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

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

Helmut Mausser 1, and  Oleksandr Romanko 1,

1. 

Quantitative Research, Risk Analytics, Business Analytics, IBM, 185 Spadina Avenue, Toronto, ON M5T2C6, Canada, Canada

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.
Keywords: optimization, Value-at-risk, regularization., conditional value-at-risk.
Mathematics Subject Classification: Primary: 91G10, 91G60, 91B30, 90C05; Secondary: 90C20, 90C11, 65K0.
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/.

show all references

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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