Optimization Approaches to Multiplicative Tariff of Rates Estimation in Non-Life Insurance

We focus on rating of non-life insurance contracts. We employ multiplicative models with basic premium levels and specific surcharge coefficients for various levels of selected risk/rating factors. We use generalized linear models (GLM) to describe the probability distribution of total losses for a contract during one year. We show that the traditional frequency–severity approaches based only on GLM with logarithmic link function can lead to estimates which do not fulfill business requirements. For example, a maximal surcharge and monotonicity of coefficient can be desirable. Moreover, our approach can handle total losses, which are based on arbitrary loss distributions, possibly decomposed into several classes, e.g., small and large or property and bodily injury. Various costs and loadings can be also incorporated into the tariff rates. We propose optimization problems for rate estimation which enable hedging against expected losses and taking into account a prescribed loss ratio and other business requirements. Moreover, we introduce stochastic programming problems with reliability type constraints which take into account individual risk of each rate cell or collective risk. In the numerical study, we apply the approaches to Motor Third Party Liability (MTPL) policies.