Log-supermodularity of weight functions and the loading monotonicity of weighted insurance premiums

TL;DR

This paper investigates the conditions under which insurance premiums increase with loading parameters, demonstrating that log-supermodularity of weight functions guarantees this monotonicity and enabling the construction of new premiums.

Contribution

It establishes the role of log-supermodularity in ensuring loading monotonicity of insurance premiums and provides a methodology for designing such premiums.

Findings

Proves log-supermodularity implies loading monotonicity.

Shows many known premiums satisfy this property.

Provides new weight functions for constructing monotonic 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.