# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018198

## A smoothing SAA algorithm for a portfolio choice model based on second-order stochastic dominance measures

 1 School of Mathematics and Computational Sciences, Xiangtan University, Xiangtan 411105, Hunan, China 2 Hunan First Normal University, Changsha 410215, Hunan, China

* Corresponding author: Liu Yang

Received  March 2017 Revised  October 2017 Published  December 2018

In this paper, we provide a smoothing sample average approximation (SAA) method to solve a portfolio choice model based on second-order stochastic dominance (SSD) measure. Introducing a second-order stochastic dominance constraint in portfolio choice is theoretically attractive since all risk-averse investors would prefer a dominating portfolio. However, how to get the best choice among SSD efficient portfolios which is based on a stochastic optimization model is a challenge. We use the sample average to approximate the expected return rate function in the model and get a linear/nonlinear programming when the benchmark has discrete distribution. Then we propose a smoothing penalty algorithm to solve this problem. Meanwhile, we investigate the convergence of the optimal value of the transformed model and show that the optimal value converges to its counterpart with probability approaching to one at exponential rate as the sample size increases. By comparing the numerical results of the smoothing SAA algorithm with the common linear programming (LP) algorithm, we find that the smoothing algorithm has better performance than the LP algorithm in three aspects: (ⅰ)the smoothing SAA method can avoid the infinite constraints in the transformed models and the size of the smoothing algorithm model will not increase as the sample grows; (ⅱ)the smoothing SAA algorithm can deal with the nonlinear portfolio models with nonlinear transaction cost function; (ⅲ) the smoothing algorithm can get the global optimal solution because the smoothing function maintains the original convexity.

Citation: Liu Yang, Xiaojiao Tong, Yao Xiong, Feifei Shen. A smoothing SAA algorithm for a portfolio choice model based on second-order stochastic dominance measures. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018198
##### References:
 [1] M. J. Akian, L. Menaldi and A. Sulem, Multi-asset porfolio selection problem with transaction cosats, Mathematics and Computers in Simulation, 38 (1995), 163-172. doi: 10.1016/0378-4754(93)E0079-K. [2] J. Ang, F. Meng and J. Sun, Two-stage stochastic linear programs with incomplete information on uncertainty, European Journal of Operational Research, 233 (2014), 16-22. doi: 10.1016/j.ejor.2013.07.039. [3] R. Bruni, F. Cesarone, A. Scozzari and F. Tardella, On exact and approximate stochastic dominance strategies for portfolio selection, European Journal of Operational Research, 259 (2017), 322-329. doi: 10.1016/j.ejor.2016.10.006. [4] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566. doi: 10.1137/S1052623402420528. [5] D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350. doi: 10.1007/s10107-003-0453-z. [6] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, Journal of Banking and Finance, 30 (2006), 433-451. [7] J. Dupa$\check{c}$ov$\acute{a}$ and M. Kopa, Robustness of optimal portfolios under risk and stochastic dominance constraints, E.J.Oper. Res., 234 (2014), 434-441. doi: 10.1016/j.ejor.2013.06.018. [8] L. F. Escudero, J. F. Monge and D. R. Morales, An SDP approach for multiperiod mixed 0-1 linear programming models with stochastic dominance constraints for risk management, Comp. Oper. Res., 58 (2015), 32-40. doi: 10.1016/j.cor.2014.12.007. [9] C. I. F$\acute{a}$bi$\acute{a}$n, G. Mitra, D. Roman and V. Zverovich, An enhanced model for portfolio choice with SSD criteria: A constructive approach, Quantitative Finance, 11 (2011), 1525-1534. doi: 10.1080/14697680903493607. [10] P. C. Fishburn, Decision and Value Theory, John Wiley and Sons, New York, 1964. [11] T. Homem-De-Mello and S. Mehrota, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM Journal of Optimization, 20 (2009), 1250-1273. doi: 10.1137/08074009X. [12] J. E. Hodder, J. C. Jackwerth and O. Kolokolova, Improved portfolio choice using second-order stochastic dominance, Review of Finance, 19 (2015), 1623-1647. [13] J. Hu, T. Homem-De-Mello and S. Mehrota, Sample average approximation of stochastic dominance constrained programs, Mathematical Programming, Series A, 133 (2012), 171-201. doi: 10.1007/s10107-010-0428-9. [14] C. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9. [15] B. Li, C. Z. Wu, H. H. Dam, A. Cantoni and K. L. Teo, A parallel low complexity zero-forcing beamformer design for multiuser MIMO systems via a regularized dual decomposition method, IEEE Transactions on Signal Processing, 63 (2015), 4179-4190. doi: 10.1109/TSP.2015.2437846. [16] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. [17] B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474. [18] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. [19] Y. Liu and H. Xu, Stability analysis of stochastic programs with second order dominance constraints, Mathematical Programming, 142 (2013), 435-460. doi: 10.1007/s10107-012-0585-0. [20] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, San Diego, 1979. [21] M. Menegatti, A note on portfolio selection and stochastic dominance, Decisions Econ. Finan., 39 (2016), 327-331. doi: 10.1007/s10203-016-0179-z. [22] R. Meskarian, H. Xu and J. Fliege, Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization, European Journal of Operational Research, 216 (2012), 376-385. doi: 10.1016/j.ejor.2011.07.044. [23] R. Meskarian, J. Fliege and H. Xu, Stochstic programming with multivariate second order stochastic dominance constraints with applications in portfolio optimization, Appl. Math. Optim., 70 (2014), 111-140. doi: 10.1007/s00245-014-9236-6. [24] J. M. Peng and Z. Lin, A non-interior continuation method for generalized linear complementarity problems, Math.Program, 86 (1999), 533-563. doi: 10.1007/s101070050104. [25] J. P. Quirk and R. Saposnik, Admissibility and measurable utility functions, Review of Economic Studies, 29 (1962), 140-146. [26] A. Shapiro, Monte Carlo sampling Methods, in:Stochastic Programming, Handbook in Operations Research and Management Science, 10 (2003), 353-425. doi: 10.1016/S0927-0507(03)10006-0. [27] H. Sun, h. Xu and Y. Wang, A smoothing penalized sample average approximation method for stochastic programs with second-order stochastic dominance constraints, Asia-Pacific Journal of Operational Research, 30 (2013), 1340002, 25 pp. doi: 10.1142/S0217595913400022. [28] 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., Ser. A, 143 (2014), 31-59. doi: 10.1007/s10107-013-0711-7. [29] X. J. Tong, L. Qi, F. Wu, et al., A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset, Applied Mathematics and Computation, 216 (2010), 1723–1740. doi: 10.1016/j.amc.2009.12.031. [30] L. Yang, Y. Chen and X. Tong, Smoothing Newton-like method for the solution of nonlinear systems of equalities and inequalities, Numerical Mathematics: Theory, Methods and Applications, 2 (2009), 224-236.

show all references

##### References:
 [1] M. J. Akian, L. Menaldi and A. Sulem, Multi-asset porfolio selection problem with transaction cosats, Mathematics and Computers in Simulation, 38 (1995), 163-172. doi: 10.1016/0378-4754(93)E0079-K. [2] J. Ang, F. Meng and J. Sun, Two-stage stochastic linear programs with incomplete information on uncertainty, European Journal of Operational Research, 233 (2014), 16-22. doi: 10.1016/j.ejor.2013.07.039. [3] R. Bruni, F. Cesarone, A. Scozzari and F. Tardella, On exact and approximate stochastic dominance strategies for portfolio selection, European Journal of Operational Research, 259 (2017), 322-329. doi: 10.1016/j.ejor.2016.10.006. [4] D. Dentcheva and A. Ruszczyński, Optimization with stochastic dominance constraints, SIAM Journal on Optimization, 14 (2003), 548-566. doi: 10.1137/S1052623402420528. [5] D. Dentcheva and A. Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical Programming, 99 (2004), 329-350. doi: 10.1007/s10107-003-0453-z. [6] D. Dentcheva and A. Ruszczyński, Portfolio optimization with stochastic dominance constraints, Journal of Banking and Finance, 30 (2006), 433-451. [7] J. Dupa$\check{c}$ov$\acute{a}$ and M. Kopa, Robustness of optimal portfolios under risk and stochastic dominance constraints, E.J.Oper. Res., 234 (2014), 434-441. doi: 10.1016/j.ejor.2013.06.018. [8] L. F. Escudero, J. F. Monge and D. R. Morales, An SDP approach for multiperiod mixed 0-1 linear programming models with stochastic dominance constraints for risk management, Comp. Oper. Res., 58 (2015), 32-40. doi: 10.1016/j.cor.2014.12.007. [9] C. I. F$\acute{a}$bi$\acute{a}$n, G. Mitra, D. Roman and V. Zverovich, An enhanced model for portfolio choice with SSD criteria: A constructive approach, Quantitative Finance, 11 (2011), 1525-1534. doi: 10.1080/14697680903493607. [10] P. C. Fishburn, Decision and Value Theory, John Wiley and Sons, New York, 1964. [11] T. Homem-De-Mello and S. Mehrota, A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM Journal of Optimization, 20 (2009), 1250-1273. doi: 10.1137/08074009X. [12] J. E. Hodder, J. C. Jackwerth and O. Kolokolova, Improved portfolio choice using second-order stochastic dominance, Review of Finance, 19 (2015), 1623-1647. [13] J. Hu, T. Homem-De-Mello and S. Mehrota, Sample average approximation of stochastic dominance constrained programs, Mathematical Programming, Series A, 133 (2012), 171-201. doi: 10.1007/s10107-010-0428-9. [14] C. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9. [15] B. Li, C. Z. Wu, H. H. Dam, A. Cantoni and K. L. Teo, A parallel low complexity zero-forcing beamformer design for multiuser MIMO systems via a regularized dual decomposition method, IEEE Transactions on Signal Processing, 63 (2015), 4179-4190. doi: 10.1109/TSP.2015.2437846. [16] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. [17] B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474. [18] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. [19] Y. Liu and H. Xu, Stability analysis of stochastic programs with second order dominance constraints, Mathematical Programming, 142 (2013), 435-460. doi: 10.1007/s10107-012-0585-0. [20] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, San Diego, 1979. [21] M. Menegatti, A note on portfolio selection and stochastic dominance, Decisions Econ. Finan., 39 (2016), 327-331. doi: 10.1007/s10203-016-0179-z. [22] R. Meskarian, H. Xu and J. Fliege, Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization, European Journal of Operational Research, 216 (2012), 376-385. doi: 10.1016/j.ejor.2011.07.044. [23] R. Meskarian, J. Fliege and H. Xu, Stochstic programming with multivariate second order stochastic dominance constraints with applications in portfolio optimization, Appl. Math. Optim., 70 (2014), 111-140. doi: 10.1007/s00245-014-9236-6. [24] J. M. Peng and Z. Lin, A non-interior continuation method for generalized linear complementarity problems, Math.Program, 86 (1999), 533-563. doi: 10.1007/s101070050104. [25] J. P. Quirk and R. Saposnik, Admissibility and measurable utility functions, Review of Economic Studies, 29 (1962), 140-146. [26] A. Shapiro, Monte Carlo sampling Methods, in:Stochastic Programming, Handbook in Operations Research and Management Science, 10 (2003), 353-425. doi: 10.1016/S0927-0507(03)10006-0. [27] H. Sun, h. Xu and Y. Wang, A smoothing penalized sample average approximation method for stochastic programs with second-order stochastic dominance constraints, Asia-Pacific Journal of Operational Research, 30 (2013), 1340002, 25 pp. doi: 10.1142/S0217595913400022. [28] 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., Ser. A, 143 (2014), 31-59. doi: 10.1007/s10107-013-0711-7. [29] X. J. Tong, L. Qi, F. Wu, et al., A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset, Applied Mathematics and Computation, 216 (2010), 1723–1740. doi: 10.1016/j.amc.2009.12.031. [30] L. Yang, Y. Chen and X. Tong, Smoothing Newton-like method for the solution of nonlinear systems of equalities and inequalities, Numerical Mathematics: Theory, Methods and Applications, 2 (2009), 224-236.
The expected returns for different smoothing parameters
Comparison of the CPU time for LP and SMOOTH
The expected returns for different transaction cost ratio
The expected returns for different penalty parameter and sample size
Expectation and variance of return rates
 code of stock 600690 713 600115 600111 600000 expectation 0.0051 0.004 0.003 0.01 0.0049 variance 0.0031 0.0042 0.005 0.006 0.0036 code of stock 600252 600011 600362 401 600267 expectation 0.0072 0.0028 0.0073 0.0052 0.005 variance 0.0061 0.0027 0.0065 0.0046 0.003
 code of stock 600690 713 600115 600111 600000 expectation 0.0051 0.004 0.003 0.01 0.0049 variance 0.0031 0.0042 0.005 0.006 0.0036 code of stock 600252 600011 600362 401 600267 expectation 0.0072 0.0028 0.0073 0.0052 0.005 variance 0.0061 0.0027 0.0065 0.0046 0.003
Comparison of the numerical results for LP and SMOOTH
 Method Problem $x$ E$[\cdot]$ LP No-cost (0, 0, 0, 0.7744, 0, 0.2165, 0, 0.0091, 0, 0) 0.0095 SMOOTH No-cost (0, 0, 0, 0.9592, 0, 0.0348, 0, 0.006, 0, 0) 0.0089 LP cost (0, 0, 0, 0.6620, 0, 0.2554, 0, 0.0642, 0, 0) 0.0075 SMOOTH cost (0, 0.0586, 0.0809, 0.1435, 0, 0, 0.4344, 0, 0, 0.2608) 0.0067
 Method Problem $x$ E$[\cdot]$ LP No-cost (0, 0, 0, 0.7744, 0, 0.2165, 0, 0.0091, 0, 0) 0.0095 SMOOTH No-cost (0, 0, 0, 0.9592, 0, 0.0348, 0, 0.006, 0, 0) 0.0089 LP cost (0, 0, 0, 0.6620, 0, 0.2554, 0, 0.0642, 0, 0) 0.0075 SMOOTH cost (0, 0.0586, 0.0809, 0.1435, 0, 0, 0.4344, 0, 0, 0.2608) 0.0067
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