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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: [email protected] (Serguey Khovansky), [email protected] (Oleksandr
Zhylyevskyy)
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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.
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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.
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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.
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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)
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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.
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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
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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,
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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
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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
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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
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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.
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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.
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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
Page 15
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
Page 16
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
Page 17
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
Page 18
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
Page 19
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
Page 20
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
Page 21
[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
Page 22
References
Altonji, J.G., Segal, L.M., 1996. Small-sample bias in GMM estimation of covariance struc-
tures. Journal of Business & Economic Statistics 14, 353–366.
Andrews, D.W.K., 2003. Cross-Section Regression with Common Shocks. Cowles
Foundation Discussion Paper 1428. Yale University, http://cowles.econ.yale.edu/P/cd/
d14a/d1428.pdf (accessed October 28, 2010).
Andrews, D.W.K., 2005. Cross-section regression with common shocks. Econometrica 73,
1551–1585.
Ang, A., Chen, J., 2007. CAPM over the long run: 1926-2001. Journal of Empirical Finance
14, 1–40.
Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2006. The cross-section of volatility and
expected returns. Journal of Finance 61, 259–299.
Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2009. High idiosyncratic volatility and low
returns: International and further U.S. evidence. Journal of Financial Economics 91,
1–23.
Bali, T.G., Cakici, N., Yan, X., Zhang, Z., 2005. Does idiosyncratic risk really matter?
Journal of Finance 60, 905–929.
Bhootra, A., Hur, J., 2011. High Idiosyncratic Volatility and Low Returns: A Prospect
Theory Based Explanation. Working Paper. California State University, Fullerton.
Black, F., Jensen, M.C., Scholes, M., 1972. The capital asset pricing model: Some em-
pirical tests, in: Jensen, M.C. (Ed.), Studies in the Theory of Capital Markets. Praeger
Publishers, Inc., New York, pp. 79–121.
Brandt, M.W., Brav, A., Graham, J.R., Kumar, A., 2010. The idiosyncratic volatility puzzle:
Time trend or speculative episodes? Review of Financial Studies 23, 863–899.
Campbell, J.Y., Lettau, M., Malkiel, B.G., Xu, Y., 2001. Have individual stocks become
more volatile? An empirical exploration of idiosyncratic risk. Journal of Finance 56, 1–43.
22
Page 23
Chabi-Yo, F., 2011. Explaining the idiosyncratic volatility puzzle using Stochastic Discount
Factors. Journal of Banking & Finance 35, 1971–1983.
Chudik, A., Pesaran, M.H., Tosetti, E., 2011. Weak and strong cross-section dependence
and estimation of large panels. Econometrics Journal 14, C45–C90.
Epstein, L.G., Schneider, M., 2008. Ambiguity, information quality, and asset pricing. Jour-
nal of Finance 63, 197–228.
Fama, E.F., French, K.R., 1992. The cross-section of expected stock returns. Journal of
Finance 47, 427–465.
Fama, E.F., French, K.R., 1997. Industry costs of equity. Journal of Financial Economics
43, 153–193.
Fama, E.F., MacBeth, J.D., 1973. Risk, return, and equilibrium: Empirical tests. Journal
of Political Economy 81, 607–636.
Fu, F., 2009. Idiosyncratic risk and the cross-section of expected stock returns. Journal of
Financial Economics 91, 24–37.
Goyal, A., Santa-Clara, P., 2003. Idiosyncratic risk matters! Journal of Finance 58, 975–
1007.
Guo, H., Savickas, R., 2010. Relation between time-series and cross-sectional effects of
idiosyncratic variance on stock returns. Journal of Banking & Finance 34, 1637–1649.
Hansen, L.P., 1982. Large sample properties of generalized method of moments estimators.
Econometrica 50, 1029–1054.
Hsiao, C., 2003. Analysis of Panel Data. Cambridge University Press, New York. 2nd edition.
Huang, W., Liu, Q., Rhee, S., Zhang, L., 2010. Return reversals, idiosyncratic risk, and
expected returns. Review of Financial Studies 23, 147–168.
Jagannathan, R., Wang, Z., 1998. An asymptotic theory for estimating beta-pricing models
using cross-sectional regression. Journal of Finance 53, 1285–1309.
23
Page 24
Jiang, G.J., Xu, D., Yao, T., 2009. The information content of idiosyncratic volatility.
Journal of Financial and Quantitative Analysis 44, 1–28.
Kahneman, D., Tversky, A., 1979. Prospect theory: An analysis of decision under risk.
Econometrica 47, 263–292.
Kim, D., 1995. The errors in the variables problem in the cross-section of expected stock
returns. Journal of Finance 50, 1605–1634.
Levy, H., 1978. Equilibrium in an imperfect market: A constraint on the number of securities
in the portfolio. American Economic Review 68, 643–658.
Lewellen, J., Nagel, S., 2006. The conditional CAPM does not explain asset-pricing anoma-
lies. Journal of Financial Economics 82, 289–314.
Lintner, J., 1965. The valuation of risk assets and the selection of risky investments in stock
portfolios and capital budgets. Review of Economics and Statistics 47, 13–37.
Malkiel, B.G., Xu, Y., 2006. Idiosyncratic Risk and Security Returns. Working Paper. Uni-
versity of Texas at Dallas, http://www.utdallas.edu/~yexiaoxu/IVOT H.PDF (accessed
October 28, 2010).
Mehra, R., Prescott, E.C., 2003. The equity premium in retrospect, in: Constantinides, G.,
Harris, M., Stulz, R. (Eds.), Handbook of the Economics of Finance. Elsevier B.V., pp.
887–936.
Merton, R.C., 1973. An intertemporal capital asset pricing model. Econometrica 41, 867–
887.
Merton, R.C., 1987. A simple model of capital market equilibrium with incomplete informa-
tion. Journal of Finance 42, 483–510.
Miller, M.H., Scholes, M., 1972. Rates of return in relation to risk: A re-examination of
some recent findings, in: Jensen, M.C. (Ed.), Studies in the Theory of Capital Markets.
Praeger Publishers, Inc., New York, pp. 47–78.
24
Page 25
Rozeff, M.S., Kinney, Jr., W.R., 1976. Capital market seasonality: The case of stock returns.
Journal of Financial Economics 3, 379–402.
Shanken, J., 1992. On the estimation of beta-pricing models. Review of Financial Studies
5, 1–33.
Sharpe, W.F., 1964. Capital asset prices: A theory of market equilibrium under conditions
of risk. Journal of Finance 19, 425–442.
Xu, Y., Malkiel, B.G., 2003. Investigating the behavior of idiosyncratic volatility. Journal
of Business 76, 613–644.
25
Page 26
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
Page 27
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
on,
min
imum
,m
axim
um
,m
edia
n,
skew
nes
s,an
dkurt
osis
ofth
ecr
oss-
sect
ional
retu
rndis
trib
uti
on,
resp
ecti
vely
;MT
M0
isth
egr
oss
retu
rnon
the
S&
PC
omp
osit
eIn
dex
;r
isth
efo
ur-
wee
kT
-Bill
rate
.
27
Page 28
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
-0.1
791
4.83
020.
8180
0.00
18
06-1
352
480.
9454
0.12
860.
2902
2.52
330.
9516
0.39
4510
.432
90.
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
0.13
240.
3152
2.28
260.
9300
1.17
5711
.194
50.
9217
0.00
11
09-1
652
441.
0376
0.18
050.
3400
4.99
911.
0223
4.00
0267
.198
31.
0401
0.00
09
10-1
752
421.
0341
0.16
770.
3083
5.12
731.
0217
4.37
4083
.669
91.
0460
0.00
07
13-2
052
430.
9750
0.14
080.
2667
3.20
000.
9716
2.41
5929
.961
10.
9821
0.00
14
14-2
152
410.
9612
0.13
000.
1787
2.47
520.
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
.455
10.
9322
0.00
40
20-2
752
300.
8388
0.13
080.
1727
1.81
150.
8411
0.03
725.
3920
0.86
150.
0042
21-2
852
300.
9014
0.13
630.
0750
3.00
000.
9142
0.71
9118
.299
90.
9848
0.00
42
22-2
952
280.
9697
0.14
680.
1627
4.94
510.
9781
4.19
0911
0.29
181.
0371
0.00
37
23-3
052
261.
0335
0.16
180.
3293
4.06
671.
0268
3.51
9054
.985
31.
0506
0.00
30
24-3
152
231.
1118
0.19
220.
4111
4.24
141.
0992
3.40
8442
.142
21.
1049
0.00
26
This
table
pre
sents
des
crip
tive
stat
isti
csfo
rgr
oss
stock
retu
rns
for
18w
eekly
retu
rnin
terv
als
inO
ctob
er20
08.
The
table
layo
ut
isan
alog
ous
toT
able
2.
28
Page 29
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
Page 30
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
Page 31
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
Page 32
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
Page 33
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
Page 34
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