Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functional

TL;DR

This paper develops a method to determine the optimal insurance indemnity that maximizes exponential utility of terminal wealth, considering premiums computed via a convex functional, with proven convergence of the numerical solution.

Contribution

It introduces a necessary condition for optimal indemnity under convex premium computation and provides a convergent numerical algorithm to find the optimal policy.

Findings

Numerical algorithm converges to the unique optimal indemnity.

Optimal indemnity maximizes exponential utility under convex premium.

Illustrative numerical examples demonstrate the method's effectiveness.

Abstract

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.