Some risk management problems for firms with internal competition and debt

Abstract

Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (Ut, Zt), where Ut is the size of the business unit and Zt is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process Xt and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for Ut and a singular control for Zt. The optimal strategies are different for the cases δ > 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that 'too many people is counter-productive'. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.