On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory

In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e, the Markowitz mean-variance problem as well as the problems based on the mean-variance utility function and the quadratic utility.Conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier, the set of mean-variance optimal portfolios. It is shown that the solutions of the Markowitz optimization problem and the quadratic utility problem are not always mean-variance efficient. The conditions for the mean-variance efficiency of the solutions depend on the unknown parameters of the asset returns. We deal with the problem of parameter uncertainty in detail and derive the probabilities that the estimated solutions of the Markowitz problem and the quadratic utility problem are mean-variance efficient. Because these probabilities deviate from one the above mentioned quadratic optimization problems are not stochastically equivalent. The obtained results are illustrated by an empirical study.Comment: Revised preprint. To appear in European Journal of Operational Research. Contains 18 pages, 6 figure

Modelling and Optimization of a Non-Constrained Multi-objective Problem having Multiple Utility Functions using Bayesian Theory

One of the multi-objective optimization methods makes use of the utility function for the objective functions. Utility function creating the most satisfaction answers for decision makers (DMs) by considering the priorities of the DMs; in an available studies; there are only one utility function for each objective function. But due to practical situation in different decision making environments in an industry or trade lead each goal has multiple utility functions. This paper presents a model of multi- objective problem in which each of the objective function has multiple utility function applying Bayesian theory. This model allows DMs to calculate the probability of these utilities using conditional probability in conditions of uncertainty. In addition, examples are given to illustrate the usefulness of this model

Objective comparisons of the optimal portfolios corresponding to different utility functions

This paper considers the effects of some frequently used utility functions in portfolio selection by comparing the optimal investment outcomes corresponding to these utility functions. Assets are assumed to form a complete market of the Black-Scholes type. Under consideration are four frequently used utility functions: the power, logarithm, exponential and quadratic utility functions. To make objective comparisons, the optimal terminal wealths are derived by integration representation. The optimal strategies which yield optimal values are obtained by the integration representation of a Brownian martingale. The explicit strategy for the quadratic utility function is new. The strategies for other utility functions such as the power and the logarithm utility functions obtained this way coincide with known results obtained from Merton's dynamic programming approach.Utility function Optimal portfolio Martingale measure Integration representation Brownian martingales