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Investment under risk tolerance constraints and non-concave utility functions: implicit risks, incentives and optimal strategies Rodriguez-Mancilla, Jose Ramon

Abstract

The objective of this thesis is to contribute in the understanding of both the induced behavior and the underlying risks of a decision maker who is rewarded through option-like compensation schemes or who is subject to risk tolerance constraints. In the first part of the thesis we consider a risk averse investor who maximizes his expected utility subject to a risk tolerance constraint expressed in terms of the risk measure known as Conditional Value-at-Risk. We study some of the implicit risks associated with the optimal strategies followed by this investor. In particular, embedded probability measures are uncovered using duality theory and used to assess the probability of surpassing a loss threshold defined by the risk measure known as Value-at-Risk. Using one of these embedded probabilities, we derive a measure of the financial cost of hedging the loss exposure associated to the optimal strategies, and we show that, under certain assumptions, it is a coherent measure of risk. In the second part of the thesis, we analyze the investment decisions that managers undertake when they are paid with option-like compensation packages. We consider two particular cases: • We study the optimal risk taking strategies followed by a fund manager who is paid through a relatively general option-like compensation scheme. Our analysis is developed in a continuous-time framework that permits to obtain explicit formulas. These are used first to analyze the incentives induced by this type of compensation schemes and, second, to establish criterions to determine appropriate parameter values for these compensation packages in order to induce specific manager's behaviors. • We consider a hedge fund manager who is paid through a simple option-like compensation scheme and whose investment universe includes options. We analyze the nature of the optimal investment strategies followed by this manager. In particular, we establish explicit optimal conditions for option investments in terms of embedded martingale measures that are derived using duality theory. Our analysis in this case is developed in a discrete-time framework, which allows to consider incomplete markets and fat-tailed distributions -such as option return distributions- in a much simpler manner than in a continuous-time framework.

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