Abstract
Consider an insurance company which is allowed to invest into a riskless and a risky asset under a constant mix strategy. The total claim amount is modeled by a non-standard renewal risk model with dependence between the claim size and the inter-arrival time introduced by a Farlie-Gumbel-Morgenstern copula. The price of the risky asset is described by an exponential Levy process. Based on some known results, the uniform asymptotic estimate for ruin probability with investment strategy is obtained with regularly varying tailed claims. Applying the asymptotic formula, we provide an approximation of the optimal investment strategy to maximize the expected terminal wealth subject to a risk constraint on the Value-at-Risk, which is defined with respect to finite-time discounted net loss. A numerical example is illustrated for the results, which demonstrates that big dependence parameter is advantageous for the insurer. We explain the reason by some inequalities.
Highlights
The renewal risk model has been playing a fundamental role in classical and modern risk theory since it was introduced by Sparre Andersen in
The reason for this result is that the intensity and average rate of claims both decrease as the dependence parameter increases
The underwriting process is modeled by a non-standard renewal risk model with dependence introduced by a FGM copula and the price of the risky asset is characterized by an exponential Lévy process
Summary
The renewal risk model has been playing a fundamental role in classical and modern risk theory since it was introduced by Sparre Andersen in. In order to measure the integrate risk, Kostadinova [ ] provided a definition of Valueat-Risk (VaR) based on infinite-time discounted net loss Note that this definition does not depend on the initial capital and the time, the solution to the optimization problem there is independent of the time period. We use a concept of VaR based on finite-time discounted net loss, which is described as follows. The use of VaRp(Vθ∗(t)) as a risk measure is explained by the fact that the insurer can prevent the maximum loss from exceeding this quantity in finite-time horizon with a sufficiently high probability – p. It could be understood as the minimal initial capital required.
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