Sparse markowitz portfolio selection by using stochastic linear complementarity approach

Qiyu Wang; Hailin Sun

doi:10.3934/jimo.2017059

Article Contents

2018, Volume 14, Issue 2: 541-559. Doi: 10.3934/jimo.2017059

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Sparse markowitz portfolio selection by using stochastic linear complementarity approach

Qiyu Wang^1, and
Hailin Sun^2, ,

1.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
2.
School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China

* Corresponding author
* Corresponding author

Received: January 2016

Early access: June 2017

Published online: June 2017

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Abstract

We consider the framework of the classical Markowitz mean-variance (MV) model when multiple solutions exist, among which the sparse solutions are stable and cost-efficient. We study a two - phase stochastic linear complementarity approach. This approach stabilizes the optimization problem, finds the sparse asset allocation that saves the transaction cost, and results in the solution set of the Markowitz problem. We apply the sample average approximation (SAA) method to the two - phase optimization approach and give detailed convergence analysis. We implement this methodology on the data sets of Standard and Poor 500 index (S & P 500), real data of Hong Kong and China market stocks (HKCHN) and Fama & French 48 industry sectors (FF48). With mock investment in training data, we construct portfolios, test them in the out-of-sample data and find their Sharpe ratios outperform the $\ell_1$ penalty regularized portfolios, $\ell_p$ penalty regularized portfolios, cardinality constrained portfolios, and $1/N$ investment strategy. Moreover, we show the advantage of our approach in the risk management by using the criteria of standard deviation (STD), Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).

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Mathematics Subject Classification: Primary: 90C15, 90C90; Secondary: 91G10.

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Figure 1. The convergence of the SAA problem in Example 4.1

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Figure 2. S & P 500 Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS100 and 1/N

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Figure 3. Hong Kong and Mainland China Cross Market Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS20, CCPS25, $\ell_p$ 0.015 and 1/N

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Figure 4. FF48 Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS18, CCPS24, $\ell_p$ 0.015 and 1/N

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Table 1. Convergence analysis of SAA sparse portfolio optimal value (STD) for Example 4.1

N	true	500	1500	3000	4500	6000	7500	9000	10000	dmissing
Val	2.489	2.490	2.487	2.487	2.487	2.489	2.488	2.490	2.486	2.496

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Table 2. S & P 500 Portfolio return, STD, Sharpe Ratio, sparsity, VaR, CVaR and distance

S & P 500	LCPSP	$\ell_1$ $0.1$	CCPS100	1/N
return	0.001	0.000823	0.0014	-0.00003
STD	0.0024	0.002287	0.0084	0.0074
Sharpe	0.3989	0.359736	0.1694	-0.0045
VaR	0.0041	0.004289	0.0115	0.0131
CVaR	0.0046	0.004292	0.0147	0.0131
sparsity	89(406)	66.25	58.6	500
distance		1.00E-05	3.50E-07

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Table 3. Hong Kong and Mainland China Cross Market Portfolio return, STD, Sharpe Ratio, sparsity, VaR and CVaR and distance

HKCHN	LCPSP	$\ell_1$ $0.1$	CCPS 20	CCPS25	$\ell_p$ 0.015	1/N
return	0.001296	0.00044	0.001348	0.001583	0.000537	-0.00171
STD	0.007345	0.007115	0.013617	0.012368	0.010248	0.006009
Sharpe	0.1764	0.061903	0.099	0.128	0.0524	-0.2841
VaR	0.013248	0.011514	0.023592	0.024182	0.014756	0.010944
CVaR	0.01408	0.011807	0.026692	0.024382	0.015791	0.010944
sparsity	27(49)	14	19.45	23.1	24.1
distance		0.0024	0.003406	0.00012	0.144709

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Table 4. FF48 Portfolio return, STD, Sharpe Ratio, VaR, CVaR, sparsity and distance

FF48	LCPSP	$\ell_1$ $0.1$	CCPS18	CCPS 24	$\ell_p$ 0.015	1/N
return	-0.1201	-0.13259	-0.6413	-0.3736	-0.3334	-0.7838
STD	5.9265	5.917113	8.5639	7.5703	5.3545	8.002
Sharpe	-0.0203	-0.0224	-0.0749	-0.0493	-0.0623	-0.098
VaR	8.5242	14.59896	12.8905	9.8635	10.0079	7.827
CVaR	10.3835	14.86594	15.514	12.8947	13.5395	10.0896
sparsity	29(48)	24.35	21.45	7.2	31.2	48
distance		0.0044	0.2232	0.5608	0.0938

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