Properties of the beta coefficient of the global minimum variance portfolio

2020 ◽  
Vol 8 (1) ◽  
pp. 11-21
Author(s):  
S. M. Yaroshko ◽  
◽  
M. V. Zabolotskyy ◽  
T. M. Zabolotskyy ◽  
◽  
...  
Keyword(s):  
Confidence Interval ◽  
Asymptotic Distribution ◽  
Global Minimum ◽  
Asset Returns ◽  
Minimum Variance ◽  
Benchmark Portfolio ◽  
Beta Coefficient ◽  
Minimum Variance Portfolio ◽  
Global Minimum Variance Portfolio ◽  
Daily Returns

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.

Mean-Variance Analysis

2017 ◽  
Author(s):  
Kerry E. Back
Keyword(s):  
Global Minimum ◽  
Minimum Variance ◽  
Variance Analysis ◽  
Affine Function ◽  
Market Portfolio ◽  
Minimum Variance Portfolio ◽  
The Mean ◽  
Global Minimum Variance Portfolio ◽  
Mean Variance

The mean‐variance frontier is characterized with and without a risk‐free asset. The global minimum variance portfolio and tangency portfolio are defined, and two‐fund spanning is explained. The frontier is characterized in terms of the return defined from the SDF that is in the span of the assets. This is related to the Hansen‐Jagannathan bound. There is an SDF that is an affine function of a return if and only if the return is on the mean‐variance frontier. Separating distributions are defined and shown to imply two‐fund separation and mean‐variance efficiency of the market portfolio.

scholarly journals An analysis of a mean-variance enhanced index tracking problem with weights constraints

2018 ◽  
Vol 15 (4) ◽  
pp. 183-192
Author(s):  
Wanderlei Lima de Paulo ◽  
Marta Ines Velazco Fontova ◽  
Renato Canil de Souza
Keyword(s):  
Asset Allocation ◽  
Global Minimum ◽  
Minimum Variance ◽  
Index Tracking ◽  
Tracking Problem ◽  
Market Index ◽  
Satisfactory Performance ◽  
Minimum Variance Portfolio ◽  
Global Minimum Variance Portfolio ◽  
Mean Variance

In this paper, the authors deal with a mean-variance enhanced index tracking (EIT) problem with weights constraints. Using a shrinkage approach, they show that constructing the constrained EIT portfolio is equivalent to constructing the unconstrained EIT portfolio. This equivalence allows to study the effect of weights constraints on the covariance matrix and on the EIT portfolio. In general, the effects of weights constraints on the EIT portfolio are different from those observed in the case of global minimum variance portfolio. Finally, the authors present a numerical asset allocation example, where the S&P 500 index is used as the market index to be tracked using a portfolio composed of ten stocks, in which the constrained EIT portfolio shows a satisfactory performance when compared to the unconstrained case.

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