Optimal control of a big financial company with debt liability under bankrupt probability constraints

Zongxia Liang,Bin Sun

doi:10.1007/s11464-011-0120-2

Abstract

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company’s bankruptcy soon. In order to protect the shareholders’ profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, partial differential equation and variational inequality approach. Moreover, we establish a riskbased capital standard to ensure the capital requirement can cover the total given risk by numerical analysis, and give reasonable economic interpretation for the results.

Highlights

In this paper, we study a model of a big financial company, which has the possibility to choose various production/business policies with different expected profits and associated risks
According to Miller Modigliani, the managements of the company choose the maximum of shareholders’ return as their goals. We see from these literatures that the optimality is achieved by using an optimal dividend payout barrier b, i.e. whenever the liquid reserve of the company goes above b, the excess is paid out to the shareholders as dividends
The optimal dividend barrier b may be too low to be acceptable because this low dividend payout barrier may result in the company’s bankruptcy soon

Summary

Introduction

We study a model of a big financial company, which has the possibility to choose various production/business policies with different expected profits and associated risks. Based on the new idea, He, Hou and Liang[16](2008) studied the linear regularsingular optimal control problem of insurance company with proportional reinsurance policy under bankrupt probability constraints as we state above. Liang, Huang and Yao[25, 26, 24](2010) gave a exact result of such a control problem by proving the bankrupt probability is continuous and strictly increasing w.r.t. the dividend barrier b These new results are mainly about the insurance company with proportional reinsurance policy. Motivated by these works, under any given bankrupt probability constraint, we are interested in a common company, such as a big financial company, insurance company, ..., facing constant liability payments, which controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. The procedure of solving the HJB equations and proofs of lemmas which are used to prove the main result will be presented in the appendix

Mathematical Model

Main Result

Risk Analysis

Numerical examples

Proof of Main Result