Optimal open-loop strategies in a debt management problem

Abstract

The paper studies optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present, which increases with the size of the debt. This induces a pool of risk-neutral lenders to charge a higher interest rate, to compensate for the possible loss of part of their investment. Solutions are interpreted as Stackelberg equilibria, where the borrower announces his repayment strategy [Formula: see text] at all future times, and lenders adjust the interest rate accordingly. This yields a highly non-standard problem of optimal control, where the instantaneous dynamics depend on the entire future evolution of the system. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as [Formula: see text].

A Debt Management Problem with Currency Devaluation

“…An analytical study of a variant of the model in [12] was performed in [5], in the case where no currency devaluation is available to the governement, and provided a semi-explicit formula for the optimal strategy in the deterministic case (i.e., when the GDP evolves deterministically).…”

A Stochastic Model of Optimal Debt Management and Bankruptcy

“…In the models studied in [7,8], the borrower has a fixed income, but large values of the debt determine a bankruptcy risk. Namely, if at a given time t the debt-to-income ratio x(t) is too big, there is a positive probability that panic spreads among investors and bankruptcy occurs within a short interval [t, t + ε].…”

“…In the economics literature, some related models of debt and bankruptcy can be found in [1,3,7,11,12]. A general introduction to Nash equilibria and differential games can be found in [5,6].…”

“…Here µ ≥ 0 is an exponential growth rate, while W denotes Brownian motion on a filtered probability space. Differently from [7,8], in [13] it is the borrower himself that chooses when to declare bankruptcy. This decision will be taken when the debt-to-income ratio reaches a certain threshold x * , beyond which the burden of servicing the debt becomes worse than the cost of bankruptcy.…”

The vanishing viscosity limit for a system of H-J equations related to a debt management problem

“…The main motivation comes from a problem of optimal debt management, in infinite time horizon, with exponential discount and in the presence of a bankruptcy risk. As in [8,9,10,16], this is modeled as a noncooperative game between a borrower and a pool of risk-neutral lenders. Here the independent variable x is the debt-toincome ratio, V is the value function for the borrower (i.e., his expected cost, under optimal play) while the function ρ(x) ≥ 0 accounts for an instantaneous bankruptcy risk, which increases with the size of the debt.…”