Dividend Strategies for Insurance risk models

Based on different objectives, various insurance risk models with adaptive polices have been proposed, such as dividend model, tax model, model with credibility premium, and so on. In this report, we will only focus the study on dividend strategies. Here are some reasons why the dividend strategy is of interest. For an insurance company, ruin occurs when a claim size is greater than its reserve, which means the insurer does not have enough assets to pay its liability. The goal for insurance company before was thus to minimize the probability of ruin. However, it is clear that, when facing the same claim risk, a senior company that starts business earlier has a smaller probability of ruin than a new company, only because more premiums have been collected. In other words, it is not necessary to require an insurance company to hold the reserves as many as possible. Also, when start a new business line, the insurance company may raise fund from its shareholders as the initial capital. Therefore, it is reasonable to pay the dividends as a return of their investment. Starting from De Finetti, researchers have shown that, in the classical risk model, the optimal strategy is the constant dividend barrier strategy in maximizing the total expected discounted dividends paid before ruin, if the distribution of the independent and identically distributed claims has a completely monotone density (see Gerber and Shiu (2006) and Loeffen (2008)). However, researchers also show that the constant dividend barrier strategy is not optimal, even by straying not too far from the above models; see, e.g., Albercher and Hartinger (2006). To be more specific, Albrecher and Hartinger (2006) discussed the non-optimality of the constant dividend barrier for the renewal risk model (Erlang-2 interclaim time) with exponential claims. Insurers may pay out the excessive part to shareholders as dividends when the reserves are greater than certain amount, which is called barrier. There are different dividend strategies of interest in the literature (see Avanzi 2008). In the constant dividend strategy, dividends are paid continuously at the same rate as the premium whenever the surplus reaches (above) a pre-fixed barrier level. The threshold strategy is similar to the constant dividend strategy, but paying the dividends at a rate smaller than the the premium rate. Other strategies include linear (or nonlinear) barrier strategy, where the barrier level adjusted with respect to time, which is a nondecreasing linear (or nonlinear) function of time.