Coherent Capital for Treaty ROE Calculations

This paper explores how a coherent risk measure could be used to determine risk-sensitive capital requirements for reinsurance treaties. The need for a risk-sensitive capital calculation arises in the context of estimating the return on equity (ROE) for several treaties or different options on one treaty. Looking at the loss random variable alone is insufficient for a complete risk analysis since this would fail to incorporate the impact of adjustable premium and ceding commission provisions on the final net risk. The paper presents a framework for systematically reflecting treaty features by viewing capital as a function of the distribution of the final net underwriting loss. To avoid negative values for indicated capital, the concept of a risk quantity variable is introduced as a non-negative monotonically increasing function of the net underwriting loss variable. Two risk quantities are discussed: one obtained by capping the net underwriting loss from below at zero, and the other by taking the excess of the net underwriting loss above its expectation. A coherent risk measure is then applied to a risk quantity to obtain indicated capital. The approach is demonstrated in simple discrete distribution examples by applying a coherent measure, the Tail Value at Risk, to the two risk quantities. Sensitivity testing on the examples is presented showing how the different measures respond to changes in premium adequacy, swing rating, sliding scale commission plans, and layering. In summary, this paper is one attempt to bridge the gap between the theoretical results of coherence theory and the practical need for methods to set risk-sensitive capital in treaty ROE analysis.