Interplay of insurance and financial risks in a discrete-time model with strongly regular variation

TL;DR

This paper models an insurance company's wealth considering both insurance and financial risks, deriving asymptotic formulas for ruin and tail probabilities under strongly regular variation assumptions without requiring dominance between risks.

Contribution

It introduces a discrete-time risk model incorporating both insurance and financial risks with strongly regular variation, providing unified asymptotic formulas for ruin probabilities.

Findings

Asymptotic formulas for ruin probabilities derived

Tail probabilities expressed as linear combinations of X and Y

No dominance condition required between risks

Abstract

Consider an insurance company exposed to a stochastic economic environment that contains two kinds of risk. The first kind is the insurance risk caused by traditional insurance claims, and the second kind is the financial risk resulting from investments. Its wealth process is described in a standard discrete-time model in which, during each period, the insurance risk is quantified as a real-valued random variable $X$ equal to the total amount of claims less premiums, and the financial risk as a positive random variable $Y$ equal to the reciprocal of the stochastic accumulation factor. This risk model builds an efficient platform for investigating the interplay of the two kinds of risk. We focus on the ruin probability and the tail probability of the aggregate risk amount. Assuming that every convex combination of the distributions of $X$ and $Y$ is of strongly regular variation, we derive some precise asymptotic formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of $X$ and $Y$. Our treatment is unified in the sense that no dominating relationship between $X$ and $Y$ is required.