Premiums And Reserves, Adjusted By Distortions

TL;DR

This paper explores the distorted premium principle in actuarial science, providing dual characterizations and proposing a novel perspective that maintains the probability measure while increasing outcomes, aiding in premium and reserve calculations.

Contribution

It introduces a new viewpoint on distorted premiums by keeping the probability measure fixed and increasing outcomes, enhancing practical premium and reserve computations.

Findings

Dual characterizations of distorted premiums are developed.

An alternative approach to distortion by increasing outcomes is proposed.

The new perspective supports time-consistent premium and reserve calculations.

Abstract

The net-premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, an insurance company, applying the net-premium principle, goes bankrupt with probability one in the long run, even if the company covers its entire costs by collecting the respective fees from its customers. It is therefore an intrinsic necessity for the insurance industry to apply premium principles, which guarantee at least further existence of the company itself; otherwise, the company naturally could not insure its clients to cover their potential, future claims. Beside this intriguing fact the underlying loss distribution typically is not known precisely. Hence alternative premium principles have been developed. A simple principle, ensuring risk-adjusted credibility premiums, is the distorted premium principle. This principle is convenient in insurance companies, as the actuary does not have to change his or her tools to compute the premiums or reserves. This paper addresses the distorted premium principle from various angles. First, dual characterizations are developed. Next, distorted premiums are typically computed by under-weighting or ignoring low, but over-weighting high losses. It is demonstrated here that there is an alternative, opposite point of view, which consists in leaving the probability measure unchanged, but increasing the outcomes instead. It turns out that this new point of view is natural in actuarial practice, as it can be used for premium calculations, as well as to determine the reserves of subsequent years in a time consistent way.