# American Institute of Mathematical Sciences

October  2014, 10(4): 1109-1127. doi: 10.3934/jimo.2014.10.1109

## CVaR proxies for minimizing scenario-based Value-at-Risk

Received  September 2012 Revised  October 2013 Published  February 2014

Minimizing VaR, as estimated from a set of scenarios, is a difficult integer programming problem. Solving the problem to optimality may demand using only a small number of scenarios, which leads to poor out-of-sample performance. A simple alternative is to minimize CVaR for several different quantile levels and then to select the optimized portfolio with the best out-of-sample VaR. We show that this approach is both practical and effective, outperforming integer programming and an existing VaR minimization heuristic. The CVaR quantile level acts as a regularization parameter and, therefore, its ideal value depends on the number of scenarios and other problem characteristics.
Citation: Helmut Mausser, Oleksandr Romanko. CVaR proxies for minimizing scenario-based Value-at-Risk. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1109-1127. doi: 10.3934/jimo.2014.10.1109
##### References:
 [1] C. Acerbi and D. Tasche, Expected Shortfall: A natural coherent alternative to Value at Risk, Economic Notes, 31 (2002), 379-388. doi: 10.1111/1468-0300.00091. [2] C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking & Finance, 26 (2002), 1487-1503. doi: 10.1016/S0378-4266(02)00283-2. [3] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A First Course in Order Statistics, SIAM Publishers, Philadelphia, PA, 2008. doi: 10.1137/1.9780898719062. [4] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068. [5] V. Brazauskas, B. L. Jones, M. L. Puri and R. Zitikis, Estimating conditional tail expectation with actuarial applications in view, Journal of Statistical Planning and Inference, 138 (2008), 3590-3604. doi: 10.1016/j.jspi.2005.11.011. [6] J. Daníelsson, B. N. Jorgensen, G. Samorodnitsky, M. Sarma and C. G. de Vries, Subadditivity re-examined: The case for Value-at-Risk, preprint, London School of Economics, 2005. [7] V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. doi: 10.1287/mnsc.1080.0986. [8] K. Høyland, M. Kaut and S. W. Wallace, A heuristic for moment-matching scenario generation, Computational Optimization and Applications, 24 (2003), 169-185. doi: 10.1023/A:1021853807313. [9] N. Larsen, H. Mausser and S. Uryasev, Algorithms for optimization of Value-at-Risk, in Financial Engineering, E-commerce and Supply Chain (eds. P. Pardalos and V. K. Tsitsiringos), Kluwer Academic Publishers, Norwell, MA, (2002), 19-46. doi: 10.1007/978-1-4757-5226-7_2. [10] J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM Journal on Optimization, 19 (2008), 674-699. doi: 10.1137/070702928. [11] H. Mausser and O. Romanko, Bias, exploitation and proxies in scenario-based risk minimization, Optimization, 61 (2012), 1191-1219. doi: 10.1080/02331934.2012.684795. [12] H. Mausser and D. Rosen, Efficient risk/return frontiers for credit risk, Journal of Risk Finance, 2 (2000), 66-78. doi: 10.1108/eb022948. [13] K. Natarajan, D. Pachamanova and M. Sim, Incorporating asymmetric distribution information in robust value-at-risk optimization, Management Science, 54 (2008), 573-585. doi: 10.1287/mnsc.1070.0769. [14] A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996. doi: 10.1137/050622328. [15] B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Computational study of a chance constrained portfolio selection problem, Optimization Online, 2008. Available from: http://www.optimization-online.org/DB_HTML/2008/02/1899.html. [16] B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Sample average approximation method for chance constrained programming: Theory and applications, Journal of Optimization Theory and Applications, 142 (2009), 399-416. doi: 10.1007/s10957-009-9523-6. [17] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. [18] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6. [19] D. Wuertz and H. Katzgraber, Precise Finite-Sample Quantiles of the Jarque-Bera Adjusted Lagrange Multiplier Test, MPRA Paper No. 19155, University of Munich, 2009. Available from: http://mpra.ub.uni-muenchen.de/19155/.

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##### References:
 [1] C. Acerbi and D. Tasche, Expected Shortfall: A natural coherent alternative to Value at Risk, Economic Notes, 31 (2002), 379-388. doi: 10.1111/1468-0300.00091. [2] C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking & Finance, 26 (2002), 1487-1503. doi: 10.1016/S0378-4266(02)00283-2. [3] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A First Course in Order Statistics, SIAM Publishers, Philadelphia, PA, 2008. doi: 10.1137/1.9780898719062. [4] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068. [5] V. Brazauskas, B. L. Jones, M. L. Puri and R. Zitikis, Estimating conditional tail expectation with actuarial applications in view, Journal of Statistical Planning and Inference, 138 (2008), 3590-3604. doi: 10.1016/j.jspi.2005.11.011. [6] J. Daníelsson, B. N. Jorgensen, G. Samorodnitsky, M. Sarma and C. G. de Vries, Subadditivity re-examined: The case for Value-at-Risk, preprint, London School of Economics, 2005. [7] V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. doi: 10.1287/mnsc.1080.0986. [8] K. Høyland, M. Kaut and S. W. Wallace, A heuristic for moment-matching scenario generation, Computational Optimization and Applications, 24 (2003), 169-185. doi: 10.1023/A:1021853807313. [9] N. Larsen, H. Mausser and S. Uryasev, Algorithms for optimization of Value-at-Risk, in Financial Engineering, E-commerce and Supply Chain (eds. P. Pardalos and V. K. Tsitsiringos), Kluwer Academic Publishers, Norwell, MA, (2002), 19-46. doi: 10.1007/978-1-4757-5226-7_2. [10] J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM Journal on Optimization, 19 (2008), 674-699. doi: 10.1137/070702928. [11] H. Mausser and O. Romanko, Bias, exploitation and proxies in scenario-based risk minimization, Optimization, 61 (2012), 1191-1219. doi: 10.1080/02331934.2012.684795. [12] H. Mausser and D. Rosen, Efficient risk/return frontiers for credit risk, Journal of Risk Finance, 2 (2000), 66-78. doi: 10.1108/eb022948. [13] K. Natarajan, D. Pachamanova and M. Sim, Incorporating asymmetric distribution information in robust value-at-risk optimization, Management Science, 54 (2008), 573-585. doi: 10.1287/mnsc.1070.0769. [14] A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996. doi: 10.1137/050622328. [15] B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Computational study of a chance constrained portfolio selection problem, Optimization Online, 2008. Available from: http://www.optimization-online.org/DB_HTML/2008/02/1899.html. [16] B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Sample average approximation method for chance constrained programming: Theory and applications, Journal of Optimization Theory and Applications, 142 (2009), 399-416. doi: 10.1007/s10957-009-9523-6. [17] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. [18] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6. [19] D. Wuertz and H. Katzgraber, Precise Finite-Sample Quantiles of the Jarque-Bera Adjusted Lagrange Multiplier Test, MPRA Paper No. 19155, University of Munich, 2009. Available from: http://mpra.ub.uni-muenchen.de/19155/.
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