×

Varying confidence levels for CVaR risk measures and minimax limits. (English) Zbl 1436.90092

Summary: Conditional value at risk (CVaR) has been widely studied as a risk measure. In this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with CVaR appearing in the objective and also the constraints. We start by considering a problem in which CVaR is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one. We make use of a consistent tail condition which ensures that the CVaR of a random function will converge uniformly to its supremum as the confidence level increases, and establish an error bound for the CVaR optimal solution under second order growth conditions. The results are extended to a minimization problem with a constraint on the CVaR value which in the limit as the confidence level approaches one coincides with a problem having semi-infinite constraints. We study the sample average approximation scheme for the CVaR constraints and establish an exponential rate of convergence for the sample averaged optimal solution. We propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to CVaR-constrained problems out of sample. Our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance.

MSC:

90C15 Stochastic programming
90C17 Robustness in mathematical programming
65K05 Numerical mathematical programming methods
91B05 Risk models (general)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Basel Committee on Banking Supervision, Fundamental review of the trading book: A revised market risk framework, Bank for International Settlements (2013)
[2] Ben-Tal, A.; El Ghaoui, L.; Nemirovski, A., Robust Optimization (2009), Princeton: Princeton University Press, Princeton · Zbl 1221.90001
[3] Polak, E., On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Rev., 29, 21-89 (1987)
[4] Soyster, Al, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21, 1154-1157 (1973) · Zbl 0266.90046
[5] Scaillet, O., Nonparametric estimation and sensitivity analysis of expected shortfall, Math. Financ., 14, 115-129 (2004) · Zbl 1097.91049
[6] Embrechts, P.; Klueppelberg, C.; Mikosch, T., Modeling Extremal Events for Insurance and Finance (1997), Berlin: Springer, Berlin · Zbl 0873.62116
[7] Chen, Sx, Nonparametric estimation of expected shortfall, J. Financ. Econ., 6, 87-107 (2008)
[8] Calafiore, G.; Campi, Mc, Uncertain convex programs: randomized solutions and confidence levels, Math. Program., 102, 25-46 (2005) · Zbl 1177.90317
[9] Ben-Tal, A.; Nemirovski, A., Robust truss topology design via semidefinite programming, SIAM J. Optim., 7, 991-1016 (1997) · Zbl 0899.90133
[10] Ben-Tal, A.; Nemirovski, A., Robust convex optimization, Math. Oper. Res., 23, 769-805 (1998) · Zbl 0977.90052
[11] El Ghaoui, L.; Lebret, H., Robust solutions to uncertain semidefinite programs, SIAM J. Optim., 9, 33-52 (1998) · Zbl 0960.93007
[12] Calafiore, G.; Campi, Mc, The scenario approach to robust control design, IEEE Trans. Autom. Control, 51, 742-753 (2006) · Zbl 1366.93457
[13] Calafiore, G., Random convex programs, SIAM J. Optim., 20, 3427-3464 (2010) · Zbl 1211.90168
[14] Campi, Mc; Garatti, S., The exact feasibility of randomized solutions of robust convex programs, SIAM J. Optim., 19, 1211-1230 (2008) · Zbl 1180.90235
[15] Campi, Mc; Garatti, S., A sampling-and-discarding approach to chance constrained optimization: feasibility and optimality, J. Optim. Theory Appl., 148, 257-280 (2010) · Zbl 1211.90146
[16] Vayanos, P.; Kuhn, D.; Rustem, B., A constraint sampling approach for multi-stage robust optimization, Automatica, 48, 459-471 (2012) · Zbl 1244.93097
[17] Rockafellar, Rt; Uryasev, S., Optimization of conditional value-at-risk, J. Risk, 2, 21-41 (2000)
[18] Pflug, G.; Römisch, W., Modeling, Measuring and Managing Risk (2007), Singapore: World Scientific, Singapore · Zbl 1153.91023
[19] Zhu, L.; Li, H., Asymptotic analysis of multivariate tail conditional expectations, N. Am. Actuar. J., 16, 350-363 (2012) · Zbl 1291.60108
[20] Rockafellar, Rt; Wets, Rj-B, Variational Analysis (1998), Berlin: Springer, Berlin · Zbl 0888.49001
[21] Rockafellar, Rt; Uryasev, S., Conditional value-at-risk for general loss distributions, J. Bank. Financ., 26, 1443-1471 (2002)
[22] Ogryczak, W.; Ruszczynski, A., Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13, 60-78 (2002) · Zbl 1022.91017
[23] Guo, S.; Xu, H.; Zhang, L., Convergence analysis for mathematical programs with distributionally robust chance constraint, SIAM J. Optim., 27, 784-816 (2017) · Zbl 1471.90101
[24] Robinson, Sm, An application of error bounds for convex programming in a linear space, SIAM J. Control Optim., 13, 271-273 (1975) · Zbl 0297.90072
[25] Shapiro, A.; Xu, H., Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation, Optimization, 57, 395-418 (2008) · Zbl 1145.90047
[26] Klatte, D.: A note on quantitative stability results in nonlinear optimization. Seminarbericht Nr. 90, Sektion Mathematik, Humboldt-Universität zu, pp. 77-86. Berlin, Berlin (1987) · Zbl 0636.90082
[27] Heyde, Cc; Kou, Sg, On the controversy over tailweight of distributions, Oper. Res. Lett., 32, 399-408 (2004) · Zbl 1075.62039
[28] Lim, Ae; Shanthikumar, Jg; Vahn, G., Conditional value-at-risk in portfolio optimization: coherent but fragile, Oper. Res. Lett., 39, 163-171 (2011) · Zbl 1219.91130
[29] Mcneil, Aj; Frey, R.; Embrechts, P., Quantitative Risk Management (2005), Princeton: Princeton University Press, Princeton · Zbl 1089.91037
[30] Brazauskas, V.; Jones, Bl; Puri, Ml; Zitikis, R., Estimating conditional tail expectation with actuarial applications in view, J. Stat. Plan. Inference, 138, 3590-3604 (2008) · Zbl 1152.62027
[31] Deme, Eh; Girard, S.; Guillou, A.; Hallin, M.; Mason, D.; Pfeifer, D.; Steinebach, J., Reduced-bias estimator of the conditional tail expectation of heavy-tailed distributions, Mathematical Statistics and Limit Theorems (2015), New York: Springer, New York
[32] Necir, A.; Rassoul, A.; Zitikis, R., Estimating the conditional tail expectation in the case of heavy-tailed losses, J. Probab. Stat., 2010, 1-17 (2010) · Zbl 1200.91142
[33] Ballestin, F.; Leus, R., Resource-constrained project scheduling for timely project completion with stochastic activity durations, Prod. Oper. Manag., 18, 459-474 (2009)
[34] Laslo, Z., Activity time-cost tradeoffs under time and cost chance constraints, Comput. Ind. Eng., 44, 365-384 (2003)
[35] Abourizk, Sm; Halpin, Dw, Statistical properties of construction duration data, J. Constr. Eng. Manag., 118, 525-543 (1992)
[36] Johnson, D., The triangular distribution as a proxy for the beta distribution in risk analisis, J. R. Stat. Soc. Ser. Stat., 46, 387-398 (1997)
[37] Zhu, S.; Fukushima, M., Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57, 1155-1168 (2009) · Zbl 1233.91254
[38] Wozabal, D., Robustifying convex risk measures for linear portfolios: a nonparametric approach, Oper. Res., 62, 1302-1315 (2014) · Zbl 1358.91116
[39] Karoui, N.E., Lim, A.E., Vahn, G.Y.: Performance-based regularization in mean-CVaR portfolio optimization. arXiv preprint arXiv:1111.2091, (2011)
[40] Markowitz, H.; Usmen, N., The likelihood of various stock market return distributions, Part 1: principles of inference, J. Risk Uncertain., 13, 207-219 (1996) · Zbl 0876.90017
[41] Markowitz, H.; Usmen, N., The likelihood of various stock market return distributions, Part 2: empirical results, J. Risk Uncertain., 13, 221-247 (1996) · Zbl 0876.90018
[42] Hu, W.; Kercheval, An, Portfolio optimization for student \(t\) and skewed \(t\) returns, Quant. Financ., 10, 91-105 (2010) · Zbl 1198.91191
[43] Platen, E.; Sidorowicz, R., Empirical evidence on student t log-returns of diversified world stock indices, J. Stat. Theory Pract., 2, 233-251 (2008) · Zbl 1427.62122
[44] Krokhmal, P.; Palmquist, J.; Uryasev, S., Portfolio optimization with conditional value-at-risk objective and constraints, J. Risk, 4, 43-68 (2002)
[45] Xu, H., Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, J. Math. Anal. Appl., 368, 692-710 (2010) · Zbl 1196.90089
[46] Tiba, D.; Zalinescu, C., On the necessity of some constraint qualification conditions in convex programming, J. Convex Anal., 11, 95-110 (2004) · Zbl 1082.49028
[47] Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K., Nonlinear Parametric Optimization (1982), Berlin: Akademine, Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.