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.

Abstract Full Text(HTML) Figure(5) / Table(5) Related Papers Cited by

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:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

| Show Table

DownLoad: CSV

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	-

| Show Table

DownLoad: CSV

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	-	-	-	-	-	-	-	-	-

| Show Table

DownLoad: CSV

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	-	-	-	-	-	-

| Show Table

DownLoad: CSV

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	-	-	-	-	-	-

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	D. Barro and E. Canestrelli, Tracking error: a multistage portfolio model, Ann. Oper. Res., 165 (2009), 47-66. doi: 10.1007/s10479-007-0308-8.
[2]	M. R. Borges, Efficient market hypothesis in European stock markets, Eur. J. Financ., 16 (2010), 711-726.
[3]	W. Chen, Artificial bee colony algorithm for constrained possibilistic portfolio optimization problem, Physica A., 429 (2015), 125-139. doi: 10.1016/j.physa.2015.02.060.
[4]	Z. Chen, Multiperiod consumption and portfolio decisions under the multivariate GARCH model with transaction costs and CVaR-based risk control, OR Spectrum., 27 (2005), 603-632.
[5]	Z. Chen and D. Xu, Knowledge-based scenario tree generation methods and application in multiperiod portfolio selection problem, Appl. Stoch. Model. Bus., 30 (2014), 240-257. doi: 10.1002/asmb.1970.
[6]	Y. W. Cheung and K. S. Lai, A search for long memory in international stock market returns, J. Int. Money. Financ., 14 (1995), 597-615.
[7]	A. Consiglio and A. Staino, A stochastic programming model for the optimal issuance of government bonds, Ann. Oper. Res., 193 (2012), 159-172. doi: 10.1007/s10479-010-0755-5.
[8]	G. B. Dantzig and G. Infanger, Multi-stage stochastic linear programs for portfolio optimization, Ann. Oper. Res., 45 (1993), 59-76. doi: 10.1007/BF02282041.
[9]	H. Davari-Ardakani, M. Aminnayeri and A. Seifi, A study on modeling the dynamics of statistically dependent returns, Physica A., 405 (2014), 35-51. doi: 10.1016/j.physa.2014.02.077.
[10]	H. Davari-Ardakani, M. Aminnayeri and A. Seifi, Hedging strategies for multi-period portfolio optimization, Sci. Iran., 22 (2015), 2644-2663.
[11]	H. Davari-Ardakani, M. Aminnayeri and A. Seifi, Multistage portfolio optimization with stocks and options, Int. Trans. Oper. Res., 23 (2016), 593-622. doi: 10.1111/itor.12174.
[12]	R. Ferstl and A. Weissensteiner, Cash management using multi-stage stochastic programming, Quant. Financ., 10 (2010), 209-219. doi: 10.1080/14697680802637908.
[13]	S. E. Fleten, K. Hoyland and S. W. Wallace, The performance of stochastic dynamic and fixed mix portfolio models, Eur. J. Oper. Res., 140 (2002), 37-49. doi: 10.1016/S0377-2217(01)00195-3.
[14]	A. Geyer, M. Hanke and A. Weissensteiner, Scenario tree generation and multi-asset financial optimization problems, Oper. Res. lett., 41 (2013), 494-498. doi: 10.1016/j.orl.2013.06.003.
[15]	N. Gülpinar, B. Rustem and R. Settergren, Simulation and optimization approaches to scenario tree generation, J. Econ. Dyn. Control., 28 (2004), 1291-1315. doi: 10.1016/S0165-1889(03)00113-1.
[16]	P. Gupta, G. Mittal and M. K. Mehlawat, Multiobjective expected value model for portfolio selection in fuzzy environment, Optim. Lett., 7 (2013), 1765-1791. doi: 10.1007/s11590-012-0521-5.
[17]	P. Gupta, G. Mittal and M. K. Mehlawat, A multi-period fuzzy portfolio optimization model with minimum transaction lots, Eur. J. Oper. Res., 242 (2015), 933-941. doi: 10.1016/j.ejor.2014.10.061.
[18]	K. Hoyland and S. W. Wallace, Generating scenario trees for multistage decision problems, Manage. Sci., 47 (2001), 295-307.
[19]	K. Hoyland, M. Kaut and S. W. Wallace, A heuristic for moment-matching scenario generation, Comput. Optim. Appl., 24 (2003), 169-185. doi: 10.1023/A:1021853807313.
[20]	B. Jacobsen, Long term dependence in stock returns, J. Eimpir. Financ., 3 (1996), 393-417.
[21]	X. Ji, Sh. Zhu, Sh. Wang and Sh. Zhang, A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming, IIE. Trans., 37 (2005), 957-969.
[22]	T. Lux, Long term stochastic dependence in financial prices: evidence from German stock market, Appl. Econ. Lett., 3 (1996), 701-706.
[23]	R. Mansini, W. Ogryczak and M. G. Speranza, Twenty years of linear programming based portfolio optimization, Eur. J. Oper. Res., 234 (2014), 518-535. doi: 10.1016/j.ejor.2013.08.035.
[24]	H. Markowitz, Advantages of multiperiod portfolio models, J. Portfolio. Manage., 29 (2003), 35-45.
[25]	J. M. Mulvey, W. R. Pauling and R. E. Madey, Portfolio selection, J. Financ., 7 (1952), 77-91.
[26]	P. Rocha and D. Kuhn, Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules, Eur. J. Oper. Res., 216 (2012), 397-408. doi: 10.1016/j.ejor.2011.08.001.
[27]	C. T. Şakar and M. Köksalan, A stochastic programming approach to multicriteria portfolio optimization, J. Global. Optim., 57 (2013), 299-314. doi: 10.1007/s10898-012-0005-2.
[28]	A. Sensoy and B. M. Tabak, Time-varying long term memory in the European Union stock markets, Physica A., 436 (2015), 147-158.
[29]	J. F. Slifker and S. S. Shapiro, The Johnson system: selection and parameter estimation, Technometrics., 22 (1980), 239-246.
[30]	N. Topaloglou, H. Vladimirou and S. A. Zenios, A dynamic stochastic programming model for international portfolio management, J. Bank. Financ., 26 (2008), 1501-1524. doi: 10.1016/j.ejor.2005.07.035.
[31]	N. Topaloglou, H. Vladimirou and S. A. Zenios, Optimizing international portfolios with options and forwards, J. Bank. Financ., 35 (2011), 3188-3201.
[32]	A. C. Worthington and H. Higgs, Random walks and market efficiency in European equity markets, Global. J. Financ. Econ., 1 (2004), 59-78.
[33]	L. Yin and L. Han, International assets allocation with risk management via multi-stage stochastic programming, Comput. Econ., (2013). doi: 10.1007/s10614-013-9365-z.
[34]	L. Yin and L. Han, Options strategies for international portfolios with overall risk management via multi-stage stochastic programming, Ann. Oper. Res., 206 (2013), 557-576. doi: 10.1007/s10479-013-1375-7.
[35]	P. Zhang, An interval mean-average absolute deviation model for multiperiod portfolio selection with risk control and cardinality constraints, Soft. Comput., 20 (2016), 1203-1212.