2.6 Mimicking curvature factors
2.6.3 A curvature factor model
We construct curvature factor model based on our mimicking tradable factors. The expected excess return of asset i from the factor model is
, , , (2.32)
where is the expected return on the market portfolio, is expected return on the mimicking curvature factor, ∈ 1, 2, 2 and
, and , are the factor loadings from the time-series regression:
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, , , , . (2.33)
Since our curvature factor is constructed to mimic the second-order risk premium, i.e., ∝ , , our theory implies that the curvature factor loading is related to the second-order risk, i.e., , ∝ , . Stocks with high curvature factor loadings, by construction, are less risky because they are more sensitive to the market variance risk premium, thereby providing hedging against market volatility risk.
We test our curvature factor model at the firm level using the cross-sectional regression:
, , , , , ,
, , , (2.34)
where the dependent variable is the monthly individual stock returns; , , and
, , are post-ranking betas estimated from the 25 portfolios formed on historical CAPM beta ( , ). , is a set of control variables, including book-to-market ratio (B/M), book-to-market capitalization (Size), past 11-month return (RET_2_12), dividend yield (YLD), and illiquidity (ILLIQ). Following the methodology of Fama and French (1992), we assign each of the 25 portfolio-level post-ranking beta estimates to the individual stocks within the portfolio at that time.
Table 2.8 reports the results for the firm-level Fama-MacBeth regressions. In Panel A, we find that is negative (−0.501) with a significant t-statistic of −3.30, reported in column [2]. Moreover, in column [3], is also negative (−0.682) with a significant t-statistic of −2.59. In Panel B, we test our model with control variables of risk-neutral moments, including VARRN, SKEWRN, and KURTRN. We find similar results in which , , and have significant negative values of −1.108 (with a statistic of −2.77), −1.307 (with a statistic of −2.75), and −1.008 (with a
t-94 statistic of −2.61), respectively.
In summary, we find significant evidence that mimicking curvature factors are priced risk factors. That is, investor required lower expected returns for holding stocks with greater exposure to the curvature factor because these assets are more sensitive to market variance risk premium, thereby providing hedge for market volatility risk. In other words, ignoring the curvature factor might omit an important source of priced risk.
2.7 Performance of the curvature factor model
2.7.1 The curvature factor model and cross-sectional volatility-return relation
We examine whether our mimicking curvature factors help explain the well-known (idiosyncratic) volatility puzzle of Ang, Hodrick, Xing, and Zhang (2006, 2009) and the MAX puzzle of Bali, Cakici, and Whitelaw (2011). Following the literature, TVOL is defined as the annualized past one month variance of daily stock returns; IVOL is defined as the annualized residual variance of the daily stock regressed on the Fama and French (1993) three factors over the past month; MAX is defined as the maximum daily stock return over the past one month. In our model, Corollary 2 and Corollary 3 imply that the second-order risk premium should help explain the cross-sectional return differentials with respect to TVOL and IVOL. If MAX, a volatility measure itself, is highly correlated to these two volatility measures, the second-order risk premium should also explain the pricing effect of MAX.
In Table 2.9, we report the Spearman correlations for second-order risks, and volatilities which includes TVOL, IVOL, and MAX. The table shows that these volatility measures are highly correlated each other with correlations ranging from 0.86 to 0.97.
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Moreover, we find that these volatility measures are highly correlated with the second-order risks as well as the curvature factor loadings. Thus, results suggest that high volatility stocks could have high exposure to market variance risk premium and therefore have low expected returns.
Table 2.10 presents the performance of portfolios formed based on TVOL, IVOL, and MAX. Stocks are sorted into quintile portfolios, from the lowest (quintile 1) to the highest
(quintile 5), and the portfolio returns are value-weighted. For each portfolio, we compute the risk-adjusted return with respect to our curvature factors as well as Fama-French (1993) and Carhart (1997) four factors (MKT, SMB, HML, and UMD) from the intercept estimate of a time-series regression. Controlling for Fama-French and Carhart four factor model, we find evidence consistent with Ang, Hodrick, Xing, and Zhang’s (2006) that the
‘5-1’ portfolios for TVOL and IVOL have significantly negative alphas of −0.93% (with a t-statistic of −4.72) and −0.96% (with a t-statistic of −5.46), respectively. Similarly, we
find that the ‘5-1’ portfolio for MAX also has a significantly negative alphas of −0.60%
with a t-statistic of −3.63.
More importantly, after controlling for market factor and our mimicking curvature factors, we find that none of the ‘5-1’ portfolios has significant abnormal returns. For example, the ‘5-1’ portfolio for TVOL has insignificant alphas of −0.10% (with a t-statistic of –0.45), −0.06% (with a t-statistic of –0.45), and 0.27% (with a t-statistic of 0.67) controlling FCUR1, FCUR2, and FCUR2RN, respectively. Furthermore, all of the ‘5-1’
portfolios have significant exposures to our mimicking curvature factors. For example, the ‘5-1’ portfolio for TVOL has significant exposures of 1.09 (with a t-statistic of 8.95), 0.77 (with a t-statistic of 22.02), and 0.92 (with a t-statistic of 11.84) to FCUR1, FCUR2, and FCUR2RN, respectively.
In summary, consistent with our model, we find that these volatility measures are
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closely related to the second-order systematic risks and the abnormal returns associated with these variables become insignificant once we control for the mimicking curvature factors. Thus, it is the systematic second-order risk premium that accounts for these volatility puzzles.
2.7.2 The curvature factor model and betting-against-beta premium
We examine whether our mimicking factors help explain the betting-against-beta premium of Frazzini and Pedersen (2014). They show that a betting against beta (BAB) factor, which is long leveraged low-beta assets and short high-beta assets, produces significant positive risk-adjusted returns, supporting their theory of margin constraint.Our model, in contrast, suggests that the betting-against-beta strategy is exposed to high order systematic risks.
Table 2.11 presents the performance of portfolios formed on betting-against-beta (BAB). At the beginning of each calendar month, stocks are ranked in ascending order on the basis of , at the end of the previous month, where , is the beta of Frazzini and Pedersen (2014). To construct the BAB factor, all stocks are assigned to one of two portfolios: low beta and high beta. Stocks are weighted by the ranked betas (lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio), and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The betting against beta factor (BAB) is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio.
The BAB factor yields a significantly positive average excess return of 0.89% with a t-statistic of 3.82. Controlling for Fama-French and Carhart four factor model, BAB still
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has a significant positive alpha of 0.58% with a t-statistic of 2.78. Thus, our results are comparable with theirs. Controlling for the market factor and our mimicking curvature factors, we find BAB still has a significant alphas of 0.71 (with a t-statistic of 2.77) using FCUR1, a significant alphas of 0.64 (with a t-statistic of 2.77) using FCUR2, and an
insignificant alphas of 0.11 (with a t-statistic of 0.23) using FCUR2RN. Thus, despite the successful performance of FCUR2RN, the market factor and our other mimicking curvature factors are insufficient to fully explain the BAB factor.
We extend the analysis using a generalized five-factor model which augments our curvature factors with Fama-French and Carhart four factors. Controlling for the five factors, the BAB premium disappears in which the BAB factor has an insignificant alphas of 0.29 (with a statistic of 1.30) using FCUR1, an insignificant alphas of 0.20 (with a t-statistic of 0.95) using FCUR2, and an insignificant alphas of 0.17 (with a t-t-statistic of 0.40) using FCUR2RN. Thus, although Fama-French and Carhart four factors are similarly insufficient to explain the BAB factor, the extended five-factor model does capture the BAB premium, indicating the importance of the second-order risk premium.
In summary, we find that the second-order risk premium helps explain the BAB premium. Thus, the second-order risk premium provides an alternative explanation for the betting-against-beta premium other than the market friction in their paper.
2.7.3 Performance of portfolios formed on historical curvature factor loadings
Table 2.12 presents the performance of portfolios formed on historical curvature factor loadings ( , , and ). In each month, we estimate our curvature factor model using the daily stock returns over the past one month. ,
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, and are the historical curvature factor loadings of FCUR1, FCUR2, and FCUR2RN, respectively. In each month, stocks are sorted into 25 portfolios from the lowest (1) to the highest (25) and the portfolio returns are equal-weighted.
The results indicate that the portfolios formed on the historical curvature factor loadings have significant risk premiums that the current Fama-French and Carhart four factor model cannot explain. Specifically, controlling for the four factor model, the abnormal returns are −1.47 (with a t-statistic of −7.69) for the ’25-1’ portfolio formed on , −1.43 (with a t-statistic of −7.01) for the ’25-1’ portfolio formed on , and
−1.12 (with a t-statistic of −2.61) for the ’25-1’ portfolio formed on . These results support the validity of our mimicking curvature factors for the second-order risk premium.
2.8 Conclusions
The negative market variance risk premium and the second-order risk appear to affect cross-sectional asset pricing. This paper presents an approximate capital asset pricing model, in which, along with the first-order co-moment risks in existing literature, higher order co-moment risks and high order risk premiums are important for pricing individual stocks. Stocks with high exposure to the second-order risk are more volatile and are capable of earning the upside variance premium provided by the increasing region of the pricing kernel implied by the negative market variance risk premium.
Our results show that the second-order risk is significantly and negatively priced and contributes to an inverse-U shaped relation between cross-sectional expected returns and systematic risks. We show that our mimicking curvature factors for the second-order risk premium well explain several volatility-related puzzles as well as the BAB premium. Our
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study provides a unified framework for better understanding of the high order risk-return tradeoff and sheds light on the role of the second-order risk premium.
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Table 2. 1: Market moments implied by SPX and SPX index options
This table presents the estimates of physical market moments, risk-neutral market moments, and their differences. The physical market moments are computed using the full sample logarithmic monthly SPX returns. The 30-day risk-neutral market moments are estimated by the model-free approach of Bakshi, Kapadia, and Madan (2003) for each day following the procedure of Chang, Christoffersen, and Jacobs (2013). We then average daily estimates from index option prices to obtain the full sample risk-neutral market moments. The first four cumulants (K1, K2, K3, and K4) are reported. The sample period is from 1996 to 2012.
Cumulants
% % % %
Physical 0.398 0.218 -0.008 0.002
Risk-neutral 0.039 0.479 -0.057 0.023 Difference 0.359 -0.260 0.049 -0.021
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Table 2. 2: Summary statistics
This table reports the mean, standard deviation, and percentile (5th , 25th, median, 75th , and 95th) statistics for variables used in this study. We first compute in each month the cross-sectional statistics for each security and then report the time-series average. Historical CAPM beta ( , ) is calculated using market model on the daily stock returns over the past month. We estimate ,
⁄ ⁄ ⁄ for the first order coskewness, , ⁄ ⁄
for the first order kurtosis, , ⁄ ⁄ ⁄| | for the second order coskewness, and , ⁄ ⁄| | for the second order cokurtosis. We estimate the co-moments divided by market variance using the residual daily data whereas the market skewness ( ) and the market kurtosis ( ) are computed using the full sample monthly market returns. We apply the model-free approach of Bakshi, Kapadia, and Madan (2003) to estimate the 30-day risk-neutral moments for each day, following the procedure outlined in Chang, Christoffersen, and Jacobs (2013). The average of daily estimates for risk-neutral variance (VARRN), risk-neutral skewness (SKEWRN), and risk-neutral kurtosis (KURTRN) are reported. The risk-neutral variance beta ( , , ) is computed by the linear relation that
, , , , , , , , , , through nonnegative least square method, where
, , is the risk-neutral variance of the stock i, and , , is the risk-neutral variance of the market. B/M is the book-to-market ratio; Size is market capitalization measured in billions of dollar;
RET_2_12 reports the average of past 11-month returns prior to last month; ILLIQ reports the average of Amihud’s (2002) illiquidity measure. The sample period is from January 1963 to December 2012.
Descriptive statistics
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Table 2. 3: The pricing of high-order risks
This table reports the estimates for the risk prices of high order risks using portfolio returns. In Panel A, we sort stocks into 25 portfolios based on the historical CAPM beta in each month and compute the equal-weighted portfolio returns. In Panel B, we adopt the Fama-French 25 value-equal-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:
, , , , , , .
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:
, , , , ,
, ,
where , , , , and , the orthogonalized high order market risks with respect to their lower order risks . Robust Newey and West (1987) t-statistics with eight lags that account for autocorrelations are presented in parentheses. The sample period is from January 1963 to December 2012.
Cross-sectional regressions
Adj. R2
Panel A: 25 portfolios formed on CAPM beta
[1] -0.161 1.215 0.927 0.711 0.655
Panel B: 25 portfolios formed on Size-B/M
[1] 0.237 0.459 3.953 0.940 0.712
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Table 2. 4: Performance of portfolios formed on the first order systematic risk
This table presents the performance of portfolios formed on the first order co-moment risks ( , , , , and , ). In each month, stocks are sorted into 25 portfolios from the lowest (1) to the highest (25). After the portfolio formation, we calculate the equal-weighted monthly stock returns for each portfolio. , is calculated using market model on the daily stock returns over the past month. Define as the residual stock return and as the demeaned market return. We compute that ,
⁄ ⁄ ⁄ and , ⁄ ⁄ , where
⁄ ⁄ and ⁄ .We estimate the co-moments divided by market
variance using the residual daily data whereas the market skewness ( ) and the market kurtosis ( ) are computed using the full sample monthly market returns. Results for , , , and , are reported in Panel A, Panel B, and Panel C, respectively. Performance of the bottom portfolios (1 and 2), the middle portfolios (12 and 13), and the top portfolios (24 and 25) are reported. The column “CUR1” refers to the curvature portfolio that longs the difference in the top portfolios (25-24) and shorts difference in the bottom portfolios (2-1). 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. Robust Newey and West (1987) t-statistics with eight lags that account for autocorrelations are presented in parentheses. The sample period is from January 1963 to December 2012.
Portfolio ranking CUR1
1 2 12 13 24 25 (25-24)-(2-1)
Panel A: Performance of the 25 portfolios formed on
Excess returns 0.23 0.72 0.79 0.79 0.48 -0.02 -1.00 (-5.79) α-CAPM -0.25 0.29 0.32 0.31 -0.28 -0.84 -1.10 (-6.38) α-FF3 -0.53 0.03 0.05 0.03 -0.41 -0.99 -1.14 (-7.12) α-FFC4 -0.40 0.16 0.13 0.13 -0.19 -0.74 -1.11 (-7.15) Panel B: Performance of the 25 portfolios formed on
Excess returns 0.04 0.60 0.74 0.73 0.56 0.14 -0.98 (-5.62) α-CAPM -0.60 -0.02 0.31 0.30 -0.04 -0.49 -1.02 (-6.01) α-FF3 -0.88 -0.27 0.05 0.04 -0.25 -0.73 -1.08 (-7.14) α-FFC4 -0.67 -0.08 0.14 0.10 -0.10 -0.52 -1.01 (-6.83) Panel C: Performance of the 25 portfolios formed on
Excess returns 0.07 0.55 0.67 0.68 0.60 0.12 -0.96 (-6.18) α-CAPM -0.56 -0.07 0.23 0.24 -0.01 -0.51 -0.99 (-6.53) α-FF3 -0.85 -0.34 -0.01 0.01 -0.20 -0.76 -1.08 (-8.02) α-FFC4 -0.66 -0.19 0.07 0.06 0.02 -0.54 -1.03 (-7.78)
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Table 2. 5: Performance of portfolios formed on the second order systematic risk
This table presents the performance of portfolios formed on the second order co-moment risks ( , and
, ). In each month, stocks are sorted into 25 portfolios from the lowest (1) to the highest (25). After the portfolio formation, we calculate the equal-weighted monthly stock returns for each portfolio. Define as the residual stock return and as the demeaned market return. We compute that ,
⁄ ⁄ ⁄| | and , ⁄ ⁄| | , where
⁄ ⁄ and ⁄ .We estimate the co-moments divided by market
variance using the residual daily data whereas the market skewness ( ) and the market kurtosis ( ) are computed using the full sample monthly market returns. Results for , and , are reported in Panel A and Panel B, respectively. Performance of the bottom portfolios (1 and 2), the middle portfolios (12 and 13), and the top portfolios (24 and 25) are reported. The column “CUR2” refers to the
variance using the residual daily data whereas the market skewness ( ) and the market kurtosis ( ) are computed using the full sample monthly market returns. Results for , and , are reported in Panel A and Panel B, respectively. Performance of the bottom portfolios (1 and 2), the middle portfolios (12 and 13), and the top portfolios (24 and 25) are reported. The column “CUR2” refers to the