On the equivalence of quadratic optimization problems commonly used in portfolio theory

• We consider three quadratic optimization problems applied in portfolio theory.
• Conditions are derived under which the solutions of these three optimization procedures are lying on the efficient frontier.
• We show that the three quadratic optimization problems are mathematically equivalent but not from a stochastic point of view.
• We derive the probabilities that the estimated solutions of the Markowitz problem and the quadratic utility problem are mean–variance efficient.

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.