Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error's asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.

► In this paper, normal approximation of total claim distribution is plugged in risk measures. ► We focus on the rate of convergence of the error relative to the number of clients. ► The relative error's asymptotic distribution is specified. ► Distortion risk measures are considered. ► Robust representation of distribution-invariant coherent risk measures is discussed.