A multistage stochastic programming framework for cardinality constrained portfolio optimization

Ardeshir Ahmadi; Hamed Davari-Ardakani

doi:10.3934/naco.2017023

Article Contents

2017, Volume 7, Issue 3: 359-377. Doi: 10.3934/naco.2017023

This issue Previous Article A type of new consensus protocol for two-dimension multi-agent systems Next Article

A multistage stochastic programming framework for cardinality constrained portfolio optimization

Ardeshir Ahmadi^1, and
Hamed Davari-Ardakani^2, ,

1.
Department of Systems Engineering, IHU University, Tehran, Iran
2.
Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

* Corresponding author
* Corresponding author

Received: December 2016

Revised: July 2017

Published: October 2017

This paper was prepared at the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), Tehran, Iran, January 25-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Assoc. Prof. A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

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Abstract

This paper presents a multistage stochastic programming model to deal with multi-period, cardinality constrained portfolio optimization. The presented model aims to minimize investor's expected regret, while ensuring achievement of a minimum expected return. To generate scenarios of market index returns, a random walk model based on the empirical distribution of market-representative index returns is proposed. Then, a single index model is used to estimate stock returns based on market index returns. Afterward, historical returns of a number of stocks, selected from Frankfurt Stock Exchange (FSE), are used to implement the presented scenario generation method, and solve the stochastic programming model. In addition, the impact of cardinality constraints, transaction costs, minimum expected return and predetermined investor's target wealth are investigated. Results show that the inclusion of cardinality constraints and transaction costs significantly influences the investors risk-return tradeoffs. This is also the case for investors target wealth.

Keywords:

Mathematics Subject Classification: 90C11, 90C15, 90C90.

Citation:

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Figure 1. The schematic representation of a scenario tree for T periods

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Figure 2. A schematic representation of the proposed scenario tree generation method

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Figure 3. Risk vs. expected return for portfolios with and without cardinality constraints (Target wealth = ＄1000000)

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Figure 4. Risk vs. expected return obtained by setting different levels of target wealth (＄1000000 and ＄1050000) for portfolios with and without cardinality constraints

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Figure 5. Investor's risk for different levels of proportional transaction costs

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Table 1. Descriptive statistics of historical CDAX returns

Mean	Standard Deviation	Median	Minimum	Maximum	Skewness	Kurtosis
0.0060	0.0569	0.0103	-0.1795	0.1745	-0.5381	1.7814

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Table 2. α_i and β_i values of the single index model for all stocks

Stock	B & A	LR81	LTEC	MZA	NEC1	N2X	OTP
Intercept	0.015231	0.0008692	-0.0028	0.039533	-3.1E-05	0.001772	-0.01099
Slope	0.756845	1.211379	0.889253	1.837928	0.644086	0.971493	1.961487
Stock	SIE	TAH	BMW	XCY	O4B	ZYT	-
Intercept	-.00063	0.006095	0.0098	0.024254	0.001565	0.003712	-
Slope	1.091311	0.292933	1.186136	0.592039	0.564903	1.498048	-

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Table 3. Investor's expected regret considering different target wealth, minimum expected return and proportional transaction costs

Target wealth	1000000			1050000			1100000
Proportional transaction cost	0	0.01	0.02	0	0.01	0.02	0	0.01	0.02
0.95	0	0	0	63429.6	85112.7	103678.9	157646.9	196749.4	223658.1
0.99	0	0	0	63429.6	85112.7	103678.9	157646.9	196749.4	223658.1
1.01	0	1290.1	2987.9	63429.6	85112.7	103678.9	157646.9	196749.4	223658.1
1.03	36.1	3953.9	9018.7	63429.6	85112.7	103678.9	157646.9	196749.4	223658.1
1.04	400.0	5567.1	12654.5	63429.6	85112.7	103678.9	157646.9	196749.4	223658.1
1.05	1142.6	7705.6	17739.6	63429.6	85112.7	104334.1	157646.9	196749.4	223658.1
1.06	2350.1	11121.8	26179.7	63429.6	85134.9	109571.7	157646.9	196749.4	223838.3
1.07	3904.2	15907.5	37152.7	63429.6	87879.1	118868.3	157646.9	198064.3	226873.5
1.08	5954.9	22562.9	49694.6	63429.6	95349.1	129104.5	157646.9	202106.9	232405.6
1.09	8525.3	36300.4	66271.2	63694.1	106415.1	140267.7	157646.9	208788.3	240209
1.10	12086.2	54243.7	88133.5	64838.1	120626.0	157811.5	157675.5	217600.2	251621.1
1.11	17279.9	74827.6	-	67119.8	138148.1	-	158591.5	228534.1	-
1.12	24358.6	98255.0	-	74520.5	159434.9	-	163337.6	242911.9	-
1.13	52656.5	-	-	104774.3	-	-	185917.5	-	-
1.14	-	-	-	-	-	-	-	-	-

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Table 4. Investor's expected regret considering different target wealth and minimum expected return with and without cardinality

	Cardinality Constraints			No Cardinality Constraints
Target wealth	1000000	1050000	1100000	1000000	1050000	1100000
0.95	0	81742.31	192944.5	0	63429.62	157646.9
0.99	0	81742.31	192944.5	0	63429.62	157646.9
1	0	81742.31	192944.5	0	63429.62	157646.9
1.01	0	81742.31	192944.5	0	63429.62	157646.9
1.02	197.428	81742.31	192944.5	0	63429.62	157646.9
1.03	893.094	81742.31	192944.5	36.097	63429.62	157646.9
1.04	2574.389	81742.31	192944.5	399.947	63429.62	157646.9
1.05	5261.672	81879.22	192944.5	1142.637	63429.62	157646.9
1.06	8917.349	82803.35	192944.5	2350.142	63429.62	157646.9
1.07	18358.44	87336.35	193443	3904.241	63429.62	157646.9
1.08	35077.99	96174.55	198126.4	5954.918	63429.62	157646.9
1.09	-	-	-	8525.318	63694.05	157646.9
1.10	-	-	-	12086.15	64838.09	157675.5
1.11	-	-	-	17279.88	67119.82	158591.5
1.12	-	-	-	24358.63	74520.5	163337.6
1.13	-	-	-	52656.51	104774.3	185917.5
1.14	-	-	-	-	-	-

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Table 5. Investor's expected regret considering different proportional transaction costs and number of assets

Number of assets	6			12
Proportional transaction cost	0	0.01	0.02	0	0.01	0.02
0.95	229314.2	275191.5	318875.4	192944.5	225372.1	258752.1
0.99	229314.2	275191.5	318875.4	192944.5	225372.1	258752.1
1.01	229314.2	275191.5	318875.4	192944.5	225372.1	258752.1
1.03	229314.2	275191.5	318875.4	192944.5	225372.1	258752.1
1.04	229314.2	275191.5	318875.4	192944.5	225372.1	258752.1
1.05	229314.2	275191.5	320958.9	192944.5	225372.1	258752.1
1.06	229314.2	277364.2	338961.2	192944.5	225372.1	258752.1
1.07	229314.2	280367.1	-	192944.5	230553.7	263452.1
1.08	229314.3	-	-	192944.5	235638.9	-
1.09	231175.9	-	-	193443.0	239987.4	-
1.10	-	-	-	198126.4	-	-
1.11	-	-	-	-	-	-

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