Recurrent neural network for dynamic portfolio selection

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Recurrent neural network for dynamic

portfolio selection

Chi-Ming Lin

a,c

, Jih-Jeng Huang

b

, Mitsuo

Gen

a

, Gwo-Hshiung Tzeng

d,e,*

a_{Graduate School of Information, Production and Systems, Waseda University,}

Kitakyushu, Japan

b_{Department of Information Management, National Taiwan University,}

Taipei, Taiwan, ROC

c

Center for General Education, College of Management, Kainan University, Taoyuan, Taiwan, ROC

d

Department of Business Administration, Kainan University, Taoyuan, Taiwan, ROC

e

Institute of Management of Technology, National Chiao Tung University, 1001 Ta-Hsuch Road, Hsinchu 300, Taiwan, ROC

Abstract

In this paper, the dynamic portfolio selection problem is considered. The Elman net-work is ﬁrst designed to simulate the dynamic security behavior. Then, the dynamic covariance matrix is estimated by the cross-covariance matrices. Finally, the dynamic portfolio selection model is formulated. In addition, a numerical example is used to dem-onstrate the proposed method and compare with the vector autoregression (VAR) model. On the basis of the numerical example, we can conclude that the proposed method out-perform to the VAR model and provide the accurate dynamic portfolio selection. Ó 2005 Elsevier Inc. All rights reserved.

* _{Corresponding author. Address: Institute of Management of Technology, National Chiao Tung}

University, 1001 Ta-Hsuch Road, Hsinchu 300, Taiwan, ROC. E-mail address:u5460637@ms16.hinet.net(G.-H. Tzeng).

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Keywords: Neural network; Dynamic portfolio selection; Elman network; Cross-covariance mat-rices; Vector autoregression (VAR)

1. Introduction

The mean–variance approach was proposed by Markowitz to deal with the

portfolio selection problem [1]. A decision-maker can determine the optimal

investing ratio to each security based on the sequent return rate. The

formula-tion of the mean–variance method can be described as follows [1–3]:

min X n i¼1 Xn j¼1 rijxixj s.t. X n i¼1 lixiP E; Xn i¼1 xi¼ 1; xiP0 8i ¼ 1; . . . ; n; ð1Þ

where li denotes the expected return rate of the ith security, rij denotes the

covariance coeﬃcient between the ith security and the jth security, and E de-notes the acceptable least rate of the expected return.

On the basis of Eq.(1), it can be seen that the conventional portfolio selec-tion problem above is considered as a static situaselec-tion. However, this assump-tion is truly against our intuiassump-tion i.e. we always vary our optimal portfolio selection with time. Although many methods including vector autoregression

(VAR) [3,4] and generalize autoregressive conditional heteroshedastic

(GARCH)[5–7]has been proposed to deal with the dynamic portfolio selection

problem, several restricted assumptions, such as stationary time series, inde-pendent variables, and the linear relationship among variables, make these models impractical. The purpose of this paper is to propose a non-parameter and non-linear method to deal with the dynamic portfolio selection problem. In this paper, a dynamic portfolio selection model is proposed by

incorpo-rating the recurrent neural network (RNN) [8,9] and the cross-covariance

matrices[4]. The dynamic expected return rate is ﬁrst derived using the Elman

network[8]. Then, the cross-covariance matrices are calculated to estimate the covariance matrix among securities.

The remainder of this paper is organized as follows. The dynamic portfolio selection model is proposed in Section2. A numerical example, which is used to illustrate the proposed method and compare with the VAR method, is

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2. Dynamic portfolio selection model

Consider a market contains n securities. The dynamic behavior of the secu-rities can be depicted as shown inFig. 1. The return rates/risk of securities in the period t 1 will aﬀect the return rates/risk of the securities in the period t, and so forth.

In order to simulate the dynamic behavior of the security market, the Elman

network is employed here. The Elman network[8]is one kind of the recurrent

neural networks and widely used to deal with time-varying problems [10,11].

Recently, the recurrent neural networks have been reported the better accuracy than the conventional timer series approaches such as autoregression

inte-grated moving-average (ARIMA), VAR, and GARCH[12,13]. In this paper,

the Elman network is constructed as shown inFig. 2.

After obtaining the forecasting expected return rates, we should calculate the covariance matrix among securities. In this paper, the cross-covariance matrices are used to derive the covariance matrix with diﬀerent periods. The contents of the cross-covariance matrices can be described as follows.

Consider the multivariate time series Zt, and the mean vector l, then the

cross-covariance matrices at the lth lag can be deﬁned as Rtþ1_l ¼ ½rtþ1 ij ¼ CovðZt;ZtlÞ ¼ E½ðZt lÞðZtl lÞ0 ¼ E z1t l1 z2t l2 .. . zkt lk 2 6 6 6 6 4 3 7 7 7 7 5½z1ðtlÞ l1; z2ðtlÞ l2; . . . ; zkðtlÞ lk ¼ rtþ1 11 ðlÞ rtþ112 ðlÞ rtþ11k ðlÞ rtþ1₂₁ ðlÞ rtþ1 22 ðlÞ r tþ1 2k ðlÞ .. . .. . ... rtþ1 k1 ðlÞ rtþ1k2 ðlÞ rtþ1kk ðlÞ 2 6 6 6 6 4 3 7 7 7 7 5. ð2Þ

Now, we can reformulate the conventional portfolio selection to consider the time-varying portfolio selection problem as follows:

min X n i¼1 Xn j¼1 rtþ1_ij xtþ1_i xtþ1_j s.t. X n i¼1 ltþ1_i xtþ1_i P R; Xn i¼1 xtþ1_i ¼ 1; xtþ1_i P0 8i ¼ 1; . . . ; n; ð3Þ

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where ltþ1

i denotes the expected return rate of the ith security in the period

t + 1, rtþ1

ij denotes the covariance coeﬃcient between the ith security and the

μnσn μ2σ2 t –1 t t +1 … … … μ1σ1 μ2σ2 μ1σ1 μ2σ2 μ1σ1 μnσn μnσn

Fig. 1. The dynamic security market network.

f1(.) f2(.) fm(.) μ2 μn γ1 γ2 γm μ1 μ2 μn μ1 Recurrent Loop Time t Time t+1 … … … …

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jth security in the period t + 1, and R denotes the acceptable least rate of the expected return.

In the next section, a numerical example is used to demonstrate the pro-posed method and compare the accuracy with the VAR model.

3. Numerical example

In this numerical example, ﬁve series with 100 data, which are collected form TaiwanÕs stock market, are used to demonstrate the proposed method. The

re-turn rates of the ﬁve stocks can be depicted as shown inFig. 3.

Next, we perform VAR(1) and the Elman network to forecast the expected return rates of the ﬁve stocks. By calculating the mean-square error (MSE) as

shown in Table 1, it can be seen that the Elman network outperform to

VAR(1). That is, the Elman network can predict the more accurate results than VAR(1).

Table 1

Mean-square error of the ﬁve stocks

MSE l1 l2 l3 l4 l5 VAR(1) 11.9409 12.4489 14.2361 14.7871 11.5802 ElmanÕs network 5.4637 9.2073 6.4914 5.1323 5.1610 0 20 40 60 80 100 -5 0 5 0 20 40 60 80 100 -5 0 5 0 20 40 60 80 100 -5 0 5 0 20 40 60 80 100 -5 0 5 0 20 40 60 80 100 -5 0 5

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We also use the Elman network and VAR(1) to predict the expected return

rates of the ﬁve stocks as shown in Table 2 to be the inputs of the dynamic

portfolio selection model.

From Table 2, we can conclude that although the expected return rates of the five stocks have the same direction, it can be seen that the values are much different. Since the tiny different results may change the optimal portfolio selec-tion, we should predict the expected return rate as accuracy as possible.

In order to calculate the covariance matrix of the ﬁve stocks, the cross-covariance matrix at the 1st lag is used to estimate the cross-covariance matrix. Note that other weighted methods like mean average or geometric average method can also be used to estimate the covariance matrix. The covariance matrix of the ﬁve stocks derived by VAR(1) and the Elman network can be shown as inTables 3 and 4.

On the basis of the results above, we can obtain the optimal dynamic

port-folio selection by solving Eq. (3). The comparison of the optimal portfolio

Table 2

The expected return rates of the ﬁve stocks

Expected return rate (%) l1 l2 l3 l4 l5

VAR(1) 0.5327 0.0484 0.0047 0.2487 0.4674

ElmanÕs network 0.3628 0.0258 0.0283 0.1395 0.3098

Table 4

The covariance matrix derived by the Elman network Covariance matrix (ElmanÕs network) l1 l2 l3 l4 l5 l1 1.0402 0.0179 1.1726 1.1004 1.0104 l2 0.0179 0.4253 0.0484 0.0280 0.0123 l3 1.1726 0.0484 1.3473 1.2557 1.1408 l4 1.1004 0.0280 1.2557 1.1736 1.0702 l5 1.0104 0.0123 1.1408 1.0702 0.9819 Table 3

The covariance matrix derived by VAR(1) Covariance matrix (VAR(1)) l1 l2 l3 l4 l5 l1 1.4915 0.0288 1.5586 1.5067 1.4402 l2 0.0288 1.5800 0.0744 0.0243 0.0343 l3 1.5586 0.0744 2.0940 1.8476 1.5259 l4 1.5067 0.02427 1.8476 1.6945 1.4689 l5 1.4402 0.03434 1.5259 1.4689 1.3953

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selection between VAR(1) and the proposed method can be presented as shown inTable 5.

FromTable 5, it can be seen that the results of the optimal dynamic port-folio selection between VAR(1) and the Elman network is quite diﬀerent. Since the accuracy of the Elman network outperform to VAR(1), we can conclude that the optimal dynamic portfolio selection of the Elman network should be the alternative.

4. Discussions and conclusions

Mean–variance is widely used in the ﬁnance area to deal with the portfolio selection problem. However, the conventional method only considers the static situation. The purpose of the mean–variance approach is to determine the t + 1 period optimal investing rate to each security based on the sequent return rate. The key is to forecast the t + 1 period return rate as accuracy as possible.

In this paper, the dynamic portfolio selection problem is considered. The El-man network is ﬁrst designed to simulate the dynamic behavior of the security market to predict the expected return rate. Next, the cross-covariance matrices are used to estimate the covariance matrix among securities. Finally, the opti-mal dynamic portfolio selection is determined by the revised model. On the ba-sis of the numerical example, we can conclude that the accuracy of the Elman network outperform to VAR(1) and provide the better solution for the dy-namic portfolio selection problems.

References

[1] H. Markowitz, Portfolio selection, Journal of Finance 7 (1) (1952) 77–91.

[2] H. Markowitz, Portfolio Selection: Eﬃcient Diversiﬁcation of Investments, Wiley, New York, 1959.

[3] H. Markowitz, Mean–Variance Analysis in Portfolio Choice and Capital Market, Basil Blackwell, New York, 1987.

[4] G.C. Tiao, G.E.P. Box, Modeling multiple time series with applications, Journal of the American Statistical Association 76 (4) (1981) 802–816.

Table 5

The optimal dynamic portfolio selection Optimal portfolio

selection

l1 l2 l3 l4 l5 Risk Acceptable least

return rate VAR(1) 0 0.5695 0.3668 0.0637 0 0.8278 0.01 ElmanÕs network 0 0.7719 0.1970 0.0311 0 0.3065 0.01 VAR(1) 0 0.5847 0.3893 0.0260 0 0.8600 0.02 ElmanÕs network 0 0.8928 0.1072 0 0 0.3498 0.02

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[8] J.L. Elman, Finding structure in time, Cognitive Science 14 (2) (1990) 179–221.

[9] M.I. Jordan, Attractor dynamics and parallelism in a connectionist sequential machine, in: Proceedings of the Eighth Annual Conference of the Cognitive Science Society, Amherst, 1986, pp. 531–546.

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