Paper_Impact of idiosyncratic volatility on stock returns: A cross-sectional study
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Paper_Impact of idiosyncratic volatility on stock returns: A cross-sectional study



NES 20th Anniversary Conference, Dec 13-16, 2012 ...

NES 20th Anniversary Conference, Dec 13-16, 2012
Article "Impact of idiosyncratic volatility on stock returns: A cross-sectional study" presented by Serguey Khovansky at the NES 20th Anniversary Conference.
Authors: Serguey Khovansky; Oleksandr Zhylyevskyy



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Paper_Impact of idiosyncratic volatility on stock returns: A cross-sectional study Paper_Impact of idiosyncratic volatility on stock returns: A cross-sectional study Document Transcript

  • Impact of idiosyncratic volatility on stock returns: A cross-sectional study Serguey Khovanskya,∗, Oleksandr Zhylyevskyyb a Graduate School of Management, Clark University, 950 Main Street, Worcester, MA 01610 b Department of Economics, Iowa State University, 460D Heady Hall, Ames, IA 50011AbstractThe paper proposes a new approach to assess effects of stock-specific idiosyncratic volatilityon stock returns. In contrast to the popular two-pass regression method, it relies on a novelGMM-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 returndata within two separate months in 2008: January and October. The results suggest anegative idiosyncratic volatility premium, and indicate an increase in average cross-sectionalidiosyncratic volatility by one-half between January and October. The approach provides afull set of estimates for every examined return interval, and enables a decomposition of theconditional expected stock return.JEL classification: G12, C21Keywords: Idiosyncratic volatility, Idiosyncratic volatility premium, Cross-section of stockreturns, Generalized Method of Moments ∗ Corresponding author. Phone: 774-232-3903, fax: 508-793-8822. Email addresses: (Serguey Khovansky), (OleksandrZhylyevskyy)
  • 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 Epsteinand 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 thepremium can be negative (e.g., Bhootra and Hur, 2011). Additional explanations for whythe 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 estimateidiosyncratic volatility effects, the existing empirical studies need to employ long time-seriesof stock return data (Shanken, 1992). As a result, their estimates may be representative moreof historical rather than current market conditions, especially if the underlying parameterssuch 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-specificidiosyncratic volatility on stock returns. More specifically, we employ a novel econometricprocedure that allows us to obtain consistent estimates, using only a single cross-section ofreturn data. Having a long time-series of data is not required. In principle, the procedurecould be implemented using returns computed over an interval of an arbitrary duration. Asan empirical illustration, we focus on weekly returns. However, the approach also can beapplied to daily and intra-daily returns, if microstructure issues are taken into account. 1 The 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 futuremarket return. However, Bali et al. (2005) show that the relationship does not hold for value-weightedvariance. Also, there is no consensus about the time-series behavior of idiosyncratic volatility. Campbell etal. (2001) report a steady increase in the volatility after 1962. However, Brandt et al. (2010) argue thatthis finding is only indicative of an episodic phenomenon. It should be noted that we do not focus on theissues 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 portfolioindex 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 geometricBrownian motion and depend on this same source of market risk (i.e., it is a common riskshared by all stocks). In addition, they are affected by stock-specific idiosyncratic risks. Wedo not take a stance on whether idiosyncratic volatility should command a premium in thecapital market equilibrium, but rather we allow for a potential effect of a stock’s idiosyncraticvolatility 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 Researchin Security Prices (CRSP). We focus on weekly returns in January and October 2008, whichare chosen because it is potentially interesting to compare parameter estimates and accuracyof estimation on data from a relatively less volatile trading period (January 2008), and aperiod 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 intervalsoverlap. 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. Forexample, the gross return for the week of January 2-9 is 0.9486; however, it would be 0.9994if idiosyncratic volatility commanded no premium. We also obtain estimates of the averagecross-sectional idiosyncratic volatility, which is an important stock market characteristic(Campbell et al., 2001; Goyal and Santa-Clara, 2003). These estimates suggest that theturbulent October 2008 was marked by an increase in the idiosyncratic volatility by one-halfof 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 mainadvantage of the procedure is that it enables consistent parameter estimation and simplestatistical inference when cross-sectional observations are dependent, due to a common shock. 3 View slide
  • In the case of cross-sectional stock returns, the source of the common shock can be marketrisk, which, to a varying degree, affects all stocks. Standard estimation methods cannotdeliver consistent estimates in this setting, because the common shock induces “too much”data dependence among the observations, in which case, conventional laws of large numbersfail to apply. To date, empirical studies of the idiosyncratic volatility premium have typically employeda version of the two-pass regression method of Fama and MacBeth (1973).2 Although actualimplementations may vary, the main idea is that in the first pass, time-series regressions arerun to estimate each stock’s idiosyncratic volatility; while in the second pass, the first-passestimates 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 thenumber of stocks) grows infinitely large (Shanken, 1992). Also, since the regressors in thesecond 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 isnot uncommon for these complexities to be ignored in practice, resulting in biased inference(Shanken, 1992; Jagannathan and Wang, 1998). Moreover, accounting for the time-varyingnature 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 additionalassumptions, 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 datalength) grows infinitely large. Thus, it is not affected by the available time-series lengthof the stock return data. Also, since it does not involve estimating individual stock betasand idiosyncratic volatilities, it is not subject to the errors-in-variables problem arising inthe two-pass regression method, and avoids the need to impose strong assumptions abouttheir time-series behavior. Instead, it relies on a parametric model describing a financial 2 Black et al. (1972), among others, contributed to the development of the two-pass methodology. 4 View slide
  • market setting, and on a distributional assumption regarding a cross-section of the betasand volatilities. While the need to make the distributional assumption might be seen asa potential limitation, practitioners can explore several alternative assumptions to checkthe robustness of the estimates to a misspecification. Overall, we believe our approach willbe an attractive alternative to the two-pass regression method when researchers need tocharacterize 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 marketmodel. Section 3 outlines the econometric approach. Section 4 describes the data used inthe 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 classicalfinance framework. We then derive expressions for gross returns and specify distributionalassumptions 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 awell-diversified stock portfolio bearing only market risk. In what follows, this asset is referredto as “the market index”. Its price at time t is denoted by Mt . All other risky assets areindividual stocks bearing the market risk and stock-specific idiosyncratic risks. We index thestocks by i, with i = 1, 2, ..., and denote the price of a stock i at time t by Sti . In addition tothe market index and the stocks, there is a default-free bond that pays interest at a risk-freerate r. The assumption of the constant risk-free interest rate is not restrictive, as we willfocus on return intervals of short duration (in the empirical illustration, we analyze weeklyreturns). The price dynamics of the market index follows a geometric Brownian motion and isdescribed by a stochastic differential equation dMt /Mt = µm dt + σm dWt , (1) 5
  • with a drift µm = r + δσm , (2)where Wt is a standard Brownian motion indicating a source of the market risk, σm > 0is the market volatility, and δ is the market risk premium. As explained in Section 3, theestimation procedure will not allow us to identify δ, because this parameter is differencedout when stock returns are conditioned on the market index return. Conditioning on themarket index return is a critical step in the econometric procedure that ensures consistencyof the estimates. The price dynamics of a stock i (i = 1, 2, ...) also follows a geometric Brownian motionand is described by a stochastic differential equation dSti /Sti = µi dt + βi σm dWt + σi dZti , (3)with a drift µi = r + δβi σm + γσi , (4)where Zti is a standard Brownian motion indicating a source of the idiosyncratic risk that onlyaffects stock i. Zti is independent of Wt and every Ztj for j = i. Also, βi is the stock’s beta,3σi > 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 commonrisk among the stocks because an innovation in Wt results in a shock that is shared by allstocks (i.e., it is a common shock). Sensitivity of individual stock prices to the commonshock is allowed to vary. Thus, βi need not be equal to βj for j = i. Also, the stocks canhave different idiosyncratic volatilities (i.e., σi need not be equal to σj for j = i). Classical finance models (e.g., Sharpe, 1964; Lintner, 1965) imply that idiosyncraticvolatility commands no equilibrium return premium. In that case, γ = 0 and the pricedynamics of our model would coincide with that of the ICAPM with a constant investmentopportunity set (Merton, 1973, Section 6).4 However, the current finance literature indi- 3 Eqs. (1) and (3) imply that βi is the ratio of the covariance between the instantaneous returns on the 2 2stock and the market index, βi σm dt, to the variance of the instantaneous return on the market index, σm dt. 4 As noted earlier, our focus is on return intervals of relatively short duration. Thus, we do not considerthe 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 theresearchers about the sign of the premium or the underlying pricing mechanism (see a reviewin 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 shouldexpect 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 (adiscussion of specific choices of the dates t = 0 and t = T used in the empirical illustrationis provided in Section 4). More specifically, the data inputs include the gross market index 1 1 2 2return, MT /M0 , and the gross returns on the stocks, ST /S0 , ST /S0 , ... . Applying Itˆ’s lemma to Eqs. (1) and (3), we derive expressions for MT /M0 : o 2 MT /M0 = exp µm − 0.5σm T + σm WT , (5) i iand ST /S0 : i i ST /S0 = exp µi − 0.5βi2 σm − 0.5σi T + βi σm WT + σi ZT , 2 2 i (6) iwhere WT and ZT are independent and identically distributed (i.i.d.) as N (0, T ), and thedrifts 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 econometricprocedure, we specify the stock betas and idiosyncratic volatilities as random coefficientsdrawn 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 estimatedsimultaneously 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 requirementhelps 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 supportof βi should not be sign-restricted. To facilitate an evaluation of the finite-sample propertiesof the econometric procedure (see details on the Monte Carlo exercise in Section 3), it is alsodesirable 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 distributionmost closely meets these criteria. More specifically, we assume that βi ∼ i.i.d. U N I [κβ , κβ + λβ ] , (7)where the lower and upper boundaries of the support are parameterized in terms of κβ andλβ > 0, and σi ∼ i.i.d. U N I [0, λσ ] , (8)where the upper boundary of the support is λσ > 0. Here, κβ , λβ , and λσ are the distributionparameters to estimate. Loosely speaking, we are effectively using non-informative priors onthe 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 . Theycould also explore many alternative sets of assumptions, in order to assess the robustnessof estimation results to a misspecification. We anticipate that some assumptions may ne-cessitate the use of simulation techniques to approximate theoretical moments, and defer aninvestigation of this possibility to future research.3. Estimation approach All individual stocks in a cross-section share the same source of market risk, which makestheir returns mutually dependent. This dependence is captured by Eq. (6), which implies 1 1 2 2that the gross returns ST /S0 , ST /S0 , ... are all affected by the same random variable WT . i i j jHence, dependence between a pair of returns ST /S0 and ST /S0 need not diminish for any 1 1 2 2i = j. Technically, the gross stock returns ST /S0 , ST /S0 , ... exhibit strong cross-sectionaldependence (for a formal discussion, see Chudik et al., 2011). In this context, conventionallaws of large numbers do not apply (these laws presume that data dependence vanishesasymptotically, which is not the case here). Therefore, standard econometric methods (OLS, 8
  • MLE, etc.) cannot deliver consistent estimates. However, it is possible to achieve consistencyusing a GMM-type econometric procedure suggested by Andrews (2003, 2005). i i Notice that except for WT , all other sources of randomness in the expression for ST /S0 iin Eq. (6) – namely, βi , σi , and ZT – are i.i.d. across the stocks. Also, WT is a continuous -1 2function of MT /M0 , since Eq. (5) implies that WT = σm [ln (MT /M0 ) − (µm - 0.5σm ) T ].Therefore, conditional on the market index return MT /M0 , the individual gross stock returns 1 1 2 2ST /S0 , ST /S0 , ... 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 stockreturns with corresponding theoretical moments that are conditioned on a realization ofthe market index return. Conditioning on the market index return is a critical step of theeconometric procedure. Intuitively, conditioning here is similar to including a special re-gressor that “absorbs” data dependence. Once data dependence is dealt with this way, weobtain estimates that are consistent and asymptotically mixed normal. The latter propertydistinguishes them from asymptotically normal estimates in the standard GMM case. How-ever, despite the asymptotic mixed normality, hypothesis testing can be implemented usingconventional 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 i i ξ i i ξ gi (ξ; θ) = ST /S0 − Eθ ST /S0 |MT /M0 , (9)where ξ is a real number and Eθ [·|·] denotes a conditional expected value when the modelparameters are set equal to θ. The function gi (·) represents a moment restriction, becauseEθ0 [gi (ξ; θ 0 ) |MT /M0 ] = 0, where θ 0 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 i i ξ(ST /S0 ) 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 distributionalassumptions, but rather is due to conditioning on the market index return. Effectively, oncethe 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 thatis needed for its estimation. Now, let ξ1 , ..., ξk be k real numbers indicating different moment orders, where k is atleast as large as the number of the model parameters to estimate, that is, k ≥ 5. We definea k × 1 vector of moment restrictions i i g ST /S0 ; θ, MT /M0 = (gi (ξ1 ; θ) , ..., gi (ξk ; θ)) ,and then write the GMM objective function as: n n −1 i i −1 −1 i i Qn (θ; Σ) = n g ST /S0 ; θ, MT /M0 Σ n g ST /S0 ; θ, MT /M0 , i=1 i=1where Σ is a k × k positive definite matrix. In this paper, the one-step GMM estimator,θ 1,n , is defined as the global minimizer of the objective function Qn (·), with Σ set equal toan identity matrix Ik : θ 1,n = arg min Qn (θ; Ik ) . θ∈Θ 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 theweighting matrix. In the second step, the objective function instead incorporates a consistentestimate of an “optimal” weighting matrix. In our case, the second step would amount toreplacing 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-stepestimator. 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 estimatortends to outperform the two-step estimator in practice (see Altonji and Segal, 1996, andreferences therein). To assess finite-sample properties of the econometric procedure and determine whether ornot we should apply a two-step (rather than a one-step) estimation approach in the empiricalillustration, we perform a Monte Carlo simulation exercise. It comprises 1,000 estimationrounds on simulated weekly stock return samples of size n = 5, 500. The sample size is setsimilar to the actual number of actively traded stocks on U.S. stock exchanges during theperiod 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 isset at θ 0 = (0.20, −2.00, 0.50, 3.00, 1.00) , which is similar to our actual estimates on weeklyreturns for January 2008. Also, we set δ0 = 0.20, which is based on the range of values of theSharpe 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 ofinitial search values and minimizing the objective function numerically with the Nelder-Meadalgorithm (a similar approach is applied in the empirical illustration). The results of the Monte Carlo exercise are summarized in Table 1. We present root meansquare errors (RMSE) of the estimates, absolute values of differences between the mediansof the estimates and the true values, and absolute values of the biases. According to thesemeasures (which, ideally, should be as small as possible), the one-step estimates tend tohave better finite-sample properties than the two-step estimates. Therefore, we will use theone-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 CRSPprovides comprehensive information on all securities listed on the New York Stock Exchange,American Stock Exchange, and NASDAQ, including daily closing prices, returns, etc. Thisinformation is supplemented with interest rate data from the Federal Reserve Bank Reportsdatabase, 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 timeframe (see more on it below). True returns on the excluded stocks are difficult to determineaccurately. We also drop shares of closed-end funds, exchange-traded funds, and financialreal 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, someoperating 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 risksassociated with different stocks. We estimate the model using data from two different months: January and October ofthe year 2008. These months are chosen because it is potentially interesting to compareparameter estimates and accuracy of estimation on the return data from trading periodsof substantially different financial market volatility. In particular, data summary statisticspresented below indicate that January was a much less volatile month than October (Octoberfeatured 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 usto separately investigate 15 cross-sections of returns in January and 18 cross-sections inOctober. Weekly stock returns are obtained by cumulating daily ex-dividend returns. Wefocus on weekly returns because they may be less affected than daily returns by variousmicrostructure effects, which are not captured by the financial market model. It should benoted, however, that the econometric method is applicable, in principle, under any returninterval 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-weightedCRSP market portfolio index. However, weekly returns on the two indices are practicallyindistinguishable during the examined months and exhibit nearly perfect correlation (thecorrelation coefficient is 99.56%).5 We approximate the risk-free interest rate by an averageof 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 MTof the gross return on the S&P Composite Index (column 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 5 Other statistical measures also indicate close similarity of the two index return series. For example, themean 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 inOctober. 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 deviationof 0.1012 in January (January 22-29). These standard deviation statistics may be indicativeof a higher average cross-sectional idiosyncratic volatility in October. Moreover, the cross-sectional distributions are leptokurtic, as illustrated by the plots of empirical densities inFig. 1 (the interval of January 22-29) and Fig. 2 (October 23-30).6 Another data featureindicated by Tables 2 and 3 is a decline in the T-Bill rate from an average of 2.75% inJanuary, to 0.25% in October.5. Results We first present selected estimates of the entire model in order to show the full outcomeof 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 theeconomic 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 ofJanuary, in order to mitigate a potential impact of the “turn-of-the-year” anomaly (Rozeffand Kinney, 1976). Correspondingly, the interval in October is also chosen close to theend of the month. To illustrate the robustness of the estimation procedure, we report theresults for two choices of moment restrictions. Panel A provides estimates under eightrestrictions, when the moment order vector ξ = (−2, −1.5, −1, −0.5, 0.5, 1, 1.5, 2) . Panel Breports estimates under six restrictions, when ξ = (−1.5, −1, −0.5, 0.5, 1, 1.5) . As can beseen by comparing the panels, numerical values of the estimates of statistically significantparameters (σm , λβ , and λσ ) are similar between the two moment restriction choices. In the 6 Empirical densities for other return intervals have similar characteristics. 13
  • case of statistically insignificant parameters (γ and κβ ), numerical values in the two panelsare somewhat different, but have the same sign. A similar pattern is observed in additionalestimations of the model using other choices of ξ (results are not reported). Thus, ourfindings appear to be robust to the choice of the restrictions. In what follows, we focus onestimates 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 annualterms) for January 22-29, and 0.0672 for October 23-30. These estimates are lower than thehistorical average of 0.2, according to Mehra and Prescott (2003), but within the range ofvalues reported in other studies (e.g., Campbell et al., 2001; Xu and Malkiel, 2003). The negative (but not statistically significant) estimates of the idiosyncratic volatilitypremium γ for January 22-29 and October 23-30 are broadly in line with the findings byAng et al. (2006) and Jiang et al. (2009). However, the magnitudes of the estimates aredifficult to compare directly, due to a substantial difference in the time frame between thedata sets (namely, January and October 2008 in our case, and the last several decades of thetwentieth century in the other studies). The following subsection shows that the estimatesof γ for the other return intervals in January and October are nearly always negative andstatistically 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 ofκβ + λβ /2, and a range of betas from κβ to κβ + λβ . According to the estimates, the averagestock 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 theindividual stock beta values from 0.3417 to 3.3892 for January 22-29, and from −0.3058 to2.5309 for October 23-30. In comparison, Fama and French (1992) report that the betas of100 size-beta stock portfolios range from 0.53 to 1.79. Expectedly, our estimates imply awider 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 annualterms). Thus, according to the estimates, annualized average cross-sectional idiosyncraticvolatilities 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 volatilityof 0.1417, and Ang et al. (2009), who report a value of 0.1472 (we convert the estimatesto the monthly frequency in order to facilitate the comparison). Our results indicate anincrease in the average cross-sectional idiosyncratic volatility by nearly two-thirds betweenJanuary 22-29 and October 23-30. Such an increase in the overall idiosyncratic volatilitylevel 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 idiosyncraticvolatility: the idiosyncratic volatility premium γ and average cross-sectional idiosyncraticvolatility λσ /2. Their estimates for all weekly return intervals in January and October 2008are presented in Tables 5 and 6, respectively. Table 5 shows that γ is estimated to be negative and statistically significant for allreturn intervals in January, except for the intervals of January 22-29 and January 23-30when the estimates are negative but not statistically significant. The estimates are, onaverage, −6.0666, with a standard deviation of 3.6396. Table 6 reports similar findingsfor October. We estimate a negative and statistically significant γ for a large majority ofreturn intervals, except for October 9-16 and October 24-31, when the estimates are positivebut not significant, and October 10-17 and October 23-30, when they are negative but notsignificant. 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 thefindings of Ang et al. (2006) and Jiang et al. (2009), who compute a negative premium byapplying 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 idiosyncraticvolatility 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 astandard deviation of 0.0519. In turn, the October estimates are, on average, 0.8372, witha standard deviation of 0.0725. Thus, the October estimates are larger than the Januaryestimates 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%, onaverage. This substantial increase in the idiosyncratic volatility level, in the wake of thefinancial 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 i iE [ST /S0 |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 marketrisk component S, and the idiosyncratic volatility component I (expressions for S and Ifollow 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 thatpays interest at the risk-free rate r. The second component is √ π x2S xS √ xS √ S= √ exp erf √ − κβ yS − erf √ − (κβ + λβ ) yS , (10) 2λβ yS 4yS 2 yS 2 yS 2 2where xS = ln (MT /M0 ) + (0.5σm − r) T , yS = 0.5σm T , and erf (·) is the error function. S isrelated to the market risk because Eq. (10) shows that it depends on the gross market indexreturn MT /M0 , the market volatility σm , and parameters κβ and λβ of the distribution of thestock betas, Eq. (7), but does not depend on parameters related to idiosyncratic volatility.The third component is exp (λσ γT ) − 1 I= , if λσ = 0 and γ = 0; and I = 1, otherwise. (11) λσ γTI is related to idiosyncratic volatility because Eq. (11) indicates that it depends only on theidiosyncratic volatility premium γ and parameter λσ of the distribution of σi , Eq. (8). This representation facilitates an analysis of the structure of conditional returns. Supposethat the effect of idiosyncratic volatility were eliminated by either ruling out the idiosyncraticvolatility 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) /Eas 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. Inthat 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 aquantity (exp (rT ) · S − exp (rT ) · I) /E as the relative magnitude of the difference betweenthe conditional expected gross return that would result from the market risk effect, and theconditional 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 inJanuary and October 2008, respectively. As can be seen, values of the conditional expectedreturn 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. Inparticular, 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 returnwere 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 deviationof 0.0348. In October, the estimates are, on average, 1.0271, with a standard deviation of0.0581. Conversely, if the effect of the market risk were eliminated, values of the conditionalreturn 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, witha 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 relativeterms. More specifically, the January estimates imply an average relative decline in theconditional 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 alsoreveal a sizable positive difference between the conditional expected gross return that wouldresult from the market risk effect (i.e., exp (rT ) · S), and the conditional return that would 7 In 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 ofthe 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 deviationof 0.0778) in October. As can be seen, the magnitudes of these average relative differencesare close to each other between the two months. Overall, the sizable estimates presentedhere suggest that the impact of stock-specific idiosyncratic volatility on the stock returns iseconomically 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 estimatedmodel to the data.6. Conclusion This paper proposes a new approach to quantify the effects of stock-specific idiosyncraticvolatility on stock returns. The estimation method requires using only a single cross-sectionof 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 orthogonalBrownian motions: one of them underlies the source of market risk and the other capturesthe stock-specific idiosyncratic risk. Parameters of the model are estimated using a novelGMM-type econometric procedure, which accounts for the cross-sectional dependence ofstock return observations, due to the common source of market risk. The procedure allowsus to consistently and simultaneously estimate model parameters and conduct statisticalinference without employing a long financial time-series, unlike in the case of the populartwo-pass regression method. The approach will aid finance researchers and practitioners in obtaining contemporaneousestimates of the idiosyncratic volatility premium and average cross-sectional idiosyncratic 8 ¯ Values of R can be found in the column M ean of Tables 2 and 3. 18
  • volatility. We provide an empirical illustration by estimating the model on actual weeklystock return data from January and October 2008. Consistent with several recent studies, ourfindings indicate a negative idiosyncratic volatility premium. The magnitude of the estimatessuggests that the impact of the stock-specific idiosyncratic volatility on the expected returnshould not be ignored. In addition, the estimates indicate an increase in the idiosyncraticvolatility level between January and October 2008, by one-half, on average. Our approachcould be used as an alternative to the two-pass regression method, especially when researchersneed to characterize a stock market using only the most current, rather than historicalinformation. 19
  • Appendix i i ξProposition 1. The conditional expected value Eθ (ST /S0 ) |MT /M0 can be expressed as ξEθ (ST /S0 ) |MT /M0 = exp [rξT ]·S ξ ln MT + i i M0 1 2 σ 2 m - r T , - 2 ξσm T ·I ξγT, 1 ξ (ξ - 1) T , 1 2 2where the function S [xS , yS ] with xS = ξ ln MT + M0 1 2 1 2 − r T and yS = − 2 ξσm T is: σ 2 m √ √ √ π x2 x x S [xS , yS ] = √ 2λβ yS exp − 4yS S erfi √S 2 yS + (κβ + λβ ) yS − erfi 2√S S + κβ yS y ifξ < 0; S = 1 if ξ = 0; √ √ √ π x2 x x S [xS , yS ] = √ 2λβ -yS exp − 4yS S erf √S 2 -yS − κβ -yS − erf √S 2 -yS − (κβ + λβ ) -ySif ξ > 0. 1 In turn, the function I [xI , yI ] with xI = ξγT and yI = 2 ξ (ξ − 1) T is: √ √ π x2 x x I [xI , yI ] = 2λσ √yI exp − 4yI I erfi 2√I I + λσ yI − erfi 2√I I y y if ξ < 0 or ξ > 1; I [xI , yI ] = 1 if ξ = 0 or if ξ = 1 and xI = 0; exp(λσ xI )−1 I [xI , yI ] = λ σ xI if ξ = 1 and xI = 0; √ π x2 x x √ I [xI , yI ] = √ 2λσ −yI exp − 4yI I erf √I 2 −yI − erf √I 2 −yI − λσ −yI if 0 < ξ < 1, z 2where erf (·) is the error function, erf (z) = √ π exp (−t2 ) dt, and erfi (·) is the imaginary 0 z 2error function, erfi (z) = √ π exp (t2 ) dt. 0 i i i i ξ ξProof. By plugging the expression for (ST /S0 ) into Eθ (ST /S0 ) |MT /M0 and using iproperties of the moment generating function of a normal random variable ZT , we obtain ξ i iEθ (ST /S0 ) |MT /M0 = Eθ exp ξ µi - 1 βi2 σm - 2 σi T + ξβi σm WT + 1 ξ 2 σi T |MT /M0 . 2 2 1 2 2 2By plugging in the expression for WT implied by Eq. (5), we obtain ξEθ (ST /S0 ) |MT /M0 = exp (rξT )×Eθ exp ξ ln MT + i i M0 1 2 σ 2 m 1 2 − r T · βi + − 2 ξσm T ×βi2 ) |MT /M0 ] × Eθ exp ξγT · σi + 1 ξ (ξ − 1) T · σi |MT /M0 . Importantly, note that this 2 2step eliminates δ. Next, we take into account distributional assumptions for βi and σi anduse Lemma 1 to show that Eθ exp ξ ln MT + M0 1 2 σ 2 m 1 2 - r T βi + - 2 ξσm T βi2 |MT /M0= S ξ ln MT + M0 1 2 σ 2 m − r T , − 1 ξσm T , and Eθ exp ξγT σi + 1 ξ (ξ − 1) T σi |MT /M0 2 2 2 2 1= I ξγT, 2 ξ (ξ − 1) T .Lemma 1. Suppose that a random variable X is uniform, X ∼ U N I [κ, κ + λ], where λ > 0. √ 2 πConsider real constants a and b. If b < 0, then E [exp (aX + bX 2 )] = √ 2λ −b exp − a × 4b 20
  • a √ a √ erf √ 2 −b − −bκ − erf √ 2 −b − −b (κ + λ) . If b = 0, then E [exp (aX + bX 2 )] =exp(a[κ+λ])−exp(aκ) √ π 2 √ aλ . If b > 0, then E [exp (aX + bX 2 )] = 2λ√b exp - a 4b a erfi 2√b + b [κ + λ] a √− erfi √ 2 b + bκ .Proof. The proof is straightforward and available from the authors on request. 21
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  • Table 1: Summary of results of Monte Carlo simulation exercise RM SE |M edian–true value| |M ean–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 1.0000This 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. Thenumber of the simulation rounds is 1, 000. In each round, we simulate a sample of weeklyreturns 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 theNelder-Mead algorithm. The vector of moment restrictions is constructed using the momentorder vector ξ = (−2, −1.5, −1, −0.5, 0.5, 1, 1.5, 2) . We present root mean square errors ofthe estimates across the simulation rounds (column RM SE), absolute values of differencesbetween the medians of the estimates and the true parameter values (|M edian–true value|),and absolute values of the biases (|M ean–true value|). 26
  • Table 2: Summary statistics for weekly gross returns, January 2008 MT Return interval n M ean M in M ax M edian Skew Kurt M0 r January 02-09 5447 0.9485 0.0914 0.3592 2.4823 0.9519 2.0689 38.2232 0.9737 0.0318 03-10 5451 0.9645 0.0900 0.3448 3.0707 0.9682 3.1210 72.5157 0.9815 0.0323 04-11 5450 0.9761 0.0860 0.3474 2.5333 0.9794 1.7908 36.5698 0.9925 0.0323 07-14 5444 0.9869 0.0876 0.3682 2.2629 0.9869 2.3954 35.8359 1.0000 0.0323 08-15 5450 0.9832 0.0855 0.3590 2.1023 0.9851 1.3806 22.4091 0.9934 0.0321 09-16 5449 0.9857 0.0890 0.2006 2.3611 0.9860 1.4820 23.9073 0.9745 0.0318 10-17 5450 0.9536 0.0862 0.2322 2.2162 0.9553 0.9329 20.7363 0.9387 0.0314 11-18 5450 0.9593 0.0851 0.1633 1.9493 0.9610 0.4871 17.2310 0.9459 0.030127 15-22 5450 0.9597 0.0809 0.0800 2.1520 0.9625 -0.0380 20.0768 0.9490 0.0276 16-23 5448 0.9772 0.0863 0.1458 1.7628 0.9755 0.2408 10.7863 0.9748 0.0255 17-24 5449 1.0107 0.0906 0.1489 2.2824 1.0027 0.9329 17.1550 1.0141 0.0236 18-25 5447 1.0160 0.0904 0.1615 2.4736 1.0063 1.8007 25.1550 1.0041 0.0216 22-29 5442 1.0515 0.1012 0.3717 2.3643 1.0385 1.9094 19.1922 1.0395 0.0207 23-30 5441 1.0289 0.0985 0.3750 2.4000 1.0165 2.3931 26.7411 1.0129 0.0204 24-31 5442 1.0351 0.0932 0.3990 2.1079 1.0252 1.5400 16.5788 1.0196 0.0196 This table presents descriptive statistics for gross stock returns for 15 weekly return intervals in January 2008. Statistics are reported at the weekly frequency; n is the number of stocks; M ean,, M in, M ax, M edian, Skew, and Kurt are the mean, standard deviation, minimum, maximum, median, skewness, and kurtosis of the cross-sectional return distribution, MT respectively; M0 is the gross return on the S&P Composite Index; r is the four-week T-Bill rate.
  • Table 3: Summary statistics for weekly gross returns, October 2008 MT Return interval n M ean M in M ax M edian Skew Kurt M0 r October 01-08 5251 0.8210 0.1208 0.1711 1.7494 0.8278 0.0516 5.5957 0.8483 0.0029 02-09 5251 0.8021 0.1277 0.1606 1.5926 0.8061 0.0362 5.0845 0.8166 0.0021 03-10 5251 0.8163 0.1311 0.2067 1.8214 0.8249 -0.1791 4.8302 0.8180 0.0018 06-13 5248 0.9454 0.1286 0.2902 2.5233 0.9516 0.3945 10.4329 0.9493 0.0019 07-14 5248 0.9851 0.1350 0.3152 1.9583 0.9866 0.3936 7.3081 1.0018 0.0017 08-15 5247 0.9351 0.1324 0.3152 2.2826 0.9300 1.1757 11.1945 0.9217 0.0011 09-16 5244 1.0376 0.1805 0.3400 4.9991 1.0223 4.0002 67.1983 1.0401 0.0009 10-17 5242 1.0341 0.1677 0.3083 5.1273 1.0217 4.3740 83.6699 1.0460 0.000728 13-20 5243 0.9750 0.1408 0.2667 3.2000 0.9716 2.4159 29.9611 0.9821 0.0014 14-21 5241 0.9612 0.1300 0.1787 2.4752 0.9591 1.4002 16.6126 0.9570 0.0023 15-22 5237 0.9846 0.1264 0.1903 2.0952 0.9867 0.5539 11.1645 0.9878 0.0032 16-23 5235 0.9302 0.1266 0.1791 1.9091 0.9320 0.3300 8.8603 0.9595 0.0037 17-24 5232 0.9018 0.1256 0.1748 2.3048 0.9057 0.4389 10.4551 0.9322 0.0040 20-27 5230 0.8388 0.1308 0.1727 1.8115 0.8411 0.0372 5.3920 0.8615 0.0042 21-28 5230 0.9014 0.1363 0.0750 3.0000 0.9142 0.7191 18.2999 0.9848 0.0042 22-29 5228 0.9697 0.1468 0.1627 4.9451 0.9781 4.1909 110.2918 1.0371 0.0037 23-30 5226 1.0335 0.1618 0.3293 4.0667 1.0268 3.5190 54.9853 1.0506 0.0030 24-31 5223 1.1118 0.1922 0.4111 4.2414 1.0992 3.4084 42.1422 1.1049 0.0026 This table presents descriptive statistics for gross stock returns for 18 weekly return intervals in October 2008. The table layout is analogous to Table 2.
  • 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 0.00This table presents estimates of the financial market model for the return intervals of January22-29 and October 23-30, 2008. To illustrate the robustness of the estimation method, thetable 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 lowerbound 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 stockvolatilities. 29
  • Table 5: Estimates of idiosyncratic volatility premium and average idiosyncratic volatility, January 2008 Idiosyncratic volatility Average idiosyncratic Return interval premium, γ 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 0.0519This table presents estimates of idiosyncratic volatility premium and average cross-sectionalidiosyncratic volatility for 15 weekly return intervals in January 2008. The results are basedon 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 areannualized. 30
  • Table 6: Estimates of idiosyncratic volatility premium and average idiosyncratic volatility, October 2008 Idiosyncratic volatility Average idiosyncratic Return interval premium, γ 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 0.0725This table presents estimates of idiosyncratic volatility premium and average cross-sectionalidiosyncratic 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 E-erT S erT (S-I) ¯ Interval E erT S I erT S erT I E 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 0.0311 0.0001 0.0337 0.0311 0.0348 0.0322 0.0346 0.0571 0.0003This table provides an analysis of the structure of conditional expected gross stock returns inJanuary 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 volatilitycomponent I. erT S is the value of the return that would result from the market risk effect.erT I is the value of the return that would result from the idiosyncratic volatility effect.E-erT S E is the value of the relative return increment, due to the idiosyncratic volatility effect. rT (S-I)e is the value of the relative magnitude of the difference between erT S and erT I. ¯ E-R E ¯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 E-erT S erT (S-I) ¯ Interval E erT S I erT S erT I E 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 0.0834 0.0000 0.0564 0.0574 0.0581 0.0591 0.0654 0.0778 0.0002This table provides an analysis of the structure of conditional expected gross stock returnsin 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 January22-29, 2008. The empirical density of the returns is represented by the dotted curve. The density of thenormal 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 October23-30, 2008. The figure layout is analogous to Fig. 1. 34