## Jeremiah Bejarano and Dr. Jeffrey Humpherys, Mathematics

In this project, my goal was to analyze the relation between marketshare incentives and risk taking in the hedge fund industry. Using the techniques of stochastic optimal control or dynamic programming, as applied in the typical Continuous-Time Consumption and Portfolio Choice model, I worked to develop a mathematical model to explore the effects of management fees and performance bonuses as well as the effects of changing market share on the fund manager’s implicit aversion towards high-risk portfolios. My continuing hypothesis is that due to fundamental differences between the investors and managers incentives, (essentially profit maximization vs. commission maximization) a fund manager will at times make sub-optimal or overly-risky investments which are not in the best interest of the investor. Over the course of my research I was able to make progress in solving the model and derived the conditions which describe the solution, but the partial differential equation that described the solution was too unruly and I am still working on a numerical solution (or at least analyzing the solution’s properties). However, I am planning on some simplifications to the model but, also, I have discovered a better method through which to model the problem. In this final report I will explain more about the problem, describe my accomplishments, and retell some of the invaluable experiences that I’ve had through the opportunity that ORCA and its donors have provided me.

### PROBLEM SUMMARY

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. However, in this proposed work, we 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 and test the hypothesis that the manager will take certain undesirable risks is perfectly rational behavior under a commission maximizing objective function.

Particularly, in the study of moral hazard among hedge fund managers’ compensation schemes, others use this model to study the effects that variations on the model have on that individual’s (the manager’s) implicit risk aversion. By slightly modifying the individual’s utility function or other model specifications, we can give ourselves a framework out of which we can study the behavior of said managers. In this project, I attempted to analyze a manager’s implicit risk aversion when the manager considers both that (1) he or she earns a percent commission of the volume of assets managed and that (2) his or her quarterly performance (reports made public only periodically) directly affects the marketshare that his or her fund commands.

Hedge fund managers typically receive a fixed management fee of about 1-5% of the assets under management as well as bonuses (often around 20% of the excess) when a fund performs above a given benchmark. These fees give the managers a strong incentive to produce returns above a certain benchmark. This decision can readily be modeled as a typical portfolio choice problem using the tools of continuous-time dynamic programming (Merton 1971) coupled with the appropriate objective function. However, in practice a fund manager must not only consider his gain from management fees and bonuses, but must consider the effect of the fund’s performance on his reputation. The fund’s performance will affect the manager’s reputation, which in turn lead to more or less investment capital down the road. The chief contribution of this project is to extend the typical portfolio choice problem by adding factors that consider the growth dynamics of the volume of assets managed (i.e. change in market share).

### ACCOMPLISHMENTS

I should emphasize that this project uses some sophisticated mathematics. Much of my time researching was devoted to learning the methods used. Also, a lot of my time was devoted to understanding the relevant literature. Keeping in mind that I was new to the field, it’s easy to understand that these two activities would take up a lot of time. But that time was very well spent and was invaluable to my academic career. At the end of the time that I had, I even made some significant progress! I was able to partially solve the mathematical model: I derived the conditions that described the solution! (This is a rather large non-linear partial differential equation. If you are interested, it can be obtained upon request.)

Also, I was able to present my work at the College of Physical and Mathematical Science’s Spring Research Conference 2011. (http://cpms.byu.edu/about/spring-research-conference/submitted-abstracts/?yr=2011&id=396) This was a good experience and taught me a lot about how the academic world works. It helped me to prepare to enter the world of research.

### EXPERIENCES

Besides what I already mentioned in the “accomplishments” section, I had many good experiences that helped to develop my career. Because I plan to go to grad school to pursue a PhD. After I graduate, it was good to have a chance to get involved in research. This helped me to understand better what my future career would be like. I was able to confirm that this is something that I really want to do. Also, being able to work with my mentor, Dr. Jeff Humpherys, was wonderful. Dr. Humpherys is an outstanding educator as well as a talented mathematician. He was able to guide me as I pursued my research goals. Also, he was able to encourage me to develop my ideas and to be ambitious but also realistic and productive. 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