Loading Monotonicity of Weighted Premiums, and Total Positivity Properties of Weight Functions

TL;DR

This paper investigates how weight functions with higher order total positivity influence the monotonicity of insurance premiums as a loading parameter varies, linking risk randomness to the required positivity degree.

Contribution

It introduces the use of higher order totally positive weight functions to establish monotonicity properties of weighted premiums, connecting risk variability to total positivity requirements.

Findings

Seven classes of weight functions analyzed for total positivity

Higher risk randomness necessitates higher order total positivity

Derived monotonicity properties of premiums based on weight functions

Abstract

We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of weighted transformed premiums. We deduce that the greater the degree of randomness of insured risks, the higher the order of total positivity that should be required for the chosen weight functions. We examine seven classes of weight functions that have appeared in the literature, and we ascertain the higher order total positivity properties of those functions.