Worst-case-optimal dynamic reinsurance for large claims

Abstract

We control the surplus process of a non-life insurance company by dynamic proportional reinsurance. The objective is to maximize expected (utility of the) surplus under the worst-case claim development. In the large claim case with a worst-case upper limit on claim numbers and claim sizes, we find the optimal reinsurance strategy in a differential game setting where the insurance company plays against mother nature. We analyze the resulting strategy and illustrate its characteristics numerically. A crucial feature of our result is that the optimal strategy is robust to claim number and size modeling and robust to the choice of utility function. This robustness makes a strong case for our approach. Numerical examples illustrate the characteristics of the new approach. We analyze the optimal strategy, e.g. in terms of the more conventional, in the insurance context, objective of minimizing the probability of ruin. Finally, we calculate the intrinsic risk-free return of the model and we show that the principle of Markowitz -don't put all your eggs in one basket -does not hold in this setting.

Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion

“…Zhang and Siu [53] investigate a reinsurance and investment problem with model uncertainty and formulate the problem as a zero-sum stochastic differential game. Lin et al [25] and Korn et al [21] discuss the optimal reinsurance problem and the optimal reinsurance-investment problem with model uncertainty by using a stochastic differential game approach. Yi et al [49] and Yi et al [50] study the robust reinsurance-investment problem for an AAI under the expected exponential utility framework and MV criterion, respectively.…”

“…The reason is that our selection of the ambiguity preference functions (23)-(24) are based on the expressions of q * 1 (t, x, l, u), q * 2 (t, x, l, u) and H(t, x, l; q * ) in equations (55), (56) and (11), such that the optimal control u deduced from the HJB equation (57) takes a linear form of variables x and l. Only with this linear form of u, we can separate equations (21) and (22) into a system of ODEs by the order of variables x and l, in other words, we can obtain analytical solutions for our problem.…”

A Note on Realistic Dividends in Actuarial Surplus Models

“…Other interesting transformations have been considered in the literature, which carry interesting properties. For instance, Korn, Menkens, and Steffensen [60] consider worst-case optimisation in the case of large non-life insurers. The main challenge of modifying current formulations of the objective is that it is often difficult to prove or disprove the existence of solutions under a new objective.…”

Worst-Case Scenario Portfolio Optimization Given the Probability of a Crash

“…Seifried [22] evolved a martingale approach for the worst-case scenario. Moreover, the worst-case scenario approach has been applied to the optimal investment problem of an insurance company (see Korn [11]) and to optimize reinsurance for an insurance company (see Korn et al [13]). Korn et al [13] show also in their setting that the worst-case scenario approach has a negative diversification effect.…”

“…Moreover, the worst-case scenario approach has been applied to the optimal investment problem of an insurance company (see Korn [11]) and to optimize reinsurance for an insurance company (see Korn et al [13]). Korn et al [13] show also in their setting that the worst-case scenario approach has a negative diversification effect. Furthermore, both portfolio optimization under proportional transaction costs (see Belak et al [4]) and the infinite time consumption problem (see Desmettre et al [7]) have been studied in a worst-case scenario optimization setting.…”