2.5.1 Cross-sectional regressions
We examine our approximate capital asset pricing model in the cross-section. In each month, we sort stocks into 25 portfolios based on the historical CAPM beta ( , ) and compute the equal-weighted portfolio returns. To achieve higher testing power, we also adopt the Fama-French 25 value-weighted portfolio returns formed on size and B/M. We first estimate the following time-series regression for each portfolio on the Fama-French (1993) and Carhart (1997) four factors:
, , ,
, , , . (2.30)
In the second stage, we use the Fama-MacBeth (1973) cross-sectional regression to estimate the prices of high order risks while controlling for common factor loadings:
, , , , , ,
, . (2.31)
where , , , , and , the orthogonalized high order market risks with respect to their lower order risks. Robust Newey and West (1987) t-statistics with eight
87 lags that account for autocorrelations are used.
Panel A of Table 2.3 reports the estimates for the risk prices of high order risks using equal-weighted portfolio returns formed on the historical CAPM beta. As reported in column [1], we find that , is positive (0.711) with a significant t-statistic of 3.49.
Furthermore, as reported in column [2], , is negative (−2.372) with a significant t-statistic of −5.69. In column [4], we find that , has a significant positive value of 1.567 (with a t-statistic of 5.40) and , yields a significant negative value of −2.116 (with a t-statistic of −4.69) whereas , and , are insignificant.
Panel B reports the estimates using Fama-French 25 value-weighted portfolio returns formed on size and B/M. The result in Panel B is similar with that of Panel A. For example, as reported in column [4], , is positive (0.925) with a significant t-statistic of 4.87 and , is negative (−0.826) with a significant t-statistic of −2.18 whereas , and , are insignificant.
In summary, we find supporting evidence for the pricing of the first two orders of market risks. The first-order risk is significantly and positively priced while the second-order risk is significantly and negatively priced. More importantly, our findings imply that cross-sectional relation between expected return and market beta should be inverse-U shaped, which constitutes the main idea of our empirical tests in the following sections.
2.5.2
Evidence on the second-order risk premiumPerformance of portfolios formed based on the first-order systematic risk
We examine the performance of portfolios formed on the first-order co-moment risks, including the market beta ( , ), the first-order coskewness ( , ) and the first-order cokurtosis ( , ). In each month, stocks are sorted into 25 portfolios from the lowest (1)
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to the highest (25). After the portfolio formation, we calculate the equally-weighted monthly stock returns for each portfolio. For each portfolio, we compute the risk-adjusted return with respect to Fama-French (1993) and Carhart (1997) four factors (MKT, SMB, HML, and UMD) from the intercept estimate of a time-series regression. Table 2.4
presents the results for , , , and , in Panel A, Panel B, and Panel C, respectively.
Consistent with an inverse-U shaped pattern implied by the model, as can be seen in Table 2.4, all of the first-order co-moment risk sorted portfolios exhibit that stocks in the bottom portfolio and in the top portfolio have lower stock returns than stocks in the middle portfolios. To exploit the economic value of the inverse-U shaped pattern, we construct a curvature portfolio (CUR1) by the sum of the twice difference for the portfolios formed on each of the first order co-moment risks. That is, for N risk-sorted portfolios, the curvature portfolio would be ∑ Δ , where Δ and Δ Δ Δ . Thus, for the 25 portfolios, CUR1 is the curvature portfolio that longs the difference in the top portfolios (25-24) and shorts difference in the bottom portfolios (2-1).
We find that CUR1 is significantly negative at −1.00% (with a t-statistic of −5.79) for , sorted portfolios, at −0.98% (with a t-statistic of −5.62) for , sorted portfolios, and at −0.96% (with a t-statistic of −6.18) for , sorted portfolios.
Controlling for the Fama-French (1993) and Carhart (1997) four factor model, CUR1 still gives a significant alpha of −1.11% with a t-statistic of −7.15 for , sorted portfolios,
−1.01% with a t-statistic of −6.83 for , sorted portfolios, and −1.03% with a t-statistic of −7.78 for , sorted portfolios.
In summary, consistent with Corollary 1, we find an inverse-U shaped pattern for
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portfolios formed on the first-order risks. We also find the curvature portfolio based on the trading strategy exploiting the inverse-U shaped pattern generates significant abnormal returns where the the Fama-French (1993) and Carhart (1997) four factor model cannot explain. Thus, the findings suggest that the second-order risk premium is statistically and economically significant.
Performance of portfolios formed on the second-order co-moment risks
We examine the performance of portfolios formed on the second-order co-moment risks, including the second-order coskewness ( , ) and the second-order cokurtosis ( , ). In each month, stocks are sorted into 25 portfolios from the lowest (1) to the highest (25) and the portfolio returns are equal-weighted. Table 2.5 presents the results for , and , in Panel A and Panel B, respectively.
Consistent with the negative risk prices for the second-order risk, as can be seen in Table 2.4, portfolio returns exhibit decreasing pattern, albeit slightly increasing initially.
To exploit the economic value of the second-order risk premium, we construct another curvature portfolio (CUR2) that longs the top portfolio and shorts the bottom portfolio for the portfolios formed on each of the second order co-moments risks.
We find that CUR2 is significantly negative at −1.27% (with a t-statistic of −4.32) for , sorted portfolios, and at −1.06% (with a t-statistic of −3.05) for , sorted portfolios. Controlling for the Fama-French (1993) and Carhart (1997) four factor model, CUR2 still gives a significant alpha of −1.47% with a t-statistic of −8.37 for , sorted portfolios, and −1.37% with a t-statistic of −6.74 for , sorted portfolios. In Panel C, we find similar results for the 15 portfolios formed on the risk-neutral variance beta ( , , ). We find that the curvature portfolio (CUR2=15-1) is significantly negative at –
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1.34% with a t-statistic of −2.21 and has a significant alpha of −1.38% with a t-statistic of −3.71 adjusted by the Fama-French (1993) and Carhart (1997) four factor model.
In summary, consistent with Corollary 1, we find negative relation between cross-sectional stock returns and the second-order risks. We also find that the curvature portfolios generate significant abnormal returns where the Fama-French (1993) and Carhart (1997) four factor model cannot explain. Thus, the findings confirm that the second-order risk premium is statistically and economically significant.