Construction of uncertainty sets for portfolio selection problems
Research, Preprint
OPEN
Wiechers, Christof
(2011)
 Publisher: Cologne: University of Cologne, Seminar of Economic and Social Statistics

Subject:
C13  G11  G32  Portfolio Optimization  C18  Portfolio Optimization,Risk Constraints,Coherent Distortion Risk Measures,Uncertainty Sets  Risk Constraints  Coherent Distortion Risk Measures  Uncertainty Sets  C61
While modern portfolio theory grounds on the tradeoff between portfolio return and portfolio variance to determine the optimal investment decision, postmodern portfolio theory uses downside risk measures instead of the variance. Prominent examples are given by the risk measures ValueatRisk and its coherent extension, Conditional ValueatRisk. When avoiding distributional assumptions on the process that generates the risky assets' returns, historical return data or expert knowledge remain the only data available to the investor. His problem is then to maximize the return of his portfolio given the risk constraint that his portfolio does not fall short of some threshold return. For the Conditional ValueatRisk, the solution is known to be achievable by a linear program. This paper extends the solution to the investor's problem whenever his risk preferences are given by any coherent distortion risk measure. More precisely, it is shown that whenever the risk constraint is given by a coherent distortion risk measure, a linear program leads to the solution. A geometric interpretation of this solution is immediate, which is related to the nonparametric description of data by socalled weightedmean trimmed regions. The connections of the solution to robust optimization and decision theory are illustrated.