Article,
Estimation for Markowitz Efficient Portfolios
Journal of the American Statistical Association, 75 (371): 544-554 (1980)
Abstract
Given a set of N assets a portfolio is determined by a set of weights x<sub>i</sub>, i = 1, 2, ..., N; ∑<sup>N</sup><sub>i = 1</sub> x<sub>i</sub> = 1 indicating the proportion of the value of the portfolio devoted to each asset. A Markowitz efficient portfolio is the vector of weights X<sub>m</sub> that minimizes the variance σ<sub>m</sub><sup>2</sup> of the total return from the portfolio, subject to the condition that the portfolio mean premium return μ<sub>m</sub> has a certain value. The estimators for the N × 1 vector X<sub>m</sub>, the return premium μ<sub>m</sub>, and the variable σ<sub>m</sub><sup>2</sup> require estimators for the mean premium return vector μ and for the covariance matrix Σ. The expectations, variances, and asymptotic distributions of the estimators of X<sub>m</sub>, μ<sub>m</sub>, and σ<sub>m</sub><sup>2</sup> are derived under the assumption that returns are normally distributed. The use of these sampling properties for statistical inference is also discussed. The derived results are also compared with results obtained from a Monte Carlo simulation for a population of 20 stocks and several sample sizes.
