Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T10:59:23.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Some risk management problems for firms with internal competition and debt

Part of: Stochastic systems and control Hamilton-Jacobi theories, including dynamic programming

Published online by Cambridge University Press:  14 July 2016

Xin Guo
Xin Guo*
Affiliation:
IBM
*
∗ Postal address: IBM T. J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA. Email address: xinguo@us.ibm.com
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Hamilton-Jacobi equationsingular controlnonlinear controldividend optimizationinternal competition

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 49L99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 55 - 69
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]. Alvarez, L. H. R., and Shepp, L. A. (1998). Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177.CrossRefGoogle Scholar
[2]. Asmussen, S., and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.CrossRefGoogle Scholar
[3]. Asmussen, S., Hojgaard, B., and Taksar, M. (2000). Optimal risk control and dividend distribution policies: example of excess of loss reinsurance. Finance Stoch. 4, 299–324.CrossRefGoogle Scholar
[4]. Beneš, V. E., Witsenhausen, H. S., and Shepp, L. A. (1980). Some solvable stochastic control problems. Stochastics 4, 39–83.Google Scholar
[5]. Chernoff, H. (1961). Sequential test for the mean of a normal distribution. Proc. 4th Berkeley Symp. Statist. Prob., Vol. 1, University of California Press, Berkeley, pp. 79–91.Google Scholar
[6]. Choulli, T., Taksar, M., and Zhou, X. Y. (2002). A diffusion model for optimal dividend distribution for a company with constraints on risk control. To appear in SIAM J. Control Opimization.Google Scholar
[7]. Fleming, W. H., and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, Berlin.Google Scholar
[8]. Guo, X., and Liu, J. (2002). Controlling risks via non-linear reinsurance policy. Unpublished manuscript.Google Scholar
[9]. Harrison, M. J. (1985). Brownian Motion and Stochastic Flow System. John Wiley, Chichester.Google Scholar
[10]. Jeanblanc-Picqué, M., and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257–277.Google Scholar
[11]. Karlin, S., and Taylor, H. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[12]. Radner, R., and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econon. Dynam. Control 20, 1373–1393.CrossRefGoogle Scholar
[13]. Revue, D., and Yor, M. (1989). Continuous Martingale and Brownian Motion. Springer, Berlin.Google Scholar
[14]. Shepp, L. and Shiryayev, A. N. (1996). Hiring and firing optimally in a corporation. J. Econom. Dynam. Control 20, 1523–1529.Google Scholar
[15]. Shreve, S., and Karatzas, I. (1998). Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
[16]. Taksar, M., and Zhou, X. Y. (1998). Optimal risk and dividend control for a company with debt liability. Insurance Math. Econom. 22, 105–122.Google Scholar