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Dominating estimators for minimum-variance portfolios. (English) Zbl 1441.62264
Summary: In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets \(d\geq 4\) and number of observations \(n\geq d+2\). The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed \(d\) but \(n\to \infty\) and \(n,d\to \infty\) but \(n/d\to q\leq \infty\) are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62G09 Nonparametric statistical resampling methods
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J07 Ridge regression; shrinkage estimators (Lasso)
91G10 Portfolio theory
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[1] Aparicio, F.; Estrada, J., Empirical distributions of stock returns: European securities markets, 1990-95, European journal of finance, 7, 1-21, (2001)
[2] Chan, L.; Karceski, J.; Lakonishok, J., On portfolio optimization: forecasting covariances and choosing the risk model, Review of financial studies, 12, 937-974, (1999)
[3] Chopra, V.; Ziemba, W., The effect of errors in means, variances, and covariances on optimal portfolio choice, Journal of portfolio management, 19, 6-11, (1993)
[4] Cogley, T.; Sargent, T., The market price of risk and the equity premium: a legacy of the great depression, Journal of monetary economics, 55, 454-476, (2008)
[5] DeMiguel, V.; Garlappi, L.; Uppal, R., Optimal versus naive diversification: how inefficient is the \(1 / N\) portfolio strategy?, Review of financial studies, 22, 1915-1953, (2009)
[6] Dimson, E.; Marsh, P.; Staunton, M., Global evidence on the equity risk premium, Journal of applied corporate finance, 15, 27-38, (2003)
[7] Elton, E.; Gruber, M., Estimating the dependence structure of share prices—implications for portfolio selection, Journal of finance, 28, 1203-1232, (1973)
[8] Frahm, G., Linear statistical inference for global and local minimum variance portfolios, Statistical papers, (2008)
[9] Frost, P.; Savarino, J., An empirical Bayes approach to efficient portfolio selection, Journal of financial and quantitative analysis, 21, 293-305, (1986)
[10] Garlappi, L.; Uppal, R.; Wang, T., Portfolio selection with parameter and model uncertainty: a multi-prior approach, Review of financial studies, 20, 41-81, (2007)
[11] Golosnoy, V.; Okhrin, Y., Multivariate shrinkage for optimal portfolio weights, The European journal of finance, 13, 441-458, (2007)
[12] Haugen, R., Building a better index: cap-weighted benchmarks are inefficient vehicles, Pensions and investments, 18, 1-3, (1990)
[13] Haugen, R.; Baker, N., The efficient market inefficiency of capitalization-weighted stock portfolios, Journal of portfolio management, 17, 35-40, (1991)
[14] Haugen, R.; Baker, N., Interpreting the evidence on risk and expected return: comment, Journal of portfolio management, 19, 36-43, (1993)
[15] Jagannathan, R.; Ma, T., Risk reduction in large portfolios: why imposing the wrong constraints helps, Journal of finance, 58, 1651-1683, (2003)
[16] Jobson, J.; Korkie, B., Improved estimation for Markowitz portfolios using james – stein type estimators, (), 279-284
[17] Jorion, P., Bayes – stein estimation for portfolio analysis, Journal of financial and quantitative analysis, 21, 279-292, (1986)
[18] Judge, G.; Bock, M., The statistical implications of pre-test and Stein-rule estimators in econometrics, (1978), North-Holland Publishing Company · Zbl 0395.62078
[19] Kan, R.; Zhou, G., Optimal portfolio choice with parameter uncertainty, Journal of financial and quantitative analysis, 42, 621-656, (2007)
[20] Kempf, A.; Memmel, C., Estimating the global minimum variance portfolio, Schmalenbach business review, 58, 332-348, (2006)
[21] Ledoit, O.; Wolf, M., A well-conditioned estimator for large-dimensional covariance matrices, Journal of multivariate analysis, 88, 365-411, (2001) · Zbl 1032.62050
[22] Ledoit, O.; Wolf, M., Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of empirical finance, 10, 603-621, (2003)
[23] Ledoit, O.; Wolf, M., Robust performance hypothesis testing with the sharpe ratio, Journal of empirical finance, 15, 850-859, (2008)
[24] Markowitz, H., Portfolio selection, Journal of finance, 7, 77-91, (1952)
[25] McNeil, A.; Frey, R.; Embrechts, P., Quantitative risk management, (2005), Princeton University Press · Zbl 1089.91037
[26] Memmel, C., 2004. Schätzrisiken in der Portfoliotheorie. Ph.D. Thesis. University of Cologne. Department of Economic and Social Statistics. Germany.
[27] Merton, R., On estimating the expected return on the market: an exploratory investigation, Journal of financial economics, 8, 323-361, (1980)
[28] Okhrin, Y.; Schmid, W., Distributional properties of portfolio weights, Journal of econometrics, 134, 235-256, (2006) · Zbl 1420.91430
[29] Politis, D., The impact of bootstrap methods on time series analysis, Statistical science, 18, 219-230, (2003) · Zbl 1332.62340
[30] Politis, D.; Romano, J., A circular block-resampling procedure for stationary data, (), 263-270 · Zbl 0845.62036
[31] Press, S., Applied multivariate analysis, (2005), Dover Publications · Zbl 1191.62093
[32] Sharpe, W., A simplified model for portfolio analysis, Management science, 9, 277-293, (1963)
[33] Srivastava, M.; Bilodeau, M., Stein estimation under elliptical distributions, Journal of multivariate analysis, 28, 247-259, (1989) · Zbl 0667.62039
[34] Stein, C., Inadmissability of the usual estimator for the Mean of a multivariate normal distribution, (), 197-206
[35] Winston, K., The efficient index and prediction of portfolio variance, Journal of portfolio management, 19, 27-34, (1993)
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