A Stochastic Model of Optimal Debt Management and Bankruptcy

TL;DR

This paper models optimal debt management as a stochastic game with bankruptcy, analyzing strategies and costs for borrowers under uncertainty and varying bankruptcy thresholds.

Contribution

It introduces a stochastic framework for debt management with bankruptcy, deriving optimal strategies and analyzing their properties and costs.

Findings

Existence of optimal feedback strategies for borrowers.

Impact of bankruptcy threshold on total expected costs.

Comparison between stochastic and deterministic models.

Abstract

A problem of optimal debt management is modeled as a noncooperative game between a borrower and a pool of lenders, in infinite time horizon with exponential discount. The yearly income of the borrower is governed by a stochastic process. When the debt-to-income ratio $x(t)$ reaches a given size $x^*$, bankruptcy instantly occurs. The interest rate charged by the risk-neutral lenders is precisely determined in order to compensate for this possible loss of their investment. For a given bankruptcy threshold $x^*$, existence and properties of optimal feedback strategies for the borrower are studied, in a stochastic framework as well as in a limit deterministic setting. The paper also analyzes how the expected total cost to the borrower changes, depending on different values of $x^*$, changes, depending on different values of $x^*$?.