Monthly Pattern and Portfolio Effect on Higher Moments of Stock Returns: Empirical Evidence from Hong Kong

Using a direct test, this paper studies the month-of-the-year effect on the higher moments of six industrial stock indices of the Hong Kong market. We also examine the portfolio effect on skewness and kurtosis across month of the year to see if such an anomaly exists. The empirical results support a weak month-of-the-year effect in higher moments of stock returns. Using a complete sample of all possible combinations for each portfolio size, we show that portfolio effect varies across month of the year for both skewness and kurtosis. In particular, our results show that diversification does not necessarily provide benefits to rational investors when the stock return distribution is non-normal, even though portfolio formation can reduce standard deviation. In June, August and October, diversification across industrial sectors results in a more negatively skewed and leptokurtic return distribution, which is not preferred by investors with risk-aversion. Two (one) possible explanations for the portfolio effect on skewness (kurtosis) are also provided. Our empirical results add new evidence to the existence of anomalies in the Hong Kong stock market. This revised version was published online in August 2006 with corrections to the Cover Date.     

Friction in the trading process and the estimation of systematic risk

This paper considers how estimates of the market model beta parameter can be biased by friction in the trading process (information, decision, and transaction costs) that (a) leads to a distinction between observed and ‘true’ returns; (b) causes observed returns to be generated asynchronously for a set of interdependent securities; and (c) thereby introduces serial cross-correlation into security returns. Several propositions are derived from which consistent estimators of beta are obtained, and the effect of differencing interval length on beta estimates is specified. The formulation is contrasted with the related analyses of Scholes-Williams (1977) and Dimson (1979).     

Modeling Conditional Skewness in Stock Returns

Abstract

In this paper, we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling skewness and kurtosis of the size typically encountered in stock return series. The need to allow for skewness can also be readily tested. The model is consistent with the volatility feedback effect in that conditional skewness is dependent on conditional variance. Compared to previously presented GARCH models allowing for conditional skewness, the model is analytically tractable, parsimonious and facilitates straightforward interpretation.Our empirical results indicate the presence of conditional skewness in the monthly postwar US stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.     

The information content of implied skewness and kurtosis changes prior to earnings announcements for stock and option returns

We use option prices to examine whether changes in stock return skewness and kurtosis preceding earnings announcements provide information about subsequent stock and option returns. We demonstrate that changes in jump risk premiums can lead to changes in implied skewness and kurtosis and are also associated with the mean and variability of the stock price response to the earnings announcement. We find that changes in both moments have strong predictive power for future stock returns, even after controlling for implied volatility. Additionally, changes in both moments predict call returns, while put return predictability is primarily linked to changes in skewness.     

Accounting Ratios and the Cross‐section of Expected Stock Returns

Under clean‐surplus accounting, the log return on a stock can be decomposed into a linear function of the contemporaneous log return on equity, the contemporaneous log dividend–price ratio (if the stock pays a dividend), and both the contemporaneous and lagged values of the log book‐to‐market equity ratio. This paper studies the implications of this decomposition for the cross‐section of conditional expected stock returns. The empirical analysis reveals that the log accounting ratios capture cross‐sectional variation in both the conditional mean and conditional variance of log stock returns, which is consistent with the decomposition. It also brings fresh insights to the relation between firm size (market equity) and conditional expected stock returns. The evidence indicates that the conditional median return increases with firm size, while the conditional return skewness decreases with firm size. Empirically, the skewness effect outweighs the median effect, leading to the well‐documented inverse relation between size and average returns. The results of out‐of‐sample tests suggest that investors could use the information provided by the observed values of the log accounting ratios to formulate more effective portfolio strategies.     

The Impact of Merger Bids on the Participating Firms' Security Holders

This paper investigates whether merger bids have an impact on the wealth of the participating firms' bondholders and stockholders. Monthly and daily bond and stock returns are calculated relative to the announcement date of a merger bid for a sample of conglomerate mergers. The results show that while the stockholders of target firms gain from a merger bid, no other securityholders either gain or lose. To provide direct evidence on the existence of “diversification effects” and “incentive effects,” we test whether the bondholders' returns are dependent upon the correlation between the returns of the merging firms and whether the size of the bondholders' and stockholders' returns in individual mergers are correlated. The results are consistent with a capital market that efficiently resolves conflicts of interest between stockholders and bondholders.     

“好”的不确定性、“坏”的不确定性与股票市场定价——基于中国股市高频数据分析

不确定性并不是都是“坏”的,“好”的不确定性也同样存在。本文采用Barndorff-Nielsen et al.(2010)提出的已实现半方差作为股票市场“好”的不确定性和“坏”的不确定性的代理指标,并在此基础上构建了相对符号变差(RSV),分析RSV对中国股市定价的影响。基于2007-2017年中国A股5分钟高频数据的实证研究发现:(1)与理论解释相一致,RSV与股票收益之间呈现负相关关系。无论是基于单变量分组、双变量分组还是公司层面的截面回归,这种影响在经济上和统计上都显著。(2)RSV是独立于已实现偏度的一个重要定价因子,且RSV对股票的定价能力强于已实现偏度的定价能力。(3)RSV对中国股市的影响是状态依存的,相对于经济景气程度高的状态,在经济景气程度低的状态下RSV定价影响更大。(4)基于RSV构建的投资组合的表现明显优于市场超额收益率组合、SMB组合和HML组合的表现。     

Do Sovereign Re‐Ratings Destabilize Equity Markets during Financial Crises? New Evidence from Higher Return Moments

This study investigates the effects of S&P's sovereign re‐ratings on the higher moments of equity market returns over recent financial crises. Using a set of intraday stock market index prices and sovereign credit ratings for a sample of 36 countries that experienced sovereign rating changes over the period from 1996 to 2013, we find that the higher moments of stock market returns are significantly more responsive to sovereign re‐ratings during financial crises, but the effects on stock markets are not the same across different financial crises. The effects during crises are, however, magnified for large downgrades and those that are associated with a loss of investment grade status. We find that there are asymmetric effects during financial crises in that downgrades are consistently more significant than upgrades in increasing realized volatility and realized kurtosis. Both upgrades and downgrades affect realized skewness in times of crises in the expected direction.     

SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

SKEWNESS AND KURTOSIS IN JAPANESE EQUITY RETURNS: EMPIRICAL EVIDENCE

In this paper, the distribution of equity returns on the Tokyo Stock Exchange is examined from 1965 to 1984, and significant and persistent skewness and kurtosis are found. The deviation of security returns from normality declines with increasing portfolio size and appears to be greater than the non-normality evidenced in U.S. security returns. Further, these deviations from normality persist even after controlling for January and firm size effects.     

Ex Ante Skewness and Expected Stock Returns

We use option prices to estimate 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 (positive) relation between ex ante volatility (kurtosis) and subsequent returns in the cross‐section, and more ex ante negatively (positively) skewed returns yield subsequent higher (lower) 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.     

The Dark Side of Circuit Breakers

Market-wide circuit breakers are trading halts aimed at stabilizing the market during dramatic price declines. Using an intertemporal equilibrium model, we show that a circuit breaker significantly alters market dynamics and affects investor welfare. As the market approaches the circuit breaker, price volatility rises drastically, accelerating the chance of triggering the circuit breaker—the so-called “magnet effect,” returns exhibit increasing negative skewness, and trading activity spikes up. Our empirical analysis supports the model's predictions. Circuit breakers can affect overall welfare negatively or positively, depending on the relative significance of investors' trading motives for risk sharing versus irrational speculation.     

The impact of the COVID-19 pandemic and Russia-Ukraine war on multiscale spillovers in green finance markets: Evidence from lower and higher order moments

This study analyzes the impact of the COVID-19 pandemic and the Russia-Ukraine war on the connectedness of lower-order moments (returns and volatility) and higher-order moments (skewness and kurtosis) in the markets of green bonds, clean energy, wind, solar, and sustainability indexes. To compare the spillover effects of these moments, we use the Diebold and Yilmaz and Barunik and Krehlik methods. Our findings show that the total spillover effect of lower-order moments is higher than that of higher-order moments in the time domain. In the frequency domain, the total return and skewness spillover are primarily concentrated in the short term, whereas the total volatility spillover is mainly concentrated in the long term. Furthermore, we observe that the spillover effect of the Russia-Ukraine war on the green finance market is mild, while the COVID-19 pandemic has a significant and unprecedented influence on the spillover of both lower- and higher-order moments in this market. Additionally, we note that before the COVID-19 outbreak, the total kurtosis spillover was irregular, but it became concentrated in the long term after the outbreak. Moreover, the continuation of COVID-19 has had an unprecedented and long-lasting impact on the kurtosis and skewness of the green bond market.     

Risk measurement distortion: an improved model of return smoothing

We examine the effects of smoothed hedge fund returns on standard deviation, skewness, and kurtosis of return and on correlation of returns using a MA(2)-GARCH(1,1)-skewed-t representation instead of the traditional MA(2) model employed in the literature. We present evidence that our proposed representation is more consistent with the behavior of hedge fund returns than the traditional MA(2) representation and that the traditional method tends to overstate the degree of smoothing observed in hedge fund returns. We examine methods for correcting the distortive effects of smoothing using our representation.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

The risk–return relations in the Singapore stock market

This paper examines the risk–return relations in the Singapore stock market for the period April 1986 to December 1998. Though beta is significantly related to realized returns, the explanatory power is low. Adding other stock characteristics such as skewness and kurtosis provides limited incremental benefits. However, when a conditional framework based on up and down markets is introduced, the explanatory power increases more than 100 times and there is a significant positive (negative) relation between beta and returns when the market excess returns are positive (negative). The same relation applies when unsystematic risk, total risk and kurtosis are added separately to the beta–return relation during up and down markets with increased explanatory power. Our results indicate that other stock characteristics in addition to beta are also important in pricing risky assets and investors do not hold diversified portfolios. Our results are also checked and compared with another conditional model with time-varying betas conditional on a set of economic variables.     

Weekly Pattern of Exchange Rate Risks: Evidence from Ten Asian-Pacific Currencies

This paper examines the day-of-the-week effect on the currency returns of ten Asian-Pacific countries and differs from previous studies in that it tests directly the effect on higher moments of currency returns. Using ten-year daily data, the results first show that currency returns are non-normally distributed, particularly with very large kurtosis. The hypothesis of equal higher moments (e.g., skewness or kurtosis or both) cannot be rejected by any pair of weekdays only for the Australian dollar. For the remaining nine currencies, the same hypothesis is rejected by at least one pair of weekdays. Six currencies reject the hypothesis in all pairs of weekdays, supporting the existence of the day-of-the-week effect on higher moments. Further analysis shows that Rogalski's effect exists on the higher moments of three currencies because the day-of-the-week effect exists only in non-January months. Sub-period analysis indicates that the weekly patterns on higher moments are quite consistent across two sub-periods for all currencies except the Taiwanese dollar. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some extensions of the CAPM for individual assets

There is ample evidence that stock returns exhibit non-normal distributions with high skewness and excess kurtosis. Experimental evidence has shown that investors like positive skewness, dislike extreme losses and show high levels of prudence. This has motivated the introduction of the four-moment capital asset pricing model (CAPM). This extension, however, has not been able to successfully explain average returns. Our paper argues that a number of pitfalls may have contributed to the weak and conflicting empirical results found in the literature. We investigate whether conditional models, whether models that use individual stocks rather than portfolios and whether models that extend both the moment and factor dimension can improve on more traditional static, portfolio-based, mean–variance models. More importantly, we find that the use of a scaled coskewness measure in cross-section regression is likely to be spurious because of the possibility for the market skewness to be close to zero, at least for some periods. We provide a simple solution to this problem.     

How does trading volume affect financial return distributions?

We assess investors' reaction to new information arrivals in financial markets by examining the relationships between trading volume and the higher moments of returns in 18 international equity and currency markets. Our volume-volatility results support extant information theories and further contribute new evidence of cross market relations between volume and volatility. We also find that the direct impact of volume on the level of negative skewness is less significant for more diversified regional portfolios. Furthermore, the negative interaction between volume and kurtosis can be explained by the differences of opinion in financial markets. We observe stronger interdependence among higher moments in reaction to significant events, but the strength is dampened by trading volume. This result is consistent with trading volume being a source of heteroskedasticity in asset returns.