Impact of idiosyncratic volatility on stock returns: A cross-sectional

study

Serguey Khovanskya,∗, Oleksandr Zhylyevskyyb

aGraduate School of Management, Clark University, 950 Main Street, Worcester, MA 01610bDepartment of Economics, Iowa State University, 460D Heady Hall, Ames, IA 50011

Abstract

The paper proposes a new approach to assess effects of stock-specific idiosyncratic volatility

on stock returns. In contrast to the popular two-pass regression method, it relies on a novel

GMM-type estimation procedure that utilizes only a single cross-section of return observa-

tions. The approach is illustrated empirically by applying it to weekly U.S. stock return

data within two separate months in 2008: January and October. The results suggest a

negative idiosyncratic volatility premium, and indicate an increase in average cross-sectional

idiosyncratic volatility by one-half between January and October. The approach provides a

full set of estimates for every examined return interval, and enables a decomposition of the

conditional expected stock return.

JEL classification: G12, C21

Keywords: Idiosyncratic volatility, Idiosyncratic volatility premium, Cross-section of stock

returns, Generalized Method of Moments

∗Corresponding author. Phone: 774-232-3903, fax: 508-793-8822.Email addresses: skk9e@virginia.edu (Serguey Khovansky), oz9a@iastate.edu (Oleksandr

Zhylyevskyy)

1. Introduction

The finance literature is presently witnessing a debate on idiosyncratic volatility “pre-

mium.” The models of Levy (1978), Merton (1987), Malkiel and Xu (2006), and Epstein

and Schneider (2008) predict a positive premium in the capital market equilibrium. How-

ever, research based on Kahneman and Tversky’s (1979) prospect theory suggests that the

premium can be negative (e.g., Bhootra and Hur, 2011). Additional explanations for why

the premium can be negative have been proposed by Guo and Savickas (2010) and Chabi-Yo

(2011).

Empirical evidence on the premium is also contradictory. Fu (2009) and Huang et al.

(2010) document a statistically significant positive premium. In contrast, Ang et al. (2006,

2009) and Jiang et al. (2009) find the premium to be negative.1 To consistently estimate

idiosyncratic volatility effects, the existing empirical studies need to employ long time-series

of stock return data (Shanken, 1992). As a result, their estimates may be representative more

of historical rather than current market conditions, especially if the underlying parameters

such as individual stock betas and idiosyncratic volatilities change over time. However,

practical business applications may require a more up-to-date characterization of the market.

This paper proposes a new estimation approach to quantify the effects of stock-specific

idiosyncratic volatility on stock returns. More specifically, we employ a novel econometric

procedure that allows us to obtain consistent estimates, using only a single cross-section of

return data. Having a long time-series of data is not required. In principle, the procedure

could be implemented using returns computed over an interval of an arbitrary duration. As

an empirical illustration, we focus on weekly returns. However, the approach also can be

applied to daily and intra-daily returns, if microstructure issues are taken into account.

1The issue of whether idiosyncratic volatility has forecasting power is debatable as well. Goyal and Santa-

Clara (2003) document a positive relationship between equal-weighted average stock variance and future

market return. However, Bali et al. (2005) show that the relationship does not hold for value-weighted

variance. Also, there is no consensus about the time-series behavior of idiosyncratic volatility. Campbell et

al. (2001) report a steady increase in the volatility after 1962. However, Brandt et al. (2010) argue that

this finding is only indicative of an episodic phenomenon. It should be noted that we do not focus on the

issues of the forecasting power and time-series behavior of idiosyncratic volatility in this paper.

2

A parametric model underlying the return data is an important component of the ap-

proach. We consider a continuous-time model comprising a well-diversified market portfolio

index and a cross-section of individual stocks. The index follows a geometric Brownian mo-

tion and is affected by a source of market risk. Individual stocks also follow a geometric

Brownian motion and depend on this same source of market risk (i.e., it is a common risk

shared by all stocks). In addition, they are affected by stock-specific idiosyncratic risks. We

do not take a stance on whether idiosyncratic volatility should command a premium in the

capital market equilibrium, but rather we allow for a potential effect of a stock’s idiosyncratic

volatility on the stock’s drift term and propose to estimate this effect, if any, from the data.

In addition, we compute average cross-sectional idiosyncratic volatility.

The approach is empirically illustrated using U.S. stock data from the Center for Research

in Security Prices (CRSP). We focus on weekly returns in January and October 2008, which

are chosen because it is potentially interesting to compare parameter estimates and accuracy

of estimation on data from a relatively less volatile trading period (January 2008), and a

period that featured a major market turmoil and financial crisis (October 2008). In total,

we analyze 15 return intervals in January, and 18 intervals in October. Some of the intervals

overlap. In line with the findings of Ang et al. (2006, 2009) and Jiang et al. (2009),

the idiosyncratic volatility premium is estimated to be negative and statistically significant.

Also, by decomposing the conditional expected stock return with respect to the risk source

(market vs. idiosyncratic), we find that the impact of the premium cannot be ignored. For

example, the gross return for the week of January 2-9 is 0.9486; however, it would be 0.9994

if idiosyncratic volatility commanded no premium. We also obtain estimates of the average

cross-sectional idiosyncratic volatility, which is an important stock market characteristic

(Campbell et al., 2001; Goyal and Santa-Clara, 2003). These estimates suggest that the

turbulent October 2008 was marked by an increase in the idiosyncratic volatility by one-half

of its January 2008 level, from about 0.16 in January to 0.24 in October in monthly terms.

The estimation is implemented using a novel Generalized Method of Moments (GMM)-

type econometric procedure suggested by Andrews (2003, 2005). Broadly speaking, the main

advantage of the procedure is that it enables consistent parameter estimation and simple

statistical inference when cross-sectional observations are dependent, due to a common shock.

3

In the case of cross-sectional stock returns, the source of the common shock can be market

risk, which, to a varying degree, affects all stocks. Standard estimation methods cannot

deliver consistent estimates in this setting, because the common shock induces “too much”

data dependence among the observations, in which case, conventional laws of large numbers

fail to apply.

To date, empirical studies of the idiosyncratic volatility premium have typically employed

a version of the two-pass regression method of Fama and MacBeth (1973).2 Although actual

implementations may vary, the main idea is that in the first pass, time-series regressions are

run to estimate each stock’s idiosyncratic volatility; while in the second pass, the first-pass

estimates are used in a cross-sectional regression to compute the idiosyncratic volatility pre-

mium. Despite its intuitive appeal, the two-pass method has several well-known limitations.

For example, it delivers consistent estimates only when time-series length (rather than the

number of stocks) grows infinitely large (Shanken, 1992). Also, since the regressors in the

second pass are measured with error, the estimator is subject to an errors-in-variables prob-

lem (Miller and Scholes, 1972), which may induce an attenuation bias in the estimates (Kim,

1995). The statistical properties of the second-pass estimator are complex. As such, it is

not uncommon for these complexities to be ignored in practice, resulting in biased inference

(Shanken, 1992; Jagannathan and Wang, 1998). Moreover, accounting for the time-varying

nature of stock betas (Fama and French, 1997; Lewellen and Nagel, 2006; Ang and Chen,

2007) and idiosyncratic volatilities (Fu, 2009) is challenging and requires imposing additional

assumptions, which further complicate statistical inference.

The approach studied in this paper aims to address these limitations. In particular,

it delivers consistent estimates as the number of stocks (rather than the time-series data

length) grows infinitely large. Thus, it is not affected by the available time-series length

of the stock return data. Also, since it does not involve estimating individual stock betas

and idiosyncratic volatilities, it is not subject to the errors-in-variables problem arising in

the two-pass regression method, and avoids the need to impose strong assumptions about

their time-series behavior. Instead, it relies on a parametric model describing a financial

2Black et al. (1972), among others, contributed to the development of the two-pass methodology.

4

market setting, and on a distributional assumption regarding a cross-section of the betas

and volatilities. While the need to make the distributional assumption might be seen as

a potential limitation, practitioners can explore several alternative assumptions to check

the robustness of the estimates to a misspecification. Overall, we believe our approach will

be an attractive alternative to the two-pass regression method when researchers need to

characterize a stock market using only the most current, rather than historical information.

The remainder of this paper proceeds as follows. Section 2 specifies the financial market

model. Section 3 outlines the econometric approach. Section 4 describes the data used in

the empirical illustration. Section 5 discusses the empirical results. Section 6 concludes.

Selected analytical formulas are derived in the appendix.

2. Financial market model

We first specify a financial market model and discuss how it relates to the classical

finance framework. We then derive expressions for gross returns and specify distributional

assumptions that help implement a GMM-type econometric procedure outlined in Section 3.

2.1. Model setup

Financial investors trade in many risky assets in continuous time. One of the assets is a

well-diversified stock portfolio bearing only market risk. In what follows, this asset is referred

to as “the market index”. Its price at time t is denoted by Mt. All other risky assets are

individual stocks bearing the market risk and stock-specific idiosyncratic risks. We index the

stocks by i, with i = 1, 2, ..., and denote the price of a stock i at time t by Sit . In addition to

the market index and the stocks, there is a default-free bond that pays interest at a risk-free

rate r. The assumption of the constant risk-free interest rate is not restrictive, as we will

focus on return intervals of short duration (in the empirical illustration, we analyze weekly

returns).

The price dynamics of the market index follows a geometric Brownian motion and is

described by a stochastic differential equation

dMt/Mt = µmdt+ σmdWt, (1)

5

with a drift

µm = r + δσm, (2)

where Wt is a standard Brownian motion indicating a source of the market risk, σm > 0

is the market volatility, and δ is the market risk premium. As explained in Section 3, the

estimation procedure will not allow us to identify δ, because this parameter is differenced

out when stock returns are conditioned on the market index return. Conditioning on the

market index return is a critical step in the econometric procedure that ensures consistency

of the estimates.

The price dynamics of a stock i (i = 1, 2, ...) also follows a geometric Brownian motion

and is described by a stochastic differential equation

dSit/Sit = µidt+ βiσmdWt + σidZ

it , (3)

with a drift

µi = r + δβiσm + γσi, (4)

where Zit is a standard Brownian motion indicating a source of the idiosyncratic risk that only

affects stock i. Zit is independent of Wt and every Zj

t for j 6= i. Also, βi is the stock’s beta,3

σi > 0 is the stock’s idiosyncratic volatility, and γ is the idiosyncratic volatility premium;

γ is not sign-restricted and may be zero. Eq. (3) implies that Wt is a source of common

risk among the stocks because an innovation in Wt results in a shock that is shared by all

stocks (i.e., it is a common shock). Sensitivity of individual stock prices to the common

shock is allowed to vary. Thus, βi need not be equal to βj for j 6= i. Also, the stocks can

have different idiosyncratic volatilities (i.e., σi need not be equal to σj for j 6= i).

Classical finance models (e.g., Sharpe, 1964; Lintner, 1965) imply that idiosyncratic

volatility commands no equilibrium return premium. In that case, γ = 0 and the price

dynamics of our model would coincide with that of the ICAPM with a constant investment

opportunity set (Merton, 1973, Section 6).4 However, the current finance literature indi-

3Eqs. (1) and (3) imply that βi is the ratio of the covariance between the instantaneous returns on the

stock and the market index, βiσ2mdt, to the variance of the instantaneous return on the market index, σ2

mdt.4As noted earlier, our focus is on return intervals of relatively short duration. Thus, we do not consider

the case of the ICAPM in which the investment opportunity set is allowed to be time-varying.

6

cates that idiosyncratic volatility can be priced, although there is no consensus among the

researchers about the sign of the premium or the underlying pricing mechanism (see a review

in the introductory section). To account for a potential premium, we include a term “+γσi”

in the specification of the stock’s drift. If idiosyncratic volatility is, indeed, priced, we should

expect to estimate a statistically significant γ.

2.2. Model solution and distributional assumptions

We focus on a cross-section of returns for a time interval from date t = 0 to t = T (a

discussion of specific choices of the dates t = 0 and t = T used in the empirical illustration

is provided in Section 4). More specifically, the data inputs include the gross market index

return, MT/M0, and the gross returns on the stocks, S1T/S

10 , S

2T/S

20 , ... .

Applying Ito’s lemma to Eqs. (1) and (3), we derive expressions for MT/M0:

MT/M0 = exp[(µm − 0.5σ2

m

)T + σmWT

], (5)

and SiT/Si0:

SiT/Si0 = exp

[(µi − 0.5β2

i σ2m − 0.5σ2

i

)T + βiσmWT + σiZ

iT

], (6)

where WT and ZiT are independent and identically distributed (i.i.d.) as N (0, T ), and the

drifts µm and µi are given by Eqs. (2) and (4), respectively.

It is impossible to consistently estimate the individual stock betas β1, β2, ... and idiosyn-

cratic volatilities σ1, σ2, ... using only a cross-section of returns, because the number of param-

eters to estimate would grow with the sample size. Therefore, to implement the econometric

procedure, we specify the stock betas and idiosyncratic volatilities as random coefficients

drawn from a probability distribution (for a survey of the econometric literature on random-

coefficient models, see Hsiao, 2003, Chapter 6). Parameters of this distribution are estimated

simultaneously with other identifiable parameters of the financial market model.

As a practical matter, sensitivity of individual stock returns to the market and idiosyn-

cratic sources of risk should be finite, since infinite stock returns are not observed in the data.

Given this consideration, it may be preferable to model the distribution of βi and σi as hav-

ing finite support. In addition to being empirically relevant, the finite support requirement

helps ensure the existence of theoretical moments of the gross stock return. The support of

7

σi must be positive. A negative βi cannot be ruled out, however; and therefore, the support

of βi should not be sign-restricted. To facilitate an evaluation of the finite-sample properties

of the econometric procedure (see details on the Monte Carlo exercise in Section 3), it is also

desirable to have a distribution that would let us express moment restrictions analytically.

We have explored a number of alternative options and found that the uniform distribution

most closely meets these criteria. More specifically, we assume that

βi ∼ i.i.d. UNI [κβ, κβ + λβ] , (7)

where the lower and upper boundaries of the support are parameterized in terms of κβ and

λβ > 0, and

σi ∼ i.i.d. UNI [0, λσ] , (8)

where the upper boundary of the support is λσ > 0. Here, κβ, λβ, and λσ are the distribution

parameters to estimate. Loosely speaking, we are effectively using non-informative priors on

the random coefficients.

Empirical finance researchers and practitioners could impose a different set of distribu-

tional assumptions, for example, a more flexible case of mutually correlated βi and σi. They

could also explore many alternative sets of assumptions, in order to assess the robustness

of estimation results to a misspecification. We anticipate that some assumptions may ne-

cessitate the use of simulation techniques to approximate theoretical moments, and defer an

investigation of this possibility to future research.

3. Estimation approach

All individual stocks in a cross-section share the same source of market risk, which makes

their returns mutually dependent. This dependence is captured by Eq. (6), which implies

that the gross returns S1T/S

10 , S

2T/S

20 , ... are all affected by the same random variable WT .

Hence, dependence between a pair of returns SiT/Si0 and SjT/S

j0 need not diminish for any

i 6= j. Technically, the gross stock returns S1T/S

10 , S

2T/S

20 , ... exhibit strong cross-sectional

dependence (for a formal discussion, see Chudik et al., 2011). In this context, conventional

laws of large numbers do not apply (these laws presume that data dependence vanishes

asymptotically, which is not the case here). Therefore, standard econometric methods (OLS,

8

MLE, etc.) cannot deliver consistent estimates. However, it is possible to achieve consistency

using a GMM-type econometric procedure suggested by Andrews (2003, 2005).

Notice that except for WT , all other sources of randomness in the expression for SiT/Si0

in Eq. (6) – namely, βi, σi, and ZiT – are i.i.d. across the stocks. Also, WT is a continuous

function of MT/M0, since Eq. (5) implies that WT = σ-1m [ln (MT/M0)− (µm - 0.5σ2

m)T ].

Therefore, conditional on the market index return MT/M0, the individual gross stock returns

S1T/S

10 , S

2T/S

20 , ... are i.i.d. This property allows us to modify the standard GMM approach

(Hansen, 1982).

More specifically, to estimate the model parameters, we match sample moments of stock

returns with corresponding theoretical moments that are conditioned on a realization of

the market index return. Conditioning on the market index return is a critical step of the

econometric procedure. Intuitively, conditioning here is similar to including a special re-

gressor that “absorbs” data dependence. Once data dependence is dealt with this way, we

obtain estimates that are consistent and asymptotically mixed normal. The latter property

distinguishes them from asymptotically normal estimates in the standard GMM case. How-

ever, despite the asymptotic mixed normality, hypothesis testing can be implemented using

conventional Wald tests (Andrews, 2003; 2005).

We collect all identifiable parameters of the financial market model in a vector θ =

(σm, γ, κβ, λβ, λσ)′, and define a function

gi (ξ;θ) =(SiT/S

i0

)ξ − Eθ

[(SiT/S

i0

)ξ |MT/M0

], (9)

where ξ is a real number and Eθ [·|·] denotes a conditional expected value when the model

parameters are set equal to θ. The function gi (·) represents a moment restriction, because

Eθ0 [gi (ξ;θ0) |MT/M0] = 0, where θ0 is the true parameter vector.

Proposition 1 in the appendix expresses the conditional expected value Eθ analytically.

Notably, it does not depend on δ. This parameter is differenced out after we condition

(SiT/Si0)ξ

on MT/M0, but before we apply the distributional assumptions, Eqs. (7) and (8),

in the derivation of Eθ. Thus, the loss of identification here is not caused by the distributional

assumptions, but rather is due to conditioning on the market index return. Effectively, once

the observation of MT/M0 is accounted for in the function gi (·) through conditioning, a

9

cross-section of individual stock returns provides no independent information about δ that

is needed for its estimation.

Now, let ξ1, ..., ξk be k real numbers indicating different moment orders, where k is at

least as large as the number of the model parameters to estimate, that is, k ≥ 5. We define

a k × 1 vector of moment restrictions

g(SiT/S

i0;θ,MT/M0

)= (gi (ξ1;θ) , ..., gi (ξk;θ))′ ,

and then write the GMM objective function as:

Qn (θ; Σ) =

(n−1

n∑i=1

g(SiT/S

i0;θ,MT/M0

))′Σ−1

(n−1

n∑i=1

g(SiT/S

i0;θ,MT/M0

)),

where Σ is a k × k positive definite matrix. In this paper, the one-step GMM estimator,

θ1,n, is defined as the global minimizer of the objective function Qn (·), with Σ set equal to

an identity matrix Ik:

θ1,n = arg minθ∈Θ

Qn (θ; Ik) .

Econometric theory recommends implementing GMM estimation as a two-step procedure.

In the first step, a GMM objective function typically utilizes an identity matrix as the

weighting matrix. In the second step, the objective function instead incorporates a consistent

estimate of an “optimal” weighting matrix. In our case, the second step would amount to

replacing Σ with a consistent estimate of the conditional variance of the moment restrictions,

matrix Eθ0 [gg′|MT/M0], and then minimizing the objective function, to obtain a two-step

estimator. Asymptotically, a two-step procedure should be more efficient. However, its finite-

sample properties are less clear. In fact, researchers find that the one-step GMM estimator

tends to outperform the two-step estimator in practice (see Altonji and Segal, 1996, and

references therein).

To assess finite-sample properties of the econometric procedure and determine whether or

not we should apply a two-step (rather than a one-step) estimation approach in the empirical

illustration, we perform a Monte Carlo simulation exercise. It comprises 1,000 estimation

rounds on simulated weekly stock return samples of size n = 5, 500. The sample size is set

similar to the actual number of actively traded stocks on U.S. stock exchanges during the

period considered in the empirical illustration. The returns are simulated by assuming an

10

annual risk-free rate of 1% and using the financial market model solution, Eqs. (5) and

(6), and the distributional assumptions, Eqs. (7) and (8). The true parameter vector is

set at θ0 = (0.20,−2.00, 0.50, 3.00, 1.00)′, which is similar to our actual estimates on weekly

returns for January 2008. Also, we set δ0 = 0.20, which is based on the range of values of the

Sharpe ratio of the U.S. stock market, as implied by Mehra and Prescott’s (2003) estimates.

In each simulation round, we compute the one- and two-step estimates by utilizing a grid of

initial search values and minimizing the objective function numerically with the Nelder-Mead

algorithm (a similar approach is applied in the empirical illustration).

The results of the Monte Carlo exercise are summarized in Table 1. We present root mean

square errors (RMSE) of the estimates, absolute values of differences between the medians

of the estimates and the true values, and absolute values of the biases. According to these

measures (which, ideally, should be as small as possible), the one-step estimates tend to

have better finite-sample properties than the two-step estimates. Therefore, we will use the

one-step estimation approach in the empirical illustration.

4. Data

Data for the empirical illustration come from the Center for Research in Security Prices

(CRSP), which is accessed through Wharton Research Data Services (WRDS). The CRSP

provides comprehensive information on all securities listed on the New York Stock Exchange,

American Stock Exchange, and NASDAQ, including daily closing prices, returns, etc. This

information is supplemented with interest rate data from the Federal Reserve Bank Reports

database, also accessed through WRDS.

We focus on regularly traded stocks of operating companies, and exclude from considera-

tion stocks of companies in bankruptcy and stocks that were delisted during the sample time

frame (see more on it below). True returns on the excluded stocks are difficult to determine

accurately. We also drop shares of closed-end funds, exchange-traded funds, and financial

real estate investment trusts. These investment vehicles pool together many individual as-

sets. As such, their price dynamics are not adequately represented by Eq. (3). Also, some

operating companies issue stocks of two or more share classes (e.g., Berkshire Hathaway,

Inc. issues “Class A” and “Class B” stocks). In that case, we only include the class with

11

the largest number of outstanding stocks. Including several stock classes of the same com-

pany would violate the model’s assumption regarding independence of idiosyncratic risks

associated with different stocks.

We estimate the model using data from two different months: January and October of

the year 2008. These months are chosen because it is potentially interesting to compare

parameter estimates and accuracy of estimation on the return data from trading periods

of substantially different financial market volatility. In particular, data summary statistics

presented below indicate that January was a much less volatile month than October (October

featured a major market turmoil and financial crisis). To fully utilize the available data,

we study all weekly return intervals during these two months. This approach allows us

to separately investigate 15 cross-sections of returns in January and 18 cross-sections in

October. Weekly stock returns are obtained by cumulating daily ex-dividend returns. We

focus on weekly returns because they may be less affected than daily returns by various

microstructure effects, which are not captured by the financial market model. It should be

noted, however, that the econometric method is applicable, in principle, under any return

interval duration, including a single business day.

We approximate the market index return using the S&P Composite Index, which is avail-

able in the CRSP database. As an alternative, we also investigated using the value-weighted

CRSP market portfolio index. However, weekly returns on the two indices are practically

indistinguishable during the examined months and exhibit nearly perfect correlation (the

correlation coefficient is 99.56%).5 We approximate the risk-free interest rate by an average

of annualized four-week T-Bill rate quotes for the return interval under consideration.

Summary statistics for cross-sections of weekly gross stock returns in January and Oc-

tober 2008 are presented in Tables 2 and 3, respectively. In the tables, we also list values

of the gross return on the S&P Composite Index (column MT

M0) and the average T-Bill rate

(r). There are slightly fewer stocks in October, with roughly 5, 240 stocks in a cross-section,

compared to 5, 450 in January. Cross-sectional means of the stock returns tend to be lower

5Other statistical measures also indicate close similarity of the two index return series. For example, the

mean of the absolute difference between them is 0.0048, and the standard deviation is 0.0051.

12

in October. Also, the returns exhibit a substantially higher cross-sectional variability in

October. In particular, the minimum cross-sectional standard deviation in October is 0.1208

(the return interval of October 1-8), which is larger than the maximum standard deviation

of 0.1012 in January (January 22-29). These standard deviation statistics may be indicative

of a higher average cross-sectional idiosyncratic volatility in October. Moreover, the cross-

sectional distributions are leptokurtic, as illustrated by the plots of empirical densities in

Fig. 1 (the interval of January 22-29) and Fig. 2 (October 23-30).6 Another data feature

indicated by Tables 2 and 3 is a decline in the T-Bill rate from an average of 2.75% in

January, to 0.25% in October.

5. Results

We first present selected estimates of the entire model in order to show the full outcome

of the econometric procedure. Next, we present estimates of the idiosyncratic volatility pre-

mium and average cross-sectional idiosyncratic volatility for every examined return interval.

We then discuss the structure of conditional expected returns, and briefly comment on the

economic significance of the findings.

5.1. Examples of full model estimates

Table 4 lists estimates of all model parameters for the return intervals of January 22-

29 and October 23-30. We choose to present estimates for an interval close to the end of

January, in order to mitigate a potential impact of the “turn-of-the-year” anomaly (Rozeff

and Kinney, 1976). Correspondingly, the interval in October is also chosen close to the

end of the month. To illustrate the robustness of the estimation procedure, we report the

results for two choices of moment restrictions. Panel A provides estimates under eight

restrictions, when the moment order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′. Panel B

reports estimates under six restrictions, when ξ = (−1.5,−1,−0.5, 0.5, 1, 1.5)′. As can be

seen by comparing the panels, numerical values of the estimates of statistically significant

parameters (σm, λβ, and λσ) are similar between the two moment restriction choices. In the

6Empirical densities for other return intervals have similar characteristics.

13

case of statistically insignificant parameters (γ and κβ), numerical values in the two panels

are somewhat different, but have the same sign. A similar pattern is observed in additional

estimations of the model using other choices of ξ (results are not reported). Thus, our

findings appear to be robust to the choice of the restrictions. In what follows, we focus on

estimates obtained under ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′ (Panel A).

The estimates of σm imply the value of the stock market volatility of 0.0537 (in annual

terms) for January 22-29, and 0.0672 for October 23-30. These estimates are lower than the

historical average of 0.2, according to Mehra and Prescott (2003), but within the range of

values reported in other studies (e.g., Campbell et al., 2001; Xu and Malkiel, 2003).

The negative (but not statistically significant) estimates of the idiosyncratic volatility

premium γ for January 22-29 and October 23-30 are broadly in line with the findings by

Ang et al. (2006) and Jiang et al. (2009). However, the magnitudes of the estimates are

difficult to compare directly, due to a substantial difference in the time frame between the

data sets (namely, January and October 2008 in our case, and the last several decades of the

twentieth century in the other studies). The following subsection shows that the estimates

of γ for the other return intervals in January and October are nearly always negative and

statistically significant.

The estimates of the parameters κβ and λβ are broadly consistent with the results re-

ported in the literature. Eq. (7) implies an average cross-sectional stock beta value of

κβ +λβ/2, and a range of betas from κβ to κβ +λβ. According to the estimates, the average

stock beta is 1.8655 for January 22-29, and 1.1126 for October 23-30. In comparison, Fu

(2009) reports an average stock beta of 1.22. Also, the estimates indicate a range of the

individual stock beta values from 0.3417 to 3.3892 for January 22-29, and from −0.3058 to

2.5309 for October 23-30. In comparison, Fama and French (1992) report that the betas of

100 size-beta stock portfolios range from 0.53 to 1.79. Expectedly, our estimates imply a

wider range of values, since we consider individual stocks rather than stock portfolios.

The estimates of the parameter λσ are also consistent with the findings in the literature.

Eq. (8) implies an average cross-sectional idiosyncratic volatility value of λσ/2 (in annual

terms). Thus, according to the estimates, annualized average cross-sectional idiosyncratic

volatilities are 0.5290 for January 22-29, and 0.8739 for October 23-30; or 0.1527 and 0.2523

14

on the monthly basis, respectively. The result for January 22-29 bears similarity to the find-

ings of Fu (2009), who computes an average cross-sectional monthly idiosyncratic volatility

of 0.1417, and Ang et al. (2009), who report a value of 0.1472 (we convert the estimates

to the monthly frequency in order to facilitate the comparison). Our results indicate an

increase in the average cross-sectional idiosyncratic volatility by nearly two-thirds between

January 22-29 and October 23-30. Such an increase in the overall idiosyncratic volatility

level in the wake of the financial crisis seems intuitive.

5.2. Estimates of idiosyncratic volatility premium and average idiosyncratic volatility

We are mainly interested in computing the model parameters related to idiosyncratic

volatility: the idiosyncratic volatility premium γ and average cross-sectional idiosyncratic

volatility λσ/2. Their estimates for all weekly return intervals in January and October 2008

are presented in Tables 5 and 6, respectively.

Table 5 shows that γ is estimated to be negative and statistically significant for all

return intervals in January, except for the intervals of January 22-29 and January 23-30

when the estimates are negative but not statistically significant. The estimates are, on

average, −6.0666, with a standard deviation of 3.6396. Table 6 reports similar findings

for October. We estimate a negative and statistically significant γ for a large majority of

return intervals, except for October 9-16 and October 24-31, when the estimates are positive

but not significant, and October 10-17 and October 23-30, when they are negative but not

significant. The estimates are, on average, −5.5748, with a standard deviation of 3.9705.

As noted earlier, our estimates of the idiosyncratic volatility premium are in line with the

findings of Ang et al. (2006) and Jiang et al. (2009), who compute a negative premium by

applying a different methodological approach (i.e., a two-pass regression method).

Tables 5 and 6 also show that the estimates of the average cross-sectional idiosyncratic

volatility λσ/2 are always statistically significant, except for the interval of January 18-25

(all listed values are annualized). In January, the estimates are, on average, 0.5452, with a

standard deviation of 0.0519. In turn, the October estimates are, on average, 0.8372, with

a standard deviation of 0.0725. Thus, the October estimates are larger than the January

estimates by 53.56%, on average. This finding is not surprising since Tables 2 and 3 imply

15

that cross-sectional standard deviations of the returns are higher in October by 57.80%, on

average. This substantial increase in the idiosyncratic volatility level, in the wake of the

financial crisis, is intuitive and may be due to the leverage effect.

5.3. Analysis of conditional expected returns

Proposition 1 in the appendix indicates that the conditional expected gross stock return

E [SiT/Si0|MT/M0] (denoted below as E) may be represented as a product E = exp (rT )·S ·I,

comprising the following three components: the risk-free component exp (rT ), the market

risk component S, and the idiosyncratic volatility component I (expressions for S and I

follow from Proposition 1, by setting ξ = 1, and are provided shortly). The first component,

exp (rT ), represents the gross return between dates 0 and T on the default-free bond that

pays interest at the risk-free rate r. The second component is

S =

√π

2λβ√yS

exp

(x2S4yS

)[erf

(xS

2√yS− κβ

√yS

)− erf

(xS

2√yS− (κβ + λβ)

√yS

)], (10)

where xS = ln (MT/M0) + (0.5σ2m − r)T , yS = 0.5σ2

mT , and erf (·) is the error function. S is

related to the market risk because Eq. (10) shows that it depends on the gross market index

return MT/M0, the market volatility σm, and parameters κβ and λβ of the distribution of the

stock betas, Eq. (7), but does not depend on parameters related to idiosyncratic volatility.

The third component is

I =exp (λσγT )− 1

λσγT, if λσ 6= 0 and γ 6= 0; and I = 1, otherwise. (11)

I is related to idiosyncratic volatility because Eq. (11) indicates that it depends only on the

idiosyncratic volatility premium γ and parameter λσ of the distribution of σi, Eq. (8).

This representation facilitates an analysis of the structure of conditional returns. Suppose

that the effect of idiosyncratic volatility were eliminated by either ruling out the idiosyncratic

volatility premium by setting γ = 0, or shutting down the source of stock-specific idiosyn-

cratic risk by setting λσ = 0 (so that σi = 0 for every i). In that case, the conditional ex-

pected gross return would depend only on the risk-free and market risk components, namely,

E = exp (rT ) · S, since I = 1. Thus, we can interpret a quantity (E − exp (rT ) · S) /E

as the relative increment in the conditional expected gross return due to the effect of id-

iosyncratic volatility. Conversely, suppose that the effect of the market risk were eliminated,

16

by either ruling out stock exposure to the source of market risk by setting κβ = λβ = 0

(so that βi = 0 for every i), or shutting down this source of risk by setting σm = 0. In

that case, the conditional expected gross return would depend only on the risk-free and id-

iosyncratic volatility components, namely, E = exp (rT ) · I.7 Therefore, we can interpret a

quantity (exp (rT ) · S − exp (rT ) · I) /E as the relative magnitude of the difference between

the conditional expected gross return that would result from the market risk effect, and the

conditional return that would result from the idiosyncratic volatility effect.

Tables 7 and 8 present estimates of the conditional expected weekly gross stock return,

its components, and the indicated relative differences for every weekly return interval in

January and October 2008, respectively. As can be seen, values of the conditional expected

return in January are, on average, 0.9890, with a standard deviation of 0.0311. In compar-

ison, the October estimates indicate lower returns and a larger variation among them. In

particular, values of the conditional return are, on average, 0.9438, with a standard devi-

ation of 0.0834. If the effect of idiosyncratic volatility on the conditional expected return

were eliminated, values of the conditional return would tend to increase. More specifically,

the January estimates of exp (rT ) · S are, on average, 1.0493, with a standard deviation

of 0.0348. In October, the estimates are, on average, 1.0271, with a standard deviation of

0.0581. Conversely, if the effect of the market risk were eliminated, values of the conditional

return would tend to decrease. In particular, values of exp (rT )·I in January are, on average,

0.9435, with a standard deviation of 0.0322. In October, they are, on average, 0.9184, with

a standard deviation of 0.0591.

Notably, estimates of the quantity (E − exp (rT ) · S) /E suggest an overall sizable neg-

ative impact of idiosyncratic volatility on the conditional expected gross return in relative

terms. More specifically, the January estimates imply an average relative decline in the

conditional return of 0.0615 (with a standard deviation of 0.0346). In turn, an average rel-

ative decline in October is 0.0929 (with a standard deviation of 0.0654). The results also

reveal a sizable positive difference between the conditional expected gross return that would

result from the market risk effect (i.e., exp (rT ) · S), and the conditional return that would

7In the case of σm = 0, the expression for E is derived directly from Eq. (6).

17

result from the idiosyncratic volatility effect (i.e., exp (rT ) · I). In particular, estimates of

the quantity (exp (rT ) · S − exp (rT ) · I) /E indicate an average relative difference of 0.1071

(with a standard deviation of 0.0571) in January, and of 0.1162 (with a standard deviation

of 0.0778) in October. As can be seen, the magnitudes of these average relative differences

are close to each other between the two months. Overall, the sizable estimates presented

here suggest that the impact of stock-specific idiosyncratic volatility on the stock returns is

economically significant.

Lastly, Tables 7 and 8 present values of the absolute difference between the conditional

return E and the cross-sectional mean R of gross stock returns in the sample.8 The dis-

crepancy between E and R is small in absolute magnitude. In January, values of∣∣E-R

∣∣ are,

on average, 0.0002, with a standard deviation of 0.0003. In October, they are, on average,

0.0002, with a standard deviation of 0.0002. These values indicate a good fit of the estimated

model to the data.

6. Conclusion

This paper proposes a new approach to quantify the effects of stock-specific idiosyncratic

volatility on stock returns. The estimation method requires using only a single cross-section

of return data. The analysis is performed in the setting of a financial market model compris-

ing a market index and many individual stocks. Each stock price is driven by two orthogonal

Brownian motions: one of them underlies the source of market risk and the other captures

the stock-specific idiosyncratic risk. Parameters of the model are estimated using a novel

GMM-type econometric procedure, which accounts for the cross-sectional dependence of

stock return observations, due to the common source of market risk. The procedure allows

us to consistently and simultaneously estimate model parameters and conduct statistical

inference without employing a long financial time-series, unlike in the case of the popular

two-pass regression method.

The approach will aid finance researchers and practitioners in obtaining contemporaneous

estimates of the idiosyncratic volatility premium and average cross-sectional idiosyncratic

8Values of R can be found in the column Mean of Tables 2 and 3.

18

volatility. We provide an empirical illustration by estimating the model on actual weekly

stock return data from January and October 2008. Consistent with several recent studies, our

findings indicate a negative idiosyncratic volatility premium. The magnitude of the estimates

suggests that the impact of the stock-specific idiosyncratic volatility on the expected return

should not be ignored. In addition, the estimates indicate an increase in the idiosyncratic

volatility level between January and October 2008, by one-half, on average. Our approach

could be used as an alternative to the two-pass regression method, especially when researchers

need to characterize a stock market using only the most current, rather than historical

information.

19

Appendix

Proposition 1. The conditional expected value Eθ

[(SiT/S

i0)ξ |MT/M0

]can be expressed as

Eθ

[(SiT/S

i0)ξ |MT/M0

]= exp [rξT ]·S

[ξ(

ln MT

M0+[12σ2m - r

]T)

, - 12ξσ2

mT]·I[ξγT, 1

2ξ (ξ - 1)T

],

where the function S [xS , yS ] with xS = ξ(

ln MT

M0+[12σ2m − r

]T)

and yS = −12ξσ2

mT is:

S [xS , yS ] =√π

2λβ√yS

exp(− x2S

4yS

) [erfi(

xS2√yS

+ (κβ + λβ)√yS

)− erfi

(xS

2√yS

+ κβ√yS

)]if

ξ < 0;

S = 1 if ξ = 0;

S [xS , yS ] =√π

2λβ√-yS

exp(− x2S

4yS

) [erf(

xS2√-yS− κβ

√-yS

)− erf

(xS

2√-yS− (κβ + λβ)

√-yS

)]if ξ > 0.

In turn, the function I [xI , yI ] with xI = ξγT and yI = 12ξ (ξ − 1)T is:

I [xI , yI ] =√π

2λσ√yI

exp(− x2I

4yI

) [erfi(

xI2√yI

+ λσ√yI

)− erfi

(xI

2√yI

)]if ξ < 0 or ξ > 1;

I [xI , yI ] = 1 if ξ = 0 or if ξ = 1 and xI = 0;

I [xI , yI ] = exp(λσxI)−1λσxI

if ξ = 1 and xI 6= 0;

I [xI , yI ] =√π

2λσ√−yI

exp(− x2I

4yI

) [erf(

xI2√−yI

)− erf

(xI

2√−yI− λσ

√−yI

)]if 0 < ξ < 1,

where erf (·) is the error function, erf (z) = 2√π

z∫0

exp (−t2) dt, and erfi (·) is the imaginary

error function, erfi (z) = 2√π

z∫0

exp (t2) dt.

Proof. By plugging the expression for (SiT/Si0)ξ

into Eθ

[(SiT/S

i0)ξ |MT/M0

]and using

properties of the moment generating function of a normal random variable ZiT , we obtain

Eθ

[(SiT/S

i0)ξ |MT/M0

]= Eθ

[exp

(ξ(µi -

12β2i σ

2m - 1

2σ2i

)T + ξβiσmWT + 1

2ξ2σ2

i T)|MT/M0

].

By plugging in the expression for WT implied by Eq. (5), we obtain

Eθ

[(SiT/S

i0)ξ |MT/M0

]= exp (rξT )×Eθ

[exp

(ξ(

ln MT

M0+[12σ2m − r

]T)· βi +

(−1

2ξσ2

mT)×

β2i ) |MT/M0] × Eθ

[exp

(ξγT · σi + 1

2ξ (ξ − 1)T · σ2

i

)|MT/M0

]. Importantly, note that this

step eliminates δ. Next, we take into account distributional assumptions for βi and σi and

use Lemma 1 to show that Eθ

[exp

(ξ(

ln MT

M0+[12σ2m - r

]T)βi +

(- 12ξσ2

mT)β2i

)|MT/M0

]= S

[ξ(

ln MT

M0+[12σ2m − r

]T),−1

2ξσ2

mT], and Eθ

[exp

(ξγTσi + 1

2ξ (ξ − 1)Tσ2

i

)|MT/M0

]= I

[ξγT, 1

2ξ (ξ − 1)T

]. �

Lemma 1. Suppose that a random variable X is uniform, X ∼ UNI [κ, κ+ λ], where λ > 0.

Consider real constants a and b. If b < 0, then E [exp (aX + bX2)] =√π

2λ√−b exp

(−a2

4b

)×

20

[erf(

a2√−b −

√−bκ

)− erf

(a

2√−b −

√−b (κ+ λ)

)]. If b = 0, then E [exp (aX + bX2)] =

exp(a[κ+λ])−exp(aκ)aλ

. If b > 0, then E [exp (aX + bX2)] =√π

2λ√b

exp(

-a2

4b

) [erfi(

a2√b

+√b [κ+ λ]

)− erfi

(a

2√b

+√bκ)]

.

Proof. The proof is straightforward and available from the authors on request. �

21

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25

Table 1: Summary of results of Monte Carlo simulation exercise

RMSE |Median–true value| |Mean–true value| True

Parameter 1-step 2-step 1-step 2-step 1-step 2-step value

σm 0.5631 0.7967 0.0916 0.1433 0.2770 0.2643 0.2000

γ 0.5337 0.7546 0.0014 0.0818 0.0223 0.1013 −2.0000

κβ 1.3163 1.8470 0.0067 0.0514 0.0209 0.0611 0.5000

λβ 2.4048 3.3329 0.3979 0.4720 0.0862 0.1034 3.0000

λσ 0.0277 0.0394 0.0039 0.0049 0.0052 0.0060 1.0000

This table compares finite-sample properties of the one-step (“1-step”) and two-step (“2-

step”) estimates of the financial market model in a Monte Carlo simulation exercise. The

number of the simulation rounds is 1, 000. In each round, we simulate a sample of weekly

returns of size 5, 500. The annual risk-free rate is 1%. The estimates are computed by utiliz-

ing a grid of initial search values and minimizing the objective function numerically with the

Nelder-Mead algorithm. The vector of moment restrictions is constructed using the moment

order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′. We present root mean square errors of

the estimates across the simulation rounds (column RMSE), absolute values of differences

between the medians of the estimates and the true parameter values (|Median–true value|),

and absolute values of the biases (|Mean–true value|).

26

Tab

le2:

Su

mm

ary

stati

stic

sfo

rw

eekly

gro

ssre

turn

s,Janu

ary

2008

Ret

urn

inte

rval

nMean

Std.dev.

Min

Max

Median

Skew

Kurt

MT

M0

r

Jan

uar

y02

-09

5447

0.94

850.

0914

0.35

922.

4823

0.95

192.

0689

38.2

232

0.97

370.

0318

03-1

054

510.

9645

0.09

000.

3448

3.07

070.

9682

3.12

1072

.515

70.

9815

0.03

23

04-1

154

500.

9761

0.08

600.

3474

2.53

330.

9794

1.79

0836

.569

80.

9925

0.03

23

07-1

454

440.

9869

0.08

760.

3682

2.26

290.

9869

2.39

5435

.835

91.

0000

0.03

23

08-1

554

500.

9832

0.08

550.

3590

2.10

230.

9851

1.38

0622

.409

10.

9934

0.03

21

09-1

654

490.

9857

0.08

900.

2006

2.36

110.

9860

1.48

2023

.907

30.

9745

0.03

18

10-1

754

500.

9536

0.08

620.

2322

2.21

620.

9553

0.93

2920

.736

30.

9387

0.03

14

11-1

854

500.

9593

0.08

510.

1633

1.94

930.

9610

0.48

7117

.231

00.

9459

0.03

01

15-2

254

500.

9597

0.08

090.

0800

2.15

200.

9625

-0.0

380

20.0

768

0.94

900.

0276

16-2

354

480.

9772

0.08

630.

1458

1.76

280.

9755

0.24

0810

.786

30.

9748

0.02

55

17-2

454

491.

0107

0.09

060.

1489

2.28

241.

0027

0.93

2917

.155

01.

0141

0.02

36

18-2

554

471.

0160

0.09

040.

1615

2.47

361.

0063

1.80

0725

.155

01.

0041

0.02

16

22-2

954

421.

0515

0.10

120.

3717

2.36

431.

0385

1.90

9419

.192

21.

0395

0.02

07

23-3

054

411.

0289

0.09

850.

3750

2.40

001.

0165

2.39

3126

.741

11.

0129

0.02

04

24-3

154

421.

0351

0.09

320.

3990

2.10

791.

0252

1.54

0016

.578

81.

0196

0.01

96

This

table

pre

sents

des

crip

tive

stat

isti

csfo

rgr

oss

stock

retu

rns

for

15w

eekly

retu

rnin

terv

als

inJan

uar

y20

08.

Sta

tist

ics

are

rep

orte

dat

the

wee

kly

freq

uen

cy;n

isth

enum

ber

ofst

ock

s;Mean

,Std.dev.,Min

,Max

,Median

,Skew

,an

dKurt

are

the

mea

n,

stan

dar

ddev

iati

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27

Tab

le3:

Su

mm

ary

stati

stic

sfo

rw

eekly

gro

ssre

turn

s,O

ctob

er2008

Ret

urn

inte

rval

nMean

Std.dev.

Min

Max

Median

Skew

Kurt

MT

M0

r

Oct

ober

01-0

852

510.

8210

0.12

080.

1711

1.74

940.

8278

0.05

165.

5957

0.84

830.

0029

02-0

952

510.

8021

0.12

770.

1606

1.59

260.

8061

0.03

625.

0845

0.81

660.

0021

03-1

052

510.

8163

0.13

110.

2067

1.82

140.

8249

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791

4.83

020.

8180

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18

06-1

352

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9454

0.12

860.

2902

2.52

330.

9516

0.39

4510

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9493

0.00

19

07-1

452

480.

9851

0.13

500.

3152

1.95

830.

9866

0.39

367.

3081

1.00

180.

0017

08-1

552

470.

9351

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240.

3152

2.28

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9300

1.17

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9217

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09-1

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3400

4.99

911.

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31.

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09

10-1

752

421.

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0.16

770.

3083

5.12

731.

0217

4.37

4083

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91.

0460

0.00

07

13-2

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9750

0.14

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2667

3.20

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9716

2.41

5929

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10.

9821

0.00

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14-2

152

410.

9612

0.13

000.

1787

2.47

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9591

1.40

0216

.612

60.

9570

0.00

23

15-2

252

370.

9846

0.12

640.

1903

2.09

520.

9867

0.55

3911

.164

50.

9878

0.00

32

16-2

352

350.

9302

0.12

660.

1791

1.90

910.

9320

0.33

008.

8603

0.95

950.

0037

17-2

452

320.

9018

0.12

560.

1748

2.30

480.

9057

0.43

8910

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10.

9322

0.00

40

20-2

752

300.

8388

0.13

080.

1727

1.81

150.

8411

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725.

3920

0.86

150.

0042

21-2

852

300.

9014

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630.

0750

3.00

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9142

0.71

9118

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9848

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42

22-2

952

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9697

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1627

4.94

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9781

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0.29

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37

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4.06

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3.51

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31.

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0.00

30

24-3

152

231.

1118

0.19

220.

4111

4.24

141.

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3.40

8442

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21.

1049

0.00

26

This

table

pre

sents

des

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oss

stock

retu

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for

18w

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als

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08.

The

table

layo

ut

isan

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ous

toT

able

2.

28

Table 4: Examples of estimates of entire financial market model

Panel A: Moment order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′

January 22-29, 2008 October 23-30, 2008

Parameter Estimate P-value Estimate P-value

σm 0.0537 0.00 0.0672 0.00

γ −2.1117 0.34 −1.2936 0.56

κβ 0.3417 0.74 −0.3058 0.74

λβ 3.0475 0.00 2.8367 0.00

λσ 1.0580 0.00 1.7478 0.00

Panel B: Moment order vector ξ = (−1.5,−1,−0.5, 0.5, 1, 1.5)′

January 22-29, 2008 October 23-30, 2008

Parameter Estimate P-value Estimate P-value

σm 0.0481 0.00 0.0572 0.00

γ −0.9814 0.75 −1.7590 0.15

κβ 0.0724 0.95 −0.1863 0.74

λβ 2.9699 0.00 2.8769 0.00

λσ 1.0682 0.00 1.7148 0.00

This table presents estimates of the financial market model for the return intervals of January

22-29 and October 23-30, 2008. To illustrate the robustness of the estimation method, the

table presents results obtained using two sets of moment restrictions. Panel A reports the re-

sults of the estimation using the moment order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2).

Panel B reports the results for the moment order vector ξ = (−1.5,−1,−0.5, 0.5, 1, 1.5).

Parameter σm is market volatility, γ is the idiosyncratic volatility premium, κβ is the lower

bound of the support of the distribution of the stock betas and λβ is the range of this sup-

port, and λσ is the upper bound of the support of the distribution of the idiosyncratic stock

volatilities.

29

Table 5: Estimates of idiosyncratic volatility premium and average idiosyncratic volatility, January 2008

Return intervalIdiosyncratic volatility

premium, γ

Average idiosyncratic

volatility, λσ/2

Estimate P-value Estimate P-value

January 02-09 −4.7251 0.00 0.5609 0.02

03-10 −5.0907 0.00 0.5370 0.00

04-11 −8.0336 0.00 0.4747 0.00

07-14 −0.8656 0.00 0.5359 0.00

08-15 −4.4627 0.00 0.5106 0.00

09-16 −9.1830 0.00 0.4816 0.00

10-17 −6.2418 0.00 0.5357 0.00

11-18 −9.2725 0.00 0.5077 0.00

15-22 −10.1691 0.00 0.6871 0.00

16-23 −8.3481 0.00 0.5983 0.00

17-24 −10.4235 0.00 0.5843 0.00

18-25 −10.2997 0.00 0.5327 0.85

22-29 −2.1117 0.34 0.5290 0.00

23-30 −0.2054 0.99 0.5683 0.00

24-31 −1.5664 0.04 0.5345 0.00

Mean −6.0666 0.5452

Std. dev. 3.6396 0.0519

This table presents estimates of idiosyncratic volatility premium and average cross-sectional

idiosyncratic volatility for 15 weekly return intervals in January 2008. The results are based

on separate estimations of the entire financial market model on each corresponding interval,

using the moment order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′. All estimates are

annualized.

30

Table 6: Estimates of idiosyncratic volatility premium and average idiosyncratic volatility, October 2008

Return intervalIdiosyncratic volatility

premium, γ

Average idiosyncratic

volatility, λσ/2

Estimate P-value Estimate P-value

October 01-08 −8.5104 0.00 0.8095 0.00

02-09 −8.4123 0.00 0.8858 0.00

03-10 −8.4921 0.00 0.8999 0.00

06-13 −7.8321 0.00 0.7532 0.00

07-14 −5.9003 0.00 0.7949 0.00

08-15 −0.8830 0.00 0.8595 0.01

09-16 1.2672 0.50 0.9863 0.00

10-17 −0.3004 0.81 0.9162 0.00

13-20 −2.2921 0.00 0.8599 0.00

14-21 −8.4223 0.00 0.7311 0.00

15-22 −8.9121 0.00 0.7179 0.00

16-23 −8.8000 0.00 0.7585 0.00

17-24 −8.7240 0.00 0.7743 0.00

20-27 −8.5196 0.00 0.8786 0.00

21-28 −9.0261 0.00 0.8394 0.00

22-29 −6.8041 0.00 0.8235 0.00

23-30 −1.2936 0.57 0.8739 0.00

24-31 1.5115 0.80 0.9085 0.00

Mean −5.5748 0.8372

Std. dev. 3.9705 0.0725

This table presents estimates of idiosyncratic volatility premium and average cross-sectional

idiosyncratic volatility for 18 weekly return intervals in October 2008. The results are ob-

tained analogously to the ones reported in Table 5.

31

Table 7: Analysis of conditional expected gross returns, January 2008

Interval E erT S I erTS erTI E-erTSE

erT (S-I)E

∣∣E-R∣∣

Jan.02-09 0.9486 1.0006 0.9988 0.9492 0.9994 0.9498 -0.0535 0.0523 0.0001

03-10 0.9647 1.0006 1.0173 0.9477 1.0179 0.9483 -0.0552 0.0722 0.0002

04-11 0.9762 1.0006 1.0512 0.9280 1.0519 0.9286 -0.0776 0.1263 0.0001

07-14 0.9870 1.0006 0.9955 0.9909 0.9961 0.9915 -0.0092 0.0047 0.0001

08-15 0.9833 1.0006 1.0277 0.9561 1.0284 0.9567 -0.0459 0.0729 0.0001

09-16 0.9857 1.0006 1.0741 0.9172 1.0748 0.9177 -0.0903 0.1593 0.0000

10-17 0.9535 1.0006 1.0175 0.9365 1.0181 0.9371 -0.0678 0.0850 0.0001

11-18 0.9590 1.0006 1.0507 0.9121 1.0513 0.9127 -0.0963 0.1446 0.0003

15-22 0.9584 1.0004 1.0681 0.8969 1.0686 0.8973 -0.1150 0.1788 0.0013

16-23 0.9767 1.0004 1.0558 0.9247 1.0562 0.9251 -0.0814 0.1342 0.0005

17-24 1.0102 1.0004 1.1106 0.9093 1.1110 0.9096 -0.0998 0.1994 0.0005

18-25 1.0160 1.0003 1.1067 0.9178 1.1071 0.9181 -0.0896 0.1860 0.0000

22-29 1.0517 1.0004 1.0747 0.9782 1.0751 0.9786 -0.0223 0.0918 0.0001

23-30 1.0290 1.0004 1.0310 0.9977 1.0314 0.9981 -0.0023 0.0324 0.0001

24-31 1.0351 1.0004 1.0520 0.9836 1.0524 0.9840 -0.0167 0.0661 0.0000

Mean 0.9890 1.0005 1.0488 0.9430 1.0493 0.9435 -0.0615 0.1071 0.0002

Std.dev. 0.0311 0.0001 0.0337 0.0311 0.0348 0.0322 0.0346 0.0571 0.0003

This table provides an analysis of the structure of conditional expected gross stock returns in

January 2008. The return may be represented as a product E = erT · S · I, with three com-

ponents: the risk-free component erT , market risk component S, and idiosyncratic volatility

component I. erTS is the value of the return that would result from the market risk effect.

erTI is the value of the return that would result from the idiosyncratic volatility effect.

E-erTSE

is the value of the relative return increment, due to the idiosyncratic volatility effect.

erT (S-I)E

is the value of the relative magnitude of the difference between erTS and erTI.∣∣E-R

∣∣is the absolute value of the difference between E and the corresponding sample mean R.

32

Table 8: Analysis of conditional expected gross returns, October 2008

Interval E erT S I erTS erTI E-erTSE

erT (S-I)E

∣∣E-R∣∣

Oct.01-08 0.8211 1.0001 0.9384 0.8750 0.9384 0.8750 -0.1429 0.0773 0.0001

02-09 0.8022 1.0000 0.9267 0.8657 0.9267 0.8657 -0.1551 0.0760 0.0002

03-10 0.8163 1.0000 0.9463 0.8626 0.9464 0.8626 -0.1593 0.1026 0.0000

06-13 0.9455 1.0000 1.0605 0.8916 1.0605 0.8916 -0.1216 0.1787 0.0001

07-14 0.9852 1.0000 1.0797 0.9125 1.0797 0.9125 -0.0959 0.1698 0.0002

08-15 0.9352 1.0000 0.9494 0.9851 0.9494 0.9851 -0.0151 -0.0382 0.0001

09-16 1.0380 1.0000 1.0124 1.0252 1.0124 1.0252 0.0246 -0.0123 0.0004

10-17 1.0344 1.0000 1.0400 0.9946 1.0400 0.9946 -0.0055 0.0439 0.0003

13-20 0.9754 1.0000 1.0140 0.9619 1.0140 0.9619 -0.0396 0.0534 0.0004

14-21 0.9614 1.0000 1.0836 0.8872 1.0837 0.8872 -0.1271 0.2043 0.0003

15-22 0.9844 1.0001 1.1146 0.8832 1.1147 0.8832 -0.1323 0.2351 0.0002

16-23 0.9301 1.0001 1.0586 0.8785 1.0587 0.8786 -0.1383 0.1936 0.0001

17-24 0.9018 1.0001 1.0279 0.8772 1.0280 0.8773 -0.1400 0.1672 0.0000

20-27 0.8389 1.0001 0.9695 0.8652 0.9696 0.8652 -0.1559 0.1245 0.0000

21-28 0.9016 1.0001 1.0438 0.8637 1.0439 0.8637 -0.1578 0.1998 0.0001

22-29 0.9703 1.0001 1.0821 0.8966 1.0821 0.8967 -0.1153 0.1911 0.0008

23-30 1.0339 1.0001 1.0572 0.9779 1.0573 0.9780 -0.0226 0.0767 0.0005

24-31 1.1122 1.0001 1.0821 1.0277 1.0822 1.0278 0.0270 0.0489 0.0004

Mean 0.9438 1.0000 1.0270 0.9184 1.0271 0.9184 -0.0929 0.1162 0.0002

Std.dev. 0.0834 0.0000 0.0564 0.0574 0.0581 0.0591 0.0654 0.0778 0.0002

This table provides an analysis of the structure of conditional expected gross stock returns

in October 2008. The table layout is analogous to Table 7.

33

Figure 1: Cross-sectional distribution of gross returns, January 22-29, 2008.

This figure plots the cross-sectional distribution of weekly gross stock returns for the interval of January

22-29, 2008. The empirical density of the returns is represented by the dotted curve. The density of the

normal distribution, with the same mean and variance, is represented by the solid curve.

Figure 2: Cross-sectional distribution of gross returns, October 23-30, 2008.

This figure plots the cross-sectional distribution of weekly gross stock returns for the interval of October

23-30, 2008. The figure layout is analogous to Fig. 1.

34