Skewness and kurtosis in portfolio analysis: modelling, estimation and test theory

The aim of this project is to make a series of theoretical and empirical contributions in the estimation of optimal portfolio weights and their risk measures under non-Gaussian stochastic models. First, we are planning to work with the mean-variance optimal portfolios derived by Harry Markowitz. The aim is to obtain the exact/asymptotic distributions of their estimated weights as well as of their estimated main characteristics, like the expected return and the variance, under the assumption that asset returns follow a skewed heavy-tailed stochastic model.

This results will be used to investigate the impact of the skewness and kurtosis on the sample distribution of optimal portfolio weights and on the corresponding testing procedure. Similar tasks will be also treated in the case of the estimated three parameters of the e_cient frontier, the set of all optimal portfolios in the mean-variance space. These finding will allow us to characterize the behaviour of all estimated optimal portfolio by analysing only three statistics.

Second, we consider more sophisticated portfolio choice problems which are based on minimizing the Value-at-risk and the conditional Value-at-Risk of a portfolio. These two risk measures have increased their popularity recently and are recommended by the Basel Committee on Banking Supervision. In this project, we are planning (i) to obtain the expression of the weights of the optimal portfolios which minimizes the portfolio Value-at-risk and the portfolio conditional Value-at-Risk under the assumption that the asset return follow a skewed and heavy-tailed distribution; (ii) provide their estimators; and (iii) to derive the exact/asymptotic distribution of the estimated weights and risk measures. We further plan to study the impact of heavy tails, skewness, and time dependency on the estimation of optimal portfolio weights and the corresponding risk measures in the stochastic model with return predictability. The theoretical results of the project will be intensively investigated via simulations and applied to real data.