Actuarial Mathematics, Risk, and Statistics

More than 100 years ago, Swedish actuaries F. Lundberg and H. Cramér proposed the following formula as a model for an insurance company's surplus. This model, known as the classical risk model or the Cramér-Lundberg model, is considered the most fundamental model in risk theory.

Here, U(t) is the amount of surplus at time t, u is the initial surplus, c is a positive constant representing the premium rate, and S(t) represents the total amount of claims paid up to time t. In other words, in this model, there is a premium income of c per unit of time, which contributes to the increase in surplus. On the other hand, claim payments affect the decrease in surplus by S(t). Furthermore, looking at the equation on the right, N(t) represents the frequency of claims up to time t, and Xi represents the amount of the i-th claim payment. This means that the total claim amount S(t) includes two types of "uncertainty": uncertainty related to frequency and uncertainty related to the payment amount. It is known as a type of stochastic model called a compound Poisson process. One area of research that applies this classical risk model is the study of the shareholder dividend problem. Specifically, we consider a new model where, for a realized value of U(t) as shown in Figure 1, a boundary value b, called a dividend barrier, is set, and the portion of U(t) that exceeds b is returned to shareholders as dividends (see Figure 2). Considering this boundary value b, a lower level of b seems advantageous as dividends can be obtained earlier. However, a lower level of b also increases the risk of U(t) < 0 occurring sooner (in this model, this phenomenon signifies the ruin of the insurance company). Conversely, a higher level of b reduces the risk of early ruin but decreases the opportunity to enjoy dividends. Amid this trade-off, De Finetti and others proposed an optimal dividend barrier as a boundary value such that the expected present value of dividends paid until the time of ruin T

is maximized.

After that rather long introduction, the objective of my research is to statistically estimate this optimal dividend barrier. The distributions of the random variables in this model, such as the claim frequency N(t) and the claim amount Xi, are generally unknown, so some form of estimation is required to calculate the optimal dividend barrier. One of my current interests is to propose an estimator for this optimal dividend barrier as a function of observable data and to evaluate the quality of that estimator (i.e., how close it is to the true optimal dividend barrier).