Bowley Insurance with Expected Utility Maximization of the Policyholders

Abstract

This article studies the Bowley solution for a sequential game within the expected utility framework. We assume that the policyholders are expected utility maximizers and there exists a representative policyholder who faces a fixed loss with given probability and no loss otherwise. This policyholder selects the optimal indemnity function in response to the pricing kernel set by the insurer. Knowing the policyholder’s choice of indemnity function, the insurer adjusts the pricing kernel to maximize its expected net profit. This pricing kernel is of our central interest in this article, and in our setting the pricing kernel can be evaluated via the safety loading factor in an expected value premium principle. For a wide class of utility functions, we show that the optimal safety loading factor increases with respect to both the policyholder’s risk aversion level and the probability of zero loss. We also show that the insurance contract corresponding to the Bowley solution is Pareto dominated in the sense that both parties’ interests can be further improved, which shows the inefficiency of the Bowley solution. Some numerical examples are presented to illustrate the main results, and it is shown that both the policyholder and insurer can strictly benefit from the Bowley solution.

ACKNOWLEDGMENTS

We thank the co-editor and two anonymous referees for their valuable suggestions that helped improving this article substantially.

Notes

1 A risk measure $\rho$ is called:

• convex if for all random variables $(Y,Z)$ and all $\lambda \in [0,1],$ $\rho (\lambda Y+(1-\lambda )Z)\le \lambda \rho (Y)+(1-\lambda )\rho (Z);$

• comonotonic-additive if $\rho (Y+Z)=\rho (Y)+\rho (Z),$ for all $(Y,Z)$ that are comonotonic, where a comonotonic pair $(Y,Z)$ is such that $\left[Y(\omega )-Y({\omega }^{\prime })\right] \left[Z(\omega )-Z({\omega }^{\prime })\right]\ge 0$ for any two states of the world $\omega$ and ${\omega }^{\prime }.$

2 A risk measure $\rho$ is called translation-invariant if $\rho (Y+c)=\rho (Y)+c,$ for all random variables Y and all $c\in \mathbb{R}.$

3 This result is related to Ohlin’s lemma (Ohlin 1969), which implies that $\mathbb{E}[u^{-1}(Z_1)]\le \mathbb{E}[u^{-1}(Z_2)]$ because $u^{-1}$ is convex. However, we needed to prove a strict inequality when u is strictly concave (and $u^{-1}$ is strictly convex).

Additional information

Funding

Wenjun Jiang acknowledges the financial support received from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-04204) and the University of Calgary.