Multivariate Insurance Portfolio Risk Retention Using the Method of Multipliers

For an insurance company insuring multiple risks, capital allocation is an important practical problem. In the capital allocation problem, the insurance company must determine the amount of capital to assign to each policy or, equivalently, the amount of premium to be collected from each policy. Doing this relates to the problem of determining the risk retention parameters for each policy within the portfolio. In this article, the insurance risk retention problem of determining the optimal retention parameters is explored in a multivariate context. Given an underlying claims distribution and premium constraint, we are interested in finding the optimal amount of risk to retain or, equivalently, which level of risk retention parameters should be chosen by an insurance company. The risk retention parameter may be deductible (d), upper limit (u), or coinsurance (c). We present a numerical approach to solving the risk retention problem using the method of multipliers and illustrate how it can be implemented. In a case study, the minimum amount of premium to be collected is used as a constraint to the optimization and the upper limit is optimized for each policyholder. A Bayesian approach is taken for estimation of the parameters in a simple model involving regional effects and individual policyholder effects for the Wisconsin Local Government Property Insurance Fund (LGPIF) data, where the parameter estimation is performed in the R computing environment using the Stan library.