Optimal quota-share reinsurance based on the mutual benefit of insurer and reinsurer
An insurer wants to buy proportional reinsurance for an exponentially distributed claim \(X\). Both, insurer and reinsurer use the same exponential utility function. The range of reinsurance is limited because both parties want to increase their expected utilities. The joint goal is to minimise the sum of some risk measure. More specifically, one looks for \[ \min_{b \in [0,1]}\{J(b X -P_I(b)-u^I) + J((1-b) X -P_R(b)-u^R)\}\;, \] where \(P_R(b) = (1+\theta) (1-b)\mathbb{E}[X]\) is the reinsurance premium, \(P_I(b) = P_0 - P_R(b)\) is the premium part of the insurer, and \(u^I\) and \(u^R\) are the initial capitals of the insurer and reinsurer, respectively. As risk measure \(J\), the following cases are treated: (1) ruin probability \(J(Y) = \mathbb{P}[Y > 0]\), (2) variance \(J(Y) = \mathrm{Var}[Y]\), (3) value-at-risk \(J(Y) = \mathrm{VaR}_\gamma = \inf\{L: \mathbb{P}[Y \le -L] \ge \gamma\}\), (4) tail-value-at-risk \(J(Y)= \mathrm{TVaR}_\gamma = (1-\gamma)^{-1} \int_\gamma^1 \mathrm{VaR}_q\;d q\), (5) generalized Dutch type I risk measure. The optimal retention levels can then be calculated. Several cases of the parameters have to be distinguished because of the expected utility constraint.
scientific article; zbMATH DE number 6887292
scientific article