# Erik Thorsén: Assessment of the uncertainty in small and large dimensional portfolio allocation

**Time: **
Wed 2019-12-11 15.15

**Location: **
Kräftriket, house 6, room 306 (Cramér-rummet)

**Doctoral student: **
Erik Thorsén
, Stockholms universitet

**Opponent: **
Thomas Holgersson, Linnaeus University, School of Business and Economics , Department of Economics and Statistics

### Abstract

Portfolio theory is a large subject with many branches. In this thesis we concern ourselves with one of these, the precense of uncertainty in the portfolio allocation problem and in turn, what it leads to. There are many forms of uncertainty, we consider two of these. The first being the optimization problem itself and optimizing what might be the wrong objective. In the classical mean-variance portfolio problem we aim to provide a portfolio with the smallest risk while we constrain the mean. However, in practice we might not assign a fixed portfolio goal but assign probabilities to the amount of return a portfolio might give and its relation to benchmarks. That is, we assign quantiles of the portfolio return distribution. In this scenario, the use of the portfolio mean as a return measure could be misleading. It does not take any quantile into account! In the first paper, we exchange the portfolio moments to quantile-based measures in the portfolio selection problem. The properties of the quantile-based portfolio selection problem is thereafter investigated with two different (quantile-based) measures of risk. We also present a closed form solution under the assumption that the returns follow an elliptical distribution. In this specific case the portfolio is shown to be mean-variance efficient.

The second paper takes on a different type of uncertainty which is classic to statistics, the problem of estimation uncertainty. We consider the sample estimators of the mean vector and of the covariance matrix of the asset returns and integrate the uncertainty these provide into a large class of optimal portfolios. We derive the sampling distribution, of the estimated optimal portfolio weights, which are obtained in both small and large dimensions. This consists of deriving the joint distribution of several quantities and thereafter specifying their high dimensional asymptotic distribution.