On the distance to the desired terminal surplus distribution under reinsurance

In this paper, we consider a pool of risks within an insurance company. The company's management would like the collective's yearly surplus to have a certain distribution. As this target is not always achievable, proportional reinsurance can be purchased in order to come to the desired distribution as close as possible. We consider two different distance measures: squared Wasserstein distance and quadratic distance with a fixed correlation. We find the optimal strategy for both cases. In the first case, the optimal strategy concentrates on the fitting the variance but also takes into account the desired expectation. In the second case, the optimal strategy takes into account the fixed expectation and the correlation. In a series of examples, we illustrate our concept on such risk measures as value-at-risk and expected shortfall, calculate the correlation between a random variable with the desired distribution and the collective's surplus for light- and heavy-tailed claims distributions. The concepts discussed hold practical relevance in risk management of an insurance company, specifically in handling surplus distribution and risk assessment within a defined time frame. Selecting a target surplus distribution, such as at an accounting period's conclusion, aims to achieve desired values of value-at-risk, expected shortfall, and expected utility simultaneously.