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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

Yanqin Bai , , Yudan Wei and  Qian Li

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.

Keywords: Mean-variance model, accelerated gradient algorithm, sensitivity of parameters, quadratic programming problem, portfolio selection.
Mathematics Subject Classification: Primary: 68Q25, 90C20; Secondary: 91B99.
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

Figure 1.  Efficient frontiers of (TMVsc) and (MV)
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Figure 2.  Efficient frontiers of (TMVsc) and (MV) under stress scenario
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Figure 3.  Out-of-sample performance related to accumulated returns
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Figure 4.  Performance of the CPU time for different value of m/n
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Table  . 
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.
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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.
Table  . 
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.
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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.
Table 1.  33 stocks from S & P 500
StockNameStockName
1MASTERCARD18MEDCO HEALTH SLTN.
2PRICELINE.COM19SOUTHWESTERN ENERGY
3MCDONALDS20CVS CAREMARK
4AUTOZONE21J M SMUCKER
5WATSON PHARMS.22URBAN OUTFITTERS
6FAMILY DOLLAR STORES23APOLLO GP.'A'
7PERRIGO24CELGENE
8STERICYCLE25INTERCONTINENTAL EX.
9INTUITIVE SURGICAL26LOCKHEED MARTIN
10EDWARDS LIFESCIENCES27ALTRIA GROUP
11GOODRICH28HORMEL FOODS
12FIDELITY NAT.INFO.SVS.29NETFLIX
13F5 NETWORKS30MICRON TECHNOLOGY
14WAL MART STORES31VARIAN MED.SYS.
15COCA COLA32BROWN-FORMAN 'B'
16BIOGEN IDEC33GAMESTOP 'A'
17TRAVELERS COS.
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StockNameStockName
1MASTERCARD18MEDCO HEALTH SLTN.
2PRICELINE.COM19SOUTHWESTERN ENERGY
3MCDONALDS20CVS CAREMARK
4AUTOZONE21J M SMUCKER
5WATSON PHARMS.22URBAN OUTFITTERS
6FAMILY DOLLAR STORES23APOLLO GP.'A'
7PERRIGO24CELGENE
8STERICYCLE25INTERCONTINENTAL EX.
9INTUITIVE SURGICAL26LOCKHEED MARTIN
10EDWARDS LIFESCIENCES27ALTRIA GROUP
11GOODRICH28HORMEL FOODS
12FIDELITY NAT.INFO.SVS.29NETFLIX
13F5 NETWORKS30MICRON TECHNOLOGY
14WAL MART STORES31VARIAN MED.SYS.
15COCA COLA32BROWN-FORMAN 'B'
16BIOGEN IDEC33GAMESTOP 'A'
17TRAVELERS COS.
Table 2.  Performance related to the ratio of mean to standard deviation
Expected return(×10−3)$\rho= 7 $$\rho=8$$\rho= 9$
meanmeanmean
standard deviationstandard deviationstandard deviation
MV0.20500.20070.1959
TMV$_{sc}(\tau=5)$0.21750.21220.2036
TMV$_{sc}(\tau=10)$0.21340.20550.1986
TMV$_{sc}(\tau=20)$0.20890.20130.1956
Optimal portfolio0.2225 0.2127 0.2006
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Expected return(×10−3)$\rho= 7 $$\rho=8$$\rho= 9$
meanmeanmean
standard deviationstandard deviationstandard deviation
MV0.20500.20070.1959
TMV$_{sc}(\tau=5)$0.21750.21220.2036
TMV$_{sc}(\tau=10)$0.21340.20550.1986
TMV$_{sc}(\tau=20)$0.20890.20130.1956
Optimal portfolio0.2225 0.2127 0.2006
Table 3.  Out-of-sample performance related to the ratio of mean to standard deviation
Expected return(×10−3)$\rho= 7 $$\rho=8$$\rho= 9$
meanmeanmean
standard deviationstandard deviationstandard deviation
MV 0.19100.18770.1832
TMV$_{sc}(\tau=5)$ 0.20160.1986 0.1923
Optimal portfolio 0.2066 0.1967 0.1874
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Expected return(×10−3)$\rho= 7 $$\rho=8$$\rho= 9$
meanmeanmean
standard deviationstandard deviationstandard deviation
MV 0.19100.18770.1832
TMV$_{sc}(\tau=5)$ 0.20160.1986 0.1923
Optimal portfolio 0.2066 0.1967 0.1874
Table 4.  Discussion of τ for S & P 500
ρ
(×10−3)		 τ Rate
(×10−3)		RateR
(%)		 Sensitivity
(×10−3)		 RateS
(%)
MV0.4086-3.9889-
$\tau=5$0.490219.951.713657.04
7$\tau=10$0.42764.651.949051.14
$\tau=15$0.42103.022.027749.17
$\tau=20$0.41862.432.069748.11
MV0.5145-4.3323-
$\tau=5$0.590814.842.495642.40
8$\tau=10$0.54445.812.815335.02
$\tau=15$0.53514.012.923232.53
$\tau=20$0.53053.113.002530.70
MV0.6511-6.1201-
$\tau=5$0.734512.823.824737.51
9$\tau=10$0.68525.254.168931.88
$\tau=15$0.67583.804.282930.02
$\tau=20$0.67263.314.340629.08
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ρ
(×10−3)		 τ Rate
(×10−3)		RateR
(%)		 Sensitivity
(×10−3)		 RateS
(%)
MV0.4086-3.9889-
$\tau=5$0.490219.951.713657.04
7$\tau=10$0.42764.651.949051.14
$\tau=15$0.42103.022.027749.17
$\tau=20$0.41862.432.069748.11
MV0.5145-4.3323-
$\tau=5$0.590814.842.495642.40
8$\tau=10$0.54445.812.815335.02
$\tau=15$0.53514.012.923232.53
$\tau=20$0.53053.113.002530.70
MV0.6511-6.1201-
$\tau=5$0.734512.823.824737.51
9$\tau=10$0.68525.254.168931.88
$\tau=15$0.67583.804.282930.02
$\tau=20$0.67263.314.340629.08
Table 5.  Discussion of τ for HK stocks
ρ
(×10−3)		 τ Rate
(×10−3)		RateR
(%)		 Sensitivity
(×10−3)		 RateS
(%)
MV0.2469-7.9021-
$\tau=10$0.281914.184.871338.35
3$\tau=20$0.26165.955.489630.53
$\tau=30$0.25955.105.657728.40
$\tau=40$0.25824.585.489626.64
MV0.3118-8.2937-
$\tau=10$0.350512.415.38635.06
3.5$\tau=20$0.33567.635.656931.79
$\tau=30$0.33086.095.915428.68
$\tau=40$0.32724.946.160225.72
MV0.4445-9.7782-
$\tau=10$0.45001.249.12076.72
4$\tau=20$0.44820.839.20275.89
$\tau=30$0.44720.619.27565.14
$\tau=40$0.44620.409.37564.43
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ρ
(×10−3)		 τ Rate
(×10−3)		RateR
(%)		 Sensitivity
(×10−3)		 RateS
(%)
MV0.2469-7.9021-
$\tau=10$0.281914.184.871338.35
3$\tau=20$0.26165.955.489630.53
$\tau=30$0.25955.105.657728.40
$\tau=40$0.25824.585.489626.64
MV0.3118-8.2937-
$\tau=10$0.350512.415.38635.06
3.5$\tau=20$0.33567.635.656931.79
$\tau=30$0.33086.095.915428.68
$\tau=40$0.32724.946.160225.72
MV0.4445-9.7782-
$\tau=10$0.45001.249.12076.72
4$\tau=20$0.44820.839.20275.89
$\tau=30$0.44720.619.27565.14
$\tau=40$0.44620.409.37564.43
Table 6.  Comparison Algorithm 4.1 and Algorithm 2.1
$\tau$$\rho~(\times 10^{-3})$OptAlgorithm 4.1Algorithm 2.1
CPU time$N_{iter}$CPU time$N_{iter}$
55.50.017196.958670.6214
560.0170151.088≥ 1000≥ 279
570.0304142.585≥ 1000≥ 300
105.50.0297104.262279.082
1060.0312123.875302.288
1070.042480.243≥ 1000≥ 327
205.50.0351173.0102627.1186
2060.0366141.084350.5105
2070.049492.152≥ 1000≥ 350
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$\tau$$\rho~(\times 10^{-3})$OptAlgorithm 4.1Algorithm 2.1
CPU time$N_{iter}$CPU time$N_{iter}$
55.50.017196.958670.6214
560.0170151.088≥ 1000≥ 279
570.0304142.585≥ 1000≥ 300
105.50.0297104.262279.082
1060.0312123.875302.288
1070.042480.243≥ 1000≥ 327
205.50.0351173.0102627.1186
2060.0366141.084350.5105
2070.049492.152≥ 1000≥ 350
Table 7.  Numerical experiments for S & P 500
nmminmaxaverageNiter
1058.227.517.633
101011.536.024.432
20515.544.834.162
201034.634.942.656
301023.495.562.178
302070.0125.3104.189
401051.9100.168.484
402074.8123.585.368
501041.859.452.064
502076.3123.095.582
1001039.6128.182.079
1002085.8277.5156.771
10050615.8830.8738.2185
20020225.5530.3405.3137
20050722.91106.8911.8165
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nmminmaxaverageNiter
1058.227.517.633
101011.536.024.432
20515.544.834.162
201034.634.942.656
301023.495.562.178
302070.0125.3104.189
401051.9100.168.484
402074.8123.585.368
501041.859.452.064
502076.3123.095.582
1001039.6128.182.079
1002085.8277.5156.771
10050615.8830.8738.2185
20020225.5530.3405.3137
20050722.91106.8911.8165
Table 8.  Numerical experiments for randomly data
nmminmaxaverageNiter
10512.814.513.325
101036.737.436.951
2057.17.47.314
201014.634.324.832
301013.215.514.520
302045.446.946.139
40109.410.09.713
402022.726.124.019
50108.610.49.513
502022.324.223.119
100105.812.89.310
1002016.628.822.613
1005047.154.951.515
2002033.339.636.517
2005050.978.166.714
Table Options

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nmminmaxaverageNiter
10512.814.513.325
101036.737.436.951
2057.17.47.314
201014.634.324.832
301013.215.514.520
302045.446.946.139
40109.410.09.713
402022.726.124.019
50108.610.49.513
502022.324.223.119
100105.812.89.310
1002016.628.822.613
1005047.154.951.515
2002033.339.636.517
2005050.978.166.714
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