Pareto-optimal insurance with an upper limit on the insurer's exposure

Abstract

We examine the problem of determining Pareto-optimal (PO) insurance contracts when the insurer imposes an ex ante upper limit on disbursement. The problem is similar in spirit to that of Cummins & Mahul (2004), but it extends it in two directions: first, we use the more general and more flexible class of distortion premium principles; and second, we allow for heterogeneity in beliefs between the insurer and the insured. We unify the settings of Ghossoub (2019a, 2019b), and we adapt the approaches therein to encompass the case of a policy limit. First, we show that PO contracts are those that result from a budget-constrained optimization problem for the DM. We then provide a closed-form characterization of optimal contracts. Our result is similar in spirit to that of Cummins & Mahul (2004), who show that when policy limits are introduced to Arrow's model, PO contracts are limited deductible contracts. While Ghossoub (2019a, 2019b) shows that variable deductible contracts are optimal, the results of the present paper indicate that limited variable deductible contracts are optimal when policy limits are present. We illustrate our results via numerical examples.

Keywords:

• Optimal insurance
• Pareto efficiency
• distortion premium principles
• heterogeneous beliefs
• upper limit

JEL Classifications:

• C02
• D86
• G22

MATHEMATICS SUBJECT CLASSIFICATION:

• 91G05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 A probability distortion function is an non-decreasing function $T:[0,1]\to [0,1]$ that satisfies $T(0)=0$ and $T(1)=1$.

2 See, e.g. Buhlmann (1980).

3 The difference between $\delta (t)$ and $m(t)$ is small enough as to be barely visible on a graph. For this reason, we do not include the plots of $\delta (t)$ and $m(t)$ in this example.

Additional information

Funding

Mario Ghossoub acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (Grant No. 2018-03961). The contribution made by Oma Coke was based on work carried out prior to and independently of his professional duties at Murex. Oma Coke is not authorized to communicate on behalf of Murex Canada Software Limited or its affiliates. The research contribution is the author's own and does not reflect the positions or views of Murex Canada Software Limited or its affiliates.