Value at risk and expected shortfall based on Gram-Charlier-like expansions

Abstract

The reliability of risk measures for financial portfolios crucially rests on the availability of sound representations of the involved random variables. The trade-off between adherence to reality and specification parsimony can find a fitting balance in a technique that "adjust" the moments of a density function by making use of its associated orthogonal polynomials. This approach rests on the Gram-Charlier expansion of a Gaussian law which, allowing for leptokurtosis to an appreciable extent, makes the resulting random variable a tail-sensitive density function. In this paper we determine the density of sums of leptokurtic normal variables duly adjusted for excess kurtosis by means of their Gram-Charlier expansions based on Hermite polynomials. The resultant density can be effectively used to represent a portfolio return and as such proves suitable for computing some risk measures such as Value at Risk and expected short fall. An application to a portfolio of financial returns is used to provide evidence of the effectiveness of the proposed approach.

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