Optimization of the Non-Life Insurance Risk Diversification in Solvency II

According to the current Solvency II standard approach, non-life risk capital charges take into account geographical diversification by adjusting volume measures using a Herfindahl-Hirschman concentration index for premiums and reserves at a line of business level. The lower the Herfindahl index the less concentrated is a portfolio and the greater is its diversification extent. The diversification factor of a portfolio of risks with respect to some risk measure is defined to be the quotient of the portfolio risk measure to the sum of the stand-alone risk measures over all risks in the portfolio. Maximum diversification is obtained by minimizing the diversification factor. According to the QIS4 proposal the minimum diversification factor is equal to 0.75. This value is not optimal. If the risk measure is proportional to the standard deviation of the risk, then the absolute minimum value of 0.707 allows for an additional diversification reduction of maximum magnitude 4.3 percent. The latter is true in the case of the value-at-risk and the conditional value-at-risk measures for the class of multivariate elliptical risk distributions. However, the current Solvency II standard approach to non-life risk relies on log-normal distributions. In this framework, the minimum diversification factor, which depends on the volatility of the portfolio, is in the average equal to 0.667, which results in an absolute diversification reduction of magnitude 8.3 percent compared to QIS4. Extending the analysis to the class of multivariate log-elliptical risk distributions, further results on the minimum diversification factor can be obtained. For the class of multivariate log-Laplace distributions, which are able to model fat tails similarly to the class of generalized Pareto distributions in Extreme Value Theory, this minimum value is in the average 0.68 resulting in an absolute reduction of lower magnitude 7 percent.

Keywords: Solvency II non-life risk, value-at-risk, conditional value-at-risk, Herfindahl-Hirschman index, diversification factor, multivariate elliptical and log-elliptical risk distributions