Downside-risk averse investors and the use of Extreme Value Theory in optimal portfolio choice

The aim of this dissertation is to investigate the optimal portfolio selection problem for a risk-averse investor who wants to ameliorate the risk of extreme adverse events which cause unexpected large losses. Instead of referring to the modern portfolio theory (MPT), this thesis builds upon the foundations of Extreme Value Theory and focusses upon solving the optimisation problem minimizing quantile-based risk measures. This choice is justified by the fact that the two main assumptions of the mean-variance approach, normally distributed returns and the use of variance as risk measure, could underestimate the extremes in presence of heavy-tailed distributions, consequently leading to an optimistic asset allocation. We take into account the potential heavy-tailedness in portfolio choice applying EVT for modelling the tails of the returns and focus on quantile-based risk measures (Value at risk and Expected shortfall), which are more appropriate for safety-first investors whose primary goal is to guard against the occurrence of unexpected large losses. We also believe that this approach better reflects the risk-based framework introduced by the new regulatory requirements for banking and insurance sectors. The identified optimal portfolio is then compared with the one obtained under the classical MPT framework. Furthermore, an analysis to quantify and to understand how the extremal dependence between markets can impact the asset allocation problem is also performed. For this purpose two extremal dependence measures from Multivariate EVT are introduced. Intuitively, the fact that the securities can exhibit asymptotic independence or asymptotic dependence, means that tail diversification and a reduction of portfolio extreme risk can better be achieved by holding pairs of assets that are asymptotically independent. The methodology proposed in this thesis is applied to the construction of portfolios of major international equity indices satisfying downside risk requirements of different tightness. Measures of extremal dependence are used to refine intuition about resulting portfolio allocations as well as separately analysing each chosen index (right and left) tails.