Optimal Reinsurance

Reinsurance is the transfer part of insurance business from a direct insurer, the cedent to the reinsurer. When the direct insurer face the huge or especial risk. It is necessary to transfer its risk through reinsurance. One of the key problems in reinsurance is the optimal Reinsurance. Which is the best, how and how much to transfer? Reinsurance is not only a critical part of the reinsurance system, but also an important tool for direct insurer to transfer its risk. By solving the optimal reinsurance problem, we can determine how to use reinsurance to transfer risk, so it is the core part in researches about reinsurance.Throughout the paper we assume that R is a nonnegative random variable defined on a given probability space (Ω, S,P) and R is a measurable function of Y. The key problem is how to choose reinsurance function. From quota share reinsurance (R = aY) to stop loss reinsurance (R = (Y — b)₊) and change loss reinsurance (R = a(Y — b)₊), which is better is just the problem which our paper concerns.Generally, risk and utility are the standards to measure the reinsurance functions. Here, we consider mostly the risk standards. The insurer is interested to purchase as much of risk protection as possible at a price not exceeding a given limit price P. So in order to find an optimal contract for insurer. We must first determine a risk measure as well as the pricing rule of the contracts. Here, we choose (?)₁, (?))2 as respect risk measure under Mean-Variance premium principles.In Chapter 1, we provide general sufficient conditions that a given contract is optimal within the class R(R₁, R₂) under a risk measure(l) and when every contract R is priced according to expected value principleand we also give the examples to explain how to use the theorem and confirm the parameters. In Chapter 2, we study the same problems when every contract R is priced according to standard deviation principleP = ER + βDR;and we also give the concrete form of reinsurance function. In Chapter 3, we provide general sufficient conditions that a given contract is optimal within the class 3J(i?i, R2) under a more general premium principle ER — f(P,DR), using the same risk measure. In this case, we studied concrete premium principle and concrete \I>i, \&2 through examples. Also give the method how to decide the parameters.In Chapter 4, we study the reinsurance about accumulated risk model. In §4.2 we derive total optimal quota-share reinsurance strategy of dependent risks-insured with dependent claim numbers under the expectation premium principle. In §4.3, we give the concrete numerical examples to show how to use the theory. In §4.4, we introduce a sort of planar utility function and study the reinsurance from point of utility view.Previous papers only considered the insurer's benefits. We consider both the insurer's benefits and reinsurer's benefits.