Maria Nilsson and Sara Jönsson

Portfolio optimization - How to maximize your expected utility?

Abstract

The foundation of modern portfolio theory is how to maximize your expected
utility. Mean-variance analysis says that this could be done only knowing
the mean and the variance of the portfolio. This theory comes from a Taylor
expansion of the two first moments of the expected utility. This however is
only true if the investor has a quadratic utility function or the
returns are normally distributed and for a short holding period. It is a well 
known fact that these assumptions are not realistic.

We want to compare this method of making a Taylor expansion with
a direct utility maximization method and a L2 projection method. How
do these methods perform when having more fat-tailed distributed returns,
different utility functions and for different holding periods?
We study the methods performances for simulated normally distributed
and Student's t-distributed returns as well for real data consisting of
28 stocks chosen from OMXS30. The utility functions that are used are the 
logarithmic and the power utility functions. The resulting portfolio is 
evaluated considering its expected utility, expected wealth, standard deviation, 
Value at Risk and Expected Shortfall with the different holding periods of two 
months, one year and five years. The conclusion is that the direct utility 
maximization gets the highest expected utility and the L2 projection model is 
also a good approximation for holding periods of two months and one year. The 
mean-variance framework however has the significantly lowest expected utility.