Optimal Reinsurance under Absolute Ruin

Emeritus Professor Yuen, Kam Chuen   (Principal Investigator (PI))

42

2021-01-01

2024-07-12

396967

Optimal Reinsurance under Absolute Ruin

Absolute Ruin, Ambiguity Aversion, Dependence, Optimal Reinsurance, Per-loss reinsurance

Others - MathematicsApplied Mathematics

Physical Sciences (P)

17306220

General Research Fund (GRF)

2020

Completed

1 In the context of optimal reinsurance, most of the research concerns about quota-share reinsurance (sometimes called proportional reinsurance), or excess-of-loss reinsurance, or a combination of the two. In the first project objective, an insurance company can purchase per-loss reinsurance. That is, there is no restriction on the form of reinsurance. It is assumed that the reinsurance premium is computed according to the mean-variance premium principle which combines the expected-value and variance premium principles. Also, the company can invest its surplus in a financial market consisting of one risky asset and one risk-free asset. There is an investment constraint that the amount invested in the risky asset cannot exceed a priori given amount. Furthermore, short-selling is prohibited but borrowing is allowed. In this setup, several insurance risk models with investment and reinsurance can be considered. Without restricting the form of reinsurance, we study the optimal reinsurance-investment problem for an insurer who wishes to minimize the probability of absolute ruin. By the technique of stochastic dynamic programming, we derive the Hamilton–Jacobi–Bellman equation and the corresponding optimal results. We then compare the results with those obtained under the objective of minimizing the probability of traditional ruin. We also investigate some properties of the optimal strategy, and carry out some numerical study to examine how the model parameters affect the optimal strategy. To make the optimization more interesting, we would like to remove the investment constraint so that the amount of risky asset is uncapped. We then re-examine the same optimal problem but without the constraint on investment. 2 In the second project objective, we incorporate model uncertainty into an insurer's controlled surplus, and solve a novel optimal robust reinsurance problem under the objective of minimizing the probability of absolute ruin. Similar to the setup of the first project objective, the insurer can purchase per-loss reinsurance with the premium computed according to the mean-variance premium principle. The risk models of study only take reinsurance into consideration. We assume that the insurer does not have perfect confidence in the drift term of the insurance surplus. In other words, the insurer suspects that the drift of the insurance risk may be misspecified. To handle the issue of misspecification, we use the idea of model ambiguity to describe this phenomenon. Similar to the robust-value approach, we consider the reference probability measure (benchmark) and a set of equivalent measures which have absolute continuous Radon-Nikodym derivatives with respect to the benchmark. To capture the insurer's ambiguity aversion, we penalize the absolute ruin probability with a term based on relative entropy, and thus a new robust value is defined by employing the criterion of minimizing the penalized probability of absolute ruin. To tackle this optimization problem, we apply the technique of stochastic dynamic programming and solve the corresponding boundary-value problem. It is expected that the corresponding optimal reinsurance strategy and the optimal drift distortion of the equivalent measure can be obtained in explicit feedback form. Finally, we provide numerical examples to illustrate the effects of ambiguity aversion on the optimal solutions and discuss their economic implications therein. 3 In recent years, the study of correlated classes of business has been one of the main topics in the actuarial literature. In view of its importance, we incorporate the so-called thinning dependence into the risk model of study, and then revisit the optimization problem stated in the first project objective. In the thinning risk model, an insurance company has several dependent classes of business; stochastic sources that may cause a claim in at least one of the dependent classes of business are classified into a number of groups; and each event occurred in anyone of the groups may cause a claim in each class of business with certain probability. It is worth noting that a special case of the thinning risk model is the frequently-used common shock risk model. Here the thinning risk model is assumed to follow a compound Poisson risk model. Together with reinsurance and investment, the resulting surplus process is a kind of jump-diffusion process. Although the thinning dependence has nice mathematical properties, the technique used in the first project objective cannot be applied directly to tackle the present optimization problem. The incorporation of the thinning dependence and the presence of the jump process definitely introduce a great deal of difficulty in the derivations of main results. In addition to the optimization problem, we would like to investigate the impact of dependence on the optimal results. Due to the complex nature of the resulting surplus process, it is expected that expressions for the optimal results are very complicated, and thus one can hardly work out the impact of dependence analytically. In view of this, we plan to examine how the dependence parameters affect the optimal strategy by means of a large scale of numerical study.