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April  2017, 13(2): 947-965. doi: 10.3934/jimo.2016055

## An optimal trade-off model for portfolio selection with sensitivity of parameters

 Department of Mathematics, Shanghai University, Shanghai 200444, China

* Corresponding author

Received  October 2015 Revised  March 2016 Published  August 2016

Fund Project: This research was supported by a grant from the National Natural Science Foundation of China (No.11371242).

In this paper, we propose an optimal trade-off model for portfolio selection with sensitivity of parameters, which are estimated from historical data. Mathematically, the model is a quadratic programming problem, whose objective function contains three terms. The first term is a measurement of risk. And the later two are the maximum and minimum sensitivity, which are non-convex and non-smooth functions and lead to the whole model to be an intractable problem. Then we transform this quadratic programming problem into an unconstrained composite problem equivalently. Furthermore, we develop a modified accelerated gradient (AG) algorithm to solve the unconstrained composite problem. The convergence and the convergence rate of our algorithm are derived. Finally, we perform both the empirical analysis and the numerical experiments. The empirical analysis indicates that the optimal trade-off model results in a stable return with lower risk under the stress test. The numerical experiments demonstrate that the modified AG algorithm outperforms the existed AG algorithm for both CPU time and the iterations, respectively.

Citation: Yanqin Bai, Yudan Wei, Qian Li. An optimal trade-off model for portfolio selection with sensitivity of parameters. Journal of Industrial & Management Optimization, 2017, 13 (2) : 947-965. doi: 10.3934/jimo.2016055
##### References:
 [1] F.A. Al-Khayyal, C. Larsen and T.V. Voorhis, A relaxation method for nonconvex quadratically constrained quadratic programs, Journal of Global Optimization, 6 (1995), 215-230.  doi: 10.1007/BF01099462.  Google Scholar [2] C. Audet, P. Hansen, B. Jaumard and G. Savard, A branch-and-cut algorithm for nonconvex quadratically constrained quadratic programming, Mathematical Programming, 87 (2000), 131-152.  doi: 10.1007/s101079900106.  Google Scholar [3] V. Boginski, S. Butenko and P.M. Pardalos, Statistical analysis of financial networks, Computational Statistics & Data Analysis, 48 (2005), 431-443.  doi: 10.1016/j.csda.2004.02.004.  Google Scholar [4] V.K. Chopra and W.T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio chocie, Journal of Portfolio Management, 19 (1993), 6-11.  doi: 10.3905/jpm.1993.409440.  Google Scholar [5] X.T. Cui, X.L. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Management Optimization, 5 (2009), 33-46.  doi: 10.3934/jimo.2009.5.33.  Google Scholar [6] X. T. Cui, Studies on Portfolio Selection Problems with Different Risk Measures and Trading Constraints, O224, Fudan University, 2013. Google Scholar [7] Z.B. Deng, Y.Q. Bai, S.C. Fang, T. Ye and W.X. Xing, A branch-and-cut approach to potofolio selection with marginal risk control in a linear cone programming framework, Journal of Systems Science and Systems Engineering, 22 (2013), 385-400.  doi: 10.1007/s11518-013-5234-5.  Google Scholar [8] S. Ghadimi and G. H. Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming (accepted), Mathematical Programming, (2015). arXiv: 1310.3787v1 Google Scholar [9] D. Goldfarb and G. Iyengar, Rubust portfolio selection problems, Mathematices of Operations Reserch, 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar [10] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2. 1, http://cvxr.com/cvx/, 2014. Google Scholar [11] P. Horst, P.M. Pardolos and N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, 1995.   Google Scholar [12] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar [13] V. Kalyagin, A. Koldanov, P. Koldanov and V. Zamaraev, Market graph and markowtiz model, Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos M. Pardalos, Springer Science (2014), 293-306. Google Scholar [14] G.H. Lan, An optimal method for stochastic composite optimization, Mathematical Programming, 133 (2012), 365-397.  doi: 10.1007/s10107-010-0434-y.  Google Scholar [15] Q. Li and Y.Q. Bai, Optimal trade-off portfolio selection between total risk and maximum relative marginal risk (accepted), Optimization Methods & Software, 31 (2016), 681-700.  doi: 10.1080/10556788.2015.1041946.  Google Scholar [16] J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Mathematical Programming, 103 (2005), 251-282.  doi: 10.1007/s10107-005-0582-7.  Google Scholar [17] H.M. Markowitz, Portfolio selection, Journal of Finace, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar [18] Y.E. Nesterov, A method for unconstrained convex minimization problem with the rate of convergence $\mathcal{O}(1/k^2)$, Doklady AN SSSR, 269 (1983), 543-547.   Google Scholar [19] Y. E. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2003. doi: 10.007/978-1-4419-8853-9.  Google Scholar [20] Y.E. Nestrov, Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar [21] U. Raber, A simplicial branch-and-bound method for solving nonconvex all-quadratic programs, Journal of Global Optimization, 13 (1998), 417-432.  doi: 10.1023/A:1008377529330.  Google Scholar [22] B. Scherer, Can rubust portfolio optimization help to build better portfolios, Journal of Asset Management, 7 (2007), 374-387.  doi: 10.1057/palgrave.jam.2250049.  Google Scholar [23] X.L. Sun, X.J. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77.  doi: 10.1007/s40305-013-0004-0.  Google Scholar [24] Y.F. Sun, A. Grace, K.L. Teo and G.L. Zhou, Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.  doi: 10.3934/jimo.2015.11.1275.  Google Scholar [25] K.L. Teo and X.Q. Yang, Portfolio selection problem with minimax type risk function, Annals of Operations Research, 101 (2001), 333-349.  doi: 10.1023/A:1010909632198.  Google Scholar [26] Y. Tian, S.C. Fang, Z.B. Deng and Q.W. Jin, Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Industrial and Management Optimization, 12 (2016), 1041-1056.  doi: 10.3934/jimo.2016.12.1041.  Google Scholar [27] S.S. Zhu, D. Li and X.L. Sun, Portfolio selection with marginal risk control, The Journal of Computational Finance, 14 (2010), 3-28.  doi: 10.21314/JCF.2010.213.  Google Scholar

show all references

##### References:
 [1] F.A. Al-Khayyal, C. Larsen and T.V. Voorhis, A relaxation method for nonconvex quadratically constrained quadratic programs, Journal of Global Optimization, 6 (1995), 215-230.  doi: 10.1007/BF01099462.  Google Scholar [2] C. Audet, P. Hansen, B. Jaumard and G. Savard, A branch-and-cut algorithm for nonconvex quadratically constrained quadratic programming, Mathematical Programming, 87 (2000), 131-152.  doi: 10.1007/s101079900106.  Google Scholar [3] V. Boginski, S. Butenko and P.M. Pardalos, Statistical analysis of financial networks, Computational Statistics & Data Analysis, 48 (2005), 431-443.  doi: 10.1016/j.csda.2004.02.004.  Google Scholar [4] V.K. Chopra and W.T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio chocie, Journal of Portfolio Management, 19 (1993), 6-11.  doi: 10.3905/jpm.1993.409440.  Google Scholar [5] X.T. Cui, X.L. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, Journal of Industrial and Management Optimization, 5 (2009), 33-46.  doi: 10.3934/jimo.2009.5.33.  Google Scholar [6] X. T. Cui, Studies on Portfolio Selection Problems with Different Risk Measures and Trading Constraints, O224, Fudan University, 2013. Google Scholar [7] Z.B. Deng, Y.Q. Bai, S.C. Fang, T. Ye and W.X. Xing, A branch-and-cut approach to potofolio selection with marginal risk control in a linear cone programming framework, Journal of Systems Science and Systems Engineering, 22 (2013), 385-400.  doi: 10.1007/s11518-013-5234-5.  Google Scholar [8] S. Ghadimi and G. H. Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming (accepted), Mathematical Programming, (2015). arXiv: 1310.3787v1 Google Scholar [9] D. Goldfarb and G. Iyengar, Rubust portfolio selection problems, Mathematices of Operations Reserch, 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar [10] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2. 1, http://cvxr.com/cvx/, 2014. Google Scholar [11] P. Horst, P.M. Pardolos and N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, 1995.   Google Scholar [12] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar [13] V. Kalyagin, A. Koldanov, P. Koldanov and V. Zamaraev, Market graph and markowtiz model, Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos M. Pardalos, Springer Science (2014), 293-306. Google Scholar [14] G.H. Lan, An optimal method for stochastic composite optimization, Mathematical Programming, 133 (2012), 365-397.  doi: 10.1007/s10107-010-0434-y.  Google Scholar [15] Q. Li and Y.Q. Bai, Optimal trade-off portfolio selection between total risk and maximum relative marginal risk (accepted), Optimization Methods & Software, 31 (2016), 681-700.  doi: 10.1080/10556788.2015.1041946.  Google Scholar [16] J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Mathematical Programming, 103 (2005), 251-282.  doi: 10.1007/s10107-005-0582-7.  Google Scholar [17] H.M. Markowitz, Portfolio selection, Journal of Finace, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar [18] Y.E. Nesterov, A method for unconstrained convex minimization problem with the rate of convergence $\mathcal{O}(1/k^2)$, Doklady AN SSSR, 269 (1983), 543-547.   Google Scholar [19] Y. E. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2003. doi: 10.007/978-1-4419-8853-9.  Google Scholar [20] Y.E. Nestrov, Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar [21] U. Raber, A simplicial branch-and-bound method for solving nonconvex all-quadratic programs, Journal of Global Optimization, 13 (1998), 417-432.  doi: 10.1023/A:1008377529330.  Google Scholar [22] B. Scherer, Can rubust portfolio optimization help to build better portfolios, Journal of Asset Management, 7 (2007), 374-387.  doi: 10.1057/palgrave.jam.2250049.  Google Scholar [23] X.L. Sun, X.J. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, Journal of the Operations Research Society of China, 1 (2013), 55-77.  doi: 10.1007/s40305-013-0004-0.  Google Scholar [24] Y.F. Sun, A. Grace, K.L. Teo and G.L. Zhou, Portfolio optimization using a new probabilistic risk measure, Journal of Industrial and Management Optimization, 11 (2015), 1275-1283.  doi: 10.3934/jimo.2015.11.1275.  Google Scholar [25] K.L. Teo and X.Q. Yang, Portfolio selection problem with minimax type risk function, Annals of Operations Research, 101 (2001), 333-349.  doi: 10.1023/A:1010909632198.  Google Scholar [26] Y. Tian, S.C. Fang, Z.B. Deng and Q.W. Jin, Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Industrial and Management Optimization, 12 (2016), 1041-1056.  doi: 10.3934/jimo.2016.12.1041.  Google Scholar [27] S.S. Zhu, D. Li and X.L. Sun, Portfolio selection with marginal risk control, The Journal of Computational Finance, 14 (2010), 3-28.  doi: 10.21314/JCF.2010.213.  Google Scholar
Efficient frontiers of (TMVsc) and (MV)
Efficient frontiers of (TMVsc) and (MV) under stress scenario
Out-of-sample performance related to accumulated returns
Performance of the CPU time for different value of m/n
 Algorithm 2.1. The AG algorithm [8] Step 0. Input $x_0\in {{\rm{R}}^n}$, $\{\alpha_k\}$ s.t. $\alpha_1=1$ and $\alpha_k\in\{0,1\}$ for any $k\geq 2$, $\{\beta_k>0\}$, $\{\lambda_k>0\}$, and the accuracy parameter $\epsilon$. Let $x_0^{ag}=x_0$, $k=1$. Step 1. Let $x_{k}^{md}=(1-\alpha_k)x_{k-1}^{ag}+\alpha_kx_{k-1}.$ Step 2.Compute $\nabla \Psi(x_k^{md})$, let ${x_k} = {\cal P}({x_{k - 1}},\nabla \Psi (x_k^{md}),{\lambda _k}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$ $x_k^{ag} = {\cal P}(x_k^{md}, \nabla \Psi (x_k^{md}), {\beta _k}).\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (3)$ Step 3. If $\|\mathcal{G}\|\leq \epsilon$, stop. Else $k=k+1$, go to Step 1.
 Algorithm 2.1. The AG algorithm [8] Step 0. Input $x_0\in {{\rm{R}}^n}$, $\{\alpha_k\}$ s.t. $\alpha_1=1$ and $\alpha_k\in\{0,1\}$ for any $k\geq 2$, $\{\beta_k>0\}$, $\{\lambda_k>0\}$, and the accuracy parameter $\epsilon$. Let $x_0^{ag}=x_0$, $k=1$. Step 1. Let $x_{k}^{md}=(1-\alpha_k)x_{k-1}^{ag}+\alpha_kx_{k-1}.$ Step 2.Compute $\nabla \Psi(x_k^{md})$, let ${x_k} = {\cal P}({x_{k - 1}},\nabla \Psi (x_k^{md}),{\lambda _k}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$ $x_k^{ag} = {\cal P}(x_k^{md}, \nabla \Psi (x_k^{md}), {\beta _k}).\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (3)$ Step 3. If $\|\mathcal{G}\|\leq \epsilon$, stop. Else $k=k+1$, go to Step 1.
 Algorithm 4.1. The modified AG algorithm Step 0. Input a feasible solution $x_0$ and the accuracy parameter $\epsilon$. Let $x_0^{ag}=x_0$, $k=1$, $\alpha_k=\frac{2}{k+1}, \beta_k= \frac{1}{2L_{\Psi}}$. Step 1. Let $x_k^{md} = (1 - {\alpha _k})x_{k - 1}^{ag} + {\alpha _k}{x_{k - 1}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 4 \right)$ Step 2. Compute $\nabla\Psi(x_k^{md})$, let $x_k^{ag} = {\cal P}(x_k^{md},\nabla \Psi (x_k^{md}),{\beta _k}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 5 \right)$ ${x_k} = x_{k - 1}^{ag} + \frac{1}{{{\alpha _k}}}(x_k^{ag} - x_{k - 1}^{ag}).\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 6 \right)$ Step 3. If $\|\mathcal{G}\|\leq \epsilon$, stop. Else $k=k+1$, go to Step 1.
 Algorithm 4.1. The modified AG algorithm Step 0. Input a feasible solution $x_0$ and the accuracy parameter $\epsilon$. Let $x_0^{ag}=x_0$, $k=1$, $\alpha_k=\frac{2}{k+1}, \beta_k= \frac{1}{2L_{\Psi}}$. Step 1. Let $x_k^{md} = (1 - {\alpha _k})x_{k - 1}^{ag} + {\alpha _k}{x_{k - 1}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 4 \right)$ Step 2. Compute $\nabla\Psi(x_k^{md})$, let $x_k^{ag} = {\cal P}(x_k^{md},\nabla \Psi (x_k^{md}),{\beta _k}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 5 \right)$ ${x_k} = x_{k - 1}^{ag} + \frac{1}{{{\alpha _k}}}(x_k^{ag} - x_{k - 1}^{ag}).\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 6 \right)$ Step 3. If $\|\mathcal{G}\|\leq \epsilon$, stop. Else $k=k+1$, go to Step 1.
33 stocks from S & P 500
 Stock Name Stock Name 1 MASTERCARD 18 MEDCO HEALTH SLTN. 2 PRICELINE.COM 19 SOUTHWESTERN ENERGY 3 MCDONALDS 20 CVS CAREMARK 4 AUTOZONE 21 J M SMUCKER 5 WATSON PHARMS. 22 URBAN OUTFITTERS 6 FAMILY DOLLAR STORES 23 APOLLO GP.'A' 7 PERRIGO 24 CELGENE 8 STERICYCLE 25 INTERCONTINENTAL EX. 9 INTUITIVE SURGICAL 26 LOCKHEED MARTIN 10 EDWARDS LIFESCIENCES 27 ALTRIA GROUP 11 GOODRICH 28 HORMEL FOODS 12 FIDELITY NAT.INFO.SVS. 29 NETFLIX 13 F5 NETWORKS 30 MICRON TECHNOLOGY 14 WAL MART STORES 31 VARIAN MED.SYS. 15 COCA COLA 32 BROWN-FORMAN 'B' 16 BIOGEN IDEC 33 GAMESTOP 'A' 17 TRAVELERS COS.
 Stock Name Stock Name 1 MASTERCARD 18 MEDCO HEALTH SLTN. 2 PRICELINE.COM 19 SOUTHWESTERN ENERGY 3 MCDONALDS 20 CVS CAREMARK 4 AUTOZONE 21 J M SMUCKER 5 WATSON PHARMS. 22 URBAN OUTFITTERS 6 FAMILY DOLLAR STORES 23 APOLLO GP.'A' 7 PERRIGO 24 CELGENE 8 STERICYCLE 25 INTERCONTINENTAL EX. 9 INTUITIVE SURGICAL 26 LOCKHEED MARTIN 10 EDWARDS LIFESCIENCES 27 ALTRIA GROUP 11 GOODRICH 28 HORMEL FOODS 12 FIDELITY NAT.INFO.SVS. 29 NETFLIX 13 F5 NETWORKS 30 MICRON TECHNOLOGY 14 WAL MART STORES 31 VARIAN MED.SYS. 15 COCA COLA 32 BROWN-FORMAN 'B' 16 BIOGEN IDEC 33 GAMESTOP 'A' 17 TRAVELERS COS.
Performance related to the ratio of mean to standard deviation
 Expected return(×10−3) $\rho= 7$ $\rho=8$ $\rho= 9$ mean mean mean standard deviation standard deviation standard deviation MV 0.2050 0.2007 0.1959 TMV$_{sc}(\tau=5)$ 0.2175 0.2122 0.2036 TMV$_{sc}(\tau=10)$ 0.2134 0.2055 0.1986 TMV$_{sc}(\tau=20)$ 0.2089 0.2013 0.1956 Optimal portfolio 0.2225 0.2127 0.2006
 Expected return(×10−3) $\rho= 7$ $\rho=8$ $\rho= 9$ mean mean mean standard deviation standard deviation standard deviation MV 0.2050 0.2007 0.1959 TMV$_{sc}(\tau=5)$ 0.2175 0.2122 0.2036 TMV$_{sc}(\tau=10)$ 0.2134 0.2055 0.1986 TMV$_{sc}(\tau=20)$ 0.2089 0.2013 0.1956 Optimal portfolio 0.2225 0.2127 0.2006
Out-of-sample performance related to the ratio of mean to standard deviation
 Expected return(×10−3) $\rho= 7$ $\rho=8$ $\rho= 9$ mean mean mean standard deviation standard deviation standard deviation MV 0.1910 0.1877 0.1832 TMV$_{sc}(\tau=5)$ 0.2016 0.1986 0.1923 Optimal portfolio 0.2066 0.1967 0.1874
 Expected return(×10−3) $\rho= 7$ $\rho=8$ $\rho= 9$ mean mean mean standard deviation standard deviation standard deviation MV 0.1910 0.1877 0.1832 TMV$_{sc}(\tau=5)$ 0.2016 0.1986 0.1923 Optimal portfolio 0.2066 0.1967 0.1874
Discussion of τ for S & P 500
 ρ (×10−3) τ Rate(×10−3) RateR(%) Sensitivity(×10−3) RateS (%) MV 0.4086 - 3.9889 - $\tau=5$ 0.4902 19.95 1.7136 57.04 7 $\tau=10$ 0.4276 4.65 1.9490 51.14 $\tau=15$ 0.4210 3.02 2.0277 49.17 $\tau=20$ 0.4186 2.43 2.0697 48.11 MV 0.5145 - 4.3323 - $\tau=5$ 0.5908 14.84 2.4956 42.40 8 $\tau=10$ 0.5444 5.81 2.8153 35.02 $\tau=15$ 0.5351 4.01 2.9232 32.53 $\tau=20$ 0.5305 3.11 3.0025 30.70 MV 0.6511 - 6.1201 - $\tau=5$ 0.7345 12.82 3.8247 37.51 9 $\tau=10$ 0.6852 5.25 4.1689 31.88 $\tau=15$ 0.6758 3.80 4.2829 30.02 $\tau=20$ 0.6726 3.31 4.3406 29.08
 ρ (×10−3) τ Rate(×10−3) RateR(%) Sensitivity(×10−3) RateS (%) MV 0.4086 - 3.9889 - $\tau=5$ 0.4902 19.95 1.7136 57.04 7 $\tau=10$ 0.4276 4.65 1.9490 51.14 $\tau=15$ 0.4210 3.02 2.0277 49.17 $\tau=20$ 0.4186 2.43 2.0697 48.11 MV 0.5145 - 4.3323 - $\tau=5$ 0.5908 14.84 2.4956 42.40 8 $\tau=10$ 0.5444 5.81 2.8153 35.02 $\tau=15$ 0.5351 4.01 2.9232 32.53 $\tau=20$ 0.5305 3.11 3.0025 30.70 MV 0.6511 - 6.1201 - $\tau=5$ 0.7345 12.82 3.8247 37.51 9 $\tau=10$ 0.6852 5.25 4.1689 31.88 $\tau=15$ 0.6758 3.80 4.2829 30.02 $\tau=20$ 0.6726 3.31 4.3406 29.08
Discussion of τ for HK stocks
 ρ (×10−3) τ Rate(×10−3) RateR(%) Sensitivity(×10−3) RateS (%) MV 0.2469 - 7.9021 - $\tau=10$ 0.2819 14.18 4.8713 38.35 3 $\tau=20$ 0.2616 5.95 5.4896 30.53 $\tau=30$ 0.2595 5.10 5.6577 28.40 $\tau=40$ 0.2582 4.58 5.4896 26.64 MV 0.3118 - 8.2937 - $\tau=10$ 0.3505 12.41 5.386 35.06 3.5 $\tau=20$ 0.3356 7.63 5.6569 31.79 $\tau=30$ 0.3308 6.09 5.9154 28.68 $\tau=40$ 0.3272 4.94 6.1602 25.72 MV 0.4445 - 9.7782 - $\tau=10$ 0.4500 1.24 9.1207 6.72 4 $\tau=20$ 0.4482 0.83 9.2027 5.89 $\tau=30$ 0.4472 0.61 9.2756 5.14 $\tau=40$ 0.4462 0.40 9.3756 4.43
 ρ (×10−3) τ Rate(×10−3) RateR(%) Sensitivity(×10−3) RateS (%) MV 0.2469 - 7.9021 - $\tau=10$ 0.2819 14.18 4.8713 38.35 3 $\tau=20$ 0.2616 5.95 5.4896 30.53 $\tau=30$ 0.2595 5.10 5.6577 28.40 $\tau=40$ 0.2582 4.58 5.4896 26.64 MV 0.3118 - 8.2937 - $\tau=10$ 0.3505 12.41 5.386 35.06 3.5 $\tau=20$ 0.3356 7.63 5.6569 31.79 $\tau=30$ 0.3308 6.09 5.9154 28.68 $\tau=40$ 0.3272 4.94 6.1602 25.72 MV 0.4445 - 9.7782 - $\tau=10$ 0.4500 1.24 9.1207 6.72 4 $\tau=20$ 0.4482 0.83 9.2027 5.89 $\tau=30$ 0.4472 0.61 9.2756 5.14 $\tau=40$ 0.4462 0.40 9.3756 4.43
Comparison Algorithm 4.1 and Algorithm 2.1
 $\tau$ $\rho~(\times 10^{-3})$ Opt Algorithm 4.1 Algorithm 2.1 CPU time $N_{iter}$ CPU time $N_{iter}$ 5 5.5 0.0171 96.9 58 670.6 214 5 6 0.0170 151.0 88 ≥ 1000 ≥ 279 5 7 0.0304 142.5 85 ≥ 1000 ≥ 300 10 5.5 0.0297 104.2 62 279.0 82 10 6 0.0312 123.8 75 302.2 88 10 7 0.0424 80.2 43 ≥ 1000 ≥ 327 20 5.5 0.0351 173.0 102 627.1 186 20 6 0.0366 141.0 84 350.5 105 20 7 0.0494 92.1 52 ≥ 1000 ≥ 350
 $\tau$ $\rho~(\times 10^{-3})$ Opt Algorithm 4.1 Algorithm 2.1 CPU time $N_{iter}$ CPU time $N_{iter}$ 5 5.5 0.0171 96.9 58 670.6 214 5 6 0.0170 151.0 88 ≥ 1000 ≥ 279 5 7 0.0304 142.5 85 ≥ 1000 ≥ 300 10 5.5 0.0297 104.2 62 279.0 82 10 6 0.0312 123.8 75 302.2 88 10 7 0.0424 80.2 43 ≥ 1000 ≥ 327 20 5.5 0.0351 173.0 102 627.1 186 20 6 0.0366 141.0 84 350.5 105 20 7 0.0494 92.1 52 ≥ 1000 ≥ 350
Numerical experiments for S & P 500
 n m min max average Niter 10 5 8.2 27.5 17.6 33 10 10 11.5 36.0 24.4 32 20 5 15.5 44.8 34.1 62 20 10 34.6 34.9 42.6 56 30 10 23.4 95.5 62.1 78 30 20 70.0 125.3 104.1 89 40 10 51.9 100.1 68.4 84 40 20 74.8 123.5 85.3 68 50 10 41.8 59.4 52.0 64 50 20 76.3 123.0 95.5 82 100 10 39.6 128.1 82.0 79 100 20 85.8 277.5 156.7 71 100 50 615.8 830.8 738.2 185 200 20 225.5 530.3 405.3 137 200 50 722.9 1106.8 911.8 165
 n m min max average Niter 10 5 8.2 27.5 17.6 33 10 10 11.5 36.0 24.4 32 20 5 15.5 44.8 34.1 62 20 10 34.6 34.9 42.6 56 30 10 23.4 95.5 62.1 78 30 20 70.0 125.3 104.1 89 40 10 51.9 100.1 68.4 84 40 20 74.8 123.5 85.3 68 50 10 41.8 59.4 52.0 64 50 20 76.3 123.0 95.5 82 100 10 39.6 128.1 82.0 79 100 20 85.8 277.5 156.7 71 100 50 615.8 830.8 738.2 185 200 20 225.5 530.3 405.3 137 200 50 722.9 1106.8 911.8 165
Numerical experiments for randomly data
 n m min max average Niter 10 5 12.8 14.5 13.3 25 10 10 36.7 37.4 36.9 51 20 5 7.1 7.4 7.3 14 20 10 14.6 34.3 24.8 32 30 10 13.2 15.5 14.5 20 30 20 45.4 46.9 46.1 39 40 10 9.4 10.0 9.7 13 40 20 22.7 26.1 24.0 19 50 10 8.6 10.4 9.5 13 50 20 22.3 24.2 23.1 19 100 10 5.8 12.8 9.3 10 100 20 16.6 28.8 22.6 13 100 50 47.1 54.9 51.5 15 200 20 33.3 39.6 36.5 17 200 50 50.9 78.1 66.7 14
 n m min max average Niter 10 5 12.8 14.5 13.3 25 10 10 36.7 37.4 36.9 51 20 5 7.1 7.4 7.3 14 20 10 14.6 34.3 24.8 32 30 10 13.2 15.5 14.5 20 30 20 45.4 46.9 46.1 39 40 10 9.4 10.0 9.7 13 40 20 22.7 26.1 24.0 19 50 10 8.6 10.4 9.5 13 50 20 22.3 24.2 23.1 19 100 10 5.8 12.8 9.3 10 100 20 16.6 28.8 22.6 13 100 50 47.1 54.9 51.5 15 200 20 33.3 39.6 36.5 17 200 50 50.9 78.1 66.7 14
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