Probabilistic Analysis of Beta-Coefficient Sample Estimator for Minimum Value-at-Risk Portfolio

Abstract

The problems of financial assets portfolio beta-coefficient diversification and evaluation play an important role among aspects of risk management in the financial institutions activities. The paper is dedicated to statistical properties of the sample estimator of the financial assets minimum Value-at-Risk portfolio beta-coefficient. We assume that the weights of the portfolio benchmark are constant and the vector of portfolio assets returns is multivariate normally distributed. We find an analytical expression for calculating the beta-coefficient if the parameters of the portfolio assets returns distribution vector are known. In practice, these parameters are usually unknown, so we make use of distribution parameters sample estimators. By replacing the unknown parameters in the expression for calculating the beta-coefficient with their sample estimators, we obtain a sample estimator of the beta-coefficient, which is a random variable. We provide an asymptotic distribution of this sample estimator. We investigate the rate of empirical distributions convergence to asymptotic ones using simulation modeling with 100,000 repetitions. We take the corresponding sample estimators as the precise value of the distribution parameters of the portfolio assets returns vector. These estimators are obtained from the data on daily stock prices of companies included in the Dow Jones list for the period from 01.07.2020 to 11.01.2022. Six portfolios with different dimensions ({5, 10, 15, 20, 25, 30} of the first companies from the Dow Jones list sorted in alphabetically order) are considered. We select the equally weighted portfolio for the benchmark portfolio. It is noted that the beta-coefficient sample estimator is significantly biased and the bias increases with the number of assets in portfolio. We propose a corrected estimator whose bias is significantly smaller and does not depend on the number of assets in portfolio. It is concluded that the obtained asymptotic results can be used in practice for modeling the behavior of the sample estimator of the beta-coefficient in the presence of moderate dimension historical data samples (500-1000 observations).