Abstract
We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, portfolio optimization is ill-posed because it leads to unbounded positions, infinitely short on some assets and infinitely long on others. As first observed by Kondor and coworkers, this phenomenon is actually a phase transition. We investigate the nature of this transition by means of a replica approach.
Acknowledgements
We thank O.C. Martin and M. Potters for useful discussions, and particularly J.P. Bouchaud for critical reading of the manuscript. S.C. is supported by the EC through the network MTR 2002-00319, STIPCO, and I.K. by the National Office of Research and Technology under grant No. KCKHA005.
Notes
§See Acerbi and Tasche (Citation2002) for the subtleties related to a discrete distribution.