# OPTIMAL DECISIONS IN ILLIQUID HEDGE FUNDS

• Hugo Ramirez Jaime

Student thesis: Phd

### Abstract

During the work of this research project we were interested in mathematical techniques that give us an insight to the following questions: How do we understand the trading decisions made by a manager of a hedge fund and what influences these decisions? In what way does an illiquid market affect these decisions and the performance of the fund? And how does the payment scheme affect the investor's decisions? Based on existing work on hedge fund management, we start with a fund that can be modelled with one risky investment and one riskless investment. Next, subject to the hedge fund special reward scheme we maximise the expected utility of wealth of the manager, by controlling the percentage invested in the risky investment, namely the portfolio. We use stochastic control techniques to derive a partial differential equation (PDE) and numerically obtain its corresponding viscosity solution, which provides a weak notion of solutions to these PDEs. This is then taken to a liquidity constrained scenario, to compare the behaviour of the two scenarios. Using the same approach as before we notice that due to the liquidity restriction we cannot use a simple model to combine the risky and riskless investments as a total amount, and hence the PDE is one order higher than before.We then model an investor who is investing in the hedge fund subject to the manager's optimal portfolio decisions, with similar mathematical tools as before. Comparisons between the investor's expected utility of wealth and the utility of having the money invested in the risk-free investment suggests that, in some cases, the investor is paying more to the manager than the return he is receiving for having invested in the hedge fund, compared to a risk-free investment. For that reason we propose a strategic game where the manager's action is to allocate the money between the two assets and the investor's action is to add money to the fund when he expects profit. The result is that the investor profits from the option to reinvest in the fund, although in some extreme cases the actions of the manager make the investor receive a negative value for having the option.
Date of Award 31 Dec 2016 English The University of Manchester Peter Duck (Supervisor) & Paul Johnson (Supervisor)

### Keywords

• investor inflows
• high-water mark
• HJB equation
• semi-Lagrangian
• finite differences
• stochastic control
• liquidity
• hedge fund investor and manager
• hedge funds
• viscosity solutions

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