تجزیه و تحلیل حساسیت از ارزش در معرض خطر

The aim of this paper is to analyze the sensitivity of Value at Risk (VaR) with respect to portfolio allocation. We derive analytical expressions for the first and second derivatives of the VaR, and explain how they can be used to simplify statistical inference and to perform a local analysis of the VaR. An empirical illustration of such an analysis is given for a portfolio of French stocks.

Value at Risk (VaR) has become a key tool for risk management of financial institutions. The regulatory environment and the need for controlling risk in the financial community have provided incentives for banks to develop proprietary risk measurement models. Among other advantages, VaR provide quantitative and synthetic measures of risk, that allow to take into account various kinds of cross-dependence between asset returns, fat-tail and non-normality effects, arising from the presence of financial options or default risk, for example. There is also growing interest on the economic foundations of VaR. For a long time, economists have considered empirical behavioural models of banks or insurance companies, where these institutions maximise some utility criteria under a solvency constraint of VaR type (see Gollier et al., 1996 and Santomero and Babbel, 1996). Similarly, other researchers have studied optimal portfolio selection under limited downside risk as an alternative to traditional mean-variance efficient frontiers (see Roy, 1952, Levy and Sarnat, 1972, Arzac and Bawa, 1977 and Jansen et al., 1998). Finally, internal use of VaR by financial institutions has been addressed in a delegated risk management framework in order to mitigate agency problems Kimball, 1997, Froot and Stein, 1998 and Stoughton and Zechner, 1999. Indeed, risk management practitioners determine VaR levels for every business unit and perform incremental VaR computations for management of risk limits within trading books. Since the number of such subportfolios is usually quite large, this involves huge calculations that preclude online risk management. One of the aims of this paper is to derive the sensitivity of VaR with respect to a modification of the portfolio allocation. Such a sensitivity has already been derived under a Gaussian and zero mean assumption by Garman, 1996 and Garman, 1997. Despite the intensive use of VaR, there is a limited literature dealing with the theoretical properties of these risk measures and their consequences on risk management. Following an axiomatic approach, Artzner et al., 1996 and Artzner et al., 1997 (see also Albanese, 1997 for alternative axioms) have proved that VaR lacks the subadditivity property for some distributions of asset returns. This may induce an incentive to disagregate the portfolios in order to circumvent VaR constraints. Similarly, VaR is not necessarily convex in the portfolio allocation, which may lead to difficulties when computing optimal portfolios under VaR constraints. Beside global properties of risk measures, it is thus also important to study their local second-order behavior. Apart from the previous economic issues, it is also interesting to discuss the estimation of the risk measure, which is related to quantile estimation and tail analysis. Fully parametric approaches are widely used by practitioners (see, e.g. JP Morgan Riskmetrics documentation), and most often based on the assumption of joint normality of asset (or factor) returns. These parametric approaches are rather stringent. They generally imply misspecification of the tails and VaR underestimation. Fully non-parametric approaches have also been proposed and consist in determining the empirical quantile (the historical VaR) or a smoothed version of it Harrel and Davis, 1982, Falk, 1984, Falk, 1985, Jorion, 1996 and Ridder, 1997. Recently, semi-parametric approaches have been developed. They are based on either extreme value approximation for the tails Bassi et al., 1997 and Embrechts et al., 1998, or local likelihood methods (Gouriéroux and Jasiak, 1999a). However, up to now the statistical literature has focused on the estimation of VaR levels, while, in a number of cases, the knowledge of partial derivatives of VaR with respect to portfolio allocation is more useful. For instance, partial derivatives are required to check the convexity of VaR, to conduct marginal analysis of portfolios or compute optimal portfolios under VaR constraints. Such derivatives are easy to derive for multivariate Gaussian distributions, but, in most practical applications, the joint conditional p.d.f. of asset returns is not Gaussian and involves complex tail dependence (Embrechts et al., 1999). The goal here is to derive analytical forms for these derivatives in a very general framework. These expressions can be used to ease statistical inference and to perform local risk analysis. The paper is organized as follows. In Section 2, we consider the first and second-order expansions of VaR with respect to portfolio allocation. We get explicit expressions for the first and second-order derivatives, which are characterized in terms of conditional moments of asset returns given the portfolio return. This allows to discuss the convexity properties of VaR. In Section 3, we introduce the notion of VaR efficient portfolio. It extends the standard notion of mean-variance efficient portfolio by taking VaR as underlying risk measure. First-order conditions for efficiency are derived and interpreted. Section 4 is concerned with statistical inference. We introduce kernel-based approaches for estimating the VaR, checking its convexity and determining VaR efficient portfolios. In Section 5, these approaches are implemented on real data, namely returns on two highly traded stocks on the Paris Bourse. Section 6 gathers some concluding remarks.

We have considered the local properties of the VaR. In particular, we have derived explicit expressions for the sensitivities of the risk measures with respect to the portfolio allocation and applied the results to the determination of VaR efficient portfolios. The empirical application points out the difference between a VaR analysis based on a Gaussian assumption for asset returns and a direct non-parametric approach. This analysis has been performed under two restrictive conditions, namely i.i.d. returns and constant portfolio allocations. These conditions can be weakened. For instance, we can introduce non-parametric Markov models for returns, allowing for non-linear dynamics, and compute the corresponding conditional VaR together with their derivatives. Such an extension is under current development. The assumption of constant holdings until the benchmark horizon can also be questioned. Indeed in practice, the portfolio can be frequently updated and a major part of the risk can be due to inappropriate updating. The effect of a dynamic strategy on the VaR can only be evaluated by Monte Carlo methods (see for instance the impulse response analysis in Gouriéroux and Jasiak, 1999b). It has also to be taken into account when determining a dynamic VaR efficient hedging strategy (see Foellmer and Leukert, 1998). Finally, let us remark that our kernel-based approach can be used to analyse the sensitivity of the expected shortfall, i.e. the expected loss knowing that the loss is larger that a given loss quantile. This is also under current development.