Risk - reward optimization; illustrated with Conditional Value-at-Risk

Abstract

An important aspect in portfolio optimization is the quantification of risk. Variance was the starting point, as proposed by Harry Markowitz in the 1950’s, but it’s obviously flawed since it measures high returns as risk. Re- search have been done from both theoretical and logic point of view to im- prove risk measures. The result is two different groups of measures defined by axioms: (financial) deviation measures capturing the uncertainty, and risk measures which attempts to measure total exposure. I will in this thesis present an overview of new ideas about measurement of risk. I focus especially on the work of Ralph T. Rockafellar and Stanislav Uryasev. This include Conditional Value-at-risk (CVaR), a measure that fulfills the axioms for a risk measure, and has the possibility to be solved by linear programming in an optimization model. I will also present the connected CVaR deviation. For even if risk and deviation measures are conceptual different, newer research shows an one to one relationship given certain conditions. Lastly I compare mean-CVaR optimization with the traditional Markowitz model1 to see if new methods actually has improved the performance of port- folio optimization. This is illustrated by optimizing the stocks in the S&P100 index for two different periods. I have also timed different solvers to show the benefits of linearization.