# American Institute of Mathematical Sciences

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

 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  November 2020 Early access  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.

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 and Management Optimization, 2020, 16 (6) : 2581-2602. doi: 10.3934/jimo.2019071
##### References:
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Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim., 14 (2003), 548-566.  doi: 10.1137/S1052623402420528. [8] D. Dentcheva and A. Ruszczyński, Semi-infinite probabilistic constraints: Optimality and convexification, Optimization, 53 (2004), 583-601.  doi: 10.1080/02331930412331327148. [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. [10] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Bank. Financ., 30 (2006), 433-451.  doi: 10.1137/S1052623402420528. [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. [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. [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. [14] M. Gugat, A parametric review on the Mangasarian-Fromovitz constraint qualification, Math. Program., 85 (1999), 643-653.  doi: 10.1007/s101070050075. [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. [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. [17] P. Jorion, Value at Risk: The New Benchmark for Controlling Market Risk, McGraw-Hill Inc., US, 2006. [18] J. E. Kelley, The cutting-plane method for solving convex programs, SIAM J. Appl. Math., 8 (1960), 703-712.  doi: 10.1137/0108053. [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. [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. [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. [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. [23] H. Markowitz, Portfolio Selection, J. Finance, (1952), 77-91. [24] H. Markowitz, Portfolio Selection, John Wiley & Sons, New York, 1959. [25] K. Mosler and M. Scarsini, (eds.), Stochastic Orders and Decision under Risk, Institute of Mathematical Statistics, Hayward, CA, 1991. [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. [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. [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. [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. [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. [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. [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. [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. [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. [35] W. E. Sharpe, The Sharpe Ratio, J. Portfolio Manage., 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501. [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. [37] G. A. Whitmore and M. C. Findlay, (eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk, D.C.Heath, Lexington, MA., 1978. [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.

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. [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. [3] K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer Texts in Statistics, Springer, New York, 2006. [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. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [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. [7] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM J. Optim., 14 (2003), 548-566.  doi: 10.1137/S1052623402420528. [8] D. Dentcheva and A. Ruszczyński, Semi-infinite probabilistic constraints: Optimality and convexification, Optimization, 53 (2004), 583-601.  doi: 10.1080/02331930412331327148. [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. [10] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, J. Bank. Financ., 30 (2006), 433-451.  doi: 10.1137/S1052623402420528. [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. [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. [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. [14] M. Gugat, A parametric review on the Mangasarian-Fromovitz constraint qualification, Math. Program., 85 (1999), 643-653.  doi: 10.1007/s101070050075. [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. [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. [17] P. Jorion, Value at Risk: The New Benchmark for Controlling Market Risk, McGraw-Hill Inc., US, 2006. [18] J. E. Kelley, The cutting-plane method for solving convex programs, SIAM J. Appl. Math., 8 (1960), 703-712.  doi: 10.1137/0108053. [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. [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. [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. [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. [23] H. Markowitz, Portfolio Selection, J. Finance, (1952), 77-91. [24] H. Markowitz, Portfolio Selection, John Wiley & Sons, New York, 1959. [25] K. Mosler and M. Scarsini, (eds.), Stochastic Orders and Decision under Risk, Institute of Mathematical Statistics, Hayward, CA, 1991. [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. [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. [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. [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. [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. [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. [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. [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. [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. [35] W. E. Sharpe, The Sharpe Ratio, J. Portfolio Manage., 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501. [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. [37] G. A. Whitmore and M. C. Findlay, (eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk, D.C.Heath, Lexington, MA., 1978. [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.
In-sample back testing with NDX data with NDX index as benchmark
NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
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
 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
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
 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
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
 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
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
 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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