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

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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.
[12]	R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
[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.
[12]	R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
[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 (TMV_sc) and (MV)

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Figure 2. Efficient frontiers of (TMV_sc) 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

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.

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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.

Table 2. 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

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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

Table 3. Out-of-sample performance related to the ratio of mean to standard deviation

Expected return(×10⁻³)	$\rho= 7 $	$\rho=8$	$\rho= 9$
	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

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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

Table 4. Discussion of τ for S & P 500

ρ (×10⁻³)	τ	Rate (×10⁻³)	Rate_R (%)	Sensitivity (×10⁻³)	Rate_S (%)
	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

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ρ (×10⁻³)	τ	Rate (×10⁻³)	Rate_R (%)	Sensitivity (×10⁻³)	Rate_S (%)
	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

Table 5. Discussion of τ for HK stocks

ρ (×10⁻³)	τ	Rate (×10⁻³)	Rate_R (%)	Sensitivity (×10⁻³)	Rate_S (%)
	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

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ρ (×10⁻³)	τ	Rate (×10⁻³)	Rate_R (%)	Sensitivity (×10⁻³)	Rate_S (%)
	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

Table 6. Comparison Algorithm 4.1 and Algorithm 2.1

$\tau$	$\rho~(\times 10^{-3})$	Opt	Algorithm 4.1		Algorithm 2.1
$\tau$	$\rho~(\times 10^{-3})$	Opt	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

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$\tau$	$\rho~(\times 10^{-3})$	Opt	Algorithm 4.1		Algorithm 2.1
$\tau$	$\rho~(\times 10^{-3})$	Opt	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

Table 7. Numerical experiments for S & P 500

n	m	min	max	average	N_iter
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

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n	m	min	max	average	N_iter
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

Table 8. Numerical experiments for randomly data

n	m	min	max	average	N_iter
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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n	m	min	max	average	N_iter
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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American Institute of Mathematical Sciences