Ex Ante Skewness and Expected Stock Returns
Summary (5 min read)
1 Introduction
- Models implying that investors consider higher moments in returns have a long history in the literature.
- The work of Barberis and Huang (2008), Brunnermeier, Gollier, and Parker (2007), and the empirical evidence presented in Mitton and Vorkink (2007) and Boyer, Mitton, and Vorkink (2010) imply that the skewness of individual securities may also influence investors’ portfolio decisions.
- These papers focus on physical moments of returns, in contrast to the risk-neutral moments that the authors examine.
- The remainder of the paper is organized as follows.
2 Data and Computing Ex Ante Risk-Neutral Moments
- The authors wish to examine the relation, if any, between features of the risk-neutral density function and the pricing of stocks.
- Data on stock returns are obtained from the Center for Research in Security Prices (again provided through Wharton Research Data Services).
- The contracts are based on Ci,t(τ ;K) and Pi,t(τ ;K), which are the time t prices of European calls and puts written on the underlying stock with strike price K and expiration τ periods from time t. Expressions for Vi,t(τ), Wi,t(τ), and Xi, (A2) and (A3).
- In Table 1, the authors present descriptive statistics for the sample estimates of volatility, skewness, and kurtosis.
3.1 Arbitrage Issues
- Under the assumption that no-arbitrage rules hold between the options market and the underlying security prices, the information set contained in both cash and derivatives markets should be the same.
- Several authors have shown that information in option prices can provide valuable forecasts of features of the payoff distributions in the underlying market.
- He also shows that fears of a crash increased immediately after the crash itself.
- A corollary to their results is that option prices and the prices of underlying stocks did not diverge during the Internet bubble and they argue that Ofek and Richardson’s results may be a consequence of misleading or stale price quotes in their options data set.
- Motivated by the Battalio and Schultz results, the authors employ additional filters to try to ensure that their results are not driven by stale or misleading prices.
3.2 Portfolio Sorts
- Each day, the authors sample the prices of out of the money calls and puts on individual securities that have expiration dates that are closest to 0.083 years (one month), 0.250 years (three months), 0.500 years (six months) and 1.000 years to maturity and midpoint bid and offer prices of $0.50 or greater.
- The puts that the authors retain in this circumstance are those that are closest to, but out of the money.
- In Table 2, the authors report results for portfolios sorted on the basis of estimated volatility, skewness, and kurtosis.
- In the next three columns of Table 2, the authors report the average firm’s risk-neutral volatility, skewness and kurtosis estimates for each of the ranked portfolios.
3.3 Multivariate sorts
- The authors estimate the relation between higher moments and subsequent returns, while controlling for variation in other higher moments, using double and triple sorts.
- In the double-sorting method, the authors sort firms into tercile portfolios based independently on volatility, skewness, and kurtosis.
- In each Panel, the number of firms in each portfolio are reported in parentheses below the returns.
- The effect is monotonic in 4 out of 6 cases and the return differential across the extreme skewness terciles varies from -19 to -100 basis points per month.
- The authors also find that securities with higher skewness have lower subsequent returns.
3.4 Factor-Adjusted Returns
- In Table 2, the authors adjust for the differences in characteristics across portfolios, following Daniel, Grinblatt, Titman, and Wermers (1997), by subtracting the return of the specific Fama-French portfolio to which an individual firm is assigned.
- The authors also adjust for differences in characteristics across their moment-sorted portfolios by estimating a time series regression of the High-Low portfolio returns for each moment on the three factors proposed in Fama and French (1993).
- Tests statistics for the null hypothesis that the coefficient is zero are presented below the point estimates.
- The patterns in the intercepts for skewness-sorted portfolios are of the same sign as the volatility-sorted alphas but larger in magnitude and statistically significant at the 5% level.
- The results in Table 4 are also informative in this regard.
3.5 Robustness checks
- As noted above, one of their concerns following the findings of Battalio and Schultz (2006) is that results might be driven by stale or misleading prices.
- As a robustness check, the authors perform other tests to examine the possibility that return differentials are driven by liquidity issues, either in the underlying equity returns or by stale or illiquid option prices.
- The authors consider alternative minimum price criteria for the options included in their sample.
- The authors also risk-adjust returns relative to an aggregate liquidity factor, as in Pástor and Stambaugh (2003).
- The results presented in this section are robust to these additional requirements, and are discussed in more detail in BLANK (2011).
4 Higher Moment Returns: Systematic and Idiosyncratic Com-
- The authors analyze the extent to which the cross-sectional relations between higher moments and returns presented in Tables 2 and 4 are due to investors seeking compensation for higher co-moment risk, rather than idiosyncratic moments.
- The authors start in subsection 4.1 with a characterization of co-skewness and co-kurtosis in the context of single factor models, inspired by the analysis in Harvey and Siddique (2000) and Bakshi, Kapadia, and Madan (2003).
- In subsection 4.2, the authors estimate the relation of risk-neutral co-moments to returns in their sample.
- In subsection 4.3 the authors decompose total moments into co-moments and idiosyncratic moments and examine the relationship of these components to subsequent returns.
- In a final subsection the authors report on various robustness checks using more general specifications.
4.1 Co-skewness, Co-kurtosis and a Single Factor Model
- Bakshi, Kapadia, and Madan (2003) suggest a procedure for computing the co-skewness of an asset with a factor.
- The authors note that if the parameters a and b are ’risk-neutralized’, equation (5) is also well-defined under the risk-neutral measure.
- Accordingly, the authors estimate bi using the procedure in Coval and Shumway (2001).
- To avoid cross-sectional biases in β related to cross-sectional variation in liquidity, the authors use the procedure in Dimson (1979) that corrects for infrequent trading; their reported results use a 1-day lead and lag of the market return as additional regressors.
4.2 Relation of risk-neutral co-moments to returns
- Given estimates of bi, the authors compute co-skewness and co-kurtosis using equations (6) and (7); note that the estimate of bi corresponds directly to a measure of risk-neutral covariance with the single factor m.
- The premium is substantially attenuated when the authors 9Coval and Shumway (2001) report that their estimates of bi following this procedure are very similar to those calculated by directly regressing option returns on the market portfolio.
- Differences in co-skewness are associated with significant negative differences in returns of 48 basis points for three-month options, and 64 basis points for 12 month options.
- Overall, their estimates of risk-neutral co-moments appear to generate dispersion in returns that are consistent with the relation between physical co-moments and returns observed by other researchers.
- In sharp contrast to the results in Table 4 on moment-sorted portfolios, all of the alphas in Table 6 are insignificant, and the R-squares for co-skewness and co-kurtosis sorted portfolios are substantially higher than the R-squares for skewness and kurtosis sorted portfolios.
4.3 Decomposing total moment return effects
- The authors decompose the return differential observed for total moment sorted portfolios in Table 2, into components related to dispersion in co-moments and dispersion in idiosyncratic moments.
- Summary statistics for these portfolios are presented in Table 7, and regressions adjusting for the contribution of Fama and French risk factors to returns are presented in Table 8.
- The results in Tables 7 and 8 mimic to a large extent the total moment results reported in Tables 2 and 4.
- In general, differences in idiosyncratic moments appear to drive most of the dispersion in total moments, and the returns differential associated with differences in idiosyncratic moments is both statistically and economically significant.
- 10Note that within the quarter, the average errors must be equal to zero; as a consequence, including the residual in the average idiosyncratic moment would not change their results.
5 Risk Neutral and Physical Probability Distributions
- Up to this point, the authors have focused on the estimation of risk-neutral moments, and the relationship with subsequent returns.
- The models, such as Barberis and Huang (2008) and Brunnermeier, Gollier, and Parker (2007) that consider the effects of skewness and fat tails in individual securities’ distributions on expected returns deal with investors’ estimates of the physical distribution.
- The purpose of this section is to analyze the relation between risk-neutral and physical moments in their sample, and whether the predictive power of riskneutral moments for subsequent returns is due to their relation to physical moments.
- In addition, the authors examine the relation between risk-neutral moments and forward-looking valuation ratios.
5.1 Risk-neutral moments, physical moments and subsequent returns
- Bakshi, Kapadia, and Madan (2003), who examine the relation between risk-neutral and physical distributions, note that under certain conditions the risk-neutral distribution can be obtained by simply exponentially ‘tilting’ the physical density, with the tilt determined by the risk-aversion of investors.
- The authors compute risk neutral skewness using the standard measure based on the third moment of returns.
- The authors use this information to compute frequency tables to assess the frequency with which the risk-neutral skew ranking is equal to the physical skew ranking.
- Particularly in the first and third terciles, there is a propensity for low risk neutral skew firms to have low physical skewness (48% and 48% for 3- and 12-month maturities), and for high risk neutral skew firms to have high physical skewness (42% and 44% for 3- and 12-month maturities).
- Overall the results in Table 9 tell us that there is an association between Q and P skewness measures, but that they are far from perfectly related.
5.2 Skewness, Valuation and the Internet Bubble
- The authors have presented evidence that risk-neutral higher moments are associated with crosssectional variation in subsequent returns, and that a significant portion of this explanatory power is due to idiosyncratic moments.
- The authors analyze the relation between higher risk-neutral moments and valuation ratios.
- As these results demonstrate, there are strong relations 13We use E/P ratios, rather than price-to-earnings, due to the prevalence of negative earnings for some firms in their sample.the authors.
- The authors further divide the sample into two subperiods, using March of 2000 as the dividing point.
6 Conclusions
- The authors explore the possibility that higher moments of the returns distribution are important in explaining security returns.
- The authors also find a positive relation between kurtosis and returns.
- The authors use several different methods to control for differences in higher co-moments, and their related compensation for risk, when estimating the relation between higher moments and returns.
- The authors find that P andQ skewness measures are strongly, but not perfectly, related.
- Risk neutral skewness is truly a market-based forward looking prediction–and the relation between risk-neutral moments and valuation ratios, for which the authors find evidence, is consistent with this interpretation.
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Citations
322 citations
Cites background from "Ex Ante Skewness and Expected Stock..."
...Conrad, Dittmar, and Ghysels (2008) and Xing, Zhang, and Zhao (2008) study the cross-sectional di¤erences in stock returns as a function of the risk-neutral skewness of individual stocks....
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309 citations
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Cites background or methods from "Ex Ante Skewness and Expected Stock..."
...…by some theoretical and empirical research (Brunnermeier et al., 2007; Mitton and Vorkink, 2007; Barberis and Huang, 2008; Boyer et al., 2010; Conrad et al., 2013). in discretionary accruals.4 Another important difference is that we employ a difference-indifferences research design that…...
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...3 Left tail risk is also important if idiosyncratic return skewness is a priced component of stock returns, as suggested by some theoretical and empirical research (Brunnermeier et al., 2007; Mitton and Vorkink, 2007; Barberis and Huang, 2008; Boyer et al., 2010; Conrad et al., 2013)....
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...Crash risk is particularly important for investors because it may not be diversifiable (Brunnermeier et al., 2007; Mitton and Vorkink, 2007; Barberis and Huang, 2008; Boyer et al., 2010; Conrad et al., 2013; Sunder, 2010)....
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267 citations
References
24,874 citations
"Ex Ante Skewness and Expected Stock..." refers background or methods in this paper
...This evidence implies that there is potentially important variation in the returns of higher moment sorted portfolios that is not captured by the Fama and French (1993) risk adjustment framework....
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...Following Fama and French (1993), we assume that earnings per share data are available no more recently than six months after the fiscal year end....
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...However, Fama and French (1993) interpret the relation between characteristics and returns as evidence of risk factors....
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...These relations persist after controlling for firm characteristics, such as beta, size, and book-to-market ratios, and adjustment for the Fama and French (1993) risk factors....
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14,171 citations
13,433 citations
6,064 citations
5,677 citations
"Ex Ante Skewness and Expected Stock..." refers methods in this paper
...The parameters are estimated in equation (16) via GMM using the sample moment restrictions...
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...Similar to our analysis of co-skewness and co-kurtosis above, our first test uses the S&P 500 as the tangency portfolio in estimating Mt using equation (16)....
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...Mt(τ) = d0 + d1 (R ∗ t (τ)) + d2 (R ∗ t (τ)) 2 + d3 (R ∗ t (τ)) 3 (16)...
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Frequently Asked Questions (4)
Q2. What are the future works mentioned in the paper "Ex ante skewness and expected stock returns" ?
The authors explore the possibility that higher moments of the returns distribution are important in explaining security returns. These methods range from a simple single-factor data-generating process, suggested in Bakshi, Kapadia, and Madan ( 2003 ) and similar to the method ( in physical returns ) used in Harvey and Siddique ( 2000 ), to a non-parametric calculation of the stochastic discount factor.
Q3. How do the authors adjust for the size- and book-to-market characteristics of securities?
When the authors adjust for the size- and book-to-market characteristics of securities, the characteristic-adjusted returns hardly change, averaging -79 and -67 basis points per month, respectively, across the two maturity bins.
Q4. What is the relation between risk neutral and physical probabilities?
The relation between risk-neutral and physical probabilities therefore depends on the price of risk; risk-neutral probabilities subsume, or incorporate, the effects of risk, since the prices from which they are calculated embed investors’ risk preferences.