Study of Relative Risk Indexes

Topics: Normal distribution, Standard deviation, Modern portfolio theory Pages: 35 (8173 words) Published: March 4, 2013
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: hallerbach@few.eur.nl 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.

Abstract

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

1.

Introduction

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 [1888] 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 [1997] and Jorion [2001] 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 [1997] 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

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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...


References: Artzner, P., F. Delbaen, J.-M. Eber and D. Heath, 1999, “Coherent Measures of Risk,” Mathematical Finance, 9/3, p. 203-228 Carroll, R.B., T. Perry, H. Yang & A. Ho, 2001, “A New Approach to Component VaR”, The Journal of Risk 3/3, Spring, pp. 57-67 Credit Suisse Financial Products, 1997, “Credit Risk+: A Credit Risk Management Framework”, London UK Duffie, D. & J. Pan, 1997, “An Overview of Value at Risk”, The Journal of Derivatives 4/3, Spring, pp.7-49 Edgeworth, F.Y., 1888, “The Mathematical Theory of Banking”, Journal of the Royal Statistical Society, 51/1, March, pp. 113-127 Embrechts P., A. McNeil & D. Straumann, 2002, “Correlation and Dependence in Risk Management: Properties and Pitfalls”, In: “Risk Management: Value at Risk and Beyond”, M.A.H. Dempster (ed.), Cambridge University Press, Cambridge, pp. 176-223 Fang, K.T., S. Kotz & K.W. Ng, 1990, “Symmetric Multivariate and Related Distributions”, Chapman and Hall, London UK Fong, G. & O.A. Vasicek, 1997, “A Multidimensional Framework for Risk Analysis”, Financial Analysts Journal July/August, pp. 51-57 Garman, M.B., 1996, “Improving on VAR”, RISK 9/5, May, pp.61-63 Garman, M.B., 1997, “Taking VAR to Pieces”, RISK 10/10, October, pp. 70-71 Gouriéroux, C., J.P. Laurent & O. Scaillet, “Sensitivity Analysis of Values at Risk”, Journal of Empirical Finance 7, pp. 225-245 Hallerbach, W.G., 1999, “Decomposing Portfolio Value-at-Risk: A General Analysis”, discussion paper 99-034/2, Tinbergen Institute Rotterdam, http://www.tinbergen.nl/ Härdle, W., 1990, “Applied Nonparametric Regression”, Econometric Society Monographs 19, Cambridge University Press, Cambridge Hull, J.C., 2000, “Options, Futures and Other Derivatives”, Prentice-Hall, Upper Saddle River NJ Jarrow, R. & A. Rudd, 1982, “Approximate Option Valuation For Arbitrary Stochastic Processes”, Journal of Financial Economics 10, Nov., pp. 347-369 Jorion, Ph., 2001, “Value at Risk: The Benchmark for Controlling Market Risk”, McGraw-Hill, Chicago Ill., 2nd ed. (1st ed. 1997) JPMorgan, 1996, “RiskMetrics™: Technical Document”, 4th ed., New York NY Kelker, D., 1970, “Distribution Theory of Spherical Distributions and a Location-Scale Parameter Generalization”, Sankhyá: The Indian Journal of Statistics: Series A 32, pp. 419430
26
Kendall, M.G. & A. Stuart, 1969, “The Advanced Theory of Statistics”, Volume 1 (Distribution Theory), Charles Griffin & Co, London UK Koyluoglu, H.U. & J. Stoker, 2002, “Honour Your Contribution”, RISK April, pp. 90-94 Litterman, R., 1997a, “Hot Spots and Hedges (I)”, RISK 10/3, March, pp. 42-45 Litterman, R., 1997b, “Hot Spots and Hedges (II)”, RISK 10/5, May, pp. 38-42 Maddala, G.S., 1977, “Econometrics”, McGraw-Hill, Auckland Martin, R., K. Thompson & C. Browne, 2001, “VAR: Who Contributes and How Much?”, RISK August, pp. 99-102 Mausser, H. & D. Rosen, 1998, “Beyond VaR: From Measuring Risk to Managing Risk”, Algo Research Quarterly 1/2, December, pp. 5-20 Owen, J. & R. Rabinovitch, 1983, “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice”, The Journal of Finance 38/3, Sept, pp. 745-752 Pagan, A. & A. Ullah, 1999,“Nonparametric Econometrics”, Cambridge University Press, Cambridge Spanos, A., 1986, “Statistical Foundations of Econometric Modelling”, Cambridge University Press, Cambridge UK Stapleton, R.C. & M.G. Subrahmanyam, 1983, “The Market Model and Capital Asset Pricing Theory: A Note”, The Journal of Finance 38/5, Dec, pp. 1637-1642 Tasche, D., 1999, “Risk Contributions and Performance Measurement”, working paper Zentrum Mathematik (SCA), Munich University of Technology, http://www.ma.tum.de/stat/ Tasche, D. & L. Tibiletti, 2001, “Approximations for the Value-at-Risk Approach to RiskReturn Analysis”, working paper Zentrum Mathematik (SCA), Munich University of Technology, http://www.ma.tum.de/stat/
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