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     

Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.     

MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.     

Asymmetric pricing of implied systematic volatility in the cross‐section of expected returns

Assuming a symmetric relation between returns and innovations in implied market volatility, Ang, A., Hodrick, R., Xing, Y., and Zhang, X. (2006) find that sensitivities to changes in implied market volatility have a cross‐sectional effect on firm returns. Dennis, P., Mayhew, S., and Stivers, C. (2006), however, find an asymmetric relation between firm‐level returns and implied market volatility innovations. We incorporate this asymmetry into the cross‐sectional relation between sensitivity to volatility innovations and returns. Using both portfolio sorting and firm‐level regressions, we find that sensitivity to VIX innovations is negatively related to returns when volatility is rising, but is unrelated when it is falling. The negative relation is robust to controls for other variables, suggesting only the increase in implied market volatility is a priced risk factor. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:34–54, 2011     

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.     

RISK HORIZON AND REBALANCING HORIZON IN PORTFOLIO RISK MEASUREMENT

This paper analyzes portfolio risk and volatility in the presence of constraints on portfolio rebalancing frequency. This investigation is motivated by the incremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision. In contrast to the standard market risk measure based on a 10‐day value‐at‐risk calculated at 99% confidence, the IRC considers more extreme losses and is measured over a 1‐year horizon. More importantly, whereas 10‐day VaR is ordinarily calculated with a portfolio’s holdings held fixed, the IRC assumes a portfolio is managed dynamically to a target level of risk, with constraints on rebalancing frequency. The IRC uses discrete rebalancing intervals (e.g., monthly or quarterly) as a rough measure of potential illiquidity in underlying assets. We analyze the effect of these rebalancing intervals on the portfolio’s profit and loss distribution over a risk‐measurement horizon. We derive limiting results, as the rebalancing frequency increases, for the difference between discretely and continuously rebalanced portfolios; we use these to approximate the loss distribution for the discretely rebalanced portfolio relative to the continuously rebalanced portfolio. Our analysis leads to explicit measures of the impact of discrete rebalancing under a simple model of asset dynamics.     

Multi‐period hedge ratios for a multi‐asset portfolio when accounting for returns co‐movement

This study presents a model to select the optimal hedge ratios of a portfolio composed of an arbitrary number of commodities. In particular, returns dependency and heterogeneous investment horizons are accounted for by copulas and wavelets, respectively. A portfolio of London Metal Exchange metals is analyzed for the period July 1993–December 2005, and it is concluded that neglecting cross correlations leads to biased estimates of the optimal hedge ratios and the degree of hedge effectiveness. Furthermore, when compared with a multivariate‐GARCH specification, our methodology yields higher hedge effectiveness for the raw returns and their short‐term components. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:182–207, 2008     

Diversification potential of Asian frontier,BRIC emerging and major developed stock markets: A wavelet-based value at risk approach

This study examines the portfolio risk and the co-movements between each of the BRIC emerging and South Asian frontier stock markets and each of the major developed stock markets (U.S., UK and Japan), using the wavelet squared coherence approach as well as the wavelet-based Value at Risk (VaR) method. The results show that the co-movements and diversification benefits between these markets vary over time and across frequencies. Additionally, the co-movements are intensified in the wake of the recent global financial crisis (GFC) and the Eurozone sovereign debt crisis (ESDC). More precisely, the wavelet-based VaR ratio indicates that including a BRIC or a South Asian (particularly Pakistan and Sri Lanka at both the short- and long-term) stock market in a portfolio of the developed stock markets reduces the resulting portfolio's VaR. Specifically, adding China in the medium term to this portfolio reduces risk in the pre- and during both the GFC and ESDC periods. By assigning optimal weights to the different market assets in the portfolio formulation, the analysis thus has implications for international investors.     

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity‐adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta‐normal VaR or RiskMetrics (J.P. Morgan, 1996).     

Mean‐variance efficiency of the market portfolio and futures trading

We derived an intertemporal capital asset pricing model in which the mean‐variance efficiency of the market portfolio is neither a necessary nor a sufficient condition. We obtained this result by modeling a frictionless, continuously open financial market in which nonredundant futures contracts are available for trade, in addition to cash assets. Introducing such contracts modifies the way investors optimally allocate their wealth. Their portfolios then comprise the riskless asset, a perturbed mean‐variance‐efficient portfolio of cash assets, and a perturbed mean‐variance‐efficient portfolio of futures contracts. Furthermore, a (3 + K) mutual fund separation is obtained, with K being the number of economic state variables, in lieu of the usual (2 + K) fund separation. Mean‐variance efficiency of the market portfolio is a necessary condition only when cash assets are the sole traded assets. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:329–346, 2001     

Does the More Risk‐Averse Investor hold a Less Risky Portfolio?*

We study the suitability of using absolute risk aversion as a measure of willingness to take risk in the Arrow–Debreu portfolio framework. We define a global measure of risk for the Arrow–Debreu portfolio, which is measured by the sensitivity of an individual's Arrow–Debreu portfolio payoff to the change in the market return. We call this measure ‘conservatism’ and show that the concept of ‘more conservative’ is stronger than that of ‘more risk‐averse.’ A higher absolute risk aversion is only necessary but not sufficient to induce a less risky Arrow–Debreu portfolio. Our results not only challenge the well‐accepted notion that a more risk‐averse investor holds a less risky portfolio, but also suggest a stronger measure – conservatism – for evaluating the riskiness of portfolio.     

Quantitative impact of correlation errors on basket options with time‐varying correlations

This study examines the quantitative impact of correlation errors on basket options with time‐varying correlations and the risk measures (conditional) value‐at‐risk (VaR) in the framework of Basel II. The results show that risk measure misestimation due to correlation errors are the largest and most asymmetric for the at‐the‐money and out‐of‐the‐money basket option. Delta hedging of the basket option reduces risk but increases size and asymmetry effects substantially. Finally, the square‐root‐of‐time rule for VaR does not adjust adequately to correlation errors and consistently underestimates risk measures, which could lead to the VaR exceedance clustering observed during the recent financial crisis. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

The design and pricing of fixed‐ and moving‐window contracts: An application of Asian‐Basket option pricing methods to the hog‐finishing sector

Asian‐Basket‐type moving‐window contracts are an increasingly used risk‐management tool in the North American hog sector. The moving‐window contract is decomposed into a portfolio of a long Asian‐Basket put and a short Asian‐Basket call option. A projected break‐even price is used to determine the floor price, and then Monte Carlo simulation methods are used to price both a moving‐ and a fixed‐window contract. These methods provide unbiased pricing of fixed‐ and moving‐window hog‐finishing contracts of 1‐year duration. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1047–1073, 2003     

VaR在投资组合应用中存在的缺陷与CVaR模型

VaR在投资组合应用中存在的两个缺陷:一是不满足一致性公理,二是尾部损失测量的非充分性,这些缺陷可能导致组合优化上的错误.当且仅当组合回报服从正态分布时,VaR才能应用于组合优化,这极大地限制了VaR在投资组合管理中的适用范围.本文最后介绍了CVaR模对VaR模型的改进及其在投资组合优化中的应用.     

Hedging long‐term commodity risk

This study focuses on the problem of hedging longer‐term commodity positions, which often arises when the maturity of actively traded futures contracts on this commodity is limited to a few months. In this case, using a rollover strategy results in a high residual risk, which is related to the uncertain futures basis. We use a one‐factor term structure model of futures convenience yields in order to construct a hedging strategy that minimizes both spot‐price risk and rollover risk by using futures of two different maturities. The model is tested using three commodity futures: crude oil, orange juice, and lumber. In the out‐of‐sample test, the residual variance of the 24‐month combined spot‐futures positions is reduced by, respectively, 77%, 47%, and 84% compared to the variance of a naïve hedging portfolio. Even after accounting for the higher trading volume necessary to maintain a two‐contract hedge portfolio, this risk reduction outweighs the extra trading costs for the investor with an average risk aversion. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:109–133, 2003     

BETTER THAN DYNAMIC MEAN‐VARIANCE: TIME INCONSISTENCY AND FREE CASH FLOW STREAM

As the dynamic mean‐variance portfolio selection formulation does not satisfy the principle of optimality of dynamic programming, phenomena of time inconsistency occur, i.e., investors may have incentives to deviate from the precommitted optimal mean‐variance portfolio policy during the investment process under certain circumstances. By introducing the concept of time inconsistency in efficiency and defining the induced trade‐off, we further demonstrate in this paper that investors behave irrationally under the precommitted optimal mean‐variance portfolio policy when their wealth is above certain threshold during the investment process. By relaxing the self‐financing restriction to allow withdrawal of money out of the market, we develop a revised mean‐variance policy which dominates the precommitted optimal mean‐variance portfolio policy in the sense that, while the two achieve the same mean‐variance pair of the terminal wealth, the revised policy enables the investor to receive a free cash flow stream (FCFS) during the investment process. The analytical expressions of the probability of receiving FCFS and the expected value of FCFS are derived.     

A modified static hedging method for continuous barrier options

This study modifies the static replication approach of Derman, E., Ergener, D., and Kani, I. (1995, DEK) to hedge continuous barrier options under the Black, F. and Scholes, M. (1973) model. In the DEK method, the value of the static replication portfolio, consisting of standard options with varying maturities, matches the zero value of the barrier option at n evenly spaced time points when the stock price equals the barrier. In contrast, our modified DEK method constructs a portfolio of standard options and binary options with varying maturities to match not only the zero value but also zero theta on the barrier. Our numerical results indicate that the modified DEK approach improves performance of static hedges significantly for an up‐and‐out call option under the BS model even if the bid–ask spreads are considered. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Do futures‐based strategies enhance dynamic portfolio insurance?

In this paper we investigate the relative performance of two approaches to dynamic portfolio insurance: the synthetic put and the Constant Proportion Portfolio Insurance (CPPI). The investigation is conducted on the Australian market, over a sample period of 59 non‐overlapping quarters from December 1987 to December 2002. Its main contribution is to provide a comprehensive assessment of the two approaches under different market conditions, and the testing of ex ante information as an input into the trading program. The major finding is that the futures‐based implementation of both synthetic put and the CPPI approach is robust to both tranquil and turbulent market conditions in preserving the desired floor. The fact that this conclusion includes the case of employing implied volatility (obtained from the options market) is highly encouraging as it suggests high implementability of the strategy. Notably, the risk‐return tradeoff shows that portfolio insurance using this volatility measure yields a return that is 64 basis points over the risk free investment. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:591–608, 2004     

Improved estimation of portfolio value‐at‐risk under copula models with mixed marginals

Portfolio value‐at‐risk (PVAR) is widely used in practice, but recent criticisms have focused on risks arising from biased PVAR estimates due to model specification errors and other problems. The PVAR estimation method proposed in this article combines generalized Pareto distribution tails with the empirical density function to model the marginal distributions for each asset in the portfolio, and a copula model is used to form a joint distribution from the fitted marginals. The copula–mixed distribution (CMX) approach converges in probability to the true marginal return distribution but is based on weaker assumptions that may be appropriate for the returns data found in practice. CMX is used to estimate the joint distribution of log returns for the Taiwan Stock Exchange (TSE) index and the associated futures contracts on SGX and TAIFEX. The PVAR estimates for various hedge portfolios are computed from the fitted CMX model, and backtesting diagnostics indicate that CMX outperforms the alternative PVAR estimators. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:997–1018, 2006