The use of risk measures and its applications in portfolio optimisation

Abstract

In this dissertation, we study the application of risk measures to portfolio optimisation. A risk measure is a functional over the set of random portfolio returns mappings . We present the various risk measures in this dissertation within an axiomatic framework. Although Value-at-Risk (VaR) has been widely used, the Conditional-Value-at-Risk (CVaR) has become the more popular risk measure since it is a coherent and convex risk measure. We solve a CVaR based optimisation model that is used for portfolio optimisation and hedging a target portfolio. Additionally, we solve a CVaR based optimisation model with cost considerations included in the objective function. Further, we include alternative risk measures such as distortion, spectral, drawdown and coherent-distortion risk measures (CDRM) and develop optimisation problems for each risk measure as either the objective function or as a constraint in a linear programming problem. Since the 2008 crisis era, it has become important to note the universal agreement that financial assets have fat tails and that financial and investment managers must be able to account for it in their risk management strategies. We present fat-tail analysis for CVaR optimisation problems and perfom emperical risk analysis on the FTSE/JSE ALSI index.