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Journal ArticleDOI

Ex Ante Skewness and Expected Stock Returns

TL;DR: This paper used a sample of option prices and the method of Bakshi, Kapadia and Madan (2003) to estimate the ex ante higher moments of the underlying individual securities' risk-neutral returns distribution.
Abstract: We use a sample of option prices, and the method of Bakshi, Kapadia and Madan (2003), to estimate the ex ante higher moments of the underlying individual securities’ risk-neutral returns distribution. We find that individual securities’ volatility, skewness, and kurtosis are strongly related to subsequent returns. Specifically, we find a negative relation between volatility and returns in thecross-section. We also find a significant relation between skewness and returns, with more negatively (positively) skewed returns associated with subsequent higher (lower) returns, while kurtosis is positively related to subsequent returns. We analyze the extent to which these returns relations represent compensation for risk. We find evidence that, even after controlling for differences in comoments, individual securities’ skewness matters. As an application, we examine whether idiosyncratic skewness in technology stocks might explain bubble pricing in Internet stocks. However, when we combine information in the risk-neutral distribution and a stochastic discount factor to estimate the implied physical distribution of industry returns, we find little evidence that the distribution of technology stocks was positively skewed during the bubble period – in fact, these stocks have the lowest skew, and the highest estimated Sharpe ratio, of all stocks in our sample.

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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Ex Ante Skewness and Expected Stock Returns
First Draft: March 2007
This Draft: January 11, 2012
Abstract
We use a sample of option prices to estimate the ex ante higher moments of the underlying individual
securities’ risk-neutral returns distribution. We find that individual securities’ risk neutral volatility,
skewness, and kurtosis are strongly related to future returns. Specifically, we find a negative relation
between past volatility and subsequent returns in the cross-section. We also find that ex ante more
negatively (positively) skewed returns are associated with subsequent higher (lower) returns, while
ex ante kurtosis is positively related to subsequent returns. We analyze the extent to which these
returns relations represent compensation for risk and find evidence that, even after controlling for
differences in co-moments, individual securities’ skewness matters.

1 Introduction
Models implying that investors consider higher moments in returns have a long history in the
literature. Researchers such as Rubinstein (1973), Kraus and Litzenberger (1976) and Kraus
and Litzenberger (1983) develop models of expected returns which incorporate skewness. In
these models, the higher moments which are relevant for individual securities are co-moments
with the aggregate market portfolio. Subsequently, empirical work provided evidence that
higher moments of the return distribution are important in pricing securities. Consistent
with the models’ focus on co-moments, the tests in these papers ask whether a security’s co-
skewness or co-kurtosis with the market is priced; historical returns data are typically used to
measure these co-moments. For example, Harvey and Siddique (2000) explore both skewness
and co-skewness and test whether co-skewness is priced, and Dittmar (2002) tests whether
a security’s co-skewness and co-kurtosis with the market portfolio might influence investors’
expected returns.
Other recent papers have suggested that additional features of individual securities’ pay-
off distribution may be relevant for understanding differences in assets’ returns. For ex-
ample, Ang, Hodrick, Xing, and Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009)
document that firms’ idiosyncratic return volatility contains important information about fu-
ture returns. The work of Barberis and Huang (2008), Brunnermeier, Gollier, and Parker
(2007), and the empirical evidence presented in Mitton and Vorkink (2007) and Boyer, Mit-
ton, and Vorkink (2010) imply that the skewness of individual securities may also influence
investors’ portfolio decisions. Xing, Zhang, and Zhao (2010) find that portfolios formed by
sorting individual securities on a measure which is related to idiosyncratic skewness gen-
erate cross-sectional differences in returns. Green and Hwang (2009) use the approach of
Zhang (2006) and find that IPOs with high expected skewness (’lottery’ stocks) experience
significantly greater first-day returns, followed by substantially greater negative abnormal
returns in the subsequent three to five years.
We therefore have two strands in the existing literature: (1) models and empirical re-
sults that emphasize the importance of higher moments as they affect stochastic discount
factors, (2) models and empirical evidence that focus on the higher moment characteristics
of individual securities. In this second strand of the literature, researchers have proposed
both behavioral and rational models. For example, Barberis and Huang (2008) argue that
investors with cumulative prospect theory preferences demand securities with highly skewed
payoffs, such as IPO stocks. Brunnermeier, Gollier, and Parker (2007) develop a model of op-
timal (as opposed to rational) beliefs which also predicts that investors will overinvest in the
most highly (right) skewed securities, with the consequence that those securities will have
1

lower subsequent average returns. They also show that, while there is a rational expectations
solution to their model, it represents a knife-edge case. Mitton and Vorkink (2007) introduce
a rational model where investors have heterogeneous preferences for skewness and show that
idiosyncratic skewness can impact prices. Chabi-Yo, Ghysels, and Renault (2010) also present
a model where heterogeneity of beliefs across rational investors can generate a pricing kernel
which depends on idiosyncratic moments.
In this paper, we examine the importance of higher moments using a new approach. We
exploit the fact that if option and stock prices reflect the same information, then it is possi-
ble to use options market data to extract estimates of the higher moments of the securities’
(risk-neutral) probability density function. Our method has several advantages. First, option
prices are a market-based estimate of investors’ expectations. Many authors, including Bates
(1991), Rubinstein (1994) and Jackwerth and Rubinstein (1996) have argued that option mar-
ket prices can capture the information of market participants. Second, the use of option prices
eliminates the need of a long time series of returns to estimate the moments of the return dis-
tribution; this is especially helpful when trying to forecast the payoff distribution of relatively
new firms or during periods where expectations, at least for some firms, may change relatively
quickly. Third, options reflect a true ex ante measure of expectations; they do not give us, as
Battalio and Schultz (2006) note, the “unfair advantage of hindsight.” As Jackwerth and Ru-
binstein (1996) state, “not only can the nonparametric method reflect the possibly complex
logic used by market participants to consider the significance of extreme events, but it also
implicitly brings a much larger set of information . . . to bear on the formulation of probability
distributions.
We begin with a sample of options on individual stocks, and test whether cross-sectional
differences in estimates of the higher moments of an individual security’s payoff extracted
from options are related to subsequent returns. Consistent with the Ang, Hodrick, Xing, and
Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009) findings for physical measures of
idiosyncratic volatility, we find a negative relation between risk-neutral volatility and subse-
quent returns. We also document a significant negative relation between firms’ risk-neutral
skewness and subsequent returns - that is, more negatively skewed securities have higher
subsequent returns. In addition, we find a significant positive relation between firms’ risk-
neutral kurtosis and subsequent returns. 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.
We examine the extent to which these relations between risk-neutral higher moments and
subsequent returns are determined by co-moments with the market portfolio. We measure
co-moments using the approaches of Harvey and Siddique (2000) and Bakshi, Kapadia, and
2

Madan (2003), and then decompose total moments into co-moments such as co-skewness
and idiosyncratic moments. We find that the relation between idiosyncratic higher moments,
particularly idiosyncratic skewness, and subsequent returns persists, even after controlling
for differences in covariance, co-skewness and co-kurtosis.
1
Our results are consistent with models such as Brunnermeier, Gollier, and Parker (2007),
and Barberis and Huang (2008), which predict that investors will trade off the benefits of
diversification and skewness, holding more concentrated positions in skewed securities, and
resulting in a negative relation between idiosyncratic skewness and expected returns. These
results are also consistent with the empirical evidence in Mitton and Vorkink (2007), who
examine the choices of investors in a sample of discount brokerage accounts and find that
investors appear to hold relatively undiversified portfolios and accept lower Sharpe ratios
for positively skewed portfolios and securities. These papers focus on physical moments of
returns, in contrast to the risk-neutral moments that we examine. Consequently, we ana-
lyze the relation between our risk-neutral estimates of skewness and estimates formed from
historical returns. We find a positive and statistically significant relation between these esti-
mates; however, we find comparatively little evidence that the relation between risk-neutral
moments and subsequent returns in our sample is driven by this relation; that is, after con-
trolling for differences in physical moments, the predictive relation between risk-neutral mo-
ments and subsequent returns continues to hold. In contrast, after controlling for differences
in risk-neutral moments, we find no clear pattern in returns for portfolios which differ in
physical skew.
The remainder of the paper is organized as follows. In section 2, we detail the method we
employ for recovering measures of volatility, skewness, and kurtosis, following Bakshi, Ka-
padia, and Madan (2003) and we discuss the data (and data filters) used in our analysis. In
Section 3 we focus on testing whether estimates of the ex ante higher moments of the payoff
distribution obtained from options data are related to the subsequent returns of the underly-
ing security. In Section 4, we analyze the extent to which the relations between option-based
ex ante higher moment sorts and subsequent returns are due to investors seeking compensa-
tion for higher co-moment risk, rather than idiosyncratic moments. We examine in Section 5
the relation between risk-neutral and physical distributions, and in particular the comparison
of portfolio sorts based on skewness under both measures. We conclude in Section 6.
1
In robustness checks, we also explore a stochastic discount factor approach and consider several alternative
specifications of the stochastic discount factor, both parametric and non-parametric. We find results similar to
those obtained from the decomposition of higher moments. These results are available in a companion document
containing supplementary material: see BLANK (2011).
3

2 Data and Computing Ex Ante Risk-Neutral Moments
We wish to examine the relation, if any, between features of the risk-neutral density func-
tion and the pricing of stocks. In this section we describe the data and the methods used to
compute ex ante estimates of volatility, skewness, and kurtosis.
Our data on option prices are from Optionmetrics (provided through Wharton Research
Data Services). We begin with daily option price data for all out-of-the-money calls and puts
for all stocks from 1996-2005. Closing prices are constructed as midpoint averages of the
closing bid and ask prices.
Data on stock returns are obtained from the Center for Research in Security Prices (again
provided through Wharton Research Data Services). We employ daily and monthly returns
from 1996-2005 for all individual securities covered by CRSP with common shares outstand-
ing. Risk free rates are the continuously compounded yield computed from the bank discount
yields on secondary market three month Treasury Bills taken from the Federal Reserve Re-
port H.15. Finally, we obtain balance sheet data for the computation of book-to-market ratios
from Compustat and compute these ratios following the procedure in Davis, Fama, and French
(2000).
We begin by calculating higher moments of firms’ risk neutral probability distributions. In-
tuitively, a risk neutral probability distribution is computed so that today’s fair (i.e. arbitrage-
free) price of an asset is equal to the discounted expected value of the future payoffs of the
asset, where the discount rate used is simply the riskfree rate. Thus, under the risk neutral
measure, all financial assets in the economy have the same expected rate of return, regard-
less of their risk. In contrast, if we use the actual (or physical) probability distribution of
the asset’s payoffs and assume that investors are risk-averse, assets which have more risk in
their distribution of payoffs should have a greater expected rate of return (and so lower prices)
than less risky assets. 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.
Like the physical density, the risk neutral density has first, second, third and fourth mo-
ments, respectively mean, variance, skewness and kurtosis. All densities are extracted from
options and are therefore conditional and for a given horizon. In a risk-neutral density, the
mean should correspond to the risk free rate at a given time with a particular maturity.
To estimate the higher moments of the (risk-neutral) density function of individual securi-
ties, we use the results in Bakshi and Madan (2000) and Bakshi, Kapadia, and Madan (2003).
Bakshi and Madan (2000) show that any payoff to a security i can be constructed and priced
4

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Abstract: The cross-section of stock returns has substantial exposure to risk captured by higher moments in market returns. We estimate these moments from daily S&P 500 index option data. The resulting time series of factors are thus genuinely conditional and forward-looking. Stocks with high sensitivities to innovations in implied market volatility and skewness exhibit low returns on average, whereas those with high sensitivities to innovations in implied market kurtosis exhibit high returns on average. The results on market skewness risk are extremely robust to various permutations of the empirical setup. The estimated premium for bearing market skewness risk is between -6.00% and -8.40% annually. This market skewness risk premium is economically significant and cannot be explained by other common risk factors such as the market excess return or the size, book-to-market, momentum, and market volatility factors. Using ICAPM intuition, the negative price of market skewness risk indicates that it is a state variable that negatively affects the future investment opportunity set.

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  • ...…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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Journal ArticleDOI
TL;DR: In this article, the relationship between average return and risk for New York Stock Exchange common stocks was tested using a two-parameter portfolio model and models of market equilibrium derived from the two parameter portfolio model.
Abstract: This paper tests the relationship between average return and risk for New York Stock Exchange common stocks. The theoretical basis of the tests is the "two-parameter" portfolio model and models of market equilibrium derived from the two-parameter portfolio model. We cannot reject the hypothesis of these models that the pricing of common stocks reflects the attempts of risk-averse investors to hold portfolios that are "efficient" in terms of expected value and dispersion of return. Moreover, the observed "fair game" properties of the coefficients and residuals of the risk-return regressions are consistent with an "efficient capital market"--that is, a market where prices of securities

14,171 citations

Journal ArticleDOI
TL;DR: Cumulative prospect theory as discussed by the authors applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses, and two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting function.
Abstract: We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses. Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting functions. A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability. Expected utility theory reigned for several decades as the dominant normative and descriptive model of decision making under uncertainty, but it has come under serious question in recent years. There is now general agreement that the theory does not provide an adequate description of individual choice: a substantial body of evidence shows that decision makers systematically violate its basic tenets. Many alternative models have been proposed in response to this empirical challenge (for reviews, see Camerer, 1989; Fishburn, 1988; Machina, 1987). Some time ago we presented a model of choice, called prospect theory, which explained the major violations of expected utility theory in choices between risky prospects with a small number of outcomes (Kahneman and Tversky, 1979; Tversky and Kahneman, 1986). The key elements of this theory are 1) a value function that is concave for gains, convex for losses, and steeper for losses than for gains,

13,433 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that standard errors of more than 3.0% per year are typical for both the CAPM and the three-factor model of Fama and French (1993), and these large standard errors are the result of uncertainty about true factor risk premiums and imprecise estimates of the loadings of industries on the risk factors.

6,064 citations

Journal ArticleDOI
TL;DR: This article analyzed how mutual fund performance relates to past performance and found evidence that differences in performance between funds persist over time and that this persistence is consistent with the ability of fund managers to earn abnormal returns.
Abstract: This paper analyzes how mutual fund performance relates to past performance. These tests are based on a multiple portfolio benchmark that was formed on the basis of securities characteristics. We find evidence that differences in performance between funds persist over time and that this persistence is consistent with the ability of fund managers to earn abnormal returns.

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)
Q1. What are the contributions in "Ex ante skewness and expected stock returns" ?

The authors analyze the extent to which these returns relations represent compensation for risk and find evidence that, even after controlling for differences in co-moments, individual securities ’ skewness matters. 

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. 

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. 

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.