The upper bounds of the ruin probability for an insurance discrete-time risk model with proportional reinsurance and investment

Abstract

In this study, the two upper bounds of the ruin probability for discrete time risk model derived by adding two controlled factors to the classical discrete time risk model: proportional reinsurance and investment are proposed. These upper bounds are derived using an inductive method and rely on a recursive form of the finite time and/or an integral equation of ultimate (infinite time) of ruin probability which is also derived in this study. Both of the upper bounds are formulated by the assumption that the retention level of reinsurance and the amount of stock investment during each time period are controlled as constant values. The first upper bound can be used with the finite time ruin probability and the ultimate ruin probability under the condition that the value of the adjustment coefficient can be found. The second upper bound is formulated by a using new worse than used distribution. This upper bound can only be used with the finite time ruin probability, and its value can be found even though the value of the adjustment coefficient does not exist. However, this upper bound has limitations on the total claims amount which the total claims amount in each time period must come from the summation of independent and identically distributed (i.i.d.) claim amounts, and the number of claims is also i.i.d. in each time period.

Two numerical examples are used to consider the characteristics of the derived upper bounds. In the first example, the total claims amount is assumed to follow an exponential distribution from which the value of the adjustment coefficient can be found to show the first derived upper bound. In the other example, the claim amounts

are set as a Pareto distribution, from which the adjustment coefficient cannot be found and is used to show some of the characteristics of the second upper bound. Moreover, real-life motor insurance claims data that fits a log-normal distribution is used to show the application of the derived upper bounds. Under the different agreements of the three aforementioned situations, it was found that the values of the two upper bounds derived in this study responded to the two additional controlled factors in the proposed risk model in the same direction.