Decomposing Portfolio Value-at-Risk: A General Analysis
Winfried G. Hallerbach *)
Associate Professor, Department of Finance Erasmus University Rotterdam POB 1738, NL-3000 DR Rotterdam The Netherlands phone: +31.10.408 1290 facsimile: +31.10.408 9165 e-mail: firstname.lastname@example.org http://www.few.eur.nl/few/people/hallerbach/
final version: October 15, 2002 forthcoming in The Journal of Risk 5/2, Febr. 2003
*) I’d like to thank Michiel de Pooter and Haikun Ning for excellent programming assistance. I appreciate the critical remarks and helpful comments of Jan Annaert, Phelim Boyle, Stephen Figlewski, Philippe Jorion (the editor), Ton Vorst, an anonymous referee and participants of the Northern Finance Association Meeting in Calgary. Of course, any remaining errors are mine.
A variety of methods is available to estimate a portfolio’s Value-at-Risk. Aside from the overall VaR there is an apparent need for information about marginal VaR, component VaR and incremental VaR. Expressions for these VaR metrics have been derived under the restrictive normality assumption. In this paper we investigate these VaR concepts in an elliptical world and in a general distribution-free (simulation) setting, and show how they can be estimated.
Keywords: Value-at-Risk, marginal VaR, component VaR, incremental VaR, nonnormality, non-linearity, simulation
JEL classification: C13, C14, C15, G10, G11
Value-at-Risk (VaR) is defined as a one-sided confidence interval on potential portfolio losses over a specific horizon. Interest in such a diagnostic metric can be traced back to Edgeworth  but the developments in this field were really spurred by the release of RiskMetrics™ by J.P.Morgan in October 1994. An intensive and still growing body of research focuses on estimating a portfolio’s VaR and various analytical or simulation-based methods have been developed (see for example Duffie & Pan  and Jorion  for an overview). Currently we observe a shift from portfolio risk measurement to detailed risk analysis and subsequent risk management. Aside from the portfolio’s overall VaR there is an apparent need for information about (i) marginal VaR (MVaR): the marginal contribution of the individual portfolio components to the diversified portfolio VaR, (ii) component VaR (CVaR): the proportion of the diversified portfolio VaR that can be attributed to each of the individual components, and (iii) incremental VaR (IVaR): the incremental effect on VaR of adding a new asset or trade to the existing portfolio. Garman [1996, 1997], Jorion  and Litterman [1997a,b] have studied these VaR measures under the assumption that returns are drawn from a multivariate normal distribution. For many trading portfolios, however, the assumption of normally distributed returns does not apply. Fat tailed distributions are rule rather than exception for financial market factors and the inclusion of non-linear derivative instruments in the portfolio gives rise to distributional asymmetries. The same applies to credit portfolios: when estimating creditVaR one must cope with skewed loss distributions. Whenever these deviations from normality are expected to cause serious biases in VaR calculations, one has to resort to either alternative distribution specifications or simulation methods. In this paper we investigate the concepts of MVaR, CVaR and IVaR in a general setting. We put the standard results derived under normality in a broader perspective and
show how they can be generalized to the wide class of elliptical distributions. In addition we present a simple procedure for estimating these VaR metrics in a simulation context. The structure of the paper is as follows. Section 2 reviews the concepts of MVaR, CVaR and IVaR and summarizes these metrics in a restrictive normal world. In section 3 we derive a general expression for MVaR and show how the total portfolio VaR can be decomposed into...
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