Economic capital management for insurance companies using conditional value at risk and a copula approach

The loss ratio (LR) for insurance companies is defined as the ratio of incurred claims and earned premiums for a specified class of insurance (CoI). The company may estimate then its capital requirement for that particular CoI by using Value at Risk (VaR) or conditional VaR (CVaR) of the loss ratio distribution at a specified probability value. The overall objective of the company is to evaluate the aggregate capital requirement through a weighted sum of marginal capital requirements for all the classes of insurance. Nevertheless, this procedure may tend to over-estimate the aggregate capital requirement because it does not take into consideration the real dependence amongst the different classes of insurance. In other words, perfect dependence does not allow considering diversification effects. In this paper, we present a model which permits to take into consideration real correlations of the several CoIs. Thanks to copula functions, we are able to generate (by Monte Carlo simulations) correlated loss ratios with known marginal distributions. This approach is described through a numerical example that used data collected from some of the most important Italian non life insurance companies. We show then the diversification benefit thus obtained. We conclude the paper building an efficient frontier on the plane LR - CVaR; the efficient frontier may be considered a useful tool to manage the global company risk.