Optimal Reinsurance-Investment Strategy for a Monotone Mean-Variance Insurer in the Cram\'er-Lundberg Model

As classical mean-variance preferences have the shortcoming of non-monotonicity, portfolio selection theory based on monotone mean-variance preferences is becoming an important research topic recently. In continuous-time Cram\'er-Lundberg insurance and Black-Scholes financial market model, we solve the optimal reinsurance-investment strategies of insurers under mean-variance preferences and monotone mean-variance preferences by the HJB equation and the HJBI equation, respectively. We prove the validity of verification theorems and find that the optimal strategies under the two preferences are the same. This illustrates that neither the continuity nor the completeness of the market is necessary for the consistency of two optimal strategies. We make detailed explanations for this result. Thus, we develop the existing theory of portfolio selection problems under the monotone mean-variance criterion

On the Optimal Stochastic Control of Dividend and Penalty Payments in an Insurance Company

In this thesis we consider the surplus of a non-life insurance company and assume that it follows either the classical Cramér-Lundberg model or its diffusion approximation. That is, we consider a continuous time model, where premiums are cashed at a constant rate and claims occur randomly with random sizes modelled by a compound Poisson process. In actuarial mathematics the risk of an insurance company is traditionally measured by the probability of ruin, where the time of ruin is defined as the first time when the surplus becomes negative. Using the ruin probability as a risk measure has been criticised because it is unrealistic to assume that an insurance company is ruined as soon as the surplus becomes negative. In this thesis, we assume that the insurer is not ruined although the surplus becomes negative. In order to avoid ruin, penalty payments occur, depending on the level of the surplus. For example, penalty payments occur if the insurance company needs to borrow money. In the first part of this thesis we consider the diffusion approximation to the Cramér-Lundberg model and we aim to determine a dividend strategy that maximises the difference between the expected discounted dividend and penalty payments, where penalty payments are either modelled by an exponential, linear or quadratic function. We show that the optimal strategy is a so-called barrier strategy and calculate the optimal barrier. The second part studies the analogous problem where the surplus process of an insurance company is given by a Cramér-Lundberg model. Here, similar results are obtained. In conclusion, we consider the problem where we have to determine an optimal investment and reinsurance strategy and the surplus follows the diffusion approximation. The insurance company can invest in several risky assets and reduce the insurance risk either by excess of loss or proportional reinsurance. The aim is to find a strategy which minimises the penalty payments that are necessary to avoid ruin. Various penalty functions are considered and closed form solutions are derived

Dividend Maximization Under a Set Ruin Probability Target in the Presence of Proportional and Excess-of-loss Reinsurance

We study dividend maximization with set ruin probability targets for an insurance company whose surplus is modelled by a diffusion perturbed classical risk process. The company is permitted to enter into proportional or excess-of-loss reinsurance arrangements. By applying stochastic control theory, we derive Volterra integral equations and solve numerically using block-by-block methods. In each of the models, we have established the optimal barrier to use for paying dividends provided the ruin probability does not exceed a predetermined target. Numerical examples involving the use of both light- and heavy-tailed distributions are given. The results show that ruin probability targets result in an improvement in the optimal barrier to be used for dividend payouts. This is the case for light- and heavy-tailed distributions and applies regardless of the risk model used

Optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes: An alternative approach

AbstractThe optimal dividend problem proposed in de Finetti [1] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined. Avram et al. [9] studied the case when the risk process is modelled by a general spectrally negative Lévy process and Loeffen [10] gave sufficient conditions under which the optimal strategy is of the barrier type. Recently Kyprianou et al. [11] strengthened the result of Loeffen [10] which established a larger class of Lévy processes for which the barrier strategy is optimal among all admissible ones. In this paper we use an analytical argument to re-investigate the optimality of barrier dividend strategies considered in the three recent papers