Density approximations and VaR computation for compound Poisson-lognormal distributions

ABSTRACT

Parametric approximations of the compound Poisson-lognormal distribution are developed and used to compute Value-at-Risk (VaR). As guidelines for finding an approximation, the skewness–kurtosis space and the tail behavior are considered. The Generalized Beta distribution of the second kind (GB2) and a mixture of lognormals are found to provide a good fit. In certain cases, the GB2 can be estimated by moment-matching, thus providing a simulation-free procedure for VaR computation. For confidence levels larger than 99%, extreme value theory approaches are developed. According to extensive Monte Carlo evidence, the proposed approximations are more efficient than crude Monte Carlo.

KEYWORDS:

• Extreme value theory
• Generalized beta distribution
• Lognormal mixture
• Probability-weighted moments
• Random sum

MATHEMATICS SUBJECT CLASSIFICATION:

• 62E17
• 62P05

Acknowledgments

The author would like to thank an anonymous referee whose valuable comments considerably improved an earlier version of this article.