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Examine of Comparative Risk Indices

Decomposing Portfolio Value-at-Risk: A General Examination

Winfried G. Hallerbach *)

Associate Professor, Department of Finance Erasmus University Rotterdam POB 1738, NL-3000 DR Rotterdam Holland phone: +31. 10. 408ps 1290 facsimile: +31. 15. 408 9165 e-mail: [email protected] eur. nl http://www.few.eur.nl/few/people/hallerbach/

final version: October 15, 2002 forthcoming inside the Journal of Risk 5/2, Febr. the year 2003

*) I'd like to thank Michiel de Pooter and Haikun Ning intended for excellent encoding assistance. My spouse and i appreciate the critical remarks and helpful remarks of By Annaert, Phelim Boyle, Sophie Figlewski, Philippe Jorion (the editor), Bunch Vorst, a great anonymous referee and members of the Upper Finance Affiliation Meeting in Calgary. Naturally , any remaining errors will be mine.

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A variety of methods is available to estimate a portfolio's Value-at-Risk. Aside from the general VaR there is an evident need for information about marginal Va, component VaR and pregressive VaR. Movement for these Va metrics have been derived underneath the restrictive normality assumption. From this paper all of us investigate these kinds of VaR concepts in an elliptical world and a general distribution-free (simulation) environment, and show how they may be approximated.

Keywords: Value-at-Risk, marginal Va, component Va, incremental Va, nonnormality, nonlinearity, simulation

JEL classification: C13, C14, C15, G10, G11

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Value-at-Risk (VaR) is defined as a one-sided self-confidence interval in potential collection losses on the specific ecart. Interest in this kind of a analysis metric can be traced back in Edgeworth [1888] but the innovations in this field were actually spurred by the release of RiskMetricsв„ў by simply J. S. Morgan in October year 1994. An intensive but still growing body of study focuses on calculating a portfolio's VaR and various analytical or simulation-based methods have already been developed (see for example Duffie & Baking pan [1997] and Jorion [2001] for an overview). At present we notice a switch from portfolio risk dimension to thorough risk examination and following risk management. Apart from the portfolio's general VaR there is an noticeable need for details about (i) limited VaR (MVaR): the minor contribution individuals portfolio components to the varied portfolio Va, (ii) aspect VaR (CVaR): the amount of the varied portfolio VaR that can be related to each of the specific components, and (iii) gradual VaR (IVaR): the gradual effect on Va of adding a new property or operate to the existing portfolio. Garman [1996, 1997], Jorion [1997] and Litterman [1997a, b] include studied these types of VaR procedures under the supposition that results are drawn from a multivariate normal circulation. For many trading portfolios, however , the assumption of normally distributed returns does not apply. Fat tailed distributions happen to be rule rather than exception pertaining to financial market factors and the inclusion of nonlinear derivative instruments in the portfolio gives rise to distributional asymmetries. The same applies to credit portfolios: when calculating creditVaR a single must handle skewed loss distributions. Whenever these deviations from normality are expected to cause serious biases in VaR measurements, one has to resort to either alternative circulation specifications or perhaps simulation methods. In this conventional paper we check out the concepts of MVaR, CVaR and IVaR in a general placing. We position the standard effects derived underneath normality within a broader point of view and

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show how they can be general to the large class of elliptical droit. In addition all of us present an easy procedure for estimating these VaR metrics in a simulation framework. The framework of the paper is as follows. Section two reviews the concepts of MVaR, CVaR and IVaR and summarizes these metrics in a restrictive normal community. In section 3 we derive a general expression to get MVaR and have absolutely how the total portfolio Va can be decomposed into...

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