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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996. doi: 10.1137/050622328.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. Google Scholar

[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.  Google Scholar

[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/. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2006), 969-996. doi: 10.1137/050622328.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. Google Scholar

[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.  Google Scholar

[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/. Google Scholar

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