Pricing general insurance with constraints

Abstract

Deterministic control theory is used to find the optimal premium strategy for an insurer in order to maximise a given objective. The optimal strategy can be loss-leading depending on the model parameters, which may result in negative premium values. In such circumstances, it is optimal to capture as much of the market as possible before making a profit towards the end of the time horizon. In reality, the amount by which an insurer can lower premiums is constrained by borrowing restrictions and the risk inherent in building up a large exposure. Consequently, the effect of constraining the pricing problem is analysed with two forms of constraint: a bounded premium and a solvency requirement. If a lower bound is placed on the premium then an analytical solution can be found, which is not necessarily a smooth function of time. The optimal premium strategy is described in qualitative terms, without recourse to specifying particular parameter values, by considering the value of the terminal optimal premium. Solvency constraints lead to an optimisation problem which is coupled to the state equations and so there is no analytical solution. Numerical results are presented for a subset of the parameter space using control parameterisation, which turns the optimisation problem into a nonlinear programming problem.

Potential Games With Aggregation in Non-Cooperative General Insurance Markets

“…Simultaneously, Emms (2007a) determined the optimal strategy for an insurer that maximises a particular objective over a fixed planning horizon and the premium by using a competitive demand model and the expected main claim size. Moreover, Emms (2007b) considered the process with different types of constrains. By assuming a deterministic control framework, the optimisation problem was solved using elements from control parameterisation.…”

Optimal strategies for a non-linear premium-reserve model in a competitive insurance market

“…Later, Emms (2011) introduces a simple parameterisation which represents the insurance market's response to an insurer adopting a pricing strategy determined via optimal control theory and claims are modelled using a lognormally distributed mean claim size rate. Analytically, a generalisation of the demand function, which has been mentioned initially in Emms (2007aEmms ( , 2007b, is assumed, which implements the impacts on the optimal premium strategy for an insurer. If there is no reaction in the market, then he finds an analytical expression for the optimal relative premium, and if there is no insurance claims, then the optimal relative premium is 0, since there is no need for insurance.…”

“…Emms et al . (2007) model markets average premium as a geometric Brownian motion , where Z is a Wiener process and the drift μ and the volatility σ are assumed to be constant. The future market average premium is lognormally distributed (and hence positive), i.e., .…”

“…In addition, using a linear discounted function for the wealth process of the company, as in Emms et al . (2007), they provide an analytical (endogenous) formula for the optimal premium strategy of the insurance company when it is expected to lose part of the market. Mathematically speaking, they create a maximisation problem for the wealth process of a company, which is solved by using a stochastic dynamic programming (see Emms, 2008).…”

Designing and pricing menus of extended warranty contracts

“…There is also a considerable amount of literature on designing insurance contracts, analyzed from the perspective of the consumer (see [7,26,28,38,39,41]), the provider (see [10,11,13,14,16,18,19,24,36]), and the market as a whole (see [1,6,40]). This work is related in that a warranty is an insurance contract.…”