Adjusting MV-efficient portfolio frontier bias for skewed and non-mesokurtic returns

This study investigates the bias adjustment for mean–variance efficient portfolio frontiers due to population mean and variance estimation error in Taiwan stock market. Although Siegel and Woodgate (2007; Management Science, 53, 1005–1015) and Kan and Smith (2008; Management Science, 54, 1364–1380) suggested two portfolio frontiers that improved upon the out-of-sample performance of a traditional sample portfolio frontier. However, this study shows that, using the copula function and Gram-Charlier series, the two frontiers are theoretically biased toward the actual frontier unless returns behave normally, and the bias is related to the return skewness and kurtosis. Indeed, the two frontiers are empirically biased to the lower-left side of the actual ones, because the Taiwan stock returns are right-skewed and highly leptokurtic. Thus, this study thus proposes revised portfolio frontiers that are closer to the actual frontier than unrevised ones. This improvement may enhance the estimation accuracy of the capital market line, and hence this study can provide an effective investment reference.

[1]  A. Siegel,et al.  Performance of Portfolios Optimized with Estimation Error , 2007, Manag. Sci..

[2]  Alex W. H. Chan.  Merton, Robert C. , 2010 .

[3]  Marno Verbeek,et al.  Selecting Copulas for Risk Management , 2007 .

[4]  G. Frankfurter,et al.  Portfolio Selection: The Effects of Uncertain Means, Variances, and Covariances , 1971, Journal of Financial and Quantitative Analysis.

[5]  H. Theil.  Introduction to econometrics , 1978 .

[6]  Peter A. Frost,et al.  An Empirical Bayes Approach to Efficient Portfolio Selection , 1986, Journal of Financial and Quantitative Analysis.

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

[8]  Chris Kirby,et al.  It’s All in the Timing: Simple Active Portfolio Strategies that Outperform Naïve Diversification , 2010, Journal of Financial and Quantitative Analysis.

[9]  Raymond Kan,et al.  The Distribution of the Sample Minimum-Variance Frontier , 2007, Manag. Sci..

[10]  Raymond Kan,et al.  Optimal Portfolio Choice with Parameter Uncertainty , 2007, Journal of Financial and Quantitative Analysis.

[11]  P. R. Fisk,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1971 .

[12]  Huei-Mei Liang,et al.  Estimating value at risk of portfolio by conditional copula-GARCH method , 2009 .

[13]  S. Rachev,et al.  Multivariate Skewed Student's t Copula in the Analysis of Nonlinear and Asymmetric Dependence in the German Equity Market , 2008 .

[14]  Y. Tse,et al.  Modeling the fat tails in Asian stock markets , 2011 .

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

[16]  M. Rockinger,et al.  The Copula-GARCH model of conditional dependencies: An international stock market application , 2006 .

[17]  Tie Su,et al.  SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES , 1996 .

[18]  Olivier Ledoit,et al.  Improved estimation of the covariance matrix of stock returns with an application to portfolio selection , 2003 .

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

[20]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[21]  Guofu Zhou,et al.  Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies ☆ , 2011 .

[22]  Matthias Fischer,et al.  An empirical analysis of multivariate copula models , 2009 .

[23]  William T. Ziemba,et al.  Covariance complexity and rates of return on assets. , 2007 .

[24]  Jinpeng Ma,et al.  Excess volatility and the cross-section of stock returns , 2014 .

[25]  Vijay S. Bawa,et al.  The effect of estimation risk on optimal portfolio choice , 1976 .

[26]  Andrew J. Patton.  Estimation of multivariate models for time series of possibly different lengths , 2006 .

[27]  Kehluh Wang,et al.  The dynamic dependence between the Chinese market and other international stock markets: A time-varying copula approach , 2011 .

[28]  Statistical properties of Taiwan Stock Index , 2007 .

[29]  Peter M. Clarkson,et al.  On the Diversification, Observability, and Measurement of Estimation Risk , 1996, Journal of Financial and Quantitative Analysis.

[30]  Hung-Hsi Huang,et al.  Portfolio selection and portfolio frontier with background risk , 2013 .

[31]  Chi-Fu Huang,et al.  Foundations for financial economics , 1988 .

[32]  K. Nam,et al.  Asymmetric and leptokurtic distribution for heteroscedastic asset returns: The SU-normal distribution , 2008 .

[33]  R. Jarrow,et al.  APPROXIMATE OPTION VALUATION FOR ARBITRARY STOCHASTIC PROCESSES , 1982 .

[34]  P. Embrechts,et al.  Correlation: Pitfalls and Alternatives , 1999 .

[35]  Andrew J. Patton,et al.  Estimation of Copula Models for Time Series of Possibly Different Lengths , 2001 .

[36]  Campbell R. Harvey,et al.  Emerging Equity Market Volatility , 1995 .

[37]  Yannick Malevergne,et al.  Testing the Gaussian copula hypothesis for financial assets dependences , 2001, cond-mat/0111310.

[38]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[39]  Philippe Jorion.  Bayes-Stein Estimation for Portfolio Analysis , 1986, Journal of Financial and Quantitative Analysis.

[40]  R. C. Merton,et al.  An Analytic Derivation of the Efficient Portfolio Frontier , 1972, Journal of Financial and Quantitative Analysis.

[41]  J. Rosenberg,et al.  A General Approach to Integrated Risk Management with Skewed, Fat-Tailed Risk , 2004 .