A model of debt with bankruptcy risk and currency devaluation

TL;DR

This paper models sovereign debt management considering bankruptcy risk and currency devaluation, using a system of Hamilton-Jacobi equations to find optimal strategies and analyze their behavior.

Contribution

It introduces a novel nonstandard optimal control framework for debt and currency devaluation, providing existence results and characterizing optimal strategies.

Findings

Existence of solutions to the Hamilton-Jacobi system.

Optimal feedback strategies for debt repayment and currency devaluation.

Behavior analysis of strategies near critical thresholds.

Abstract

The paper studies a system of Hamilton-Jacobi equations, arising from a stochastic optimal debt management problem in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. In this model, the borrower is a sovereign state that can decide how much to devaluate its currency and which fraction of its income should be used to repay the debt. Moreover, the borrower has the possibility of going bankrupt at a random time and must declare bankruptcy if the debt reaches a threshold x*. When bankruptcy occurs, the lenders only recover a fraction of their capital. To offset the possible loss of part of their investment, the lenders buy bonds at a discounted price which is not given a priori. This leads to a nonstandard optimal control problem. We establish an existence result of solutions to this system and in turn recover optimal feedback payment strategy $u*(x)$ and currency devaluation $v*(x)$. In addition, the behavior of $(u*,v*)$ near $0$ and x* is studied.