VaR Criteria for optimal limited change-loss and truncated change-loss reinsurance

摘要

Reinsurance can provide an effective way for insurer to manage its risk exposure. In this paper, we further analyze the optimal reinsurance models recently proposed by J. Cai and K. S. Tan [Astin Bulletin, 2007, 37(1): 93-112]. With the criteria of minimizing the value-at-risk (VaR) risk measure of insurer's total loss exposure, we derive the optimal values of sharing proportion a, retention d, and layer l of two reinsurance treaties: the limited change-loss f(x) = a{(x - d)+ - (x - l)+} and the truncated change-loss f(x) = a(x - d)+I_((x≤l)). Both of the reinsurance plans have been considered to be more realistic and practical in the real business. Our solutions have several appealing features: (ⅰ) there is only one condition to verify for the existence of optimal limited change-loss reinsurance while there always exists an optimal truncated change-loss reinsurance, (ⅱ) the resulting optimal parameters have simple analytic forms which depend only on assumed loss distribution, reinsurer's safety loading, and insurer's risk tolerance, (ⅲ) the optimal retention d for limited change-loss reinsurance is the same as that by Cai and Tan while the optimal value is smaller for truncated change-loss, (ⅳ) the optimal sharing proportion and layer are always the same for both reinsurance plans, (ⅴ) minimized VaR are strictly lower than the values derived by Cai and Tan, (ⅵ) the optimization results reveal possible drawbacks of VaR-based risk management that a heavy tail risk exposure may be expressed by lower VaR.