Statistical Analysis of Portfolio Characteristics for Different Risk Measures
Final Report Abstract
In this project, the problem of estimation risk was incorporated in portfolio strategies which are mainly based on minimizing the VaR (CVaR). The application of quantilebased risk measures possesses several advantages with respect to the usage of the variance. First, the quantile function contains all relevant information about the portfolio return. Even if a single quantile is used in practice, the portfolio selection procedure based on the VaR (CVaR) is more flexible than both the mean-variance analysis and the approach based on optimizing higher moments of the asset returns. The reason is that in contrast to the uniquely defined variance, the investor has the choice between several quantiles. This provides more flexibility for modeling the investor's attitude towards risk. Second, in contrast to the variance, the VaR and the CVaR are one-sided risk measures. While the variance is influenced by positive and negative values, large positive returns do not change the VaR and the CVaR. Third, since the recommendation of the VaR as a standard tool for banking supervision, the VaR is widely applied in practice. It has become very popular because of its simple and easy understandable representation of high losses. We considered the sample estimators for the weights, the expected return, the variance, the value-at-risk (VaR), and the conditional VaR (CVaR) of the minimum VaR and the minimum CVaR portfolios. The exact distributions of the estimators were derived assuming that the asset returns are independently and multivariate normally distributed, whereas the corresponding asymptotic results were obtained for stationary processes. These expressions were used for studying the distributional properties of the estimated characteristics. For example, we presented confidence regions for the nnnimum VaR portfolio and the minimum CVaR portfolio in the mean-variance space as well as in the mean-VaR (mean-CVaR) space. Furthermore, we studied the portfolio selection problem based on the VaR utility function when general linear constraints are imposed on the portfolio weights. Since in case of normally distributed asset returns, the application of the variance, the VaR, and the CVaR, as a risk measure, leads to the same portfolio allocation strategy, wo investigated the influence of the parameter uncertainty on the portfolio selection problems which are based on different quadratic utility functions. It turned out that under some conditions they are mathematically equivalent but not always mean-variance efficient. Conditions were derived under which the solutions of the considered quadratic optimization problems lay on the efficient frontier. These conditions, however, cannot be checked in a practical situation because they all depend on the unknown parameters of the asset returns. We provided a comprehensive treatment of this problem by deriving the exact expressions of the probabilities that the estimated solutions of the Markowitz problem and the one obtained by maximizing the quadratic utility are mean-variance efficient. Another important problem, which was treated in the project, is the influence of the violation of the normality assumption on the performance of optimal portfolios. It can happen that the application of the mean-variance analysis could lead to a portfolio which departs significantly from the optimal one if a large departure from the normal distribution is present. We derived the distribution theory for studying the performance of the estimated portfolio weights when the distribution of the asset returns is skewed. As a model for the sample of the asset returns, the matrix variate closed skew normal distribution was assumed. This distribution has already been applied for modeling the skewness in data as well as it does not assume that the data consists of independent observations and also allows for heavy tails in the data. Assuming that the sample of the asset returns follows a matrix variate closed skew normal distribution, we derived the exact distribution of the estimated GMV portfolio weights and the test for the equality of linear combinations of the GMV portfolio weights to a known vector of constants in the small sample case. These results were used to study the influence of skewness on the distributional properties of the estimated GMV portfolio weights. Finally, we provided a methodology for determining a bound on the risk aversion coefficient, which separates portfolios that are equivalent or significantly different from the GMV portfolio and concluded that investing in the GMV portfolio is statistically justified for investors with a very wide range of the risk aversion coefficients. The distribution of the estimated weights of the optimal portfolios from the efficient frontier were derived as well. The possible applications of the obtained results are diverse. They are of fundamental interest in practice and find a number of applications in many scientific disciplines.
Publications
- (2011). On the exact distribution of the estimated expected utility portfolio weights: Theory and applications. Statistics & Risk Modeling, 28, 319-342
Bodnar, T. and W. Schmid
- (2012). Minimum VaR and minimum CVaR optimal portfolios: estimators, confidence regions, and tests. Statistics & Risk Modeling, 29, 281-314
Bodnar, T., Schmid, W. and T. Zabolotskyy
- (2013). Boundaries of the risk aversion coefficient: Should we invest in the global minimum variance portfolio. Applied Mathematics and Computation, 219, 5440-5448
Bodnar, T. and Y. Oklirin
- (2013). On the equivalence of quadratic optimization problems commonly used in portfolio theory. European Journal of Operational Research, 229, 637-644
Bodnar, T., Parolya, N. and W. Schmid
