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Optimal incentive-compatible insurance with background risk. (English) Zbl 1478.91163

An insurer faces two risks, \(X\) and \(Y\), where \(X \ge 0\). The risk \(X\) can be reinsured, the risk \(Y\) not. For example, \(Y\) could be an investment risk the insurer faces. It is assumed that the second moments exist. Possible reinsurance treaties are \(I(x)\), satisfying \(0 \le I(x_1) - I(x_2) \le x_1 - x_2\) for all \(x_2 \le x_1\). That is, the reinsurer carries \(I(x)\), the insurer \(x - I(x)\) of a loss of size \(x\), and both function \(I(x)\) and \(x - I(x)\) are non-decreasing. Thus, the risk left to the insurer becomes \[ T(X;Y) = X - I(X) + \pi(I(X)) + Y\;,\] where \(\pi(I(X))\) is the reinsurance premium. The premium principle should only depend on the expected value \(\mathbb{E}[I(X)]\). Hence, \(\pi(I(X)) = P(\mathbb{E}[I(X)])\) for some function \(P\). It is assumed that \(P\) is differentiable with \(P(0) = 0\) and \(P'(x) \ge 1\). The goal is to minimise a function \(M(\mathbb{E}[T(X;Y)], \mathrm{Var}[T(X;Y)])\) that is increasing in both of the variables.
Denote by \(\psi(x) = \mathbb{E}[Y \mid X = x]\) the conditional expectation of \(Y\) given \(X\). The idea is then to reinsure \(X + \psi(X)\), because the variance of \(T(X;Y)\) depends on \(X + \psi(X)\) only. A classical theorem by Arrow-Ohlin shows that the optimal reinsurance is the excess of loss reinsurance. This implies that, disregarding the constraint on \(I(x)\), the optimal reinsurance is of the form \[ I_\kappa(x) = \min\{(x + \psi(x) - \kappa)_+, x\}\] for some \(\kappa \in \mathbb{R}\).
One now assumes further that the support of \(X\) is \([0,\infty]\) and that \(\psi\) is continuously differentiable. The support is then divided into intervals where either \(\psi'(x) \ge 0\), \(\psi'(x) \in (-1,0)\) or \(\psi'(x) \le -1\). It is further assumed, without mentioning it, that it is possible to divide the support in finitely many intervals, \(m\) intervals, say. A new reinsurance treaty \(\tilde I\) is then constructed such that \(\mathbb{E}[\tilde I(X); X \in S] = \mathbb{E}[I(X); X \in S]\) in each of the intervals \(S\) of the partition. It is then shown that the resulting treaty fulfils the constraint on possible reinsurance treaties and that the variance of the new treaty is not larger than the original one. Since the construction depends on \(X + \psi(X)\) only, \(\tilde I(X)\) must be optimal. The cases \(m=1\) and \(m=2\) are discussed in particular.

MSC:

91G05 Actuarial mathematics
91B05 Risk models (general)
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