Reinsurance Decision Making and Expected Utility: Comment
评论Samson和Thomas文章中的再保险最大保费结果，指出其与风险理论定理矛盾，原因是计算中使用的积分实际上不存在。
Results contained in an article by Danny Samson and Howard Thomas that recently appeared in this Journal contradict well-known theorems of risk theory concerning maximum premiums for proportional and non-proportional reinsurance. ' If we compare line 1 in Table 1 with line 3 in Table 2, we come to the conclusion that the direct insurer is prepared to pay a much higher maximum premium for proportional (342.8) than for non-proportional reinsurance (239.5) given about the same expected values (219.9 and 216.9 respectively) and the same risk aversion kD = 0.0002. These surprising results are an apparent contradiction with theory. They are caused by the fact that neither the integral in the numerator in equation ( 17) nor the integrals in equation ( 18) actually exist, as the integrand tends towards infinity when X is increasing. It is understandable that this was not noticed by the authors during numerical integration, as the integrand in the given example drops down to very small values (approximately 10-15) before tending towards infinity. On the other hand, this integral is at the same time the moment-generating function of the lognormal distribution at kD. It is easy to prove that this function does not exist for positive arguments.