ABSTRACT
 Top of page
 ABSTRACT
 I. Pricing Systematic Volatility in the CrossSection
 II. Pricing Idiosyncratic Volatility in the CrossSection
 III. Conclusion
 REFERENCES
We examine the pricing of aggregate volatility risk in the crosssection of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, booktomarket, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility.
It is well known that the volatility of stock returns varies over time. While considerable research has examined the timeseries relation between the volatility of the market and the expected return on the market (see, among others, Campbell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the crosssection of expected stock returns has received less attention. Timevarying market volatility induces changes in the investment opportunity set by changing the expectation of future market returns, or by changing the riskreturn tradeoff. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the crosssection of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns.
The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the crosssection of expected stock returns. We want to both determine whether the volatility of the market is a priced risk factor and estimate the price of aggregate volatility risk. Many option studies have estimated a negative price of risk for market volatility using options on an aggregate market index or options on individual stocks.^{1} Using the crosssection of stock returns, rather than options on the market, allows us to create portfolios of stocks that have different sensitivities to innovations in market volatility. If the price of aggregate volatility risk is negative, stocks with large, positive sensitivities to volatility risk should have low average returns. Using the crosssection of stock returns also allows us to easily control for a battery of crosssectional effects, such as the size and value factors of Fama and French (1993), the momentum effect of Jegadeesh and Titman (1993), and the effect of liquidity risk documented by Pástor and Stambaugh (2003). Option pricing studies do not control for these crosssectional risk factors.
We find that innovations in aggregate volatility carry a statistically significant negative price of risk of approximately −1% per annum. Economic theory provides several reasons why the price of risk of innovations in market volatility should be negative. For example, Campbell (1993, 1996) and Chen (2002) show that investors want to hedge against changes in market volatility, because increasing volatility represents a deterioration in investment opportunities. Riskaverse agents demand stocks that hedge against this risk. Periods of high volatility also tend to coincide with downward market movements (see French, Schwert, and Stambaugh (1987) and Campbell and Hentschel (1992)). As Bakshi and Kapadia (2003) comment, assets with high sensitivities to market volatility risk provide hedges against market downside risk. The higher demand for assets with high systematic volatility loadings increases their price and lowers their average return. Finally, stocks that do badly when volatility increases tend to have negatively skewed returns over intermediate horizons, while stocks that do well when volatility rises tend to have positively skewed returns. If investors have preferences over coskewness (see Harvey and Siddique (2000)), stocks that have high sensitivities to innovations in market volatility are attractive and have low returns.^{2}
The second goal of the paper is to examine the crosssectional relationship between idiosyncratic volatility and expected returns, where idiosyncratic volatility is defined relative to the standard Fama and French (1993) model.^{3} If the Fama–French model is correct, forming portfolios by sorting on idiosyncratic volatility will obviously provide no difference in average returns. Nevertheless, if the Fama–French model is false, sorting in this way potentially provides a set of assets that may have different exposures to aggregate volatility and hence different average returns. Our logic is the following. If aggregate volatility is a risk factor that is orthogonal to existing risk factors, the sensitivity of stocks to aggregate volatility times the movement in aggregate volatility will show up in the residuals of the Fama–French model. Firms with greater sensitivities to aggregate volatility should therefore have larger idiosyncratic volatilities relative to the Fama–French model, everything else being equal. Differences in the volatilities of firms' true idiosyncratic errors, which are not priced, will make this relation noisy. We should be able to average out this noise by constructing portfolios of stocks to reveal that larger idiosyncratic volatilities relative to the Fama–French model correspond to greater sensitivities to movements in aggregate volatility and thus different average returns, if aggregate volatility risk is priced.
While high exposure to aggregate volatility risk tends to produce low expected returns, some economic theories suggest that idiosyncratic volatility should be positively related to expected returns. If investors demand compensation for not being able to diversify risk (see Malkiel and Xu (2002) and Jones and RhodesKropf (2003)), then agents will demand a premium for holding stocks with high idiosyncratic volatility. Merton (1987) suggests that in an informationsegmented market, firms with larger firmspecific variances require higher average returns to compensate investors for holding imperfectly diversified portfolios. Some behavioral models, like Barberis and Huang (2001), also predict that higher idiosyncratic volatility stocks should earn higher expected returns. Our results are directly opposite to these theories. We find that stocks with high idiosyncratic volatility have low average returns. There is a strongly significant difference of −1.06% per month between the average returns of the quintile portfolio with the highest idiosyncratic volatility stocks and the quintile portfolio with the lowest idiosyncratic volatility stocks.
In contrast to our results, earlier researchers either find a significantly positive relation between idiosyncratic volatility and average returns, or they fail to find any statistically significant relation between idiosyncratic volatility and average returns. For example, Lintner (1965) shows that idiosyncratic volatility carries a positive coefficient in crosssectional regressions. Lehmann (1990) also finds a statistically significant, positive coefficient on idiosyncratic volatility over his full sample period. Similarly, Tinic and West (1986) and Malkiel and Xu (2002) unambiguously find that portfolios with higher idiosyncratic volatility have higher average returns, but they do not report any significance levels for their idiosyncratic volatility premiums. On the other hand, Longstaff (1989) finds that a crosssectional regression coefficient on total variance for sizesorted portfolios carries an insignificant negative sign.
The difference between our results and the results of past studies is that the past literature either does not examine idiosyncratic volatility at the firm level, or does not directly sort stocks into portfolios ranked on this measure of interest. For example, Tinic and West (1986) work only with 20 portfolios sorted on market beta, while Malkiel and Xu (2002) work only with 100 portfolios sorted on market beta and size. Malkiel and Xu (2002) only use the idiosyncratic volatility of one of the 100 beta/size portfolios to which a stock belongs to proxy for that stock's idiosyncratic risk and, thus, do not examine firmlevel idiosyncratic volatility. Hence, by not directly computing differences in average returns between stocks with low and high idiosyncratic volatilities, previous studies miss the strong negative relation between idiosyncratic volatility and average returns that we find.
The low average returns to stocks with high idiosyncratic volatilities could arise because stocks with high idiosyncratic volatilities may have high exposure to aggregate volatility risk, which lowers their average returns. We investigate this conjecture and find that this is not a complete explanation. Our idiosyncratic volatility results are also robust to controlling for value, size, liquidity, volume, dispersion of analysts' forecasts, and momentum effects. We find the effect robust to different formation periods for computing idiosyncratic volatility and for different holding periods. The effect also persists in bull and bear markets, recessions and expansions, and volatile and stable periods. Hence, our results on idiosyncratic volatility represent a substantive puzzle.
The rest of this paper is organized as follows. In Section I, we examine how aggregate volatility is priced in the crosssection of stock returns. Section II documents that firms with high idiosyncratic volatility have very low average returns. Finally, Section III concludes.
II. Pricing Idiosyncratic Volatility in the CrossSection
 Top of page
 ABSTRACT
 I. Pricing Systematic Volatility in the CrossSection
 II. Pricing Idiosyncratic Volatility in the CrossSection
 III. Conclusion
 REFERENCES
The previous section examines how systematic volatility risk affects crosssectional average returns by focusing on portfolios of stocks sorted by their sensitivities to innovations in aggregate volatility. In this section, we investigate a second set of assets sorted by idiosyncratic volatility defined relative to the FF3 model. If market volatility risk is a missing component of systematic risk, standard models of systematic risk, such as the CAPM or the FF3 model, should misprice portfolios sorted by idiosyncratic volatility because these models do not include factor loadings measuring exposure to market volatility risk.
A. Estimating Idiosyncratic Volatility
A.1. Definition of Idiosyncratic Volatility
Given the failure of the CAPM to explain crosssectional returns and the ubiquity of the FF3 model in empirical financial applications, we concentrate on idiosyncratic volatility measured relative to the FF3 model
 (8)
We define idiosyncratic risk as in equation (8). When we refer to idiosyncratic volatility, we mean idiosyncratic volatility relative to the FF3 model. We also consider sorting portfolios on total volatility, without using any control for systematic risk.
A.2. A Trading Strategy
To examine trading strategies based on idiosyncratic volatility, we describe portfolio formation strategies based on an estimation period of L months, a waiting period of M months, and a holding period of N months. We describe an L/M/N strategy as follows. At month t, we compute idiosyncratic volatilities from the regression (8) on daily data over an Lmonth period from month t−L−M to month t−M. At time t, we construct valueweighted portfolios based on these idiosyncratic volatilities and hold these portfolios for N months. We concentrate most of our analysis on the 1/0/1 strategy, in which we simply sort stocks into quintile portfolios based on their level of idiosyncratic volatility computed using daily returns over the past month, and we hold these valueweighted portfolios for 1 month. The portfolios are rebalanced each month. We also examine the robustness of our results to various choices of L, M, and N.
The construction of the L/M/N portfolios for L > 1 and N > 1 follows Jegadeesh and Titman (1993), except our portfolios are value weighted. For example, to construct the 12/1/12 quintile portfolios, each month we construct a valueweighted portfolio based on idiosyncratic volatility computed from daily data over the 12 months of returns ending 1 month prior to the formation date. Similarly, we form a valueweighted portfolio based on 12 months of returns ending 2 months prior, 3 months prior, and so on up to 12 months prior. Each of these portfolios is value weighted. We then take the simple average of these 12 portfolios. Hence, each quintile portfolio changes 1/12th of its composition each month, where each 1/12th part of the portfolio consists of a valueweighted portfolio. The first (fifth) quintile portfolio consists of 1/12th of the lowest valueweighted (highest) idiosyncratic stocks from 1 month ago, 1/12th of the valueweighted lowest (highest) idiosyncratic stocks from 2 months ago, etc.
B. Patterns in Average Returns for Idiosyncratic Volatility
Table VI reports average returns of portfolios sorted on total volatility, with no controls for systematic risk, in Panel A and of portfolios sorted on idiosyncratic volatility in Panel B.^{12} We use a 1/0/1 strategy in both cases. Panel A shows that average returns increase from 1.06% per month going from quintile 1 (low total volatility stocks) to 1.22% per month for quintile 3. Then, average returns drop precipitously. Quintile 5, which contains stocks with the highest total volatility, has an average total return of only 0.09% per month. The FF3 alpha for quintile 5, reported in the last column, is −1.16% per month, which is highly statistically significant. The difference in the FF3 alphas between portfolio 5 and portfolio 1 is −1.19% per month, with a robust tstatistic of −5.92.
Table VI. Portfolios Sorted by Volatility We form valueweighted quintile portfolios every month by sorting stocks based on total volatility and idiosyncratic volatility relative to the Fama–French (1993) model. Portfolios are formed every month, based on volatility computed using daily data over the previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) volatilities. The statistics in the columns labeled Mean and Std. Dev. are measured in monthly percentage terms and apply to total, not excess, simple returns. Size reports the average log market capitalization for firms within the portfolio and B/M reports the average booktomarket ratio. The row “51” refers to the difference in monthly returns between portfolio 5 and portfolio 1. The Alpha columns report Jensen's alpha with respect to the CAPM or Fama–French (1993) threefactor model. Robust Newey–West (1987)tstatistics are reported in square brackets. Robust joint tests for the alphas equal to zero are all less than 1% for all cases. The sample period is July 1963 to December 2000. Rank  Mean  Std. Dev.  % Mkt Share  Size  B/M  CAPM Alpha  FF3 Alpha 

Panel A: Portfolios Sorted by Total Volatility 
1  1.06  3.71  41.7%  4.66  0.88  0.14  0.03 
[1.84]  [0.53] 
2  1.15  4.48  33.7%  4.70  0.81  0.13  0.08 
[2.14]  [1.41] 
3  1.22  5.63  15.5%  4.10  0.82  0.07  0.12 
[0.72]  [1.55] 
4  0.99  7.15  6.7%  3.47  0.86  −0.28  −0.17 
[−1.73]  [−1.42] 
5  0.09  8.30  2.4%  2.57  1.08  −1.21  −1.16 
[−5.07]  [−6.85] 
51  −0.97   −1.35  −1.19 
[−2.86]   [−4.62]  [−5.92] 

Panel B: Portfolios Sorted by Idiosyncratic Volatility Relative to FF3 
1  1.04  3.83  53.5%  4.86  0.85  0.11  0.04 
[1.57]  [0.99] 
2  1.16  4.74  27.4%  4.72  0.80  0.11  0.09 
[1.98]  [1.51] 
3  1.20  5.85  11.9%  4.07  0.82  0.04  0.08 
[0.37]  [1.04] 
4  0.87  7.13  5.2%  3.42  0.87  −0.38  −0.32 
[−2.32]  [−3.15] 
5  −0.02  8.16  1.9%  2.52  1.10  −1.27  −1.27 
[−5.09]  [−7.68] 
51  −1.06   −1.38  −1.31 
[−3.10]   [−4.56]  [−7.00] 
We obtain similar patterns in Panel B, where the portfolios are sorted on idiosyncratic volatility. The difference in raw average returns between quintile portfolios 5 and 1 is −1.06% per month. The FF3 model is clearly unable to price these portfolios since the difference in the FF3 alphas between portfolio 5 and portfolio 1 is −1.31% per month, with a tstatistic of −7.00. The size and booktomarket ratios of the quintile portfolios sorted by idiosyncratic volatility also display distinct patterns. Stocks with low (high) idiosyncratic volatility are generally large (small) stocks with low (high) booktomarket ratios. The risk adjustment of the FF3 model predicts that quintile 5 stocks should have high, not low, average returns.
The findings in Table VI are provocative, but there are several concerns raised by the anomalously low returns of quintile 5. For example, although quintile 5 contains 20% of the stocks sorted by idiosyncratic volatility, quintile 5 is only a small proportion of the value of the market (only 1.9% on average). Are these patterns repeated if we only consider large stocks, or only stocks traded on the NYSE? The next section examines these questions. We also examine whether the phenomena persist if we control for a large number of crosssectional effects that the literature has identified either as potential risk factors or anomalies. In particular, we control for size, booktomarket, leverage, liquidity, volume, turnover, bid–ask spreads, coskewness, dispersion in analysts' forecasts, and momentum effects.
C. Controlling for Various CrossSectional Effects
Table VII examines the robustness of our results with the 1/0/1 idiosyncratic volatility portfolio formation strategy to various crosssectional risk factors. The table reports FF3 alphas, the difference in FF3 alphas between the quintile portfolios with the highest and lowest idiosyncratic volatilities, together with tstatistics to test their statistical significance.^{13} All the portfolios formed on idiosyncratic volatility remain value weighted.
Table VII. Alphas of Portfolios Sorted on Idiosyncratic Volatility The table reports Fama and French (1993) alphas, with robust Newey–West (1987)tstatistics in square brackets. All the strategies are 1/0/1 strategies described in Section II.A for idiosyncratic volatility computed relative to FF3, but control for various effects. The column “51” refers to the difference in FF3 alphas between portfolio 5 and portfolio 1. In the panel labeled “NYSE Stocks Only,” we sort stocks into quintile portfolios based on their idiosyncratic volatility, relative to the FF3 model, using only NYSE stocks. We use daily data over the previous month and rebalance monthly. In the panel labeled “Size Quintiles,” each month we first sort stocks into five quintiles on the basis of size. Then, within each size quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility. In the panels controlling for size, liquidity volume, and momentum, we perform a double sort. Each month, we first sort stocks based on the first characteristic (size, booktomarket, leverage, liquidity, volume, turnover, bid–ask spreads, or dispersion of analysts' forecasts) and then, within each quintile we sort stocks based on idiosyncratic volatility relative to the FF3 model. The five idiosyncratic volatility portfolios are then averaged over each of the five characteristic portfolios. Hence, they represent idiosyncratic volatility quintile portfolios controlling for the characteristic. Liquidity represents the Pástor and Stambaugh (2003) historical liquidity beta, leverage is defined as the ratio of total book value of assets to book value of equity, volume represents average dollar volume over the previous month, turnover represents volume divided by the total number of shares outstanding over the past month, and the bid–ask spread control represents the average daily bid–ask spread over the previous month. The coskewness measure is computed following Harvey and Siddique (2000) and the dispersion of analysts' forecasts is computed by Diether et al. (2002). The sample period is July 1963 to December 2000 for all controls with the exceptions of liquidity (February 1968 to December 2000), the dispersion of analysts' forecasts (February 1983 to December 2000), and the control for aggregate volatility risk (January 1986 to December 2000). All portfolios are value weighted.  Ranking on Idiosyncratic Volatility  51 

1 Low  2  3  4  5 High 

NYSE Stocks Only  0.06  0.04  0.02  −0.04  −0.60  −0.66 
[1.20]  [0.75]  [0.30]  [−0.40]  [−5.14]  [−4.85] 
Size Quintiles  Small 1  0.11  0.26  0.31  0.06  −0.43  −0.55 
[0.72]  [1.56]  [1.76]  [0.29]  [−1.54]  [−1.84] 
2  0.19  0.20  −0.07  −0.65  −1.73  −1.91 
[1.49]  [1.74]  [−0.67]  [−5.19]  [−8.14]  [−7.69] 
3  0.12  0.21  0.03  −0.27  −1.49  −1.61 
[1.23]  [2.40]  [0.38]  [−3.36]  [−10.1]  [−7.65] 
4  0.03  0.22  0.17  −0.03  −0.82  −0.86 
[0.37]  [2.57]  [2.47]  [−0.45]  [−6.61]  [−4.63] 
Large 5  0.09  0.04  0.03  0.14  −0.17  −0.26 
[1.62]  [0.72]  [0.51]  [1.84]  [−1.40]  [−1.74] 
Controlling for Size  0.11  0.18  0.09  −0.15  −0.93  −1.04 
[1.30]  [2.49]  [1.35]  [−1.99]  [−6.81]  [−5.69] 
Controlling for BooktoMarket  0.61  0.69  0.71  0.50  −0.19  −0.80 
[3.02]  [2.80]  [2.49]  [1.47]  [−0.48]  [−2.90] 
Controlling for Leverage  0.11  0.11  0.08  −0.24  −1.12  −1.23 
[2.48]  [2.20]  [1.19]  [−2.45]  [−7.81]  [−7.61] 
Controlling for Liquidity  0.08  0.09  −0.01  −0.16  −1.01  −1.08 
[1.71]  [1.53]  [−0.09]  [−1.62]  [−8.61]  [−7.98] 
Controlling for Volume  −0.03  0.02  −0.01  −0.39  −1.25  −1.22 
[−0.49]  [0.39]  [−0.32]  [−7.11]  [−10.9]  [−8.04] 
Controlling for Turnover  0.11  0.03  −0.11  −0.49  −1.34  −1.46 
[2.49]  [0.58]  [−1.79]  [−6.27]  [−11.0]  [−10.7] 
Controlling for Bid–Ask Spreads  −0.07  −0.01  −0.09  −0.49  −1.26  −1.19 
[−1.21]  [−0.18]  [−1.14]  [−5.36]  [−9.13]  [−6.95] 
Controlling for Coskewness  −0.02  −0.00  0.01  −0.37  −1.40  −1.38 
[−0.32]  [−0.02]  [0.08]  [−2.30]  [−6.07]  [−5.02] 
Controlling for Dispersion in Analysts' Forecasts  0.12  −0.07  0.11  0.01  −0.27  −0.39 
[1.57]  [−0.76]  [1.12]  [0.09]  [−1.76]  [−2.09] 
C.1. Using Only NYSE Stocks
We examine the interaction of the idiosyncratic volatility effect with firm size in two ways. First, we rank stocks based on idiosyncratic volatility using only NYSE stocks. Excluding NASDAQ and AMEX has little effect on our results. The highest quintile of idiosyncratic volatility stocks has an FF3 alpha of −0.60% per month. The 51 difference in FF3 alphas is still large in magnitude, at −0.66% per month, with a tstatistic of −4.85. While restricting the universe of stocks to only the NYSE mitigates the concern that the idiosyncratic volatility effect is concentrated among small stocks, it does not completely remove this concern because the NYSE universe still contains small stocks.
C.2. Controlling for Size
Our second examination of the interaction of idiosyncratic volatility and size uses all firms. We control for size by first forming quintile portfolios ranked on market capitalization. Then, within each size quintile, we sort stocks into quintile portfolios ranked on idiosyncratic volatility. Thus, within each size quintile, quintile 5 contains the stocks with the highest idiosyncratic volatility.
The second panel of Table VII shows that in each size quintile, the highest idiosyncratic volatility quintile has a dramatically lower FF3 alpha than the other quintiles. The effect is not most pronounced among the smallest stocks. Rather, quintiles 24 have the largest 51 differences in FF3 alphas, at −1.91%, −1.61%, and −0.86% per month, respectively. The average market capitalization of quintiles 24 is, on average, 21% of the market. The tstatistics of these alphas are all above 4.5 in absolute magnitude. In contrast, the 51 alphas for the smallest and largest quintiles are actually statistically insignificant at the 5% level. Hence, it is not small stocks that are driving these results.
The row labeled “Controlling for Size” averages across the five size quintiles to produce quintile portfolios with dispersion in idiosyncratic volatility, but which contain all sizes of firms. After controlling for size, the 51 difference in FF3 alphas is still −1.04% per month. Thus, market capitalization does not explain the low returns to high idiosyncratic volatility stocks.
In the remainder of Table VII, we repeat the explicit doublesort characteristic controls, replacing size with other stock characteristics. We first form portfolios based on a particular characteristic, then we sort on idiosyncratic volatility, and finally we average across the characteristic portfolios to create portfolios that have dispersion in idiosyncratic volatility but contain all aspects of the characteristic.
C.3. Controlling for BooktoMarket Ratios
It is generally thought that high booktomarket firms have high average returns. Thus, in order for the booktomarket effect to be an explanation of the idiosyncratic volatility effect, the high idiosyncratic volatility portfolios must be primarily composed of growth stocks that have lower average returns than value stocks. The row labeled “Controlling for BooktoMarket” shows that this is not the case. When we control for booktomarket ratios, stocks with the lowest idiosyncratic volatility have high FF3 alphas, and the 51 difference in FF3 alphas is −0.80% per month, with a tstatistic of −2.90.
C.4. Controlling for Leverage
Leverage increases expected equity returns, holding asset volatility and asset expected returns constant. Asset volatility also prevents firms from increasing leverage. Hence, firms with high idiosyncratic volatility could have high asset volatility but relatively low equity returns because of low leverage. The next line of Table VII shows that leverage cannot be an explanation of the idiosyncratic volatility effect. We measure leverage as the ratio of total book value of assets to book value of equity. After controlling for leverage, the difference between the 51 alphas is −1.23% per month, with a tstatistic of −7.61.
C.5. Controlling for Liquidity Risk
Pástor and Stambaugh (2003) argue that liquidity is a systematic risk. If liquidity is to explain the idiosyncratic volatility effect, high idiosyncratic volatility stocks must have low liquidity betas, giving them low returns. We check this explanation by using the historical Pástor–Stambaugh liquidity betas to measure exposure to liquidity risk. Controlling for liquidity does not remove the low average returns of high idiosyncratic volatility stocks. The 51 difference in FF3 alphas remains large at −1.08% per month, with a tstatistic of −7.98.
C.6. Controlling for Volume
Gervais et al. (2001) find that stocks with higher volume have higher returns. Perhaps stocks with high idiosyncratic volatility are merely stocks with low trading volume? When we control for trading volume over the past month, the 51 difference in alphas is −1.22% per month, with a tstatistic of −8.04. Hence, the low returns on high idiosyncratic volatility stocks are robust to controlling for volume effects.
C.7. Controlling for Turnover
Our next control is turnover, measured as trading volume divided by the total number of shares outstanding over the previous month. Turnover is a noisy proxy for liquidity. Table VII shows that the low alphas on high idiosyncratic volatility stocks are robust to controlling for turnover. The 51 difference in FF3 alphas is −1.19% per month, and it is highly significant with a tstatistic of −8.04. Examination of the individual turnover quintiles (not reported) indicates that the 51 differences in alphas are most pronounced in the quintile portfolio with the highest, not the lowest, turnover.
C.8. Controlling for BidAsk Spreads
An alternative liquidity control is the bid–ask spread, which we measure as the average daily bid–ask spread over the previous month for each stock. In order for bid–ask spreads to be an explanation, high idiosyncratic volatility stocks must have low bid–ask spreads and corresponding low returns. Controlling for bid–ask spreads does little to remove the effect. The FF3 alpha of the highest idiosyncratic volatility portfolio is −1.26%, while the 51 difference in alphas is −1.19% and remains highly statistically significant with a tstatistic of −6.95.
C.9. Controlling for Coskewness Risk
Harvey and Siddique (2000) find that stocks with more negative coskewness have higher returns. Stocks with high idiosyncratic volatility may have positive coskewness, giving them low returns. Computing coskewness following Harvey and Siddique (2000), we find that exposure to coskewness risk is not an explanation. The FF3 alpha for the 51 portfolio is −1.38% per month, with a tstatistic of −5.02.
C.10. Controlling for Dispersion in Analysts' Forecasts
Diether, Malloy, and Scherbina (2002) provide evidence that stocks with higher dispersion in analysts' earnings forecasts have lower average returns than stocks with low dispersion of analysts' forecasts. They argue that dispersion in analysts' forecasts measures differences of opinion among investors. Miller (1977) shows that if there are large differences in stock valuations and short sale constraints, equity prices tend to reflect the view of the more optimistic agents, which leads to low future returns for stocks with large dispersion in analysts' forecasts.
If stocks with high dispersion in analysts' forecasts tend to be more volatile stocks, then we may be finding a similar anomaly to Diether et al. (2002). Over Diether et al.'s sample period, 1983–2000, we test this hypothesis by performing a characteristic control for the dispersion of analysts' forecasts. We take the quintile portfolios of stocks sorted on increasing dispersion of analysts' forecasts (Table VI of Diether et al. (2002, p. 2128)) and within each quintile sort stocks on idiosyncratic volatility. Note that this universe of stocks contains mostly large firms, where the idiosyncratic volatility effect is weaker, because multiple analysts usually do not make forecasts for small firms.
The last two lines of Table VII present the results for averaging the idiosyncratic volatility portfolios across the forecast dispersion quintiles. The 51 difference in alphas is still −0.39% per month, with a robust tstatistic of −2.09. While the shorter sample period may reduce power, the dispersion of analysts' forecasts reduces the noncontrolled 51 alpha considerably. However, dispersion in analysts' forecasts cannot account for all of the low returns to stocks with high idiosyncratic volatility.^{14}
D. A Detailed Look at Momentum
Hong, Lim, and Stein (2000) argue that the momentum effect documented by Jegadeesh and Titman (1993) is asymmetric and has a stronger negative effect on declining stocks than a positive effect on rising stocks. A potential explanation behind the idiosyncratic volatility results is that stocks with very low returns have very high volatility. Of course, stocks that are past winners also have very high volatility, but loser stocks could be overrepresented in the high idiosyncratic volatility quintile.
In Table VIII, we perform a series of robustness tests of the idiosyncratic volatility effect to this possible momentum explanation. In Panel A, we perform 5 × 5 characteristic sorts first over past returns, and then over idiosyncratic volatility. We average over the momentum quintiles to produce quintile portfolios sorted by idiosyncratic risk that control for past returns. We control for momentum over the previous 1 month, 6 months, and 12 months. Table VIII shows that momentum is not driving the results. Controlling for returns over the past month does not remove the very low FF3 alpha of quintile 5 (−0.59% per month), and the 51 difference in alphas is still −0.66% per month, which is statistically significant at the 1% level. When we control for past 6month returns, the FF3 alpha of the 51 portfolio increases in magnitude to −1.10% per month. For past 12month returns, the 51 alpha is even larger in magnitude at −1.22% per month. All these differences are highly statistically significant.
Table VIII. Alphas of Portfolios Sorted on Idiosyncratic Volatility Controlling for Past Returns The table reports Fama and French (1993) alphas, with robust Newey–West (1987)tstatistics in square brackets. All the strategies are 1/0/1 strategies described in Section II.A, but control for past returns. The column “51” refers to the difference in FF3 alphas between portfolio 5 and portfolio 1. In the first three rows labeled “Past 1month” to “Past 12months,” we control for the effect of momentum. We first sort all stocks on the basis of past returns, over the appropriate formation period, into quintiles. Then, within each momentum quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility, relative to the FF3 model. The five idiosyncratic volatility portfolios are then averaged over each of the five characteristic portfolios. Hence, they represent idiosyncratic volatility quintile portfolios controlling for momentum. The second part of the panel lists the FF3 alphas of idiosyncratic volatility quintile portfolios within each of the past 12month return quintiles. All portfolios are value weighted. The sample period is July 1963 to December 2000.  Ranking on Idiosyncratic Volatility  51 

1 Low  2  3  4  5 High 

Panel A: Controlling for Momentum 
Past 1 month  0.07  0.08  0.09  −0.05  −0.59  −0.66 
[0.43]  [0.94]  [1.26]  [−0.47]  [−3.60]  [−2.71] 
Past 6 months  −0.01  −0.12  −0.28  −0.45  −1.11  −1.10 
[−0.20]  [−1.86]  [−3.60]  [−5.20]  [−9.35]  [−7.18] 
Past 12 months  0.01  −0.05  −0.28  −0.64  −1.21  −1.22 
[0.15]  [−0.76]  [−3.56]  [−6.95]  [−11.5]  [−9.20] 

Panel B: Past 12Month Quintiles 
Losers 1  −0.41  −0.83  −1.44  −2.11  −2.66  −2.25 
[−1.94]  [−3.90]  [−6.32]  [−9.40]  [−10.6]  [−7.95] 
2  −0.08  −0.24  −0.64  −1.09  −1.70  −1.62 
[−0.49]  [−1.58]  [−4.40]  [−6.46]  [−8.90]  [−7.00] 
3  −0.06  −0.11  −0.26  −0.48  −1.03  −0.97 
[−0.52]  [−1.16]  [−2.15]  [−3.49]  [−7.93]  [−5.85] 
4  0.15  0.07  0.23  −0.03  −0.65  −0.80 
[1.57]  [0.65]  [2.27]  [−0.29]  [−4.76]  [−4.89] 
Winners 5  0.45  0.85  0.71  0.52  −0.03  −0.48 
[3.52]  [5.44]  [3.97]  [2.63]  [−0.13]  [−2.01] 
In Panel B, we closely examine the individual 5 × 5 FF3 alphas of the quintile portfolios sorted on past 12month returns and idiosyncratic volatility. Note that if we average these portfolios across the past 12month quintile portfolios, and then compute alphas, we obtain the alphas in the row labeled “Past 12months” in Panel A of Table VIII. This more detailed view of the interaction between momentum and idiosyncratic volatility reveals several interesting facts.
First, the low returns to high idiosyncratic volatility are most pronounced for loser stocks. The 51 differences in alphas range from −2.25% per month for the loser stocks, to −0.48% per month for the winner stocks. Second, the tendency for the momentum effect to be concentrated more among loser, rather than winner, stocks cannot account for all of the low returns to high idiosyncratic volatility stocks. The idiosyncratic volatility effect appears significantly in every past return quintile. Hence, stocks with high idiosyncratic volatility earn low average returns, no matter whether these stocks are losers or winners.
Finally, the momentum effect itself is also asymmetric across the idiosyncratic volatility quintiles. In the first two idiosyncratic volatility quintiles, the alphas of losers (winners) are roughly symmetrical. For example, for stocks with the lowest idiosyncratic volatilities, the loser (winner) alpha is −0.41% (0.45%). In the second idiosyncratic volatility quintile, the loser (winner) alpha is −0.83% (0.85%). However, as idiosyncratic volatility becomes very high, the momentum effect becomes highly skewed towards extremely low returns on stocks with high idiosyncratic volatility. Hence, one way to improve the returns to a momentum strategy is to short past losers with high idiosyncratic volatility.
E. Is It Exposure to Aggregate Volatility Risk?
A possible explanation for the large negative returns of high idiosyncratic volatility stocks is that stocks with large idiosyncratic volatilities have large exposure to movements in aggregate volatility. We examine this possibility in Table IX. The first row of Panel A reports the results of quintile sorts on idiosyncratic volatility, controlling for β_{ΔVIX}. This is done by first sorting on β_{ΔVIX} and then on idiosyncratic volatility, and then averaging across the β_{ΔVIX} quintiles. We motivate using past β_{ΔVIX} as a control for aggregate volatility risk because we have shown that stocks with past high β_{ΔVIX} loadings have high future exposure to the FVIXmimicking volatility factor.
Table IX. The Idiosyncratic Volatility Effect Controlling for Aggregate Volatility Risk We control for exposure to aggregate volatility using the β_{ΔVIX} loading at the beginning of the month, computed using daily data over the previous month following equation (3). We first sort all stocks on the basis of β_{ΔVIX} into quintiles. Then, within each β_{ΔVIX} quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility, relative to the FF3 model. In Panel A, we report FF3 alphas of these portfolios. We average the five idiosyncratic volatility portfolios over each of the five β_{ΔVIX} portfolios. Hence, these portfolios represent idiosyncratic volatility quintile portfolios controlling for exposure to aggregate volatility risk. The column “51” refers to the difference in FF3 alphas between portfolio 5 and portfolio 1. In Panel B, we report ex post FVIX factor loadings from a regression of each of the 25 β_{ΔVIX}× idiosyncratic volatility portfolios onto the Fama–French (1993) model augmented with FVIX as in equation (6). Robust Newey–West (1987)tstatistics are reported in square brackets. All portfolios are value weighted. The sample period is from January 1986 to December 2000. Panel A: FF3 Alphas 

 Ranking on Idiosyncratic Volatility  51 

1 Low  2  3  4  5 High 

Controlling for Exposure to Aggregate Volatility Risk  0.05  0.01  −0.14  −0.49  −1.14  −1.19 
[0.83]  [0.09]  [−1.14]  [−3.08]  [−5.00]  [−4.72] 

Panel B: FVIX Factor Loadings 
 Ranking on Idiosyncratic Volatility 
 1 Low  2  3  4  5 High 

β_{ΔVIX} Quintiles  Low 1  −6.40  −1.98  −0.55  8.80  7.51 
[−3.82]  [−0.78]  [−0.23]  [2.16]  [2.31] 
2  −2.66  −3.21  0.06  −3.04  5.37 
[−2.27]  [−2.06]  [0.05]  [−2.00]  [1.80] 
3  −6.51  −2.74  −1.93  −0.31  7.25 
[−4.50]  [−2.41]  [−1.14]  [−0.29]  [3.37] 
4  5.65  3.73  3.50  1.33  8.22 
[2.31]  [2.08]  [2.83]  [0.87]  [3.97] 
High 5  7.53  2.46  8.60  7.53  5.79 
[5.16]  [1.16]  [3.72]  [2.53]  [1.65] 
Panel A of Table IX shows that after controlling for aggregate volatility exposure, the 51 alpha is −1.19% per month, almost unchanged from the 51 quintile idiosyncratic volatility FF3 alpha of −1.31% in Table VI with no control for systematic volatility exposure. Hence, it seems that β_{ΔVIX} accounts for very little of the low average returns of high idiosyncratic volatility stocks. Panel B of Table IX reports ex post FVIX factor loadings of the 5 × 5 β_{ΔVIX} and idiosyncratic volatility portfolios, where we compute the postformation FVIX factor loadings using equation (6). We cannot interpret the alphas from this regression, because FVIX is not a tradeable factor, but the FVIX factor loadings give us a picture of how exposure to aggregate volatility risk may account for the spreads in average returns on the idiosyncratic volatility sorted portfolios.
Panel B shows that in the first three β_{ΔVIX} quintiles, we obtain almost monotonically increasing FVIX factor loadings that start with large negative ex post β_{FVIX} loadings for low idiosyncratic volatility portfolios and end with large positive ex post β_{FVIX} loadings. However, for the two highest past β_{ΔVIX} quintiles, the FVIX factor loadings have absolutely no explanatory power. In summary, exposure to aggregate volatility partially explains the puzzling low returns to high idiosyncratic volatility stocks, but only for stocks with very negative and low past loadings to aggregate volatility innovations.
F. Robustness to Different Formation and Holding Periods
If risk cannot explain the low returns to high idiosyncratic volatility stocks, are there other explanations? To help disentangle various stories, Table X reports FF3 alphas of other L/M/N strategies described in Section II.A. First, we examine possible contemporaneous measurement errors in the 1/0/1 strategy by setting M= 1. Allowing for a 1month lag between the measurement of volatility and the formation of the portfolio ensures that the portfolios are formed only with information definitely available at time t. The top row of Table X shows that the 51 FF3 alpha on the 1/1/1 strategy is −0.82% per month, with a tstatistic of −4.63.
Table X. Quintile Portfolios of Idiosyncratic Volatility for L/M/N Strategies The table reports Fama and French (1993) alphas, with robust Newey–West (1987)tstatistics in square brackets. The column “51” refers to the difference in FF3 alphas between portfolio 5 and portfolio 1. We rank stocks into quintile portfolios of idiosyncratic volatility, relative to FF3, using L/M/N strategies described in Section II.A. At month t, we compute idiosyncratic volatilities from the regression (8) on daily data over an L month period from months t−L−M to month t−M. At time t, we construct valueweighted portfolios based on these idiosyncratic volatilities and hold these portfolios for N months, following Jegadeesh and Titman (1993), except our portfolios are value weighted. The sample period is July 1963 to December 2000. Strategy  Ranking on Idiosyncratic Volatility  51 

1 low  2  3  4  5 High 

1/1/1  0.06  0.04  0.09  −0.18  −0.82  −0.88 
[1.47]  [0.77]  [1.15]  [−1.78]  [−4.88]  [−4.63] 
1/1/12  0.03  0.02  −0.02  −0.17  −0.64  −0.67 
[0.91]  [0.43]  [−0.37]  [−1.79]  [−5.27]  [−4.71] 
12/1/1  0.04  0.08  −0.01  −0.29  −1.08  −1.12 
[1.15]  [1.32]  [−0.08]  [−2.02]  [−5.36]  [−5.13] 
12/1/12  0.04  0.04  −0.02  −0.35  −0.73  −0.77 
[1.10]  [0.54]  [−0.23]  [−2.80]  [−4.71]  [−4.34] 
A possible behavioral explanation for our results is that higher idiosyncratic volatility does earn higher returns over longer horizons than 1 month, but shortterm overreaction forces returns to be low in the first month. If we hold high idiosyncratic volatility stocks for a long horizon (N= 12 months), we might see a positive relation between past idiosyncratic volatility and future average returns. The second row of Table X shows that this is not the case. For the 1/1/12 strategy, we still see very low FF3 alphas for quintile 5, and the 51 difference in alphas is still −0.67% per month, which is highly significant.
By restricting the formation period to L= 1 month, our previous results may just be capturing various shortterm events that affect idiosyncratic volatility. For example, the portfolio of stocks with high idiosyncratic volatility may be largely composed of stocks that have just made, or are just about to make, earnings announcements. To ensure that we are not capturing specific shortterm corporate events, we extend our formation period to L= 12 months. The third row of Table X reports FF3 alphas for a 12/1/1 strategy. Using one entire year of daily data to compute idiosyncratic volatility does not remove the anomalous high idiosyncratic volatilitylow average return pattern: The 51 difference in alphas is −1.12% per month. Similarly, the patterns are robust for the 12/1/12 strategy, which has a 51 alpha of −0.77% per month.
G. Subsample Analysis
Table XI investigates the robustness of the low returns to stocks with high idiosyncratic volatility over different subsamples. First, the effect is observed in every decade from the 1960s to the 1990s. The largest difference in alphas between portfolio 5 and portfolio 1 occurs during the 1980s, with an FF3 alpha of −2.23% per month, and we observe the smallest magnitude of the FF3 alpha of the 51 portfolio during the 1970s, during which time the FF3 alpha is −0.77% per month. In every decade, the effect is highly statistically significant.
Table XI. The Idiosyncratic Volatility Effect over Different Subsamples The table reports Fama and French (1993) alphas, with robust Newey–West (1987)tstatistics in square brackets. The column “51” refers to the difference in FF3 alphas between portfolio 5 and portfolio 1. We rank stocks into quintile portfolios of idiosyncratic volatility, relative to FF3, using the 1/0/1 strategy described in Section II.A and examine robustness over different sample periods. The stable and volatile periods refer to the months with the lowest and highest 20% absolute value of the market return, respectively. The full sample period is July 1963 to December 2000. Subperiod  Ranking on Idiosyncratic Volatility  51 

1 Low  2  3  4  5 High 

Jul 1963–Dec 1970  0.06  0.03  0.09  −0.36  −0.94  −1.00 
[1.23]  [0.42]  [0.73]  [−2.18]  [−5.81]  [−5.62] 
Jan 1971–Dec 1980  −0.24  0.32  0.19  0.03  −1.02  −0.77 
[−2.53]  [3.20]  [1.55]  [0.21]  [−5.80]  [−3.14] 
Jan 1981–Dec 1990  0.15  0.08  −0.16  −0.66  −2.08  −2.23 
[2.14]  [1.07]  [−1.25]  [−4.82]  [−10.1]  [−9.39] 
Jan 1991–Dec 2000  0.16  −0.01  0.14  −0.48  −1.39  −1.55 
[1.34]  [−0.08]  [0.77]  [−2.41]  [−3.31]  [−3.19] 
NBER Expansions  0.06  0.02  0.08  −0.33  −1.19  −1.25 
[1.26]  [0.25]  [1.01]  [−3.18]  [−7.07]  [−6.55] 
NBER Recessions  −0.10  0.64  −0.01  −0.34  −1.88  −1.79 
[−0.65]  [3.58]  [−0.04]  [−1.32]  [−3.32]  [−2.63] 
Stable Periods  0.05  −0.02  −0.11  −0.62  −1.66  −1.71 
[0.44]  [−0.25]  [−1.07]  [−4.06]  [−6.56]  [−4.75] 
Volatile Periods  −0.04  0.24  0.32  0.18  −0.93  −0.89 
[−0.29]  [1.69]  [2.32]  [0.55]  [−2.40]  [−2.02] 
A possible explanation for the idiosyncratic volatility effect may be asymmetry of return distributions across business cycles. Volatility is asymmetric (and larger with downward moves), so stocks with high idiosyncratic volatility may have normal average returns during expansionary markets, and their low returns may mainly occur during bear market periods, or recessions. We may have observed too many recessions in the sample relative to what agents expected ex ante. We check this hypothesis by examining the returns of high idiosyncratic volatility stocks conditioning on NBER expansions and recessions. During NBER expansions (recessions), the FF3 alpha of the 51 portfolio is −1.25% (−1.79%). Both the expansion and recession differences in FF3 alphas are significant at the 1% level. There are more negative returns to high idiosyncratic volatility stocks during recessions, but the fact that the tstatistic in NBER expansions is −6.55 shows that the low returns from high idiosyncratic volatility also thrive during expansions.
A final possibility is that the idiosyncratic volatility effect is concentrated during the most volatile periods in the market. To test for this possibility, we compute FF3 alphas of the difference between quintile portfolios 5 and 1 conditioning on periods with the lowest or highest 20% of absolute moves of the market return. These are ex post periods of low or high market volatility. During stable (volatile) periods, the difference in the FF3 alphas of the fifth and first quintile portfolios is −1.71% (−0.89%) per month. Both the differences in alphas during stable and volatile periods are significant at the 5% level. The most negative returns of the high idiosyncratic volatility strategy are earned during periods when the market is stable. Hence, the idiosyncratic volatility effect is remarkably robust across different subsamples.
III. Conclusion
 Top of page
 ABSTRACT
 I. Pricing Systematic Volatility in the CrossSection
 II. Pricing Idiosyncratic Volatility in the CrossSection
 III. Conclusion
 REFERENCES
Multifactor models of risk predict that aggregate volatility should be a crosssectional risk factor. Past research in option pricing has found a negative price of risk for systematic volatility. Consistent with this intuition, we find that stocks with high past exposure to innovations in aggregate market volatility earn low future average returns. We use changes in the VIX index constructed by the Chicago Board Options Exchange to proxy for innovations in aggregate volatility.
To find the component of market volatility innovations that is reflected in equity returns, we construct a factor to mimic innovations in market volatility following Breeden et al. (1989) and Lamont (2001). We first form portfolios on the basis of their past sensitivity to first differences in the VIX index. Then, we project innovations in VIX onto these portfolios to produce a factor that mimics aggregate volatility risk, which we term FVIX. This portfolio of basis assets is maximally correlated with the realized aggregate volatility innovations. Portfolios constructed by ranking on past betas to first differences in VIX also exhibit strong patterns in postformation FVIX factor loadings. In particular, the ex post increasing patterns in FVIX factor loadings correspond to decreasing Fama–French (1993) alphas over the same period that the alphas are computed.
We estimate a crosssectional price of volatility risk of approximately −1% per annum, and this estimate is robust to controlling for size, value, momentum, and liquidity effects. Hence, the decreasing average returns to stocks with high past sensitivities to changes in VIX is consistent with the crosssection of returns pricing aggregate volatility risk with a negative sign. However, despite the statistical significance of the negative volatility risk premium, its small size and our relatively small sample mean that we cannot rule out a potential Peso problem explanation. Since the FVIX portfolio does well during periods of market distress, adding another volatility spike like October 1987 or August 1998 to our sample would change the sign of the price of risk of FVIX from negative to positive. Nevertheless, our estimate of a negative price of risk of aggregate volatility is consistent with a multifactor model or Intertemporal CAPM. In these settings, aggregate volatility risk is priced with a negative sign because riskaverse agents reduce current consumption to increase precautionary savings in the presence of higher uncertainty about future market returns. Our results are also consistent with the estimates of a negative price of risk for aggregate volatility estimated by many option pricing studies.
We also examine the returns of a set of test assets that are sorted by idiosyncratic volatility relative to the Fama–French (1993) model. We uncover a very robust result. Stocks with high idiosyncratic volatility have abysmally low average returns. In particular, the quintile portfolio of stocks with the highest idiosyncratic volatility earns total returns of just −0.02% per month in our sample. These low average returns to stocks with high idiosyncratic volatility cannot be explained by exposures to size, booktomarket, leverage, liquidity, volume, turnover, bidask spreads, coskewness, or dispersion in analysts' forecasts characteristics. The effect also persists in bull and bear markets, NBER recessions and expansions, volatile and stable periods, and is robust to considering different formation and holding periods as long as 1 year. Although we argue that aggregate volatility is a new crosssectional, systematic factor, exposure to aggregate volatility risk accounts for very little of the anomalous low returns of stocks with high idiosyncratic volatility. Hence, the crosssectional expected return patterns found by sorting on idiosyncratic volatility present something of a puzzle.