Risk Measures of the ERNB Distribution Generated by G-NB Family

Abstract

This paper provides VaR and CVaR risk measures, calculated for the Erlang-Negative Binomial (ERNB) distribution. The Erlang and negative binomial distributions are given and then ERNB distribution is obtained using a family of univariate distributions which is called G-Negative Binomial (G-NB). It is defined as compounding the negative binomial distribution (NB) with any continuous distribution (G). Here, we use the ERNB distribution obtained as taking Erlang distribution instead of G. In this paper, we focus on the estimation of VaR and CVaR risk measures for this distribution in closed form and the explicit expressions are also presented for some parameter values. The results are portrayed in the figures. In additionally, numerical examples are given to illustrate changing of the risk measure according to some parameters on a real data set of automobile insurance policies.