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We show that the prices of risk for factors that are nonlinear in the market return can be obtained using index option prices. The price of coskewness risk corresponds to the market variance risk premium, and the price of cokurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An analysis of factor models with coskewness risk indicates that the new estimates of the price of risk improve the models’ performance compared with regression-based estimates.

The aims of this study are twofold. First, to determine the sign and magnitude of the skewness risk premium (SRP) in the Italian index option market using two procedures: (i) skewness swap contract...

This study investigates the cross-country impact of U.S. equity market skewness risk. We find that a large decrease in the U.S. market skewness significantly predicts higher future returns on international equity markets. The predictability remains significant after controlling for a set of U.S. and local forecasting variables. Furthermore, we find strong predictability in- an out-of-sample setting and the predictability delivers a large economic value. The U.S. market skewness also forecasts U.S. economic recessions and international market conditions, consistent with the international three-moment capital asset pricing model (three-moment CAPM) and the intertemporal capital asset pricing model (ICAPM).

Using an equilibrium asset and option pricing model in a production economy under jump diffusion, we derive an analytical link between the equity premium, risk aversion and the systematic variance and skewness risk premium. In an empirical application of the model using more than 20 years of data on S&P500 index options, we find that, in line with theory, risk-averse investors demand risk-compensation for holding equity when the systematic skewness risk premium is high. However, when we differentiate between market conditions proxied by investor sentiment, we find that in up-markets (high sentiment) risk aversion is low, while in down-markets (low sentiment) risk aversion is high. We show that in line with theory, the skewness-risk-premium-return relationship only holds when risk aversion is high. In periods of low risk aversion, investors demand lower risk compensation, thus substantially weakening the skewness-risk premium-return trade off. Therefore, we also provide new evidence that helps to disentangle sentiment from risk aversion.

This paper investigates the role of realized and implied and their risk premia (variance and skewness) for commodities’ future returns. We estimate these moments from high frequency and commodity futures option data that results in forward-looking measures. Risk premia are computed as the difference between implied and realized moments. We highlight, from a cross-sectional and time series perspective, the strong positive relation between commodity returns and implied skewness. Moreover, we emphasize the high performance of skewness risk premium. Additionally, we show that their portfolios exhibit the best risk-return tradeoff. Most of our results are robust to other factors such as the momentum and roll yield.

Based on intraday data for a large cross-section of individual stocks and exchange traded funds, we show that short-term as well as long-term fluctuations of realized market and average idiosyncratic higher moments risks are priced in the cross-section of asset returns. Specifically, we find that market and average idiosyncratic volatility and kurtosis are significantly priced by investors mainly in the long-run even if controlled by market moments and other factors, while skewness is mostly short-run phenomenon. A conditional pricing model capturing the time-variation of moments confirms downward-sloping term structure of skewness risk and upward-sloping term structure of kurtosis risk, moreover the term structures connected to market skewness risk and average idiosyncratic skewness risk exhibit different dymanics.

Despite half a century of research, we still do not know the best way to model skewness of financial returns. We address this question by comparing the predictive ability and associated portfolio performance of several prominent skewness models in a sample of ten international equity market indices. Models that employ information from the option markets provide the best outcomes overall. We develop an option-based model that accounts for the skewness risk premium. The new model produces the most informative forecasts of future skewness, the lowest prediction errors and the best portfolio performance in most of our tests.

Utility is the fundamental variable thought to underlie economic choices. In particular, utility functions are believed to reflect preferences toward risk, a key decision variable in many real-life situations. To assess the validity of utility representations, it is therefore important to examine risk preferences. In turn, this approach requires formal definitions of risk. A standard approach is to focus on the variance of reward distributions (variance-risk). In this study, we also examined a form of risk related to the skewness of reward distributions (skewness-risk). Thus, we tested the extent to which empirically derived utility functions predicted preferences for variance-risk and skewness-risk in macaques. The expected utilities calculated for various symmetrical and skewed gambles served to define formally the direction of stochastic dominance between gambles. In direct choices, the animals' preferences followed both second-order (variance) and third-order (skewness) stochastic dominance. Specifically, for gambles with different variance but identical expected values (EVs), the monkeys preferred high-variance gambles at low EVs and low-variance gambles at high EVs; in gambles with different skewness but identical EVs and variances, the animals preferred positively over symmetrical and negatively skewed gambles in a strongly transitive fashion. Thus, the utility functions predicted the animals' preferences for variance-risk and skewness-risk. Using these well-defined forms of risk, this study shows that monkeys' choices conform to the internal reward valuations suggested by their utility functions. This result implies a representation of utility in monkeys that accounts for both variance-risk and skewness-risk preferences.

In this study, we investigate the skewness risk premium in the financial market under a general equilibrium setting. Extending the long-run risks (LRR) model proposed by Bansal and Yaron (J Financ 59:1481–1509, 2004) by introducing a stochastic jump intensity for jumps in the LRR factor and the variance of consumption growth rate, we provide an explicit representation for the skewness risk premium, as well as the volatility risk premium, in equilibrium. On the basis of the representation for the skewness risk premium, we propose a possible reason for the empirical facts of time-varying and negative risk-neutral skewness. Moreover, we also provide an equity risk premium representation of a linear factor pricing model with the variance and skewness risk premiums. The empirical results imply that the skewness risk premium, as well as the variance risk premium, has superior predictive power for future aggregate stock market index returns, which are consistent with the theoretical implication derived by our model. Compared with the variance risk premium, the results show that the skewness risk premium plays an independent and essential role for predicting the market index returns.

We investigate the volatility and skewness risk premium spillovers among the U.S., U.K., German, and Japanese stock markets. We define risk premia as the difference between risk-neutral and realized moments. Our findings highlight that during periods of stress, cross-market and cross-moment spillovers increase and that these increases are mirrored by a decrease in within-market effects. We document strong bidirectional spillovers between volatility and skewness risk premia and emphasize the prominent role played by the volatility risk premium. Finally, we show that several announcements drive the time-varying risk premium spillovers.

We propose a model-free method for measuring the jump skewness risk premium via a trading strategy. We find that in the S&P 500 option market, the premium is positive and greater in absolute terms than the variance premium. The trading strategy allows for examining the premium in different holding periods: daytime, when markets are open, and overnight, outside of trading hours. We demonstrate that both premia vary considerably between trading and non-trading periods, and also increase in periods of market distress. The daytime return on jump skewness is not spanned by other systematic risk factors, suggesting it is a systematic risk factor itself. Outside of trading hours, skewness risk does not seem to be distinguishable from variance risk. We also decompose total return skewness into a jump component, and a signed variance component, and demonstrate that only the jump component is priced. Our work also sketches a new set of stylized facts about variance and jump skewness premia that option pricing models should match.

We find that realized skewness is a significant indicator of returns across a range of assets from different asset classes, namely commodities, government bonds, equity indices and currencies. Taking on skewness risk is broadly compensated within, but more substantially across asset classes. Portfolios in these four asset classes with long positions on most negatively (or least positively) skewed assets and short positions on least negatively (or most positively) skewed assets generate on average a Sharpe ratio of 0.35 between 1990 and 2017. We find little evidence of a common risk driver among these portfolios, to the extent that their combination benefits substantially from diversification, delivering a Sharpe ratio of 0.72. The patterns are not subsumed by other known factors that drive returns, such as value, momentum or carry factors and, consequently, mean-variance efficient multi-factor portfolios assign a positive weight to skewness. Our results remain robust to different measures of skewness and across sub-samples.

We provide a new methodology to empirically investigate the respective roles of systematic and idiosyncratic skewness in explaining expected stock returns. Using a large number of predictors, we forecast the cross-sectional ranks of systematic and idiosyncratic skewness which are easier to predict than their actual values. Compared to other measures of ex ante systematic skewness, our forecasts create a significant spread in ex post systematic skewness. A predicted systematic skewness risk factor carries a significant risk premium that ranges from 7% to 12% per year and is robust to the inclusion of downside beta, size, value, momentum, profitability, and investment factors. In contrast to systematic skewness, the role of idiosyncratic skewness in pricing stocks is less robust. Finally, we document how the determinants of systematic and idiosyncratic skewness differ.

We show that exposure to the risk of kurtosis in oil market drives the cross-section of stock returns from 1996 to 2014. The average monthly difference between the return of portfolio of stocks with low exposure and high exposure to the risk of kurtosis is -0.37%, showing that higher exposure to oil’s kurtosis risk will be penalized by lower average returns. We are able to confirm the significance of kurtosis risk within the statistical framework of Carhart 4-factor model. In contrast to the skewness risk, which is only a significant player in some of the sub-periods, kurtosis risk is keeping its significance through all sub-periods, as well as after taking market moments into account and within different maturities. The significance of the risk of skewness gets more evident moving from shorter to longer maturities. The risk of volatility, which has been shown to be a significant-priced risk in the cross-section of stock returns in literature, loses its significance after controlling for the third and fourth moments.

Using the model-independent approaches of Trolle and Schwartz (2008) and Kozhan et al (2013), we estimate the Variance Risk Premium and Skew Risk Premium for oil market. After estimation, the contribution of the paper is twofold. First, we try to figure out which variables can describe the variation in variance and skew risk premium and the risk-neutral Skewness. The results show that we are able to describe a considerable portion of the variation in all three variables using speculative and macroeconomic factors and most of the variation is being described by by the latter group of variables. We also try to capture the effect of fundamentals of oil market on these variables. The results show that the fundamentals are able to describe big portion of variation in these three variables. Second, we try to predict the return of delta-hedged and delta-vega hedged option portfolios and also oil futures return. The results show that variance risk premium is a significant predictor of delta-hedged portfolio cumulative return with a positive sign while skew risk premium can describe the cumulative return of the delta-Vega Hedged portfolio with negative sign. The results also show that both variance and skew risk premia are significant predictors of cumulative futures return with different signs.

In this article, we show how to take into account skewness risk in portfolio allocation. Until recently, this issue has been seen as a purely statistical problem, since skewness corresponds to the third statistical moment of a probability distribution. However, in finance, the concept of skewness is more related to extreme events that produce portfolio losses. More precisely, the skewness measures the outcome resulting from bad times and adverse scenarios in financial markets. Based on this interpretation of the skewness risk, we focus on two approaches that are closely connected. The first one is based on the Gaussian mixture model with two regimes: a normal regime and a turbulent regime. The second approach directly incorporates a stress scenario using jump-diffusion modeling. This second approach can be seen as a special case of the first approach. However, it has the advantage of being clearer and more in line with the experience of professionals in financial markets: skewness is due to negative jumps in asset prices. After presenting the mathematical framework, we analyze an investment portfolio that mixes risk premia, more specifically risk parity, momentum and carry strategies. We show that traditional portfolio management based on the volatility risk measure is biased and corresponds to a short-sighted approach to bad times. We then propose to replace the volatility risk measure by a skewness risk measure, which is calculated as an expected shortfall that incorporates a stress scenario. We conclude that constant-mix portfolios may be better adapted than actively managed portfolios, when the investment universe is composed of negatively skewed financial assets.

This paper characterizes the equilibrium demand and risk premiums in the presence of skewness risk. We extend the classical mean-variance two-fund separation theorem to a three-fund separation theorem. The additional fund is the skewness portfolio, i.e. a portfolio that gives the optimal hedge of the squared market return; it contributes to the skewness risk premium through co-variation with the squared market return and supports a stochastic discount factor that is quadratic in the market return. When the skewness portfolio does not replicate the squared market return, a tracking error appears; this tracking error contributes to risk premiums through kurtosis and pentosis risk if and only if preferences for skewness are heterogeneous. In addition to the common powers of market returns, this tracking error shows up in stochastic discount factors as priced factors that are products of the tracking error and market returns.

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. The empirical observation on skewness research suggests that derivative professionals may also desire to hedge beyond volatility risk and there exists the need to hedge higher-moment market risks, such as skewness and kurtosis risks. We study two derivative contracts – skewness swap and kurtosis swap – which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the both skewness and kurtosis risk premiums are significantly negative.

In this study, I implement a novel methodology to extract crash risk premia from options and stock markets. I document a dramatic increase in crash risk premia after the 2008/2009 financial crisis, indicating that investors are willing to pay high insurance to hedge against crashes in individual stocks. My results apply to all sectors but are most pronounced for the financial sector. At the same time, crash risk premia on the market index remained at pre-crisis levels. I theoretically explain this puzzling feature in an economy where investors face short-sale constraints. Under short-sale constraints, prices are less informationally efficient which can explain the increase in downside risk in individual stocks. In the data, I document a strong link between proxies of short-sale constraints and crash risk premia.

This study investigates the cross-country impact of U.S. equity market skewness risk. We find that a large decrease in the U.S. market skewness significantly predicts high future returns on international equity markets. The predictability remains significant after controlling for a set of U.S. and local forecasting variables. Furthermore, we find strong predictability in an out-of-sample setting and the predictability delivers a large economic value. The U.S. market skewness also forecasts U.S. economic recessions and international market conditions, consistent with the international three-moment capital asset pricing model (three-moment CAPM) and intertemporal capital asset pricing model (ICAPM).

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 the cross-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. To analyze the extent to which these returns relations represent compensation for risk, we use data on index options and the underlying index to estimate the stochastic discount factor over the 1996-2005 sample period, and allow the stochastic discount factor to include higher moments. We find evidence that, even after controlling for differences in co-moments, individual securities’ skewness matters. However, when we combine information in the risk-neutral distribution and the 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.

The cross section of stock returns has substantial exposure to risk captured by higher moments of market returns. We estimate these moments from daily Standard & Poor's 500 index option data. The resulting time series of factors are genuinely conditional and forward-looking. Stocks with high exposure to innovations in implied market skewness exhibit low returns on average. The results are robust to various permutations of the empirical setup. The market skewness risk premium is statistically and 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, or by firm characteristics.

This article examines the role of nonnormality risks in explaining the momentum puzzle of equity returns. It shows that momentum profits are not normally distributed and, relatedly, that the momentum profitability is partly a compensation for systematic negative skewness risk in line with market efficiency. This finding is pervasive across nine trading strategies that combine different holding and ranking periods and is reinforced when time dependencies in abnormal returns and risks are explicitly modelled. The analysis also reveals that the market and skewness risks of momentum portfolios evolve over the business cycle in a manner that is consistent with market timing and risk aversion. While nonnormality risks matter, a large proportion of the momentum profits remains unexplained which may provide comfort to behavioural theorists.

This paper examines the relationship between currency option's implied skewness and its future realized skewness, where the difference is known as the skewness risk premium (SRP). The SRP indicates whether investors pay a premium to be insured against future crash risk. Past investigations about implied and realized skewness within currency markets showed that both measures are loosely connected or even exhibit a negative relationship that cannot be rationalized by no-arbitrage arguments. Therefore, this paper studies time-series of future and option contract positions data in order to explain the disconnection in terms of investor's position-induced demand pressure. While demand pressures on options do not sufficiently contribute to the disconnection, there is evidence that, surprisingly, demand pressure in currency future markets have the power to explain this market anomaly. Furthermore, currency momentum also plays an important role, which leads to a strong cyclical demand for OTM calls in rising or OTM puts in declining markets. In order to exploit the disconnection of skewness, a simple skew swap trading strategy proposed by Schneider (2012) has been set up. The resulting skew swap returns are relatively high, but the return distribution is extremely fat-tailed. To appropriately compare different skew swap strategy returns, this paper proposes a Higher Moment Sharpe Ratio that also takes higher moments into account.

This article develops a model that takes into account skewness risk in risk parity portfolios. In this framework, asset returns are viewed as stochastic processes with jumps or random variables generated by a Gaussian mixture distribution. This dual representation allows us to show that skewness and jump risks are equivalent. As the mixture representation is simple, we obtain analytical formulas for computing asset risk contributions of a given portfolio. Therefore, we define risk budgeting portfolios and derive existence and uniqueness conditions. We then apply our model to the equity/bond/volatility asset mix policy. When assets exhibit jump risks like the short volatility strategy, we show that skewness-based risk parity portfolios produce better allocation than volatility-based risk parity portfolios. Finally, we illustrate how this model is suitable to manage the skewness risk of long-only equity factor portfolios and to allocate between alternative risk premia., 48 pages, 23 figures, 21 tables

This paper provides new insights in the skewness risk premium in the stock market. By building strategies which take position in the individual skewness of the constituents of the SP100, we show that the skewness risk premium becomes positive and significant for almost all the stocks after the 2007-2009 financial crisis. We find that this is due to a drastic increase (in absolute value) in the price of the skewness, while we do not find any significant change in the realised skewness of the returns. Consistently, we find that the shape of the average implied volatility smile across stocks becomes steeper after the crisis. Moreover, we find that this pre/post crisis structural change does not apply to the market skewness risk premium, computed as the skew premium of the index SP500.

We develop a one-period model of investor asset holdings where investors have heterogeneous preference for skewness. Introducing heterogeneous preference for skewness allows the model's investors, in equilibrium, to underdiversify. We find suppport for our model's three key implications using a dataset of 60,000 individual investor accounts. First, we document that the portfolio returns of underdiversified investors are substantially more positively skewed than those of diversified investors. Second, we show that the apparent mean-variance inefficiency of underdiversified investors can be largely explained by the fact that investors sacrifice mean-variance efficiency for higher skewness exposure. Furthermore, contrary to the asset-pricing predictions of models that incorporate return skewness in the context of full diversification, we show that idiosyncratic skewness, and not just coskewness, can impact equilibrium prices. Third, the underdiversification of investors does not appear to be coincidentally related to skewness. Stocks most often selected by underdiversified investors have substantially higher average skewness -- especially idiosyncratic skewness -- than stocks most often selected by diversified investors.

We show that the prices of risk for factors that are nonlinear in the market return are readily obtained using index option prices. The price of co-skewness risk corresponds to the market variance risk premium, and the price of co-kurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An out-of-sample analysis of factor models with co-skewness and co-kurtosis risk indicates that the new estimates of the price of risk improve the models' performance.

In literature, many researchers focus on information contained in stochastic volatility dynamics, such as CBOE VIX index and its risk premium. However, there are relatively fewer studies on stochastic skewness dynamics. Simple linear regression indicates that stochastic volatility and stochastic skewness have different information about economic fundamentals: regressing CBOE VIX index on CBOE Skewness index, ranging from 1990 to 2013, yields R2 only 2%. Motivated by such striking yet simple empirical result and advances in reduced-form option pricing literature, we build one discrete-time equilibrium option pricing model with non-normal innovation. In our framework, market return skewness is driven by a stochastic process independent of volatility. Thus volatility and skewness dynamics possess different information about stochastic investment opportunity. Specifically, we use Variance Gamma distribution to generate economic growth shocks. Variance gamma innovation is able to be decomposed into two shocks, one of them is strictly positive, and the other is strictly negative. Our skewness risk factor controls the frequency and magnitude of these two shocks, therefore, controls the switching behavior of economic condition, e.g. from good to bad state. We show in our model that stochastic skewness risk should be priced, provided that investors have recursive utility function. Empirically, using Fama-Macbeth two pass regression and a panel of S&P 500 index option returns, we find that stochastic skewness has positive, statistically significant risk premium, and its sign does not change across our sample period. Moreover, our results show that skewness risk has superior explanation power for index option returns, compared with performance of volatility risk and Merton-type of jump risk. The shock of skewness risk premium is able to predict short-term market excess returns, the R2 is as high as 7%.

Previous studies on the cross-sectional market moments’ risk premia find significantly negative risk premia for the market volatility and the market skewness risks and a positive premium for the market kurtosis risk. However, a significantly negative price of risk for the market skewness and a positive price of risk for the market kurtosis move against the ICAPM intuition. We refer to these counter-intuitive prices of risk as the anomalies in the market moments’ risk premia, and show that these anomalies only exist in low risk-aversion periods. Once investors’ risk-aversion rise, these anomalies vanish. Moreover, we find that Baker and Wurgler’s (2006) index of investor sentiment is strongly negatively affected by the past realizations in investor risk-aversion. Consequently, our empirical results can also be replicated by analyzing periods of high and low sentiment, separately.

If asset returns have systematic skewness, expected returns should include rewards for accepting this risk. We formalize this intuition with an asset pricing model that incorporates conditional skewness. Our results show that conditional skewness helps explain the cross-sectional variation of expected returns across assets and is significant even when factors based on size and book-to-market are included. Systematic skewness is economically important and commands a risk premium, on average, of 3.60 percent per year. Our results suggest that the momentum effect is related to systematic skewness. The low expected return momentum portfolios have higher skewness than high expected return portfolios.

We propose a reliable new estimator for realized skewness which is robust to microstructure noise at ultra-high frequency level. Asymptotic theory for the new estimator has been de- rived. Simulation example veries its superior performance. We apply the new estimator to tick data of the S&P 500 index for forecasting equity premium in the U.S. market from 1990-2011 and nd that it has signicant forecast-ability both in-sample and out-of-sample. We also show that the new skewness measure plus the variance risk premium provides right decomposition for the skewness risk. We thus provide evidence that realized skewness links to conditional market premium.

The finance literature has found it challenging to explain empirically observed option returns, most notably in the case of out-of-the-money (OTM) equity index puts. I propose liquidation risk in options markets, defined as the possibility of forced selling of speculative positions following a liquidity shock, as a major driver of the relative pricing of OTM put vs. call options (option-implied skewness) as well as their relative returns (skewness risk premium). Employing size and sign of speculative net long positions (OSP) in a panel of commodity futures options markets as a key proxy for liquidation risk, I find that both implied skewness as well as the skewness risk premium are positively correlated with OSP and that these effects are stronger when funding conditions deteriorate. I also provide direct evidence of the price effects during such liquidation events. Trading strategies designed to theoretically exploit the skewness premium yield up to 2.5 percent per month and load significantly on risk factors related to the ease of funding for financial intermediaries.

We infer conditional swap rate moments model independently from swaption cubes. Conditional volatility and skewness exhibit systematic variation across swap maturities and option expiries (conditional kurtosis less so), with conditional skewness sometimes changing sign. Conditional skewness displays some relation to the level and volatility of swap rates but is most consistently related to the conditional correlation between swap rates and swap rate variances. From realized excess returns on synthetic variance and skewness swap contracts, we infer that variance and (to a lesser extent) skewness risk premia are negative and time varying. For the most part, results hold true in both the USD and EUR markets and in both precrisis and crisis subsamples. We design and estimate a dynamic term structure model that captures much of the dynamics of conditional swap rate moments.

Idiosyncratic security risks are modelled as following a joint spherical distribution characterized by a mean vector and a generalized covariance matrix. Skewness is generated by a single factor for the whole economy, but upon which different securities have different loadings. This results in three-fund separation—two funds to span the spherical risk and one more fund to span the additional skewness risk. A three-parameter normative portfolio analysis that allows short sales restrictions is developed. In addition, a three-parameter capital asset pricing model is provided.

We use a comprehensive database of inter-dealer quotes to conduct the first empirical analysis of the dynamics of the swaption cube. Using a model independent approach, we establish a set of stylized facts regarding the cross-sectional and time-series variation of conditional volatility and skewness of the swap rate distributions implied by the swaption cube. We then develop and estimate a dynamic term structure model that is consistent with these stylized facts, and use it to infer volatility and skewness of the risk-neutral and physical swap rate distributions. Finally, we investigate the fundamental drivers of these distributions. In particular, we find that volatility, volatility risk premia, skewness, and skewness risk premia are significantly related to the characteristics of agents’ belief distributions for the macroeconomy, with GDP beliefs the most important factor in the USD market, and inflation beliefs the most important factor in the EUR market. This is consistent with differences in monetary policy objectives in the two markets.

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. The empirical observation on skewness research suggests that derivative professionals may also desire to hedge beyond volatility risk and there exists the need to hedge higher‐moment market risks, such as skewness and kurtosis risks. We study two derivative contracts – skewness swap and kurtosis swap – which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the both skewness and kurtosis risk premiums are significantly negative.

Returns of hedge funds generally exhibit non-normality. It is well documented that if asset returns have systematic skewness, expected returns should include rewards for accepting this risk. This skewness risk premium should be controlled for in any estimate of performance. To investigate this issue we specify the Residual Augmented Least Squares (RALS) estimator, designed to exploit non-normality in a time series’ distribution. Specifying a linear factor model, we provide robust estimates of hedge fund performance, demonstrating the increase in efficiency of RALS relative to OLS estimation. Our evidence suggests that measuring performance using OLS alphas is inefficient, understating the performance of some hedge funds and overstating the performance of others. We then examine the source of the OLS mispricing. We find that the level of mispricing is positively related to estimates of skewness in the historical fund returns. We conclude that when estimated by OLS the performance of managers who pursue a strategy exhibiting positive skewness is understated and those whose strategy exhibits negative skewness is overstated. Estimation by RALS overcomes this. Our findings are robust to the biases in hedge fund databases.

Hedge funds are fundamentally exposed to equity volatility, skewness, and kurtosis risks based on the systematic pattern and significant spread in alphas from the existing models that do not control for the higher-moment risks. The spread and pattern in alphas do not disappear with bootstrap simulation, Bayesian analysis to account for potential estimation error, adjustment for backfilling bias, and the inclusion of additional systematic factors. Significant cross-sectional variation in higher-moment exposures is observed across fund styles with equity-oriented styles displaying more extreme exposures. Investable higher-moment factors explain the time series behavior of returns of a large number of Managed Futures, Event Driven, and Long/Short Equity hedge funds. Average exposure sensitivities for higher-moment factors are statistically significant in an estimation that accounts for style fixed effects and fund random effects.

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.

Motivated by the Internet bubble in the late 1990’s and early years of this century, we explore the possibility that higher moments of the returns distribution may be important in explaining security returns. Using a sample of option prices from 1996-2005, we estimate individual securities’ volatility, skewness and kurtosis using the method of Bakshi, Kapadia and Madan (2003). We flnd that higher moments are strongly related to returns, even after controlling for difierences in size and book-to-market. Consistent with Ang, Hodrick, Xing and Zhang (2006), we flnd a negative relation between cross-sectional volatility and returns. We also flnd a signiflcant relation between skewness and returns, with more negatively skewed returns associated with subsequent positive returns. Kurtosis has a strong positive relation with returns. In an examination of industry portfolios, we flnd that technology stocks have strikingly higher exposure to skewness risk, and, in a cross-sectional test, we flnd evidence that higher moments are priced.

Human beings want to believe that good outcomes in the future are more likely, but also want to make good decisions that increase average outcomes in the future. We consider a general equilibrium model with complete markets and show that when investors hold beliefs that optimally balance these two incentives, portfolio holdings and asset prices match six observed patterns: (i) because the cost of biased beliefs are typically second-order, investors typically hold biased assessments of probabilities and so are not perfectly diversified according to objective metrics; (ii) because the costs of biased beliefs temper these biases, the utility costs of the lack of diversification are limited; (iii) because there is a complementarity between believing a state more likely and purchasing more of the asset that pays off in that state, investors over-invest in only one Arrow-Debreu security and smooth their consumption well across the remaining states; (iv) because different households can settle on different states to be optimistic about, optimal portfolios of ex ante identical investors can be heterogeneous; (v) because low-price and low-probability states are the cheapest states to buy consumption in, overoptimism about these states distorts consumption the least in the rest of the states, so that investors tend to overinvest in the most skewed securities; (vi) finally, because investors with optimal expectations have higher demand for more skewed assets, ceteris paribus, more skewed asset can have lower average returns.

The paper examines the role of non-normality risks in explaining the momentum puzzle of equity returns. It shows that momentum profits are not normally distributed and, relatedly, that the momentum profitability is partly a compensation for systematic negative skewness risk in line with market efficiency. This finding is pervasive across nine trading strategies that combine different holding and ranking periods and is reinforced when time dependencies in abnormal returns and risks are explicitly modeled. The analysis also reveals that the market and skewness risks of momentum portfolios evolve over the business cycle in a manner that is consistent with market timing and risk aversion. While non-normality risks matter, a large proportion of the momentum profits remains unexplained which may provide comfort to behavioural theorists.

This article provides several new insights into the economic sources of skewness. First, we document the differential pricing of individual equity options versus the market index, and relate it to variations in return skewness. Second, we show how risk aversion introduces skewness in the risk-neutral density. Third, we derive laws that decompose individual return skewness into a systematic component and an idiosyncratic component. Empirical analysis of OEX options and 30 stocks demonstrates that individual risk-neutral distributions differ from that of the market index by being far less negatively skewed. This paper explains the presence and evolution of risk-neutral skewness over time and in the cross-section of individual stocks.

We present a framework for modeling and estimating dynamics of variance and skewness from time-series data using a maximum likelihood approach assuming that the errors from the mean have a non-central conditional t distribution. We parameterize conditional variance and conditional skewness in an autoregressive framework similar to that of GARCH models and estimate the parameters in a conditional noncentral t distribution. The likelihood function has two time-varying parameters, the degrees of freedom and the noncentrality parameter. We apply this methodology to daily and monthly equity returns data from the U.S., Germany and Japan, concurrently estimating conditional mean, variance and skewness. We find that there is significant conditional skewness. We then use this model to understand how the inclusion of conditional skewness affects some of the well-known stylized facts about conditional variance and the relation between returns and conditional variance. Two important stylized facts about conditional variance we examine are persistence and asymmetry in variance. Persistence refers to the tendency where high conditional variance is followed by high conditional variance. Asymmetry in variance, i.e., the observation that conditional variance depends on the sign of the innovation to the conditional mean has been documented in asymmetric variance models used in Glosten, Jagannathan, and Runkle (1993) and Engle and Ng (1993). We find that the evidence of asymmetric variance is really just conditional skewness. Inclusion of conditional skewness also impacts the persistence in conditional variance. However, we also find that there are significant seasonalities in variance and the results also depend on how the seasonal effects are accommodated in the estimation methodology. We also examine the relation between expected returns and volatility.

We infer conditional swap rate moments model independently from swaption cubes. Conditional volatility and skewness exhibit systematic variation across swap maturities and option expiries (conditional kurtosis less so), with conditional skewness sometimes changing sign. Conditional skewness displays some relation to the level and volatility of swap rates but is most consistently related to the conditional correlation between swap rates and swap rate variances. From realized excess returns on synthetic variance and skewness swap contracts, we infer that variance and (to a lesser extent) skewness risk premia are negative and time varying. For the most part, results hold true in both the USD and EUR markets and in both precrisis and crisis subsamples. We design and estimate a dynamic term structure model that captures much of the dynamics of conditional swap rate moments.