SKEWNESS‐AWARE ASSET ALLOCATION: A NEW THEORETICAL FRAMEWORK AND EMPIRICAL EVIDENCE

This paper presents a new measure of skewness, skewness‐aware deviation, that can be linked to prospective satisficing risk measures and tail risk measures such as Value‐at‐Risk. We show that this measure of skewness arises naturally also when one thinks of maximizing the certainty equivalent for an investor with a negative exponential utility function, thus bringing together the mean‐risk, expected utility, and prospective satisficing measures frameworks for an important class of investor preferences. We generalize the idea of variance and covariance in the new skewness‐aware asset pricing and allocation framework. We show via computational experiments that the proposed approach results in improved and intuitively appealing asset allocation when returns follow real‐world or simulated skewed distributions. We also suggest a skewness‐aware equivalent of the classical Capital Asset Pricing Model beta, and study its consistency with the observed behavior of the stocks traded at the NYSE between 1963 and 2006.     

Hedging and value at risk: A semi‐parametric approach

The non‐normality of financial asset returns has important implications for hedging. In particular, in contrast with the unambiguous effect that minimum‐variance hedging has on the standard deviation, it can actually increase the negative skewness and kurtosis of hedge portfolio returns. Thus, the reduction in Value at Risk (VaR) and Conditional Value at Risk (CVaR) that minimum‐variance hedging generates can be significantly lower than the reduction in standard deviation. In this study, we provide a new, semi‐parametric method of estimating minimum‐VaR and minimum‐CVaR hedge ratios based on the Cornish‐Fisher expansion of the quantile of the hedged portfolio return distribution. Using spot and futures returns for the FTSE 100, FTSE 250, and FTSE Small Cap equity indices, the Euro/US Dollar exchange rate, and Brent crude oil, we find that the semiparametric approach is superior to the standard minimum‐variance approach, and to the nonparametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently generates hedge portfolios that in most cases have lower VaR and CVaR. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:780–794, 2010     

Hedging and value at risk

In this article, it is shown that although minimum‐variance hedging unambiguously reduces the standard deviation of portfolio returns, it can increase both left skewness and kurtosis; consequently the effectiveness of hedging in terms of value at risk (VaR) and conditional value at risk (CVaR) is uncertain. The reduction in daily standard deviation is compared with the reduction in 1‐day 99% VaR and CVaR for 20 cross‐hedged currency portfolios with the use of historical simulation. On average, minimum‐variance hedging reduces both VaR and CVaR by about 80% of the reduction in standard deviation. Also investigated, as an alternative to minimum‐variance hedging, are minimum‐VaR and minimum‐CVaR hedging strategies that minimize the historical‐simulation VaR and CVaR of the hedge portfolio, respectively. The in‐sample results suggest that in terms of VaR and CVaR reduction, minimum‐VaR and minimum‐CVaR hedging can potentially yield small but consistent improvements over minimum‐variance hedging. The out‐of‐sample results are more mixed, although there is a small improvement for minimum‐VaR hedging for the majority of the currencies considered. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:369–390, 2006     

The Effect of Diversification on Tail Risk: Evidence from US Equity Mutual Fund Portfolios

This paper examines the effect of diversification on the tail risk of US equity mutual fund portfolios by utilizing classical higher‐moment measures and robust tail weight measures. Empirical results show that market standard portfolios based on the mean‐variance framework are exposed to greater tail risk than benchmark portfolios are and diversification further intensifies this exposure.     

Timing the Market with a Combination of Moving Averages

A combination of simple moving average trading strategies with several window lengths delivers a greater average return and skewness as well as a lower variance and kurtosis compared with buying and holding the underlying asset using daily returns of value‐weighted US decile portfolios sorted by market size, book‐to‐market, momentum, and standard deviation as well as more than 1000 individual US stocks. The combination moving average (CMA) strategy generates risk‐adjusted returns of 2% to 16% per year before transaction costs. The performance of the CMA strategy is driven largely by the volatility of stock returns and resembles the payoffs of an at‐the‐money protective put on the underlying buy‐and‐hold return. Conditional factor models with macroeconomic variables, especially the market dividend yield, short‐term interest rates, and market conditions, can explain some of the abnormal returns. Standard market timing tests reveal ample evidence regarding the timing ability of the CMA strategy.     

Portfolio Optimization and Martingale Measures

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b ≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b = 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numéraire portfolio.     

MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set‐valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set‐valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set‐valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set‐valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set‐valued risk measures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).     

A Markowitz Optimization of Commodity Futures Portfolios

We examine the diversification benefits of using individual futures contracts instead of simply a commodity index. We determine the ex‐ante, ex‐post, and stability results for optimal Markowitz portfolios, investigate the instability between the ex‐ante and ex‐post results, and compare our results to traditional and naïve portfolios. The ex‐ante complete futures portfolio dominates the traditional and naive portfolios and the ex‐post portfolio outperforms the naïve portfolio. The instability between the ex‐ante and ex‐post results is primarily driven by the time‐varying returns of the individual assets rather than by risk. Finally, the Sharpe portfolio results are essentially identical to the Markowitz results. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:343‐368, 2013     

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

贝塔系数是风险的正确度量吗

在经典的投资组合理论中 ,假设所有资产的报酬率服从对数正态分布 ,因而只需要用收益的方差来度量风险就足够了 ,忽略了偏度的影响。资产收益的分布往往不是对称的 ,偏度是客观存在的 ,而且投资者具有正偏度的爱好。所以必须用方差和偏度来共同度量投资的风险 ,在这种情况下 ,贝塔系数不再是风险的正确度量 ,采用有效的修正方法 ,可以用来对资产进行正确的定价     

From carry trades to curvy trades

Traditional carry trade strategies are based on differences in short-term interest rates, neglecting any other information embedded in yield curves. We derive return distributions of currency portfolios, where the signals to buy and sell currencies are based on summary measures of the yield curve. We find that a strategy based on the relative curvature factor, the curvy trade, yields higher Sharpe ratios and a smaller return skewness than traditional carry strategies. Curvy trades build less upon the typical carry currencies and are hence less susceptible to crash risk. In line with that, standard pricing factors of traditional carry returns fail to explain curvy trade returns.     

Dealing with downside risk in a multi‐commodity setting: A case for a “Texas hedge”?

This study analyzes the problem of multi‐commodity hedging from the downside risk perspective. The lower partial moments (LPM₂)‐minimizing hedge ratios for the stylized hedging problem of a typical Texas panhandle feedlot operator are calculated and compared with hedge ratios implied by the conventional minimum‐variance (MV) criterion. A kernel copula is used to model the joint distributions of cash and futures prices for commodities included in the model. The results are consistent with the findings in the single‐commodity case in that the MV approach leads to over‐hedging relative to the LPM₂‐based hedge. An interesting and somewhat unexpected result is that minimization of a downside risk criterion in a multi‐commodity setting may lead to a “Texas hedge” (i.e. speculation) being an optimal strategy for at least one commodity. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:290–304, 2010     

Dynamic hedging with futures: A copula‐based GARCH model

In a number of earlier studies it has been demonstrated that the traditional regression‐based static approach is inappropriate for hedging with futures, with the result that a variety of alternative dynamic hedging strategies have emerged. In this study the authors propose a class of new copula‐based GARCH models for the estimation of the optimal hedge ratio and compare their effectiveness with that of other hedging models, including the conventional static, the constant conditional correlation (CCC) GARCH, and the dynamic conditional correlation (DCC) GARCH models. With regard to the reduction of variance in the returns of hedged portfolios, the empirical results show that in both the in‐sample and out‐of‐sample tests, with full flexibility in the distribution specifications, the copula‐based GARCH models perform more effectively than other dynamic hedging models. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1095–1116, 2008     

Risk management with weighted VaR

This article studies the optimal portfolio selection of expected utility‐maximizing investors who must also manage their market‐risk exposures. The risk is measured by a so‐called weighted value‐at‐risk (WVaR) risk measure, which is a generalization of both value‐at‐risk (VaR) and expected shortfall (ES). The feasibility, well‐posedness, and existence of the optimal solution are examined. We obtain the optimal solution (when it exists) and show how risk measures change asset allocation patterns. In particular, we characterize three classes of risk measures: the first class will lead to models that do not admit an optimal solution, the second class can give rise to endogenous portfolio insurance, and the third class, which includes VaR and ES, two popular regulatory risk measures, will allow economic agents to engage in “regulatory capital arbitrage,” incurring larger losses when losses occur.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

Emerging markets,downside risk and the asset allocation decision

This study examines the use of downside risk measures in the construction of an optimal international portfolio, with particular reference to the estimated allocations in emerging markets and the out-of-sample performance of the optimal portfolios. The use of downside risk measures is assessed due to the problems of using a conventional mean-variance analysis approach in the presence of the non-normality often found to be present in emerging market data. The data set used consists of the MSCI indices for developed equity markets and the IFC data set on emerging markets. The primary component of the paper consists of the construction of optimal portfolios under both mean-variance and downside risk frameworks. In addition, the use of Bayes–Stein estimators is also assessed, in an attempt to reduce estimation error. The resulting estimated allocations are then used to assess the out-of-sample performance of the optimal portfolios. The results indicate that for risk-averse investors the use of downside risk measures can result in significant improvements in performance.     

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.     

MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN INCOMPLETE MARKET

In this paper, for a process S , we establish a duality relation between K_p , the     - closure of the space of claims in     , which are attainable by "simple" strategies, and     , all signed martingale measures     with     , where   p ≥ 1, q ≥ 1  and     . If there exists a     with     a.s., then K_p consists precisely of the random variables     such that ϑ is predictable S -integrable and     for all     . The duality relation corresponding to the case   p = q = 2  is used to investigate the Markowitz's problem of mean–variance portfolio optimization in an incomplete market of semimartingale model via martingale/convex duality method. The duality relationship between the mean–variance efficient portfolios and the variance-optimal signed martingale measure (VSMM) is established. It turns out that the so-called market price of risk is just the standard deviation of the VSMM. An illustrative example of application to a geometric Lévy processes model is also given.     

Market Timing With Moving Averages

I present evidence that a moving average (MA) trading strategy has a greater average return and skewness as well as a lower variance compared to buying and holding the underlying asset using monthly returns of value‐weighted US decile portfolios sorted by market size, book‐to‐market, and momentum, and seven international markets as well as 18,000 individual US stocks. The MA strategy generates risk‐adjusted returns of 3–7% per year after transaction costs. The performance of the MA strategy is driven largely by the volatility of stock returns and resembles the payoffs of an at‐the‐money protective put on the underlying buy‐and‐hold return. Conditional factor models with macroeconomic variables, especially the default premium, can explain some of the abnormal returns. Standard market timing tests reveal ample evidence regarding the timing ability of the MA strategy.     

COOPERATIVE GAMES WITH GENERAL DEVIATION MEASURES

Cooperative games with players using different law‐invariant deviation measures as numerical representations for their attitudes towards risk in investing to a financial market are formulated and studied. As a central result, it is shown that players (investors) form a coalition (cooperative portfolio) that behaves similar to a single player (investor) with a certain deviation measure. An explicit formula for that deviation measure is obtained. An approach to optimal risk sharing among investors is developed, and a “fair” division of the cooperative portfolio expected gain, belonging to the core of a corresponding cooperative game, is suggested.