Moral Hazard in Hedge Funds: An Approach Using Stochastic Differential Games

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Jeremiah Bejarano and Dr. Jeffrey Humpherys, Department of Mathematics

This project is a continuation of a line of research that I have been working on over the last two years, with the object of analyzing the relationship incentive structure present in the hedge fund industry and identifying any potential for moral hazard. In economics theory, “moral hazard” is a situation in which one party is more willing to take on a risk due to the fact that he or she knows that the potential costs or burden of that risk will be borne, in part or in whole, by others. In this project, my goal was to develop a mathematical model to explore the effects of management fees and performance bonuses as well as changing market share on the fund manager’s implicit aversion towards high-risk portfolios. Because hedge fund’s trading data is typically unavailable for analysis, an empirical approach is out of the question. Rather, we can make this argument deductively, by showing that overly-risky behavior is actually optimal from the fund manager’s perspective. In essence, can we show mathematically that, given the commission-based compensation structure of the fund and the dynamics of market share, the fund manager has an incentive to take on excessively risky investments? In this report, I will detail my approach to this problem, including details of my previous attempt, and will report my experiences in working with my advisor, Professor Jeff Humpherys, and the outcomes of opportunity afforded by the ORCA grant program.

Problem Summary

The central point of interest in this project is moral hazard. As I described, moral hazard is a situation in which one party is more willing to take on a risk due to the fact that he or she knows that the potential costs or burden of that risk will be borne, in part or in whole, by others. The most common example given to illustrate moral hazard relates to the 2008 financial crisis. It is suspected that many of the originators of subprime loans did so knowing that the borrowers would not be able to maintain their payments in the long term. After the loans were pooled and sold, the originators no longer bore any of the risk of the loans and, therefore, had little or no motivation to inspect the long-term sustainability of the loans in the first place. In this project, the object was to examine the existence of this kind of misalignment of incentives in the hedge fund industry.

The use of mathematical models to study moral hazard among such funds is not unique; however, the mathematical model in this project is novel. Related work includes a paper by Carpenter (2000) that explores the effect of incentive fees on the optimal investment strategy of a risk-averse fund manager. Also, Kouwenberg and Ziemba (2007) solve a similar problem in a framework including management fee, bonuses, and a portion of the manager’s own money invested in the fund with the utility function of prospect theory in the place of HARA. Each of these papers successfully gives results similar to what I wish to show. However, my wish was to build on these studies so as to take into account the growth dynamics and turnover of market share (the fact that a manager’s performance directly affects the number of investors that invest with the firm, the volume of assets managed and thus the manager’s commission). Doing so, I wanted to test the hypothesis that a manager willing take certain undesirable risks is perfectly rational behavior under a commission maximizing objective function.


Throughout the time that I spent on this project, I gained research experience and learned to use some powerful tools in economics and mathematics. This project was a good one and there is definitely some merit to further study. However, after working on the problem for several months, it seems that some of the mathematical obstacles were too tough to overcome. The two most important problems are the inherent discontinuities in the model. If we are to model how the fund manager and the investors interact over multiple periods (financial quarters or years), there results an irreconcilable discontinuity between each period. The second problem is that most of the existing literature on stochastic differential games deals with cooperative differential games. It was my misunderstanding that cooperative differential games would be suitable for the situation I desired to model. Non-cooperative games are always more difficult, but their results are much more true to life than the framework that cooperative games provide. Cooperative games are useful in more specialized situations. Non-cooperative games are those in which the individuals do not explicitly cooperate in selecting their strategies. Due to these issues, my advisor wisely suggested that I reconsider my approach: simplify the problem! During my collaboration with Professor Humpherys, due to his guidance, I nonetheless was able to achieve some important accomplishments.


Although I was not able to complete my project the way that I had planned (I have a working paper, but it is incomplete and the remaining problems seem presently insurmountable), I was able to achieve a few important accomplishments. One of the main purposes of this grant was to gain experience doing research that would help prepare me to get accepted to a top graduate school. Happily, it did. This March I was accepted into the University of Chicago’s Economics Ph.D. program and will be entering the program this fall (2013). I was told by the admissions committee that they were particularly impressed by my research experiences with Professor Humpherys, his IMPACT program, and by my other research in the economics department here at BYU. One of the important parts of my collaboration with Professor Humpherys, including this research project, has been his encouragement to pursue other research interests. When progress on this project slowed, he encouraged me to pursue other projects. In particular, he persuaded me to pursue studying High Performance Computing. Because of this encouragement I was able to gain a summer research opportunity at Stanford University over the last summer. Because of his encouragement, I have developed and published online a considerable amount of material. I have a working paper on application of High Performance Computing (HPC) to economics, an online manual and tutorial on learning HPC, and several YouTube videos. These have been moderately popular, receiving views in the thousands.

In conclusion, working with my advisor, Professor Humpherys, has been such a blessing and one of the most important parts of my undergraduate career. I am grateful for that opportunity and grateful to the ORCA grant program which has made that possible. Having had the guidance of a good mentor during my ORCA experience will be a positive influence that I will remember for the rest of my career.