Properties of the beta coefficient of the global minimum variance portfolio

: pp. 11–21
Received: January 16, 2020
Revised: August 28, 2020
Accepted: October 02, 2020
Lviv Polytechnic National University
Ivan Franko National University of Lviv
Ivan Franko National University of Lviv

The paper is devoted to the investigation of statistical properties of the sample estimator of the beta coefficient in the case when the weights of benchmark portfolio are constant and for the target portfolio, the global minimum variance portfolio is taken.  We provide the asymptotic distribution of the sample estimator of the beta coefficient assuming that the asset returns are multivariate normally distributed.  Based on the asymptotic distribution we construct the confidence interval for the beta coefficient.  We use the daily returns on the assets included in the DAX index for the period from 01.01.2018 to 30.09.2019 to compare empirical and asymptotic means, variances and densities of the standardized estimator for the beta coefficient.  We obtain that the bias of the sample estimator converges to zero very slowly for a large number of assets in the portfolio.  We present the adjusted estimator of the beta coefficient for which convergence of the empirical variances to the asymptotic ones is not significantly slower than for a sample estimator but the bias of the adjusted estimator is significantly smaller.

  1. Markowitz H.  Portfolio selection.  Journal of finance. 7, 77–91 (1952).
  2. Merton R. C.  An analytical derivation of the efficient frontier.  Journal of financial and quantitative analysis. 7 (4), 1851–1872 (1972).
  3. Okhrin Y., Schmid W.  Distributional properties of optimal portfolio weights.  Journal of econometrics. 134 (1), 235–256 (2006).
  4. Okhrin Y., Schmid W.  Estimation of optimal portfolio weights.  International journal of theoretical and applied finance. 11 (3), 249–276 (2008).
  5. Ingersoll J. E.  Theory of financial decision making.  New York, Rowman & Littlefield Publishers (1987).
  6. Sharpe W. F.  The Sharpe ratio.  The journal of portfolio management. 21 (1), 49–58 (1994).
  7. Schmid W., Zabolotskyy T.  On the existence of unbiased estimators for the portfolio weights obtained by maximizing the Sharpe ratio.   ASTA - Advances in Statistical Analysis. 92, 29–34 (2008).
  8. Bodnar T., Zabolotskyy T.  Maximization of the Sharpe ratio of an asset portfolio in the context of risk minimization.  Economic annals - ХХІ. 11-12 (1), 110–113 (2013).
  9. Bodnar T., Zabolotskyy T.  How risky is the optimal portfolio which maximizes the Sharpe ratio?  ASTA - Advances in statistical analysis. 101 (1), 1–28 (2017).
  10. Alexander G. J., Baptista M. A.  Economic implication of using a mean-VaR model for portfolio selection: a comparison with mean-variance analysis.  Journal of economic dynamics & control. 26 (7-8), 1159–1193 (2002).
  11. Alexander G. J., Baptista M. A.  A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model.  Management Science. 50 (9), 1261–1273 (2004).
  12. Bodnar T., Schmid W., Zabolotskyy T.  Minimum VaR and Minimum CVaR optimal portfolios: estimators, confidence regions, and tests.  Statistics & Risk Modeling. 29 (4), 281–314 (2012).
  13. Bodnar T., Schmid W.  A test for the weights of the global minimum variance portfolio in an elliptical model.  Metrika. 67, 12–143 (2008).
  14. Bodnar T., Mazur S., Okhrin Y.  Bayesian estimation of the global minimum variance portfolio.  European journal of operational research. 256 (1), 292–307 (2017).
  15. Kan R., Smith D. R.  The distribution of the sample minimum-variance frontier.  Management. 54 (7), 1364–1380 (2008).
  16. Chan L. K. C., Karceski J., Lakonishok J.  On portfolio optimization: forecasting and choosing the risk model.  The review of financial study. 12 (5), 937–974 (1999).
  17. Bodnar T., Schmid W.  Econometrical analysis of the sample efficient frontier.  The European journal of finance. 15 (3), 317–335 (2009).
  18. Bodnar T., Gupta A. K., Vitlinskiy V., Zabolotskyy T.  Statistical inference for the $\beta$ coefficient.  Risks. 7 (2), 56 (2019).
  19. Markowitz H.  Foundations of portfolio theory.  Journal of finance. 7, 469–477 (1991).
  20. Ling S., McAleer M.  Asymptotic theory for a vector ARMA-GARCH model.  Econometric theory. 19 (2), 280–310 (2003).
  21. Harville D. A.  Matrix algebra from a statistician's perspective.  New York, Springer Science+Business Media (2008).
  22. Brockwell P. J., Davis R. A.  Time series: theory and methods.  New York, Springer Science+Business Media (2006).
  23. DasGupta A.  Asymptotic theory of statistics and probability.  New York, Springer (2008).
  24. Muirhead R. J.  Aspects of multivariate statistical theory.  New York, Wiley (1982).
Mathematical Modeling and Computing, Vol. 8, No. 1, pp. 11–21 (2021)