Some mathematical aspects of price optimisation

Abstract

Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely a) construction of an optimal renewal tariff subject to business and technical constraints, and b) determination of an optimal allocation of certain premium loadings. We consider both continuous and discrete optimisation and then present several algorithmic sub-optimal solutions. Additionally, we explore some simulation techniques. Several illustrative examples show both the complexity and the importance of the optimisation approach.

Price Optimisation for New Business

“…The optimisation problems are quite similar to the ones defined in Hashorva et al [2] for the renewal business.…”

“…The simplest choice for π j is by considering the continuous version of the step function defined above. Referring to [2], we assume that for each customer j,…”

“…In this section, we consider the logit function, as defined in (3.3), commonly used to model the elasticity of a customer due to price changes. In Hashorva et al [2], the latter was used to model the probability of renewal after premium change and in [13], the lapses observed in the insurance industry.…”

“…The simplest choice for π j is by considering the continuous version of the step function defined above. Referring to [2], we assume that for each customer j, π j (P j , δ j ) = α j + β j δ j , where α j and β j are two constants to be estimated in applications. In this case, (2.1) is a quadratic optimisation problem subject to linear constraints, see [12].…”

“…Price optimisation for new business has already been discussed in brief, see [1] and for renewal business see [2] for more details. However, in the literature, price optimisation is more focused on the regulations and ethical points of view, see [3,4].…”