Bayesian Versus Michaud Resampling Methods in Portfolio Optimization

Bayesian versus Michaud resampling methods in portfolio optimization.

UNIVERSITY OF BOLOGNA

Bayesian versus Michaud resampling methods in portfolio optimization.

A Comparative Analysis

Abel Zerihun Shiferaw- ID No. 0000701625

6/5/2015

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Contents

ABSTRACT ...................................................................................................................................................... 2

1. INTRODUCTION ..................................................................................................................................... 3

2. MEAN-VARIANCE EFFICIENCY MODEL .................................................................................................. 5

2.1. Theoretical Framework ................................................................................................................. 5

2.2. Limitation of Mean-Variance Model ........................................................................................... 11

3. RESAMPLED EFFICIENCY FRONTIERS .................................................................................................. 11

3.1. Merits and demerits of resampled portfolio .............................................................................. 13

4. BAYESIAN PORFOLIO ANALYSIS .......................................................................................................... 14

4.1. Theoretical Framework ........................................................................................................... 15

5. BAYESIAN VS. MARKOWITZ MEAN-VARIANCE METHODS .................................................................. 19

6. CONCLUSION ....................................................................................................................................... 20

REFERENCES ................................................................................................................................................ 22

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ABSTRACT

There have been increasing number of empirical studies that conduct a comparison analysis of methods for handling uncertain inputs to a mean-variance analysis, specifically, Michaud's resampled frontier and Bayesian inference with diffuse prior. Studies like the one by Markowitz and Usmen (2003) found that the investment performance of Resampled Efficiency optimized portfolios (Michaud 1998) is superior to that of Markowitz (1959) mean-variance (MV) optimized portfolios with sophisticated Bayesian estimates of risk and return. By contrast, other studies like the one by Harvey, Liechty and Liechty (2008) have performed a similar comparison analysis of the performance of Bayesian methods for determining portfolio weights with Monte Carlo based resampling approach advocated in Michaud (1998) and found that the Bayesian method always has a better performance. In this paper, I discuss the theoretical underpinnings of Michaud's resampled frontier and Bayesian inference with diffuse prior. I also present different empirical studies that are conducted on the efficiency of the two competing models.

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1. INTRODUCTION

In investment portfolio construction and optimization, an investor or analyst has to determine before investing, which asset classes, such as domestic or foreign fixed income, domestic or foreign equity to invest in and what proportion of the total portfolio should be allocated for each asset class. Harry Markowitz(1952, 1959) pioneered in tackling this problem and laid the foundation for Modern Portfolio Theory. He described a method for constructing a portfolio with optimal risk/return characteristics. His portfolio optimization method finds the minimum risk portfolio with a given expected return. Nonetheless, since the Markowitz or Mean-Variance Efficient Portfolio is calculated from the sample mean and covariance, instead of that of the population, they are likely different from the population mean and covariance. This may lead to allocation of too much weight to assets with better estimated than true risk/return characteristics. In order to account for this uncertainty in the sample estimates, Michaud(1998) proposed a new optimization method in which the investor can create many alternative efficient frontier based on resampled versions of the data. Each resampled dataset will result in a different set of Markowitz efficient portfolios. These efficient frontiers of portfolios can then averaged to create a resampled efficient frontier. The appropriate compromise between the investor's risk aversion and desired return will then guide the investor to choose a portfolio from the set of resampled efficient frontier portfolios. Since such a portfolio is different from the Markowitz efficient portfolio it will have suboptimal risk/return characteristics with respect to the sample mean and covariance, but optimal characteristics when averaged over the many possible values of the unknown true mean and covariance.

An alternative and competing method of portfolio optimization is also Bayesian method. Bayesian methods are widely used methods of portfolio optimization. Bayesian methods are statistical techniques optimally designed for improving estimates of statistical parameters.

Avramov and Zhou(2009 ) reviewed Bayesian studies of portfolio analysis and identified three of its important features. First, it can employ useful prior information about quantities of interest. Second, it accounts for estimation risk and model uncertainty. Third, it facilitates the use of fast, intuitive, and easily implementable numerical algorithms in which to simulate otherwise complex economic quantities. Avramov and Zhou(2009 ) further pointed out that there are three building blocks underlying Bayesian portfolio analysis. The first is the formation of prior beliefs, which are

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typically represented by a probability density function on the stochastic parameters underlying the stock return evolution. The prior density can reflect information about events, macroeconomic news, asset pricing theories, as well as any other insights relevant to the dynamics of asset returns. The second is the formulation of the law of motion