سیاست های بیمه اتکایی و سرمایه‌گذاری زمان سازگار مطلوب برای بیمه‌گر میانگین – واریانس

This paper investigates the optimal time-consistent policies of an investment-reinsurance problem and an investment-only problem under the mean-variance criterion for an insurer whose surplus process is approximated by a Brownian motion with drift. The financial market considered by the insurer consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. A general verification theorem is developed, and explicit closed-form expressions of the optimal polices and the optimal value functions are derived for the two problems. Economic implications and numerical sensitivity analysis are presented for our results. Our main findings are: (i) the optimal time-consistent policies of both problems are independent of their corresponding wealth processes; (ii) the two problems have the same optimal investment policies; (iii) the parameters of the risky assets (the insurance market) have no impact on the optimal reinsurance (investment) policy; (iv) the premium return rate of the insurer does not affect the optimal policies but affects the optimal value functions; (v) reinsurance can increase the mean-variance utility.

Insurers can control their risks by means of some business activities, such as investing in a financial market, purchasing reinsurance, and acquiring new business (acting as a reinsurer for other insurers). As a result, there have arisen many optimization problems with various objectives in insurance risk management. This topic has been extensively investigated in the literature. For example, Browne (1995) obtains the optimal investment strategies for an insurer who maximizes the expected utility of the terminal wealth or minimizes the ruin probability, where the surplus process of the insurer is modeled by a drifted Brownian motion. Yang and Zhang (2005)

In this paper we study two mean-variance optimization problems with time-consistent policies for an insurer. One is an investment-reinsurance problem and the other is an investmentonly problem. In the first problem the insurer is allowed to invest in a financial market and purchase proportional reinsurance/acquire new business. In the second problem the insurer is only allowed to invest in a financial market but not allowed to purchase reinsurance/acquire new business. The surplus process of the insurer is assumed to follow a DA model and the financial market consists of one risk-free asset and multiple risky assets whose price processes are governed by geometric Brownian motions. A verification theorem is developed for a more general optimization problem including the two problems as special cases. And explicit closed-form expressions for the optimal time-consistent policies and the optimal value functions of the two problems are derived