurement. Even if those concepts could be quantified, it is often impossible to collect such

data in order to perform a quantitative confirmation of the developed theory, which could

limit the generality of the developed model and its confirmability.

Another difficulty in implementing qualitative research in the field of asset pricing is

the large number of biases found in investors. This problem is not unique to finance, but

nevertheless seems more acute in the field of asset pricing, possibly because the financial

impact of decisions is easier to quantify ex post and asset managers need to make routinely

predictive decisions in a random environment, which incites the activation of psychological

defence mechanisms to handle disappointments. Therefore, while overconfidence may be a

common trait in humans, it is likely to be more pronounced in the financial decision-making

process. For example, Aronson (2007), Chapter 2, considers eleven psychological biases and

mechanisms that give investors illusory confidence over the validity of their statements: (i)

overconfidence, optimism bias; (ii) self-attribution bias; (iii) illusion of control; (iv)

knowledge illusion; (v) biased second-hand knowledge; (vi) representativeness heuristic bias;

(vii) sample size neglect; (viii) illusory trends and patterns; (ix) hindsight bias; (x)

confirmation bias; (xii) illusory correlation. Other classifications also exist, and their findings

are similar. Such behavioural biases are also recognised by industry practitioners and

associations. For example, Pompian (2016) recognises a total of 20 behavioural biases

affecting investors, grouped into behavioural persistence, information processing, and

emotional biases. The presence of significant behavioural biases in investors would make very

difficult the separation of actual behaviours from biased rationalisations, and would be a

formidable obstacle in generalising text-based qualitative information into a test of an asset

pricing relation.

In view of the interpretative difficulties of verbal information and limitations on result

generalisation, we opt not to pursue a grounded theory study where methods like depth

interviews are employed to collect data on how investors assess stock risk and whether and

74

how they incorporate idiosyncratic risk into their decision-making process, and the collected

information is then generalised into a theory of the link between idiosyncratic risk and

investment allocation. In the data collection step, the interviewees may tend to demonstrate

overconfidence and may produce misleading results. In the theory-development step, we

should recognise that the stock market needs to clear so that the prevailing prices should

equate supply and demand, so that our theory would need to take into account how aggregate

demand for all assets is affected by the developed model.58 Furthermore, a small number of

interviewees would limit the ability to generalise the results to the whole market and sample

period, and could undermine the goal of this research to uncover the causes for the mixed

evidence on the correlation between idiosyncratic risk and returns.

Returns and portfolio allocations are essentially quantified in terms of continuous

measures, which suggests that quantitative research might be well-suited in explaining the

phenomena at hand. Moreover, the field of asset pricing has been extensively researched in

the past sixty years, which, combined with the extensive availability of data on stock returns,

could allow conducting tests of existing theories on the entire population, rather than just on a

small sample. This is an important consideration because those theories predict a stochastic

causal relation between risk taken and return earned. The probabilistic nature of the causality

suggests that there is substantial probability that a flawless decision-making process could

nevertheless result in loss-making investments purely by chance, and therefore general

conclusions about the link between idiosyncratic risk and returns require confirmation in a

large sample.

Overall, in this study we choose to use quantitative methods. Our first argument for

that is that the predicted phenomenon (market returns) is inherently quantitative in nature and

can be evaluated with high precision. Secondly, we are concerned about the reliability of

many qualitative methods in view the evidence of a number of psychological biases of

investors, documented in existing research. A decision to choose quantitative over qualitative

research approach, however, comes at the cost of loss of detail. Considerations about the

specific decision-making process and multiplicity of investor objectives that could be

uncovered or examined through qualitative research are sacrificed. We recognise those

58 If investors chose to reduce their investment in, say, stock A, they still need to invest the

remaining amount into other stocks, say stock B, and the risk-free asset. This illustrates a core

proposition of general equilibrium models where changes in the demand for one asset (stock

A) impacts all the prices of all other assets (stock B and the risk-free rate, in this simple

example).

75

limitations of our research design, but having in mind our objective to revisit the empirical

support of existing theories and reconcile the conflicting evidence in previous research, we

have opt for the use of quantitative research design.

3.2.4. Statistical methods

In the last subsection we reasoned that quantitative methods are better suited to the

intended deductive approach in a manner that could allow reaching general conclusions about

the validity of the theoretical model. There is an array of methods that can be used in

quantitative research, and in this section we shall motivate the use of the quantitative methods

employed in our study.

There are various classifications of quantitative methods, e.g. descriptive, exploratory,

experimental, or statistical. Descriptive research aims to identify the characteristics of the

explored phenomenon. For example, it could employ surveys and questionnaires to gather

information about how investors assess the risk of their holdings and make allocation

decisions. A study of this type could mix both quantitative and qualitative information in

order to answer such a question. Information on risk assessment methodologies and the

incorporation of idiosyncratic risk into portfolio allocation decisions could be collected

through questionnaires. That method also has the advantage that the questionnaire could be

structured to collect both qualitative information (e.g. through open questions) and

quantitative data. However, the method is exposed to problems similar to those of depth

interviews and qualitative research: the presence of behavioural biases would suggest the use

of long questionnaires that could detect inconsistencies and behavioural biases, which would

limit the response rate and representativeness of results. Furthermore, in view of the evidence

of changing investment patterns presented by McLean and Pontiff (2016) it would be difficult

to generalise the results of that study to explain patterns in the past. Finally, it would be

difficult to reconcile the conflicting evidence from previous studies in that approach,

inasmuch as their differences are only indirectly related to information that could be gathered

in such a manner, e.g. number of assets in portfolios and attitudes towards portfolio

diversification. The dependent characteristic that needs explanation is return, and descriptive

methods would not be able to estimate reliably what premia investors require for assuming

different exposures to idiosyncratic risk.

Exploratory research could also be employed in a related context to classify

investments by risk and return. For example, we could collect or estimate various

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characteristics associated with idiosyncratic risk, e.g. frequency and quality of disclosures,

variability of cash flows or stock prices, co-movement with market, sector, indebtedness,

company age (life cycle), or any other characteristic. We can then seek to classify companies

in various categories, seeking to explore how each category differs from the others in terms of

characteristics and returns. For example, we could explore whether NYSE-listed companies

on average earn more than Nasdaq-listed companies. Various methods could be employed in

such a study, e.g. clustering (for unsupervised grouping of companies into clusters for the

purposes of exploring how those clusters differ in terms of average returns), neural networks

(for predicting growing companies from a set of characteristics), analysis of variance

(ANOVA, for comparison of returns for various categorical dimensions, e.g. sector or market.

Such an approach could be useful to develop new hypotheses for the factors affecting returns.

On the other hand, such an approach could be questioned in terms of its external validity on

the same grounds as data mining that we discussed in the inductive research section.

Experimental designs offer another possible approach for addressing the research

question. In the present context a number of experimental settings could be envisaged. For

example, one can devise various lotteries and offer them to participants in the experiment in

order to confirm whether choices are consistent with those of the expected utility theory and

identify behavioural patterns, e.g. preference for positive skewness. Alternatively, one may

set up a virtual trading platform and explore how participants form their portfolios in a

controlled setting. Yet another possibility could be to perform simulations that re-create the

theoretical setting of the different models and explore how premia change with a number of

securities in individual portfolios, with the relaxation of some of the assumptions, with

inclusion of parameter uncertainty (e.g. via Bayesian learning), etc. For example, Allais

(1953) demonstrates in experimental setting a violation of the independence axiom.59 Smith

59 The Allais (1953) paradox is a well-documented violation of the independence axiom. In

his experiment individuals are asked to select between two pairs of lotteries. The first lottery

𝑣1 promises a certain payoff of 100 million francs, whereas the alternative lottery 𝑣2 offers payoffs of 500mn, 100mn and 0 with respective probabilities 0.1, 0.89, and 0.01. He finds that

most people prefer 𝑣1 over 𝑣2, 𝑣1 ≽ 𝑣2. The other pair of lotteries are 𝑢1, paying 100 mn and 0 with probabilities 0.11 and 0.89, and 𝑢2, paying 500mn and 0 with probabilities 0.10 and 0.90. He finds that most people prefer 𝑢2 over 𝑢1, 𝑢1 ≼ 𝑢2. The paradox was that there is no utility function 𝑈(∙) that would satisfy these choices. The first pair implies that 𝑈(100) > 0.10 𝑈(500) + 0.89 𝑈(100) + 0.01 𝑈(0), so rearranging the terms we obtain that 0.11 𝑈(100) > 0.10 𝑈(500) + 0.01 𝑈(0). On the other hand, the second pair of lotteries suggested that 0.11 𝑈(100) + 0.89 𝑈(0) < 0.10 𝑈(500) + 0.90 𝑈(0), and thus after a rearrangement yields 0.11 𝑈(100) < 0.10 𝑈(500) + 0.01 𝑈(0), contradicting the inequality from the first pair. This shows that the independence axiom may be violated in

77

et al. (1988) observe trades in a series of simulated asset markets and document the

emergence of bubbles in most of them, although the probability of bubbles decreases with

trader experience. Asparouhova et al. (2016) simulate the equilibrium in an extension of the

Lucas tree economy60 and find that trading substantially improves intertemporal consumption

smoothing, but prices remain volatile, similarly to real markets. Such studies could provide

valuable information on the topic, but also come with a set of challenges. Experimental

designs that elucidate deviations of the axioms of the expected utility theory are questioned as

concentrating on deviations that emerge in contrived settings or small lotteries and do not

necessarily translate into similar behaviour in larger, real-life behaviours. At any rate

deviations from the assumptions of decision-making under uncertainty would have

implications far beyond the problem at hand, and such deviations would need to be confirmed

in a range of other settings, that go well outside the scope of this study.

Similarly, simulations from the model economy would ultimately be determined by

the quality of the underlying model and its operationalisation, and assessment of their validity

would necessitate development of new approaches for their validation, e.g. in terms of

response to changes of volatility in a simulated economy and in actual markets. Such

exploration could be a useful contribution to experimental designs, but again goes beyond the

goals of this thesis and its scope.

In the same vein, the effect of parameter uncertainty could be significantly

contributing to the actual outcomes, and in fact we find evidence in favour of the hypothesis.

We find that whereas one-month idiosyncratic volatility forecasts are not useful predictors of

returns, the mean-reverting ones are. This could be consistent with investor uncertainty over

future volatility, augmented with longer holding periods or trading costs that make portfolio

rebalancing costly. This highlights the difficulties in employing simulation designs and

generalising their results for the whole market: the outcomes would depend on many

practice, and the choice between lotteries could depend on their context; in particular, the first

pair offers the choice between a certain win and a speculative bet, and most people tend to go

for the certain sum. The second pair offers the choice between two speculative lotteries and

most people tend to go for the one with the larger payoff in the unlikely event of winning.

Following Allais (1953), many more tests were conducted and these confirmed that in various

combinations of lotterys people tend to make choices that violate the independence axiom,

like certainty and isolation effects (Kahneman and Tversky, 1979) and the context (framing)

effect (Hershey and Schoemaker, 1980). 60 Lucas (1978) proposes a model of exchange economy populated by an infinite number of

identical individuals, each of which is endowed with a never-perishing tree that yields a

random crop of apples (dividends) each year. In that setting he demonstrates that the price of

the trees equals the present value of future crops discounted with a stochastic discount factor.

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assumptions that are incorporated in the simulation – e.g. holding period, transaction costs,

parameter learning – and the external validity of the results is therefore difficult to ascertain.

In this study we employ statistical methods in order to address the research questions

at hand. Such methods can be used with any type of numerical data, but are particularly well

suited for studies that investigate continuous data in terms of correlation and stochastic

(probabilistic) causation. Returns are in practice a continuous random variable, therefore such

methods allow the estimation of expected mean returns or return quantiles conditional on a set

of categorical, ordinal, or numerical variables. This could allow the estimation of spreads in

returns between different clusters of companies in a manner that recognises the correlation of

the explanatory factors and estimates the contribution of each independent variable on the

conditional return moments and quantiles. For example, idiosyncratic risk could be associated

with higher beta, lower liquidity, younger company age, smaller size, or smaller book value of

equity. Some of the simple descriptive or exploratory methods do not allow control for such a

correlation and consequently it is difficult to attribute observed differences in returns to each

of the correlated factors. Other methods (e.g. neural networks or some non-parametric

methods) may be better suited to handle non-linear dependencies,61 but the cost is obscuring

the contribution of each independent variable on stock returns, making the model behave like

a “black box”. Instead, we choose to conduct our analyses using linear regression models,

where dependent variables are hypothesised to be linear functions of a set of explanatory

variables. That approach offers transparent interpretation of regression parameters and

directions of co-movement, and still could accommodate complicated models, including

polynomial ones.

Therefore, in our study we chose to use regression models to infer the relationship

between returns and risk characteristics like beta, size, liquidity, and idiosyncratic risk. The

choice is closely related with the choice of deductive approach, and in particular our concerns

about external validity, generalizability and truth-preservation. In this way we can expect that

our conclusions will be stable in time, markets, institutional settings, market infrastructures,

and our conclusions would not be overturned or be significantly amended from new evidence.

Similarly, our objective to prevent data mining and model overfitting compelled us to use

regression analysis instead of non-linear or non-parametric models.

61 The correlation is essentially a measure of linear dependence. It is possible for two

variables to be dependent in a non-linear fashion, and yet their correlation could be zero.

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3.2.5. Regression models

3.2.5.1. Correlation and causation

The use of statistical models requires a set of explanatory variables (categorical,

ordinal, or numerical) that are hypothesised to be affecting the conditional distribution of the

dependent variable. The concerns about external validity and how well the results generalise

out of our sample require the use of as large a sample as possible. This consideration forced

us to leave out characteristics, which could predict the risk profile of the company, but were

not publicly available or were available for only a fraction of all companies or subsets of

years. Therefore, we had to give up on use of some measures of idiosyncratic risk and make

do with what could be readily obtained for a sufficient number of companies. For example,

the variability of net income or sales could be a better proxy of idiosyncratic risk that is not

affected by “noise trading”, but due to the absence of such information for a representative

sample of companies (especially for the smaller ones), we could not implement such an

approach. Similar considerations do not allow the use of option-implied volatilities, because

these are available only for recent years and for parts of the market (e.g. larger companies)

that may not be representative for the entire cross-section. Similarly, availability of

information on individual transactions could provide better measure of breadth of the investor

base, and hence better measure of undiversified idiosyncratic risk, but such information was

not available to us. Information on other characteristics that could affect the conditional mean

returns like investor profile (e.g., the share of institutional ownership) could be useful in

clarifying whether those investors were likely holding undiversified portfolios and would be

seeking a risk premium for idiosyncratic risk, but again such information was not available.

Therefore, the operationalisation of the theoretical model for statistical testing requires

the estimation of risk characteristics using mostly publicly available data: stock prices,

capitalisation, book value of equity, traded volume. Other characteristics had to be estimated

from this limited set of data, which is clearly one of the limitations of the approach.

Idiosyncratic risk is estimated from the limited available information on the total stock returns

(daily and monthly) and the available information of market excess returns, the Fama–French

factors, and, in some cases, market momentum. Thus, idiosyncratic risk is reduced to just the

changes of prices that were not attributable to some market trend (market return, Fama–

French factors, or momentum). This definition of idiosyncratic risk is much narrower that

what could be considered in other designs, but has the advantage that it can be estimated for

almost all securities and thus enhances the external validity (generalisability) of our results.

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Statistical analyses can be performed in terms of correlation, but our context requires

more – establishing the causation from risk to returns. For example, we can note that if some

characteristic (e.g., size) is correlated with higher average return, this does no imply that

necessarily size is a characteristic that is valued by investors. Instead, it might be the case that

smaller companies have less mature business models and diversified cash flows, which makes

them more vulnerable to economic shocks. In that case, the correlation between size and

returns would not imply that size is ‘causing’ higher returns, only that they are correlated. In

general, causality is a concept that is difficult to implement in statistical studies. Probably the

most popular statistical test of causation is the Granger (1969) causality test. In his approach,

one time series (𝑋𝑡) Granger-causes another one (𝑌𝑡), if past history of the first series (𝑋𝑖, 𝑖 =

0… (𝑡 − 1)) together with the past series of the predicted series itself (𝑌𝑖 , 𝑖 = 0… (𝑡 − 1))

predicts 𝑌𝑡. Nevertheless, it does not follow that 𝑋𝑡 truly causes 𝑌𝑡. 62 Therefore, our design

has certain limitations in establishing causality per se, without the context provided by the

theoretical models, from which the relationship is developed – Levy (1978) and Merton

(1987). To address these concerns, as well as look-ahead bias, we use not the realised returns

in a given month, but the expected values based on the past history. Nevertheless, it does not

follow that those statistical analyses necessarily establish causality between idiosyncratic risk

and volatilities. Instead, they establish correlation in a setting that aims to reduce look-ahead

bias. The causation is only inferred from the fact that this correlation pattern is predicted by

the theoretical model; it is still possible, however, that idiosyncratic volatilities serve as a

proxy for company exposure to some other risk factor, and the underlying model is overall not

valid. We aim to mitigate those concerns by implementing a host of robustness checks, but the

risk cannot be eliminated altogether.

3.2.5.2. Factor model estimation

The choice of specific statistical methods depends on a number of factors, like the

types of variables available, the types of hypotheses tested, assumptions and powers of the

various tests. Most of the explanatory variables available to us in this study (either retrieved

from external data sources, or calculated by us) were continuous, which allowed significant

flexibility in selecting statistical methods. In this phase we have to make three principal

choices. The first one is the selection of methods to estimate idiosyncratic returns (also

interchangeably referred to as shocks or innovations) and methods to forecast the expected

62 “Post hoc ergo propter hoc” fallacy (Lat. “after this, therefore because of this”).

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idiosyncratic risk. The second one is selecting methods to compare the predictive accuracy of

the alternative forecasts. The last one is a method to measure the impact of idiosyncratic

volatilities on realised excess returns. In this subsection we shall explain the principal options

available to us, while the details of their implementation are provided in the remaining

sections of this chapter.

The first part of the decision concerned what sort of model could be used to forecast

returns. Idiosyncratic risk is an abstract concept that is difficult to define. In the model of

Merton (1987) it is simply a random deviation of end-of-period production of each firm that is

uncorrelated with the rest of the investees. In our setting this concept requires an

operationalisation, where idiosyncratic risk is the difference between the observed actual

return on the stocks and the expected return on the stock, estimated from some model, the

parameters of which are estimated from past periods. It is possible to select some non-linear

model for stock selection, e.g. Levin (1995), and use its forecasts to separate actual returns

into expected and unexpected components, we acknowledged that such an approach is rarely

pursued in literature and would obscure comparability of our results with other research in the

field. At any rate it should be emphasized that linear asset pricing models can accommodate

many non-linear situations63 and that assumption should not be considered a substantial

limitation.

Ross (1976) proposes an alternative to the CAPM reviewed previously – the Arbitrage

Pricing Theory (APT). He assumes that 𝑁 factors drive returns in linear fashion, i.e.

𝑟𝑖,𝑡 = 𝛼 + ∑ 𝑁 𝑛=1 𝛽𝑛,𝑖𝑓𝑛,𝑡 + 𝜀𝑖,𝑡,

where 𝜀𝑖,𝑡 are mutually uncorrelated and have expected value zero and finite variance, i.e.

𝔼𝜀𝑖,𝑡 = 0, 𝔼 < ∞, 𝔼𝜀𝑖,𝑡𝜀𝑗,𝑡 = 0. Furthermore, it is assumed that the expected values of

factors equals zero (𝔼𝑓𝑛,𝑡 = 0), so that prices are affected only by the unexpected factor

realisations. Notably, the model does not assume that the factors are mutually independent

(𝔼𝑓𝑛,𝑡𝑓𝑚,𝑡 need not be zero), and even does not require factors to be independent with the

errors (𝔼𝑓𝑛,𝑡𝜀𝑖,𝑡 need not be zero) or have finite variances. Ross argues that in an efficient

market one should not be able to construct an arbitrage zero-investment portfolio.64 In

63 E.g. it is allowed that one factor is a square of another one or product of two others. 64 To prove the existence of an arbitrage-free equilibrium he assumes the following: 1) There

is at least one asset with bounded losses; 2) There exists at least one investor who is uniformly

less risk-averse than some constant relative risk-averse agents, and who believes that the

assets are generated by the linear model stated above, and who is not asymptotically

negligible; 3) All agents are risk-averse and hold the same expectations; 4) the aggregate

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empirical applications the factor model takes the form

𝐸𝑖 − 𝜌 ≈ 𝛽1,𝑖𝛾1 + 𝛽2,𝑖𝛾2 +⋯+ 𝛽𝑁,𝑖𝛾𝑁,

where 𝛾𝑗 is the risk premium on a portfolio that has unit exposure only to factor 𝑗; if 𝐸𝑗 is

the expected return on a portfolio with unit loading only to factor 𝑗 and zero loadings on all

other factors 𝑓𝑖,𝑖≠𝑗, then 𝛾𝑗 = 𝐸𝑗 − 𝜌.

The APT does not stipulate a list of factors; it is therefore useful to identify the criteria

that should be met by candidate factors. The structure of the model presents immediately two

eligibility criteria: firstly, the assumption of zero expected factor value implies that factors

affect prices through their unexpected realisations. Secondly, the factor should be pervasive in

the sense of affecting sufficiently many securities through the linear relationship above, or

else randomly constructed portfolios would tend to diversify it away, producing portfolios

with no exposure to that factor. Many macroeconomic factors appear suitable candidates, and

Chen et al. (1986) investigate how various macroeconomic factors could explain asset returns.

They conclude that the following factors significantly affect stock returns: industrial

production (led by one period to make it contemporaneous with asset prices); changes in risk

premium (the spread between the yield on portfolio of ‘Baa’ and lower grade bonds, and

long-term government bonds); twists of the yield curve (the spread between a short and a long

point of government yield curve); unanticipated inflation (the difference between actual

inflation and expected inflation estimated by macro-econometric model); changes in expected

inflation; the last two inflation-related factors were reported to be more significant in more

turbulent periods characterised with volatile inflation.

Macroeconomic models are useful for modelling macroeconomic influences on

portfolio values, and various hybrid models (combining macroeconomic series with other

data) are implemented in the financial industry. For example, Northfield (2013) combines the

macroeconomic factors of Chen et al. (1986) with other series like oil prices, housing starts,

exchange rates, and five statistical factors in order to model portfolio macroeconomic risks.

Macroeconomic factor models are occasionally criticised for poor explanatory

power65. Furthermore, the classic model of Chen et al. (1986) highlights the implementation

difficulties. Thus, while stock prices are available at intra-day frequencies, many economic

series (e.g. industrial production, inflation, gross national product) are available only at

demand for all assets is non-negative; 5) expectations are uniformly bounded. Under those

assumptions he proves that there exist 𝜌 and 𝛽𝑖 such that ∑ ∞ 𝑖=1 (𝐸𝑖 − 𝜌 − 𝛽𝑖𝛾) <∞, with

𝜌 equal to the risk-free rate, if a risk-free asset exists in the market. 65 See, for example, Connor (1995)

83

monthly or even quarterly frequencies. The flow series may entail a lag (hence industrial

production needs to lead one period or more); moreover, information when the values of the

macroeconomic indicators are released to the public and incorporated in asset prices as well

as any subsequent revisions of the indicator are usually hard to obtain. What matters for APT

are unanticipated factor realisations, hence one needs to implement a model of the rationally

expected values (like the expected inflation, in order to derive surprise inflation), which

inherently entails model risk. Furthermore, part of the information of the economic

environment would be acquired by investors either by direct observation, or through other

proxy variables, so it is unclear what part of the effect of the economic datum should be

realised during the period of the slowdown and what part should be realised upon the release

of the economic data. Finally, the impact of macroeconomic releases need not be flat, and

large surprises may trigger more than proportional response relative to small surprises.

Because of these limitations, arbitrage models with trading factors are often preferred over

macroeconomic models.

Therefore, in our study we employ a linear factor model with traded factors in order to

separate the observed returns into systematic and idiosyncratic components. In different

contexts we use a varying number of factors, depending on the context. The exact list of

factors used is discussed in the next section.

Estimation of the parameters of the linear factor models is deceptively straightforward.

The common practice in the literature is to use the Ordinary Least Squares (‘OLS’) estimator

that minimises the sum of the squared errors. The OLS estimator has the desired property of

being the best linear unbiased estimator of model parameters, however, since the objective

norm is squared error, differences between the observed and fitted values are squared and thus

outliers could significantly affect the estimates. Another possible approach is to employ a

robust estimator of regression coefficients. Such estimators are, for example, the Least

Absolute Deviation (‘LAD’)66, the M-estimator67, or the Trimmed Regression Quantile68, to

66 The LAD estimator minimises the sum of the absolute values of the errors, i.e. solves

min𝛽𝑖 ∑𝑡 |𝜀𝑖𝑡| 67 The M-estimators minimise the sum of errors scaled by some factor 𝜎 and weighted through some function 𝜌, i.e. they solve min𝛽𝑖 ∑𝑡 𝜌(𝜀𝑖𝑡/𝜎). There are many choices for 𝜌,

for example the function for the Huber’s M-estimator is

𝜌(𝑥) = {

1

2 𝑥2 if|x| < 𝑐

𝑐|𝑥| − 1

2 𝑐2 if|x| ≥ c

.

84

name just a few of the options.69 In practice, however, non-OLS estimators for factor models

of asset returns are exceptions rather than the rule, possibly because the asymptotic theory for

these estimators is less developed compared to estimators like OLS and Maximum

Likelihood, which hinders hypothesis testing. Concerning unbiasedness of the LAD estimator,

Gray et al. (2013) observe that LAD beta estimates are systematically below the

corresponding OLS estimates. They estimate the CAPM regression for all securities in the

Australian market from January 1976 to May 2012 using both the OLS and the LAD

estimator. They observe that if an estimator of beta is unbiased, then the value-weighted

average of individual betas with respect to a specially constructed index with constant weights

should equal 1 when average is taken for all constituents of the index. They find that while

that was indeed the case for the OLS estimator, the LAD averages below 1, thus suggesting

possible downward bias of the LAD estimator.

Having regard for concerns that robust estimators might be biased and in order to

facilitate comparisons with the existing literature, we opted for using the OLS estimator to

calculate the parameters of the linear factor models. Nevertheless, we believe that the topic of

the estimation method could be a worthwhile direction for future research. We note that in

some cases the betas estimated through OLS are well outside the corridor 0 to 3, which in

itself is already quite wide. This indicates that in some cases there may exist larger

idiosyncratic shocks in one single month, which then result in the factor model attempting to

reconcile the factor loadings with the extreme return. In the case of simple OLS idiosyncratic

volatility70, that could result in OLS volatility being lower than the volatility when a robust

model was being used.

Wrapping up the discussion in this subsection, we choose to use traded factor models

for asset returns in order to split returns into systematic and idiosyncratic components. The

models shall be fitted using Ordinary Least Squares (OLS) estimator.

3.2.5.3. Selection of volatility models

The estimation of volatilities was a principal methodological decision in our research.

68 The TRQ estimator is a weighted average of different beta quantiles. In general, 𝛽𝛼 = 1

1−2𝛼 ∫ 1−𝛼

𝛼 𝛽(θ)d𝜃. In practice this works by fitting some finite number of quantile regressions

and then calculating a weighted average of the estimates. 69 See Chan and Lakonishok (1992) 70 We define the idiosyncratic volatility measures later in the chapter. In this case the OLS

volatility is simply the standard deviation of the error term from the fitted factor model.

85

We hypothesise that differences in precision across alternative forecasts of volatilities result

in the conflicting findings in the literature. Therefore, we aim to use forecasts that meet three

criteria. Firstly, we want our forecasts to be consistent with existing research, so that our

study could reconcile the conflicting evidence, rather than contribute to the confusion by

offering yet another estimator. Secondly, we aim our estimators to be transparent inasmuch as

possible, so that the obtained results could be intuitively interpreted. We want to avoid a

situation where the models operate as black boxes and the results can be interpreted

principally in terms of statistical significance. Finally, we want our estimators to be

sufficiently representative for the available forecasting methods, rather than compare variants

of the same or very similar methods.

The simplest estimator of idiosyncratic risk variance is the OLS estimator, which is

simply the standard deviation of the residuals from some factor model:

𝜎𝑜𝑙𝑠,𝑡 2 = ∑𝑇𝑘=1

(𝜀𝑖,𝑡−𝑘) 2

𝑇−𝑑𝑓 ,

where 𝜀𝑖,𝑡−𝑘 denotes the idiosyncratic residual from the factor model for stock 𝑖 in month

(𝑡 − 𝑘) obtained from a regression estimated over the period (𝑡 − T) until (𝑡 − 1). 𝑇 is

the actual number of months for which information is available; thus 𝑇 is between 60 and

30 (the minimum number of months required for estimation); 𝑑𝑓 was the residual degrees

of freedom for the estimated model and equals the number of estimated parameters. The

underlying assumption is that OLS variances do not change much from period to period as the

estimates of month 𝑡 volatility differs from month (𝑡 + 1) by just one return as one return

exiting the rolling window and being replaced by a new one as the window rolls forward.

Thus, the realisation at period 𝑡 does not affect the estimated expected volatility, i.e. the

expected value is that as at the end of period 𝑡 − 1, or equivalently, at the start of period 𝑡.

The benefit of that approach is the simplicity of the estimator and the filtering of random

idiosyncratic innovations. On the other hand, in cases where idiosyncratic volatility

permanently changed to a different value, that change would filter quite slowly in the

volatility forecast.

Variances may change significantly from month to month. In that respect, it is useful

to note another interpretation of the OLS variances and their relation to idiosyncratic returns.

Observe that squared idiosyncratic return is in fact an estimator of the unobserved

idiosyncratic variance.71 From that perspective, if idiosyncratic volatility consists of some

71 By construction (as residuals from OLS regression), the idiosyncratic residuals have mean

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long-term, constant volatility and random deviations from the long-term volatility, then the

moving average of the squared monthly idiosyncratic returns could be interpreted as a filter

that removes the random deviations and extracts the long-term level.

However, the moving average filter also has a drawback – if idiosyncratic volatility is

permanently increased at some point (a step increase), it would take five years (60 months)

for that increase to filter entirely through the moving average filter. Therefore, a more

sophisticated method for estimating idiosyncratic volatility could yield better results.

The OLS estimator was used in the early studies (in the 1960s and 1970s). Since then

it has fallen out of favour as too unresponsive to changes. Nevertheless, we shall use it as one

of our estimators for two reasons. Firstly, it serves as a benchmark that could provide

perspective on how other models perform and give some intuitive feeling of how rigid the

forecasts produced by alternative methodologies are. Secondly, in the foregoing paragraphs

we interpreted the estimator as a filter, and in that sense it can be viewed as similar to other

methods that can operate as filter as well. For example, the “exponentially weighted moving

average”, which is a type of Integrated GARCH(1,1) and was used in the RiskMetrics

methodology of J.P.Morgan/Reuters (1996), can be seen as an exponential filter, and thus an

incremental improvement over the OLS filter. In our case it is useful to have the OLS

benchmark, because it gives empirical perspective on the performance of GARCH(1,1)

model.

The works of Engle (1982) and Bollerslev (1986) brought significant improvement of

flexibility in terms of modelling of volatility evolution. Engle (1982) introduced the

Autoregressive Conditional Heteroscedasticity (ARCH) model, and his formulation was

subsequently generalised by Bollerslev (1986), who formulated the Generalised

Autoregressive Conditional Heteroscedasticity (GARCH(𝑝, 𝑞) model). The simplest, but also

remarkably successful version, is the GARCH(1,1) model:

𝜎𝑡 2 = 𝜔 + 𝛼𝜀𝑡−1

2 + 𝛽𝜎𝑡−1 2 . (4)

The model always yields positive predicted variance when the three parameters are positive

(𝜔) or non-negative (𝛼, 𝛽), i.e. 𝜔 > 0, 𝛼, 𝛽 ≥ 0. When 𝛼 + 𝛽 < 1, the model is covariance

stationary72 and its unconditional73 volatility is 𝜎2 = 𝜔/(1 − 𝛼 − 𝛽). When 𝛼 + 𝛽 = 1, the

zero. Then 𝑉𝑎𝑟(𝜀) = 𝔼(𝜀2) − (𝔼(𝜀))2 = 𝔼(𝜀2). 72 A real-valued stochastic process {𝑋𝑡} is covariance-stationary if its expectation 𝔼𝑋𝑡 does not depend on 𝑡, and its 𝑘-th order autocovariance 𝔼[(𝑋𝑡 − �̅�)(𝑋𝑡+𝑘 − �̅�)] is finite and depends only on 𝑘, but not on 𝑡. 73 The unconditional volatility is simply the variance of the process, 𝑣𝑎𝑟(𝑋𝑡) . The

87

model reduces to the Integrated GARCH(1,1) model of Nelson (1990). The motivation for

that model is the observation that for financial time series, the sum of the two parameters (𝛼 +

𝛽) is often quite close to 1. The unconditional variance of IGARCH is infinite, and implies a

random walk of variances (i.e. every shock on volatility has permanent impact and never

decays). IGARCH has the benefit of being a parsimonious specification and one that ensured

that the estimated parameters are close to reality – certainly an asset if data is scarce. For this

reason it is also the model that underlies the volatility modelling (exponentially-weighted

moving average) of the successful RiskMetrics risk quantification specification of

J.P.Morgan/Reuters (1996). However, the evidence of Fu (2009) of stationarity of variances

for about 90% of all series convinces us to abstain from employing IGARCH as the principal

specification of our study.74

Since the introduction of the original GARCH(𝑝, 𝑞 ), a number of alternative

specifications were proposed, although GARCH(1,1) specification proved to be difficult to

outperform out of the sample for one-step-ahead forecasts, as documented by Hansen and

Lunde (2005). However, the model that actually gained traction in idiosyncratic risk tests is

the Exponential GARCH model introduced by Nelson (1991). The model allows an

asymmetric response to positive and negative innovations. Such asymmetric reactions are

common in financial time series, hence the preference for that model in recent literature on

idiosyncratic risk; it is also the forecasting model used in the studies of Fu (2009) and Malkiel

and Xu (2004). A couple of slightly different specifications for the model exist. Using one of

the specifications and changing the mean equation with the FFC gives:

(𝑟𝑖,𝑡 − 𝑟𝑓,𝑡) = 𝛽0 + 𝛽1(𝑟𝑀,𝑡 − 𝑟𝑓,𝑡) + 𝛽2𝑟𝑆𝑀𝐵,𝑡 + 𝛽3𝑟𝐻𝑀𝐿,𝑡 + 𝛽4𝑟𝑀𝑂𝑀,𝑡 + 𝜀𝑖,𝑡, (5)

𝜀𝑖,𝑡 = √𝜎𝑖,𝑡 2 𝜈𝑖,𝑡, (6)

log(𝜎𝑖,𝑡 2 ) = 𝜔𝑖 + ∑

𝑝 𝑘=1 𝛿𝑘log(𝜎𝑡−𝑘

2 ) + ∑𝑞𝑚=1 𝛼𝑚{𝜃𝜈𝑖,𝑡−𝑚 + [|𝜈𝑖,𝑡−𝑚| − E|𝜈𝑖,𝑡−𝑚|]}, (7)

where 𝜔𝑖, 𝛿𝑘, 𝛼𝑚, 𝜃, 𝛾 are the parameters of the EGARCH(𝑝, 𝑞) model, 𝜈𝑖,𝑡−𝑚 is an

i.i.d. random variable with mean zero and unit variance.

Fu preferred to employ the Exponential GARCH (EGARCH) model for modelling

volatilities. The advantages of EGARCH over GARCH were that it ensured positive

unconditional variance at time 𝑡 does not depend in the variance at time 𝑡 − 1. 74 We also calculated the expected forecasts from IGARCH but we found that GARCH(1,1)

produced slightly more accurate results, which was an additional consideration in favour of

GARCH(1,1) even through scarcity of data was an argument in favour of IGARCH.

88

variance75 and allowed asymmetric response to shocks.

One downside of EGARCH models is that they have lighter tails compared to regular

GARCH. If the GARCH(1,1) process is stationary, Mikosch and Starica (2000) showed that

for distributions in the Frechet domain of attraction (Paretian tails),76 the extremes of the

GARCH process are heavy-tailed with extremal index 𝜃 ∈ (0,1), with expected length of

clusters of high volatility in the normalised sequence equal to 1/𝜃.77 Thus, Mikosch and

Starica demonstrate that GARCH models allow for volatility clusters to exist and derive the

corresponding conditions. In contrast, Lindner and Meyer (2003) study the extremal

behaviour of EGARCH processes and prove the extrema of ln(𝜎𝑡 2) were light-tailed (in the

Weibul domain of attraction), which includes distributions it exponential tails like the

Gaussian distribution, and consequently that ln(𝜎𝑡 2) would not show cluster behaviour. A

further downside of the EGARCH specification for us is that it depends on more parameters,

the estimation of which could be more problematic at low frequencies like monthly data, and

that the parameter uncertainty problem could be exacerbated by the exponentiation of the

variance in EGARCH specification, which could be especially problematic given the use of

individual securities as assets in the cross-sectional tests.

75 The GARCH model ensures that by posing non-negative coefficients, but in practice in

some situation an unconstrained optimisation suggests that negative parameter estimates

result in better in-sample fit. EGARCH model would yield positive volatility even if some of

the parameters are negative. 76 Let 𝑀𝑛 be the maximum of a sample of 𝑛 draws from some distribution 𝑃(𝑋 ≤ 𝑥) = 𝐹(𝑥). As the sample grows large 𝑛 →∞, the distribution of maxima becomes degenerate as 𝑃(𝑀𝑛 ≤ 𝑥) = [𝐹(𝑥)]

𝑛 . However, the Fisher-Tippet theorem proves that if there exist

normalising series of constants 𝑎𝑛, 𝑏𝑛 such that the distribution of the normalised (scaled)

maxima 𝑃 ( 𝑀𝑛−𝑎𝑛

𝑏𝑛 ≤ 𝑥) is non-degenerate, then the limiting distribution of extremes can be

either a Gumbel distribution (for light-tailed distributions), a Weibul distribution (for

distributions with finite support), or a Frechet distribution (for medium and heavy-tailed

distributions) (Embrechts et al. (2003)). If a distribution belongs to a Frechet domain of

attraction, then the probability of exceeding a certain threshold P(X > 𝑥) would have the form 𝑥−𝛼L(𝑥), where 𝛼 is a positive constant and L(𝑥) is some slowly varying function,

i.e. lim 𝑥→∞

L(t 𝑥)

L(𝑥) = 1, ∀𝑡 > 0 , and so the tail of the distribution decays approximately like

Pareto distribution (Gnedenko (1943)). A number of known distributions do not belong to any

of these domains of attraction (i.e. they are not max-stable); two important examples are the

Gaussian and the exponential distribution. 77 The standard tail index results from the previous note were derived under assumption of

independent draws, which is clearly not the case with GARCH models, and so the 𝜃 referred here is the tail index of the extremes of the volatility series, {𝜎𝑖} The link between the extremes of the process {𝑋𝑖} and the related process {𝜎𝑖} are given by Theorem 4.1 on p. 1437 in Mikosch and Starica (2000)

89

Overall, each approach has its advantages and disadvantages. The EGARCH removes

the difficulties associated with corner solutions to the GARCH optimisation and allows for an

asymmetric volatility response to shocks, but it also limits the ability of the model to produce

volatility clustering and at monthly frequency could increase parameter uncertainty risk on

account of the larger number of parameters to be estimated. Tests with IGARCH and GARCH

suggested that IGARCH model did not result in more stable estimates but the accuracy of

GARCH was somewhat better. Together with the counter-factual assumption of permanent

shocks on variance, we opted for GARCH(1,1) as the baseline model of volatility in this

study. We impose the stationarity assumption and specifically constrain the domain for the

model parameters to 𝛼 + 𝛽 < 1 and 𝛼, 𝛽 > 0.

The distribution of the innovations – 𝜈 in the EGARCH model (6)-(7) – is another

significant choice. Often in related studies 𝜈 is assumed to be distributed either as standard

normal distribution78 or as generalised error distribution.79 Both Fu (2009) and Spiegel and

Wang (2005) use normal distribution for 𝜈 in their EGARCH specifications. Furthermore,

Fu allows variable lags for 𝑝 and 𝑞 between 1 and 3; among the nine resulting models for

each stock and date he selects the one with highest value of Akaike Information Criterion

(AIC).80 In the GARCH specification the choice of distribution might be a more crucial

choice because by design that model reacts symmetrically to innovations and the imposition

of a symmetric distribution like the standard normal in the presence of asymmetric

innovations81 might impact the predictive performance of the model. Therefore, we opted for

a non-symmetric distribution of shocks. There are a couple of different specifications of the

Skew Generalised Error Distribution (SGED). In our study we implemented

78 The standard normal (Gaussian) distribution with zero mean and unit variance has density

𝑓(𝑥) = 1

√2𝜋 𝑒 1

2 𝑥2 .

79 The Generalised Error Distribution has the following three-parameter density:

𝑓(𝑥) = 𝜅𝑒

− 1 2 | 𝑥−𝛼 𝛽 | 𝜅

21+1/𝜅𝛽Γ(1/𝜅) ,

where 𝛼 is location parameter, 𝛽 is scale parameter, and 𝜅 is shape parameter. This specification needs to be additionally standardised to zero mean and unit variance. When 𝜅 = 2, the distribution reduces to normal. 80 The Akaike’s Information Criterion (AIC) is an heuristic derived from information theory

that compares competing models in terms of their likelihood (𝐿) on a specific data set and the number of parameters (𝑘), with 𝐴𝐼𝐶 = 2𝑘 − 2ln (𝐿). 81 For the full sample skewness of idiosyncratic shocks (arithmetic returns) was positive at

about 2.3.

90

𝑓(𝑧) = 𝜅exp{−

1

2 |√2−2/𝜅

Γ(1/𝜅)

Γ(3/𝜅) 𝑧|}

√2−2/𝜅 Γ(1/𝜅)

Γ(3/𝜅) 21+1/𝑘Γ(1/𝜅)

(8)

𝑓(𝑧|𝜉) = 2

𝜉+𝜉−1 [𝑓(𝜉𝑧)𝐻(−𝑧) + 𝑓(𝜉−1𝑧)𝐻(𝑧)], (9)

where 𝐻(𝑧) is the Heaviside step function82, 𝜉 is the parameter controlling skewness, and

𝜅 is the shape parameter (when 𝜅 increases, the distribution gets flatter).

Additional distributions for the innovations were also considered, in particular the

Negative Inverse Gaussian, Skew Hyperbolic distribution, Generalised Hyperbolic

Skew-Student distribution, and Skew Student-𝑡 distribution. We found that SGED tended to

produce more robust and accurate forecasts. An intuitive reason could be that the SGED has

lighter (exponential) tails; consequently, a strong innovation was less likely compared to the

fatter-tailed alternatives, which resulted in quicker adjustment of volatility forecasts upon

stronger shock while nevertheless allowing medium tails83 and asymmetric shocks.

Wrapping up the discussion, we use GARCH-family as one of our selected models

because it is used in many volatility studies, and by some of the related studies that lend

support to the research hypothesis. We chose GARCH(1,1) for one-step forecasts in view of

the evidence of its good empirical performance, the scarcity of data at monthly frequency, and

the intuitive interpretation of its parameters. We compensate some of the loss of flexibility

relative to the EGARCH(p,q) model with varying order by employing a more flexible

distribution for the error term (the SGED distribution).

The simple random walk without drift has the form

ℎ𝑡 = ℎ𝑡−1 + 𝜖𝑡, 𝜖𝑡~𝑁(0, 𝜎 2),

82 For practical purposes,

𝐻(𝑧) = { 1 𝑖𝑓 𝑧 > 0 0 𝑖𝑓 𝑧 < 0

.

83 It might be appropriate to note that there are at least three different approaches to

discussing tail fatness (heaviness): one approach is to consider the kurtosis of the distribution

and to label distributions with kurtosis over 3 as heavy-tailed (3 is the kurtosis of the Gaussian

distribution); another approach is based on the previously mentioned concept of max-domains

of attraction, with distributions in the Frechet domain being labelled as fat-tailed; finally,

heavy-tails could also describe sub exponential distributions, for which the distribution of a

sum of sub exponential random variables being asymptotically equal to the distribution of the

maximum of the individual variables (i.e. the probability that sum would exceed some

threshold is asymptotically distributed as the probability that any single summand would

exceed that threshold). The concept of fat tails is discussed in Haas and Pigorsch (2009); in

that connection one should note that for example GARCH(1,1) with Normal innovations

produces fat-tailed stochastic process in the sense that its kurtosis is over 3. Thus light tails of

the shocks do not necessarily mean light tails of the fitted series.

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where ℎ𝑡 denotes the stochastic process and 𝜖𝑡 is the random shock at time 𝑡, which is

assumed to be independently and identically distributed (i.i.d.) with mean 0. A recursive

substitution shows that ℎ𝑡 = ℎ0 +∑ 𝜖𝑖 𝑡 𝑖=1 and that past shocks never die out but have

permanent effect on the level of ℎ𝑡. If the volatility process is a random walk, then the

expected value of ℎ𝑡 conditional on the information at time (𝑡 − 1) is just ℎ𝑡−1, which is

the known last realisation of the process and reflects all past shocks 𝜖𝑖, 𝑖 = 1… (𝑡 − 1), i.e.

𝔼𝑡−1ℎ𝑡 = 𝔼𝑡−1ℎ𝑡−1 + 𝔼𝑡−1𝜖𝑡 = ℎ𝑡−1. Since the impact of shocks never fades, the variance of

the process increases unboundedly as 𝑡 increases.

Therefore, if volatilities follow random walk, then we could use the volatility in the

last month as the expectation of the volatility for the next month. This approach of using

(𝑡 − 1) volatility ℎ𝑡−1 in place of expected volatility 𝔼𝑡−1ℎ𝑡 is employed by Ang et al.

(2009, 2006). Even if volatilities are stationary, this does not invalidate entirely the approach

of Ang et al.84 because persistence of volatilities is a well documented phenomenon. For

example, Engle and Patton (2001) recognise volatility persistence as one of the stylised facts

of asset returns. Similarly, it is consistent with the approach of Integrated GARCH (Nelson,

1990), which assumes that 𝛼 + 𝛽 = 1 in equation (4). Therefore, if monthly volatility is

sufficiently persistent, or if the estimated parameters values of the GARCH(1,1) model are

such that 𝛼 + 𝛽 is sufficiently close to 1, then the approach of Ang et al could result in

superior forecasts for ℎ𝑡 85. Therefore, ℎ𝑡−1 cannot be discarded as a predictor of 𝔼𝑡−1ℎ𝑡

even in the face of the evidence rejecting the unit root of volatilities.

We include this measure in our study also because it is the one that produced the

prediction of a negative correlation between risk and returns, and therefore it is crucial that

the evidence is revisited and the finding is confirmed or rejected.

One of the problems with the methodology of Ang et al was that it did not recognise

that in the presence of mean reversion, ℎ𝑡−1 is not an unbiased predictor of 𝔼𝑡−1(ℎ𝑡). Fu

suggests that GARCH offers a superior solution on the problem. However, there is a subtle

difference between the GARCH model and the approach of Ang et al.: the GARCH model

uses squared idiosyncratic residuals as a proxy for the realised volatility, whereas the

approach of Ang et al. (2006) estimates the volatility in month (t − 1) from

higher-frequency data (daily returns). The approach of estimating lower-frequency volatilities

84 In their study of 2006 they never claimed that volatilities followed random walk. Instead,

they defined their volatility measure to equal the volatility of the previous month. 85 Such superiority could be the result from lower noise in the volatility estimates from daily

data compared to the squared residuals used in the standard GARCH(1,1).

92

from higher-frequency data is theoretically justified by Merton (1980), who demonstrates that

volatility could be estimated with arbitrary precision using sufficiently high-frequency data.

Therefore, the estimation of monthly volatility from daily data could be a completely sound

approach and in fact could outperform squared monthly return as a proxy.

Therefore, instead of abandoning the history of monthly volatilities estimated from

daily data altogether in favour of GARCH, it might be more justified to attempt to forecast

𝔼𝑡−1(ℎ𝑡) from the available history. One such approach was pursued by Huang, Liu, Rhee

and Zhang (2012) who use ARIMA model fitted on the series of ℎ𝑡−𝑖, 𝑖 = 1,… ,24 (i.e. a

rolling window design using the last 24 months of data) in order to forecast 𝔼𝑡−1(ℎ𝑡).

One representation of the Autoregressive Moving Average (ARMA) process with lags

1, ARMA (1,1), is as follows:

ℎ𝑡 −𝑚 = 𝜙(ℎ𝑡−1 −𝑚) + 𝜃𝜀𝑡−1 + 𝜀𝑡,

where 𝑚 , 𝜙 and 𝜃 are the parameters of mode. Parameter 𝑚 is also called the

mean-revering level of volatility because when 𝜙 ∈ (0,1), the forecasted values of ℎ would

converge to 𝑚, given enough time. In general, the values of 𝜙, 𝜃 ∈ (−1,1) ensure that the

ARMA process is invertible.86 The unconditional expected value of the process equals 𝑚,

𝔼(ℎ𝑡) = 𝑚, and 𝜙 controls the speed of mean-reversion. An alternative parametrisation for

the process uses 𝜇 = 𝑚(1 − 𝜙) instead of 𝑚 , giving the alternative form ℎ𝑡 = 𝜇 +

𝜙ℎ𝑡−1 + 𝜃𝜀𝑡−1 + 𝜀𝑡.

Thus a principal difference between using the ARMA(1,1) and GARCH(1,1) in the

present context is the proxy they use for the realised volatilities in past months. Indeed,

substituting 𝜈𝑡 2 = 𝜀𝑡

2 − 𝜎𝑡 2 in 𝜎𝑡

2 = 𝜔 + 𝛼𝜀𝑡−1 2 + 𝛽𝜎𝑡−1

2 we obtain 𝜀𝑡 2 − 𝜈𝑡

2 = 𝜔 + 𝛼𝜀𝑡−1 2 +

𝛽(𝜀𝑡−1 2 − 𝜈𝑡−1

2 ), which after re-arrangement yields

𝜀𝑡 2 = 𝜔 + (𝛼 + 𝛽)𝜀𝑡−1

2 + 𝜈𝑡 2 − 𝛽𝜈𝑡−1

2 ,

which is an ARMA(1,1) process in squared shocks (idiosyncratic returns).

Therefore, we include ARMA(1,1) in our study in order to allow the controversial

measure of Ang et al. (2006) to be refined in a forward-looking manner, so as to explore how

the introduction of forward-looking expectations amends the conclusions reached by Ang et

al. (2006) and Ang et al. (2009).

A principal limitation of historical volatilities is that they incorporate only historical

86 The formulation of the ARMA process means that all observed values of ℎ𝑡 are function of the history of unobserved errors. If the process is invertible, then these errors can be

represented as weighted sums of the observed realisations.

93

information and are not, in fact, truly forward-looking. It could be expected that the use of

implied volatilities should be producing more a conclusive test of the models of Levy (1978)

and Merton (1987). However, derivatives are written only on some of the larger stocks, so

that approach could not be pursued.

3.2.5.4. Comparison of volatility forecasts

There are many methods that can be used to evaluate alternative volatility forecasts.

These differ in terms of the forecasting scheme, the measure of forecast quality, the estimator

of the true latent volatility, and the approach for testing the significance of differences of

accuracy among the alternative forecasts.

Concerning the forecasting schemes, Violante and Laurent (2012) highlight three

principal schemes: fixed, rolling, and recursive. In the fixed scheme, parameters of the model

are estimated using a set of fixed length, and then all future forecasts are made without

re-estimation of parameters. The rolling scheme is implemented using a rolling window of

fixed length. Finally, a recursive scheme is implemented using all past information to make

each forecast, with parameters being re-estimated for each period. In our situation we are in

fact unable to pick one of these, because the different forecasts used in the studies are

essentially estimated with different approaches. Thus, the OLS estimator and the estimator of

Ang et al. (2006) are estimated using rolling windows (in the latter case – of length just one),

while GARCH and ARMA forecast are based on expanding windows (recursive scheme). In

general, Violante and Laurent (2012) note that the fixed and the rolling schemes have certain

advantages over the rolling one when comparing nested models. Firstly, the fixed scheme

could be useful in situations where parameter estimation is difficult. Secondly, the rolling

scheme could accommodate situations where the estimated parameters change over time.

Finally, there are difficulties in implementing statistical tests in the recursive scheme due to

the complex asymptotic distribution of the test statistics as sample size grows with each

recursive estimation. In our case, however, the use of non-linear models requires larger

samples, and therefore the rolling scheme cannot be implemented for some of the volatility

estimators (GARCH and ARMA). Furthermore, such an implementation, if possible, would

render the produced forecasts incompatible with those in other studies.

The second problem which we face in the tests is that variances are latent

(unobservable), and therefore need to be estimated. Violante and Laurent (2012) note that

there are three typical estimators: the squared returns, the realised volatility, and the realised

kernel. The squared returns are an unbiased estimator when expected return is zero, as from

94

probability theory we have 𝑣𝑎𝑟(𝑋) = 𝔼(𝑋2) − (𝔼𝑋)2, so when 𝔼𝑋 = 0, the variance of 𝑋

equals the expected value of 𝑋2. The realised variance can also be calculated from the returns

of higher frequency. For example, the monthly variance can be estimated from daily returns

as 𝑣𝑎𝑟(𝑋𝑀) = ∑ 𝑋𝑑 2

𝑑∈𝑀 . Even with unchanged daily variance, monthly variances will change

depending on the number of days in the month. In cases where such changes were a concern,

we averaged volatilities by the number of days in month to calculate the daily variance, and

then scaled to a standardised month of 21 trading days. The last estimator of true volatility –

the realised kernel, developed by Barndorff-Nielsen et al. (2008) – is used mostly in

high-frequency trading where microstructure noise is a significant problem. The method

essentially employs kernel smoothing of the higher-frequency volatilities in order to smooth

out microstructure noise. The method is rarely used for estimation of monthly volatilities.

Moreover, the parameters of the smoother would need to be tuned, which would increase the

complexity of estimates with unclear benefit. Instead, studies like Bali and Cakici (2008) and

Spiegel and Wang (2005) filter the daily volatility series through a GARCH model which

smooths out noise from daily returns and then calculate the monthly realised volatility from

the daily filter volatilities. We follow that approach as well.

We consider three estimators of the true variance. The first is the traditional but noisy

squared idiosyncratic shock, i.e. ℎ𝑡 = 𝐼𝑅𝑒𝑡𝑡 2. In order to obtain a more accurate estimate we

also estimate an EGARCH(1,1) model using all available daily data. The in-sample estimates

for idiosyncratic variance are averaged by months (we require estimated variances for at least

10 days in each month). Thus ℎ𝑡 = 1

𝜏𝑡 ∑𝜏𝑡𝑖=1 𝜎𝑖

2, where 𝜏𝑡 is the number of days in month 𝑡.

EGARCH(𝑝, 𝑞) is defined by the following volatility equation:

log(𝜎𝑡 2) = 𝜔 + ∑𝑝𝑘=1 𝛽𝑖log(𝜎𝑡−𝑘

2 ) +

∑ 𝑞 𝑚=1 𝛼𝑚 {𝜃 (

𝜀𝑡−1

𝜎𝑡−1 ) + 𝛾 [|

𝜀𝑡−1

𝜎𝑡−1 | − E (

𝜀𝑡−1

𝜎𝑡−1 )]}. (10)

The distribution of 𝜀𝑡−1/𝜎𝑡−1 is assumed to be standard Gaussian, so 𝔼(𝜀𝑡−1/𝜎𝑡−1) =

(2/𝜋)1/2. The specification allows an asymmetric response of volatility, a phenomenon

observed in equity index returns. When 𝛾 < 0 in the above model, a negative shock 𝜀𝑡−1

increases volatility more than a positive shock of equal magnitude would have done. The

mean equation for the volatility model is again the Fama–French–Carhart four-factor model

estimated using OLS. Finally, as a control version we also filter monthly volatilities from the

GARCH(1,1) model with SGED innovations, estimated on all available monthly idiosyncratic

returns.

95

There are two principal methodologies for comparison of volatility forecast that were

employed in existing studies: the loss function approach, and the Mincer-Zarnowitz

regressions. The former (loss function approach) is useful in comparison of alternative

volatility estimators estimated from identical target frequencies, i.e. not requiring scaling from

one frequency to another. An example could be alternative GARCH specifications, e.g. the

implications of different distributions of errors on the produced GARCH(1,1) forecasts.

Mincer and Zarnowitz (1969) regressions, however, can be employed to compare forecasts

that come from different frequencies, in order to avoid the problem of data scaling.

The loss function approach compares forecasts based on cost functions that assign

weight on the distance between the volatility forecast and the estimate of the true (ex post)

volatility. Spiegel and Wang (2005) utilise that approach to compare the predictive

performance of EGARCH and OLS idiosyncratic variance estimators.87 Hansen and Lunde

(2005) analyse the performance of various cost functions and recommend two of the options:

the mean squared error (‘MSE’) and the quasi-likelihood cost function (‘QLIKE’):

𝑀𝑆𝐸𝑖 = 1

𝑇 ∑𝑇𝑡=1 (ℎ𝑖,𝑡 − �̂�𝑖,𝑡

2 ) 2

𝑄𝐿𝐼𝐾𝐸𝑖 = 1

𝑇 ∑𝑇𝑡=1 (ln(�̂�𝑖,𝑡

2 ) + ℎ𝑖,𝑡

�̂�𝑖,𝑡 2 )

2

,

where ℎ𝑖,𝑡 is the observed true (ex post) variance and �̂�𝑖,𝑡 2 is the ex ante conditional expected

variance. In the baseline case the true variance is inferred from squared returns, ℎ𝑖,𝑡 = 𝑟𝑖,𝑡 2 .

Another possible measure used in the field is the mean absolute error (‘MAE’)88:

𝑀𝐴𝐸𝑖 = 1

𝑇 ∑𝑇𝑡=1 |ℎ𝑖,𝑡 − �̂�𝑖,𝑡

2 |.

The estimated values for the cost functions depend on the noisy estimate for the true

variance (the squared idiosyncratic shocks, ℎ𝑖,𝑡 = 𝑟𝑖,𝑡 2 ). Differences in MSE or QLIKE do not

necessarily imply that one forecasting method outperforms another. Various parametric and

non-parametric tests can be employed to test forecast performance, e.g. Diebold and Mariano

(1995), Meese and Rogoff (1988), or Granger and Newbold (1986) tests. Inference based on

those tests relies on an asymptotic distribution of the test statistics, which could be

87 Spiegel and Wang (ibid.) report that EGARCH halves the mean absolute error of the

estimated variance compared to the variance of the residuals (OLS) from the mean equation,

although we were unable to confirm such an improvement in the full sample. 88 An advantage of the MAE metric is that it does not assume that the second moment

(variance) of the tested series exists (point on p. 12 in Meese and Rogoff (1983)), which

would be the case if variances follow random walk. We do not think this would be the case

here, as the unit root hypothesis is rejected by Fu (2009) for 90% of the series using the

Dickey-Fuller test, and our calculations yield very similar results.

96

unreasonable in cases where we have few observations. Therefore, we follow the suggestion

of Diebold and Mariano (1995) to employ the sign test or the Wilcoxon’s signed rank test

when a comparison of loss functions or other measures of prediction accuracy is required.

The sign test is based on the observation that if the difference between two estimators

is not systematic, then the median difference between the two series would zero. One could

then test if that is the case by performing binomial test with probability equal to 0.5. We can

calculate those differences, for example, using the differences between squared or absolute

values:

𝑑𝑖,𝑡 = (ℎ𝑖,𝑡 − �̂�1,𝑖,𝑡 2 )

2 − (ℎ𝑖,𝑡 − �̂�2,𝑖,𝑡

2 ) 2 ,

or

𝑑𝑖,𝑡 = |ℎ𝑖,𝑡 − �̂�1,𝑖,𝑡 2 | − |ℎ𝑖,𝑡 − �̂�2,𝑖,𝑡

2 |.

Therefore, for a given security, the median of the two series above should equal zero.89 The

same idea could be applied also if we compare loss functions (MSE, QLIKE, MAE) across

securities by calculating the differences 𝑑𝑖 = LF(ℎ𝑖, �̂�1,𝑖 2 ) − LF(ℎ𝑖, �̂�2,𝑖

2 ), where

LF(ℎ𝑖, �̂�k,𝑖 2 ), 𝑘 = 1,2 is the loss function for security 𝑖 calculated using its available history.

The test statistic then becomes simply

𝑆𝑠𝑟 =∑1𝑑𝑖>0

𝑁

𝑖=1

~𝐵𝑖𝑛𝑜𝑚(𝑁, 1 2⁄ ),

where 1𝑑𝑖>0 denotes the indicator function. In large samples the normal approximation could

be employed.90

The same idea of differences could be employed to calculate the Wilcoxon’s signed

rank test statistic, which equals the sum of the ranks of the absolute differences for the cases

where the difference is positive:91

𝑆𝑤 =∑1𝑑𝑖>0 𝑅𝑎𝑛𝑘(|𝑑𝑖|)

𝑁

𝑖=1

.

Besides the tabulated critical values for the test, normalised approximation could also be

89 This imposes the implicit assumption of pairwise complete forecasts, so that cases where

an estimate from one estimator was available while for the other series was unavailable would

not participate in the joint test. 90 i.e. for 𝑁 large,

𝑆𝑠𝑟−0.5𝑁

√0.25𝑁 ~𝑁(0,1).

91 Since ∑ 𝑖𝑁𝑖=1 = 𝑁(𝑁+1)

2 , we expect that the sum of the ranks of the positive differences

would be one half of the amount.

97

employed.92

Another approach to assess goodness of fit relies on estimating Mincer and Zarnowitz

(1969) regressions of the type

ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡 2 + 𝑒𝑖,𝑡, (11)

where ℎ𝑖,𝑡 is the estimate for the true (realised) idiosyncratic variance for security 𝑖 at

month 𝑡, and �̂�𝑖,𝑡 2 is the ex ante estimate for that variance. If one model predicts volatilities

correctly, we should expect to fail to reject the joint null hypothesis that 𝑎𝑖 = 0, 𝑏𝑖 = 1, so in

practice we test whether (ℎ𝑖,𝑡 − �̂�𝑖,𝑡 2 ) has mean 0. When applied to GARCH models, a

common finding is that the Mincer-Zarnowitz regressions have quite low explanatory power

as measured by their 𝑅2’s. Andersen and Bollerslev (1998) prove that the reason for the low

𝑅2’s is not a failure of the GARCH models, but rather the fact that squared returns are a noisy

estimator of volatility. In particular they show that the 𝑅2’s for GARCH(1,1) models are

bounded above by 1/𝜅 , where 𝜅 is the kurtosis of the underlying noise distribution;

therefore, for normally-distributed noise, 𝑅2 < 1/3, and for heavy-tailed distributions the

bound is even lower.

Running the Mincer-Zarnowitz regressions in variances is sometimes criticised as

placing undue weight on high volatilities episodes, or being affected by the skewness of the

volatility distribution. Thus Bali and Cakici (2008) use 𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

and a rolling average of

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

but instead of running the regression in variance, they run it into volatilities (standard

deviations), while Pagan and Schwert (1990) also report the regressions for the logs of

variances. Therefore we address such concerns by employing all three regression models, in

order to examine the predictive performance of volatility forecasts:

ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡 2 + 𝑒𝑖,𝑡,

√ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡 + 𝑒𝑖,𝑡,

ln(ℎ𝑖,𝑡 2 ) = 𝑎𝑖 + 𝑏𝑖ln(�̂�𝑖,𝑡

2 ) + 𝑒𝑖,𝑡,

where ℎ𝑡 is the realised variance and �̂� 2 is the forecast variance.

The usual OLS estimator is sensitive to extremes. In order to mitigate possible

concerns that differences in the average 𝑅2statistic are driven by few larger outliers, we also

92 i.e. for 𝑁 large, 𝑆𝑤−

𝑁(𝑁+1)

2

√ 𝑁(𝑁+1)(2𝑁+1)

24

~𝑁(0,1).

98

estimate the parameters of the three models specified above using quantile regression93.

Unlike the OLS regression, which minimised the sum of squared errors and estimated the

conditional mean of the explained variable, the quantile regression estimates some conditional

quantile 𝜏 of the explained variable. In this case we aim to predict the conditional median

(𝜏 = 0.5), which we accomplish by minimising the sum of the absolute differences, i.e. 𝑉𝜏 =

min𝛽 ∑𝑖 𝜌𝜏(𝑦𝑖 − 𝑥𝑖′𝛽), where 𝜌𝜏(𝑢) = 𝑢(𝜏 − 𝐼𝑢<0). In the case of forecasting of the median

(𝜏 = 0.5), the problem reduces to 𝑉0.5 = min𝛽 ∑𝑖 |𝑦𝑖 − 𝑥𝑖′𝛽|. An obvious constraint would

be that 𝑅2 is not directly applicable to this model. Instead, following Koenker and Machado

(1999) we compute its analogue for quantile regression – 𝑅𝜏 1, which is defined as 𝑅𝜏

1 = 1 −

𝑉𝜏/𝑉𝜏 𝑐, where 𝑉𝜏 is the optimal cost function for the evaluated model, and 𝑉𝜏

𝑐 is the optimal

cost function for a constrained model where all predictors (other than the intercept) are

constrained to equal zero. In this way 𝑅𝜏 1 measures how much the model improves the

prediction of the conditional median relative to the model of constant median. Thus, 𝑅𝜏 1 is a

direct analogue of 𝑅2 in the domain of quantile regression, and we use it to evaluate how the

competing predictors of the volatility perform.

Overall, we compare volatility forecasts in terms of Mincer and Zarnowitz (1969)

regressions run in variances, standard deviations, and logarithms. Forecast accuracy is

compared using Wilcoxon’s signed rank test for the pairwise 𝑅2 from each predictive

regression ran on security by security basis.

3.2.5.5. Assessing the correlation between expected idiosyncratic volatility and

returns

There are at least three types of tests of asset pricing models which are based on

somewhat different formulations of the asset prices. The stochastic discount factor (SDF)

approach proposes that prices equal present values of the end-of-period prices and dividends,

discounted by SDF (𝑚𝑡+1), so that 𝑝𝑡 = 𝔼𝑡(𝑚𝑡+1𝑥𝑡+1) (the pricing equation), and hence

𝔼𝑡(𝑚𝑡+1(1 + 𝑅𝑡+1) − 1) = 0, where 𝑅𝑡+1 = 𝑥𝑡+1

𝑝𝑡 − 1 =

𝑝𝑡+1+𝑑𝑡+1

𝑝𝑡 − 1 denotes the return on

the asset, 𝑝𝑡+1 is the future price, and 𝑑𝑡+1 is the future dividend. Thus, in this framework,

the valuation of prices requires estimation of the stochastic discount factor, 𝑚𝑡+1. There

could be different ways to impose structure on the SDF. One approach, e.g. used in the

Consumption CAPM (CCAPM) and the Lucas exchange economy, assumes utility function of

93 See Koenker and Gilbert Bassett (1978)

99

specific type to deduce the structure of the SDF. For example, assuming additive lifetime

utility function in consumption of the form 𝑈 = ∑ 𝛽𝑡𝔼0[𝑢(𝑐𝑡)] ∞ 𝑡=0 , so that the utility is

time-separable and lifetime utility equals the present value of the utility of the future

consumption stream, CCAPM concludes that 𝑚𝑡+1 = 𝛽𝑢′(𝑐𝑡+1)

𝑢′(𝑐𝑡) , where 𝛽 is the intertemporal

discount factor, and 𝑢′(𝑐𝑡) is the marginal utility of consumption. Another approach assumes

that 𝑚𝑡+1 can be approximated linearly by a set of factors, so that 𝑚𝑡+1 = 𝑎 + 𝑏 ′𝑓𝑡+1,

where 𝑎 is parameter vector, 𝑏 is parameter matrix, and 𝑓𝑡+1 is the vector of factors.

The beta representation of the pricing equation follows from the transformation

𝔼𝑡(𝑚𝑡+1(1 + 𝑅𝑡+1)) = 1 , so that 𝔼𝑡(𝑚𝑡+1)𝔼𝑡(1 + 𝑅𝑡+1) + 𝑐𝑜𝑣(𝑚𝑡+1, (1 + 𝑅𝑡+1)) = 1 .

Dividing by 𝔼𝑡(𝑚𝑡+1) > 0, the approach yields 𝔼𝑡(𝑅𝑡+1) = 1

𝔼𝑡(𝑚𝑡+1)

+ 𝑐𝑜𝑣(𝑚𝑡+1,𝑅𝑡+1)

𝑣𝑎𝑟(𝑚𝑡+1) (−

𝑣𝑎𝑟(𝑚𝑡+1)

𝔼𝑡(𝑚𝑡+1) ). The term 𝛽 =

𝑐𝑜𝑣(𝑚𝑡+1,𝑅𝑡+1)

𝑣𝑎𝑟(𝑚𝑡+1) is the slope of the regression of

returns on the SDF (and thus by the linear approximation of SDF in terms of factors, betas are

related to factors), while 𝜆 = − 𝑣𝑎𝑟(𝑚𝑡+1)

𝔼𝑡(𝑚𝑡+1) is the price of risk. Extensive treatment of these

derivations and the relationships between SDF, factor models, and beta representations is

available in Gospodinov and Robotti (2013), as well as Chapters 5 and 6 in Cochrane (2005).

For the purposes of the present discussion it is important to note that these relationships can

be tested in alternative forms, and these forms are related by the alternative specifications

above. In particular, time-series tests can be implemented by regressing excess returns on the

factor realizations, i.e. (𝑅𝑡 − 𝑅) = 𝛼 + 𝛽1𝑓1,𝑡 + 𝛽2𝑓2,𝑡 +⋯+ 𝛽𝑘𝑓𝑘,𝑡 . This approach is

essentially employed by us to split total returns into systematic and idiosyncratic returns using

three (Fama–French) or four (Fama–French–Carhart) factors. However, that approach is not

useful to perform the test whether idiosyncratic volatility predicts returns because volatilities

are not factors; furthermore, as we shall see in the chapter on results, some of the forecasts are

non-stationary, which also prevents running the time-series regressions in volatilities.

Another approach that is much more widely used in finance tests asset pricing models

using the Generalised Method of Moments (GMM). There is significant body of literature on

the use of those methods, but the principal idea is to find parameter values that would

minimize the distances between the moments of the left-hand side and the right-hand side of

the pricing equation 𝑝𝑡 = 𝔼𝑡(𝑚𝑡+1𝑥𝑡+1). The mean pricing error in a sample of size 𝑇 then

has the form 𝑔𝑇 = 1

𝑇 ∑ (𝑚𝑡+1𝑥𝑡+1 − 𝑝𝑡) 𝑇 𝑡=1 = 𝔼(𝑚𝑡+1𝑥𝑡+1 − 𝑝𝑡), and the GMM parameters

are estimated by minimizing the weighted errors, 𝑔𝑇 ′𝑊𝑔𝑇, where 𝑊 is the matrix of weights

100

(in case of equal weights, 𝑊 = 𝐼, where 𝐼 is the identity matrix). The method allows the

estimation of general non-linear, arbitrage-free asset pricing models. Nevertheless, it also has

the disadvantage that the choice of base instruments or parameterization could result in

unstable estimates. Compounded with the greater complexity of estimation, the method is not

often used in the related literature.94

The third method used in literature is based on cross-sectional regressions, and it is the

method used in this study. A principal advantage over the time-series estimation is that the

cross-sectional methodology can easily accommodate the addition of characteristics or factors

that are not returns.

The original methodology for the cross-sectional tests is proposed by Fama and

MacBeth (1973). They treat the coefficients from the cross-section regressions as random

variables and employ the 𝑡-test to examine the statistical significance of the different

characteristics. Thus, Fama and Macbeth estimate the cross-sectional regression separately for

each month, instead of averaging returns and betas in the testing period. The cross-sectional

regressions has the form:

𝑅𝑖,𝑡 = 𝛾0,𝑡 + 𝛾1,𝑡𝛽𝑖,𝑡 + 𝛾2,𝑡𝐼𝑉𝑂𝐿𝑖,𝑡 +⋯+ 𝜀𝑖,𝑡, (12)

where 𝐼𝑉𝑂𝐿 is some of the measures of idiosyncratic volatility employed in this study, and

𝐵𝑒𝑡𝑎 is the beta for security 𝑖.95 Many more characteristics could be added to the regression

as required for hypothesis testing, e.g. size, liquidity, lagged return, momentum, etc. The

significance of the hypothesised factor loadings (𝛾𝑖) could be tested using the 𝑡-statistic:

𝑡(�̅�𝑖) = �̅̂�𝑖

𝑠(�̂�𝑖)/√𝑛 ,

where 𝑛 is the number of cross-sectional regressions, 𝑠(𝛾𝑖) is the standard error of the

regression coefficient, and �̅�𝑖 is the mean value of the respective factor loadings, averaged

across all monthly cross-sectional regressions. Essentially, the test treats the coefficients of

the cross-sectional regression as independent random variables and performs a standard test

whether the mean realisation is statistically different from zero. Newey and West (1987)

propose a refinement of the 𝑡-test that allows to incorporate the autocorrelation of the

coefficients from the individual cross-sectional regressions. Following the prevailing practice

in the related literature, in this thesis we report the Newey–West standard errors with four

lags.

94 A notable exception is Khovansky and Zhylyevskyy (2013) 95 We discuss the estimation of beta later in this chapter.

101

3.2.5.6. Securities as assets

The literature on testing whether higher idiosyncratic returns are associated with

higher return usually employs two strategies: cross-sectional regressions, and portfolio

formation. In this study we employ the former strategy. In our view there are two advantages

of that strategy over the latter one: firstly, it allows tests based on individual securities as

assets (as opposed to portfolios as assets), which is suggested to result in superior efficiency

of the performed tests96. Secondly, portfolios are created in two dimensions, one of which was

the hypothesised dimension (idiosyncratic risk) and the other was the control dimension (beta,

size, liquidity, etc). In the tests of idiosyncratic risk, however, there are at least four

significantly correlated characteristics, each with its own set of theoretical justifications.

Those characteristics are: beta with market, size, liquidity, and idiosyncratic risk. Larger

companies empirically tend to have lower beta, higher liquidity, and lower idiosyncratic risk.

The tests based on portfolio formation aim to demonstrate that higher-idiosyncratic-risk

portfolios would earn higher expected return irrespective of the control variable used for the

other dimension. Nevertheless, one could not eliminate the risk that the observed positive

returns were due to some of the other two non-controlled characteristics. This is supported for

example by the study of Fan et al. (2015) who demonstrate a significant link between market

anomalies and idiosyncratic risk. Therefore, in this study we opt to test the link between

idiosyncratic risk and return principally in the cross-section of returns rather than through

portfolio formation.

The cross-sectional regressions, however, can also be implemented using portfolio

returns and volatilities as assets, rather than individual securities. In this study we have chosen

the second option (individual securities as assets). There are a couple of motivations for this

choice. Firstly, the use of portfolios as assets could result in loss of efficiency of the tests. The

portfolio strategy was originally conceived to address errors in variables in the estimation of

betas, which can be estimated with significant errors. However, most controls can be

estimated with high precision, including size, price/book values, return momentum, lagged

return, and idiosyncratic volatility. Therefore, Fama and French (1992) argue that the use of

portfolios is not justified, apart from the case of beta. The conjecture that use of portfolios

leads to loss of efficiency is confirmed by Ang et al. (2010), who point out that even betas

should not be averaged by portfolios. Secondly, as pointed out by Levy (2012), the Fama and

MacBeth (1973) methodology “employs portfolios rather than individual assets; therefore, it

96 Ang et al. (2010)

102

has the advantage of minimising the measurement errors in beta and the disadvantage of not

testing asset pricing of individual assets. Thus, in the case of supporting the CAPM, one

cannot generalise it to individual risky assets.” (p. 200). In our study we aim to evaluate

whether idiosyncratic volatilities are priced, and therefore averaging volatilities by portfolios

would likely result in loss of efficiency. Nevertheless, we also run robustness checks using

portfolios as assets to confirm our principal findings.

3.3. Control variables

The statistical tests for the cross-sectional correlation between idiosyncratic risk and

returns necessitate the identification of a set of variables that are hypothesized to explain the

stock returns both as time series and as cross-section. Thefore, we need to identify firstly the

variables that we shall employ to split the time series of actual returns into systematic and

idiosyncratic components. Then we shall identify control variables for the cross-sectional

regressions. At first sight the two lists could be hypothesised to be identical, i.e. we use a list

of factors to split returns into systematic and cross-sectional components, and then use the

loading on those factors in the cross-sectional regression, e.g. estimate asset betas with market

excess returns, and then use the betas as explanatory variables in the cross-section. In such a

setting, the time series tests and the cross-sectional tests would be testing the same hypothesis,

so arguably there would not even be any need for two steps. However, we noted previously

that the time-series regressions require the use of factors as explanatory variables, so some of

the variables that we aim to use in our test, including idiosyncratic volatility, cannot be used

in the time series regression. Therefore, we consider a list of factors in order to split returns

into systematic and non-systematic (idiosyncratic) components, and then we use the beta

together with characteristics that measure the individual asset’s exposure to the hypothesised

factors, as well as other characteristics considered relevant.

3.3.1. Splitting total return into systematic and idiosyncratic returns

The Capital Asset Pricing Model of Sharpe (1964) and Lintner (1965b) predicts that

there is a single factor that explains the mean excess returns of individual stocks, and that

factor is the excess return on the market portfolio, i.e.

𝜇𝑖,𝑡 = 𝑟𝑓 + 𝛽𝑖(𝜇𝑚,𝑡 − 𝑟𝑓).

103

The deviation of the actual return from the expected value is due to idiosyncratic factors,

𝜀𝑖,𝑡 = 𝑟𝑖,𝑡 − 𝜇𝑖,𝑡, but those idiosyncratic innovations are diversified away and are not priced by

the investors. The tests of CAPM are challenged by Roll (1977), who shows that the market

portfolio to which CAPM is referring is the portfolio of all assets in the economy, including

human capital, real estate, privately held businesses, overseas assets. That portfolio is not

observable, and its replacement by equity index introduces an error-in-variable problem,

which could account for observed anomalies detected by some of the empirical tests.

Fama and French (1992, 1993, 1996) isolate three factors that explain a significant

part of cross-sectional returns: market excess returns (from the CAPM), a size factor and a

value factor. The size (SMB) and value factors (HML) are estimated by splitting the stocks

into six sub-portfolios in two dimensions by ranking the portfolios in terms of size (three

groups) and in terms of book-to-market value (two groups). The realisation of the SMB factor

is estimated by considering a portfolio that is long in small shares and short in large shares.

Similarly, the realisation of the HML factor is estimated by using the returns to a no-cost

portfolio that is long in value stocks (those with high book-to-market ratio) and short in

growth shares (those with low book-to-market ratio). The approach of Fama and French is

justified in terms of construction of the predictive indices, but it is criticised for not providing

theoretical motivations as to why investors should value these factors. The SMB factor is

sometimes interpreted as a measure of probability of distress – smaller companies are less

diversified and with more limited access to funding, which increases their probability of

distress, which is priced by the market. The reasoning behind the value factor, however, is

less conclusive.

More recently, Fama and French (2007) investigate the contribution of the

sub-components of returns for value and for growth stocks. In the case of value stocks they

find that the positive returns stem from increases of price-to-book value ratio, while the

amount of equity is broadly stable. In the case of growth companies, returns came from the

strong increase of equity that is enough to outpace the corresponding decline of the

price-to-book value ratios. They suggest that eventually after the stocks are allocated to the

corresponding portfolio, the market causes the differences in performance of value and

growth portfolio to blur. In the case of growth stocks this is due to the gradual exhaustion of

the high-profit opportunities and strengthening competition, while in the case of value

companies it comes on the back of the urge to improve profitability, which for value

companies is below that of growth companies. Consequently some of the growth stocks

migrate to the value group and some of the value stocks migrate to the growth portfolio.

104

One stock market anomaly that the Fama–French specification does not account for is

stock momentum. Jegadeesh and Titman (1993) document a tendency for portfolio that buys

past winners and sells past losers to outperform the market over the sample period 1965 to

1989. To examine that effect, they calculate the return on each share over the past 𝐽 months.

Then they allocate shares in ten decile portfolios ranked by their past performance, and track

those portfolios 𝐾 months ahead. They call ‘winners’ the top decile portfolio (the one that

contains 10% of shares with the highest performance in the preceding 𝐽 months), and ‘losers’

– the bottom decile portfolio (the shares with poorest performance in the past 𝐽 months).

Examining various buy/sell strategies for various values of the past return calculation window

𝐽 and holding period 𝐾, they find that strategies that buy past winners and sell past losers

earn significant premium. For example, the strategy which selects stocks based on their

performance in the past 𝐽 = 6 months and holds them for 𝐾 = 6 on average earns a

compounded excess return of 12.01% p.a. The results are also found to hold when portfolios

are created from sub-samples formed on rankings on systemic risk (beta) and size

(capitalisation), so momentum effect is not due to systemic risk or size factor. Furthermore,

the buy-winners/sell-losers strategy earns abnormal return for horizons up to 36 months after

buying, although the last 24 months tend to reverse the gains from the first 12 months.

They find that the observed momentum could not be explained by lagged response to

common factors but is consistent with lagged response to firm-specific information, with

performance systematically strong in the first year following buying, except in the first month.

Scowcroft and Sefton (2005) investigate the sources of momentum and decompose it into

country momentum, sector (industry) momentum and the residual idiosyncratic momentum.

They find that for small-capitalisation companies momentum is primarily idiosyncratic, while

for large-capitalisation companies it is mostly industry specific.

Bondt and Thaler (1985) report that portfolios of past ‘losers’ 97 significantly

outperform the portfolio of past ‘winners’98 and find that the over-performance lasts for up to

36 months after portfolio formation. They attribute the observed differences to market

over-reaction that is reversed subsequently, albeit very slowly, as evidenced by the horizon

over which the discrepancies persist.

In general, there is no constraint on what factor model is selected for the calculation of

97 The top 35 securities/top 50 securities/the top decile with lowest cumulative abnormal

(residual) return over the 36 months preceding portfolio formation 98 The same number of securities with the highest cumulative abnormal return over the same

period as for the identification of past ‘losers’

105

idiosyncratic return, as long as we remain alert that different models result in different

residuals. Thus idiosyncratic residuals need not necessarily reflect pure idiosyncratic risk but

could reflect return on omitted factors99, e.g. return on human capital100. The early tests of

CAPM define idiosyncratic risk relative to the single-factor CAPM model. Among the recent

studies, Malkiel and Xu (2004) report results for idiosyncratic risk from the CAPM (they also

report results from the three-factor model). The one-factor specification is also used by Bali

and Cakici (2008) in their sorts by idiosyncratic volatility, although performance is assessed

based on alphas from the Fama–French specification discussed below.

More recent studies tend to base their calculation of idiosyncratic return on the CAPM

extended with the two additional Fama–French factors (this specification we shall abbreviate

as ‘FF-3’). The preference for FF-3 in recent finance literature is driven by the good empirical

performance of that specification. Most of the reviewed studies on the link between

idiosyncratic risk and the cross-section of returns define idiosyncratic risk relative to the FF-3

specification, e.g. Spiegel and Wang (2005), Fu (2009), Brockman et al. (2009), Guo et al.

(2014), Fu and Schutte (2010). More recently, Huang, Liu, Rhee and Wu (2012) add

momentum (‘MOM’) factor when calculating idiosyncratic returns. This four-factor

specification is usually referred as the Fama–French–Carhart (‘FFC’) specification by the

name of Carhart (1997), who originally added the momentum factor.

In this study we shall employ the four-factor FFC specification to separate monthly

returns into systematic and idiosyncratic components, i.e.:

(𝑟𝑖,𝑡 − 𝑟𝑓,𝑡) = 𝛽0 + 𝛽1(𝑟𝑀,𝑡 − 𝑟𝑓,𝑡) + 𝛽2𝑟𝑆𝑀𝐵,𝑡 + 𝛽3𝑟𝐻𝑀𝐿,𝑡 + 𝛽4𝑟𝑀𝑂𝑀,𝑡 + 𝜀𝑖,𝑡, (13)

where (𝑟𝑖,𝑡 − 𝑟𝑓,𝑡) is the excess return of asset 𝑖 over the risk-free rate 101, (𝑟𝑀,𝑡 − 𝑟𝑓,𝑡) is

the excess return on the market value-weighted portfolio, and 𝑟𝑆𝑀𝐵,𝑡 and 𝑟𝐻𝑀𝐿,𝑡 are the

returns on the Fama–French factors, and 𝑟𝑀𝑂𝑀,𝑡 is the return on the momentum factors. Note

that 𝜀𝑖,𝑡 are the errors of the regression, but are not the idiosyncratic innovations used in this

study. The equation above is used to forecast next period systematic returns from the factor

99 “Since volatilities, especially idiosyncratic volatilities, are unobservable, most empirical

studies estimate them using residuals from fitting a market model. Empirically, however, it is

very difficult to interpret the residuals from the CAPM or even a multi-factor model as solely

reflecting idiosyncratic risk. One can always argue that these residuals simply represent

omitted factors. Therefore, we can only assert that the residuals from a market model measure

idiosyncratic risk in the context of that model.” (p. 19 in Malkiel and Xu (2004)) 100 As proposed in Eiling (2013) 101 We use the yield on the three-month constant maturity government bonds as a measure of

the risk-free rate.

106

realisations, and then the difference to the total return would be the idiosyncratic risk.

The choice of FFC over FF-3 is motivated by two considerations. Firstly, the evidence

in favour of the momentum factor is now reasonably well established, and we have previously

referred to some of the papers documenting that finding. Secondly, as documented by Fan et

al. (2015), idiosyncratic volatility significantly correlates with return momentum, and

therefore it may be argued that its omission results in idiosyncratic residuals containing

momentum information. On the other hand, the impact of one factor specification over

another should not be overstated. For example, the reviewed study by Malkiel and Xu uses

both CAPM and FF-3 specifications but does not find significant differences in their results.

There Malkiel and Xu (2004) commented that “we use idiosyncratic volatility estimates both

from a market model and from the above Fama-French three-factor model. Since residual

volatility is a second moment, we view this approach as an indirect control for other factors.”

(p. 19). Inasmuch as the two-factor model specifications did not provide qualitatively

different results, the study can be viewed as offering evidence that the addition of extra

factors is not a driver of the obtained results. At any rate the pass-through from omitted factor

variable to idiosyncratic volatility would be limited, as pointed out by Malkiel and Xu (2004)

in connection with the impact of omitted liquidity factor: “If liquidity is indeed priced,

residuals from any asset pricing model that excludes liquidity factor will reflect it. However,

since idiosyncratic volatility is a second moment, it can only indirectly capture some of the

liquidity effect.”(p. 32)

We recognise that there may be arguments to include even more factors in the

time-series regression above. For example, Fama and French (2015) propose a five-factor

model that besides size and value, also adds profitability and investment patterns. However,

Kan and Zhang (1999b) point out that the inclusion of an irrelevant factor in the model makes

the covarionce matrix non-invertible, and the asymptotic properties of the two-step regression

tests are adversely affected.102 In view of the risks in adding extra factors to the regressions,

the fairly small history used for model estimation, as well as the limited accounting

information, we decided against adding more factors in the time-series regressions. We

nevertheless include a specific robustness test with statistical factor in order to verify our

findings.

102 Kan and Zhang (1999a) find a similar situation with the use of the generalized method of

moments.

107

3.3.2. Control variables in the cross-section regressions

We now turn to the problem of selection of control variables for the cross-sectional

regression. At first glance the most direct approach could seem to use the estimated loadings

on the four factors as regressors in the cross-sectional regressions, together with other control

variables, in particular – idiosyncratic volatilities. Such an approach would be consistent from

a theoretical perspective but is not actually pursued in the literature because of the uncertainty

of the estimation of betas. Therefore, betas of individual securities are usually replaced by

betas of portfolios;103 imposing such a contraint on the value of beta that is different from that

estimated from the time-series regression means that the other factor loadings are no longer

guaranteed to be unbiased, and therefore necessitates the use of other measures of size, value

and momentum, other than the slopes from the time-series regressions.

The CAPM justifies the use of asset’s beta with the market as an explanatory variable.

However, in view of the measurement error problem in its estimation, we shall follow the

process proposed in Fama and French (1992), and we shall form size-beta portfolios, and

replace individual betas with those of the portfolio, to which the asset is assigned in the given

period. Likewise, the exposures to size and value factors shall be measured more directly in

terms of (the logs of) capitalisation for the given security and the market/book value ratios.

Exposure to the momentum factor can be estimated using the cumulative return over a

six-month period. However, the value of the last return would have a specific significance,

because of a mild negative autocorrelation in the data (negative returns are more likely to be

followed by a positive return), and therefore the six-month period ends at the penultimate

month, rather than at the last month. Finally, we also include a liquidity variable as it tends to

be correlated with idiosyncratic volatilities (Spiegel and Wang, 2005) and could be

hypothesized to be the cause of the predictive significance of idiosyncratic volatilities.

Additional control variables were also considered, e.g. measures of tail fatness or

return skewness, but proved insignificant predictors of the cross-section. In view of the

aforementioned risk of confirming spuriously significant premium for insignificant factors,

explored by Kan and Zhang (1999b), throughout this study we shall emphasise not only

statistical significance, but also the stability of the estimates across difference specifications

and subsamples. If a certain variable is a significant predictor, we would expect that its point

103 This is considered as inefficient by Ang et al. (2010), but thus far the prevailing practice

has been to use portfolio betas, and we follow the other studies in that respect.

108

estimate remains fairly stable, and changes in its direction can be reconciled with economic

intuition.

3.4. Data sources, transformations, and summary statistics

3.4.1. Data sources

In this section we describe the data set that we use for our empirical analysis, as well

as the precautions and procedures taken to ensure the quality and integrity of the data set.

The primary data source for the stock prices and the economic time series used in this

study is Thompson Reuters Datastream and save for the Fama–French factors, all other data

items are sourced from there.

The data set covers all shares (Datastream instrument type = “equities”). Studies in the

field, e.g. Campbell et al. (2000), Ang et al. (2006), use stocks listed on NYSE, AMEX and

NASDAQ, and we have followed suit and included all of them in our sample. Specifically we

require that the included equities have a primary listing on the New York Stock Exchange

(henceforth “NYSE”), the NASDAQ Stock Market104 (henceforth “NASDAQ”), or the

NYSE MKT exchange (henceforth “NYSE MKT” or “AMEX”105). NYSE, NASDAQ and

NYSE MKT were respectively the first, the second and the third largest American stock

exchanges.

For all securities we require that the market should be the United States of America

and the currency of the issue should be the United States Dollar. The limitation of the scope to

the United States is intended to facilitate comparison with our reference studies, and also to

alleviate problems with country-specific factors of assets returns stemming from

country-specific policy and industry developments, as well as in recognition of the lower

penetration of stock exchanges in the economies of the European countries.

In terms of sectors we allow all sectors to our sample except the Nonequity Investment

Instruments, which are a separate sector in Datastream, and Financials, which in such tests are

usually excluded as a sector that pools together the risks of other sectors.

104 "NASDAQ" originally stood for National Association of Securities Dealers Automated

Quotations 105 The former name of NYSE MKT was the American Exchange (AMEX)

109

To recapitulate, the filters employed in Datastream to construct the list of companies

used in this study where as follows: “Market” = United States; “Currency” = US dollar;

“Exchange” = New York or NASDAQ or NYSE MKT; “Instrument Type” = equity; “Sector”

= all sectors except “Nonequity Investment Instruments” and “Financials”.

Available price data start from the beginning of 1973. Values for many of the features

are missing, and roughly three quarters of the data is lost due to lack of information on one of

the following items: unadjusted price (UP), number of shares (NOSH), price to book value

(PTBV). In particular, series for price-to-book value start from January 1980.

Market capitalisation for each security is calculated as the product of unadjusted price

(UP) and number of shares issued (NOSH).

A number of securities in Datastream have reported prices but apparently are not

traded actively. Their presence in the dataset could affect the estimated return distributions. In

order to mitigate the problem of presence of non-traded securities and bearing in mind that the

likelihood the price of a traded share to be exactly the same as at the beginning of the period

is very close to nil106, we excluded all monthly returns where the price at the end of the period

equals exactly the price at the start of the period unless there is reported positive traded

volume of that security (Datastream code ’VO’) in that month, i.e. 𝑉𝑂 > 0 (cases where VO

is missing were treated as zero volume).

106 For a random variable with continuous distribution function the probability of some

specific value occurring is exactly zero. Stock prices are discrete, but the step is very small so

they are close to continuous, hence the probability of exactly the same price occurring at the

end of month is rather small (even if the expected return for the month is zero), albeit not zero

as in the continuous variable case. However, as long as such incidences of removal of returns

are few and can be assumed to occur at random, they should not affect our conclusions, and so

this procedure seems reasonable in order to ensure quality of the data set.

110

Table 4: Average sector weights, Pearson correlations of sectors with the market, and cross-sector correlations

The table shows average weights (“Avg. Weight”), Pearson correlation with overall market (“Corr. with market”) and correlation matrix between the sectors of the stock issuers.

The data set includes securities from NYSE, AMEX and NASDAQ. The sub-indices are value-weighted.

Avg. Weight

Corr. with

market Basic

Materials Consumer

Goods Consumer

Services Financials Healthcare Industrials Oil and

Gas Other and

Unclassi-

fied

Techno-

logy Telecom-

munications Utilities

Basic

Materials 0.057 0.824 1 0.723 0.731 0.733 0.572 0.859 0.655 0.61 0.595 0.451 0.447

Consumer

Goods 0.079 0.851 1 0.831 0.819 0.782 0.816 0.496 0.631 0.548 0.535 0.594

Consumer

Services 0.083 0.892 1 0.822 0.69 0.872 0.424 0.69 0.703 0.592 0.453

Financials 0.057 0.875 1 0.7 0.839 0.522 0.615 0.58 0.56 0.568

Healthcare 0.1 0.788 1 0.715 0.442 0.479 0.588 0.487 0.472

Industrials 0.047 0.95 1 0.612 0.711 0.765 0.562 0.503

Oil and Gas 0.125 0.664 1 0.373 0.42 0.34 0.513

Other and

Unclassified 0.006 0.676 1 0.495 0.407 0.414

Technology 0.119 0.81 1 0.498 0.234

Telecommu

nications 0.252 0.648 1 0.468

Utilities 0.076 0.561 1

Source: author’s calculations

111

3.4.2. Calculated covariates

Excess returns (𝐸𝑅𝑖,𝑡 ) for security 𝑖 at time 𝑡 are calculated as the difference

between the arithmetic return for the month and the interest on 3-month Treasury bills –

middle rate (Datastream code FRTBS3M). The 3-month series is preferred to 1-month T-bills

on practical grounds – the latter is available only for dates starting July 31, 2001. The formula

for calculation of excess returns therefore is:

𝐸𝑅𝑖,𝑡 = 𝑃𝑖,𝑡+1−𝑃𝑖,𝑡

𝑃𝑖,𝑡 − 𝑟𝑓𝑡,

where 𝑃𝑖,𝑡 is the adjusted price at time 𝑡 for asset 𝑖, and 𝑟𝑓𝑡 is the risk-free rate from

Kenneth French’s data library.

Return indices are calculated using value-weights for all securities in our sample for

which there is information about market capitalisation. Table 4 lists the Pearson pairwise

correlations across constituent sectors, and the average weight of each sectors. Because of the

inclusion of NASDAQ in our sample, the weight of telecommunications and technology is

high, with a quarter of all companies in the telecommunications business and more than 11%

in technology stocks.

As should be expected, the calculated value-weighted market excess return co-varied

closely with the excess returns on the commonly used market indices. The correlation with

S&P 500 is 0.9900, with Dow Jones Industrials – 0.9318, with NASDAQ Composite –

0.9035, with NYSE Composite – 0.9793, and with MSCI USA – 0.9876. These values are

consistent with expectations: the highest correlation was obtained for S&P 500, which is a

value-weighted index of 500 leading stocks, and sometimes used as a proxy for the market

index as a whole; similarly, the correlation with MSCI USA is also very high, consistent with

the fact that it is a free float adjusted market capitalisation index, geared towards large and

mid-cap US equities. The correlation with the Dow Jones Industrial Average, on the other

hand, is somewhat lower, consistent with the significant differences in coverage and

calculation (Dow Jones is a price-weighted index of 30 leading stocks). The correlation with

NASDAQ Composite is lower as that index includes both US and non-US stocks (and

equity-like instruments like ADRs, REITs, limited partnership interests, etc.) listed at the

NASDAQ market.107 The correlation between the value-weighted excess returns calculated

by us and those in Kenneth R. French Data Library2015) is 0.9981.

107 We discarded NASDAQ non-US equities from our sample.

112

Most studies in the field employ the CRSP prices augmented with Compustat financial

data. The use of a single data set is known to introduce risk of data mining (data snooping). In

that respect, the use of an alternative data source adds value to this study. On the other hand,

the data coverage of the Datastream database raised some concerns in terms of poorer

coverage in the earlier period as well as data errors. Ince and Porter (2006) investigate such

differences between Datastream and CRSP/Compustat, identifying the strengths and

weaknesses of Datastream data. They also propose filters in order to improve the quality of

the estimates. In line with their suggestions, we implement those and some further measures

in our sample. With appropriate filtering, Ince and Porter (2006) demonstrate that returns and

moments based on Datastream close prices are very similar in magnitude to, and correlate

highly with, those based on CRSP data. Nonetheless, we use market excess returns from

Kenneth French’s online data library in order to limit differences caused by different coverage

in the initial sample period. In our study we include only equities; all non-equity instruments

like American depository receipts (ADRs), non-equity investment instruments, real estate

investment trusts (REITs), shares of beneficial interest, preferred shares, and other non-equity

instrument types are excluded. To limit the impact of data errors we have discarded from our

sample returns exceeding 300%.

Market capitalisation for each security is calculated as the product of unadjusted price

(UP Datastream series) and number of shares issued (NOSH series). ln(𝐵/𝑀) is the natural

logarithm of the ratio of book value of equity (M/B) at the end of the preceding month to the

market price of equity108 , and is calculated from the price-to-book value series from

Datastream (PTBV) where book values of equity are taken at a lag of six months to ensure

that they are known to investors.

𝑅𝑒𝑡𝑡 is the raw month-on-month return for each security at month 𝑡, and 𝑋𝑅𝑒𝑡𝑡 is

the excess return for each security, calculated as the difference between 𝑅𝑒𝑡 and the risk-free

rate for the respective month as retrieved from Kenneth French’s data library. 𝐼𝑅𝑒𝑡𝑡 is the

idiosyncratic return calculated as the residual corresponding to month 𝑡 from the four-factor

Fama–French–Carhart model estimated with monthly data from (𝑡 − 60) to (𝑡 − 1).

108 Practitioners usually use the reciprocal ratio of market capitalization to book value

(price-to-book value). Since book value could be very close to zero, we follow to custom in

academic literature to place book value in the numerator. We have also excluded cases with

negative book value of equity, because for them the logarithm is not defined.

113

Table 5: Descriptive statistics, 1/1980–3/2013

The table summarises the descriptive statistics for the sample used in the tests. ‘Ret (%)’ is the arithmetic return on the stock; ‘XRet (%)’ is the excess return on the stocks over

the risk-free rate for the respective month as retrieved from Kenneth French’s data library; ‘IRet (%)’ is the idiosyncratic return calculated as the residual corresponding to

month t from the four-factor Fama–French–Carhart model estimated with monthly data from (t-60) to (t-1).

‘Beta’ is the stock beta calculated by portfolios constructed as in Fama and French (1992). ‘ln(Cap)’ is the natural logarithm of market capitalisation calculated as number of

shares times the unadjusted price. ‘ln(B/M)’ is the natural logarithm of book value of equity as available at the end of the preceding month to market price of equity (B/M), and

is calculated from the price-to-book value series from Datastream (PTBV) where book values of equity are taken at a lag of six months to ensure that they are known to

investors. ‘Ret(-2, -7)’ is the cummulative return for the six months from (t-7) to (t-2); (t-1) is not included in order to control for return reversals. ‘Roll’ is the bid-ask spread

calculated Roll’s model; Roll (1984).

‘Mean (EW)’ and ‘Mean (VW)’ are the equally-weighted and the value-weighted values of the respective indicators. ‘St.dev.’ is the standard deviation; ‘Median’, ‘Q1’ and ‘Q3’

are the median and the first and third quartiles of the sample. ‘Skewness’ is the skewness coefficient for the sample. ‘Obs’ is the number of rows for which data is available.

Variables Mean (EW) Mean (VW) St.dev. Median Q1 Q3 Skewness Obs

Ret (%) 1.39 0.58 14.54 0.51 -5.79 7.34 1.75 863,999

XRet (%) 1.05 0.32 14.54 0.19 -6.15 7.01 1.75 863,999

IRet (%) -0.18 -0.64 12.32 -0.64 -6.29 5.07 1.51 863,993

Beta 1.20 1.00 0.31 1.18 0.98 1.44 0.08 863,999

ln(𝐶𝑎𝑝) 5.87 9.81 1.91 5.76 4.45 7.15 0.31 863,999

ln(𝐵/𝑀) -0.71 -1.25 0.77 -0.63 -1.12 -0.22 -0.78 863,999

Ret(-2, -7) 1.09 1.09 0.41 1.04 0.88 1.23 3.23 862,210

Roll 6.88 4.28 4.24 5.81 4.08 8.4 2.74 863,970

Source: author’s calculations

114

The CAPM proposes that the only factor explaining the cross-section of market

returns is the market beta. The estimation of beta, however, is fraught with problems, and a

principal one among those is the measurement errors in betas. In this study we measure

systemic risk (𝐵𝑒𝑡𝑎) following the procedure proposed by Fama and French (1992) with

minor modifications. In each month we calculate the betas using the previous 60 months

(but no less than 30 months) of data, not including the current month. Stocks are then

assigned to five quintile-size portfolios. Within each quintile-size portfolio, ten decile beta

portfolios are formed. For calculation of the quintile breakpoints for size sorting and the

decile breakpoints for beta sorting we exclude the NASDAQ and NYSE MKT shares (i.e.,

NYSE breakpoints). The beta for each of the 50 size-beta portfolios is calculated running

full-period regression of the equally-weighted average monthly excess returns on the current

and previous period market excess returns. Then 𝐵𝑒𝑡𝑎 was calculated as the sum of the

slopes of the two market returns, which is intended to correct for non-synchronous trading,

and that beta is assigned to all securities in those portfolios.

Jegadeesh and Titman (1993) report that past returns are predictors of current

performance (momentum effect), and following Fu (2009) we construct a return proxy equal

to the cumulative gross return from months 𝑡 − 7 till 𝑡 − 2 inclusive, i.e. 𝑅𝑒𝑡(−2,−7) =

∏7𝑖=2 (1 + 𝑅𝑡−𝑖), where 𝑅 is the gross return; thus a cumulative decline of 20 per cent over

the six-month period is recorded as 𝑅𝑒𝑡(−2,−7) = 0.8. Return in 𝑡 − 1 is not included, in

order to ensure that the variable significance is not due to return reversals. Instead, return in

month 𝑡 − 1 is used as a separate independent variable.

Asset liquidity is discussed in the context of the cross-section of stock returns. Spiegel

and Wang (2005) demonstrate that liquidity is inversely correlated with idiosyncratic risk.

There is no single measure of volatility, and a number of measures are used in the literature.

For example, Amihud and Mendelson (1986) propose that liquidity measured in terms of the

bid-ask spread is desired by investors and therefore the less liquid shares should earn a

premium. Fu (2009) measures idiosyncratic risk in terms of rolling average trading volume, as

well as the coefficient of variation of traded volume. We estimate liquidity using Roll (1984)

model-based estimator (𝑅o𝑙𝑙) of the spread, because the spread between the buy and sell sides

of the market might be a less ambiguous measure of liquidity compared to traded volume,

where arguably the free float instead of the total number of shares should be in the

denominator. Roll (1984) proposes that the spread (𝑆 ) could be estimated from the

115

auto-covariance of prices as they bounced back and forth between the sell side and the buy

side of the market:

𝑆 = {2√−𝐶𝑜𝑣(Δ𝑃𝑡, Δ𝑃𝑡−1) 𝑖𝑓 𝐶𝑜𝑣(Δ𝑃𝑡, Δ𝑃𝑡−1) < 0

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ,

where 𝐶𝑜𝑣(. ) is the first order return covariance. The estimator is biased when the sample

size is small and the frequency is low (less frequent than daily); therefore we estimate the

spread using the daily returns from the past year.

The descriptive statistics reported in Table 4 overall are similar to those reported in

other studies like Fama and French (1992) and Fu (2009), which confirms that our sample is

representative for the market.

Earlier studies such as that of Fama and MacBeth (1973) identify idiosyncratic risk as

the standard deviation of the residuals from the fitted market models:

𝜎𝑜𝑙𝑠,𝑡 2 =∑

𝑇

𝑘=1

(𝜀𝑖,𝑡−𝑘) 2

𝑇 − 𝑑𝑓 ,

where 𝜀𝑖,𝑡−𝑘 denotes the idiosyncratic residual from the FFC model for stock 𝑖 in month

(𝑡 − 𝑘) obtained from a regression estimated over the period (𝑡 − 60) until (𝑡 − 1). 𝑇 is

the actual number of months for which information is available; thus 𝑇 is between 60 and

30 (the minimum number of months required for estimation); 𝑑𝑓 was the residual degrees

of freedom for the estimated model and equals the number of estimated parameters, so that

𝑑𝑓 = 5 (four factors and an intercept). It is immediately clear from the specification that the

return for period 𝑡 plays no role in the estimation of OLS idiosyncratic variance,

𝜎𝑜𝑙𝑠,𝑡 2 .Clearly, the resulting estimates are strongly autocorrelated as the estimates are obtained

using nearly-identical data sets – in two consecutive months there would be one month

leaving the sample and one month entering the sample (rolling window design), or one period

added to the sample (expanding window design). In our study idiosyncratic volatility

(standard deviation) is obtained from the residuals of the rolling regressions with data ending

at period (𝑡 − 1) and including 60 to 30 months of return (as available). These volatility

estimates are denoted by 𝐼𝑉𝑡−1 𝑂𝐿𝑆; to avoid unnecessarily cluttering the notation we sometimes

omit the time subscript, which is set to (𝑡 − 1) to remind us that the return at period 𝑡 was

not incorporated in that estimate.

The second measure of expected idiosyncratic volatility is estimated using

GARCH(1,1) model with SGED innovations. The forecasts from that model are denoted by

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

. The GARCH model is fitted using an expanding window containing all available

116

data up to the beginning of the month (i.e. end of 𝑡 − 1); similarly to Fu (2009), at least 60

months of data109 were required in order to estimate the model and yield a forecast for

month-𝑡 idiosyncratic volatility. No information from month-𝑡 is used in the estimation of

any parameters of the model. Variance forecasts are obtained from the model fitted until (𝑡 −

1). Essentially this means that if idiosyncratic volatility is above its mean-reverting level, the

model forecasts for month 𝑡 would be lower compared to the fitted variance for month (𝑡 −

1); the opposite would hold if month-(𝑡 − 1) variance is below the mean-reverting level. The

existence of the mean-reverting level is ensured by the parameter constraints on the GARCH

model (𝛼 + 𝛽 < 1, 𝛼 > 0, 𝛽 > 0). Thus our model is specified as follows:

(𝑟𝑖,𝑡 − 𝑟𝑓,𝑡) = 𝛽0 + 𝛽1(𝑟𝑀,𝑡 − 𝑟𝑓,𝑡) + 𝛽2𝑟𝑆𝑀𝐵,𝑡 + β3𝑟𝐻𝑀𝐿,𝑡 + 𝛽4𝑟𝑀𝑂𝑀,𝑡 + 𝜀𝑖,𝑡,

𝜀𝑖,𝑡 = √𝜎𝑖,𝑡 2 𝜈𝑖,𝑡,

𝜈𝑖,𝑡~𝑆𝐺𝐸𝐷, 𝔼𝜈𝑖,𝑡 = 0, 𝔼𝜈𝑖,𝑡 2 = 1

𝜎𝑖,𝑡 2 = 𝜔 + 𝛼𝜀𝑡−1

2 + 𝛽𝜎𝑖,𝑡−1 2 ,

𝛼 > 0, 𝛽 > 0, 𝛼 + 𝛽 < 1.

The estimation of the parameters of the above model proceeds sequentially. First, the

mean equation is estimated using OLS over the period from month 1 to month (𝑡 − 1), if at

least 60 months of continuous data were available (expanding window design). The

estimated series of residuals ( 𝜀𝑖,𝑡 , 𝑡 = 1… (𝑡 − 1) ) is then used to estimate the

maximum-likelihood parameters of the volatility model. The fitted model produces the

forecasts for month 𝑡 using the last idiosyncratic innovation for month (𝑡 − 1) and the

fitted variance for month (𝑡 − 1).

109 In cases where a stock was not traded in a particular month, identified by either trading

volume equal to zero or by zero change of the price in the month. Such deletions resulted in

gaps of the series, and thus some infrequently traded securities were excluded from the

criterion of 60 months of contiguous trading. This alleviates concerns that the finding of

significance of idiosyncratic risk was driven by few securities with thin trading and are not

representative for the market.

117

Table 6: Descriptive statistics for idiosyncratic volatility forecasts, 1/1980–3/2013

The table reports the descriptive statistics for the calculated volatility forecasts. ‘𝐼𝑉𝑂𝐿𝑆’ is the volatility of the residual of rolling monthly regressions using thirty to sixty months

of data, as available. ‘𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

’ is the idiosyncratic volatility from the daily data in the preceding month calculated using the residuals from Fama-French regressions with the

daily data for the preceding month; for convenience the estimates are scaled to monthly frequency using √𝑇. ‘𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

’ is the forecasts from GARCH(1,1) model with SGED

innovations estimated using expanding window comprising of at least sixty months of continuous data. ‘𝐼�̂�𝑎𝑟𝑚𝑎’ is expected volatility from ARMA(1,1) model fitted on the

available series of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

. ‘𝑚’ is the mean-reverting level of volatility implied by the fitted ARMA model. ‘Mean (EW)’ and ‘Mean (VW)’ are the equally-weighted and the value-weighted values of the respective indicators. ‘St.dev.’ is the standard deviation; ‘Median’, ‘Q1’ and ‘Q3’

are the median and the first and third quartiles of the sample. ‘Skewness’ is the skewness coefficient for the sample. ‘Obs’ is the number of rows for which estimates are

available.

Variables Mean (EW) Mean (VW) St.dev. Median Q1 Q3 Skewness Obs

𝐼𝑉𝑂𝐿𝑆 (%) 12.22 7.64 6.72 10.54 7.51 15.19 1.63 863,993

ln(𝐼𝑉𝑂𝐿𝑆) 2.37 1.94 0.50 2.36 2.02 2.72 0.18 863,993

𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

(%) 10.98 6.41 7.81 8.82 5.84 13.63 2.39 729,628

ln(𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

) 2.20 1.70 0.62 2.18 1.76 2.61 0.15 729,628

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

(%) 11.39 7.56 5.67 10.08 7.34 14.08 1.57 746,953

ln(𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

) 2.32 1.94 0.46 2.31 1.99 2.64 0.16 746,953

𝐼�̂�𝑎𝑟𝑚𝑎 (%) 11.61 6.58 6.45 9.97 7.00 14.52 1.64 812,920

ln(𝐼�̂�𝑎𝑟𝑚𝑎) 2.32 1.79 0.51 2.30 1.95 2.68 0.17 812,920

𝑚 (%) 13.41 7.68 6.93 11.84 8.25 16.90 1.43 812,793

ln(𝑚) 2.48 1.95 0.49 2.47 2.11 2.83 0.13 812,793

Spread -0.20 -0.21 0.49 -0.24 -0.56 0.02 1.12 812,732

Source: author’s calculations

118

𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

is the previous month idiosyncratic volatility, calculated as the standard

deviation of daily idiosyncratic residuals from the three-factor Fama–French model110 fitted

on daily data in the previous calendar month (𝑡 − 1). We require at least 15 non-zero

returns in the calculation month, save for September 2001, where only 12 trading days are

required. Zero-return days are discarded in order to reduce the impact of infrequent trading on

the volatility measure. 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

is scaled to monthly frequency by multiplying by square root

of the number of trading days in the respective month in order to ease comparison of

regression slopes across volatility measures.

The last measure of expected idiosyncratic volatility are the forecasts from the history

of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

produced using ARMA(1,1) model:

ℎ𝑡 −𝑚 = 𝜙(ℎ𝑡−1 −𝑚) + 𝜃𝜀𝑡−1 + 𝜀𝑡,

where 𝑚 , 𝜙 and 𝜃 are the parameters of mode. Parameter 𝑚 is also called the

mean-revering level of volatility because when 𝜙 ∈ (0,1), the forecasted values of ℎ would

converge to 𝑚, given enough time. In general, the values of 𝜙, 𝜃 ∈ (−1,1) ensure that the

ARMA process is invertible.111 The unconditional expected value of the process equals 𝑚,

𝔼(ℎ𝑡) = 𝑚, and 𝜙 controls the speed of mean-reversion. An alternative parametrisation for

the process uses 𝜇 = 𝑚(1 − 𝜙) instead of 𝑚 , giving the alternative form ℎ𝑡 = 𝜇 +

𝜙ℎ𝑡−1 + 𝜃𝜀𝑡−1 + 𝜀𝑡.

As particular cases the specification admits constant expected volatility (𝜙 = 0, 𝜃 =

0) and random-walk volatility (𝜙 = 1, 𝜃 = 0). Stationarity requires that the parameters 𝜙, 𝜃

of ARMA(1,1) be in the interval (−1,+1) . However, a negative value of 𝜙 (the

mean-reversion parameter) implies that expected volatility oscillates around the

mean-reverting level. Therefore we enforce the stricter limit for the mean-reversion

parameter: 𝜙 ∈ [0,1) . A downside of the ARMA(1,1) approach is that the forecasted

volatility may become negative. We discard from our sample those forecasts where volatility

is non-positive (109 records) or exceeds 200% on a monthly basis (8 records), which is

0.01% of the sample total of 812,920 forecasts.

110 The preference to the three-factor model was based on two considerations. Firstly, to be

consistent with the approach of Ang et al. Secondly, in each month we have at most 23 daily returns, which warrants more parsimonious specification. 111 The formulation of the ARMA process means that all observed values of ℎ𝑡 are function of the history of unobserved errors. If the process is invertible, then these errors can be

represented as weighted sums of the observed realisations.

119

Overall, the measures of idiosyncratic risk that we considered in this study are fairly

representative of the prevailing practice. Thus 𝐼𝑉𝑂𝐿𝑆 represents the filter-based measures;

other examples of this class could be the Hodrick-Prescott filter used by Cao (2010) and Cao

and Xu (2010), or 𝐼𝑉𝑚𝑜𝑛𝑡ℎ𝑙𝑦 – the moving average of 𝐼𝑉𝑑𝑎𝑖𝑙𝑦, employed by Bali and Cakici

(2008). GARCH(1,1) could be construed as representative of the GARCH-based models.

𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

, the measure used by Ang et al. (2009, 2006) is a measure based on assumption of

random-walk volatility, and in that respect also related to the Integrated GARCH model.

Finally, 𝐼�̂�𝑎𝑟𝑚𝑎 is representative of the mean-reversion volatility hypothesis (stationary

GARCH also falls in this category). We consider these four measures as forecasts for period-𝑡

idiosyncratic variance. Table 5 provides summary statistics for the idiosyncratic volatility

measures.

The covariates employed in the studies of the cross-section of returns tend to be

correlated. Such correlation is not only empirical regularity, but is also predicted from various

theories. In Table 6 we provide summary statistics for the key idiosyncratic risk covariates

broken down by the ten decile beta portfolios and the five quintile capitalisation portfolios.

The table confirms that stocks with higher capitalisation have lower betas and lower

idiosyncratic risk, and indeed the mean idiosyncratic risk of the high-capitalisation stocks was

roughly half that of the low-capitalisation stocks.

120

Table 7: Mean volatilities by portfolios sorted by beta and capitalisation, 1980/7–2013/3

The table reports the average volatilities of ten decile portfolios sorted by market beta (Panel A) and five quintile portfolios formed by market capitalisation (Panel B). ‘𝐼𝑉𝑂𝐿𝑆’

is the volatility of the residual of rolling monthly regressions using 30 to 60 months of data, as available. ‘𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

’ is the idiosyncratic volatility from the daily data in the

preceding month calculated using the residuals from Fama-French regressions with the daily data for the preceding month. ‘𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

’ is the forecasts from GARCH(1,1) model

with SGED innovations estimated using expanding window comprising of at least sixty months of continuous data. ‘𝐼�̂�𝑎𝑟𝑚𝑎’ is expected volatility from ARMA(1,1) model

fitted on the available series of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

.

The table demonstrates the significant correlation between market capitalisation, beta and the various volatility forecasts. Thus, the volatility of the stocks in the highest-beta

decile portfolio is between 5.45 and 7.77 percentage points higher compared to the lowest-beta portfolio. Similarly, the volatility of the lowest-capitalisation quintile portfolio is

between 7.08 and 9.08 percentage points higher than the volatility of the highest-capitalisation stocks.

Portfolio 𝑋𝑅𝑒𝑡 𝐵𝑒𝑡𝑎 ln(𝐶𝑎𝑝) 𝐼𝑉𝑂𝐿𝑆 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

𝐼𝑉𝑡−1 𝐴𝑛𝑔

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎

Panel A: Portfolios sorted by Beta

Low-beta 0.69 0.77 12.72 10.42 9.60 9.52 9.98

2 0.86 0.82 12.78 9.46 9.14 9.21 9.65

3 0.94 0.97 12.88 9.51 9.18 9.45 9.75

4 0.95 1.03 12.89 9.94 9.48 9.80 10.12

5 0.92 1.10 12.81 10.59 10.10 10.38 10.74

6 1.14 1.17 12.83 11.11 10.15 10.77 10.88

7 1.18 1.26 12.83 11.72 10.71 11.32 11.36

8 1.02 1.33 12.82 12.43 11.30 11.85 11.92

9 1.20 1.44 12.77 13.83 12.06 13.01 12.80

High-beta 1.36 1.70 12.61 18.19 14.42 16.00 15.43

121

Portfolio 𝑋𝑅𝑒𝑡 𝐵𝑒𝑡𝑎 ln(𝐶𝑎𝑝) 𝐼𝑉𝑂𝐿𝑆 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

𝐼𝑉𝑡−1 𝐴𝑛𝑔

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎

Panel B: Portfolios sorted by capitalisation

Low-cap 1.46 1.29 11.19 15.60 14.81 14.46 15.74

2 0.92 1.22 12.68 11.82 10.40 11.14 10.78

3 0.79 1.18 13.46 10.19 9.00 9.78 9.21

4 0.71 1.11 14.36 8.85 7.91 8.71 8.03

High-cap 0.52 0.99 15.90 7.27 6.62 7.38 6.66

Source: author’s calculations

122

3.5. Classification of volatility regimes

The last methodological aspect that we would like to explore in this chapter is the

identification of high-volatility episodes. The purpose of such classification of time periods is

to examine the robustness of our results. For example, some of the relationships could hold

only in low-volatility state, or the significance of some results could be due to outliers

occurring in a high-volatility environment.

The approach we take to identify such episodes is based on Fink et al. (2010), who

identify high-volatility episodes in their sample by using a Markov chain with two states,

where volatility in each state is assumed to be normally distributed with unknown mean and

variance.

Formally, let Π be a right stochastic112 matrix of transition probabilities whose

element 𝜋𝑖,𝑗 is the probability of transition from state 𝑖 to state 𝑗 conditional on the system

being in state 𝑖. The list of states is complete, i.e. there is no other state in which the system

might be, hence the rows of the transition matrix should sum to 1. They assume that when the

market volatility is in state 𝑖, market volatility is independently and identically normally

distributed with some unknown parameters (that are to be estimated), i.e. 𝜎𝑚𝑎𝑟𝑘𝑒𝑡𝑁(𝜇𝑖, 𝜎𝑖).

The parameters of that system (the matrix Π and the pairs (𝜇𝑖, 𝜎𝑖), ∀𝑖) could be estimated

using the Baum–Welch algorithm (Baum et al.,1970), and the sequence of states of the market

(the Viterbi path) at any point in time could be estimated using the Viterbi (1967) algorithm.

Firstly, we estimate the above model with 2 states. 113 We calculate monthly

volatilities of the DataStream Return Index (all sectors) from daily data as 𝜎𝑚,𝑡 2 =

21

𝑁 ∑𝑖∈𝑡 𝑅𝑖

2,

where 𝑖 ran over the days in month 𝑡, and 𝑁 was the number of days in month 𝑡. The

calculated volatility of the index is displayed in Figure 2; the mean volatility is 0.04267,

while the first, second and the third quartiles respectively equal 0.02868, 0.03637, and

0.04809 . The maximum volatility of 0.2419 is observed in October 1987, and the

second-highest volatility of 0.2259 occurred in October 2008.

112 A square matrix is right stochastic matrix if all its elements are non-negative real numbers,

with each row summing to 1 113 We implemented these calculations using the ‘RHmm’ package of R (Taramasco and

Bauer (2013))

123

Figure 3: Realised volatility of the market and volatility of the Hidden Markov Model

with Three States

Source: author’s calculations

The estimates from the Hidden Markov Model with 2 states and normal distribution

of volatilities are given in Table 8. The results suggest that the mean volatility in the

high-volatility state (state 1) is more than twice the market volatility in the low-volatility

state (state 2 ). Furthermore, the dispersion in the high-volatility state is in order of

magnitudes higher than in the low volatility state. This result is consistent with the observed

high peaks in the volatility chart, which imply that either higher number of states might be

appropriate, or conditional distributions with heavier tails might be justified.

We further note that out of 484 months used in the estimation (from January 1973

until April 2013), the market is in the high-volatility state for 125 months, which constitutes

more than a quarter of that period. Therefore, to further narrow down the list of highly volatile

episodes, we estimate the same model with three states. The results are presented in Table 8;

the high-volatility state prevails in 41 of the months, and the medium-volatility state – in

another 181 months. The mean volatility in the high-volatility state is more than thrice the

mean volatility in the low-volatility state and nearly twice as high as the volatility in the

medium state. Interestingly, the transition matrix suggests that one should not expect

transitions directly from state 1 to state 3, or from state 3 to state 1; such transitions can

occur over the course of at least two months.

month

v o

la ti lit

y

1 9 8

0 −

0 1

1 9 8

2 −

0 1

1 9 8

4 −

0 1

1 9 8

6 −

0 1

1 9 8

8 −

0 1

1 9 9

0 −

0 1

1 9 9

2 −

0 1

1 9 9

4 −

0 1

1 9 9

6 −

0 1

1 9 9

8 −

0 1

2 0 0

0 −

0 1

2 0 0

2 −

0 1

2 0 0

4 −

0 1

2 0 0

6 −

0 1

2 0 0

8 −

0 1

2 0 1

0 −

0 1

2 0 1

2 −

0 1

0.05

0.10

0.15

0.20

0.25

124

Table 8: Parameter estimates for Hidden Markov Model with two states and Normal

distribution of volatilities in each state

The table reports the parameters of a Hidden Markov Model fitted on the series of total market returns. The

models assumed the existence of two volatility regimes – high-volatility (regime 1) and low-volatility (regime

2). In each regime volatility is assumed to be independently and identically normally distributed with mean and

variance of 𝜇i and 𝜎i 2. The transition matrix is Π = [𝜋i,j], where 𝜋i,j is the probability of transition from state

i to state j.

‘Estimate’ is the point estimate of the respective parameter, while ‘Std. Error’, ‘t value’ and ‘𝑃𝑟(> |𝑡|)’ are the standard error, the t-statistic and the p-value of the estimate. Parameters are estimated using the Baum–Welch

algorithm.

The result shows the existence of at least two clearly separated volatility regimes, as evidenced by the significant

difference between 𝜇1 and 𝜇2 compared to the standard errors of the two estimates, and with significant volatility persistence as measured by 𝜋i,i, which range between 89.71% and 92.26%.

Estimate Std. Error t value 𝑃𝑟(> |𝑡|)

𝜋1,1 0.8971 0.0345 26.012 0.0000

𝜋1,2 0.1029 0.0345 2.985 0.0028

𝜋2,1 0.0374 0.0119 3.146 0.0017

𝜋2,2 0.9626 0.0119 81.059 0.0000

𝜇1 0.0686 0.0033 20.514 0.0000

𝜎1 2 0.0011 0.0000 238.109 0.0000

𝜇2 0.0333 0.0006 59.535 0.0000

𝜎2 2 0.0001 0.0000 92.141 0.0000

Log Likelihood: 1363.13

BIC Criterion: -2682.98

AIC Criterion: -2712.26

Source: author’s calculations

125

Table 9: Parameter estimates for Hidden Markov Model with three states and Normal

distribution of volatilities in each state

The table reports the parameters of a Hidden Markov Model fitted on the series of total market returns. The

models assumed the existence of three volatility regimes – high-volatility (regime 1) and low-volatility (regime

3). In each regime volatility is assumed to be independently and identically normally distributed with mean and

variance of 𝜇i and 𝜎i 2. The transition matrix is Π = [𝜋i,j], where 𝜋i,j is the probability of transition from state

i to state j.

‘Estimate’ is the point estimate of the respective parameter, while ‘Std. Error’, ‘t value’ and ‘𝑃𝑟(> |𝑡|)’ are the standard error, the t-statistic and the p-value of the estimate. Parameters are estimated using the Baum–Welch

algorithm.

The result shows the existence of three clearly separated volatility regimes, as evidenced by the significant

difference between 𝜇1, 𝜇2 and 𝜇3 compared to the standard errors of the estimates, and with significant volatility persistence as measured by 𝜋i,i, which range between 72.80% for the high-volatility regime and 93.03% in the low-volatility regime.

Estimate Std. Error t value 𝑃𝑟(> |𝑡|)

𝜋1,1 0.7280 0.1185 6.144 0.0000

𝜋1,2 0.2720 0.0897 3.032 0.0024

𝜋1,3 0.0000 0.1495 0.000 1.0000

𝜋2,1 0.0798 0.0244 3.270 0.0011

𝜋2,2 0.8228 0.0364 22.602 0.0000

𝜋2,3 0.0975 0.0279 3.498 0.0005

𝜋3,1 0.0000 0.0150 0.000 0.9998

𝜋3,2 0.0697 0.0193 3.611 0.0003

𝜋3,3 0.9303 0.0199 46.659 0.0000

𝜇1 0.0919 0.0074 12.404 0.0000

𝜎1 2 0.0015 0.0000 86.926 0.0000

𝜇2 0.0466 0.0008 59.801 0.0000

𝜎2 2 0.0001 0.0000 71.917 0.0000

𝜇3 0.0297 0.0005 56.319 0.0000

𝜎3 2 0.0000 0.0000 70.808 0.0000

Log- Likelihood: 1440.51

BIC Criterion: -2794.47

AIC Criterion: -2853.02

Source: author’s calculations

126

Table 10: Periods with high market volatility

The table summarises the episodes of high market volatility as identified by the three-state hidden Markov chain

model. The model is estimated with the Baum–Welch algorithm, and the sequence of states is estimated using

the Viterbi algorithm. Months with high volatility have estimated mean volatility of 9.19% with standard

deviation of 3.87%. Brief comments on the market events that were associated with the market turbulence are

added for clarity.

Start End Comments

July 1974 October 1974 The crisis of 1973-1974 followed shortly after the termination of

the convertibility of US dollars to gold, one of the tenets of the

Bretton Woods system, and the start of the first oil crisis in

October 1973. The stock market crash started in November 1973

with DJIA losing 14% in that month and S&P 500 losing 7%; the

fall lasted 12 month ending October 1974, with DJIA and S&P 500

losing 30.4% and 36.8%.114

October 1987 January 1988 On October 19, 1987, the Dow Jones recorded the largest fall in a

single day, falling 22.6%. S&P 500 fell 12.1% in October and

12.5% in November. The fall resulted in brokers extending

significant credits to their customers to finance margin calls. The

Fed stepped in to support the financial system by lending to banks,

expanding the monetary base. Throughout the remainder of the

year stock prices remained volatile.115

October 1997 October 1997 A short volatility episode related to the Asian financial crisis.

Consequently, on October 27, 1997, DJIA fell 7.18%, scoring its

twelfth biggest daily percentage loss. The shock was short-lived,

however, and on the next day DJIA recovered more than 60% of

the previous day loss.

August 1998 October 1998 On August 17, 1998 the government of the Russian Federation

devalued the Ruble, defaulted on domestic debt, and declared a

90-day moratorium on external debt repayment. At the end of

August, DJIA fell 11.5% in three days, and the market remained

depressed until being stimulated by a series of interest rate cuts in

October

114 Mishkin and White (2002) 115 Ibid.

127

Start End Comments

January 2000 May 2000 The dot-com bubble of 1997–2000 reached its peak on March 10,

when NASDAQ Composite peaked at more than twice its value a

year earlier. The rapid decrease in value of dot-com shares

continued until 2001 as many tech companies that had spent their

cash but were failing to make a profit found it difficult to raise

more funds on the stock exchange.

October 2000 April 2001 The period was marked by continued decline of tech stocks

coupled with a slowing economy that moved into recession in

March 2001

September

2001

September

2001

A short-term volatility following the terrorist attacks on September

1, 2001

July 2002 October 2002 The episode was a continuation of the bear market that started in

2000, with both NASDAQ and DJ scoring three consecutive years

of losses. In July 2002 volatility surged when DJ fell for 11 of 12

consecutive days, and then slid to 7700, then recovered to just

above 9000 in August, and then fell back to below 7300 in

October.

September

2008

May 2009 In September 2008 the subprime financial crisis intensified; in

early September the US federal government had to bail out Fannie

Mae and Freddie Mac, guarantors for many sub-prime mortgages.

On September 15, Lehman Brothers, an investment bank with large

exposure to subprime assets, filed for bankruptcy, prompting

worldwide financial panic. By the end of the month, two more

American banks collapsed – Washington Mutual and Wachovia.116

May 2010 June 2010 Continuing problems of Greece prompted its downgrade by

Moody’s (to ’A3’) and Standard & Poor’s (to ’BB-’); at the start of

May the Eurozone agreed on a large bailout package for Greece in

return for structural reforms and austerity measures

116 Kingsley (2012)

128

Start End Comments

August 2011 November

2011

Continuing problems in the Eurozone (bailout of Portugal in May,

second bailout of Greece in July, concerns over risks for contagion

to Spain and Italy) and the US (downgrade of the sovereign rating)

Source: the author

The transition probabilities reported in Table 9 provide a good illustration of two of

the stylised facts on volatility forecasting formulated by Engle and Patton (2001). For

example, the largest entries in the transition matrix are located along the main diagonal (𝜋1,1,

𝜋2,2, 𝜋3,3). Therefore, the current state is likely to persist into the following period. At the

same time, the probability of extreme transitions (from high volatility to low volatility, 𝜋1,3,

or from low volatility to high volatility regime, 𝜋3,1) is negligibly low. This structure of the

transition matrix provides a demonstration of the persistence property of volatilities.

Similarly, volatilities are said to be mean-reverting, and that can be seen in the transition

matrix as well. Indeed, if we identify the medium state as the mean level of volatility (and it is

the closest of the three states to the unconditional mean), then we see that the second highest

transition probabilities of the high- and low- states are those for transition towards the

medium state (𝜋1,2 , 𝜋3,2). Another common observation concerning typical patterns of

volatility dynamics is also evident in the transition matrix: the periods of low volatility are

more persistent than higher volatility states, which in our case results in 𝜋3,3 > 𝜋2,2 > 𝜋1,1.

Such a finding is consistent with the observations of Friedman and Liabson (1989), who argue

that price movements comprise of ordinary and extraordinary movements, and that their

evidence concerning volatility persistence suggests that it is due to the ordinary component of

returns.

Table 10 lists the high-volatility periods identified by the hidden Markov model with

three states, together with short comments on market events that unfolded in the respective

periods and could explain the triggers for the respective turbulence episode.

3.6. Conclusions

In this chapter we motivated our choice of methodology and we described the

methodology for performing the tests implemented for this study. The research problem could

be tackled by various approaches using a wide range of available methods. We motivated our

129

choice of a deductive, quantitative, statistical methodology. The choice of a deductive method

was motivated by the significant risk of data mining is a larger set of companies were

involved in the study, as well as concerns of the external validity of our results in case of a

deeper focus on a small set of companies. The focus on external validity, combined with the

continuous nature of the explained variable (stock returns), motivated also our choice of

statistical methods; the limitation of that approach was the loss of detailed information of the

diverse aspects of risk and the risk assessment workflow of institutional and individual

investors. On the other hand, that allowed our study to cover a substantial part of the

investment universe and to span a period of over three decades.

As a primary data source we use Thomson Reuters Datastream, augmented with factor

returns from Kenneth French’s database. Idiosyncratic returns are measured relative to the

four-factor Fama–French–Carhart model estimated on a rolling window of length 60 months

(but not less than 30 months), ending at month (𝑡 − 1) in order to avoid look-ahead bias.

Idiosyncratic volatilities are calculated in four different methods, based on the prevailing

practice, with certain refinements that were motivated in this chapter. In particular, we use the

following volatility estimators: (i) idiosyncratic return from monthly OLS regressions; (ii)

GARCH(1,1) with Skew Generalised Error Distribution estimated on an expanding window

of length at least 60 months and ending at month (𝑡 − 1); (iii) daily idiosyncratic volatility

from the previous month (Ang et al); (iv) forward-looking estimates of future daily volatility

using the past history of estimator (iii).

The predictive accuracy of idiosyncratic volatility measures is evaluated primarily

through Mincer-Zarnowitz regressions to avoid scaling between different time frequencies

impacting the ranking of predictive accuracy. As true volatilities we use the squared monthly

returns and the sum of daily squared returns filtered from a daily EGARCH model fitted on

the entire available series for each security.

The Fama–Macbeth methodology is selected to test whether idiosyncratic volatilities

predict the cross-section of returns. We use individual securities as assets in the Fama and

MacBeth (1973) regressions. Standard errors are calculated using the Newey and West (1987)

adjustment with four lags.

Overall, in this chapter we set out our methodology to quantify idiosyncratic risk and

to measure its correlation with market returns. The next chapter presents the results from the

tests described in this chapter. In particular, it will compare alternative volatility forecasts in

terms of their predictive accuracy, the correlation between idiosyncratic risk and stock

returns, and robustness tests.

130

4. Idiosyncratic Risk and the Cross-Section of

Stock Returns: Empirical Findings

4.1. Introduction

In this chapter we present our empirical finding concerning the link between

idiosyncratic risk and stock returns using the methods presented in Chapter 3. Firstly, we shall

examine which of the four idiosyncratic risk estimators from Section 3.4 is the best predictor

of future (next-period) idiosyncratic volatility. In the last chapter we suggested that the

estimators using measurements from daily returns should offer superior performance, a

hypothesis that we test in Section 4.2. Additionally, we shall also test the stationarity of

volatility processes. This was previously tested by Fu (2009); our goal in this section is to

confirm his findings, and more importantly, to compare qualitatively the forecasting methods.

If volatilities are overwhelmingly stationary, we should expect that good future forecasts

should share that property. If they do not, then it would seem that the forecasts do not follow

faithfully changes in volatilities over time.

In Section 4.3 we shall examine how the various volatility estimators perform as

predictors of the cross-section of returns. If expected idiosyncratic risk truly predicts returns,

then we should expect that the better predictors of future volatility should result in more

reliable forecasting of returns. On the other hand, the empirical tests need to operationalize

the concept of idiosyncratic risk from the model economy to the actual financial markets. In

the model economy idiosyncratic volatility is fixed and known to all participants. In the

financial markets, it is neither constant, nor known. We abstract from the problem how

investors learn the volatilities of securities and what the impact of differences in forecasts

(non-homogeneous beliefs) is on equilibrium returns. Instead, we focus on how the

stationarity of volatilities and their reversion to their means might be incorporated in the

investor decision-making and, hence, in returns. In Section 4.3 we shall test if expected

volatility for the month predicts the cross-section. From a certain perspective, those tests

could be based on the counterfactual assumption that volatilities have unit root and our best

forecast of all future volatilities is the next-period volatility, i.e. these tests do not incorporate

any information on future volatility other than the one-step forecast.

131

In Section 4.4 we build on the conflicting evidence from the different models (reported

in Section 4.3) and explore how the mean-reverting level of volatility explains the

cross-section of returns. In the simple autoregressive context the expected future path of

volatilities could be summarised in terms of just three parameters: the starting value of

volatility, the mean-reverting level, and the speed of mean-reversion. In Section 4.3 we

examine how next-period volatility (which is essentially the starting point of the expected

volatility path) predicts the cross-section of returns, and in Section 4.4 we examine the

significance of the mean level. It might be supposed that the speed of mean-reversion could

also play some part in explaining the cross-section, but our exploratory analysis (not reported

here) did not support that hypothesis. Therefore, in Section 4.4 we report how the

mean-reverting level of volatility explains the cross-section of returns. In the section we also

report various robustness tests for our result. Indeed, an argument could be made that the

predictive performance of the mean-reverting level is due solely to some particular subsection

of the population, which experiences some short-lived positive or negative momentum.

Therefore, we examine whether our results are robust across various subsections of the

population, thereby eliminating some hypothesised explanations of our results. Section 4.5

extends those tests further by performing additional tests for omitted factor, using portfolio

alphas, as well as daily data. These tests are somewhat different from the other tests reported

in this study in terms of methodology. Therefore, they do not rely solely on the methodology

laid out in Chapter 3, but also employ additional models relevant only for those sections (e.g.

statistical factor analysis and Component GARCH); the specific information on these tests is

developed in the section where the test is performed.

Finally, Section 4.6 concludes the presentation of empirical findings.

4.2. Comparison of volatility forecasts

The first question that we address is the quality of forecasts yielded by the selected

methods for volatility forecasting in the domain of idiosyncratic risk. Such an analysis

requires two inputs: estimates of the ex ante expected returns, and measurements of the ex

post realised volatility.

The rationale and the methodology for calculation of the four selected volatility

forecasts is described in the preceding chapter. Let 𝐼𝑉𝑜𝑙𝑠 denote the standard deviation of the

residuals from the Fama–French–Carhart model fitted on a rolling window of 60 (but no less

132

than 30) months ending at (𝑡 − 1). Let 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

denote the expected volatility estimated

using GARCH(1,1) with SGED (skewed generalised error distribution), where parameters are

estimated on expanding windows containing all available data up to the beginning of the

month; at least 60 months of data are required. Let 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

denote the previous month

idiosyncratic volatility, calculated as the standard deviation of daily returns in the previous

calendar month, requiring at least 15 non-zero returns in the preceding month.117 Let

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 denote the forecast from the ARMA(1,1) model calculated using the available history

of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

; the corresponding mean-reverting level implied by the ARMA model is denoted

by 𝑚. The summary statistics for 𝐼𝑉𝑂𝐿𝑆, 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

, 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

, 𝐼�̂�𝑎𝑟𝑚𝑎 and 𝑚, together with

their less-skewed log-transforms, are provided in Table 6 on p. 117.

As an initial exploratory analysis we calculate the cross-sectional correlation between

the idiosyncratic volatility forecasts and the proxies for the realised (ex post) variance. As

proxies for the unobservable true variance we use: (i) the squared idiosyncratic return

(𝐼𝑅𝑒𝑡𝑖,𝑡 2 ); (ii) the in-sample volatility (𝐼𝑉𝑖,𝑡

𝑔𝑎𝑟𝑐ℎ ) filtered from the full-sample GARCH(1,1)

model with SGED shocks, estimated with monthly data; and (3) the mean idiosyncratic

volatility filtered in-sample from daily data (𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

) and averaged by months.118 For each

month we calculate the Spearman’s rank-correlation coefficient between each pair of proxies

of true (ex post) volatility and expected (ex ante) volatility.119 The preference to the

Spearman’s rank-correlation over the standard Pearson’s coefficient120 is driven by a desire to

mitigate the possible impact of outliers on the estimated correlation coefficients.

117 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

is scaled to monthly frequency by multiplying by square root of the number of

trading days in the respective month in order to ease the comparison of coefficients across

volatility measures. 118 Expected (forecasted) values are marked by a hat, e.g. 𝐼�̂�, while the true (ex post) realised volatilities have no hat, i.e. 𝐼𝑉. To avoid cluttering the notation, sometimes we omit the hats as well as security- and time-subscripts if there is no risk of misunderstanding. 119 The Spearman’s rank-correlation coefficient equals the Pearson’s correlation coefficient

calculated in the ranks instead of the values. In case of no ties, the co-efficient can also be

written as:

𝜌 = 1 − 6∑𝑛𝑖=1𝑑𝑖

2

𝑛(𝑛2−1) ,

where 𝑑𝑖 = 𝑥𝑖 − 𝑦𝑖 is the difference between the ranks of the two variables. The coefficient measures the degree of linear dependence between the variables and the coefficient ranged

between −1 and +1. 120 𝜌 =

𝑐𝑜𝑣(𝑥,𝑦)

𝜎𝑥𝜎𝑦

133

Table 11: Average cross-sectional Spearman rank correlations of idiosyncratic

volatilities

The table reports the correlation coefficients between three measures of true volatility and five estimates of

expected volatility. |𝐼𝑅𝑒𝑡𝑖,𝑡| measures realised volatility as the absolute value of idiosyncratic returns. 𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

infers realised volatility from filtered values from GARCH(1,1) using the full series of monthly returns for each

stock. 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

infers realised volatility from filtered values from EGARCH(1,1) using the full series of daily

returns for each stock.

‘𝐼𝑉𝑂𝐿𝑆’ is the volatility of the residual of rolling monthly regressions using 30 to 60 months of data, as available.

‘𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

’ is the idiosyncratic volatility from the daily data in the preceding month calculated using the residuals

from Fama-French regressions with the daily data for the preceding month. ‘𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

’ is the forecasts from

GARCH(1,1) model with SGED innovations estimated using expanding window comprising of at least sixty

months of continuous data. ‘𝐼�̂�𝑎𝑟𝑚𝑎’ is expected volatility from ARMA(1,1) model fitted on the available series

of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

.

The table suggests that both ex ante (expected) and ex post (realised) volatility tend to cluster together based on

their calculation frequency. Thus, 𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

calculated from monthly data correlates more closely with 𝐼�̂�𝑡 𝑜𝑙𝑠

and 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

than with the other estimator of realised volatility - 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

. Likewise, 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

, calculated

from daily returns, correlates more closely with 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 and 𝐼�̂�𝑡−1

𝑑𝑎𝑖𝑙𝑦 .

𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

𝐼�̂�𝑡 𝑜𝑙𝑠 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ 𝐼�̂�𝑡−1

𝑑𝑎𝑖𝑙𝑦 𝐼�̂�𝑡

𝑎𝑟𝑚𝑎 𝑚

|𝐼𝑅𝑒𝑡𝑖,𝑡| 0.35 0.37 0.34 0.33 0.27 0.32 0.29

𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

0.81 0.86 0.90 0.67 0.81 0.76

𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

0.78 0.79 0.83 0.91 0.81

𝐼�̂�𝑡 𝑜𝑙𝑠 0.92 0.65 0.81 0.81

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

0.66 0.82 0.81

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

0.85 0.66

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 0.85

Source: author’s calculations

Spearman’s rank correlations for the proxies of the true volatility and the volatility

forecasts are reported in Table 11. Consistent with the assertion that squared monthly

idiosyncratic returns are a noisy proxy of monthly variance121, we find that they correlate

poorly with the rest of the proxies of true volatility. We find that |𝐼𝑅𝑒𝑡𝑖,𝑡| correlation with

the other two measures of true (ex post) volatility – 𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

and 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

– is consequently

low and stood at just over 1/3. 𝐼�̂�𝑡 𝑜𝑙𝑠, 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ and 𝐼�̂�𝑡

𝑎𝑟𝑚𝑎 all have fairly close correlation

coefficients with monthly returns (correlations in the range 0.32 - 0.34 ), while the

121 Since 𝐼𝑅𝑒𝑡𝑖,𝑡 2 is an estimator of variance, |𝐼𝑅𝑒𝑡𝑖,𝑡| is an estimator of volatility (standard

deviation).

134

correlation with lagged volatility (𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

, the estimator of Ang et al. (2006)) scores worst

(0.27).

The other two estimators of true ex post volatility tend to be more tightly correlated

(average correlation of 0.81) – certainly a high correlation as such, but an even higher

correlation could be expected for two ex post estimates of the same unobservable variable –

the idiosyncratic volatility. 𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

, the ex post volatility from GARCH(1,1), in fact

correlates more closely with the historical OLS volatility and forecasts from GARCH(1,1)

models (mean cross-sectional correlations of 0.86 and 0.90 respectively) than with the

other EGARCH(1,1) ex post measurement (0.81). One possible explanation could point to the

persistence of GARCH volatilities, especially when series lengths are below 10 years – 120

monthly observations, which is reflected in the long half-life122 of many of the estimates. The

median half-life for the shares in our sample123 is 35.25 months, i.e. almost 3 years. This

shows that for many companies GARCH(1,1) with monthly data produces persistent volatility

forecasts. This is in contract with the typical range of persistence of daily volatility series.

This latter point is also demonstrated by the wider interquartile distance for EGARCH(1,1)

with daily data (7.79 percentage points) compared to the interquartile distance for the true

volatilities estimated from GARCH(1,1) with monthly data ( 6.24 p.p.). Against that

backdrop, it should not come as a surprise that the correlation of the EGARCH-measure of

true volatilities correlates more closely with the forecasts from ARMA(1,1) and 𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

, for

the two of which monthly volatilities are inferred from higher-frequency (daily) returns

(correlations of 0.91 and 0.83 respectively), than with the other GARCH measure of true

volatility that employed only monthly returns (correlation 0.81). Overall, while the two

measures of true ex post volatilities yield high average cross-sectional correlations, each of

these measures tends to favour a forecasting method that was estimated from data with the

same frequency as the data underlying the measure of true volatility. A similar pattern is

observed for the correlations of volatility forecasts. The forecasts that are based on monthly

returns (the OLS historical estimator, and the GARCH(1,1) forecasts) tend to correlate more

closely with each other than with the forecasts based on estimates from daily data (𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

122 The half-life (the number of months necessary for volatility to close one half of the spread

between current volatility and the unconditional mean of the process) for the GARCH(1,1)

𝜎𝑡 2 = 𝜔 + 𝛼𝜀𝑡−1

2 + 𝛽𝜎𝑡−1 2 , where 𝛼 + 𝛽 < 1 and 𝛼, 𝛽 > 0 equals

ln (1 2⁄ )

ln(𝛼+𝛽) .

123 All companies having the same weight irrespective of the number of monthly returns

available

135

and the ARMA(1,1) forecasts). The correlation pattern for the mean-reverting level of

volatility is also interesting. Mean-reverting volatility correlates most closely with

ARMA(1,1) forecasts. This reflects a material time-variation of the mean-reverting

volatilities, which are updated with arrival of new information. Yet mean-reverting volatility

correlates much closer with the monthly-frequency volatility forecasts (0.81) than with Ang’s

historical volatility (0.66), consistent with the hypothesis that the mean-reverting level

estimates the equilibrium to which volatility would revert, provided that there was no change

in the scale of the distribution of idiosyncratic innovations.

Spearman’s rank-correlation coefficient provides an insight into the degree of

similarity of the ranking of shares by the different estimators of volatility. However, they

provide limited insight into how accurately the ex ante forecasts predicted the ex post realised

returns. In order to avoid the complications of scaling volatilities estimated from

higher-frequencies to lower-frequencies, we choose to employ Mincer-Zarnowitz regressions

of the form

ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡 2 + 𝑒𝑖,𝑡, (14)

where ℎ𝑖,𝑡 was some of the measures of realised variance (squared volatility), and �̂�𝑖,𝑡 2 is the

tested variance forecasting method. A valid test of the model above pre-supposes that ℎ𝑖,𝑡

and �̂�𝑖,𝑡 2 are stationary.

136

Table 12: Dickey–Fuller tests for ex ante and ex post volatility estimates

The table reports the results from Dickey-Fuller test statistics for estimators of realised (ex post) volatility -

𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

and 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

- and expected (ex ante) volatilities - 𝐼�̂�𝑡 𝑜𝑙𝑠, 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ , 𝐼�̂�𝑡−1

𝑑𝑎𝑖𝑙𝑦 , 𝐼�̂�𝑡

𝑎𝑟𝑚𝑎, and 𝑚. The

tests are performed separately for each security, for which there are at least 24 months of data for the respective

indicator. Both stationarity of the volatilities (Panel A) and natural logarithm of volatilities (Panel B) are tested.

For each series, we have calculated the estimate of the coefficient of the autoregressive term (𝛾(∙)) and the Dickey-Fuller test statistic (𝑡(𝛾)). For each set of tests we report the number of stocks that were tested (N), as well as the mean (‘Mean’), median (‘Median’), and first and third quartiles (‘Q1’ and ‘Q3’) for all calculated

values of 𝛾(∙) and 𝑡(𝛾). The share of series for which the unit root null hypothesis is rejected at 1% confidence level is reported in ‘UR reject (%)’.

The table demonstrates that the unit root hypothesis is rejected for about half of all realised volatilities 𝐼𝑉𝑖,𝑡 𝑔𝑎𝑟𝑐ℎ

and 𝐼𝑉𝑖,𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

. A likely reason is that these estimates are filtered from actual data series and some of the

volatility is smoothed out. The most direct measure of volatility that does not involve such smoothing - 𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

– shows that the unit root hypothesis is rejected for 95.37% of all stocks in Panel A and 93.96% of all stocks in

Panel B. The forecasts obtained from monthly data - 𝐼�̂�𝑡 𝑜𝑙𝑠 and 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ - demonstrate persistence of volatility

forecasts and for these estimators the unit root hypothesis is rejected for a low share of series. Such a mismatch

with the persistence properties of the true series also suggests that these forecasts could be inferior predictors of

true volatility and may smooth month-on-month changes in volatility.

Variable 𝑵 Mean Median Q1 Q3 UR reject (%)

Panel A: 𝐼𝑉𝑡+1 − 𝐼𝑉𝑡 = 𝛾0 + 𝛾𝐼𝑉𝑡 + 𝜀𝑡

𝛾(𝐼𝑉𝑡 𝑔𝑎𝑟𝑐ℎ

) 4489 -0.06 -0.03 -0.08 -0.00 54.24

𝑡(𝛾) < −10 -3.94 < −10 -2.10

𝛾(𝐼𝑉𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

) 5507 -0.28 -0.19 -0.40 -0.09 53.50

𝑡(𝛾) -4.12 -3.65 -5.18 -2.55

𝛾(𝐼�̂�𝑡 𝑂𝐿𝑆) 5556 -0.03 -0.01 -0.04 -0.00 1.82

𝑡(𝛾) -1.11 -1.13 -1.74 -0.49

𝛾(𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

) 4735 -0.14 -0.10 -0.19 -0.05 27.79

𝑡(𝛾) -2.86 -2.55 -3.65 -1.79

𝛾(𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

) 5294 -0.65 -0.64 -0.79 -0.51 95.37

𝑡(𝛾) -7.35 -7.05 -8.97 -5.51

𝛾(𝐼�̂�𝑡 𝑎𝑟𝑚𝑎) 5398 -0.24 -0.15 -0.31 -0.08 43.53

𝑡(𝛾) -3.60 -3.25 -4.42 -2.38

𝛾(�̂�𝑡 𝑎𝑟𝑚𝑎) 5398 -0.09 -0.03 -0.09 -0.01 14.80

𝑡(𝛾) -2.16 -1.80 -2.76 -1.03

137

Variable 𝑵 Mean Median Q1 Q3 UR reject (%)

Panel B: ln(𝐼𝑉𝑡+1) − ln(𝐼𝑉𝑡) = 𝛾0 + 𝛾ln(𝐼𝑉𝑡) + 𝜀𝑡

𝛾(𝐼𝑉𝑡 𝑔𝑎𝑟𝑐ℎ

) 4489 -0.06 -0.02 -0.07 -0.00 41.68

𝑡(𝛾) < −10 -2.75 < −10 -1.27

𝛾(𝐼𝑉𝑡 𝑒𝑔𝑎𝑟𝑐ℎ

) 5507 -0.26 -0.18 -0.36 -0.09 48.25

𝑡(𝛾) -3.82 -3.41 -4.90 -2.35

𝛾(𝐼�̂�𝑡 𝑂𝐿𝑆) 5556 -0.03 -0.01 -0.04 -0.01 1.48

𝑡(𝛾) -1.07 -1.12 -1.69 -0.48

𝛾(𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

) 4735 -0.13 -0.08 -0.16 -0.04 22.94

𝑡(𝛾) -2.61 -2.39 -3.35 -1.67

𝛾(𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

) 5294 -0.61 -0.59 -0.73 -0.46 93.96

𝑡(𝛾) -6.84 -6.59 -8.32 -5.12

𝛾(𝐼�̂�𝑡 𝑎𝑟𝑚𝑎) 5398 -0.22 -0.13 -0.28 -0.06 35.61

𝑡(𝛾) -3.32 -2.94 -4.08 -2.12

𝛾(�̂�𝑡 𝑎𝑟𝑚𝑎) 5398 -0.08 -0.03 -0.09 -0.01 12.88

𝑡(𝛾) -2.01 -1.70 -2.56 -0.93

Source: author’s calculations

In Table 12 we provide results from the Dickey and Fuller (1979) test for unit root.124

The null hypothesis is presence of unit root (i.e., 𝛾 = 0). If the process has unit root, then the

errors have permanent impact on idiosyncratic volatilities, i.e. they did not die out over time.

The table highlights some salient features of the various estimators of ex ante and ex post

volatilities.

We find values of 𝛾(𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

) that are quite similar to those reported by Fu (2009). In

our sample we reject the null hypothesis in 95% of all shares, compared to approx. 90% in Fu,

which may be due to the different composition and time-span of the two samples. On the

other hand, the other two measures of ex post volatility – the one based on GARCH with

monthly data and the other using EGARCH with daily data estimated for the full length of the

data series – resulted in volatility measures that reject the unit root hypothesis for only

53-54% of all stocks. At first glance these numbers are surprising as due to the constraints

124 The null hypothesis of the test is that 𝜌 = 0 in the autoregressive model 𝑦𝑡 = 𝜌𝑦𝑡−1 + 𝜖.

138

on the parameters of the GARCH model (𝛼 + 𝛽 < 1), the GARCH estimates are always with

finite variance and mean-reverting. However, if the sum of the two parameters 125 is

sufficiently close to unity, the half-life of the model could increase without bound as the sum

approaches one. Indeed, this is consistent with our observations in our sample, where the

median half-life is 35.25 months, and the third quartile is 377.8 months. Hence, while in

theory the associated model is stationary, in a finite sample of monthly data such

mean-reversion could be difficult to estimate and ultimately dependent not on the true

properties of the underlying process but rather on the constraints imposed on the parameter

estimation. Another possible explanation could be the great parameter uncertainty in the

estimation of GARCH parameters due to the scarcity of monthly data. However, such an

explanation is contradicted by the similar results obtained by using the EGARCH(1,1) model

with aggregated daily data. Or one might hypothesise that there is a volatility trend, e.g. due

to the decline of volatility of each security as the company matures and its cash flows become

more stable and predictable. However, this does not appear to be the case, as the

Dickey-Fuller test with a drift term and trend rejects the unit root hypothesis for an even

lower share of companies.

The OLS estimator is at the opposite extreme – for it, the null hypothesis is rejected

for less than 2% of all shares. Such behaviour is in line with the interpretation of the OLS

estimator as a moving average filter of idiosyncratic shocks. In that case, a unit impulse at the

entry of the 60-month moving average filter would be dampened by the filter, but the impulse

would have lasting impact on filtered volatility during the 60 months until the shock exits the

rolling window. Hence, in short samples we could expect that the unit root null hypothesis

would be difficult to reject, which is consistent with what we find in the present sample.

The impact of shocks on the other three volatility forecasts (GARCH forecasts,

ARMA forecasts, and ARMA mean-reverting level of volatility) are more difficult to

interpret. The GARCH forecasts shared a similar issue as the full-sample ex post volatility

estimation, namely – the long half-life of the forecasts. However, similarly to the other two

measures, significant shocks on volatility could have dual impact: on the one hand, they could

result in an increased volatility expectation for the next period (keeping the model parameters

fixed). On the other hand, new information also updates the parameter estimates. This latter

effect is probably more pronounced for the mean-reverting level of volatility, where the unit

125 The bound on the sum of the two parameter, 𝛼 + 𝛽 < 1, refers only to GARCH(1,1) model.

139

root hypothesis is rejected for less than 15% of all securities.

Overall, we note that the various volatility estimates have markedly different

time-series properties, even though they all tended to produce not too dissimilar rankings of

securities in the cross-section. At the one end is the quite static OLS estimator, where large

shocks produced long-lasting impact on volatility estimates. On the other extreme is the

measure used by Ang et al. (2006) (𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

), where the volatility in each month is determined

solely by the returns in that period. Somewhere in the middle is the (E)GARCH estimator

where month-𝑡 volatilities are also affected by prior-months’ returns, so that the variability of

filtered or predicted volatilities is somewhat smoothed at the cost of more durable impact of

volatility shocks on estimated volatilities.

We now move to the question of which idiosyncratic risk estimators produce more

accurate estimates of future (one-period ahead) volatilities. As discussed in the methodology

chapter, that approach could not be properly addressed using the loss function approach

because some of the ex ante and ex post measures are based on daily data that are

subsequently scaled to monthly frequency, and the scaling method could obscure comparisons

with those measures that are estimated directly from monthly data. Thus, the loss function

approach that is used, for example, by Spiegel and Wang (2005) cannot be applied in our

setting. Instead, as discussed in the chapter on methodology, we follow Mincer and Zarnowitz

(1969) and Pagan and Schwert (1990) and we compare volatility forecasts using the standard

predictive regressions.

140

Table 13: Predictive accuracy from Mincer-Zarnowitz regressions

The table reports aggregated results from comparison of the accuracy of volatility forecasts using Mincer-Zarnowitz regressions, where ℎ𝑖,𝑡 is some measure of realised

variance (either monthly volatility filtered from EGARCH(1,1) with daily returns, or squared monthly idiosyncratic return), and �̂�𝑖,𝑡 2 is the tested variance forecasting method

- 𝐼�̂�𝑡 𝑜𝑙𝑠, 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ , 𝐼�̂�𝑡−1

𝑑𝑎𝑖𝑙𝑦 , 𝐼�̂�𝑡

𝑎𝑟𝑚𝑎. Panels A, B and C test three alternative specifications that are run for each available security with more than 24 months of data. Parameter

estimates for each specification are obtained using two methods – ordinary least squares (OLS), or the more robust median regression. For each regression we calculate a

measure of goodness of fit - either 𝑅2 (for OLS regressions), or its analogue for quantile regression - 𝑅1(0.5), proposed by Koenker and Machado (1999). For each specification we report the mean (‘mean’) and median (‘median’) values of the set of goodness-of-fit measures (𝑅2 or 𝑅1(0.5)). ‘# of firms’ reports the number of firms, for which the respective Mincer-Zarnowitz regression was estimated. ‘% of pos.’ reports the share of cases where the goodness-of-fit measure for the respective case was superior

compared to the corresponding goodness-of-fit measure with 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 as predictor variable. The results show that 𝐼�̂�𝑎𝑟𝑚𝑎 is vastly superior to the other volatility forecasts,

especially when true volatility is filtered from EGARCH, followed by 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

. 𝐼�̂�𝑡 𝑜𝑙𝑠 and 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ on the other hand outperformed 𝐼�̂�𝑡

𝑎𝑟𝑚𝑎 is less than 13% of all cases,

when realised volatility was based on EGARCH. Thus, a simple binomial test implies that 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 is a superior predictor of volatility against all other tested estimators and in

all specifications.

Variance ℎ𝑡 from EGARCH ℎ𝑡 = 𝐼𝑅𝑒𝑡𝑡 2

forecast mean median # of firms % of pos. mean median # of firms % of pos.

Panel A: ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡 2

OLS regression – 𝑅2

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 41.07 42.96 5562 – 4.70 2.34 5591 –

𝐼�̂�𝑡 𝑜𝑙𝑠 12.38 6.14 5562 13.90 2.60 1.00 5591 36.02

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

37.87 39.71 5374 32.71 4.17 1.38 5394 33.30

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

16.49 10.50 4833 11.75 2.50 1.12 4857 34.90

141

Variance ℎ𝑡 from EGARCH ℎ𝑡 = 𝐼𝑅𝑒𝑡𝑡 2

forecast mean median # of firms % of pos. mean median # of firms % of pos.

Median regression – 𝑅1(0.5)

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 29.09 29.06 5564 – 1.67 0.88 5591 –

𝐼�̂�𝑡 𝑜𝑙𝑠 8.09 4.23 5564 10.73 1.09 0.48 5591 36.56

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

25.15 25.68 5376 29.63 1.64 0.77 5394 40.01

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

10.68 7.20 4835 9.06 1.12 0.56 4857 38.07

Panel B: √ℎ𝑖,𝑡 = 𝑎𝑖 + 𝑏𝑖�̂�𝑖,𝑡

OLS regression – 𝑅2

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 45.06 48.40 5564 – 5.43 3.24 5591 –

𝐼�̂�𝑡 𝑜𝑙𝑠 13.77 7.17 5564 12.29 3.13 1.41 5591 33.11

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

44.82 48.90 5376 39.55 4.73 2.66 5394 34.80

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

18.24 12.71 4835 9.82 3.25 1.63 4857 32.55

Median regression – 𝑅1(0.5)

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 31.06 31.41 5564 – 2.55 1.53 5591 –

𝐼�̂�𝑡 𝑜𝑙𝑠 8.84 4.79 5564 10.32 1.82 0.85 5591 36.38

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

29.48 30.97 5376 37.07 2.30 1.38 5394 37.91

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

11.47 7.83 4835 8.17 1.82 0.95 4857 36.52

142

Variance ℎ𝑡 from EGARCH ℎ𝑡 = 𝐼𝑅𝑒𝑡𝑡 2

forecast mean median # of firms % of pos. mean median # of firms % of pos.

Panel C: ln(ℎ𝑖,𝑡 2 ) = 𝑎𝑖 + 𝑏𝑖ln(�̂�𝑖,𝑡

2 )

OLS regression – 𝑅2

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 47.02 50.85 5539 – 3.63 2.13 5564 –

𝐼�̂�𝑡 𝑜𝑙𝑠 14.63 7.96 5564 11.36 2.57 1.13 5591 35.83

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

47.72 52.20 5376 42.06 3.28 1.93 5394 37.60

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

19.00 13.46 4835 9.02 2.58 1.26 4857 36.11

Median regression – 𝑅1(0.5)

𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 31.89 32.23 5539 – 2.42 1.47 5564 –

𝐼�̂�𝑡 𝑜𝑙𝑠 9.24 5.06 5564 10.26 1.79 0.83 5591 36.04

𝐼�̂�𝑡−1 𝑑𝑎𝑖𝑙𝑦

30.75 32.25 5376 39.58 2.18 1.32 5394 37.30

𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

11.78 7.97 4835 8.31 1.77 0.91 4857 36.09

Source: author’s calculations

143

The Mincer-Zarnowitz regressions are estimated individually for each security. We

require at least 24 months of pairwise-complete data. The mean and median 𝑅2’s are reported

in Table 13. The first important result from that exercise is the poor performance of the

forecasts from GARCH(1,1) in explaining either of the two proxies of the true volatility. With

either of the two measures of true volatility employed in this study, the results for 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

are closer to that of the variance of the residual from the OLS regression (𝐼𝑉𝑂𝐿𝑆) than to the

forecasts from ARMA(1,1) (𝐼�̂�𝑎𝑟𝑚𝑎) or the estimator used by Ang et al. (2006) and Ang et al.

(2009) (𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

). Under either of the two proxies of true volatility, the estimator used by Ang

et al for period-𝑡 volatility in fact outperforms the forecast from GARCH(1,1).

The magnitude of the differences found in the comparison is indicative of the

significance of the difference. At each row we provide the percentage share of securities for

each the 𝑅2 (for OLS regression) or 𝑅1 (for median regression) that have outperformed the

respective measure (𝑅2 or 𝑅1) for ARMA(1,1); that percentage is provided in column “% of

pos.”. In case the respective measure outperforms ARMA(1,1) we would see that share

exceeding 50%. The table does not include results from statistical tests of the observed

differences, however the significance of the results can be gauged from the small number of

positive differences. Indeed, we could consider the numbers of positive and negative

differences as the outcome of Bernoulli trials and calculate the significance using binomial

test.126 For an average of about 5,000 securities, the two-sided 99% confidence interval

would be between 0.4817 and 0.5183 . In all cases in Table 13 the percentages are

materially outside that interval, hence we should conclude that with high confidence the

observed under-performance of all models relative to ARMA(1,1) is not mere happenstance.

The second important feature is the low values of 𝑅2’s when squared idiosyncratic

returns are used as dependent variable. The low value of 𝑅2 in Mincer-Zarnowtz regressions

is well documented. The previously referred study of Andersen and Bollerslev (1998) proves

that the reason for the low 𝑅2s is not a failure of the GARCH models but rather the fact that

squared returns are a noisy estimator of volatility. In particular they show that the 𝑅2s for

GARCH(1,1) models are bounded above by 1/𝜅, where 𝜅 is the kurtosis of the underlying

126 If two competing models have the same predictive performance as measured by 𝑅2, then we would expect that the difference of the 𝑅2s from the competing Mincer-Zarnowitz regressions would be positive in half of the cases, with a number of positive or negative

differences following binomial distribution. The benefit of that test is that it is exact and does

not require knowledge of the distribution of the differences of 𝑅2, including no assumption of symmetry of the distribution of the differences.

144

noise distribution. In this case, however, the 𝑅2s are particularly low. A plausible reason is

that the four factors in the mean equation account for much of the predictable variability, and

the remaining idiosyncratic component is unpredictable and not persistent (so that past values

are not useful for forecasting). Alternatively, it is possible that EGARCH significantly

improves forecasts compared to the simpler GARCH(1,1) model. However, the sharp contrast

with the results where true volatility is proxied by the average volatility from the

EGARCH(1,1) model suggested that GARCH forecasts in the present context are

under-performing materially the estimates from the ARMA(1,1) and the lagged volatility

𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

.

Considering the previous observation that the parameters of a typical GARCH model

in our sample implies a half-life of about 3 years, and observing the very similar predictive

performance of GARCH(1,1) with the 24-60 months rolling average (𝐼𝑉𝑂𝐿𝑆 ), a likely

explanation of the results emerges: the low number of months available for GARCH

estimation and the inherent noise contained in monthly idiosyncratic returns results in

GARCH models with monthly data forecasting some mean, longer-term component of

volatility and failing to track with satisfactory accuracy the month-on-month variation of

volatilities.

Overall, the results in Table 13 suggested that forecasts of monthly volatilities from

ARMA(1,1) outperforms the remaining estimators studied here. Moreover, the historical

measure of Ang et al (𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

) outperforms GARCH estimates obtained from monthly

returns; in fact, for many series the latter are closely correlated to the simple OLS volatilities.

This suggests that the significance of GARCH forecasts in explaining the cross-section of

returns that is observed by Fu might be misleading and in fact the direction of correlation

could be negative, as argued by Ang et al. In the following section we shall examine this issue

and compare and contrast the power of the alternative volatility forecasts to explain the

cross-section of returns.

4.3. Idiosyncratic volatility and the cross-section of stock

returns: results from Fama–Macbeth cross-sectional regressions

We evaluate the significance of idiosyncratic volatility in explaining the cross-section

of market returns using the Fama and MacBeth (1973) cross-sectional regressions. At each

145

calendar date in our sample for which there is available information on capitalisation (𝐶𝑎𝑝),

CAPM beta (𝐵𝑒𝑡𝑎), and positive book-to-market ratio (𝐵/𝑀), we estimate the following

cross-sectional regressions

𝑅𝑖𝑡 = 𝛾0𝑡 + 𝛾1𝑡𝛽1𝑖𝑡 +⋯+ 𝛾𝑚𝑡𝛽𝑚𝑖𝑡 + 𝜀𝑖𝑡,

where 𝛽𝑘𝑖𝑡 is the loading of stock 𝑖 = 1…𝑁 on factor 𝑘 = 1…𝐾 at time 𝑡 = 1…𝑇, and

𝛾𝑘𝑡 are the parameters to be estimated. The estimates 𝛾𝑘𝑡 are treated as random variables

and the mean of the estimates from the individual cross-sectional regressions is used as an

estimator of the true value (𝛾𝑘), i.e. 𝛾𝑘 = 1

𝑇 ∑𝑇𝑡=1 𝛾𝑘𝑡. The standard error of the estimates is

calculated using the Newey and West (1987) autocorrelation-corrected coefficients with a lag

of 4.

The method of Fama–Macbeth could be employed using portfolios or individual

securities as base assets in the test. Apart from market betas, the explanatory factors in our

sample, including capitalisation, B/M, idiosyncratic volatility (a second moment), and to a

somewhat lesser extent, the liquidity spread based on auto-covariances, could be estimated

with reasonable accuracy. Nevertheless, in order to mitigate the impact of extreme values of

the explanatory factors on the cross-sectional regressions, each month the explanatory

variables are censored at the 0.005 and 0.995 quantiles.

Securities with low prices often have greater noise in their returns, which is related to

the minimum step by which prices move, causing abrupt discrete changes in share prices. To

mitigate this concern, studies usually remove from the sample securities with an unadjusted

price below some selected threshold value. We opt for a low threshold of $1 for most

regressions. Some studies127 opt for a higher threshold of $10. The choice of threshold could

affect conclusions. For example, Brandt et al. (2010: 881) document that the level of

institutional ownership and market capitalisation are significantly lower for low-priced stocks

(defined as those in price deciles 1 through 3). The low-priced stocks typically have a price

below $10, market capitalisation below $100 million, and institutional ownership below 10

per cent. Therefore, setting a threshold of $10 would discard proportionately more securities

with high retail ownership, where investors are typically less diversified and thus more

sensitive to idiosyncratic risk. Hence, we employ a baseline threshold of $1, but key

relationships were also tested for robustness against higher thresholds. The threshold (filter)

applied to each regression model is reported in the tables.

There are a number of control variables in the Fama-Macbeth cross-sectional

127 e.g. Bali and Cakici (2008)

146

regressions. In order to facilitate presentation, we split the results into two tables: one

containing the results for OLS and GARCH volatilities (Table 14), and another one – for Ang

et al and ARMA(1,1) (Table 15).

The results from the Fama–MacBeth regressions with OLS and GARCH volatilities

are presented in Table 14 on p. 148. Models 1 through 5 list the cross-sectional estimates for

the CAPM specification as well as the CAPM augmented with the three characteristics used

as a baseline explanation of the cross-section of returns. In particular, model 1 corresponds to

the standard CAPM model. The significance of the CAPM beta varies in the different

specifications, but in view of its strong theoretical justification, we keep it in all specifications

even when insignificant, in order to avoid omitted variable bias in our results. Model 2

corresponds to the CAPM with the two Fama and French factors. Model 3 adds to the

three-factor Fama–French specification the return momentum, calculated as the return

between (𝑡 − 7) and (𝑡 − 2). Model 4 further adds the liquidity spread, and model 5 adds

previous month return in order to account for return reversal. The results for models 1 through

5 are consistent with the existing body of literature and confirmed that the two Fama–French

factors, liquidity, momentum, and lagged return are all significant predictors of the

cross-section, while beta is an unreliable predictor of the cross-section of returns.

Specifications 6 through 8 introduce the monthly idiosyncratic volatility from the OLS

regressions. In all three specifications we find that 𝐼𝑉𝑜𝑙𝑠 is a significant predictor of the

cross-section. Moreover, its magnitude and significance do not change substantially between

the three compared models, suggesting that the significance of 𝐼𝑉𝑜𝑙𝑠 is not a result of its

correlation with another predictor.

Idiosyncratic volatilities usually exhibit significant skewness (cf Table 6 on p. 117). In

order to examine whether the significance of idiosyncratic volatilities is a result of their

skewness, in models 9 through 11 we also add the natural logarithm of the idiosyncratic

volatility, ln(𝐼𝑉𝑜𝑙𝑠), to the model specification. As seen in Table 6, the log-transform of

volatilities substantially reduces the skewness of the variable and thus mitigates the concerns

that the significance of the regression coefficient is an artefact purely of few high-volatility

securities. The results support the significance of 𝐼𝑉𝑜𝑙𝑠 as a predictor of the cross-section, as

the magnitude of the log of volatilities remains numerically stable and significant in the tested

specifications.

In specifications 12 through 17 we repeat the same exercise, but instead of the OLS

volatilities, we employ the one-month-ahead forecasts from the GARCH(1,1) model. We

147

previously observed that these forecasts have long half-lives and correlated with the OLS

volatilities. The long half-lives suggest that shocks on volatilities have long-lasting,

nearly-permanent effect, i.e. in finite samples volatilities could behave similarly to random

walk. On the other hand, the high correlation with the OLS volatilities suggests that such

shocks that result in material quasi-permanent change of volatility are infrequent and

volatility of each share tends to remain in a narrow band for relatively longer periods. This is

consistent with the experience of the present author from the inspection of individual series of

some high-profile stocks when volatility forecasts are obtained from GARCH(1,1) models

estimated with monthly data. The latter qualification (‘with monthly data’) suggests that for

short periods (less than a month up to a couple of months) the volatility of individual shares

may vary materially but quickly reverts to some average level, and hence there are not enough

months of exceptionally high- or low-magnitude shocks that would result in the upward or

downward revision of the volatility forecasts from GARCH(1,1).

Having regard for the foregoing discussion, we should expect that 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

would be

a significant predictor of the cross-section, and that its sign and magnitude would be similar to

that of 𝐼𝑉𝑜𝑙𝑠. Indeed, in general the results in Table 14 are consistent with that expectation.

Both volatilities and their log-transform remain significant predictors of the cross-section,

although the 𝑡 statistics are palpably lower compared with the significance of the

corresponding specifications involving 𝐼𝑉𝑜𝑙𝑠. Increasing the threshold of small-value stocks

from US$ 1 to US$ 10 decreases the significance of the idiosyncratic volatilities; in some

specifications, esp. 13a–15a, idiosyncratic volatilities are no longer significant predictors of

the cross-section. The change of significance depending on the threshold is consistent with the

explanation of different investor profiles for securities with smaller prices, which are

preferred by individual investors, who are also known for being less diversified, and hence

those securities earn a higher risk-premium for idiosyncratic volatility. On the other hand, our

results are consistent with those obtained by Bali and Cakici (2008) who find that the

correlation between idiosyncratic volatility and returns is not robust and changes across

specifications.

Finally, we note that even when idiosyncratic volatility is an insignificant predictor,

the point estimate does not change sign, suggesting that longer series or higher frequencies

through the associated larger sample sizes might offer more conclusive evidence on

idiosyncratic volatility significance.

148

Table 14: Fama–Macbeth cross-sectional regressions with OLS and GARCH forecasts

The table reports results from Fama-Macbeth cross-sectional regressions. Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). Tested variables are the OLS measure of idiosyncratic volatility (‘𝐼𝑉

𝑜𝑙𝑠’) and from GARCH(1,1)

(‘𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

’), and the natural logarithms of these measures. ‘𝑅2’ reports the averaged R-squared statistics from the cross-sectional regressions. Column ‘Filter’ indicates the subsample used for estimating the cross-sectional regressions: ‘all’ in case the entire sample is used; ‘UP>10’ indicates specifications estimated using only securities with

unadjusted price of over USD 10. Newey-West t-statistics are reported in parenthesis. The results suggest that 𝐼𝑉𝑜𝑙𝑠 and 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

are robust statistically significant predictors

of the cross-section of returns, with sign and magnitude stable across specifications. The regression coefficient tends to be somewhat lower in the specifications that involve

only stocks with unadjusted price above USD 10, suggesting that idiosyncratic risk premium is likely higher among stocks with lower unadjusted price – a segment preferred by

individuals due to lower transaction costs.

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼𝑉 𝑜𝑙𝑠 ln (𝐼𝑉𝑜𝑙𝑠) 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ln (𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ) 𝑅2

1 all 0.84 1.81

(-2.37)

2 all 0.69 -0.10 0.79 3.45

(2.21) (-2.50) (7.16)

3 all 0.63 -0.09 0.90 0.01 4.16

(2.08) (-2.48) (9.51) (4.83)

4 all 0.48 -0.01 0.93 0.01 0.05 4.66

(1.83) (-0.45) (10.83) (4.68) (2.42)

5 all 0.42 0.00 0.85 0.01 0.06 -0.04 5.26

(1.57) (-0.13) (10.23) (4.05) (2.57) (-8.94)

6 all 0.41 0.05 2.65

(1.62) (2.90)

149

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼𝑉 𝑜𝑙𝑠 ln (𝐼𝑉𝑜𝑙𝑠) 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ln (𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ) 𝑅2

6a UP>10 0.31 0.02 3.48

(1.41) (1.12)

7 all 0.30 -0.02 0.90 0.06 4.02

(1.24) (-0.51) (9.28) (3.98)

7a UP>10 0.27 -0.01 0.58 0.04 4.82

(1.24) (-0.24) (6.59) (2.19)

8 all 0.24 0.02 0.9 0.01 0.04 -0.04 0.04 5.52

(1.05) (0.53) (11.27) (4.01) (1.87) (-9.15) (3.45)

8a UP>10 0.24 -0.02 0.62 0.01 -0.03 -0.03 0.03 6.62

(1.13) (-0.49) (8.18) (3.96) (-1.17) (-6.36) (2.66)

9 all 0.24 0.67 2.71

(1.06) (3.05)

9a UP>10 0.19 0.31 3.49

(0.95) (1.45)

10 all 0.16 0.00 0.90 0.81 4.12

(0.75) (-0.06) (9.19) (3.82)

10a UP>10 0.15 0.01 0.59 0.51 4.88

(0.74) (0.16) (6.46) (2.51)

11 all 0.14 0.03 0.9 0.01 0.03 -0.04 0.56 5.62

(0.66) (0.80) (11.21) (4.14) (1.75) (-9.18) (3.44)

150

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼𝑉 𝑜𝑙𝑠 ln (𝐼𝑉𝑜𝑙𝑠) 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ln (𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ) 𝑅2

11a UP>10 0.14 -0.01 0.63 0.01 -0.04 -0.03 0.5 6.68

(0.68) (-0.23) (8.14) (3.98) (-1.66) (-6.33) (3.47)

11b UP>10 0.18 -0.01 0.68 0.01 -0.06 0.50 6.08

(0.93) (-0.17) (8.61) (4.78) (-2.38) (3.42)

12 all 0.46 0.03 2.79

(1.83) (2.01)

12a UP>10 0.40 0.00 3.51

(1.79) (0.22)

13 all 0.39 -0.02 0.84 0.04 4.21

(1.60) (-0.74) (8.73) (2.34)

13a UP>10 0.38 -0.02 0.53 0.01 4.92

(1.66) (-0.56) (5.96) (0.81)

14 all 0.30 0.01 0.85 0.01 0.04 -0.04 0.02 5.74

(1.27) (0.43) (10.66) (3.61) (2.10) (-9.82) (1.47)

14a UP>10 0.33 -0.02 0.58 0.01 -0.03 -0.03 0.01 6.76

(1.55) (-0.63) (7.79) (3.38) (-0.94) (-6.70) (1.01)

15 all 0.32 0.56 2.83

(1.37) (2.57)

15a UP>10 0.30 0.17 3.5

(1.42) (0.83)

151

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼𝑉 𝑜𝑙𝑠 ln (𝐼𝑉𝑜𝑙𝑠) 𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ln (𝐼�̂�𝑡

𝑔𝑎𝑟𝑐ℎ ) 𝑅2

16 all 0.29 -0.01 0.84 0.59 4.28

(1.28) (-0.31) (8.71) (2.80)

16a UP>10 0.28 -0.01 0.55 0.30 4.93

(1.30) (-0.20) (6.02) (1.52)

17 all 0.24 0.02 0.85 0.01 0.04 -0.04 0.36 5.81

(1.09) (0.57) (10.62) (3.70) (1.83) (-9.84) (2.17)

17a UP>10 0.26 -0.01 0.59 0.01 -0.04 -0.03 0.32 6.79

(1.30) (-0.46) (7.84) (3.42) (-1.55) (-6.74) (2.20)

17b UP>10 0.31 -0.01 0.63 0.01 -0.05 0.28 6.18

(1.51) (-0.46) (8.26) (4.14) (-2.00) (1.94)

Source: author’s calculations

152

The results in Table 14 provide some support for the hypothesis that idiosyncratic

volatility is a significant predictor of the cross-section. Nevertheless, we should also recall

that in a preceding section we argued that 𝐼𝑉𝑜𝑙𝑠 and 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

are inferior predictors of

expected volatility compared to 𝐼𝑉𝑡−1 and 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎. Against that backdrop it is important to

examine how those arguably more accurate predictors of next-period volatility predict the

cross-section of returns. If next-period idiosyncratic volatility is a true explanatory

characteristic, then we should expect superior significance of those superior estimators. The

results from those tests are presented in Table 15.

The results in Table 15 come as a surprise in view of the encouraging results obtained

with 𝐼𝑉𝑜𝑙𝑠 and 𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

. Specifications 1 through 3 explore the correlation between the

last-period idiosyncratic volatility and returns. Consistent with the findings of Ang et al.

(2009, 2006), in specifications 2 and 3 we find a statistically-significant negative correlation

between returns and past-month idiosyncratic volatility, 𝐼𝑉𝑡−1. The result remains true when

we control for the unadjusted price of the securities in our portfolio, i.e. when we allow only

securities with unadjusted price above US$ 10 in the test sample. In specifications 4, 5 and 6

we employ log-transform of idiosyncratic volatility in order to reduce the skewness of the

explanatory variable. This also allows us to reduce the impact on the regression coefficient of

the high-volatility securities in each month; Fu (2009) argues that the reversals of returns of

the highest-quintile stocks are a probable reason for the significant negative correlation

observed in the data. We find that the negative correlation documented by Ang et al. is not

robust in our sample. Only specification 6 offers a strong negative correlation, yet that

evidence is substantially weakened by the results reported in 6a, which documents no

significant correlation. Thus, the evidence appears to be consistent with the results of Fu and

suggests that much of the explanatory power of 𝐼𝑉𝑡−1 is placed in the right tail of

cross-sectional volatility distribution and that the negative correlation is detected primarily

among high-volatility, low-price stocks.

Fu points out that last-period volatility is not a forward-looking estimator of

next-period volatility, unless volatilities follow random walk, a hypothesis that he rejects for

the overwhelming majority of his sample, and which we confirm in our sample as well. We

address this deficiency of 𝐼𝑉𝑡−1 by using 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 to estimate next-period expected

volatility. Furthermore, in the preceding section we reasoned that that measure (𝐼�̂�𝑡 𝑎𝑟𝑚𝑎)

offered the highest predictive accuracy among the four estimators studied in this chapter.

153

Therefore, we should expect that 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 would offer the most relevant evidence on the

correlation between next-month volatility and returns. In specifications 7 through 12 we

explore whether idiosyncratic volatilities from our best predictor of next-period monthly

volatility explain the cross-section of next-month realised returns. In those specifications we

find that idiosyncratic volatilities are not significantly correlated with returns. This is

evidenced both by the small values of the 𝑡-statistic in most regressions, as well as the

changes of sign of the point estimates across specifications.

154

Table 15: Fama–Macbeth cross-sectional regressions with historical and ARMA forecasts

The table reports results from Fama-Macbeth cross-sectional regressions. Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). Tested variables are Ang’s previous-month idiosyncratic volatility (‘𝐼�̂�𝑡−1’) and ARMA(1,1) (‘𝐼�̂�𝑡

𝑎𝑟𝑚𝑎’),

and the natural logarithms of these measures. ‘𝑅2’ reports the averaged R-squared statistics from the cross-sectional regressions. Column ‘Filter’ indicates the subsample used for estimating the cross-sectional regressions: ‘all’ in case the entire sample is used; ‘UP>10’ indicates specifications estimated using only securities with unadjusted price of

over USD 10. Newey-West t-statistics are reported in parenthesis.

The results suggest that 𝐼�̂�𝑡−1 and 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 are not robust predictors of the cross-section of returns. Cross-sectional regressions involving 𝐼�̂�𝑡−1 confirm the findings of

previous studies of negation correlation with returns. However, using log-transformed volatility shows that the negative coefficients are result principally to the skewness of the

explanatory variable and the negative correlations does not generalise to the whole sample. 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 is not significant predictor either and its coefficient changes sign and

significance between specifications even though previous analyses suggest that it is the best predictor of volatility among the examined ones.

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼�̂�𝑡−1 ln (𝐼�̂�𝑡−1) 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 ln (𝐼�̂�𝑡

𝑎𝑟𝑚𝑎) 𝑅2

1 all 0.64 0.00 2.75

(2.12) (0.12)

1a UP>10 0.46 -0.01 3.17

(1.68) (-1.27)

2 all 0.58 -0.13 0.73 -0.02 4.20

(2.03) (-3.45) (6.49) (-1.98)

2a UP>10 0.46 -0.07 0.5 -0.02 4.64

(1.71) (-2.17) (4.66) (-1.82)

3 all 0.33 -0.05 0.81 0.01 0.09 -0.03 -0.04 5.93

(1.35) (-1.37) (9.10) (3.91) (3.18) (-7.23) (-5.35)

3a UP>10 0.33 -0.05 0.58 0.01 0.01 -0.02 -0.02 6.77

(1.50) (-1.56) (7.09) (3.85) (0.25) (-5.18) (-2.32)

155

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼�̂�𝑡−1 ln (𝐼�̂�𝑡−1) 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 ln (𝐼�̂�𝑡

𝑎𝑟𝑚𝑎) 𝑅2

4 all 0.53 0.22 2.76

(1.88) (1.49)

4a UP>10 0.41 -0.01 3.18

(1.56) (-0.12)

5 all 0.53 -0.11 0.74 0.00 4.22

(1.95) (-2.93) (6.52) (0.00)

5a UP>10 0.42 -0.07 0.51 -0.05 4.66

(1.64) (-1.91) (4.73) (-0.43)

6 all 0.35 -0.05 0.80 0.01 0.06 -0.03 -0.2 5.93

(1.46) (-1.34) (9.03) (4.19) (2.32) (7.51) (-2.58)

6a UP>10 0.33 -0.05 0.58 0.01 -0.01 -0.02 -0.04 6.75

(1.51) (-1.53) (7.07) (3.98) (-0.25) (-5.34) (-0.51)

7 all 0.40 0.03 2.91

(1.46) (2.21)

7a UP>10 0.39 0.00 3.55

(1.64) (-0.23)

8 all 0.51 -0.01 0.83 0.02 4.26

(1.94) (-0.20) (8.17) (1.22)

8a UP>10 0.44 -0.03 0.54 -0.01 4.99

(1.88) (-0.85) (5.81) (-0.34)

156

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝐼�̂�𝑡−1 ln (𝐼�̂�𝑡−1) 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 ln (𝐼�̂�𝑡

𝑎𝑟𝑚𝑎) 𝑅2

9 all 0.40 0.01 0.85 0.01 0.05 -0.04 -0.01 5.75

(1.60) (0.22) (10.10) (3.69) (2.19) (-8.83) (-0.63)

9a UP>10 0.37 -0.03 0.59 0.01 -0.02 -0.03 0.00 6.77

(1.69) (-0.83) (7.59) (3.68) (-0.68) (-6.03) (-0.08)

10 all 0.30 0.54 2.90

(1.21) (2.53)

10a UP>10 0.33 0.09 3.51

(1.44) (0.41)

11 all 0.44 0.00 0.83 0.39 4.32

(1.85) (0.00) (8.15) (1.69)

11a UP>10 0.39 -0.02 0.55 0.09 4.99

(1.74) (-0.54) (5.73) (0.43)

12 all 0.39 0.01 0.85 0.01 0.05 -0.04 0.07 5.83

(1.65) (0.29) (10.09) (3.89) (2.00) (-8.87) (0.45)

12a UP>10 0.33 -0.02 0.60 0.01 -0.05 -0.03 0.28 6.80

(1.55) (-0.47) (7.62) (3.75) (-1.69) (-6.07) (1.83)

Source: author’s calculations

157

In this section we confirmed the conflicting evidence concerning the sign of the

correlation between idiosyncratic volatilities and returns. We found that the correlation was

fairly robust when we forecast idiosyncratic volatility through the residual of monthly OLS

regressions or through GARCH(1,1) forecasts, yet the sign turned to negative or became

insignificant when we employed last-month volatility or volatility forecasts from ARMA

model. This result was puzzling given the results from the previous section, where we found

that the latter two estimators of volatility were better forecasts for the unobservable true

volatility compared to the former two. If the more accurate forecasts of volatilities were not

significantly linked to returns, then there was some other information in the former estimators

(𝐼�̂�𝑡 𝑔𝑎𝑟𝑐ℎ

and 𝐼�̂�𝑡 𝑜𝑙𝑠) that was helpful in explaining the cross-section and improving the

accuracy of predicting volatility would offer limited contribution to explaining the

cross-section of returns. We dedicate the next section to exploring this problem.

4.4. Mean-reverting level of volatility

The last section posed an interesting problem: we found that the more accurate

prediction of next-period volatilities did not result in better forecasts of cross-sectional returns

as ARMA forecasts generally were not significantly correlated with returns. The standard

CAPM model is of limited help in this case, as it assumes that volatilities are fixed. Fixed

volatilities are a reasonable assumption that allows examination of equilibrium prices and

returns in the CAPM model, which emphasises the more elusive beta, whereas variance could

be estimated with higher precisions; in reality, however, volatilities do vary from period to

period.128 Furthermore, our ability to predict changes in future volatilities is inherently

limited: the GARCH models essentially assume that next-month volatility can be predicted

from its values in the preceding months, but rarely and with limited success are able to

incorporate any fundamental explanatory variables, be that macroeconomic indicators or

security characteristics. The simple GARCH(1,1) model offered the most transparent

128 A particular difficulty in incorporating stochastic volatility in the CAPM model is that the

model assumes complete markets. If volatility of stocks changes randomly, then there may not

exist an unique risk-neutral measure, and the same contingent claim would have different

values under different risk-neutral measures; for example of conditions for market

completeness under stochastic volatility see Davis (2004).

158

demonstration of this fact as next-period volatility is just the weighted sum of the

previous-period forecast and the previous-period squared return (a measure of the realised

volatility). Thus volatility forecasts are formed from past forecasts that are continuously

updated with a proxy of the last-period realised volatility. The update rule could be symmetric

as in the simple GARCH model, but could also be asymmetric, as is the case with the

EGARCH model. Nonetheless, fundamental variables (e.g. interest rates, economic growth,

credit risk spreads, etc) are rarely present in such specifications; essentially, volatility

forecasts are function solely of the past history of volatility forecasts and the distribution of

realised returns, especially its variance and skewness. Therefore, if the GARCH forecasts and

the OLS forecasts contain information about expected returns that is not present in the ARMA

forecasts or lagged ( 𝐼𝑉𝑡−1 ) volatilities, it is reasonable to conjecture that such useful

information was contained in some other characteristic of the volatility process. Such a

characteristic could be the mean-reverting level of volatility from the ARMA model. Indeed,

if volatility was random walk, then such mean reversion would not exist. However, as we

confirmed in the previous section, the unit root hypothesis for 𝐼𝑉𝑡−1 was rejected for the

overwhelming majority of securities. The ARMA model produced best forecasts among the

examined models, and these forecasts were based on the history of 𝐼𝑉𝑡−1, which was found to

be stationary. We argued that the superior predictive power of ARMA was due to its reliance

on a less noisy proxy of realised volatility (average daily volatility scaled up to monthly

frequency) to update the next period forecast compared to the GARCH model with monthly

data that used squared monthly return as a proxy of realised volatility in the respective month.

Therefore, we also examined the predictive performance of the mean-reverting level of

volatility estimated from the ARMA model for explaining the cross-section of returns.129 In

the case of ARMA(1,1) model, 𝑋𝑡 = 𝜇 + 𝜙𝑋𝑡−1 + 𝜀𝑡 + 𝜃𝜀𝑡−1, and the mean-reverting level

was the natural explanatory variable. Indeed, we estimated 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 using the history of

𝐼𝑉𝑡−1 and neither 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 (the conditional expectation), nor 𝐼𝑉𝑡−1 offered a significant

prediction of the cross-section, which leaves the unconditional mean of the process as the

natural alternative explanatory variable (𝐸(𝑋𝑡) = 𝜇

1−𝜙 = 𝑚).

129 Let 𝑋𝑡 be a stationary stochastic process that has ARMA(1,1) representation, 𝑋𝑡 = 𝜇 + 𝜙𝑋𝑡−1 + 𝜀𝑡 + 𝜃𝜀𝑡−1. Then:

𝐸(𝑋𝑡) = 𝜇

1−𝜙 = 𝑚,

𝑣𝑎𝑟(𝑋𝑡)) = 1+2𝜙𝜃+𝜃2

1−𝜙2 𝜎𝜀 2,

𝑐𝑜𝑣(𝑋𝑡, 𝑋𝑡−1)) = (𝜙+𝜃)(1+𝜙𝜃)

1−𝜙2 𝜎𝜀 2.

159

The results from those tests are summarised in Table 16. The table shows that the

mean-reverting level of volatility is indeed a robust predictor of the cross-section. The point

estimates remained stable and significant when both the mean-reverting level and its log are

used as explanatory variables. Moreover, the estimated slopes are very similar to those found

in OLS and GARCH(1,1) tests reported previously, which supports the explanation that those

measures had significant predictive performance due to their correlation with the

mean-reverting level, rather than next-period volatility. Filtering only securities with

unadjusted price over 10 US dollars did not materially affect either the slopes, or their level of

significance (except in specifications 1a and 4a, which include only beta as control variable).

160

Table 16: Fama–Macbeth cross-sectional regressions with mean-reverting volatility

The table reports results from Fama-Macbeth cross-sectional regressions. Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). Tested variables is the mean-reverting level of volatility (‘𝑚’) and its natural logarithm. ‘𝑅

2’ reports the

averaged R-squared statistics from the cross-sectional regressions. Column ‘Filter’ indicates the subsample used for estimating the cross-sectional regressions: ‘all’ in case the

entire sample is used; ‘UP>10’ indicates specifications estimated using only securities with unadjusted price of over USD 10. Newey-West t-statistics are reported in

parenthesis.

The results suggest that 𝑚 and ln𝑚 are significant predictors of the cross-section of returns. Sign, coefficient estimates and significance levels remain stable between across alternative specifications.

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

1 all 0.36

0.04

2.76

(1.26)

(3.06)

1a UP>10 0.24

0.02

3.37

(0.97)

(1.48)

2 all 0.41 0.03 0.88

0.05

4.07

(1.50) (0.90) (8.97)

(3.62)

2a UP>10 0.29 0.03 0.6

0.04

4.75

(1.19) (0.92) (6.73)

(2.57)

3 all 0.31 0.04 0.88 0.01 0.02 -0.04 0.03

5.78

(1.25) (1.25) (11.05) (3.61) (0.71) (-8.91) (2.89)

3a UP>10 0.28 0.01 0.63 0.01 -0.06 -0.03 0.04

6.78

(1.27) (0.33) (8.53) (3.63) (-1.79) (-6.15) (3.63)

161

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

4 all 0.26

0.62 2.78

(0.94)

(3.27)

4a UP>10 0.16

0.34 3.40

(0.66)

(1.81)

5 all 0.30 0.06 0.90

0.76 4.14

(1.16) (1.64) (9.26)

(3.76)

5a UP>10 0.19 0.06 0.63

0.60 4.81

(0.80) (1.70) (6.95)

(3.02)

6 all 0.23 0.07 0.9 0.01 0.01 -0.04

0.53 5.84

(0.97) (1.93) (11.44) (3.63) (0.55) (-8.94)

(3.61)

6a UP>10 0.19 0.04 0.65 0.01 -0.08 -0.03

0.67 6.82

(0.90) (1.24) (8.79) (3.64) (-2.34) (-6.14)

(4.77)

Source: author’s calculations

162

We also examine the robustness of our results in various subsamples of the original

data set; these results are reported in Table 18 on p. 170. As a first test we compare the

significance of idiosyncratic volatility in different market environments (volatility regimes).

We calculate the mean monthly volatility of the Thomson Reuters’ DataStream Total Market

Return Index from daily data. Following Fink et al. (2010), we classify the volatility regimes

in the market using a Markov chain with three states, where volatility in each state is assumed

to be normally distributed with unknown mean and variance. The transition matrix and the

parameters of each volatility state are estimated using the Baum–Welch algorithm (Baum et

al., 1970), and the sequence of states of the market (the Viterbi path) is estimated using the

Viterbi (1967) algorithm. The model identified three states with mean volatilities of 9.25

(the high-volatility state), 4.68 (the medium-volatility state), and 2.97 (the low-volatility

state). The high-volatility state includes only 40 months, while the medium- and

low-volatility states include 150 and 210 months, respectively (starting from January

1980).130

The results from that analysis are reported as specifications ‘a’, ‘b’ and ‘c’ in Table

18. We find that the mean-reverting level idiosyncratic volatility is a significant predictor of

the cross-section in both medium- and low-volatility episodes. The coefficient of the

mean-reverting volatility is not materially different between the low-volatility and the

medium-volatility state. In contrast, during episodes of market turbulence, idiosyncratic

volatility made little difference for expected returns. However, it should be remarked that the

number of high-volatility months (only forty) means that the results from that sub-sample

should be interpreted with caution because the error of the estimates could be higher.

130 See Chapter 3. Research Methodology and Data Sources, section “3.5. ” (p. 144) for

further details. The parameters of the Markov chain were provided in Table 9 on p. 147.

163

Table 17: Descriptive statistics by principal market, 1/1980 - 3/2013

The table summarises the descriptive statistics for the sample used in the tests, split into principal exchange where stocks are traded as assigned by Datastream.

‘Beta’ is the stock beta calculated by portfolios constructed as in Fama and French (1992). ‘ln(Cap)’ is the natural logarithm of market capitalisation calculated as number of

shares times the unadjusted price. ‘ln(B/M)’ is the natural logarithm of book value of equity as available at the end of the preceding month to market price of equity (B/M),

and is calculated from the price-to-book value series from Datastream (PTBV) where book values of equity are taken at a lag of six months to ensure that they are known to

investors. ‘Roll’ is the bid-ask spread calculated Roll’s model; Roll (1984). ‘𝐼�̂�𝑎𝑟𝑚𝑎’ is expected volatility from ARMA(1,1) model fitted on the available series of 𝐼𝑉𝑡−1 𝑑𝑎𝑖𝑙𝑦

.

‘𝑚’ is the mean-reverting level of volatility implied by the fitted ARMA model. ‘Mean (EW)’ and ‘Mean (VW)’ are the equally-weighted and the value-weighted values of the respective indicators. ‘St.dev.’ is the standard deviation; ‘Median’, ‘Q1’ and

‘Q3’ are the median and the first and third quartiles of the sample. ‘Skewness’ is the skewness coefficient for the sample. ‘Obs’ is the number of rows for which data is

available.

The table demonstrates the differences between the Nasdaq and the non-Nasdaq sub-samples. The Nasdaq stocks are characterised with higher beta, lower liquidity (higher

bid-ask spread), higher volatility, and smaller capitalisation.

Variables Mean (EW) Mean (VW) St.dev. Median Q1 Q3 Skewness Obs

Panel A: Nasdaq stocks

Beta 1.30 1.19 0.30 1.28 1.06 1.50 -0.01 370,266

ln(𝐶𝑎𝑝) 5.22 9.59 1.69 5.09 4.02 6.26 0.55 370,266

ln(𝐵/𝑀) -0.78 -1.48 0.82 -0.71 -1.25 -0.23 -0.63 370,266

Roll 8.67 5.65 4.57 7.62 5.64 10.46 2.55 370,244

𝐼�̂�𝑎𝑟𝑚𝑎 14.87 8.74 6.85 13.57 10.00 18.26 1.31 347,293

𝑚 17.42 11.25 6.67 16.31 12.68 20.85 1.22 347,248

164

Panel B: Non-Nasdaq stocks

Beta 1.13 0.94 0.30 1.11 0.95 1.33 0.16 493,733

ln(𝐶𝑎𝑝) 6.36 9.87 1.92 6.37 4.97 7.68 0.10 493,733

ln(𝐵/𝑀) -0.66 -1.19 0.73 -0.58 -1.03 -0.21 -0.89 493,733

Roll 5.53 3.90 3.40 4.69 3.47 6.52 3.66 493,726

𝐼�̂�𝑎𝑟𝑚𝑎 9.18 5.98 4.88 7.97 6.01 10.90 2.37 465,627

𝑚 10.42 6.69 5.44 9.09 6.92 12.20 2.53 465,545

Source: author’s calculations

165

In order to examine whether idiosyncratic risk explains returns for all markets and not

only of NASDAQ, which is characterised with smaller stocks with higher idiosyncratic risk,

we split our sample into two sub-samples based on the exchange where the stock has primary

listing131: NASDAQ and non-NASDAQ (NYSE and AMEX). The NASDAQ sub-sample

includes 3,375 securities, while the non-NASDAQ sub-sample includes 3,103 securities.

The results for those splits are reported as specifications ‘d’ and ‘e’ in Table 4.3. We find that

the mean-reverting volatility is significant in both specifications, however the magnitude of

the coefficient for non-NASDAQ securities is between one-third and one-half that for

NASDAQ shares, suggesting that the differences between the two sub-samples are

economically significant. In our view this finding is consistent with the prediction of the

underlying theoretical model of Merton (1987), as the stocks traded at NYSE are typically

larger and better-known issuers; this point is illustrated well by Table 17, which shows that

Nasdaq stocks are on average more volatile and less liquid, and also characterised by higher

beta. In this setting higher slope in the Fama-Macbeth regressions for Nasdaq stocks conforms

with the predictions of the theoretical models that less known stocks should earn higher

premium for idiosyncratic risk. According to Merton’s model, idiosyncratic risk of more

widely-invested stocks should earn lower premium, which in our case is reflected in the

differential between the coefficients from the Fama–MacBeth regressions on NASDAQ and

non-NASDAQ sub-samples.

Another split that we perform aims to examine whether the significance of

idiosyncratic volatility in explaining the cross-section holds both in periods of economic

growth and during recessions. To that end we split our sample into two subsamples using the

business cycle breakpoints of the National Bureau of Economic Research (NBER). These

breakpoints are determined informally by the NBER’s Business Cycle Dating Committee

based on a broad set of economic indicators and served as a benchmark for the state of the US

economy. As customary in business cycle studies, we assumed that the peak month is

classified as the last month of an expansion period, and the through month is considered as the

last month of a recession period. The results are reported under specifications ‘f’ and ‘g’ in

Table 18. During both expansion and recession episodes we find that idiosyncratic volatility is

a significant predictor of the cross-section of stock returns, although in specification ‘3g’ the

131 Primary listing refers to the exchange where the stock was traded most actively. The

allocation to a specific exchange is the one determined by Thomson-Reuters.

166

confidence level is somewhat lower. The differences in the point estimates for idiosyncratic

risk between expansion and contraction are particularly striking – in specifications ‘3’ and ‘6’

the difference was between four- and five-fold. In general, the direction of the difference is

consistent with the economic intuition that in expansion episodes investors are less risk averse

and would be willing to assume more risk (in this case – idiosyncratic risk) for premium

compared to contraction episodes. Nonetheless, that explanation also has caveats: firstly, the

magnitude of the difference does not match the magnitude of risk aversion changes; secondly,

as a baseline case one would expect that a change of risk aversion would have similar effect

for the different types of risk, which is not reflected in the values of liquidity premium or beta.

We consider more reasonable to attribute those results to the much smaller sample of

contraction (recession) months in our test sample – only 50 months – which resulted in higher

risk of the result being affected by pricing adjustments during recessions, which also overlap

more with high-volatility months. Moreover, high-volatility months, for which we previously

reported no correlation between idiosyncratic risk and return, are more likely to occur during

recessions. Thus, 11 out of 41 high-volatility months fall during recessions, while recessions

account for 56 out of 400 months, showing that a turmoil (high-volatility) month is more

likely to occur during recession. Nevertheless, this difference in slopes between growth and

recession is worth exploring in future studies, as it may suggest that the observed positive link

between idiosyncratic risk and returns might be due to high-volatility growth companies,

which earn more during a period of economic expansion, and not necessarily due to the

premium on exposure to undiversified idiosyncratic risk.

Next, we investigate if the significance of idiosyncratic risk is due to previous

accumulated gains or losses. The model of Merton derives the equilibrium in essentially a

neo-classical framework with the added behavioural assumption that investors are constrained

from holding a well-diversified portfolio. The prospect theory of investing suggests that

decisions could depend on whether the position has an accumulated loss or gain relative to

some reference value. Unfortunately, there is no information concerning the value used as

reference by investors. Bhootra and Hur (2014), for example, employ the Capital Gains

Overhang, where the reference price is measured as a weighted average of all prices in the

preceding three years with weights depending on the traded volume. Benartzi and Thaler

(1995) suggest that the annual period is consistent with the tax return frequency, while

Malkiel and Xu (2004) point out that some institutional investors are required to disclose

individual loss-making positions on a quarterly basis. Therefore, there appears to be no clear

guideline as to how to evaluate the reference price used by investors. In order to investigate if

167

differences in accumulated gains and losses might have affect decisions, we split our sample

into two sub-samples based on the accumulated gains over the six months preceding the

previous month, i.e. based on our 𝑅𝑒𝑡(−7,−2) variable, with one sub-sample including all

securities with cumulative net gain over that period (𝑅𝑒𝑡(−7,−2) > 1), while the other

sub-sample including the remaining non-profit-making positions (𝑅𝑒𝑡(−7,−2) ≤ 1). These

results are reported in specifications ‘h’ and ‘i’ in Table 18. The mean-reverting level of

idiosyncratic volatility remains a significant predictor of the cross-section, and its magnitude

does not appear to be significantly affected by whether the six-month cumulative return as

measured by 𝑅𝑒𝑡(−7,−2) is in the red or not. Nevertheless, we emphasise that these tests

are intended to confirm the robustness of the mean-reverting level of volatility as a predictor

of the cross-section of returns, and are not intended as a test of the prospect theory of

investing as such. In particular, other measures of the reference price might yield different

conclusions.

Another concern that we wish to address in this section is the possibility that

idiosyncratic volatility serves as a proxy for default risk. In principle, default risk could be

estimated more directly by using some combination of market and accounting data to estimate

the default probabilities or at least credit risk scores. There is a plethora of models that could

be useful. One of the earliest ones is Altman’s z-score (Altman, 1968), where the score Z is a

linear combination of five accounting ratios, i.e.: 𝑍 = 1.2 𝑋1 + 1.4 𝑋2 + 3.3 𝑋3 + 0.6 𝑋4 +

1.0 𝑋5, where 𝑋1 is the ratio of working capital to total assets, 𝑋2 is the ratio of retained

earnings to total assets, 𝑋3 is the ratio of earnings before interest and taxes (EBIT) to total

assets, 𝑋4 is the ratio of market value of equity to book value of total liabilities, and 𝑋5 is

the ratio of sales to total assets. Other models build on the structural model of Merton (1974)

that views the value of the firm as a stochastic process and the value of equity as a European

call option on the value of the firm with exercise price equal to the value of company debt,

with default occurring whenever the value process fell below the level of debt. The works of

Altman and Merton inspired a number of other contributions, but due to lack of sufficient data

to implement those models we shall again use filters to exclude some segments from the

sample that are exposed to higher default risk. For example, Damodaran (2004) suggests (p.

256-57) that stocks that have lost substantial value over the previous year are often riskier

than the remaining stocks. He points out that this is due both to the empirical regularity that

low-priced stocks are more volatile, as well as to the increased leverage132 and financial risk

132 From a valuation perspective leverage should be calculated with market values of debt and

168

when market value of equity declines substantially. In his examination he therefore excludes

stocks with price below US$ 5, annualised volatility over 80%, beta over 1.25, and debt to

market capital over 80%. For our test we implement a similar filter aimed to leave in the test

sample only stocks with lower default probability: we exclude stocks with Beta over 1.25,

last-month volatility ( 𝐼𝑉𝑡−1 ) or expected volatility ( 𝐼�̂�𝑡 𝑎𝑟𝑚𝑎 ) over 23.09% 133 , and

price/earnings (P/E) ratio between 12.0 and 26.0. The motivation for the use of P/E stems

from its link to past earnings; when past earnings are negative, P/E is not defined, so those

rows are excluded; 12 and 26 are correspondingly the first and the third quartile of the P/E

data. The rationale for exclusion of low P/E securities is that the low values could signal low

expected future growth or low quality of earning. The rationale for exclusion of high P/E

stocks was that these may have very low earnings, which could be a signal of distress, e.g.

like the securities in the dot-com bubble.134 The cumulative application of these criteria

results in a quite small subset of the original data set – only 269,685 rows remain, i.e. about

31.2% of the original data set. That subset is flagged as the “low default” subset in the table of

results (specification “j”). We find that the significance and magnitude of the coefficients of

the cross-sectional regression are unaffected by the restriction to the low-default-risk

subsample, and in fact the coefficients are marginally higher than those of the full sample.

Therefore, these tests do not provide evidence that the significance of idiosyncratic volatility

is driven by its possible correlation with probabilities of default.

The impact of extreme observations should also be a concern for econometric tests.

For example, Walkshäusl (2013) finds that low-volatility stocks earn abnormally high return,

which the authors explain as “quality premium”, while Fu (2009) suggests that the negative

equity. Whereas market value of debt is often close to accounting value, the value of equity

could drift significantly away from the accounting value, which is reflected in the price/book

value ratio. 133 This corresponded roughly to annual volatility of 80% since 80 √12 ≈ 23.09⁄ . 134 One way to think about the expected values of P/E was through the Gordon’s growth

model, which proposed that the price P of a security with required rate of return (𝑟) and

growth rate (𝑔 ) of dividends 𝐷1 would be 𝑃0 = 𝐷1

𝑟−𝑔 = 𝐸𝑃𝑆1(1−𝑏)

𝑟−𝑔 , where 𝑏 was the

retention rate (the part of earnings not paid out as dividends). Noting that 𝑔 = 𝑏 × 𝑅𝑂𝐸, where ROE was the return on equity, the forward price/earnings ratio took the form 𝑃/𝐸 = 𝑃

𝐸𝑃𝑆 = 1−𝑔/𝑅𝑂𝐸

𝑟−𝑔 . In that setting, a low P/E ratio could correspond to either high required rate of

return (a risky business profile), or low growth perspective g. Similarly, a high P/E could be

seen as result of high expected growth (low 𝑟 − 𝑔), so that the price of the security was undepinned by growth expectations rather than by strong earnings. (Gordon and Shapiro

(1956))

169

link documented by Ang et al. (2006) is due to return reversals of high-volatility stocks. In

order to examine if our results are driven by the lowest-volatility or highest-volatility stocks,

in each month we fitted the cross-sectional regression using only the volatilities between the

0.25 and 0.75 quartile in the respective month grouped by the mean-reverting level of

volatility; thus, we used the middle 50% of the observation in each month, discarding the 25%

with lowest volatility and the 25% with highest volatility. These results are reported as

specifications “k” in Table 18. Contrary to the hypothesis that the results are driven by low- or

high-volatility stocks, we find that the regression coefficients remain statistically significant

and actually increase, which could suggest that the extreme volatilities add more noise to the

estimates instead of driving coefficient significance.

We also explore whether the mean-reverting level of volatility could be exploited by

arbitrageurs. If abnormal returns accrued only to companies that experience short episode of

high volatility and expected returns quickly revert to the mean, then the transaction costs

associated with exploiting differences in idiosyncratic volatilities could potentially offset the

higher expected returns. For example, Li et al. (2014) reported that “in extending our tests to

include holding periods beyond the first two months following portfolio formation, we found

that current-month IVOL has no meaningful relationship with stock returns in periods beyond

the second month. Empirically then, the excess returns of the IVOL anomaly all occur in the

first month or two following portfolio formation. In short, we found that the IVOL effect is

short lived, effectively requiring traders to adjust portfolio holdings at least every other month

to have a reasonable chance at producing alpha. Such frequent rebalancing naturally raises

questions about the impact of transaction costs and liquidity constraints” (p. 56-7). In order to

examine this point, we use lagged values of mean-reverting volatilities in the cross-section of

returns. If such lagged values are significant for a sufficiently long lag, then it would follow

that investors could exploit the higher returns accruing to high-volatility stocks for

sufficiently long periods. We used securities with unadjusted price over US$ 10 for lags of 1,

2, 3 and 6 months. The results from that exercise are reported in Table 19 (p. 178). We find

that past expected mean-reverting volatility levels remain significant predictors of the

cross-section of returns for at least six months. On the other hand, the estimated coefficients

for lags of six month are about one half of the coefficients of the contemporaneous

coefficients.

170

Table 18: Fama–Macbeth cross-sectional regressions with mean-reverting volatility – robustness checks

The table reports results from Fama-Macbeth cross-sectional regressions. Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). Tested variables are the mean-reverting level of volatility (‘𝑚’) and its natural logarithm. ‘𝑅

2’ reports the

averaged R-squared statistics from the cross-sectional regressions.

Column ‘Filter’ indicates the subsample used for estimating the cross-sectional regressions: ‘all’ in case the entire sample is used; ‘high vol’, ‘medium vol’ and ‘low vol’

indicate that cross-sectional regressions are estimated using only months that fall in the respective market volatility regime as classified by the Viterbi algorithms. ‘NASDAQ’

and ‘non NASDAQ’ indicate that the sample includes only securities with principal listing on NASDAQ or other (NYSE or Amex) stock exchange. ‘contraction’ and

‘expansion’ indicate that the cross-sectional regressions are estimated using months with economic contraction or expansion as classified by the National Bureau of Economic

Research (NBER). ‘growing stocks’ and ‘falling stocks’ considers only stocks with cumulative growth or decline for six months based on 𝑅𝑒𝑡(−7,−2) in order to examine possible behavioural differences depending on stock momentum. ‘low default’ indicates that the cross-sectional regressions are estimated using stocks with lower beta (below

1.25), price/earnings ratio between 12 and 26, and lower values of last-month and current-month ARMA forecasts below 23%.

The results suggest that 𝑚 and ln𝑚 are significant predictors of the cross-section of returns. Sign, coefficient estimates and significance levels remain stable across alternative specifications. Premium on NASDAQ stocks, however, tends to be notably higher. However, premium during contraction episodes appears to be significantly higher

compared to periods of expansion – a difference that cannot be explained solely as changes in risk tolerance.

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

1 all 0.36 0.04 2.76

(1.26) (3.06)

1a high vol -1.70 -0.03 5.12

(-1.46) (-0.49)

1b medium vol 1.45 0.06 2.99

(2.81) (2.64)

1c low vol 0.01 0.04 2.15

(0.05) (2.77)

1d NASDAQ 0.35 0.06 1.85

(0.95) (3.29)

171

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

1e non-NASDAQ 0.37 0.04 2.86

(1.34) (3.29)

1f contraction 0.44 0.08 3.41

(0.47) (2.02)

1g expansion 0.35 0.04 2.66

(1.25) (2.55)

1h growing stocks 0.48 0.06 3.07

(1.62) (4.31)

1i falling stocks 0.14 0.04 2.68

(0.46) (2.38)

1j low default 0.27

0.04

2.18

(1.01)

(2.47)

1k low default 0.28

0.06

1.66

(0.98)

(2.48)

2 all 0.41 0.03 0.88 0.05 4.07

(1.50) (0.90) (8.97) (3.62)

2a high vol -1.84 0.08 1.58 -0.02 7.36

(-1.69) (0.71) (3.54) (-0.30)

2b medium vol 1.44 0.05 0.82 0.06 4.52

(2.96) (0.90) (4.48) (2.79)

172

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

2c low vol 0.13 0.00 0.79 0.05 3.14

(0.47) (0.08) (8.43) (3.21)

2d NASDAQ 0.59 0.20 1.29 0.09 3.32

(1.72) (2.90) (10.19) (4.28)

2e non-NASDAQ 0.27 0.00 0.72 0.04 4.15

(1.05) (-0.12) (7.96) (3.29)

2f contraction 0.29 0.02 0.97 0.08 4.88

(0.33) (0.24) (2.63) (2.09)

2g expansion 0.43 0.03 0.87 0.04 3.95

(1.57) (0.86) (8.91) (3.01)

2h growing stocks 0.54 -0.04 0.71 0.05 4.39

(1.86) (-1.04) (7.30) (4.15)

2i falling stocks 0.12 0.06 1.13 0.05 3.98

(0.42) (1.63) (11.94) (3.06)

2j low default 0.23 0.06 0.36

0.05

3.50

(0.82) (1.64) (4.07)

(3.01)

2k middle 50% 0.24 0.10 0.90

0.09

3.33

(0.93) (2.18) (8.35)

(3.19)

173

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

3 all 0.31 0.04 0.88 0.01 0.02 -0.04 0.03 5.78

(1.25) (1.25) (11.05) (3.61) (0.71) (-8.91) (2.89)

3a high vol -1.72 0.06 1.16 -0.01 -0.13 -0.05 0.01 11.28

(-1.77) (0.52) (3.53) (-0.94) (-1.17) (-2.99) (0.28)

3b medium vol 1.15 0.07 0.87 0.01 0.07 -0.04 0.03 6.31

(2.71) (1.11) (5.85) (3.09) (1.52) (-5.41) (1.78)

3c low vol 0.12 0.02 0.84 0.01 0.02 -0.03 0.03 4.37

(0.48) (0.52) (9.54) (4.77) (0.50) (-7.67) (2.39)

3d NASDAQ 0.52 0.20 1.25 0.02 0.00 -0.05 0.07 5.94

(1.69) (3.01) (10.21) (4.66) (-0.11) (-7.58) (2.82)

3e non-NASDAQ 0.12 0.00 0.73 0.01 0.03 -0.03 0.02 6.26

(0.53) (0.07) (9.94) (2.55) (1.04) (-6.93) (2.20)

3f contraction 0.17 0.04 0.93 0.00 -0.14 -0.05 0.11 7.4

(0.20) (0.39) (4.31) (0.14) (-1.48) (-3.62) (3.10)

3g expansion 0.33 0.04 0.88 0.01 0.04 -0.03 0.02 5.55

(1.34) (1.16) (10.13) (5.36) (1.52) (-8.29) (1.84)

3h growing stocks 0.38 -0.03 0.77 0.01 -0.04 -0.02 0.04 5.94

(1.47) (-0.94) (8.92) (6.09) (-1.17) (-3.17) (3.55)

3i falling stocks -0.01 0.10 1.00 0.00 0.06 -0.05 0.04 6.04

(-0.02) (2.49) (12.18) (0.96) (1.65) (-10.97) (2.30)

174

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

3j low default 0.16 0.05 0.36 0.00 -0.02 -0.05 0.05

6.01

(0.60) (1.20) (4.38) (1.69) (-0.43) (-9.46) (3.02)

3k middle 50% 0.11 0.11 0.92 0.01 -0.02 -0.04 0.09

5.31

(0.49) (2.46) (10.44) (3.02) -(0.62) -(8.06) (3.77)

4 all 0.26 0.62 2.78

(0.94) (3.27)

4a high vol -1.66 -0.41 5.16

(-1.56) (-0.41)

4b medium vol 1.26 0.93 3.04

(2.58) (2.99)

4c low vol -0.06 0.60 2.14

(-0.23) (3.15)

4d NASDAQ 0.32 0.86 1.73

(0.88) (3.35)

4e non-NASDAQ 0.25 0.59 2.91

(0.94) (3.75)

4f contraction 0.25 1.07 3.43

(0.28) (2.14)

4g expansion 0.26 0.55 2.68

(0.97) (2.72)

175

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

4h growing stocks 0.30 0.88 3.09

(1.08) (4.61)

4i falling stocks 0.11 0.51 2.65

(0.36) (2.43)

4j low default 0.19

0.42 2.25

(0.71)

(2.65)

4k middle 50% 0.28

0.73 1.65

(0.98)

(2.48)

5 all 0.30 0.06 0.90 0.76 4.14

(1.16) (1.64) (9.26) (3.76)

5a high vol -1.8 0.06 1.57 -0.31 7.58

(-1.83) (0.50) (3.42) (-0.30)

5b medium vol 1.23 0.11 0.86 1.16 4.62

(2.71) (1.65) (4.86) (3.02)

5c low vol 0.06 0.02 0.80 0.68 3.16

(0.21) (0.54) (8.71) (3.58)

5d NASDAQ 0.51 0.22 1.30 1.40 3.20

(1.57) (3.05) (10.33) (4.44)

5e non-NASDAQ 0.16 0.02 0.74 0.60 4.23

(0.65) (0.74) (8.18) (3.83)

176

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

5f contraction 0.11 0.08 1.01 1.21 4.96

(0.14) (0.71) (2.72) (2.16)

5g expansion 0.33 0.05 0.88 0.69 4.02

(1.28) (1.42) (9.25) (3.10)

5h growing stocks 0.37 0.00 0.74 0.89 4.46

(1.41) (0.00) (7.86) (4.34)

5i falling stocks 0.09 0.07 1.13 0.66 4.01

(0.33) (1.60) (11.83) (2.87)

5j low default 0.12 0.08 0.37

0.64 3.54

(0.43) (1.93) (4.19)

(3.27)

5k middle 50% 0.25 0.10 0.90

1.01 3.32

(0.94) (2.16) (8.32)

(3.03)

6 all 0.23 0.07 0.90 0.01 0.01 -0.04 0.53 5.84

(0.97) (1.93) (11.44) (3.63) (0.55) (-8.94) (3.61)

6a high vol -1.71 0.05 1.15 -0.01 -0.13 -0.05 0.09 11.43

(-1.86) (0.37) (3.42) (-0.92) (-1.16) (-3.00) (0.16)

6b medium vol 1 0.12 0.91 0.01 0.05 -0.04 0.74 6.38

(2.49) (1.75) (6.32) (3.12) (1.22) (-5.45) (2.74)

6c low vol 0.07 0.04 0.85 0.01 0.02 -0.03 0.48 4.4

(0.28) (1.00) (9.85) (4.81) (0.55) (-7.71) (2.77)

177

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝑅 2

6d NASDAQ 0.48 0.22 1.27 0.02 0.00 -0.05 1.04 5.79

(1.64) (3.13) (10.33) (4.67) (0.07) (-7.45) (3.45)

6e non-NASDAQ 0.05 0.03 0.75 0.01 0.02 -0.03 0.43 6.32

(0.21) (0.83) (10.19) (2.56) (0.79) (-6.97) (3.07)

6f contraction -0.02 0.10 0.97 0.00 -0.14 -0.05 1.55 7.45

(-0.02) (0.84) (4.46) (0.14) (-1.59) (-3.64) (3.01)

6g expansion 0.26 0.06 0.89 0.01 0.04 -0.03 0.39 5.6

(1.13) (1.66) (10.54) (5.41) (1.39) (-8.32) (2.57)

6h growing stocks 0.26 0.00 0.80 0.01 -0.04 -0.02 0.69 5.98

(1.07) (0.03) (9.44) (6.10) (-1.42) (-3.27) (4.54)

6i falling stocks -0.02 0.11 1.00 0.00 0.06 -0.05 0.43 6.05

(-0.08) (2.50) (12.24) (0.97) (1.64) (-11.07) (2.13)

6j low default 0.07 0.06 0.37 0.01 -0.02 -0.05

0.63 5.99

(0.25) (1.48) (4.51) (1.79) (-0.62) (-9.55)

(3.47)

6k middle 50% 0.11 0.10 0.92 0.01 -0.02 -0.04

0.98 5.30

(0.50) (2.44) (10.41) (3.02) -(0.60) -(8.05)

(3.69)

Source: author’s calculations

178

Table 19: Fama-Macbeth cross-sectional regressions with mean-reverting volatility – return persistence

The table reports results from Fama-Macbeth cross-sectional regressions using lagged values of the mean-reverting level of volatility (‘𝑚−𝑙’) and its natural logarithm. If the mean-reverting level allows construction of tradable strategies, we would like the predictive performance of the variable to remain stable in order to reduce the costs of portfolio

rebalancing. The lag used in the test is indicated in column ‘Lag (𝑙)’. Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). ‘𝑅

2’ reports the

averaged R-squared statistics from the cross-sectional regressions.

The results suggest that 𝑚 and ln𝑚 are significant predictors of the cross-section of returns for lags up to 6 months. The regression coefficient remains statistically significant, but its value declines with the increase of the lag.

# Lag (𝑙) 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚−𝑙 ln𝑚−𝑙 𝑅 2

1 𝑙 = 1 0.22

0.03

3.43

(0.87)

(1.78)

𝑙 = 2 0.28

0.02

3.48

(1.11)

(1.16)

𝑙 = 3 0.29

0.02

3.52

(1.16)

(0.92)

𝑙 = 6 0.28

0.01

3.59

(1.15)

(0.72)

2 𝑙 = 1 0.27 0.04 0.62

0.05

4.81

(1.10) (1.17) (6.88)

(2.95)

𝑙 = 2 0.32 0.03 0.60

0.04

4.86

(1.31) (0.86) (6.79)

(2.18)

𝑙 = 3 0.34 0.02 0.60

0.03

4.92

(1.37) (0.73) (6.70)

(1.80)

179

# Lag (𝑙) 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚−𝑙 ln𝑚−𝑙 𝑅 2

𝑙 = 6 0.33 0.02 0.59

0.03

5.01

(1.36) (0.73) (6.53)

(1.56)

3 𝑙 = 1 0.26 0.02 0.64 0.01 -0.07 -0.03 0.05

6.84

(1.22) (0.57) (8.68) (3.61) -(1.97) -(6.21) (4.11)

𝑙 = 2 0.31 0.01 0.63 0.01 -0.05 -0.03 0.03

6.91

(1.40) (0.31) (8.62) (3.69) -(1.58) -(6.33) (2.74)

𝑙 = 3 0.31 0.01 0.63 0.01 -0.05 -0.03 0.03

6.99

(1.42) (0.22) (8.49) (3.80) -(1.36) -(6.50) (2.18)

𝑙 = 6 0.30 0.01 0.62 0.01 -0.05 -0.03 0.02

7.11

(1.40) (0.21) (8.25) (3.98) -(1.50) -(6.61) (1.94)

4 𝑙 = 1 0.13

0.39 3.46

(0.56)

(2.00)

𝑙 = 2 0.20

0.28 3.50

(0.85)

(1.44)

𝑙 = 3 0.22

0.23 3.54

(0.94)

(1.17)

𝑙 = 6 0.24

0.18 3.62

(1.01)

(0.87)

180

# Lag (𝑙) 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚−𝑙 ln𝑚−𝑙 𝑅 2

5 𝑙 = 1 0.16 0.07 0.64

0.68 4.88

(0.68) (1.96) (7.11)

(3.27)

𝑙 = 2 0.23 0.05 0.62

0.54 4.93

(0.98) (1.53) (6.96)

(2.55)

𝑙 = 3 0.25 0.05 0.62

0.47 4.98

(1.08) (1.34) (6.85)

(2.17)

𝑙 = 6 0.27 0.04 0.60

0.39 5.07

(1.17) (1.12) (6.56)

(1.75)

6 𝑙 = 1 0.17 0.05 0.67 0.01 -0.09 -0.03

0.76 6.88

(0.81) (1.52) (8.96) (3.63) -(2.52) -(6.19)

(5.05)

𝑙 = 2 0.23 0.04 0.65 0.01 -0.07 -0.03

0.56 6.95

(1.10) (1.08) (8.83) (3.70) -(2.07) -(6.32)

(3.69)

𝑙 = 3 0.24 0.03 0.65 0.01 -0.06 -0.03

0.47 7.02

(1.16) (0.88) (8.68) (3.80) -(1.77) -(6.48)

(3.00)

𝑙 = 6 0.26 0.02 0.63 0.01 -0.06 -0.03

0.38 7.15

(1.24) (0.68) (8.32) (4.00) -(1.79) -(6.60)

(2.42)

Source: author’s calculations

181

4.5. Further tests of robustness

4.5.1. Was there an omitted factor?

The evidence presented thus far suggests that the mean-reverting level of idiosyncratic

volatility is a significant predictor of the cross-section of returns. Nevertheless, as discussed

previously, one measures idiosyncratic risk with respect to some specific factor model – in

our case, the one of Fama–French–Carhart. It may happen that idiosyncratic volatility serves

only as a proxy for the loading on some other, omitted factor. For example, we found that the

premium on idiosyncratic risk was lower during an economic downturn. Then we may

hypothesise that our model lacks some economic factor, and the inclusion of the loading on

that factor in the cross-sectional regressions could make the idiosyncratic risk insignificant.

On the other hand, if the Merton model is correct, investors should dislike idiosyncratic risk

purely because of under-diversification and not because high-volatility stocks have greater

exposure to some systematic factor.

In this section we address that problem by estimating the principal omitted factor and

adding its loading as an explanatory variable in the cross-sectional regressions. If the principal

factor is the reason for the significance of the idiosyncratic risk variable, then we expect that

the addition of the new loading will make the slope of idiosyncratic risk insignificant. Firstly,

we use statistical factor analysis135 in order to estimate the returns on the omitted factor(s).

We then use the loadings of the individual securities on that factor (or, possible, on a number

of factors) in our cross-sectional regressions.

More specifically, we use the heteroscedastic factor analysis of Jones (2001) in order

to extract a set of 𝐾 common factors underlying observed idiosyncratic returns (𝐼𝑅𝑒𝑡𝑖,𝑡). The

common factors are extracted from the full data set, comprising all available time series of

idiosyncratic innovations for the studied period. The number of factors can be selected using

the 𝑃𝐶𝑝1 and 𝑃𝐶𝑝2 criteria discussed by Bai and Ng (2002). Then, factor loadings are

estimated as the coefficients of ordinary least squares (OLS) regressions of monthly returns

on each stock on the set of factors identified at the preceding stage of the analysis.

Let 𝑅𝑛 denote the 𝑛 × 𝑇 matrix of observed idiosyncratic returns (𝐼𝑅𝑒𝑡𝑖,𝑡), 𝐻 - the

135 For a review refer to Ch.4 in Connor et al. (2010)

182

matrix of (unobservable) factor realisations, 𝐵𝑛 - the matrix of factor loading, and 𝐸𝑛 -

residual returns. These residual returns could be viewed as the remaining idiosyncratic

innovations after accounting for the omitted factors of idiosyncratic returns. Then the model

of idiosyncratic returns is assumed to be in the form:

𝑅𝑛 = 𝐵𝑛𝐻 + 𝐸𝑛.

Let 𝐹 ≡ 𝑀−1/2𝐻 stand for the matrix of rotated factor realisations, introduced to simplify

notation, and let 𝐷 denote the diagonal matrix of average idiosyncratic variances. Jones

(2001) proves that the average variance (1/𝑛)𝑅𝑛′𝑅𝑛 converges to 𝐹′𝐹 + 𝐷 and could be

estimated using Jöreskog’s procedure, i.e.:

1. Compute 𝐶 = (1/𝑛)𝑅𝑛′𝑅𝑛;

2. Guess an initial 𝐷, e.g. 𝐷 = 0.5𝐶;

3. Obtain the 𝐾 largest eigenvalues of 𝐷−1/2𝐶𝐷−1/2 and create a diagonal matrix 𝛬

having the largest eigenvalues along its main diagonal; then create a matrix 𝑉 of their

corresponding eigenvectors;

4. Estimate the factor matrix as 𝐹 = 𝐷1/2𝑉(𝛬 − 𝐼)1/2;

5. Compute a new estimate of 𝐷 = 𝐶 − 𝐹′𝐹 and return to 3 until the algorithm

converged;

6. Estimate factor loading using OLS regression of observed excess returns on estimated

factors and obtain residual idiosyncratic errors 𝜖.

An important issue in factor analysis is the choice of an appropriate number of factors.

Remembering that we are analysing residuals (idiosyncratic returns) obtained from a factor

model that already has three factors, we perform this test assuming only one omitted common

factor. That factor would be the one with the highest contribution in explaining the common

pattern of idiosyncratic residuals. The methodology could be readily expanded further with

the inclusion of more than one factor.136

Summary statistics for the recovered factor and the loading of idiosyncratic returns on

the extracted factor (𝑓ℎ𝑓𝑎) are given in Table 20. Factor loadings (𝛽ℎ𝑓𝑎) are estimated using

rolling regressions with monthly idiosyncratic returns (𝐼𝑅𝑒𝑡𝑖,𝑡) from the last 24 to 60 months

preceding the current month, as available (i.e. from (𝑡 − 1) until (𝑡 − 𝑘), 𝑘 = 24…60).

136 There is no commonly accepted approach to selecting the number of factors. In general,

such approaches aim to measure whether an additional factor has additional explanatory

power (e.g. Kandel and Stambaugh (1989), Connor and Korajczyk, and Bai and Ng). For

further details see Kandel and Stambaugh (1989); Connor and Korajczyk (1993); Bai and Ng

(2002)

183

Table 20: Summary statistics of the first common factor of idiosyncratic returns and

loadings on that factor, 07/1982-03/2013

The table reports summary statistics of the latent (statistical) factor (𝑓ℎ𝑓𝑎) estimated using the heteroscedastic

factor analysis of Jones (2001) and the loading (𝛽ℎ𝑓𝑎) on the statistical factor. ‘Mean (EW)’ and ‘Mean (VW)’

are equally-weighted and value-weighted averages, ‘Std.dev’ stands for the standard deviation of the sample.,

and ‘Median’, ‘Q1’and ‘Q3’ stand for the second, first and third quartiles of the sample. ‘Skew’ is the skewness

coefficient of the sample, and ‘Count’ is the number of records in the sample.

Mean (EW) Mean (VW) Std.dev Median Q1 Q3 Skew Count

𝑓ℎ𝑓𝑎 -0.16 - 5.05 -0.06 -3.25 3.14 0.00 393 𝛽ℎ𝑓𝑎 0.05 0.03 0.48 -0.01 -0.23 0.26 0.79 713,860

Source: author’s calculations

Table 21 reports the results from Fama–Macbeth cross-sectional regressions involving

the loading (𝛽ℎ𝑓𝑎) on the statistical factor (𝑓ℎ𝑓𝑎) explaining asset returns. We find that

although the coefficient of the loading is always negative, it is not significant at conventional

levels in any of the specifications. On the other hand, the coefficients for both the

mean-reverting level of volatility (𝑚) and its natural logarithm (ln𝑚) are significant and

positive, consistent with the results from the previous section. The results support the

conclusion that the mean-reverting level of volatility is a significant predictor of the

cross-section of returns and its significance is not attributable to correlation with the loading

on some unknown factor that affects idiosyncratic returns but was omitted from the Fama–

French–Carhart specification.

184

Table 21: Fama–Macbeth cross-sectional regressions with loading on the principal factor affecting idiosyncratic returns, 07/1982 –

03/2013

The table reports the results from Fama-Macbeth cross-sectional regressions involving the loading on the statistical factor (𝛽ℎ𝑓𝑎).

Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). ‘𝑅

2’ reports the

averaged R-squared statistics from the cross-sectional regressions. ‘𝑚’ and ‘ln𝑚’ denote the mean-reverting level of volatility and its natural logarithm. The table shows that the statistical factor is insignificant in explaining the cross-section beyond the contribution of the volatility forecast, and albeit its value remains negative

and reasonably stable, it is found to be statistically insignificant.

# 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝑚 ln𝑚 𝛽ℎ𝑓𝑎 𝑅 2

1 0.91

-0.06 2.73

(2.55)

(-0.37)

2 0.82 -0.04 0.87

-0.09 4.24

(2.53) (-1.14) (8.21)

(-0.53)

3 0.54 0.03 0.88 0.01 0.06 -0.04

-0.18 6.01

(1.93) (0.92) (10.65) (2.87) (2.60) (-9.58)

(-1.14)

4 0.47 0.06 0.90 0.01 0.03 -0.04 0.02

-0.17 6.42

(1.79) (1.76) (11.41) (2.59) (1.29) (-9.66) (2.22)

(-1.04)

5 0.39 0.08 0.91 0.01 0.03 -0.04

0.44 -0.18 6.47

(1.55) (2.33) (11.69) (2.59) (1.11) (-9.67)

(3.12) (-1.12)

Source: author’s calculations

185

4.5.2. Evidence from daily data

We have previously pointed out that the Capital Asset Pricing Model (CAPM) and its

extension – Merton’s model – are not linked to any specific time frequency. Consequently,

results that hold on daily data should also hold at lower frequencies, e.g. weekly, monthly,

quarterly, or annually. The opposite is also true: if the mean-reverting level of volatility is

explaining the cross-section of returns at monthly frequency, then a similar effect should be

observed at daily frequency.

For the purposes of the test, the change of frequency, however, is not a trivial task.

Models that are suited for monthly frequency by accommodating the shorter time-series and

the higher noise in the observed proxies of the underlying, unobservable idiosyncratic

volatility, could be unsuitable for higher frequencies. For example, the previously referenced

study of Hansen and Lunde (2005) documents the strong empirical performance of the simple

GARCH(1,1) for one-day forecasts. On the other hand, the simple model also has certain

drawbacks, which invited the development of a host of GARCH extensions that accommodate

phenomena observed in various financial time series.137 For example, the observation that

volatility reacts differently to positive and negative shocks invited the development of the

Exponential GARCH model138. The idea of different volatility regimes led to the development

of the Regime-Switching ARCH model, where parameter values depend on the volatility

regime.139 The observation of slow decay of the auto-covariance function of empirical

volatility processes prompted the development of the Fractionally Integrated GARCH

model.140

In our study we emphasise the role of the mean-reverting level of idiosyncratic

volatility, to which current volatility is expected to converge gradually over time. We also

found that short-horizon forecasts (one-month-ahead) were insignificant predictors of the

cross-section of returns, which in the present context also suggests that we should aim to

employ a model that has long memory in volatility and which will not be unduly affected by

short-term volatility outbursts. Such considerations could suggest the use of the FIGARCH or

RS-(G)ARCH models, both of which offer interesting approaches for capturing the

137 see Pagan and Schwert (1990) for a comparison of competing volatility forecasting

techniques 138 see Nelson (1991) 139 see Hamilton and Susmel (1994) 140 Baillie et al. (1996)

186

mean-reverting level of volatility. Structurally, however, we think that the Component

GARCH (CGARCH) model of Lee and Engle (1999) offers a more direct and transparent

route to capturing the mean-reverting volatility while allowing the mean-reverting level to

evolve over time. When we used monthly data, such specification was not feasible due to

scarcity of data, but at the daily frequency it offers a more direct implementation of our

approach.

Lee and Engle propose to model volatility as the sum of two components: a permanent

one and a transitory one. The permanent component has persistence close to unity, whereas

the transient component has an expected value of zero and decays more quickly. More

specifically, we employ Component GARCH(1,1), which has the following specification:

𝜎𝑡 2 = 𝑞𝑡 + 𝛼(𝜀𝑡−1

2 − 𝑞𝑡−1) + 𝛽(𝜎𝑡−1 2 − 𝑞𝑡−1),

𝑞𝑡 = 𝜔 + 𝜌𝑞𝑡−1 + 𝜑(𝜀𝑡−1 2 − 𝜎𝑡−1

2 ),

where 𝑞𝑡 is the permanent component of volatility, while (𝜎𝑡 2 − 𝑞𝑡) is the transitory

component of volatility. The 𝑛-step expected values from the CGARCH(1,1) are found to be

as follows:

𝔼𝑡−1(𝜎𝑡+𝑛 2 ) = 𝔼𝑡−1(𝑞𝑡+𝑛) + (𝛼 + 𝛽)

𝑛(𝜎𝑡 2 − 𝑞𝑡),

and

𝔼𝑡−1(𝑞𝑡+𝑛) = 1 − 𝜌𝑛

1 − 𝜌 𝜔 + 𝜌𝑛𝑞𝑡,

and when 𝑛 → ∞, the unconditional expected values became 𝔼𝑡−1(𝜎𝑡+𝑛 2 ) = 𝔼𝑡−1(𝑞𝑡+𝑛) =

𝜔

1−𝜌 .

These formulae allow us to test our results from the previous section. Firstly, we can

calculate the forecasted volatility for one month ahead, which we can take to be 21 days, so

that the one-month forecast would be 𝔼𝑡−1(𝜎𝑡+21 2 ), which would be an equivalent of the

volatility forecasts from the previous section. We can also calculate the unconditional

expectation of the permanent component, towards which volatility is expected to revert as

𝑛 → ∞, i.e. 𝔼𝑡−1(𝑞𝑡+∞) = 𝜔 (1 − 𝜌)⁄ . We could then use these two volatility forecasts to

analyse the cross-section of monthly141 returns. Based on our results thus far we expect that

𝔼𝑡−1(𝜎𝑡+21 2 ) would not be a significant predictor of the cross-section of returns, while

𝔼𝑡−1(𝑞𝑡+∞) would be significant.

141 Of course nothing prevents the use of the same approach to test its performance in

explaining other frequencies, e.g. daily or weekly returns. At any rate, the use of daily returns

should also take into account the possible impact of market microstructure effects.

187

More specifically, we proceed as follows. We select a random sample of 2100

securities from the full sample of available securities.142 For each security before the start of

each month we calculate daily idiosyncratic returns from the Fama–French–Carhart model

using the last five years of data, as available, but not less than 250 returns.143 We estimate a

CGARCH(1,1) model using the available history from the rolling window, and forecast the

expected volatility at the end of the month (𝔼𝑡−1(𝜎𝑡+21 2 )), as well as the unconditional

expected value of the mean level (𝔼𝑡−1(𝑞𝑡+∞)). 144

Table 22: Summary statistics of expected volatilities from daily data

The table reports summary statistics for estimates of next-month volatility (‘𝔼𝑡−1(𝜎𝑡+21 2 )’) and of unconditional

volatility (‘𝔼𝑡−1(𝑞𝑡+∞)’) estimated directly from daily returns using Component GARCH(1,1). Estimates are calculated using expanding window design for a random sample of 2100 securities. Volatilities are scaled to

monthly frequency. The table reports values for equally-weighted and value-weighted averages (‘Mean (EW)’

and ‘Mean (VW)’), standard deviation (‘St.dev.’), median (‘Median’), first and third quartiles (‘Q1’ and ‘Q3’),

skewness coefficient (‘Skewness’) and number of observations (‘Obs’).

Variables Mean (EW) Mean (VW) St.dev. Median Q1 Q3 Skewness Obs

𝔼𝑡−1(𝜎𝑡+21 2 ) 14.41 8.11 8.83 12.06 8.32 17.99 2.13 278,301

ln 𝔼𝑡−1(𝜎𝑡+21 2 ) 2.52 1.99 0.54 2.49 2.12 2.89 0.23 278,301

𝔼𝑡−1(𝑞𝑡+∞) 18.41 9.68 16.66 13.73 8.93 22.10 4.09 278,301

ln 𝔼𝑡−1(𝑞𝑡+∞) 2.66 2.11 0.68 2.62 2.19 3.10 0.29 278,301

Source: author’s calculations

Our results are presented in Table 23. Consistent with the results from the preceding

sections, we find that expected volatility for the next month, 𝔼𝑡−1(𝜎𝑡+21 2 ), is not a significant

predictor of the cross-section of returns. On the other hand, the mean-reverting level

𝔼𝑡−1(𝑞𝑡+∞) is a significant predictor. The slopes of 𝔼𝑡−1(𝑞𝑡+∞) and ln 𝔼𝑡−1(𝑞𝑡+∞) are

somewhat lower than those estimated from monthly ARMA(1,1), which were reported

142 The use of sampling is motivated by the high computational burden of calculating

expected volatilities while preventing look-ahead bias. A small sample results in a lower

number securities available in each month, and hence high errors of the cross-sectional

coefficients, which in turn translates into higher likelihood of failing to reject the null

hypothesis that the coefficients of the cross-sectional regressions are insignificant. 143 We excluded daily returns below -25% or above 300% as possible data errors, as

suggested in Ince and Porter (2006) 144 For easier comparison with the coefficients from the preceding section, the daily

volatilities are scaled to monthly frequency by multiplying by a constant factor of √21.

188

previously in this thesis. Part of the reason could be the scaling from daily to monthly

frequency, which is evident in the higher average expected idiosyncratic volatilities reported

in Table 6 (p. 117), compared with the summary statistics of the series obtained directly from

daily data, reported in Table 22. Nevertheless, the magnitude of the difference is such that it

could not be explained solely by scaling. One plausible explanation could be the lower

persistence of daily volatilities compared to monthly volatilities, which may somewhat dilute

the predictive performance of the mean-reverting level estimated from daily data. On the

other hand, the predictive performance of the mean-reverting volatility remains robust in the

alternative model specifications, which confirmed its relevance in explaining the cross-section

of returns.

189

Table 23: Cross-sectional Fama–Macbeth regressions with volatilities calculated from daily data using Component GARCH(1,1)

The table reports results from Fama-Macbeth cross-sectional regressions using volatility forecasts obtained from Component GARCH(1,1) model fitted on expanding window

of daily returns. ‘𝔼𝑡−1(𝜎𝑡+21 2 )’ is the estimate of next-month volatility, and ‘𝔼𝑡−1(𝑞𝑡+∞)’ denotes the corresponding unconditional volatility for the model.

Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), return in the previous month (‘𝑅𝑒𝑡𝑡−1’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). ‘𝑅

2’ reports the

averaged R-squared statistics from the cross-sectional regressions. ‘𝑚’ and ‘ln𝑚’ denote the mean-reverting level of volatility and its natural logarithm. The results are consistent with those obtained from monthly data. One-month forecasts are not statistically significant once momentum (‘𝑅𝑒𝑡(−7,−2)’), liquidity (‘𝑅𝑜𝑙𝑙’) and return reversals (‘𝑅𝑒𝑡𝑡−1’) are added to the model. On the other hand, unconditional volatility remains a significant predictor of the cross-section. However, the estimate of the coefficient for 𝔼𝑡−1(𝑞𝑡+∞) is markedly below the corresponding value when forecasts from ARMA(1,1) are used, highlighting the room for improvement of the specification.

# 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝔼𝑡−1(𝜎𝑡+212 ) ln𝔼𝑡−1(𝜎𝑡+212 ) 𝔼𝑡−1(𝑞𝑡+∞) ln𝔼𝑡−1(𝑞𝑡+∞) 𝑅2

1 0.47

0.03

2.83

(1.63)

(3.02)

2 0.58 -0.01 0.90

0.03

4.35

(2.07) (-0.21) (7.67)

(2.24)

3 0.49 0.01 0.92 0.01 0.07 -0.03 0.00

6.11

(0.25) (0.04) (0.08) (0.00) (0.03) (-0.00) (0.01)

4 0.34

0.64

2.76

(1.27)

(3.04)

5 0.48 0.00 0.90

0.54

4.34

(1.88) (0.05) (7.67)

(2.36)

6 0.47 0.01 0.91 0.01 0.06 -0.03

0.11

6.15

(1.91) (0.34) (9.04) (4.06) (2.28) (-7.90)

(0.56)

7 0.60

0.02

2.37

(1.86)

(3.58)

190

# 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑅𝑒𝑡𝑡−1 𝔼𝑡−1(𝜎𝑡+212 ) ln𝔼𝑡−1(𝜎𝑡+212 ) 𝔼𝑡−1(𝑞𝑡+∞) ln𝔼𝑡−1(𝑞𝑡+∞) 𝑅2

8 0.60 -0.02 0.92

0.02

3.96

(1.96) (-0.49) (7.37)

(4.49)

9 0.42 0.03 0.94 0.01 0.05 -0.03

0.01

5.99

(1.58) (0.84) (9.34) (3.86) (1.77) (-7.77)

(3.44)

10 0.44

0.47 2.43

(1.47)

(3.77)

11 0.47 0.01 0.94

0.52 3.99

(1.64) (0.27) (7.73)

(4.48)

12 0.36 0.05 0.96 0.01 0.05 -0.04

0.30 5.99

(1.38) (1.21) (9.58) (3.87) (1.84) (-7.83)

(3.70)

Source: author’s calculations

191

4.5.3. Portfolios as assets

The preceding section utilised the Fama–MacBeth methodology using individual

securities as assets. The rationale for that choice stems from the superior efficiency compared

to portfolios145 and is consistent with the approaches of Fama and French (1992) and Fu

(2009). From a practical perspective, however, it is also useful to examine the role of

idiosyncratic volatility using suitably constructed portfolios. This allows us to drill down the

results and identify situations where the mean-reverting volatility is likely to be useful and

situations where it may not be appropriate.

We form double-sorted portfolios based on capitalisation and the expected

mean-reverting level of volatility. We first form ten decile portfolios based on market

capitalisation, and then split each of these into five quintile portfolios based on the

mean-reverting level of volatility. For each portfolio we calculate the simple (equal-weighted)

average return for the next one month, after which the double-sorting procedure is repeated.

In order to prevent the numerous small-capitalisation stocks listed at NASDAQ from asserting

undue influence on our estimates we employ breakpoints calculated only from NYSE

securities. The choice of the number of portfolios is driven by pragmatic considerations: the

significant correlation between the explanatory variables (beta, size, liquidity, and

idiosyncratic volatility) necessitates a higher number of portfolios, so that using only

twenty-five portfolios appears unjustified. On the other hand, using decile NYSE breakpoints

for the mean-reverting level of volatility results into too few (fewer than ten) securities in

some portfolios in the earlier months of the sample. Therefore, we opt for using fifty

portfolios for this test.

145 Ang et al. (2010)

192

Table 24: Fama–Macbeth regressions with quintile portfolios as assets

The table reports results from Fama-Macbeth cross-sectional regressions calculated on a total of 50 double-sorted portfolios formed each month by capitalisation (ten decile

portfolios) and idiosyncratic risk (five quintile portfolios) using breakpoints for NYSE stocks only. For each portfolio we calculate the simple (equal-weighted) average return

for the next month, after which the double-sorting procedure is repeated.

Control variables are beta with market (‘𝐵𝑒𝑡𝑎’), natural logarithms of market capitalisation (‘ln (𝐶𝑎𝑝)’) and book-to-market value (‘ln (𝐵/𝑀)’), momentum of returns measured as cumulative return from (t-7) until (t-1) (‘𝑅𝑒𝑡(−7,−2)’), liquidity measured in terms of Roll’s estimator (‘𝑅𝑜𝑙𝑙’). Control variables are averaged by portfolios. Tested variables are the mean-reverting level of volatility (‘𝑚’) and its natural logarithm. ‘𝑅2’ reports the averaged R-squared statistics from the cross-sectional regressions. Column ‘Filter’ indicates the subsample used for estimating the cross-sectional regressions: ‘all’ in case the entire sample is used; ‘UP>10’ indicates that cross-sectional

regressions are estimated using only stocks with unadjusted price above USD 10; ‘NASDAQ’ and ‘non NASDAQ’ indicate that the sample includes only securities with

principal listing on NASDAQ or other (NYSE or Amex) stock exchange.

The results suggest that 𝑚 and ln𝑚 are significant predictors of the cross-section of returns. However, the results for non-NASDAQ stocks do not support significance of idiosyncratic risk, consistent with the underlying economic model that predicts that premium for more widely-followed securities should be lower or insignificant; however, the

small number of portfolios invites further analysis of those results.

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑚 ln𝑚 𝑅2

1 All 1.35 0.02 0.52

37.39

(2.86) (0.48) (3.11)

2 All 1.10 0.04 0.66 0.00

40.95

(2.47) (1.13) (4.88) (0.89)

3 All 0.47

0.04

28.98

(1.02)

(2.16)

4 All -0.05

0.66 28.95

(-0.10)

(2.74)

5 All 0.77 0.06 0.67

0.04

41.93

(1.71) (1.76) (4.65)

(2.34)

6 All 0.15 0.08 0.66

0.72 41.35

(0.34) (1.93) (4.39)

(3.31)

193

# Filter 𝐵𝑒𝑡𝑎 ln (𝐶𝑎𝑝) ln (𝐵/𝑀) 𝑅𝑒𝑡(−7,−2) 𝑅𝑜𝑙𝑙 𝑚 ln𝑚 𝑅2

7 All 0.55 0.07 0.71 0.00

0.04

44.81

(1.30) (2.15) (5.52) (0.40)

(2.67)

8 All 0.00 0.09 0.74 0.00

0.69 44.36

(0.01) (2.42) (5.67) (0.66)

(3.33)

9 UP>10 0.28 0.06 0.63 0.01

0.04

42.27

(0.67) (1.65) (5.02) (1.14)

(2.38)

10 UP>10 -0.11 0.08 0.70 0.01

0.58 41.96

(-0.27) (1.97) (5.47) (1.36)

(3.00)

11 non-NASDAQ 0.61 0.04 0.78 0.01

0.03

38.30

(1.38) (0.94) (6.13) (1.66)

(2.09)

12 non-NASDAQ 0.18 0.05 0.79 0.01

0.51 38.20

(0.42) (1.22) (6.23) (1.52)

(2.65)

13 All 0.61 0.07 0.66 0.00 0.02 0.04

47.25

(1.46) (2.20) (5.36) (0.83) (0.36) (1.63)

14 All 0.07 0.07 0.67 0.00 0.00

0.67 46.98

(0.18) (2.01) (5.55) (0.51) (-0.01)

(2.69)

15 UP>10 0.37 0.06 0.59 0.01 -0.01 0.04

44.59

(0.96) (1.60) (4.60) (1.57) (-0.16) (1.59)