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.