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13/09/2005, 16:00 — 17:00 — Room P4.35, Mathematics Building

Yarema Okhrin, *Department of Statistics, University of Frankfurt (Oder), Germany*

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Distributional properties and estimation of optimal portfolio

The Markowitz theory of portfolio selection is a classical part of asset allocation. Under the assumption of Gaussian asset returns and investor’s preferences given by the quadratic utility function, we can present the optimal portfolio weights as a function of the first two moments of asset returns. The true moments are unknown to the investor and should be estimated from a sample. Because of this practical applications often suffer from very large or negative portfolio weights. The aim of this project is to assess the distributional properties of estimated portfolio weights and to develop improved estimation procedures. Okhrin and Schmid (2005a) consider the maximum-likelihood estimation of the moments of asset returns. They provide expression for the mean and variance of the estimated portfolio weights of four different types. It appears that the estimated weights are heavily biased in small samples and have very large variance. This explains the empirical evidence from practical applications. It is also shown that the estimated global minimum variance portfolio weights follow multivariate t-distribution, what is of special interest in testing problems. For the portfolio weights that maximize the Sharpe ratio it appears that the moments of order equal or greater than one do not exist. This questions the usefulness of such estimator and makes the results untractable. A classical approach to decrease the volatility of an estimator is shrinkage technique. Using the result of Stein, Jorion (1986) first applied the shrinkage estimation of the expected asset returns to portfolio selection. Recently Ledoit and Wolf (2003, 2004) constructed a shrinkage estimator of the covariance matrix, which is robust against the singularity of sample covariance matrix. Okhrin and Schmid (2005b) applied the shrinkage methodology directly to the optimal portfolio weights by shrinking the classical portfolio weights to the weights obtained from a linear factor model. The optimal shrinkage intensity is derived to minimize the mean-square error. It appears, that the shrinkage estimator is also very successful in the reduction of the variance of portfolio return. Additionally, a new estimator is constructed by using predictive moments from a Bayesian framework with zero-mean prior distribution for the slopes of the factor model.