The Cross-Section of Volatility and Expected Returns

 Andrew Ang, Robert J. Hodrick, Yuhang Xing, Xiaoyan Zhang. "The Cross-Section of Volatility and Expected Returns." Journal of Finance (2006): 259-299.

In this paper, the authors seek to examine the effects of aggregate volatility on the cross section of stock returns. They hypothesize that while the time-series relation of volatility and expected market return has been thoroughly researched, further consideration with respect to the cross section of stock returns should show that the volatility of the market return is a systematic risk factor and should be priced in the cross section of stock returns. With this, they then suggest that stocks with different sensitivities to aggregate volatility innovations should have different expected returns. To test this hypothesis, the authors look to show whether the aggregate market volatility is a priced risk factor and, if so, determine the price of this factor. They also examine the cross-sectional relationship between idiosyncratic volatility and expected returns to test whether such a risk factor is orthogonal to existing factors.

Theoretical Motivation

The model that they look to test is of the following form,

where (in order from left to right) the loading on the excess market return, asset sensitivity to volatility risk, and loadings of K other factors are regressed on excess stock returns.

They then simplify the above to arrive at the following with respect to expected returns,


Empirical Test

Because we do not yet know the true set of risk factors and, as a result, cannot observe such loadings of factors, the authors design a practical empirical framework to test their hypothesis.

As a measure of innovations in aggregate volatility, the authors use daily changes in the VIX index. This is represented in their pre-formation regression,

which is used to test the difference in average returns of stocks with different sensitivities to innovations in aggregate volatility. Quintile portfolios are made by sorting from lowest to highest the regressed results of the loadings on the volatility factor. The stocks of each portfolio are then value-weighted. They measure the difference of average returns between portfolios with the highest and lowest volatility coefficients and find a difference of -1.04% per month that is statistically significant at the 1% level. This can be seen in the mean column of Table 1 below,


Factor Risk Explanation

The authors then look to create a factor from aggregate volatility innovations to suggest a factor risk-based explanation for the above results. This factor, called FVIX, is then used to measure ex post exposure to aggregate volatility risk. The following regression is used for the creation of FVIX by estimating the coefficient of the returns on the base assets X,

where the previous quintile portfolios are used for the base assets.

After constructing FVIX, the authors then substitute FVIX for the daily change in VIX in their pre-formation regression to get the following cross-sectional regression,

however, to test this model a contemporaneous relationship between factor loadings and average returns must be shown. The authors do this by observing that the pre-formation FVIX factor loadings and pre-formation VIX factor loadings are very similar. This can be seen in the second panel of Table 1 as follows,

Post-formation factor loadings are also shown in Table 1. It is the full sample post-formation FVIX betas that are then used to examine ex post factor exposure to aggregate volatility risk. To control for market, size, and value factors from the Fama French 3 Factor model, the authors use the following regression,

From Table 1, it can be seen that the full sample post-formation loadings on FVIX are significantly different across the quintile portfolios.

Pricing Aggregate Volatility Risk

The full regression specification used to estimate the unconditional price of the aggregate volatility risk fact is as follows,

which includes momentum and liquidity factors UMD and LIQ, respectively.

The Fama-MacBeth regression results of the above are,


Regression I from above shows the estimated price of volatility risk is -0.08% per month. The effects of volatility can be measured by multiplying the -0.08 from above with the 13.13 ex post spread in FVIX betas from Table 1. This results in a difference in average returns of -1.05%, which is almost the same as the -1.04% difference in the raw average returns, suggesting the difference in raw average returns can be attributed to exposure to aggregate volatility risk.

Idiosyncratic Volatility

The authors then sort portfolios by idiosyncratic volatility to observe its effects on cross-sectional average returns. They define idiosyncratic volatility relative to the Fama French 3 Factor model as the square root of the variance of the residual. This is to control for systematic risk, however, they also sort portfolios by total volatility. As seen by Table VI below, average returns for portfolios sorted by total volatility and idiosyncratic volatility are very similar and the difference in highest and lowest volatility portfolios are statistically significant.



Robustness tests suggest these results cannot be explained by exposures to size, book-to-market, leverage, liquidity, volume, turnover, bid-ask spreads, coskewness, or dispersion in analysts' forecasts characteristics.

Conclusion

The authors conclude that aggregate market volatility is a risk factor that can be observed in the cross-section of stock returns, and that the price of this risk factor is negative and statistically significant.

Conditional Skewness in Asset Pricing Tests

Conditional Skewness in Asset Pricing Tests

By Campbell Harvey and Akhtar Siddique

The single factor CAPM has been proven not to be a reliable tool in predicting stock returns.  Because of this, some have created factors to help explain more of the variation in stock returns such as the SMB and HML factors created by Fama and French. Harvey and Siddique examine the linkage between the empirical evidence on these additional factors and systematic coskewness. Since there is considerable evidence that returns cannot b e adequately characterized by mean and variance alone, this leads to the inclusion of the next moment – skewness.  The hypothesis is that investors will attempt to avoid stocks that are left-skewed due to risk aversion, and these stocks should command higher expected returns as a result.

            To illustrate the relationship between skewness and expected returns, Figure 1 shows the trade-offs between mean, variance, and skewness.  Panel B includes the risk-free rate. Any points that are tangent to the risk-free plane are considered efficient portfolios.  The figure shows that expected return should increase as variance and skewness increase.  Also, the portfolios that are tangent to the risk-free plane are when skewness and variance are the highest.

The formula for finding the coskewness beta is

 

Using these betas, Harvey and Siddique create three portfolios.  The 30 percent of stocks with the most negative coskewness fall in the S- portfolio.  The 30 percent of stocks with the most positive coskewness fall in the S+ portfolio.  Based on the hypothesis and figure 1, the S- portfolio should have higher expected returns. They find the average annualized spread between the returns on S- and S+ portfolios is 3.60 percent from July 1963 to December 1993.  This result is statistically significant.  They compute the coskewness for a risky asset from its beta with the spread between the returns on the S- and S+ portfolios and call this measure B_SKS.  Using the summary stats in table 1, it is apparent that coskewness plays a role in explaining the cross section of asset returns.  Table 2 shows that conditional coskewness can explain a significant part of the variation in returns even when factors based on size and book/market like SMB and HML are added to the asset pricing model.  There is a significant correlation between the pricing errors of these factor models and the S- portfolio.

            Table 3 shows the R-squares of several regressions. It first shows the R-squared of the traditional CAPM and the 3-factor model.  Then it incorporates the S- portfolio and the SKS portfolio.  Both a cross-sectional regression (CSR) approach using rolling betas and a full-information maximum likelihood (FIML) method using constant betas are used.  Overall, when the S- or SKS portfolio are incorporated with the CAPM or 3-factor model, the R-squared increases.  This shows that including coskewness in the model can help explain stock returns over and above what the 3-factor model can.

            They move on to show that the relevance of the SMB and HML factors appears to be dependent on how old the stock is.  SMB is only significant for stocks with less than 60 months of returns.  This could be an IPO effect, where factors other than market (like SMB and HML) may be more useful in predicting the returns on firms with a short return history.  Skewness on the other hand, remains significant across almost all return length groups.

            In table 5, Harvey and Siddique show the momentum factors is related to systematic skewness.  They consider several different definitions of momentum where they vary the horizon (36-2, 24-2, 12-2, 6-2, and 3-2 months) and holding period (1, 3, 6, 12, 24, and 36 months).  All the portfolios in table 5 show a clear relation between average return and skewness.  The portfolios with the higher average returns also have the more negative skewness.

Figure 3 shows the necessity of negative skewness to have higher mean returns when using momentum strategies. The y-axis shows the mean annualized return from the strategy while the length of the line on the x-axis shows the difference in skewness between the winners and losers.  The negative slope of each line indicates that in a momentum-based trading strategy, buying the winner and selling the loser requires acceptance of substantial negative skewness.

            These findings support the hypothesis that skewness helps explain variation in asset pricing and supports the theory that having negative skewness should command higher abnormal returns because investors will naturally avoid those assets because of risk aversion.  Overall, coskewness provides us with some insight into why variables like size and book-to-market value are important and that the momentum effect is related to systematic skewness.  Harvey and Siddique concede that measurement of ex ante skewness may be difficult due to data limitations.