Statistical Analysis of the Optimal Portfolio Weights and their Risk Measures

The problem of finding the distributions of the mean-variance portfolio weights and their risk measures were only discussed for the non-singular covariance of the vector of returns and when the sample size of assets is larger than the portfolio size. Our goal is to fill this gap and to provide results, when the small sample size and the (non-) singular covariance matrix are both present. One important reason for considering the singular covariance matrix case in the portfolio theory is that often in for a given set of assets, there maybe strong stochastic dependence between them. For example, valuation of assets within a specific industry branch often are highly correlated. If the dimension of portfolio is relatively large there is a possibility of (approximate) singularity and the problem needs to be addressed in the theory. Equally important aspect is the dynamic correlation which may lead to changing singularity at different time periods. It is well documented in the literature that the realized correlation can significantly vary over the long time one which may lead to from non-singular model to a singular one. It is then important to have methodology that can address all possible situations that can be encountered during asset reallocation over longer period of time. Thus the aim of this project is to provide an effective tools for a such asset allocation problems.