Bayesian Versus Michaud Resampling Methods in Portfolio Optimization

...

governing the evolution of asset returns. Then, one could recover the predictive distribution of future asset returns, analytically or numerically, incorporating prior information, law of motion, as well as estimation risk and model uncertainty. The predictive distribution, which integrates out the parameter space, characterizes the entire uncertainty about future asset returns. The Bayesian optimal portfolio rule is obtained by maximizing the expected utility with respect to the predictive distribution.

In this paper, I have made a detailed review of outstanding literatures in portfolio optimization, mainly focusing on resampled portfolio frontiers and Bayesian portfolio optimization methods. I have put together from different empirical studies the distinguishing features, advantages and disadvantages of the two methods.

This paper is organized as follows. The next section presents the mean-variance efficiency portfolio optimization method and overall background to modern portfolio theory. The third section introduced and explains the resampled efficiency frontier method. Bayesian method, its theoretical framework and mathematical derivation are presented in the fourth section. The fifth section, compares and contrasts the resampled efficiency and Bayesian method. The last section provides concluding remarks.

5

2. MEAN-VARIANCE EFFICIENCY MODEL

Markowitz (1952, 1959) laid the foundation for Modern Portfolio Theory. Markowitz first formulated the portfolio selection problem as a tradeoff between expected return and risk, represented by variance of a portfolio. Keeping constant variance and maximizing expected return or keeping return as constant and minimizing variance led to the efficient frontier of the portfolio from where the investor could choose his portfolio mix depending upon his risk attitude. This has led to the Modern Portfolio Theory. An important implication is that an asset should not be chosen only on the basis of its individual return and risk characteristics but when it is selected along with other assets, its co-movement with respect to other assets can lead to same return but can reduce the risk of the entire portfolio.

2.1. Theoretical Framework

Under the Markowitz MV framework portfolio selection involves two fundamental problems: one, identifying a set of efficient portfolios, and two, selecting among them the one portfolio that has the optimal combination of return and risk.

As generating MV inputs often relies on historical data, the principal problems with MV optimization are posed by the input data. Using historical data as MV optimization inputs is predicated on the assumption that the reruns in the different periods are independent, they are drawn from the same distribution and the available data represent a sample of this distribution—assumptions which may not be always true.

Assuming that there are no transaction costs, and the portfolio will be chosen from K securities, If