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November  2020, 16(6): 2581-2602. doi: 10.3934/jimo.2019071

Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk

Meng Xue 1, , Yun Shi 2, and  Hailin Sun 3,,

1. 

School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3. 

Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

* Corresponding author: Hailin Sun

Received  December 2017 Revised  March 2019 Published  July 2019

Fund Project: The work is supported by National Natural Science Foundation of China grant 11871276, 11571178 and 11571056

A portfolio optimization model with relaxed second order stochastic dominance (SSD) constraints is presented. The proposed model uses Conditional Value at Risk (CVaR) constraints at probability level $ \beta\in(0,1) $ to relax SSD constraints. The relaxation is justified by theoretical convergence results based on sample average approximation (SAA) method when sample size $ N\to\infty $ and CVaR probability level $ \beta $ tends to 1. SAA method is used to reduce infinite number of inequalities of SSD constraints to finite ones and also to calculate the expectation value. The proposed relaxation on the SSD constraints in portfolio optimization problem is achieved when the probability level $ \beta $ of CVaR takes value less than but close to 1, and the model can then be solved by cutting plane method. The performance and characteristics of the portfolios constructed by solving the proposed model are tested empirically on three sets of market data, and the experimental results are analyzed and discussed. Furthermore, it is shown that with appropriate choices of CVaR probability level $ \beta $, the constructed portfolios are sparse and outperform the portfolios constructed by solving portfolio optimization problems with SSD constraints, with either index portfolios or mean-variance (MV) portfolios as benchmarks.

Keywords: Stochastic optimization, portfolio selection, second order dominance, CVaR approximation, sample average approximation.
Mathematics Subject Classification: 91G10; 90C15.
Citation: Meng Xue, Yun Shi, Hailin Sun. Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2581-2602. doi: 10.3934/jimo.2019071
References:
[1]

G. J. Alexander and A. M. Baptista, A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model, Manag. Sci., 50 (2004), 1261-1273.   Google Scholar

[2]

E. Anderson, H. Xu and D. Zhang, Confidence levels for CVaR risk measures and minimax limits, manuscript, The University of Sydney, 2014. Available from: http://hdl.handle.net/2123/9943. Google Scholar

[3]

K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer Texts in Statistics, Springer, New York, 2006.  Google Scholar

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J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[6]

N. A. Canakgoz and J. E. Beasley, Mixed-integer programming approaches for index tracking and en-hanced indexation, European J. Oper. Res., 196 (2009), 384-399.  doi: 10.1016/j.ejor.2008.03.015.  Google Scholar

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D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim., 14 (2003), 548-566.  doi: 10.1137/S1052623402420528.  Google Scholar

[8]

D. Dentcheva and A. Ruszczyński, Semi-infinite probabilistic constraints: Optimality and convexification, Optimization, 53 (2004), 583-601.  doi: 10.1080/02331930412331327148.  Google Scholar

[9]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Math. Program., 99 (2004), 329-350.  doi: 10.1007/s10107-003-0453-z.  Google Scholar

[10]

D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Bank. Financ., 30 (2006), 433-451.  doi: 10.1137/S1052623402420528.  Google Scholar

[11]

D. Dentcheva and A. Ruszczyński, Inverse cutting plane methods for optimization problems with second-order stochastic dominance constraints, Optimization, 59 (2010), 323-338.  doi: 10.1080/02331931003696350.  Google Scholar

[12]

D. Dentcheva and A. Ruszczyński, Risk-averse portfolio optimization via stochastic dominance constraints, in Handbook of Quantitative Finance and Risk Management (eds. C. Lee, A. Lee and J. Lee), Springer, New York, (2015), 2281-2302. Google Scholar

[13]

C. Fábián, G. Mitra and D. Roman, Processing second-order stochastic dominance models using cutting-plane representations, Math. Program., 130 (2011), 33-57.  doi: 10.1007/s10107-009-0326-1.  Google Scholar

[14]

M. Gugat, A parametric review on the Mangasarian-Fromovitz constraint qualification, Math. Program., 85 (1999), 643-653.  doi: 10.1007/s101070050075.  Google Scholar

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T. Homem-de-Mello and S. Mehrota, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM J. Optim., 20 (2009), 1250-1273.  doi: 10.1137/08074009X.  Google Scholar

[16]

J. Hu, T. Homen-De-Mello and S. Mehrotra, Sample average approximation of stochastic dominance constrained programs, Math. Program., 133 (2012), 171-201.  doi: 10.1007/s10107-010-0428-9.  Google Scholar

[17]

P. Jorion, Value at Risk: The New Benchmark for Controlling Market Risk, McGraw-Hill Inc., US, 2006. Google Scholar

[18]

J. E. Kelley, The cutting-plane method for solving convex programs, SIAM J. Appl. Math., 8 (1960), 703-712.  doi: 10.1137/0108053.  Google Scholar

[19]

P. Kolm, R. Tütüncü and F. Fabozzi, 60 Years of portfolio optimization: Practical challenges and current trends, European J. Oper. Res., 234 (2014), 356-371.  doi: 10.1016/j.ejor.2013.10.060.  Google Scholar

[20]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manag. Sci., 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.  Google Scholar

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H. Konno, H. Shirakawa and H. Yamazaki, A mean-absolute deviation-skewness portfolio optimization model, Ann. Oper. Res., 45 (1993), 205-220.  doi: 10.1007/BF02282050.  Google Scholar

[22]

Y. Liu and H. Xu, Stability and sensitivity analysis of stochastic programs with second order dominance constraints, Math. Program., 142 (2013), 435-460.  doi: 10.1007/s10107-012-0585-0.  Google Scholar

[23]

H. Markowitz, Portfolio Selection, J. Finance, (1952), 77-91.   Google Scholar

[24]

H. Markowitz, Portfolio Selection, John Wiley & Sons, New York, 1959. Google Scholar

[25]

K. Mosler and M. Scarsini, (eds.), Stochastic Orders and Decision under Risk, Institute of Mathematical Statistics, Hayward, CA, 1991.  Google Scholar

[26]

W. Ogryczak and A Ruszczyński, From stochastic dominance to mean-risk models: Semideviations as risk measures, Euro. J. Oper. Res., 116 (1999), 33-50.  doi: 10.1016/S0377-2217(98)00167-2.  Google Scholar

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W. Ogryczak and A Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13 (2002), 60-78.  doi: 10.1137/S1052623400375075.  Google Scholar

[28]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[29]

D. Roman, K. Darby-Dowman and G. Mitra, Portfolio construction based on stochastic dominance and target return distributions, Math. Program., 108 (2000), 541-569.  doi: 10.1007/s10107-006-0722-8.  Google Scholar

[30]

D. Roman, G. Mitra and V. Zverovich, Enhanced indexation based on second-order stochastic dominance, European J. Oper. Res., 228 (2013), 273-281.  doi: 10.1016/j.ejor.2013.01.035.  Google Scholar

[31]

G. Rudolf and A. Ruszczyński, Optimization problems with second order stochastic dominance constraints: Duality, compact formulations, and cut generation methods, SIAM J. Optim., 19 (2008), 1326-1343.  doi: 10.1137/070702473.  Google Scholar

[32]

H. Sun, H. Xu and Y. Wang, A smoothing penalized sample average approximation method for stochastic programs with second order stochastic dominance constraints, Asia-Pac. J. Oper. Res., 30 (2013), 1340002 (25 pages). doi: 10.1142/S0217595913400022.  Google Scholar

[33]

H. Sun, H. Xu, R. Meskarian and Y. Wang, Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints, SIAM J. Optim., 23 (2013), 602-631.  doi: 10.1137/110850815.  Google Scholar

[34]

H. Sun and H. Xu, Convergence analysis of stationary points in-sample average approximation of stochastic programs with second order stochastic dominance constraints, Math. Program., 143 (2014), 31-59.  doi: 10.1007/s10107-013-0711-7.  Google Scholar

[35]

W. E. Sharpe, The Sharpe Ratio, J. Portfolio Manage., 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[36]

F. A. Sortino and L. N. Price, Performance measurement in a downside risk framework, J. Invest., 3 (2009), 59-64.  doi: 10.3905/joi.3.3.59.  Google Scholar

[37]

G. A. Whitmore and M. C. Findlay, (eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk, D.C.Heath, Lexington, MA., 1978. Google Scholar

[38]

F. Xu, M. Wang, Y. H. Dai and D. Xu, A sparse enhanced indexation model with chance and cardinality constraints, J. Glob. Optim., 70 (2018), 5-25.  doi: 10.1007/s10898-017-0513-1.  Google Scholar

show all references

References:
[1]

G. J. Alexander and A. M. Baptista, A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model, Manag. Sci., 50 (2004), 1261-1273.   Google Scholar

[2]

E. Anderson, H. Xu and D. Zhang, Confidence levels for CVaR risk measures and minimax limits, manuscript, The University of Sydney, 2014. Available from: http://hdl.handle.net/2123/9943. Google Scholar

[3]

K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer Texts in Statistics, Springer, New York, 2006.  Google Scholar

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.  Google Scholar

[6]

N. A. Canakgoz and J. E. Beasley, Mixed-integer programming approaches for index tracking and en-hanced indexation, European J. Oper. Res., 196 (2009), 384-399.  doi: 10.1016/j.ejor.2008.03.015.  Google Scholar

[7]

D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim., 14 (2003), 548-566.  doi: 10.1137/S1052623402420528.  Google Scholar

[8]

D. Dentcheva and A. Ruszczyński, Semi-infinite probabilistic constraints: Optimality and convexification, Optimization, 53 (2004), 583-601.  doi: 10.1080/02331930412331327148.  Google Scholar

[9]

D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Math. Program., 99 (2004), 329-350.  doi: 10.1007/s10107-003-0453-z.  Google Scholar

[10]

D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Bank. Financ., 30 (2006), 433-451.  doi: 10.1137/S1052623402420528.  Google Scholar

[11]

D. Dentcheva and A. Ruszczyński, Inverse cutting plane methods for optimization problems with second-order stochastic dominance constraints, Optimization, 59 (2010), 323-338.  doi: 10.1080/02331931003696350.  Google Scholar

[12]

D. Dentcheva and A. Ruszczyński, Risk-averse portfolio optimization via stochastic dominance constraints, in Handbook of Quantitative Finance and Risk Management (eds. C. Lee, A. Lee and J. Lee), Springer, New York, (2015), 2281-2302. Google Scholar

[13]

C. Fábián, G. Mitra and D. Roman, Processing second-order stochastic dominance models using cutting-plane representations, Math. Program., 130 (2011), 33-57.  doi: 10.1007/s10107-009-0326-1.  Google Scholar

[14]

M. Gugat, A parametric review on the Mangasarian-Fromovitz constraint qualification, Math. Program., 85 (1999), 643-653.  doi: 10.1007/s101070050075.  Google Scholar

[15]

T. Homem-de-Mello and S. Mehrota, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM J. Optim., 20 (2009), 1250-1273.  doi: 10.1137/08074009X.  Google Scholar

[16]

J. Hu, T. Homen-De-Mello and S. Mehrotra, Sample average approximation of stochastic dominance constrained programs, Math. Program., 133 (2012), 171-201.  doi: 10.1007/s10107-010-0428-9.  Google Scholar

[17]

P. Jorion, Value at Risk: The New Benchmark for Controlling Market Risk, McGraw-Hill Inc., US, 2006. Google Scholar

[18]

J. E. Kelley, The cutting-plane method for solving convex programs, SIAM J. Appl. Math., 8 (1960), 703-712.  doi: 10.1137/0108053.  Google Scholar

[19]

P. Kolm, R. Tütüncü and F. Fabozzi, 60 Years of portfolio optimization: Practical challenges and current trends, European J. Oper. Res., 234 (2014), 356-371.  doi: 10.1016/j.ejor.2013.10.060.  Google Scholar

[20]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manag. Sci., 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.  Google Scholar

[21]

H. Konno, H. Shirakawa and H. Yamazaki, A mean-absolute deviation-skewness portfolio optimization model, Ann. Oper. Res., 45 (1993), 205-220.  doi: 10.1007/BF02282050.  Google Scholar

[22]

Y. Liu and H. Xu, Stability and sensitivity analysis of stochastic programs with second order dominance constraints, Math. Program., 142 (2013), 435-460.  doi: 10.1007/s10107-012-0585-0.  Google Scholar

[23]

H. Markowitz, Portfolio Selection, J. Finance, (1952), 77-91.   Google Scholar

[24]

H. Markowitz, Portfolio Selection, John Wiley & Sons, New York, 1959. Google Scholar

[25]

K. Mosler and M. Scarsini, (eds.), Stochastic Orders and Decision under Risk, Institute of Mathematical Statistics, Hayward, CA, 1991.  Google Scholar

[26]

W. Ogryczak and A Ruszczyński, From stochastic dominance to mean-risk models: Semideviations as risk measures, Euro. J. Oper. Res., 116 (1999), 33-50.  doi: 10.1016/S0377-2217(98)00167-2.  Google Scholar

[27]

W. Ogryczak and A Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13 (2002), 60-78.  doi: 10.1137/S1052623400375075.  Google Scholar

[28]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[29]

D. Roman, K. Darby-Dowman and G. Mitra, Portfolio construction based on stochastic dominance and target return distributions, Math. Program., 108 (2000), 541-569.  doi: 10.1007/s10107-006-0722-8.  Google Scholar

[30]

D. Roman, G. Mitra and V. Zverovich, Enhanced indexation based on second-order stochastic dominance, European J. Oper. Res., 228 (2013), 273-281.  doi: 10.1016/j.ejor.2013.01.035.  Google Scholar

[31]

G. Rudolf and A. Ruszczyński, Optimization problems with second order stochastic dominance constraints: Duality, compact formulations, and cut generation methods, SIAM J. Optim., 19 (2008), 1326-1343.  doi: 10.1137/070702473.  Google Scholar

[32]

H. Sun, H. Xu and Y. Wang, A smoothing penalized sample average approximation method for stochastic programs with second order stochastic dominance constraints, Asia-Pac. J. Oper. Res., 30 (2013), 1340002 (25 pages). doi: 10.1142/S0217595913400022.  Google Scholar

[33]

H. Sun, H. Xu, R. Meskarian and Y. Wang, Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints, SIAM J. Optim., 23 (2013), 602-631.  doi: 10.1137/110850815.  Google Scholar

[34]

H. Sun and H. Xu, Convergence analysis of stationary points in-sample average approximation of stochastic programs with second order stochastic dominance constraints, Math. Program., 143 (2014), 31-59.  doi: 10.1007/s10107-013-0711-7.  Google Scholar

[35]

W. E. Sharpe, The Sharpe Ratio, J. Portfolio Manage., 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[36]

F. A. Sortino and L. N. Price, Performance measurement in a downside risk framework, J. Invest., 3 (2009), 59-64.  doi: 10.3905/joi.3.3.59.  Google Scholar

[37]

G. A. Whitmore and M. C. Findlay, (eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk, D.C.Heath, Lexington, MA., 1978. Google Scholar

[38]

F. Xu, M. Wang, Y. H. Dai and D. Xu, A sparse enhanced indexation model with chance and cardinality constraints, J. Glob. Optim., 70 (2018), 5-25.  doi: 10.1007/s10898-017-0513-1.  Google Scholar

Figure 1.  In-sample back testing with NDX data with NDX index as benchmark
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Figure 2.  NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
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Figure 3.  NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
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Figure 4.  S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
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Figure 5.  S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
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Figure 6.  FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
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Figure 7.  FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
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Table 1.  NDX: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio
mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0009 0.0088 0.1030 0.1389
SSD 0.0032 0.0118 0.2717 0.4501
$ CVaR_ {\beta=0.9} $ 0.0033 0.0118 0.2784 0.4670
$ CVaR_{\beta=0.8} $ 0.0032 0.0117 0.2724 0.4486
$ CVaR_{\beta=0.7} $ 0.0033 0.0119 0.2810 0.4652
Benchmark: MV 0.0005 0.0078 0.0658 0.0841
SSD 0.0026 0.0102 0.2545 0.4129
$ CVaR_ {\beta=0.9} $ 0.0027 0.0118 0.2696 0.4442
$ CVaR_{\beta=0.8} $ 0.0030 0.0117 0.2931 0.4916
$ CVaR_{\beta=0.7} $ 0.0030 0.0102 0.2943 0.5004
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mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0009 0.0088 0.1030 0.1389
SSD 0.0032 0.0118 0.2717 0.4501
$ CVaR_ {\beta=0.9} $ 0.0033 0.0118 0.2784 0.4670
$ CVaR_{\beta=0.8} $ 0.0032 0.0117 0.2724 0.4486
$ CVaR_{\beta=0.7} $ 0.0033 0.0119 0.2810 0.4652
Benchmark: MV 0.0005 0.0078 0.0658 0.0841
SSD 0.0026 0.0102 0.2545 0.4129
$ CVaR_ {\beta=0.9} $ 0.0027 0.0118 0.2696 0.4442
$ CVaR_{\beta=0.8} $ 0.0030 0.0117 0.2931 0.4916
$ CVaR_{\beta=0.7} $ 0.0030 0.0102 0.2943 0.5004
Table 2.  S&P 500: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio
mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0004 0.0079 0.0534 0.0705
SSD 0.0017 0.0112 0.1490 0.2264
$ CVaR_ {\beta=0.9} $ 0.0018 0.0111 0.1606 0.2444
$ CVaR_{\beta=0.8} $ 0.0016 0.0114 0.1442 0.2171
$ CVaR_{\beta=0.7} $ 0.0017 0.0115 0.1477 0.2241
Benchmark: MV 0.0003 0.0060 0.0421 0.0573
SSD 0.0009 0.0110 0.0802 0.1103
$ CVaR_ {\beta=0.9} $ 0.0012 0.0110 0.1086 0.1517
$ CVaR_{\beta=0.8} $ 0.0013 0.0107 0.1196 0.1702
$ CVaR_{\beta=0.7} $ 0.0015 0.0110 0.1383 0.2001
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mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0004 0.0079 0.0534 0.0705
SSD 0.0017 0.0112 0.1490 0.2264
$ CVaR_ {\beta=0.9} $ 0.0018 0.0111 0.1606 0.2444
$ CVaR_{\beta=0.8} $ 0.0016 0.0114 0.1442 0.2171
$ CVaR_{\beta=0.7} $ 0.0017 0.0115 0.1477 0.2241
Benchmark: MV 0.0003 0.0060 0.0421 0.0573
SSD 0.0009 0.0110 0.0802 0.1103
$ CVaR_ {\beta=0.9} $ 0.0012 0.0110 0.1086 0.1517
$ CVaR_{\beta=0.8} $ 0.0013 0.0107 0.1196 0.1702
$ CVaR_{\beta=0.7} $ 0.0015 0.0110 0.1383 0.2001
Table 3.  FTSE 100: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio
mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0012 0.0112 0.1080 0.1685
SSD 0.0017 0.0158 0.1094 0.1848
$ CVaR_ {\beta=0.9} $ 0.0017 0.0155 0.1099 0.1836
$ CVaR_{\beta=0.8} $ 0.0020 0.0157 0.1254 0.2131
$ CVaR_{\beta=0.7} $ 0.0021 0.0164 0.1269 0.2185
Benchmark: MV 0.0018 0.0095 0.1901 0.3418
SSD 0.0023 0.0141 0.1606 0.2925
$ CVaR_ {\beta=0.9} $ 0.0021 0.0134 0.1568 0.2747
$ CVaR_{\beta=0.8} $ 0.0021 0.0136 0.1578 0.2755
$ CVaR_{\beta=0.7} $ 0.0024 0.0141 0.1703 0.3058
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mean std Sharpe Ratio Sortino Ratio
Benchmark: index 0.0012 0.0112 0.1080 0.1685
SSD 0.0017 0.0158 0.1094 0.1848
$ CVaR_ {\beta=0.9} $ 0.0017 0.0155 0.1099 0.1836
$ CVaR_{\beta=0.8} $ 0.0020 0.0157 0.1254 0.2131
$ CVaR_{\beta=0.7} $ 0.0021 0.0164 0.1269 0.2185
Benchmark: MV 0.0018 0.0095 0.1901 0.3418
SSD 0.0023 0.0141 0.1606 0.2925
$ CVaR_ {\beta=0.9} $ 0.0021 0.0134 0.1568 0.2747
$ CVaR_{\beta=0.8} $ 0.0021 0.0136 0.1578 0.2755
$ CVaR_{\beta=0.7} $ 0.0024 0.0141 0.1703 0.3058
Table 4.  Average, minimum and maximum of daily traded basket sizes of different models with both benchmarks in three data sets
NDX (100) Index MV
avg. min. max. avg. min. max.
SSD 4.60 3 9 5.55 3 9
$ CVaR_{\beta = 0.9} $ 4.65 3 10 5.53 2 9
$ CVaR_{\beta = 0.8} $ 4.65 3 10 5.51 2 8
$ CVaR_{\beta = 0.7} $ 4.64 3 9 5.50 3 8
FTSE (100) Index MV
avg. min. max. avg. min. max.
SSD 4.05 2 9 5.16 2 9
$ CVaR_{\beta = 0.9} $ 3.98 2 9 5.11 2 9
$ CVaR_{\beta = 0.8} $ 3.90 2 8 5.01 2 8
$ CVaR_{\beta = 0.7} $ 3.91 3 9 5.17 2 9
S&P (500) Index MV
avg. min. max. avg. min. max.
SSD 5.87 3 10 6.62 4 11
$ CVaR_{\beta = 0.9} $ 6.07 3 11 6.49 3 12
$ CVaR_{\beta = 0.8} $ 6.07 3 12 6.70 4 12
$ CVaR_{\beta = 0.7} $ 6.30 3 11 6.74 4 12
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NDX (100) Index MV
avg. min. max. avg. min. max.
SSD 4.60 3 9 5.55 3 9
$ CVaR_{\beta = 0.9} $ 4.65 3 10 5.53 2 9
$ CVaR_{\beta = 0.8} $ 4.65 3 10 5.51 2 8
$ CVaR_{\beta = 0.7} $ 4.64 3 9 5.50 3 8
FTSE (100) Index MV
avg. min. max. avg. min. max.
SSD 4.05 2 9 5.16 2 9
$ CVaR_{\beta = 0.9} $ 3.98 2 9 5.11 2 9
$ CVaR_{\beta = 0.8} $ 3.90 2 8 5.01 2 8
$ CVaR_{\beta = 0.7} $ 3.91 3 9 5.17 2 9
S&P (500) Index MV
avg. min. max. avg. min. max.
SSD 5.87 3 10 6.62 4 11
$ CVaR_{\beta = 0.9} $ 6.07 3 11 6.49 3 12
$ CVaR_{\beta = 0.8} $ 6.07 3 12 6.70 4 12
$ CVaR_{\beta = 0.7} $ 6.30 3 11 6.74 4 12
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