Estimation of confidence level for Value-at-Risk: Statistical analysis

Abstract

The paper investigates the problem of estimation of the confidence level for Value-at-Risk to get the minimum VaR portfolio with a predefined level of expected return. The equation which describes the relation between the confidence level and the rate of the expected return depends on the unknown parameters of distribution of asset returns which should be estimated. The classical sample estimators for unknown parameters are used. The author has examined the properties of the estimator for the confidence level in considerable detail. Under the assumption that the asset returns are multivariate, we find the asymptotic distribution of the estimator for the confidence level. Moreover, we extend this result to the case of elliptically contoured distributed asset returns. Based on the distributional properties, the confidence interval for the confidence level for VaR is constructed and the test procedure whether the resulting portfolio is statistically different from the global minimum variance portfolio is provided. Using a simulation study, we demonstrate that our results give a good approximation even in the case of moderate sample sizes n=250, n=500 not only in the case of normally distributed asset returns, but also when asset returns follow the elliptically countered distribution. We have concluded that investors can use the results of the paper with regard to all sectors of the economy. We used monthly asset returns of five stocks included into Dow Jones Index, namely: McDonald's, Johnson&Johnson, Procter&Gamble, AT&T, and Verizon Communications from 01 October 2010 to 01 September 2015 to give numerical illustration of our fundamental results.