Decomposing Portfolio Value-at-Risk: A General

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 the willingness to make restrictive assumptions on the underlying statistical distributions, a variety of analytical methods and simulation-based methods are available. Aside from the total portfolio’s VaR, there is a growing need for information about (i) the marginal contribution of the individual portfolio components to the diversified portfolio VaR, (ii) the proportion of the diversified portfolio VaR that can be attributed to each of the individual components consituting the portfolio, and (iii) the incremental effect on VaR of adding a new instrument to the existing portfolio. Expressions for these marginal, component and incremental VaR metrics have been derived by Garman [1996a, 1997a] under the assumption that returns are drawn from a multivariate normal distribution. For many portfolios, however, the assumption of normally distributed returns is too stringent. Whenever these deviations from normality are expected to cause serious distortions in VaR calculations, one has to resort to either alternative distribution specifications or historical and Monte Carlo simulation methods. Although these approaches to overall VaR estimation have received substantial interest in the literature, there exist to the best of our knowledge no procedures for estimating marginal VaR, component VaR and incremental VaR in either a non-normal analytical setting or a Monte Carlo / historical simulation context. This paper tries to fill this gap by investigating these VaR concepts in a general distribution-free setting. We derive a general expression for the marginal contribution of an instrument to the diversified portfolio VaR ? whether this instrument is already included in the portfolio or not. We show how in a most general way, the total portfolio VaR can be decomposed in partial VaRs that can be attributed to the individual instruments comprised in the portfolio. These component VaRs have the appealing property that they aggregate linearly into the diversified portfolio VaR. We not only show how the standard results under normality can be generalized to non-normal analytical VaR approaches but also present an explicit procedure 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 intuitive appeal with computational efficiency. We evaluate various alternative estimation methods in an application example and conclude that the proposed approach displays an astounding accuracy and a promising outperformance.