Optimal Investment for Executive Stockholders with Exponential Utility

Abstract

The scope of this paper is to enhance the model for the owncompany stockholder (given in Desmettre, Gould and Szimayer (2010)), who can voluntarily performance-link his personal wealth to his management success by acquiring stocks in the own-company whose value he can directly influence via spending work effort. The executive is thereby characterized by a parameter of risk aversion and the two work effectiveness parameters inverse work productivity and disutility stress. We extend the model to a constant absolute risk aversion framework using an exponential utility/disutility set-up. A closedform solution is given for the optimal work effort an executive will apply and we derive the optimal investment strategies of the executive. Furthermore, we determine an up-front fair cash compensation applying an indifference utility rationale. Our study shows to a large extent that the results previously obtained are robust under the choice of the utility/disutility set-up.2000 MSC Subject Classification: 49L20, 91B28, 93E20 Key Words: portfolio choice, executive stockholder, work effort, exponential utility * Department of Financial Mathematics, Fraunhofer ITWM, FraunhoferPlatz 1, 67663 Kaiserslautern, Germany, Tel.:+49-631-31600-4453, Email: sascha.desmettre@itwm.fraunhofer.de 1 IntroductionIt is widely supported that the remuneration of managers should be linked to performance, see, e.g., Ross (1973), Jensen and Meckling (1976), Holmstrom (1979) and others, for the fundamentals of agency theory, and the summaries of Murphy (1999) and Core, Guay and Larcker (2003). Stemming from that theory, many papers dealing with executive compensation have arisen. In recent research, the motivation for individuals in an executive position to voluntarily performance-link their private wealth by acquiring stocks in the own-company was considered. For example, Desmettre, Gould and Szimayer (2010) proposed a model that allows an executive to invest in a riskless asset, a diversified market portfolio and the own-company's stock. Thereby the executive has the possibility to influence the own-company's stock's value by applying work effort. He then maximizes the expected value of the utility of final wealth and the negative utility of integrated work effort over a fixed time horizon. They give closed-form solutions for the optimal investment strategies and the work effort which the executive will apply in a constant relative risk aversion framework. Other technical papers that consider similar dynamic pincipal-agent problems are Cadenillas, Cvitanić and Zapatero (2004) or Ou-Yang (2003), for example.The scope of this paper is to extend the work of Desmettre, Gould and Szimayer (2010) to a constant absolute risk aversion setting and thus to investigate the robustness of the model proposed by them. We consider an executive who is characterized by an absolute risk aversion parameter (η) and the two work effectiveness parameters inverse work productivity (κ) and disutility stress (α). The executive is endowed with an initial wealth incl...