# Idiosyncratic volatility covariance and expected stock returns.

Given that the idiosyncratic volatility (IDVOL) of individual stocks co-varies, we develop a model to determine how aggregate idiosyncratic volatility (AIV) may affect the volatility of a portfolio with a finite number of stocks. In portfolio and cross-sectional tests, we find that stocks whose returns are more correlated with AIV innovations have lower returns than those that are less correlated with AIV innovations. These results are robust to controlling for the stock's own IDVOL and market volatility. We conclude that risk-averse investors pay a premium for stocks that pay well when AIV is high, consistent with our model.**********

Previous studies document that aggregate idiosyncratic volatility (AIV) fluctuates substantially over time. This implies that innovations in the idiosyncratic volatility (IDVOL) of individual stocks co-vary rather than cancel each other out. While many studies examine how AIV predicts future market returns, few have investigated the asset pricing implications of the covariance of IDVOL for the cross-section of returns of individual stocks. The purpose of this study is to examine these implications.

The motivation for our empirical hypothesis is straight forward. The variance of a portfolio with a finite number of stocks increases as AIV increases, ceteris paribus. Consequently, a risk-averse investor would prefer to hold stocks that are expected to pay off relatively well when AIV is high. This preference increases the prices of these stocks today, lowering expected future returns. Thus, stocks whose returns are more correlated with innovations in AIV should have lower expected returns than stocks whose returns are less correlated with AIV innovations.

We empirically test this hypothesis by constructing a common empirical measure of AIV and estimating the correlation between an individual stock's returns and innovations in AIV. We call this correlation a stock's "IDVOL beta." We find that portfolios with high IDVOL betas have lower average returns than stocks with low IDVOL betas. Zero investment portfolios that long the highest 20% of IDVOL beta stocks and short lowest 20% of IDVOL beta stocks have abnormal returns of about -0.27% per month. This is an economically and statistically significant amount. The negative correlation is confirmed in cross-sectional regressions. This relationship is robust to a variety of standard control variables, the stock's own IDVOL, and market volatility. These findings are consistent with investors preferring stocks that pay off well when AIV is high.

I. Hypothesis Development

One well documented feature of AIV is that it fluctuates substantially over time. (1) This implies that the IDVOL of individual stocks covaries. While several studies have used AIV to forecast future market returns, we investigate the importance of AIV for the returns of individual assets. (2) To do this, we construct an illustrative model of returns and volatility showing why investors may price innovations in AIV. We begin, following Campbell, Lettau, Malkiel, and Xu (2001), by assuming a simple one-factor model for individual stock returns:

[r.sub.it] = [[beta].sub.i] [r.sub.st] + [u.sub.it], (1)

where [r.sub.it] is the return for stock i at time t, [[beta].sub.i] is its beta that we assume is unity for the average stock, [r.sub.st] is the systematic component of the return, and [u.sub.it] is an idiosyncratic shock to returns that has a mean of zero and a time-varying variance [[sigma].sup.2.sub.it]. Following the basic intuition of traditional asset pricing models, such as the capital asset pricing model, we assume that investors care about the variance of their total portfolio of assets, not just the variance of individual assets. Thus, we form an equal-weighted portfolio p with n assets. Given Equation (1), the time t return of the portfolio, [r.sub.pt], is:

[r.sub.pt] = [bar.[beta]] (n) [r.sub.st] + [1/n] [n.summation over (i=1)] [u.sub.it], (2)

where [bar.[beta]] (n)s the average of the betas of the n stocks in the portfolio.

To further evaluate the variance of the portfolio return, we make the key assumption that the variances of the idiosyncratic shocks co-vary across stocks, but the shocks themselves do not. This is an important distinction because if the shocks themselves are correlated, the source of the correlation would necessarily be another systematic component to returns. Therefore, the time-varying variance of the portfolios return [[PI].sub.pt] is:

[[PI].sub.pt] = [bar.[beta]] [(n).sup.2] [[PI].sub.st] + [(1/n).sup.2] [n.summation over (i=1)] [[sigma].sup.2.sub.it] (3)

where [[PI].sub.st] is the time-varying variance of the systematic component of returns.

To understand how AIV may affect returns, we provide structure for the variance of the idiosyncratic shocks that includes covariance between the IDVOL of individual stocks. For illustration, we choose the simplest structure that includes this covariance. We assume the following:

[[sigma].sup.2.sub.it] = [[rho].sub.i] [[mu].sub.t] + [[epsilon].sub.it], (4)

where [[mu].sub.t] is the common component of the IDVOL of all individual stocks, [[rho].sub.i] captures the covariance between the IDVOL of individual stock i and the common component, and [[epsilon].sub.it] is a mean-zero random shock. To simplify our illustration, we assume that the correlation parameter, [[rho].sub.i], for the average stock is unity and that the common component [[mu].sub.t] is the time t cross-sectional average IDVOL or AIV. (3) Substituting Equation (4) into Equation (3) provides the following relation between [[PI].sub.pt] and [[mu].sub.t]:

[[PI].sub.pt] = [bar.[beta]] [(n).sup.2] [[PI].sub.st] + [1/n] [bar.[rho]] (n) [[mu].sub.t] + [1/n] [bar.[epsilon]] [(n).sub.t], (5)

where [bar.[beta]] (n)s the cross-sectional average of the beta parameter from Equation (1), [bar.[rho]] (n)s the cross-sectional average of the IDVOL covariance parameter from Equation (4), and [bar.[epsilon]] [(n).sub.t]s the time t cross-sectional average of the random shocks to the IDVOL of individual stocks from Equation (4). From Equations (1) and (4), the average beta of the portfolio and the IDVOL covariance parameter should both be approximately unity and the average of IDVOL shocks should be approximately zero. Substituting these values into Equation (5) and taking the first difference, we see that the time t innovation in the time-varying variance of the portfolio is:

[DELTA][[PI].sub.pt] = [DELTA][[PI].sub.pt] + [1/n] [DELTA][[mu].sub.t] (6)

Thus, for a finite number of stocks, increases in AIV ([[mu].sub.t]) proportionally increase the variance of the portfolio, ceteris paribus. The basic intuition of asset pricing is that investors prefer to hold assets that pay off well when the risk to their total portfolio is greatest. From Equation (6), increases in AIV increase this risk. A risk-averse investor would prefer to hold stocks whose returns have a high correlation with AIV innovations rather than stocks with returns having a low correlation with AIV innovations. This preference would give high correlation stocks high prices and low expected returns relative to low correlation stocks. This is our empirically testable hypothesis.

II. Analysis of Aggregate Idiosyncratic Volatility

Our empirical hypothesis is that stocks with returns that are highly correlated with AIV innovations should have low returns relative to stocks with low return correlations with AIV innovations. To empirically test this, we need a measure of AIV. As a result, before we analyze the correlation between returns and AIV, we discuss the construction of AIV and analyze its properties.

A. Empirical Measure of Aggregate Idiosyncratic Volatility

To calculate AIV, we use the common measure from studies such as Goyal and Santa-Clara (2003) and Xu and Malkiel (2003), which is the standard deviation of the difference between the cross-sectional average of the total return variance of all individual stocks in the market and the market return variance. Each month, we estimate an individual stock's total monthly return variance from its daily returns. The total variance of stock i in month t, [v.sub.it], is:

[[upsilon].sub.it] = [[D.sub.t].summation over (d=1)] ([r.sup.2.sub.id] + 2[r.sub.id] [r.sub.id-1]), (7)

where [r.sub.id] is stock i's return on day d, and there are [D.sub.t] days in month t. This common construction of a monthly variance from daily returns assumes that the mean daily return is approximately zero (French, Schwert, and Stambaugh, 1987). We calculate the month t cross-sectional average of total variance across all N stocks in the market, [V.sub.t], as:

[V.sub.t] = [N.summation over (i = 1)] [x.sub.it][[upsilon].sub.it], (8)

where [v.sub.it] is the total variance of stock i in month t as defined in Equation (7) and [x.sub.it] is the weight of stock i in the month t average. We calculate the average with both equal and value weighting.

To calculate the market volatility, we first define the market return for day d as:

[r.sub.md] = [N.summation over (i=1)] [x.sub.id][r.sub.id], (9)

where [x.sub.id] is the weight of stock i on day d. We again use equal and value weighting. The estimate of the monthly market return variance, [V.sub.mt], is:

[V.sub.mt] = [[D.sub.t].summation over (d=1)] ([r.sup.2.sub.md] + 2[r.sub.md][r.sub.md-1]). (10)

Finally, our empirical estimate of the AIV in month t, [Z.sub.t] [Z.sub.t]. is:

[Z.sub.t] = [square root of ([V.sup.t] - [V.sub.mt])]. (11)

Since [Z.sub.t][Z.sub.t] is a measure &volatility and, as such, non-negative and positively skewed, we use the natural log of [Z.sub.t] [Z.sub.t] in our empirical tests. Given that we use two weighting methods in Equations (8) and (9), we actually have two estimates of Equation (11). When we use equal weights, we call [Z.sub.t] the equal-weighted AIV factor or EW AIV. When we use value weights, we call [Z.sub.t] the value-weighted AIV factor or VW AIV.

B. Properties of Aggregate Idiosyncratic Volatility

In Figure 1, we present the time trends of AIV from 1926 to 2010. We present EW AIV in the first panel and VW AIV in the second. These measures are defined as [Z.sub.t] in Equation (11). The time trends include the raw level of AIV (the light grey line) and the six-month moving average (the black line). These graphs are consistent with our key assumption that the IDVOL of individual stocks co-varies. If innovations of the IDVOL of individual stocks are independent, we would expect that the level of AIV would be approximately constant. Figure 1 reports that there are many periods where AIV increases, most notably during the Great Depression (the first half of the 1930s), during the inflation of the "Tech Bubble" (through the early 2000s), and during the financial crisis that began in late 2007. Outside of these three important periods, however, we see that the raw level of both EW AIV and VW AIV fluctuates around the moving average. Consistent with Equation (4), there seems to be a common component to the IDVOL of individual stocks.

We further investigate the properties of AIV. In Table I, we present the summary statistics of AIV, as defined by [Z.sub.t] in Equation (11). In Panel A, we report the summary statistics for the level of EW AIV and VW AIV. EW AIV is, on average, greater than VW AIV and is much more volatile. The mean value of the monthly EW AIV is 18.87% vs. 9.58% for the VW AIV. As such, AIV seems quite large. However, for our hypotheses, we are more interested in the innovations in AIV whose summary statistics we report in Panel B. Innovations in AIV are, on average, close to zero. The mean innovations are 0.02% and -0.01% for EW AIV and VW AIV, respectively. The standard deviations of these innovations, though, are quite large relative to the standard deviations of the levels. Further, the minimum and maximum innovations are very large. For EW AIV, the standard deviation is 13.82% with a minimum (maximum) of-67.31% (79.05%). For VW AIV, the standard deviation is 15.68% with a minimum (maximum) of-82.35% (99.07%). This indicates that AIV can move substantially in a month. Thus, the risk to an investor of a portfolio with a finite number of stocks can shift dramatically from one month to the next due to the potentially large shifts in AIV. As such, this investor should prefer stocks with returns highly correlated to AIV. Correlations of innovations in AIV with various risk measures are presented in Panel C. They are fairly small except when associated with market volatility. Later, in our robustness section, we demonstrate that IDVOL betas may explain returns after controlling for market volatility betas.

III. Methodology and Data for Asset Pricing Tests

Our hypothesis states that investors prefer stocks with returns that are highly correlated with AIV innovations. This preference causes higher prices today and lower future returns. Put another way, we anticipate that a stock's expected return is negatively related to its return correlation with AIV innovations. To empirically test this hypothesis, we need an empirical measure of the correlation of a stock's returns with AIV innovations. We call this empirical measure the stock's "IDVOL beta."

A. Construction of IDVOL Betas

We measure the correlation between a stock's returns and innovations in AIV by estimating an IDVOL beta, [[beta].sup.IDVOL.sub.i], from the regression:

[R.sub.it] = [[alpha].sub.i] + [[beta].sup.MKT.sub.i] [MKT.sub.t] + [[beta].sup.IDVOL.sub.i] [DELTA]ln[Z.sub.t] + [[mu].sub.it], (12)

where [R.sub.it] is excess returns, which is the return of stock i in month t in excess of the risk-free rate, [MKT.sub.t] is the month t market return in excess of the risk-free rate, [Z.sub.t] is as previously defined in Equation (11), and [DELTA] is the first difference operator. (4) We follow Ang et al.'s (2006) investigation of market volatility and use innovations in volatility rather than levels.

To allow for time variation in the IDVOL beta, we estimate Equation (12) in a 36-month rolling window with a minimum of 24 months of returns required. (5) Since we have two estimates of AIV (EW and VW) for each stock in a given month, we estimate two IDVOL betas. When using EW AIV, we estimate the equal-weighted IDVOL beta or the "EW IDVOL beta." When using VW AIV, we estimate the value-weighted IDVOL beta or the "VW IDVOL beta."

B. Test Specifications

We hypothesize a negative correlation between expected returns and IDVOL betas. Initially, we form portfolios based on IDVOL betas. Our general procedure is to sort stocks at the beginning of each month based on IDVOL betas estimated over the prior 36 months, and construct zero investment portfolios by shorting low IDVOL beta stocks and using the proceeds to purchase high IDVOL beta stocks. Based on our hypothesis, this portfolio is long stocks with low expected returns and short stocks with high expected returns. Therefore, these portfolios should have negative abnormal returns. We examine returns for one month after the portfolio formation.

We use the alpha from an augmented four-factor model (hereafter augmented FF4), that includes the Fama and French (1993) three factors and the Carhart (1997) momentum factor, as our measure of abnormal returns. The augmentation is a dummy variable that is equal to one for January returns and zero otherwise. As demonstrated in Tinic and West (1986), George and Hwang (2010), and Peterson and Smedema (2011), there is a turn-of-year effect in the correlation between returns and IDVOL. We allow for a possible January seasonal relation between returns and IDVOL betas. The estimated intercept (alpha) from this augmented FF4 regression of excess portfolio returns is the abnormal return for the other eleven months, while the estimated coefficient on the January dummy is the difference in the abnormal return in January versus the rest of the year. We estimate Newey-West (1987) t-statistics with six lags to test whether the alpha is significantly different from zero.

Since we use two measures of [Z.sub.t] to estimate Equation (12), we use two different weighting schemes when forming the zero investment portfolios, forming portfolios that are equal-weighted (EW) and value-weighted (VW), respectively.

In addition to our predictions regarding portfolio alphas, our hypothesis has cross-sectional implications. In the cross section, we expect that stocks with higher IV betas will have lower excess returns. We estimate the cross-sectional regression:

[R.sub.it] = [[gamma].sub.0t] + [[gamma].sub.IDVOL,t] [[beta].sup.IDVOL.sub.it] + Controls + [[epsilon].sub.it]. (13)

By our hypothesis, we expected a negative [gamma][IDVOL,.sub.t]. The "controls" in Equation (13) are other variables used to explain the cross-section of stock returns. These are the market beta, size, book-to-market ratio, and prior returns. We use Fama and MacBeth (1973) procedures to estimate Equation (13) with both EW and VW IDVOL betas. We estimate Newey-West (1987) t-statistics with six lags to test whether [[gamma].sub.IV,t] is, on average, negative. (6) We present the results for all months and for all months excluding January.

C. Data

The primary source of our data is Center for Research in Security Prices (CRSP). We use the daily file to construct estimates of the monthly volatility measures in Equations (7) and (10). We use the return data from the monthly CRSP data files to estimate our IDVOL betas, calculate our portfolio returns, and perform our cross-sectional regressions. For accounting data, we use two sources. When available, we use accounting data from the Compustat annual files. For older observations, we use the annual historical book value of equity data available in Kenneth French's library. (7) This older data was collected from Moody's industrial manuals back to 1929 and used in Davis, Fama, and French (2000). The Fama and French (1993) three factors, the Carhart (1997) momentum factor, and the risk-free rate are from Kenneth French's data library.

For our cross-sectional regressions, we follow Fama and French (2008) for variable construction. We choose the following variables as control variables: 1) the log of size (lnSize), 2) the market beta, 3) the previous month's return ([Ret.sub.t-1]), 4) the six-month cumulative returns ending two months prior to the regression month t ([Ret.sub.t-7,t-2]), and 5) the log of the book-to-market ratio (lnBM). lnSize is the CRSP price (PRC) times the number of shares outstanding (SHROUT) at the end of December of year y - 1, which we assign to the firm's stock returns for July of year y through June of year y + 1. Market betas are estimated from Equation (12) at the same time as the IDVOL betas. As such, whenever we include an IDVOL beta in a cross-sectional regression, we include its matching market beta. [Ret.sub.t-1] and [Ret.sub.t-y,t-2] are updated monthly.

lnBM for July of year y through June of y + 1 is the natural logarithm of the ratio of the book value of equity that is publically available at the end of June of year y over the market capitalization at the end of December of year y - 1. We measure the publically available book value of equity two ways. For observations from 1963 on, we follow Fama and French (1992) and use Compustat data for the fiscal year that ends in calendar year y - 1. The book value of equity is total assets (AT) minus total liabilities (LT) plus deferred taxes and investment tax credits (TXDITC), if available, minus the preferred stock liquidating value (PSTKL) or redemption value (PSTKRV), if either is available. (8) For the pre-1963 observations, we use data from Kenneth French's library and the assumption from Davis et al. (2000) that the data published in the Moody's industrial manuals before the end of June of year y is publically available.

Finally, we exclude utility (Standard Industrial Classification (SIC) codes 4900-4999) and financial (SIC codes 6000-6999) firms and only include common stock (SHRCD 10 or 11) and shares that trade on the NYSE, Nasdaq, or American Stock Exchange (EXCHCD 1-3). This leaves us with 1,750,238 monthly observations of 13,369 unique firms that range from July 1928 to December 2010.

IV. Empirical Results

Our hypothesis, that investors prefer stocks whose returns are highly correlated with AIV innovations, can be seen as a negative correlation between a stock's expected returns and its IDVOL beta. In this section, we report the results of our main tests.

A. Portfolio Returns

The first main test involves forming portfolios based on IDVOL betas to determine whether investors pay a premium for high IDVOL beta stocks relative to low IDVOL beta stocks. This premium leads to low subsequent returns for high IDVOL beta relative to low IDVOL beta stocks. From our hypothesis, the expected return of a zero investment portfolio that shorts stocks with low IDVOL betas and buys stocks with high ]DVOL betas should be negative. Our test statistic is the augmented FF4 alpha for these zero investment portfolios. We construct two zero investment portfolios: 1) one that is long the highest IDVOL beta quintile and short the lowest quintile (H-L), and 2) one that is long the two highest IDVOL beta quintiles and short the two lowest quintiles [(H + 4) - (2 + L)]/2. (9) We form portfolios using EW IDVOL betas and VW IDVOL betas and we equal and value weight each of those portfolios. This gives us four zero investment portfolios to test our hypothesis using the augmented FF4 alphas. We report these alphas in Table II.

Consistent with our hypothesis, in non-January months, high IDVOL beta stocks have smaller alphas than low IDVOL beta stocks. The alphas of the EW portfolios decrease monotonically from the L portfolio to the H portfolios. The VW portfolios largely do also, save for the L portfolios. The VW portfolios decrease monotonically from the 2 portfolio to the H portfolio. More importantly, the alphas for the zero investment portfolios are all negative, ranging from -0.46% to -0.15% per month, and five are statistically significant. 10 The economic significance of the alpha differences also seems large with annualized values ranging from over 1.66% to over 5%.

It is interesting that the correlation of IDVOL betas with returns is strongest when portfolio returns are equal-weighted. This is different than the relationship of realized IDVOL with returns, which Ang et al. (2006) and Bali and Cakici (2008) find is strong when returns are value-weighted and non-existent when returns are equal-weighted. The IDVOL beta effect may be more closely linked to small firm returns, while the IDVOL effect may be more closely linked to large firm returns. This may also explain why results with VW IDVOL betas have alphas closer to zero than the corresponding EW IDVOL betas (compare Row 1(2) in Panel A with Row 1(2) in Panel B). VW IDVOL betas are closely tied to large firm returns, which do not have as strong a link to IDVOL betas as small firm returns do. Thus, VW IDVOL betas may contain less information than EW IDVOL betas.

B. Covariance Structure of IDVOL Beta-Sorted Portfolios

One important diagnostic of portfolio returns is the covariance structure of the variable sorted portfolios. We want to be sure that these portfolios, formed on out-of-sample estimates of IDVOL betas, maintain their characteristics during the holding period. Following Daniel and Titman (1997) and Ang et al. (2006), we examine the covariance structure of the individual stocks with IDVOL betas. We examine the covariance structure in two ways. First, we estimate the post-formation IDVOL betas for the IDVOL beta sorted portfolios by estimating Equation (12) with the portfolios formed for Table II. This examines the persistence of IDVOL betas and tests whether the dispersion of IDVOL betas is the same during the portfolio holding period. Next, we calculate the standard deviation of the portfolio returns formed from lagged IDVOL betas. We form portfolios at month t using IDVOL betas estimated from as far back as month t-60. The intuition of the tests is that if IDVOL betas are unstable, then as portfolios are formed from IDVOL betas estimated farther and farther away from the current time, the stocks should lose their common covariance with the underlying factor. In turn, this should cause the portfolio standard deviation to fall. If IDVOL betas are stable, however, then standard deviations should not fall. (11)

In Panel A of Table III, we report the post-formation IDVOL betas. The post-formation IDVOL betas demonstrate a consistent dispersion during the portfolio holding period. All of the H-L and [(H + 4)-(2 + L)]/2 portfolio IDVOL beta differences are significantly positive. The EW portfolios exhibit a monotonic increase in post-formation IDVOL betas and the increase is nearly monotonic for the VW portfolios. Thus, IDVOL betas maintain their dispersion during the holding periods. This is consistent with the dispersion of IDVOL betas driving the negative alphas reported in Table II.

In Panel B of Table III, we report the results of the second test of covariances. Here, we form [(H+4)-(2+L)]/2 portfolios based on lagged estimates of IDVOL betas. Lags of 6, 12, 24, 36, 48, and 60 months are examined. We calculate the return standard deviation of these lagged portfolios and compare them with the current standard deviation from the sorting reported in Table II. The time t standard deviation of the portfolio (from Table II) is only slightly greater than the portfolios formed using the estimated IDVOL beta from up to 60 months previous. This indicates that IDVOL betas are reasonably stable through time.

C. Cross-Sectional Regressions

Next, we test our hypothesis by estimating the cross-sectional association between expected returns and IDVOL betas. Following analogously from our portfolio results, expected returns should decrease as IDVOL betas increase. Following Fama and MacBeth (1973) procedures, each month, we regress excess stock returns on IDVOL betas and control variables and test our hypotheses with the time series average and Newey-West (1987) corrected standard errors. We present the results in Table IV for the full sample and a sample excluding January observations. In Panel A, we use EW IDVOL betas in our regressions. In the first two columns, we use all monthly observations. In the next two, we exclude January observations. We estimate the regressions with just the IDVOL beta and then with all of the controls included. In Panel B, we repeat the analysis with VW IDVOL betas.

We find evidence consistent with the portfolio results from Table II. In Panel A of Table IV, when using the EW IDVOL beta, in all four models, the cross-sectional correlation between excess returns and IDVOL beta is negative and statistically significant. In Panel B, all of the estimated relations are, again, negative, but weaker than the results in Panel A. Only one coefficient on IDVOL beta is significant. As in Table II, the results with EW IDVOL betas are stronger. Taken with the portfolio results from Table II, these findings are consistent with our hypothesis that investors prefer stocks with high correlations to AIV innovations.

V. Robustness Tests

In this section, we analyze the robustness of our previous results. The first set of robustness tests controls for the stock's own IDVOL. The second set of tests controls for market volatility rather than market returns.

A. IDVOL Betas versus a Stock's Own IDVOL

Since AIV is constructed from the idiosyncratic volatility of individual stocks and given that many studies have demonstrated that a stock's IDVOL matters for returns for reasons such as under-diversification (Merton, 1987), we investigate whether the results in Tables II and IV are robust to controlling for IDVOL. Particularly, we control for the effect of the Ang et al. (2006, 2009) realized IDVOL (RIDVOL) and the Fu (2009) expected IDVOL (EIDVOL). (12) For each individual stock, RIDVOL for month t is the standard deviation of the residuals of a Fama and French (1993) three-factor model using the daily stock returns in month t-1. We require a minimum of 17 daily returns within a month. In constructing RIDVOL, we require daily Fama-French (1993) factors that are only available back to July, 1963 in the library. For observations prior to July, 1963, we construct our own daily size and book-to-market factors using our sample of stocks and following the methodology of Fama and French (1993). (13) We estimate a stock's EIDVOL with the following method. For each stock, using all observations, we estimate nine three-factor models of monthly returns with an EGARCH(p,q) error term. The nine models correspond to all permutations of the two parameters p,q = (1,2,3). This gives us nine monthly estimates of conditional volatility. Since we do not know what the true order of the EGARCH terms are, for each firm, we use as our measure of EIDVOL, the monthly conditional volatility of the best fitting of the nine models for that firm, based on the Akaike Information Criteria.

We control for RIDVOL and EIDVOL in portfolio returns using the neutralizing approach from Ang et al. (2006), among others. Each month, we sort stocks into five groups based on RIDVOL and then, within each group, we sort stocks into five groups based on IDVOL betas. We then form EW and VW portfolios by calculating the equal-weighted and value-weighted, respectively, average returns of stocks in the same IDVOL beta group, regardless of the RIDVOL grouping. By using the sequential grouping, we ensure the five IDVOL beta portfolios have similar average RIDVOL. When we construct our zero investment portfolios, the RIDVOL of the long and short sides of the portfolio will be equal and their dispersion in the zero investment portfolios will be neutralized. The augmented FF4 alpha will be driven by IDVOL betas, not the individual IDVOL. We repeat this procedure replacing RIDVOL with EIDVOL. We then reverse the order of the sorting procedure and examine whether differences in RIDVOL and EIDVOL have explanatory power after controlling for IDVOL betas.

Another way we examine whether the previous results are affected by individual stock IDVOL measures and IDVOL betas is the cross-sectional approach. We replicate our estimation of [[gamma].sub.IDVOL] from Equation (13), but include the IDVOL of the stock in the regression. By including both the IDVOL beta and IDVOL, we derive an estimate of the relation of excess returns with both IDVOL beta and IDVOL, while controlling for the effect of the other. We employ RIDVOL and EIDVOL.

We report the augmented FF4 alphas for portfolios that are IDVOL beta-sorted with RIDVOL (EIDVOL) neutralized in Table V, Panel A (B). We provide the augmented FF4 alphas for portfolios that are RIDVOL- (EIDVOL-) sorted with IDVOL betas neutralized in Panel C (D). We use the quintile portfolios and the two zero investment portfolios, [(H - L) and (H + 4) - (2 + L)/2], in these tests.

In Panels A and B of Table V, we note the alphas from Table II are robust to controls for the stock's own IDVOL. All 16 monthly alpha differences are negative, ranging from -0.37% to -0.09% and nine of the 16 are statistically significant. Economically, the alpha differences tend to be slightly less than those in Table II. This evidence is consistent with the hypothesis that a stock's return correlation with AIV innovations matters and that the correlation is not an alternate measure of the stock's IDVOL. Finally, similar to Table II, alpha differences are stronger for EW portfolios than for VW portfolios.

In Panel C of Table V, the importance of RIDVOL is robust to the inclusion of the IDVOL beta. All eight of the zero investment portfolios are negative and significant, consistent with Ang et al. (2006, 2009) and others. Similar to the results reported by Ang et al. (2006, 2009) and Bali and Cakici (2008), the RIDVOL effect is stronger for VW portfolios than EW portfolios. In Panel D, all alpha differences are positive, which is similar to prior studies. Only the EW portfolios have positively significant alpha differences. The strength of the EIDVOL results is somewhat mitigated when compared to the findings in Fu (2009) and Huang, Liu, Rhee, and Zhang (2010), probably because the latter two studies do not control for IDVOL beta.

To further these conclusions, we analyze the importance of IDVOL betas related to the stock's own IDVOL by performing Fama-MacBeth (1973) cross-sectional regressions. We report these results in Table VI. In Panel A, we use our full sample of cross-sections (each month from July 1928 to December 2010), while in Panel B, we exclude January. We use the EW IDVOL (VW IDVOL) beta in the regression results reported in the first (last) four columns in each panel. The (1) specification includes just IDVOL beta and RIDVOL, while the (2) specification includes just IDVOL beta and EIDVOL. The (3) specification includes just IDVOL beta, RIDVOL, and EIDVOL. The (4) specification includes IDVOL beta, RIDVOL, EIDVOL, and all of the other control variables included in the Table IV cross-sectional regressions.

The coefficients on IDVOL betas in Table VI are similar to those in Table II. All coefficients are negative and those for EW IDVOL betas are highly significant. The results are weaker when using VW IDVOL betas, where only one of the regression specifications has a statistically significant negative coefficient. Consistent with the portfolio results in Table V, the cross-sectional tests confirm the robustness of RIDVOL and EIDVOL. For all 16 of the regressions, RIDVOL (EIDVOL) has a statistically significant negative (positive) correlation with returns. This is consistent with our previous conclusion that return correlations with AIV innovations matter, while a stock's own IDVOL is still important, perhaps due to the limits to diversification.

B. IDVOL Betas versus Market Volatility Betas

The final robustness test we perform involves replacing the market return in Equation (12) with the market volatility measure, [V.sub.mt], in Equation (10). This is an important test for two reasons. First, since by construction AIV is a function of market volatility, we want to be sure that it is IDVOL and not market volatility that contains the important information for expected returns. Additionally, Guo and Savickas (2006) determine that market volatility has an important impact on the relationship between market returns and AIV We estimate Equation (13) with a "market volatility" beta instead of the market beta and report the results in Table VII. Panel A provides full sample results, while January months are excluded in Panel B.

We find evidence consistent with our previous results. The coefficients on IDVOL betas are all negative and mostly statistically significant, and all are significant using EW IDVOL. The VW IDVOL coefficients are generally significant when the market volatility beta is included. As with prior results, the EW IDVOL beta is more important than the VW IDVOL beta. Controlling for market volatility does not affect the importance of IDVOL betas when explaining the cross-section of expected returns.

VI. Conclusion

We conclude that investors prefer stocks whose returns are highly correlated with AIV innovations. We build intuition as to how AIV might matter to an investor holding a finite number of stocks. We construct AIV and test to see whether it is priced by estimating the covariance between individual stock returns and AIV innovations. We call this covariance the stocks IDVOL beta. We estimate the alphas of IDVOL beta-sorted portfolios and the cross-sectional relationship between expected excess returns and the stock's IDVOL beta. Both sets of tests suggest a negative correlation between expected returns and IDVOL betas. Zero investment portfolios' long stocks with low IDVOL betas and short stocks with high IDVOL betas consistently produce significantly negative abnormal returns. The relation estimated from the cross-sectional regressions is negative and largely statistically significant, consistent with the portfolio alphas. From this evidence, we conclude that risk-averse investors pay a premium for stocks that pay off well when AIV is high.

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We thank Marc Lipson (Editor) and two anonymous reviewers for helpful feedback and criticism. We also thank seminar participants at the University of Northern Iowa for helpful comments. Adam Smedema acknowledges that a portion of the research was done in fulfillment of requirements for his dissertation "Two Essays on Idiosyncratic Volatility" while at Florida State University. He thanks' his dissertation committee for help with this research.

(1) Examples of studies that demonstrate that AIV fluctuates over time include Campbell et al. (2001), Xu and Malkiel (2003), Bali, Cakici, and Levy (2008), Bekaert, Hodrick, and Zhang (2010), and Brandt, Brav, Graham, and Kumar (2010).

(2) Examples of studies that link AIV, as measured as the cross-sectional average IDVOL, to future aggregate market returns include Goyal and Santa-Clara (2003), Guo and Savickas (2006, 2008), and Jiang and Lee

(2006).

(3) Our hypothesis does not depend upon this assumption. We provide it to link this illustration to our subsequent empirical tests that use the common measure of AIV, which is the cross-sectional mean IDVOL. See Footnote 2 for lists of studies that empirically link the cross-sectional mean IDVOL with AIV.

(4) Note that we use an upper case "R" to represent monthly returns in excess of the risk-free rate, while we use a lower case "r" to define raw daily returns.

(5) As a robustness check, we estimate IDVOL betas in an alternative way. We follow the portfolio method from Fama and French (1992) to estimate a second stage IDVOL beta. Each month, we sort stocks into 25 groups based on five ranks of IDVOL beta and five ranks of the corresponding market beta. Then, within each of the 25 groups, we sort stocks into three ranks based on their market capitalization. We form 75 equal-weighted portfolios. We estimate Equation (12) using the full sample for each of these 75 portfolios. We assign the IDVOL beta of a portfolio to all of the constituent stocks. We derive similar results with this beta estimation technique.

(6) As a robustness check, we also consider the panel structure of the standard errors. We estimate the cross-sectional model (13) with a panel model with two dimensional (firm and time) clustered standard errors. We find results consistent with the Fama-MacBeth (1973) methodology. They are available on request.

(7) The data library is found at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. We thank Kenneth French for making this data available.

(8) If unavailable, the variables are set to zero.

(9) We follow the methods used by Fama and French (1993) for creating the SMB and HML factors and short each of the bottom two portfolios an amount equal to half our wealth and invest half that amount in each of the top two portfolios.

(10) We also performed this test using equal-weighted portfolios that excluded all stocks with prices less than $5. The results are qualitatively the same as the EW and VW portfolio results posted here.

(11) See Daniel and Titman (1997) for additional discussion of this test.

(12) Other studies reporting empirical evidence supporting the importance of IDVOL include Tinic and West (1986), Lehmann (1990), Malkiel and Xu (2006), Diavatopoulos, Doran, and Peterson (2008), Huang, Liu, Rhee, and Zhang (2010), and Peterson and Smedema (2011).

(13) See Kenneth French's data library for more details at mba.tuck.dartmouth.edu/pages/faculty/ken.french.

David R. Peterson and Adam R. Smedema *

* David R. Peterson is the Wells Fargo Professor of Finance in the College of Business at Florida State University" in Tallahassee, Florida. Adam R. Smedema is an Assistant Professor in the Department of Finance, College of Business Administration at University of Northern Iowa in Cedar Falls, IA.

Table I. Attributes of Aggregate Idiosyncratic Volatility We report several attributes of monthly aggregate idiosyncratic volatility (AIV). We use two AIV measures: 1) the equal-weighted measure (EW AIV) and 2) value-weighted measure (VW AIV). They are calculated as Z, from Equation (11). See Figure I for details regarding the construction of these volatilities. In Panel A, we present the basic summary statistics for levels of EW AIV and VW AIV In Panel B, we report the basic summary statistics for innovations in EW AIV (A EW AIV) and VW AIV (A VW AIV). The innovations are the first-difference of the log levels of AIV In Panel C, we provide the correlations of the innovations in EW and VW AIV with several risk factors and innovations in other volatility measures. The risk factors we include are the Fama-French (1993) three factors (MKT, SMB, and HML) and the Carhart (1997) momentum factor UMD. As a volatility measure, we include the value-weighted market volatility (MKT VOL). See Figure I for details concerning the construction of this volatility. We calculate world volatility (WRLD VOL) for a given month as the standard deviation of the Fama and French (2011) global market risk premium over the previous 12 months. VIX is the implied volatility on the S&P 500 index options and ISKEW is the Bakshi, Kapadia, and Madan (2003) option implied skew from S&P 500 futures options. For all volatility measures, the innovations are the first-difference of the natural log of the level of volatility. Panel A. Summary Statistics of AIV Mean (%) Std Dev (%) Skew EW AIV 18.87 8.64 1.56 VW AIV 9.58 3.84 2.50 Panel B. Summary Statistics of Innovations in AN [DELTA] EW AIV 0.02 13.82 0.47 [DELTA] VW AIV -0.01 15.68 0.58 Panel C. Correlations of Innovations in AN with Risk Factors and Other Volatility Measures EW AIV MKT SMB HML -0.09 *** -0.04 -0.02 MKT VOL VIX WRLD VOL 0.37 *** 0.10 0.10 VW AIV MKT SMB HML -0.16 *** -0.13 *** 0.03 MKT VOL VIX WRLD VOL 0.59 *** 0.28 *** 0.12 * Panel A. Summary Statistics of AIV Min (%) Max (%) EW AIV 6.94 67.37 VW AIV 4.45 40.80 Panel B. Summary Statistics of Innovations in AN [DELTA] EW AIV -67.31 79.05 [DELTA] VW AIV -82.35 99.07 Panel C. Correlations of Innovations in AN with Risk Factors and Other Volatility Measures EW AIV UMD 0.01 ISKEW 0.01 VW AIV UMD -0.02 ISKEW 0.01 *** Significant at the 0.01 level. * Significant at the 0.10 level. Table II. Augmented Four-Factor Alphas (%) for IDVOL Beta-Sorted Quintile Portfolios In this table, we report the alphas for IDVOL beta-sorted portfolios from a model with the Fama and French (1993) three factors, the Carhart (1997) momentum factor, and a January dummy. The January dummy takes a value of one for return observation from the month of January and zero otherwise. The sample period is July 1928-December 2010. In Panel A, we use the IDVOL betas estimated relative to the equal-weighted aggregate idiosyncratic volatility (EW AIV). In Panel B, we use the IDVOL betas estimated relative to the value-weighted aggregate idiosyncratic volatility (VW AIV). We construct AIV as follows. We calculate the monthly total volatility of the individual stocks as the sum of the square daily returns within the month (including a correction for serial correlation in daily returns). We calculate the cross-sectional average of the individual total volatility. Then, we calculate the monthly market volatility as the sum of the square of daily market returns (including a correction for serial correlation in daily returns). The AIV is the difference in the average total volatility of individual stocks and market volatility. For EW AIV, we use the equal weighted cross-sectional average and the equal-weighted market return volatility. For VW AIV, we use the value-weighted cross-sectional average and the value-weighted market return volatility. For each month, we estimate the IDVOL beta by regressing returns on innovations in AIV and the market return over the previous 36 months (requiring a minimum of 24 months). For each IDVOL beta, we construct portfolios that are equal-weighted (EW) and value-weighted (VW). We report the Newey-West (1987) t-statistics, with six lags, for the zero investment portfolios in parentheses. Panel A. Equal-Weighted IDVOL Betas Portfolio Weights L 2 M 4 EW [alpha] 0.37 0.25 0.15 0.06 t([alpha]) VW [alpha] 0.10 0.15 0.10 0.00 t([alpha]) Panel B. Value-Weighted IDVOL Betas EW [alpha] 0.24 0.23 0.19 0.12 t([alpha]) VW [alpha] 0.06 0.16 0.08 0.02 t([alpha]) Panel A. Equal-Weighted IDVOL Betas Portfolio [(H + 4) - Weights H (H - L) (2 + L)]/2 EW -0.09 -0.46 *** -0.32 *** (-5.03) (-5.00) VW -0.10 -0.20 -0.18 * (-1.29) (-1.79) Panel B. Value-Weighted IDVOL Betas EW -0.03 -0.27 *** -0.19 *** (-2.44) (-2.77) VW -0.09 -0.15 -0.15 (-1.16) (-1.58) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table III. Covariance Structure of IDVOL Beta-Sorted Portfolios We report the results from tests of the covariance structure of IDVOL beta-sorted portfolios. In Panel A, we present the realized IDVOL betas (times 100) for IDVOL beta-sorted portfolios. The sample period is July 1928-December 2010. The realized IDVOL betas are from a regression of the IDVOL beta-sorted portfolios on innovations in the log of aggregate idiosyncratic volatility (AIV) and the excess market return. We sort stocks into the quintile portfolios based on their IDVOL betas estimated over the prior 36 months. We construct two zero investment portfolios. (H-L) is long stocks in the highest quintile (H) portfolio and short stocks in the lowest quintile (L) portfolio. [(H + 4)-(2 + L)]/2 is long the two highest quintile portfolios and short the two lowest quintile portfolios. In the first two rows of Panel A, we use the IDVOL betas estimated relative to EW AIV In the bottom two rows, we use the IDVOL betas estimated relative to VW AIV See Table II for more details regarding the construction of AIV For each IDVOL beta, we construct portfolios that are equal-weighted (EW) and value-weighted (VW). We report the ordinary least square (OLS) t-statistics for the zero investment portfolios in brackets. In Panel B, we provide the standard deviation of [(H + 4) (2 + L)]/2 portfolios formed on stocks' lagged IDVOL betas. In the first column (t), we form portfolios based on the contemporaneous IDVOL beta. For columns labeled t-k, the portfolios are based on the IDVOL beta from k months previous. Panel A. Post-Formation IDVOL Betas (x 100) for IDVOL Beta-Sorted Quintile Portfolios AIV Portfolio Weight Weight L 2 M 4 EW EW -1.73 * -1.1 V -0.56 0.07 (-1.79) (-1.76) (-0.92) (0.11) VW -1.28 ** 0.03 0.74 ** 0.18 (2.52) (0.10) (2.51) (0.51) VW EW -1.86 ** -1.21 ** -0.82 -0.03 (-2.23) (-2.11) (-1.48) (-0.04) VW -1.45 *** -0.35 0.75 *** 0.38 (-3.10) (-1.10) (2.64) (1.21) Panel B. Standard Deviations of Lagged IDVOL Beta-Sorted Portfolios AIV Portfolio Weight Weight t t-6 t-12 t-24 EW EW 1.87 1.75 1.58 1.54 VW 2.53 2.29 2.17 2.21 VW EW 1.76 1.80 1.53 1.52 VW 2.56 2.41 2.27 2.29 AIV Portfolio [(H+4) - Weight Weight H (H - L) (2+L)]12 EW EW 1.02 2.75 *** 1.96 *** (1.08) (4.87) (5.05) VW 1.56 *** 2.84 *** 1.49 ** (2.86) (3.41) (2.58) VW EW 1.01 *** 2.87 *** 2.03 *** (1.09) (5.22) (5.41) VW 1.59 *** 3.05 *** 1.89 (3.43) (4.15) (3.61) Panel B. Standard Deviations of Lagged IDVOL Beta-Sorted Portfolios AIV Portfolio Weight Weight t-36 t-48 t-60 EW EW 1.48 1.58 1.79 VW 2.24 2.41 2.29 VW EW 1.64 1.41 1.38 VW 2.29 2.20 2.27 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table IV. Cross-Sectional Regressions of Excess Returns (%) on IDVOL Beta We report the results of cross-sectional regressions of the monthly excess returns of individual stocks, expressed as a percentage, on IDVOL beta ([[beta].sup.IDVOL]) and a variety of control variables. The sample period is July 1928-December 2010. We estimate [[beta].sup.IDVOL] by regressing excess returns on innovations in aggregate idiosyncratic volatility and excess market returns using a 36-month rolling window (24-month minimum). We use the coefficient estimate on excess market returns from this rolling regression as our estimate of [[beta].sup.MKT]. InSize is the natural log of market capitalization (in millions). We measure InSize annually in June and assign it to monthly returns for July through the following June. 1nBM is the natural log of the ratio of the book value of equity publicly known at the end of June over the market value of equity from the end of the preceding December. We assign this ratio to all observations for July through the following June. For observations where we use book value data from Compustat, we assume that the publicly available book value of equity is for the fiscal year ending the previous calendar year. The book value of equity is the book value of assets plus deferred taxes, less the book value of equity and the book value of preferred stock. For older observations, we follow Davis et al. (2000) and use the book value of equity data from the Moody's industrial manuals published by the end of June. We measure return reversals, [Ret.sub.t-1] as the return over the previous month. We measure momentum, [Ret.sub.t-2,t-7] as the cumulative return over the six-month period ending two months prior. In Panel A, we use the EW IDVOL beta, while in Panel B, we use the VW IDVOL beta. We report the Newey-West (1987) t-statistics with six lags in parentheses. Panel A. EW IDVOL Beta Full Full Non- Non- Sample Sample January January [[beta].sup.IDVOL] -1.00 *** -0.37 ** 4.88*** 4.46 ** (-4.75) (-1.99) (-4.37) (-2.59) [[beta].sup.MKT] 0.00 -0.06 (0.04) (-0.51) InSize -0.14 *** -0.04 (-3.58) (-0.91) InBM 0.26 *** 0.21- (4.44) (3.56) [Ret.sub.t-1] -8.31 *** -7.17 *** (-16.58) (-15.20) [Ret.sub.t-2,t,7] 0.34 0.84 *** (1.49) (3.35) Panel B. VW IDVOL Beta Full Full Non- Non Sample Sample January January [[beta].sup.IDVOL] -0.31 -0.28 -0.29 -0.36 * (-1.39) (-1.18) (-1.30) (-1.72) [[beta].sup.MKT] 0.00 -0.05 (0.04) (-0.46) InSize -0.14 *** -0.04 (-3.53) (-0.89) InBM 0.26 *** 0.21 *** (4.42) (3.54) [Ret.sub.t-1] -8.26 *** -7.12 *** (-16.61) (-15.20) [Ret.sub.t-2,t,7] 0.32 0.83 *** (1.41) (3.29) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table V. Augmented Four-Factor Alphas (%) of IDVOL Beta and IDVOL-Sorted Portfolios We present the alphas for neutralized portfolios from a model with the Fama and French (1993) three factors, the Carhart (1997) momentum factor, and a January dummy. The January dummy takes a value of one for observations from January and zero otherwise. In Panel A, we form portfolios with IDVOL betas and neutralize realized IDVOL (RIDVOL). RIDVOL for month t is the standard deviation of the residuals from a Fama and French (1993) three-factor model using the daily stock returns in month t-1. We require a minimum of 17 daily returns within a month to estimate RIDVOL. To neutralize RIDVOL, we use the following procedure. Each month, we sort stocks into five groups based on RIDVOL and then, within each group, sort stocks into five groups based on their IDVOL beta. We then neutralize RIDVOL by constructing portfolios calculating the average returns of stocks in the same IDVOL beta group. In Panel B, we form portfolios based on IDVOL betas and neutralize expected IDVOL (EIDVOL) as in Fu (2009). EIDVOL is estimated relative to a Fama and French (1993) three-factor model with an EGARCH(p,q) model for the error term. We estimate every permutation of p,q = 1,2,3. For each stock, we use all observations to estimate the parameters of the models. For each month, EIDVOL is the conditional expected IDVOL from the best fitting of the nine models, based on the AIC. In Panel C (Panel D), we form portfolios based on RIDVOL (EIDVOL) and neutralize the effect of the IDVOL betas on returns. In the first four rows of each panel, we use IDVOL betas estimated using equal-weighted (EW) aggregate idiosyncratic volatility (AIV). In the bottom four, we use the IDVOL betas estimated with value-weighted (VW AIV). We report the results from EW and VW portfolios. We provide the Newey-West (1987) t-statistics, with six lags for the zero investment portfolios, in parentheses. Panel A. IDVOL Beta-Sorted Portfolios, Neutralizing RIDVOL Portfolio AIV Weight Weight L 2 M 4 EW EW 0.34 0.24 0.14 0.05 VW 0.06 0.14 0.13 -0.01 VW EW 0.26 0.21 0.20 0.10 VW 0.07 0.17 0.13 -0.02 Panel B. IDVOL Beta-Sorted Portfolios, Neutralizing EIDVOL EW EW 0.20 0.33 0.25 0.14 VW 0.02 0.17 0.11 0.05 VW EW 0.16 0.28 0.26 0.21 VW 0.07 0.12 0.12 0.03 Panel C. RIDVOL-Sorted Portfolios, Neutralizing IDVOL Betas EW EW 0.19 0.26 0.26 0.16 VW 0.13 0.13 0.03 -0.03 VW EW 0.17 0.26 0.27 0.17 VW 0.14 0.12 0.07 -0.06 Panel D. EIDVOL-Sorted Portfolios, Neutralizing IDVOL Betas Portfolio AIV Weight Weight L 2 M 4 EW EW -0.24 -0.16 --0.18 -0.24 VW 0.11 0.02 0.04 -0.08 VW EW -0.23 -0.17 -0.22 -0.21 VM 0.11 0.01 -0.01 -0.09 Panel A. IDVOL Beta-Sorted Portfolios, Neutralizing RIDVOL Portfolio [(H+4)- AIV Weight Weight H (H-L) (2+L)]12 EW EW -0.01 -0.36 *** -0.27 *** (-4.12) (-4.65) VW -0.04 -0.10 -0.12 (-0.83) (-1.44) VW EW 0.00 -0.25 *** -0.18 *** (-2.88) (-3.04) VW -0.07 -0.13 -0.16 ** (-1.27) (-2.02) Panel B. IDVOL Beta-Sorted Portfolios, Neutralizing EIDVOL EW EW -0.12 -0.32 *** -0.26 *** (-3.90) (-4.55) VW -0.07 -0.09 -0.11 (-0.81) (-1.34) VW EW -0.10 -0.25 *** -0.16 *** (-2.93) (2.80) VW -0.08 -0.15 -0.12 (-1.34) (-1.56) Panel C. RIDVOL-Sorted Portfolios, Neutralizing IDVOL Betas EW EW -0.07 -0.26 ** -0.18 ** (-1.97) (-2.05) VW -0.42 -0.55 *** -0.36 *** (-4.75) (-4.64) VW EW -0.07 -0.25 * -0.17 * (-1.79) (-1.86) VW -0.34 -0.49 *** -0.33 *** (-4.06) (-4.31) Panel D. EIDVOL-Sorted Portfolios, Neutralizing IDVOL Betas Portfolio [(H+4)- AIV Weight Weight H (H-L) (2+L)]12 EW EW 0.89 1.13 *** 0.52 (7.11) (5.67) VW 0.22 0.11 0.01 (0.94) (0.11) VW EW 0.90 1.13 *** 0.54 (7.05) (5.58) VW 0.31 0.19 0.04 (1.52) (0.52) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VI. Cross-Sectional Regressions of Excess Returns (%) on IDVOL Beta, RIDVOL, and EIDVOL We report the results of cross-sectional regressions of the monthly excess returns of individual stocks on IDVOL beta ([[beta].sup.IDVOL]), RIDVOL, and EIDVOL, and a variety of other control variables. See Table V for a description of RIDVOL and EIDVOL. For space, we only report the coefficients for [[beta].sup.IDVOL], RIDVOL, and EIDVOL. The market beta, InSize, InBM, [Ret.sub.t-1], and [Ret.sub.t-1,t-7] are control variables. In Panel A, we use our full sample, while in Panel B, we use the non-January subsample. See Table IV for details on [[beta].sup.IDVOL] and the control variables. In the first four columns of results of each panel, we use the EW IDVOL beta. In the last four, we use the VW IDVOL beta. Specification (4) includes all of the control variables from Table IV. We report the six-lag Newey-West (1987) t-statistics in parentheses. Panel A. Full Sample EW IDVOL Beta (1) (2) (3) (4) [[beta]. sub.IDVOL] -0.77 *** -0.79 *** -0.75 *** -0.34 ** (-4.73) (-4.63) (-4.49) (-2.03) RIDVOL -4.60 -44.48 *** -31.56 *** (-1.06) (-15.69) (-12.15) EIDVOL 17.88 *** 23.07 *** 22.95 *** (11.67) (17.15) (18.32) Panel B. Non-January Sample [[beta]. sub.IDVOL] -0.61 *** -0.64 *** -0.59 *** -0.37 ** (-3.57) (-3.51) (-3.31) (-2.18) RIDVOL -17.57 *** -53.13 *** -36.51 *** (-3.98) (-18.54) (-14.46) EIDVOL 14.06 *** 20.52 *** 21.25 *** (9.0l) (14.83) (16.81) Panel A. Full Sample VW IDVOL Beta (1) (2) (3) (4) [[beta]. sub.IDVOL] -0.37 * -0.31 -0.31 -0.14 (-1.94) (-1.62) (-1.64) (-0.67) RIDVOL -4.62 -44.45 *** -31.45 *** (-1.06) (-15.46) (-12.06) EIDVOL 17.85 *** 23.07 *** 22.97 *** (11.58) (17.10) (18.34) Panel B. Non-January Sample [[beta]. sub.IDVOL] -0.32 -0.29 -0.27 -0.19 (-1.63) (-1.38) (-1.38) (-0.94) RIDVOL -17.61 *** -53.12 *** 36.42 *** (-3.95) (-18.28) (-14.34) EIDVOL 14.03 *** 20.51 *** 21.27 *** (8.95) (14.80) (16.84) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VII. Cross-Sectional Regressions of Excess Returns (%) on IDVOL Beta and Market Volatility Beta In this table, we report the results of cross-sectional regressions of the monthly excess returns of individual stocks on IDVOL betas ([[beta].sup.IDVOL]) market volatility betas ([beta].sup.IDVOL]), and a variety of other control variables. For space, we only report the coefficients for [[beta].sup.IDVOL] and [[beta].sup.MVOL]. lnSize, lnBM, [Ret.sub.t-1], and [Ret.sub.t-1,1-7] are control variables. In Panel A, we use our full sample, while in Panel B, we use the non-January subsample. Specification (1) includes just [[beta].sup.MVOL], while Specification (2) includes just [[beta].sup.MVOL]. Specification (3) is comprised of [[beta].sub.i.sup.IDVOL] and [[beta].sup.IDVOL] while Specification (4) consists of [[beta].sup.IDVOL], [[beta].sup.MVOL] and the other control variables. See Table IV for details regarding [[beta].sup.IDVOL] and the other control variables. The market volatility measure, [V.sub.mt], is derived in Equation (10) and then replaces the market return in Equation (12). Equation (13) is re-estimated with [[beta].sup.MVOL], an additional explanatory variable. In the first four columns of each panel, we use the EW IDVOL and market volatility betas from equal weighting [V.sub.mt]. In the last four, we use the VW IDVOL and market volatility betas from value weighting [V.sub.mt]. We report the six-lag Newey-West (1987) t-statistics in brackets. Panel A. Full Sample EW IDVOL and Market Volatility Betas (1) (2) (3) (4) [[beta].sup.IDVOL] -1.02 *** -1.22 *** -0.39 ** (-4.70) (-4.37) (-2.09) [[beta].sup.MVOL] 0.12 0.15 0.01 (1.07) (1.22) (0.14) Panel B. Non-January Sample [[beta].sup.IDVOL] -0.85 *** -1.02 *** -0.46 ** (-4.11) (-3.72) (2.50) [[beta].sub.MVOL] -0.02 0.02 -0.05 (-0.14) (0.12) (-0.44) Panel A. Full Sample VW IDVOL and Market Volatility Betas (1) (2) (3) (4) [[beta].sup.IDVOL] -0.28 -0.74 ** -0.29 (-1.20) (-2.54) (-1.25) [[beta].sup.MVOL] 0.10 0.14 0.02 (0.90) (1.18) (0.16) Panel B. Non-January Sample [[beta].sup.IDVOL] -0.26 -0.66 ** -0.39 * (-1.07) (2.43) (-1.79) [[beta].sub.MVOL] -0.04 0.02 -0.04 (-0.32) (0.15) (-0.36) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level.

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Author: | Peterson, David R.; Smedema, Adam R. |
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Publication: | Financial Management |

Article Type: | Statistical data |

Geographic Code: | 1USA |

Date: | Sep 22, 2013 |

Words: | 10782 |

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