GRADUATE SCHOOL OF SCIENCES & ENGINEERING
INDUSTRIAL ENGINEERING AND OPERATIONS MANAGEMENT
PhD THESIS DEFENSE BY NURİ ŞENSOY
Title: Portfolio Selection under Cumulative Prospect Theory
Speaker: Nuri Şensoy
Time: January 3, 2018, 10:30
Place: ENG 208
Rumeli Feneri Yolu
Thesis Committee Members:
Prof. Dr. Süleyman Özekici (Advisor, Koç University)
Prof. Dr. Fikri Karaesmen (Koç University)
Asst. Prof. Dr. Uğur Çelikyurt (Koç University)
Prof. Dr. Refik Güllü (Boğaziçi University)
Asst. Prof. Dr. Ethem Çanakoğlu (Bahçeşehir University)
Decision making under risk has always been a very important issue in portfolio selection for many years. There are several theories which try to explain the attitude of the decision maker under risk. Utility theory is one of the most important and widely used theories in portfolio selection. However, Allais and Ellsberg paradoxes, for example, show that utility theory does not exactly represent investors’ behavior when they are faced with uncertainty. As a result, Prospect Theory proposed by Kahneman and Tversky in the 1970s has become very popular in recent years. The main features of prospect theory suggest that investors make decisions based on change of wealth with respect to a reference point rather than the total wealth, that preferences are S-shaped rather than uniformly concave, and that probabilities are subjectively distorted. The main purpose of our research is to investigate the optimal portfolio selection problem for an investor who behaves according to Cumulative Prospect Theory (CPT).
Firstly, we formulate and study the single-period portfolio selection problem under CPT featuring a reference point in wealth, piecewise value functions with loss-aversion, and probability weighting functions. We focus on a class of value functions that involve combinations of linear, exponential and logarithmic functions. We introduce a new measure of loss-aversion for small payoffs, called the small loss-aversion degree (SLAD), and show that it plays a critical role on the investment decisions of the investors. We explicitly derive the optimal solutions for single risky asset cases in which the reference point is the risk-free return and in which it is not. For multiple risky assets which follow an elliptical distribution, we obtain the explicit solution of the problem defined by the piecewise linear-exponential value function for the risk-free reference point.
Secondly, we formulate and study the single-period portfolio selection problem under CPT using the piecewise exponential (PE) value function. The reference point coincides with the risk-free return and the probability weighting functions are ignored. Under the assumption of normally distributed returns of multiple risky assets, we obtain the optimal portfolio explicitly and provide several characterizations. In particular, we investigate the connection of the optimal PE portfolio to those within the mean-variance framework of Markowitz. As a special case, we analyze the single risky asset model and show that the objective function is quasi-concave.
Thirdly, we formulate and study a multi-period portfolio selection problem under CPT featuring a reference point in wealth and a piecewise power value function with loss-aversion. The main feature of the model is that returns of the risky assets and the value function all depend on an external stochastic process that represents the regime-switching market. The states of the market describe the prevailing economic, financial, social, political and other conditions that affect the deterministic and probabilistic parameters of the model. We suppose that the random changes in the market states are depicted by a Markov chain. Dynamic programming is used to derive the optimal policies analytically. We first analyze a model with a single risky asset, and identify the well-posedness condition to obtain the optimal portfolio explicitly. We then consider the model with multiple risky assets that follow the elliptical distribution, and identify a threshold for the well-posedness of the problem and obtain an explicit solutions of the portfolio optimization problem.
Finally, we identify the optimal portfolios and analyze the structure of the objective functions in a number of numerical illustrations. Also, we perform sensitivity analysis on the optimal portfolios with respect to parameters of the value function and the probability weighting function.