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January  2013, 9(1): 191-204. doi: 10.3934/jimo.2013.9.191

## Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk

 1 Centre for Industrial and Applied Mathematics, University of South Australia, Mawson Lakes, SA 5095, Australia 2 CSIRO Mathematics, Informatics and Statistics, North Ryde, NSW, Australia

Received  November 2011 Revised  June 2012 Published  December 2012

A problem of minimization of $L_1$-penalized conditional value-at-risk (CVaR) is considered. It is shown that there exists a non-negative threshold value of the penalty parameter such that the optimal value of the penalized problem is unbounded if the penalty parameter is less than the threshold value, and it is bounded if the penalty parameter is greater or equal than this value. It is established that the threshold value can be found via the solution of a linear programming problem, and, therefore, readily computable. Theoretical results are illustrated by numerical examples.
Citation: Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial and Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191
##### References:
 [1] S. Alexander, T. F. Coleman and Y. Li, Derivative portfolio hedging based on CVaR, in "Risk Measures for the 21st Century" (eds. G. Szego), London: Wiley, (2004), 339-363. [2] S. Alexander, T. F. Coleman and Y. Li, Minimizing CVaR and VaR for a portfolio of derivatives, Journal of Banking and Finance, 30 (2006), 583-605. doi: 10.1016/j.jbankfin.2005.04.012. [3] K. A. Boyle, T. F. Coleman and Y. Li, Hedging a portfolio of derivatives by modeling cost, in "IEEE Proceedings of the 2003 International Conference on Computational Intelligence for Financial Engineering (CIFE 2003)". [4] Z. G. Cao, R. D. F. Harris and J. Shen, Hedging and value at risk: A semi-parametric approach, Journal of Futures Markets, 30 (2010), 780-794. [5] G. B. Dantzig, "Linear Programming and Extensions," Princeton: Princeton University Press, 1963. [6] C. I. Fabian, Handling CVaR objectives and constraints in two-stage stochastic models, European Journal of Operational Research, 191 (2008), 888-911. doi: 10.1016/j.ejor.2007.02.052. [7] R. D. F. Harris and J. Shen, Hedging and value at risk, Journal of Futures Markets, 26 (2006), 369-390. doi: 10.1002/fut.20195. [8] D. Huang, S. Zhu, F. J Fabozzi and M. Fukushima, Portfolio selelction under distributional uncertainty: a relative robust CVaR approach, European Journal of Operational Research, 203 (2010), 185-194. doi: 10.1016/j.ejor.2009.07.010. [9] H. Mausser and D. Rosen, Beyond VaR: From measuring risk to managing risk, ALGO Research Quarterly, 1 (1998), 5-20. [10] R. T. Rockafellar and S. Uryasev, Optimization of conditional value at risk, Journal of Risk, 2 (2000), 21-41. [11] R. T. Rockafellar and S. Uryasev, Conditional value at risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6. [12] K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, Journal of Industrial and Management Optimization, 8 (2012), 343-362. doi: 10.3934/jimo.2012.8.343. [13] T. Tarnopolskaya, J. Tabak and F. R. de Hoog, L-curve for hedging instrument selection in CVaR-minimising portfolio hedging, in "18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation" (eds. R. S. Anderssen et al), (2009), 1559-1565. [14] N. Topaloglou, H. Vladimirou and S. A. Zenios, CVaR models with selective hedging for international asset allocation, Journal of Banking and Finance, 26 (2002), 1535-1561. doi: 10.1016/S0378-4266(02)00289-3. [15] S. P. Uryasev and R. T. Rockafellar, Conditional value-at-risk: Optimization approach, Stochastic Optimization: Algorithms and Applications, 54 (2001), 411-435.

show all references

##### References:
 [1] S. Alexander, T. F. Coleman and Y. Li, Derivative portfolio hedging based on CVaR, in "Risk Measures for the 21st Century" (eds. G. Szego), London: Wiley, (2004), 339-363. [2] S. Alexander, T. F. Coleman and Y. Li, Minimizing CVaR and VaR for a portfolio of derivatives, Journal of Banking and Finance, 30 (2006), 583-605. doi: 10.1016/j.jbankfin.2005.04.012. [3] K. A. Boyle, T. F. Coleman and Y. Li, Hedging a portfolio of derivatives by modeling cost, in "IEEE Proceedings of the 2003 International Conference on Computational Intelligence for Financial Engineering (CIFE 2003)". [4] Z. G. Cao, R. D. F. Harris and J. Shen, Hedging and value at risk: A semi-parametric approach, Journal of Futures Markets, 30 (2010), 780-794. [5] G. B. Dantzig, "Linear Programming and Extensions," Princeton: Princeton University Press, 1963. [6] C. I. Fabian, Handling CVaR objectives and constraints in two-stage stochastic models, European Journal of Operational Research, 191 (2008), 888-911. doi: 10.1016/j.ejor.2007.02.052. [7] R. D. F. Harris and J. Shen, Hedging and value at risk, Journal of Futures Markets, 26 (2006), 369-390. doi: 10.1002/fut.20195. [8] D. Huang, S. Zhu, F. J Fabozzi and M. Fukushima, Portfolio selelction under distributional uncertainty: a relative robust CVaR approach, European Journal of Operational Research, 203 (2010), 185-194. doi: 10.1016/j.ejor.2009.07.010. [9] H. Mausser and D. Rosen, Beyond VaR: From measuring risk to managing risk, ALGO Research Quarterly, 1 (1998), 5-20. [10] R. T. Rockafellar and S. Uryasev, Optimization of conditional value at risk, Journal of Risk, 2 (2000), 21-41. [11] R. T. Rockafellar and S. Uryasev, Conditional value at risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6. [12] K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, Journal of Industrial and Management Optimization, 8 (2012), 343-362. doi: 10.3934/jimo.2012.8.343. [13] T. Tarnopolskaya, J. Tabak and F. R. de Hoog, L-curve for hedging instrument selection in CVaR-minimising portfolio hedging, in "18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation" (eds. R. S. Anderssen et al), (2009), 1559-1565. [14] N. Topaloglou, H. Vladimirou and S. A. Zenios, CVaR models with selective hedging for international asset allocation, Journal of Banking and Finance, 26 (2002), 1535-1561. doi: 10.1016/S0378-4266(02)00289-3. [15] S. P. Uryasev and R. T. Rockafellar, Conditional value-at-risk: Optimization approach, Stochastic Optimization: Algorithms and Applications, 54 (2001), 411-435.
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