Bootstrapping a distance test for stochastic dominance analysis

A major impediment to measuring portfolio performance under stochastic dominance has been the lack of test statistics for orders of stochastic dominance above first degree. In this article, the Bootstrap method, introduced by Efron (1979), is used to estimate critical values for distance statistics in order to test the null hypothesis of no dominance, under second- and third-degree stochastic dominance, for several samples of stock returns. These test statistics, suggested by Whitmore (1978), are analogous to the Kolmogorov-Smirnov distance statistics that can be used to test for first-degree stochastic dominance. Stochastic dominance is shown to accurately assess portfolio performance of sample distributions when the population distributions are controlled and Bootstrap statistics are employed in the analysis. In addition, second- and third-degree stochastic dominance analysis of the smallfirm January anomaly indicates that, over the 23-year time period 1964 to 1986, small firms statistically dominate a diversified market index in only one calendar year.     

Stochastic dominance and portfolio analysis

The principle of stochastic dominance is used to characterize the optimal efficient sets when the distributions of the random prospects belong to a family. Most of the well-known distributions are considered. In each case, the optimal efficient sets are characterized by easily verifiable conditions on the parameters of the distributions. These optimal efficient sets are then compared with the corresponding mean-variance (MV) efficient set. It is often found that the optimal efficient sets are proper subsets of the MV efficient set. Thus, the MV criterion is a proper efficiency criterion, but the MV efficient set can be excessively large compared to the optimal efficient set.     

The effect of estimation risk on optimal portfolio choice

This paper determines the effect of estimation risk on optimal portfolio choice under uncertainty. In most realistic problems, the parameters of return distributions are unknown and are estimated using available economic data. Traditional analysis neglects estimation risk by treating the estimated parameters as if they were the true parameters to determine the optimal choice under uncertainty. We show that for normally distributed returns and ‘non-informative’ or ‘invariant’ priors, the admissible set of portfolios taking the estimation uncertainty into account is identical to that given by traditional analysis. However, as a result of estimation risk, the optimal portfolio choice differs from that obtained by traditional analysis. For other plausible priors, the admissible set, and consequently the optimal choice, is shown to differ from that in traditional analysis.     

Downside risk and asset pricing

We analyze if the value-weighted stock market portfolio is stochastic dominance (SD) efficient relative to benchmark portfolios formed on size, value, and momentum. In the process, we also develop several methodological improvements to the existing tests for SD efficiency. Interestingly, the market portfolio seems third-order SD (TSD) efficient relative to all benchmark sets. By contrast, the market portfolio is inefficient if we replace the TSD criterion with the traditional mean–variance criterion. Combined these results suggest that the mean–variance inefficiency of the market portfolio is caused by the omission of return moments other than variance. Especially downside risk seems to be important for explaining the high average returns of small/value/winner stocks.     

Speculative trading and stock returns: A stochastic dominance analysis of the Chinese A-share market

The pricing of A-shares in China has long puzzled financial economists. This paper applies recent tests of stochastic dominance (SD) to examine whether differences in the return distributions of A- and B-shares in China are consistent with market efficiency. As SD is nonparametric, market efficiency can be examined without the joint test problem arising from misspecifications in the asset pricing benchmark. Our results show A-shares have second-order dominated B-shares from 1996 to 2005. This dominance was most significant during the market segmentation period, but has continued, albeit to a lesser extent even after the B-share market was opened to local investors in 2001. Our results are robust to using residual returns from an international asset pricing model instead of raw returns. We conclude that the superior performance of A-shares cannot be attributed to risk. The results are more likely due to a return bias caused by intense speculation among retail individuals under limited arbitrage.     

Optimal rules for ordering uncertain prospects

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm to obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in finance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule to others previously advocated in the literature, including the more popular mean-variance rule as well as the semi-variance rule.     

The wealth effects of portfolio rebalancing in emerging equity markets

This paper analyzes the wealth effects of alternative portfolio rebalancing strategies for equity investments in nine emerging markets for the period from 1976 to 1998. The choice of rebalancing intervals has a large effect on wealth accumulation and the geometric mean return. The difference between no rebalancing and semi-annual rebalancing is 5.87 percentage points per year. Surprisingly, semi-annual rebalancing, which was optimal for this data set, was also 2.62 percentage points per year better than monthly rebalancing. Positive first- and second-degree autocorrelation among the monthly returns appears to account for the decrease in returns for rebalancing more frequently than semi-annually.     

Forecasting the performance of hedge fund styles

This article predicts the relative performance of hedge fund investment styles using time-varying conditional stochastic dominance tests. These tests allow for the construction of dynamic trading strategies based on nonparametric density forecasts of hedge fund returns. During the recent financial turmoil, our tests predict a superior performance for the Global Macro investment style compared with the other strategies of ‘Directional Traders’. The Dedicated Short Bias investment style is stochastically dominated by the other directional styles. These results are confirmed by simple nonparametric tests constructed from realized excess returns. Further, by utilizing a cross-validation method for optimal bandwidth parameter selection, we discover the factors that have predictive power regarding the density of hedge fund returns. We observe that different factors have forecasting power for different regions of the returns distribution and, more importantly, that the Fung and Hsieh factors have power not only for describing the risk premium but also, if appropriately exploited, for density forecasting.     

Refining the bootstrap method of stochastic dominance analysis: The case of the January effect

This study investigates the effect of sample size and population distribution on the bootstrap estimated sampling distributions for stochastic dominance (SD) test statistics. Bootstrap critical values for Whitmore's (1978) second- and third-degree stochastic dominance test statistics are found to vary with both data sample size and variance of the population distribution. The results indicate the parametric nature of the statistics and suggest that the bootstrap method should be used to estimate a sampling distribution each time a new data sample is drawn. As an application of the bootstrap method, the January small firm effect is examined. The results conflict with the SD results of others, and indicate that not all investors would prefer to hold just a portfolio of small capitalization firms in January.     

Is Bitcoin a better portfolio diversifier than gold? A copula and sectoral analysis for China

This paper aims to compare Bitcoin with gold in the diversification of Chinese portfolios using daily data over the 2010–2020 period. We propose a new development of copula-based joint distribution function of returns to simulate the Value-at-Risk and expected shortfall of portfolios including Bitcoin (or gold) and those without it. The stochastic dominance method is also used to compare the return distributions of the three types of portfolios. Empirical results show that gold is a better portfolio diversifier than Bitcoin as it helps better reduce the risk of portfolios. On the other hand, Bitcoin better increases the return but also increases the risk. The stochastic dominance results further show that portfolios diversified by gold dominate those diversified by Bitcoin. Based on these findings, we conclude that in China, gold is a better portfolio diversifier than Bitcoin for risk-averse investors. However, for risk-seeking investors, Bitcoin can be a better choice. This result is found to be robust to the time, frequency and currency effects.     

Return Characteristics of State‐Owned and Non‐State‐Owned Chinese A Shares

This study examines and compares stock returns and volatilities between state‐owned (SO) and non‐state‐owned (NSO) firms on the Shanghai and Shenzhen stock exchanges. Results vary significantly by exchange. Returns for both firm types, on both exchanges, exhibit negative skewness and high kurtosis inconsistent with a normal distribution. Returns display significant autocorrelation, even after the removal of lower‐order effects. Granger causality tests reveal that Shenzhen returns significantly lead Shanghai returns. Within both exchanges, SO firms lead NSO firms. Neither SO nor NSO firm shares are dominated in terms of second‐order stochastic dominance.     

Estimation and Evaluation of Conditional Asset Pricing Models

We find that several recently proposed consumption‐based models of stock returns, when evaluated using an optimal set of managed portfolios and the associated model‐implied conditional moment restrictions, fail to capture key features of risk premiums in equity markets. To arrive at these conclusions, we construct an optimal Generalized Method of Moments (GMM) estimator for models in which the stochastic discount factor (SDF) is a conditionally affine function of a set of priced risk factors, and we show that there is an optimal choice of managed portfolios to use in testing a null model against a proposed alternative generalized SDF.     

Investment Decision Making with Derivative Securities

This paper examines strategies employing stock options, index options, index futures options, and index futures contracts in an effort to establish under what conditions a portfolio manager should diversify into these derivative assets. The results show that futures option call writing and put buying were dominated by third-order stochastic dominance when compared to similar index options. Thus, when covered call writing or protective put buying are being considered, index options appear to be the better choice.     

Time Dominance Efficiency Analysis

Building on the stochastic dominance framework, time dominance efficiency analysis provides similar rules for a partial ordering of temporal prospects. Time dominance does not require any quantitative information about temporal preferences for screening decision alternatives according to their net present values. A binary time dominance proposition extends recent sufficient conditions and adds necessity. The paper's main contribution is the development of set time dominance. By eliminating binary undominated projects which no one would choose, set time dominance minimizes time efficient sets without imposing further preference assumptions.     

Bounding the generalized convex call price

The present article introduces the concept of generalized calls (options whose value at expiry can be any function of the difference between the price of the underlying security and the striking price) and presents some of the properties of such options through the use of absence of stochastic dominance arguments. It deals with bounding relations of call premium applied to generalized options of the convex type, i.e. nonlinear convex options. These relations are obtained from the hypothesis of absence of second-order stochastic dominance between comparable strategies and without any hypothesis on the underlying security's distribution. The article presents economic justification of this method, some classical lemmas about stochastic dominance, and some bounds for convex calls.     

Do Managerial Skills Vary Across Fund Managers? Results Using European Mutual Funds

This paper uses stochastic dominance techniques to examine whether managerial skills vary across fund managers in European equity funds. The use of these techniques allows us to compare different investment alternatives in an uncertain setting under very simple assumptions regarding investor behaviour. The results for style-adjusted returns are consistent with varying degrees of managerial skill and cannot be explained by the impact of the choice of style-adjustment procedure, country, fee policy, and fund age, fund size or survivorship bias. This result allows us to conclude that style-adjusted returns may provide a reliable guide for selecting European investment funds.     

Optimal investment under multi-factor stochastic volatility

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with a multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that investors who invest myopically, ignore derivative assets, model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.     

Beyond the Sharpe ratio: An application of the Aumann–Serrano index to performance measurement

We propose a performance measure that generalizes the Sharpe ratio. The new performance measure is monotone with respect to stochastic dominance and consistently accounts for mean, variance and higher moments of the return distribution. It is equivalent to the Sharpe ratio if returns are normally distributed. Moreover, the two performance measures are asymptotically equivalent as the underlying distributions converge to the normal distribution. We suggest a parametric and a non-parametric estimator for the new performance measure and provide an empirical illustration using mutual funds and hedge funds data.     

MARGINAL CONDITIONAL STOCHASTIC DOMINANCE,STATISTICAL INFERENCE,AND MEASURING PORTFOLIO PERFORMANCE

A simple statistical test is developed for marginal conditional stochastic dominance (MCSD). The MCSD is an extension of second‐degree stochastic dominance. As such, without specification of the return‐generating process, it can rank securities according to marginal changes of return distributions conditionally to the distribution of the market proxy, thereby proving a powerful technique for measuring portfolio performance. Although the MCSD test is asymptotic and conservative, under both the hypotheses of homoskedasticity and heteroskedasticity, it has power to detect the dominance alternative for samples with more than 300 observations. For an illustration, the MCSD test is applied to international equity markets. The test is able to show that nine of twenty‐eight equity markets are dominated by the world market. JEL classification: G11, C49     

An empirical analysis of marginal conditional stochastic dominance

Stochastic dominance is a more general approach to expected utility maximization than the widely accepted mean–variance analysis. However, when applied to portfolios of assets, stochastic dominance rules become too complicated for meaningful empirical analysis, and, thus, its practical relevance has been difficult to establish. This paper develops a framework based on the concept of Marginal Conditional Stochastic Dominance (MCSD), introduced by Shalit and Yitzhaki (1994), to test for the first time the relationship between second order stochastic dominance (SSD) and stock returns. We find evidence that MCSD is a significant determinant of stock returns. Our results are robust with respect to the most popular pricing models.