INSTABILITY OF PORTFOLIO OPTIMIZATION UNDER COHERENT RISK MEASURES Imre Kondor Collegium Budapest – Institute for Advanced Study, Szenth´aroms´ag u. 2, H-1014 Budapest, Hungary Department of Physics of Complex Systems, E¨otv¨os University, P´azm´ any P´ eter s´ et´any 1/A, H-1117 Budapest, Hungary Parmenides Center for the Study of Thinking, Kardinal Faulhaber Strasse 14a, Munich, D-80333, Germany email: kondor@colbud.hu Istv´ an Varga-Haszonits Department of Physics of Complex Systems, E¨otv¨os University, P´azm´ any P´ eter s´ et´any 1/A, H-1117 Budapest, Hungary Analytics Department of Fixed Income Division, Morgan Stanley Hungary Analytics, De´ak Ferenc u. 15, H-1052 Budapest, Hungary email: Istvan.Varga-Haszonits@morganstanley.com Abstract It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other one in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered on the special ex- ample of Expected Shortfall which is used here both as an illustration and as a springboard for generalization. Key Words: Coherent Risk Measures, Portfolio Optimization, Expected Shortfall, Estimation of Risk JEL Classification: G11, C13, D81