Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T13:06:21.731Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Portfolio Choice with Parameter Uncertainty

Published online by Cambridge University Press:  06 April 2009

Raymond Kan  and
Guofu Zhou
Raymond Kan
Affiliation:
kan@chass.utoronto.ca, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario M5S 3E6, Canada
Guofu Zhou
Affiliation:
zhou@wustl.edu, Olin School of Business, Washington University, St. Louis, MO 63130.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we analytically derive the expected loss function associated with using sample means and the covariance matrix of returns to estimate the optimal portfolio. Our analytical results show that the standard plug-in approach that replaces the population parameters by their sample estimates can lead to very poor out-of-sample performance. We further show that with parameter uncertainty, holding the sample tangency portfolio and the riskless asset is never optimal. An investor can benefit by holding some other risky portfolios that help reduce the estimation risk. In particular, we show that a portfolio that optimally combines the riskless asset, the sample tangency portfolio, and the sample global minimum-variance portfolio dominates a portfolio with just the riskless asset and the sample tangency portfolio, suggesting that the presence of estimation risk completely alters the theoretical recommendation of a two-fund portfolio.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 42 , Issue 3 , September 2007 , pp. 621 - 656
Copyright
Copyright © School of Business Administration, University of Washington 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aït-Sahalia, Y., and Brandt, M.. “Variable Selection for Portfolio Choice.” Journal of Finance, 56 (2001), 1297–1351.CrossRefGoogle Scholar
Barberis, N. “Investing for the Long Run when Returns are Predictable.” Journal of Finance, 55 (2000), 225–264.CrossRefGoogle Scholar
Bawa, V. S.; Brown, S. J.; and Klein, R. W.. Estimation Risk And Optimal Portfolio Choice. Amsterdam: North-Holland (1979).Google Scholar
Berger, J. O.Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag (1985).CrossRefGoogle Scholar
Brandt, M. “Portfolio Choice Problems.” In Handbook of Financial Econometrics, Ait-Sahalia, Y. and Hansen, L. P., eds. Amsterdam: Elsevier (2005).Google Scholar
Brown, S. J. “Optimal Portfolio Choice under Uncertainty.” Ph.D. Diss., University of Chicago (1976).Google Scholar
Brown, S. J. “The Portfolio Choice Problem: Comparison of Certainty Equivalence and Optimal Bayes Portfolios.” Communications in Statistics—Simulation and Computation, 7 (1978), 321–334.CrossRefGoogle Scholar
Chen, R., and Kan, R.. “Finite Sample Analysis of Two-Pass Cross-Sectional Regressions.” Working Paper, University of Toronto (2004).Google Scholar
Chopra, V. K., and Ziemba, W. T.. “The Effects of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice.” Journal of Portfolio Management, 19 (1993), 6–11.CrossRefGoogle Scholar
Fama, E. F., and French, K. R.. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics, 33 (1993), 3–56.CrossRefGoogle Scholar
Fourdrinier, D.; Philippe, A.; and Robert, C. P.. “Estimation of a Non-Centrality Parameter under Stein-Type-Like Losses.” Journal of Statistical Planning and Inference, 87 (2000), 43–54.CrossRefGoogle Scholar
Frost, P. A., and Savarino, J. E.. “An Empirical Bayes Approach to Efficient Portfolio Selection.” Journal of Financial and Quantitative Analysis, 21 (1986), 293–305.CrossRefGoogle Scholar
Garlappi, L.; Uppal, R.; and Wang, T.. “Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach.” Review of Financial Studies, 20 (2007), 41–81.CrossRefGoogle Scholar
Haff, L. R. “An Identity for the Wishart Distribution with Applications.” Journal of Multivariate Analysis, 9 (1979), 531–544.CrossRefGoogle Scholar
Harvey, C. R.; Liechty, J.; Liechty, M.; and Müller, P.. “Portfolio Selection with Higher Moments.” Working Paper, Duke University (2003).Google Scholar
Jobson, J. D.; Korkie, B.; and Ratti, V.. “Improved Estimation for Markowitz Portfolios Using James Stein Type Estimators.” Proceedings of the American Statistical Association, Business and Economics Statistics Section, 41 (1979), 279–284.Google Scholar
Jorion, P. “Bayes-Stein Estimation for Portfolio Analysis.” Journal of Financial and Quantitative Analysis, 21 (1986), 279–292.CrossRefGoogle Scholar
Jorion, P. “Bayesian and CAPM Estimators of the Means: Implications for Portfolio Selection.” Journal of Banking and Finance, 15 (1991), 717–727.CrossRefGoogle Scholar
Kacperczyk, M. “Asset Allocation under Distribution Uncertainty.” Working Paper, University of British Columbia (2004).Google Scholar
Kandel, S., and Stambaugh, R. F.. “On the Predictability of Stock Returns: An Asset-Allocation Perspective.” Journal of Finance, 51 (1996), 385–424.Google Scholar
Klein, R. W., and Bawa, V. S.. “The Effect of Estimation Risk on Optimal Portfolio Choice.” Journal of Financial Economics, 3 (1976), 215–231.CrossRefGoogle Scholar
Kubokawa, T.; Robert, C. P.; and Saleh, A. K. Md. E.. “Estimation of Noncentrality Parameters.” Canadian Journal of Statistics, 21 (1993), 45–57.CrossRefGoogle Scholar
Litterman, B.Modern Investment Management: An Equilibrium Approach. New York: Wiley (2003).Google Scholar
Lutgens, F. “Robust Portfolio Optimization.” Ph.D. Diss., University of Maastricht (2004).Google Scholar
Markowitz, H. M. “Mean-Variance Analysis in Portfolio Choice and Capital Markets.” Journal of Finance, 7 (1952), 77–91.Google Scholar
Meucci, A.Risk and Asset Allocation. New York: Springer-Verlag (2005).CrossRefGoogle Scholar
Mori, H. “Finite Sample Properties of Estimators for the Optimal Portfolio Weight.” Journal of Japan Statistical Society, 34 (2004), 27–46.CrossRefGoogle Scholar
Muirhead, R. J.Aspects of Multivariate Statistical Theory. New York: Wiley (1982).CrossRefGoogle Scholar
Pástor, L. “Portfolio Selection and Asset Pricing Models.” Journal of Finance, 55 (2000), 179–223.CrossRefGoogle Scholar
Pástor, L., and Stambaugh, R. F.. “Comparing Asset Pricing Models: An Investment Perspective.” Journal of Financial Economics, 56 (2000), 335–381.CrossRefGoogle Scholar
Rukhin, A. L. “Estimation of the Noncentrality Parameter of an F-Distribution.” Journal of Statistical Planning and Inference, 35 (1993), 201–211.CrossRefGoogle Scholar
Stambaugh, R. F. “Analyzing Investments Whose Histories Differ in Length.” Journal of Financial Economics, 45 (1997), 285–331.CrossRefGoogle Scholar
Stein, C. “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution.” Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1 (1956), 197–206.Google Scholar
ter Horst, J. R.; de Roon, F. A.; and Werkerzx, B. J. M.. “Incorporating Estimation Risk in Portfolio Choice.” Working Paper, Tilburg University (2002).Google Scholar
Tu, J., and Zhou, G.. “Data-Generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?” Journal of Financial Economics, 72 (2003), 385–421.CrossRefGoogle Scholar
Xia, Y. “Learning about Predictability: The Effect of Parameter Uncertainty on Dynamic Asset Allocation.” Journal of Finance, 56 (2001), 205–246.CrossRefGoogle Scholar
Zellner, A., and Chetty, V. K.. “Prediction and Decision Problems in Regression Models from the Bayesian Point of View.” Journal of the American Statistical Association, 60 (1965), 608–616.CrossRefGoogle Scholar