Optimal Insurance with Limited Commitment in a Finite Horizon

TL;DR

This paper addresses a finite horizon optimal contracting problem involving a risk-neutral principal and a risk-averse agent with stochastic income, transforming the complex constraints into a series of solvable optimal stopping problems and providing analytical and numerical solutions.

Contribution

It introduces a novel approach by converting the finite horizon dual problem into an infinite series of optimal stopping problems with explicit solutions.

Findings

Analytic solutions for each optimal stopping problem.

Verification theorem linking the value functions.

Numerical simulations of optimal contracts.

Abstract

We study a finite horizon optimal contracting problem of a risk-neutral principal and a risk-averse agent who receives a stochastic income stream when the agent is unable to make commitments. The problem involves an infinite number of constraints at each time and each state of the world. Miao and Zhang (2015) have developed a dual approach to the problem by considering a Lagrangian and derived a Hamilton-Jacobi-Bellman equation in an infinite horizon. We consider a similar Lagrangian in a finite horizon, but transform the dual problem into an infinite series of optimal stopping problems. For each optimal stopping problem we provide an analytic solution by providing an integral equation representation for the free boundary. We provide a verification theorem that the value function of the original principal's problem is the Legender-Fenchel transform of the integral of the value functions of the optimal stopping problems. We also provide some numerical simulation results of optimal contracting strategies