Financial Applications based on Gram-Charlier Expansions

The reliability of risk measures of 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 essentially 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 via their Gram-Charlier expansions based on Hermite polynomials. The aforesaid density can be properly used to compute some risk measures such as the Value at Risk and the expected short fall. An application to a portfolio of financial returns provides evidence of the effectiveness of the proposed approach.