Pareto distribution is seen everywhere in natural and economic phenomena. Nevertheless, Pareto distribution does not always have expected value. If actual claim data derives such Pareto distribution, actuaries have a lot of difficulty in evaluating premium. For example, provided that power degree of generalized Pareto distribution is smaller than 2 and larger than 1, the average value of the random variable cannot be evaluated directly without upper and lower limit. However, with some measure transformation we can calculate expected value. Moreover it is shown that the method of calculation is suitable to the natural extension of the general economic premium principle treated by Hans Bühlmann in 1980 and 1984. The numerical example of this calculation is introduced using the data on the seismic hazard map which is provided by National Research Institute for Earth Science and Disaster Prevention.
This paper investigates the optimal entry and exit decisions under a mean-reverting process over a finite horizon. Many theoretical studies on real options assume that an underlying risk follows a geometric Brownian motion over an infinite-time horizon. This assumption is not always practical, especially in discussing realistic investment strategies. In this paper, we examine effects of the mean-reverting process on both entry and exit decisions over a finite horizon. We focus on deriving the optimal boundaries of entry and exit decisions under a mean-reverting process, and compare the effects of the underlying risk process and length of the project horizon on the optimal decisions. Numerical examples in this paper demonstrate that the length of horizon could have a significant impact on the boundaries of the optimal decisions, and hence on project values, particularly under a mean-reverting process.