Non-smooth optimization methods for computation of the Conditional Value-at-risk and portfolio optimization

Abstract

We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters.

Keywords:

• Non-smooth optimization
• Conditional Value-at-risk
• Expected shortfall
• Portfolio optimization
• Numerical optimization

Keywords:

• Mathematics Subject Classifications 2000: 90C05
• 90C30
• 90C56
• 91B28

Acknowledgements

The authors wish to express their gratitude to J.E. Monsalve Tobon for the help in performing numerical experiments and acknowledge the partial support by the Victorian Partnership for Advanced Computing (VPAC), under e-Research scheme. The research by the second author was supported by the Australian Research Council. The authors would also like to thank the two anonymous referees whose valuable comments helped in improving this article.