Decomposing Portfolio Value-at-Risk: A General Analysis

An intensive and still growing body of research focuses on estimating a portfolio’s Value-at-Risk.Depending on both the degree of non-linearity of the instruments comprised in the portfolio and thewillingness to make restrictive assumptions on the underlying statistical distributions, a variety of analyticalmethods and simulation-based methods are available. Aside from the total portfolio’s VaR, there is agrowing need for information about (i) the marginal contribution of the individual portfolio components tothe diversified portfolio VaR, (ii) the proportion of the diversified portfolio VaR that can be attributed toeach of the individual components consituting the portfolio, and (iii) the incremental effect on VaR ofadding a new instrument to the existing portfolio. Expressions for these marginal, component and incremental VaR metricshave been derived by Garman [1996a, 1997a] under the assumption that returns are drawnfrom a multivariate normal distribution. For many portfolios, however, the assumption of normally distributedreturns is too stringent. Whenever these deviations from normality are expected to cause seriousdistortions in VaR calculations, one has to resort to either alternative distribution specifications orhistorical and Monte Carlo simulation methods. Although these approaches to overall VaR estimation have receivedsubstantial interest in the literature, there exist to the best of our knowledge no procedures for estimatingmarginal VaR, component VaR and incremental VaR in either a non-normal analytical setting or a MonteCarlo / historical simulation context.This paper tries to fill this gap by investigating these VaR concepts in a general distribution-freesetting. We derive a general expression for the marginal contribution of an instrument to the diversifiedportfolio VaR ? whether this instrument is already included in the portfolio or not. We show how in a mostgeneral way, the total portfolio VaR can be decomposed in partial VaRs that can be attributed to theindividual instruments comprised in the portfolio. These component VaRs have the appealing property thatthey aggregate linearly into the diversified portfolio VaR. We not only show how the standard results undernormality can be generalized to non-normal analytical VaR approaches but also present an explicitprocedure for estimating marginal VaRs in a simulation framework. Given the marginal VaR estimate,component VaR and incremental VaR readily follow. The proposed estimation approach pairs intuitiveappeal with computational efficiency. We evaluate various alternative estimation methods in an applicationexample and conclude that the proposed approach displays an astounding accuracy and a promisingoutperformance.