Distributional properties of portfolio weights

In this paper, we prove several distributional properties for optimal portfolio weights. The weights are estimated by replacing the parameters with the sample counterparts. All results for finite samples are made assuming normally distributed returns. We calculate the exact covariances for the weights obtained by the expected quadratic utility. Additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio. We obtain the conditional density for the Sharpe ratio optimal weights and show that the expectations of the Sharpe ratio optimal weights do not exist. Moreover, we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary Gaussian process.

[1]  Y. L. Tong.  The multivariate normal distribution , 1989 .

[2]  Vasyl Golosnoy,et al.  EWMA Control Charts for Monitoring Optimal Portfolio Weights , 2007 .

[3]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[4]  N. Barberis.  Investing for the Long Run When Returns are Predictable , 2000 .

[5]  J. Jobson,et al.  A Performance Interpretation of Multivariate Tests of Asset Set Intersection, Spanning, and Mean-Variance Efficiency , 1989, Journal of Financial and Quantitative Analysis.

[6]  Yadolah Dodge,et al.  Statistical data analysis and inference , 1992 .

[7]  Dennis E. Logue,et al.  Foundations of Finance. , 1977 .

[8]  M. Best,et al.  On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results , 1991 .

[9]  J. Cochrane,et al.  Portfolio Advice for a Multifactor World , 1999 .

[10]  W. Ziemba,et al.  The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice , 1993 .

[11]  E. J. Hannan,et al.  Multiple time series , 1970 .

[12]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[13]  S. Rachev,et al.  Modeling asset returns with alternative stable distributions , 1993 .

[14]  P. Samuelson.  The Fundamental Approximation Theorem of Portfolio Analysis in terms of Means, Variances and Higher Moments , 1970 .

[15]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[16]  J. Jobson,et al.  Estimation for Markowitz Efficient Portfolios , 1980 .

[17]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[18]  George P. H. Styan.  THREE USEFUL EXPRESSIONS FOR EXPECTATIONS INVOLVING A WISHART MATRIX AND ITS INVERSE , 1989 .

[19]  Mark Britten-Jones,et al.  The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights , 1999 .

[20]  R. C. Merton,et al.  On Estimating the Expected Return on the Market: An Exploratory Investigation , 1980 .

[21]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[22]  K. French,et al.  Investor Diversification and International Equity Markets , 1991 .

[23]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[24]  M. Osborne.  Brownian Motion in the Stock Market , 1959 .

[25]  Harry M. Markowitz,et al.  Foundations of Portfolio Theory , 1991 .

[26]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[27]  René M. Stulz,et al.  International Portfolio Choice and Asset Pricing: An Integrative Survey , 1994 .

[28]  Andrew Ang,et al.  International Asset Allocation With Regime Shifts , 2002 .

[29]  Chris Kirby,et al.  The Economic Value of Volatility Timing , 2000 .

[30]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics (Revised Edition) , 1999 .

[31]  J. D. Jobson,et al.  Performance Hypothesis Testing with the Sharpe and Treynor Measures , 1981 .

[32]  A. M. Mathai.  Quadratic forms in random variables , 1992 .

[33]  Andrew H. Chen,et al.  Investigations of Nonstationarity in Prices , 1974 .