Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-07T08:50:56.669Z Has data issue: false hasContentIssue false

Bayes-Stein Estimation for Portfolio Analysis

Published online by Cambridge University Press:  06 April 2009

Abstract

In portfolio analysis, uncertainty about parameter values leads to suboptimal portfolio choices. The resulting loss in the investor's utility is a function of the particular estimator chosen for expected returns. So, this is a problem of simultaneous estimation of normal means under a well-specified loss function. In this situation, as Stein has shown, the classical sample mean is inadmissible. This paper presents a simple empirical Bayes estimator that should outperform the sample mean in the context of a portfolio. Simulation analysis shows that these Bayes-Stein estimators provide significant gains in portfolio selection problems.

Type
Research Article
Copyright
Copyright © School of Business Administration, University of Washington 1986

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

REFERENCES

[1]Barry, C. B.Portfolio Analysis under Uncertain Means, Variances and Covariances.” Journal of Finance, 29 (05 1974), 515522.CrossRefGoogle Scholar
[2]Bawa, V. S.; Brown, S. J.; and Klein, R. W.. Estimation Risk and Optimal Portfolio Choice. Amsterdam: North Holland (1979).Google Scholar
[3]Berger, J.Minimax Estimation of a Multivariate Normal Mean under Polynomial Loss.” Journal of Multivariate Analysis, 8 (06 1978), 173180.CrossRefGoogle Scholar
[4]Berger, J.Statistical Decision Theory. New York: Springer-Verlag (1980).CrossRefGoogle Scholar
[5]Berger, J.Selecting a Minimax Estimator of a Multivariate Normal Mean.” The Annals of Statistics, 10 (03 1982), 8192.CrossRefGoogle Scholar
[6]Berger, J., and Bock, M. E.. “Eliminating Singularities of Stein Type Estimators of Location Vectors.” Journal of the Royal Statistical Society, 38 (1976), 166170.Google Scholar
[7]Blume, M.Betas and Their Regression Tendencies.” Journal of Finance, 30 (06 1975), 785795.CrossRefGoogle Scholar
[8]Brown, L. D.On the Admissibility of Invariant Estimators of One or More Location Parameters.” Annals of Mathematical Statistics, 37 (08 1966), 10871136.CrossRefGoogle Scholar
[9]Brown, L. D.Estimation with Incompletely Specified Loss Functions.” Journal of the American Statistical Association, 70 (06 1975), 417427.CrossRefGoogle Scholar
[10]Brown, L. D.A Heuristic Method for Determining Admissibility of Estimators—with Applications.” Annals of Statistics, 7 (09 1979), 960994.CrossRefGoogle Scholar
[11]Brown, S. J. “Optimal Portfolio Choice under Uncertainty: A Bayesian Approach.” Ph.D. Diss., Univ. of Chicago (1976).Google Scholar
[12]Brown, S. J.The Effect of Estimation Risk on Capital Market Equilibrium.” Journal of Financial and Quantitative Analysis, 14 (06 1979), 215220.CrossRefGoogle Scholar
[13]Dickenson, J. P.The Reliability of Estimation Procedures in Portfolio Analysis.” Journal of Financial and Quantitative Analysis, 9 (09 1979), 447462.CrossRefGoogle Scholar
[14]Efron, B., and Morris, C.. “Stein's Estimation Rule and its Competitors—An Empirical Bayes Approach.” Journal of the American Statistical Association, 68 (03 1973), 117130.Google Scholar
[15]Efron, B., and Morris, C.. “Data Analysis Using Stein's Estimator and its Generalizations.” Journal of the American Statistical Association, 70 (06 1975), 311319.CrossRefGoogle Scholar
[16]Efron, B., and Morris, C.. “Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution.” The Annals of Statistics, 4 (01 1976), 1121.CrossRefGoogle Scholar
[17]Frankfurter, G. M.; Phillips, H. E.; and Seagle, J. P.. “Portfolio Selection: the Effects of Uncertain Means, Variances and Covariances.” Journal of Financial and Quantitative Analysis, 6 (09 1971), 12511262.CrossRefGoogle Scholar
[18]James, W., and Stein, C.. “Estimation with Quadratic Loss.” Proceedings of the 4th Berkeley Symposium on Probability and Statistics I. Berkeley: Univ. of Calif. Press (1961), 361379.Google Scholar
[19]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), 279284.Google Scholar
[20]Jobson, J. D., and Korkie, B.. “Estimation for Markowitz Efficient Portfolios.” Journal of the American Statistical Association, 75 (09 1980), 544554.CrossRefGoogle Scholar
[21]Jorion, P.International Portfolio Diversification with Estimation Risk.” Journal of Business, 58 (07 1985), 259278.CrossRefGoogle Scholar
[22]Klein, R. W., and Bawa, V. S.. “The Effect of Estimation Risk on Optimal Portfolio Choice.” Journal of Financial Economics, 3 (06 1976), 215231.CrossRefGoogle Scholar
[23]Lindley, D. V.Discussion on Professor Stein's Paper.” Journal of Royal Statistical Society, 24 (1962), 285287.Google Scholar
[24]Lindley, D. V., and Smith, A. F. M.. “Bayes Estimates for the Linear Model.” Journal of the Royal Statistical Society, 34 (1972), 141.Google Scholar
[25]Markowitz, H. M.Portfolio Selection: Efficient Diversification of Investments. New York: Wiley and Sons (1959).Google Scholar
[26]Merton, R. C.An Analytic Derivation of the Efficient Portfolio Frontier.” Journal of Financial and Quantitative Analysis, 7 (09 1972), 18511872.CrossRefGoogle Scholar
[27]Merton, R. C.On Estimating the Expected Return on the Market.” Journal of Financial Economics, 8 (12 1980), 323361.CrossRefGoogle Scholar
[28]Morris, C. N.Parametric Empirical Bayes Inference: Theory and Applications.” Journal of the American Statistical Association, 78 (03 1983), 4755.CrossRefGoogle Scholar
[29]Stein, C. “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution.” Proceedings of the 3rd Berkeley Symposium on Probability and Statistics 1. Berkeley: Univ. of Calif. Press (1955), 197206.Google Scholar
[30]Stein, C.Confidence Sets for the Mean of a Multivariate Normal Distribution.” Journal of the Royal Statistical Society, 24 (1962), 265296.Google Scholar
[31]Vasicek, O.A Note on Using Cross-Sectional Information on Bayesian Estimation of Security Betas.” Journal of Finance, 28 (12 1973), 12331239.Google Scholar
[32]Zellner, A.An Introduction to Bayesian Inference in Econometrics. New York: Wiley and Sons (1971).Google Scholar
[33]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 (06 1965), 608615.Google Scholar
[34]Zellner, A., and Vandaele, W.. “Bayes-Stein Estimators for k–Means Regression and Simultaneous Equations Models.” In Studies in Bayesian Econometrics and Statistics, Fienberg, S. and Zellner, A., eds. Amsterdam: North Holland (1974).Google Scholar