Noureddine El Karoui, UC Berkeley
Abstract: Over the last few years, there has been some interest in the statistics community for questions involving large dimensional random matrices. The paradigm considered there can be used to ask questions about the behavior of
classical problems of applied mathematics, such as the Markowitz problem in Finance, in the high-dimensional setting where we have many assets and many observations.
I will discuss several aspects of this problem, and discuss questions such as efficient frontier estimation, correct risk assessment etc… A simple take-away message is that naive estimators lead to risk underestimation for Markowitz portfolios. The severity of the risk underestimation depends on the dimensionality of the problem.
Another important feature of the problem is the robustness of the results to various distributional assumptions. While much has been made of ``universality" results in random matrix theory, we will see that standard probabilistic assumptions impose very restrictive geometric conditions on the datasets of interest and that most results are not robust to a departure from these geometric assumptions. I will explain how to account for that, too, and to what extent we can circumvent that problem.
The talk is based on three (now old) papers including a joint work with Holger Koesters from Bielefeld (Germany).