An Asymptotic Result on Catastrophe Insurance Losses

Abstract

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

ACKNOWLEDGMENTS

The authors thank the two anonymous referees for their careful reading and valuable comments, which have helped to greatly improve the quality of the article. In particular, the authors are indebted to one referee for pointing out an interpretation of (1.4)(1.4) $$\begin{array}{c}\begin{array}{cc}{W}_T\hfill & =x{Q}_T+{\int }_0^T\frac{Q_T}{Q_t}d\left(P_t-C_t\right)\hfill \end{array}\\ \begin{array}{cc}=Q_T\left(x+{\int }_0^T{e}^{-R_t}d\left(P_t-C_t\right)\right)\hfill & \hfill \\ =Q_T\left(x+{\int }_0^T{e}^{-R_t}d{P}_t-\sum _{k=1}^{N_T}{X}_k{e}^{-R_{{\tau }_k}}\right),\hfill & \hfill \end{array}\end{array}$$(1.4) as the wealth process of an entirely equity-financed insurer.

Additional information

Funding

Yiqing Chen's research was supported by a Summer Research Grant from the College of Business and Public Administration at Drake University. Jiajun Liu's research was supported by the National Natural Science Foundation of China (NSFC No. 12201507).