Title: On approximations of Value at Risk and Expected Shortfall involving kurtosis,
Abstract: We derive new approximations for the Value at Risk and the Expected Shortfall
at high levels of loss distributions with positive skewness and excess
kurtosis, and we describe their precisions for notable ones such as for
exponential, Pareto type I, lognormal and compound (Poisson) distributions. Our
approximations are motivated by extensions of the so-called Normal Power
Approximation, used for approximating the cumulative distribution function of a
random variable, incorporating not only the skewness but the kurtosis of the
random variable in question as well. We show the performance of our
approximations in numerical examples and we also give comparisons with some
known ones in the literature.