Log-Supermodularity of Weight Functions and the Loading Monotonicity of Weighted Insurance Premiums

Abstract

The paper is motivated by a problem concerning the monotonicity of insurance premiums with respect to their loading parameter: the larger the parameter, the larger the insurance premium is expected to be. This property, usually called loading monotonicity, is satisfied by premiums that appear in the literature. The increased interest in constructing new insurance premiums has raised a question as to what weight functions would produce loading-monotonic premiums. In this paper we demonstrate a decisive role of log-supermodularity in answering this question. As a consequence, we establish – at a stroke – the loading monotonicity of a number of well-known insurance premiums and offer a host of further weight functions, and consequently of premiums, thus illustrating the power of the herein suggested methodology for constructing loading-monotonic insurance premiums.

Keywords: Insurance premium, Weighted premium, Weighted distribution, Log-supermodularity, Supermodularity, Submodularity, Monotonicity, Esscher premium, Conditional tail expectation, Kamps premium, Wang premium, Distortion premium, Chebyshev's inequality, Decision under uncertainty

JEL Classification: C02, C44, C51, D81