Optimal Loss Financing Under Bonus-Malus Contracts

Abstract

The paper analyses the question: Should an insurance customer carry an occurred loss himself, or should he make a claim to the insurance company? This question is important within bonus-malus contracts with individual experience adjustments of the premium. The analysis model includes a bonus hunger strategy where the customers prefer the most profitable financial alternative, that is, the alternative which represents the lowest rate of interest. Hence the loss of bonus after a claim is calculated as a rate of interest paid from the customer to the insurer. Within this model the paper outlines the existence of a true compensation function and a relative cost function for each customer. A set of properties for bonus-malus contracts are presented and discussed. A concrete example of a bonus-malus system and an insurance compensation function illustrates the theoretical framework in a practical manner.

Optimal Insurance Coverage under Bonus-Malus Contracts

“…An important statement which partly follows from (1) and (2) is that independent of the contractual compensation function, the true compensation function has always an individual deductible; see proposition 2 in Holtan (2001). As we discuss later, this statement explains much of the optimal coverage characteristics of bonus-malus contracts outlined in this paper.…”

“…We recapitulate briefly the main features of a general bonus-malus insurance contract as outlined in Holtan (2001). Consider an insurance buyer representing a risk of loss X, where X is a stochastic variable with probability density function f{x) and x > 0.…”

“…(s + t) are assumed to be continuous non-stochastic for all / > 0. See Holtan (2001) for a more detailed and complete description of bonus-malus effects on an insurance contract.…”

“…A contractual compensation with a deductible can, however, not be Pareto optimal within the standard risk-exchange model. • Proposition 2 in Holtan (2001) states that independent of the contractual compensation function, the true compensation function has always an individual deductible under bonus-malus contracts. Hence, given the standard risk-exchange model, it is intuitively correct that the Pareto optimal statement for standard contracts with a deductible is valid in general for bonus-malus contracts.…”

“…The aim of this paper is to take this bonus hunger mechanism into account within the framework of optimal insurance coverage under bonus-malus contracts. The paper is an extension of another paper by the author, Holtan (2001), where the necessary insight to -and concepts of -bonus-malus contracts are outlined.…”