Hedef Programlama ve En Küçük Kar Farkı Yaklaşımları ile Optimal Reasürans

Abstract

The aim of this study is to contribute to the optimal reinsurance studies, which have a considerable role in the actuarial literature, by considering the situation from a perspective that takes into account the insurer and the reinsurer together. For this purpose, firstly, the solution of the analytical model which minimizes “the VaR of the absolute value of the difference between the profits of the insurer and the reinsurer” is obtained. Secondly, the models which both taking into account both sides and multi objective / constrained, are examined by using goal programming.

In the literature review, optimal reinsurance studies are discussed under 4 main headings; studies on optimal reinsurance for insurer, studies on optimal reinsurance for reinsurer, studies on optimal reinsurance for insurer and reinsurer, studies on optimal reinsurance with multiple criteria. In addition, basic information about reinsurance, types of reinsurance, principles of premiums and risk measurements are given.

In order to illustrate the necessity of taking into account both the insurer and the reinsurer company as the parties of a reinsurance contract when calculating the optimal retention, this thesis examines a prior study of optimal reinsurance which only considers the insurer’s point of view. Subsequently discussed is a model which determines the optimal retention from both points of view of insurer and reinsurer with the corresponding simulations used to justify this model. This forms the preliminary work this thesis builds upon. This model assumes that the claim numbers are Poisson distributed and the claim sizes are exponential, lognormal and Pareto distributed. The results are obtained separately for both the stop-loss and the excess-of-loss reinsurance. As the premium principle, both standard deviation premium principle and expected value premium principle are used. The results are compared with the tables and figures for these two studies.

This thesis seeks to contribute to the existing literature by taking into account both the insurer and the reinsurer. An analytical model is set up and makes use of “the VaR of the absolute value of the difference between the profits of the insurer and the reinsurer” as a risk measure, the solution of which is obtained and presented in this thesis. The results of this analytical model are compared with the results of the preliminary work of this thesis and with the optimal reinsurance study which is examined in this thesis and has only considered the insurer point of view by considering the real world examples and using the numerical examples. We assume the aggregate loss is exponential and Pareto distributed and the premiums of both the insurer and the reinsurer are calculated using the expected value premium principle with stop-loss reinsurance.

The second aim of this thesis, is to contribute to the literature by addressing multi objective/constrained models using goal programming. First a general definition of the goal programming method used, subsequently the terminology and the variants of goal programming are summarized. For this application, 11 different multi objective/constrainted optimal reinsurance models have been constructed and their solutions have been investigated making use of goal programming. The constraints in the models are as follows: the value at risk of the absolute value of the difference between the insurer's profit and the reinsurer's profit; the absolute value of the difference between the standard deviation of insurer’s expected profit; the standard deviation of reinsurer’s expected profit; the expected utility function of insurer (with exponential utility and logaritmic utility); the expected utility function of reinsurer; the expected profit of reinsurer; the expected profit of insurer; the value at risk of the total cost of insurer; the value at risk of the total cost of reinsurer. Mathematical representations of the models are given by using the goal programming model and optimal retention levels and deviation variables are obtained using the stop-loss reinsurance are presented in the tables. The tables are prepared for both the expected value and the standard deviation premium principle and assumptions that the losses are distributed by Pareto, exponential and lognormal distributions. Comparisons between the models are made. In addition, how the results change with different premium loading coefficients and different initial wealth is shown in the tables.

As a result of this study, these models are interpreted from the joint perspective of insurer and reinsurer and examined in terms of their contributions and innovations to the literature. The difference between unilateral optimal reinsurance studies and optimal reinsurance studies taking both sides into account is demonstrated in terms of their real-world suitability and acceptability for both sides. In addition, the results of the multi objective reinsurance study and the advantages of using goal programming method in the solution of this study are also revealed.