Portfolio Selection Model with the Measures of Information Entropy- Incremental Entropy-Skewness

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1 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness Portfolio Selection Model with the Measures of Information Entroy- Incremental Entroy-Skewness 1,2 Rongxi Zhou, 3 Xiuguo ang, 4* Xuefan Dong, 5 Ze Zong 1, first author School of Economics and Management, Beijing University of Chemical Technology, Beijing , China, zhourx@buct.edu.cn 2 School of Banking and Finance, University of ew South ales, Sydney, S 2052, Australia, zhourx@buct.edu.cn 3, School of Alied Mathematics, Central University of Finance and Economics, Beijing , China 4, Corresonding Author School of Business, University of ottingham ingbo China, ingbo , China 5, School of Economics and Management, Beijing University of Chemical Technology, Beijing , China Abstract In this aer, we develo a ortfolio selection model with the measures of information entroyincremental entroy-skewness (EESM) in which the risk of the ortfolio is measured by information entroy, and the exected return is exressed by incremental entroy to indicate incremental seed of caital, and the risk of higher moment is measured by skewness. Then, we use the fuzzy rogramming technique to solve it. Finally, through a variety of emirical data sets from the Shanghai Stock Exchange in China, we evaluate the erformance of the EESM in terms of several ortfolio erformance measures. The obtained results show that the EESM erforms well relative to traditional ortfolio selection models. Keywords: Portfolio Selection, Information Entroy, Incremental Entroy, Skewness, Fuzzy Programming Technique 1. Introduction Since the roosition of Markowitz s famous mean-variance (MV) model, a great deal of studies have made a variety of significant contributions to imroving the erformance of the ortfolio selection model. However, the MV model often leads ortfolio to highly concentrate on only a few securities, which is contrary to the rincile of diversification. Recently, some scholars have alied the theory of entroy into the ortfolio selection [1-9]. By utilizing a develoed methodology of noise level estimation that makes use of roerties of the coarsegrained entroy, Krzysztof and Hołyst analyzed the noise level for the Dow Jones index and a few stocks from the ew York Stock Exchange [1]. Jana et al. [2] first roosed a mean-variance-skewness model for ortfolio selection and next added another entroy objective function to generate welldiversified asset ortfolio within otimal asset allocation. Bera and Sung [3] rovided an alternative aroach by introducing cross-entroy measure which can automatically cature the degree of imrecision of inut estimates, and Qin et al. [4] also discussed a cross-entroy minimization model in fuzzy environment. Huang [5] roosed two fuzzy mean-entroy models based on credibility and resented a hybrid intelligent algorithm. Rödder et al. [6] resented a new theory to determine the ortfolio weights by a rule-based inference mechanism under both maximum entroy and minimum relative entroy. Xu et al. [7] investigated ortfolio selection roblems by utilizing the hybrid entroy to estimate the asset risk caused by both randomness and fuzziness. Zhang et al. [8] investigated a multi-eriod ortfolio selection roblem in fuzzy environment, and used the ossibilistic semi-variance to araise the ortfolio risk, and the ossibilistic entroy to measure the diversification degree of ortfolio. A stochastic model of the investment ortfolio based on maximization of entroy was roosed in Ref. [9]. In all, entroy has been increasingly alied to the ortfolio selection as a Advances in information Sciences and Service Sciences(AISS) Volume5, umber8, Aril 2013 doi: /aiss.vol5.issue

2 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness measure of diversity because it is known that the greater the value of the entroy measure for ortfolio weights, the higher the ortfolio diversification is. On the other hand, there exists not only the variance risk but also the skewness risk in the ortfolio in Ref. [10, 11]. Moreover, the resence of negative skewness would increase the ossibility of return on assets decline much larger than the one on assets rise. hile the existence of excess eak will enable the likelihood of extreme events to increase greatly, both of which are known as the higher moment risk. These higher moment risks will affect the investment decisions of investors significantly, and great attention have already been given to the higher moment ortfolio roblems by some scholars. In articular, Prakash et al. [12], and Sunh and Yan [13] used the multiobjective rogramming method to deal with the otimization roblem of three objectives: maximize the exected return, skewness and minimize the variance to select the ortfolio with skewness risk resectively. These studies also showed that the skewness can enable the return of investors to be higher. Jondeau and Rockinger [14] used the exected utility function with Taylor exansions to discuss the roblem of the asset allocation under non-normal conditions, and found that the risk of skewness and kurtosis have a significant effect on the financial investment decisions. Several models which involve the factor of skewness are roosed by some academics due to these facts above. For examle, Bhattacharyya et al. [15] roosed a mean-variance-skewness ortfolio model, and Kerstens et al. [16] comared this model with the mean-variance model based on the idea of shortage function. hile, Usta and Kantar [17] added entroy measure into the mean-variance-skewness model to generate a well-diversified ortfolio. However, in all the above aers, the mean returns of ortfolio were measured by the arithmetic mean adoted by Markowitz which can not reflect the incremental seed of caital. Ou [18] used incremental entroy, one of the generalized entroies, to otimize ortfolios and found the new ortfolio theory based on incremental entroy carries on some asects of Markowitz s theory [18]. Indeed, the incremental seed of caital is a more objective criterion for assessing ortfolio. This aer resents a new ortfolio selection model with the measure of information entroyincremental entroy-skewness model (EESM). In this model, we relace arithmetic mean return with geometric mean one, and use the corresonding incremental entroy as a criterion for assessing a ortfolio. Meanwhile, we use information entroy to redict the risk of ortfolio and consider effects of skewness in the model. In order to reflect the subjective intention of investors in the ortfolio more exactly and to enable the model to be more effective and feasible, we use the fuzzy rogramming technique to solve the model and built some emirical comarisons. The reminder of this aer is organized as follows: two classical ortfolio selection models are resented briefly in Section 2. The EESM is introduced in Section 3. In Section 4 the emirical study is conducted to evaluate the erformance of EESM with two classical ortfolio selection Models. Finally, the conclusion is given in Section Two classical ortfolio selection models This section briefly resents Markowitz s mean-variance model (MVM) and the mean-varianceskewness model (MVSM) Mean-Variance Model (MVM) In the MVM, mean exresses the exected return and variance measures the risk. Both of them are used to evaluate the ortfolio s value. The model is given as follows: T Min X CX xr i i c i1 s.t. (1) xi 1; i 1,2,..., i1 where the yield vector of securities R=(r 1, r 2,, r ) T. The wealth fraction invested in the securities 854

3 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness X=(x 1, x 2,, x ) T, and the covariance matrix C. Additionally, c reresents the given exected return Mean-Variance-Skewness model (MVSM) The model based on the mean, the variance and the skewness can be formulated as T Min X CX T Max X S( X X ) xr i i c i1 s.t. xi 1 (2) i1 xi 0, i 1, 2,..., where r, X, c and C are similar to the ones in model (1). The second objective function in the model (2) can be exressed by n n n i j k ijk i1 j1 k1 T T T 3 X S( X X) E[ X R- E[ X R ]] x x x s where denotes for the Kronecker roduct and s ijk reresents the coskewness between the returns of asset i, j and k for ( i, j, k) [1,2,..., n]. s E[( R E[ R])( R E[ R ])( R E[ R ])] (4) ijk i i j j k k 3. Information entroy-incremental entroy-skewness model (EESM) 3.1. Incremental Entroy Incremental entroy is a kind of generalized entroy which can be defined as ( i)log k ik i1 k1 H x qr (5) where assuming that the market consists of securities and security k has n k kinds of ossible value for k= 1, 2,...,. Therefore, the amount of all rice combinations is =n 1 n 2 n, where ith one is x i =(x i1, x i2,, x i ), i = 1, 2,..., ; q k, k=1, 2,..., is the wealth fraction invested in security k; R ik is the outut ration of security k in ith rice combination. In addition, R ik =1+r ik where r ik reresents the rate of return. e also assume that no short selling is allowed, which means q k 0,k=1, 2,,. Additionally, the equation (5) can be converted to the formula as follows: where R q R. e denote i k ik k 1 ( xi ) H log Ri (6) EMBED Equation.DSMT4 i1 i i1 ( xi ) i (3) R R (7) where R is the geometric outut ratio which can reflect the incremental seed of caitals, and i the mean utilizes the arithmetic average outut ratio to measure the earnings of ortfolio. However, it is too roortional for q and to reflect the cumulative gain of fund. Therefore, the incremental entroy is a better measure than the mean to reresent the investment income. 855

4 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness 3.2. Minimum information entroy-maximum incremental entroy-maximum skewness model In this section, we relace the variance of the MVM with the information entroy to measure the risk of ortfolio, relace the mean with the incremental entroy to measure the exected return of ortfolio, and also consider effects of skewness in the ortfolio. The model is formulated as the following: Min S iln i i1 Max H logq R Max U i k ik i1 k1 k11k21k31 q q q s k1 k2 k3 k1k2k3 i 1, i 0, i 1, 2,..., i1 s.t. qk 1, qk 0, k 1, 2,..., (8) k 1 qk 0; k 1,2,..., ; j 1,2,3 j where S means the information entroy which is used to reresent the risk of ortfolio, H indicates the incremental entroy which can redict the ortfolio s return, U denotes the skewness, q k is the wealth fraction invested in security k, R ik is the outut ration of security k in ith rice combination, and s k1k2k3 reresents the coskewness between the returns of asset k 1, k 2 and k 3. In addition, i is the robability of occurrence of ith rice combination, but it can not be redicted. Obviously, the model is a multiobjective rogramming question. Therefore, the otimum roortion of ortfolio under the conditions of minimum information entroy, maximum incremental entroy and maximum skewness can be obtained. For the model (8), we can address it by the following stes: Ste 1, we assume that it is in a continuous eriod of time excet holidays and vacations, r k for k=1, 2,, reresents the closing rice of kth stock. Then, the outut ratio security it can be defined as rk rk 1 Rk 1 (9) rk 1 Ste 2, according to the actual situations of market in China that there is a 10% limitation about fluctuations of increase and decrease of normal stocks rices er day, we can divide the interval of [- 10%, 10%] into 20 isotonic subintervals and denote ith subinterval as a i for i=1, 2,, 20. Ste 3, we can calculate n ki which reresents the frequency of the yield rate of kth stock on ith subinterval. Then, we define that if R k -1-10%, the yield rate is on the subinterval [-10%, -9%), and if R k -1 10%, it is on the subinterval (9%, 10%]. Ste 4, we can calculate ρ ki which reresents the frequency rate of kth stock on the ith subinterval, and nki ki.e can define that k is equal to ρ ki, this means that the frequency can be alied to d reflect the robability distribution of the yield rate of single stock. In order to adat the synchronous analysis of multile stocks, we define i as follows: q /( q ) (10) i k ki k k k k1 k1 k1 where i reresents ith subinterval for i=1, 2,, 20, k reresents kth stock in the investment ortfolio for k=1, 2,, 20, ki reresents the frequency of kth stock on ith subinterval, resectively, so 856

5 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness n d follows: ki ki, and k reresents correlation coefficient of kth stock in the ortfolio which is defined as 1 kk1 k1 k 1 k1 k1 where μ kk+1 is the correlation coefficient between kth stock and (k+1)th stock. As a result, the ortfolio otimization model can be trasformed as follows: Min S Max H Max U q kk1 k ki k k ki k k1 k1 ln i1 qk k qk k k1 k1 k1 k1 q q k ki k k1 k1 log k i1 k1 qkk k1 k1 k11k21k31 q q q s k1 k2 k3 k1k2k3 ki 1, i 0, i1, 2,..., ; k 1, 2,..., i1 s.t. qk 1, qk 0, k 1, 2,..., (12) k 1 qk 0, k 1,2,..., ; j 1,2,3 j Moreover, in this model, the exlanations of S, H, U, q k, R ik and s k1k2k3 are similar to the ones in model (8) Fuzzy rogramming technique to solve EESM The model (12) is a multi-objective otimization question, and then we aly fuzzy rogramming technique to solve it. e denote R as income from investment, f as investment risk and u as skewness. e choose three fuzzy sets A 1, A 2, A 3 and the corresonding membershi functions A 1, A 2, from A 3 domain {R, f, u} to show the qualities of R, f and u. e choose the membershi function as follows: 0, R r R r A ( R ), 1 r R R (13) R r 1, R R 0, f f f f A ( f ), f f f 2 (14) f f 1, f f qr ik (11) 857

6 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness 0, u u u u A ( u), u u u 3 u u 1, u u where f_ is the minimum value of risk, f - is the maximum value of risk, R (15) is the maximum value of exectant income, r is the required rate of return, u_ is the minimum value of skewness and u is the maximum value of skewness. Therefore, we can transform the model (12) by using fuzzy rogramming technique to the one as follows: Max H r R r f S s.t. f f U u u u (16) As a result, it is easier to obtain the otimal solution of model (16) which is also the efficient solution of model (12). 4. The emirical study In this section, we introduce the various ortfolio erformance measures and rolling window rocedure to evaluate the erformance of the EESM relative to the MVM and MVSM Portfolio Performance Measures In order to evaluate the erformance of ortfolio models, a number of alternative erformance measures have been roosed in the Ref. [19-22]. In this study, we consider some of these erformance measures. First, there is no doubt that the Share ratio (SR) is one of the most crucial classic metric indicators of ortfolio utility. The Share ratio is equal to the risk remium of the ortfolio divided by the standard deviation, namely the formula as follows: ER ( ) Rf SR (17) where E(R ) is the ortfolio exected rate of return, R f is the risk-free rate, and σ is the standard deviation of the ortfolio. In addition, what should be claimed that the higher the Share ratio, the better the ortfolio. However, since the SR is based on the mean-variance theory, it is only valid for normally distributed returns. Particularly, the SR can lead to misleading conclusions when the return distributions are skewed or resent heavy tails. Several alternatives to the SR for otimal ortfolio selection have been roosed in the literature. Some of these alternatives are resented as the following: The adjusted skewness Share ratio (ASR), which takes into accounts the skewness of ortfolio, is defined as follows: 858

7 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness SK( R ) ASR SR 1 SR (18) 3 The mean absolute deviation ratio (MADR), which considers the risk as mean absolute deviation, is given as follows: ER ( ) MADR (19) E ( R E ( R )) The Sortino-Satchell ratio (SSR) and Farinelli and Tibiletti ratio (FTR), which are erformance measures based on the artial moments and their formulas are given as follows, resectively: ER ( ) SSR 2 (20) E(max( R,0) ) where E(max(-R, 0) 2 )is the lower artial moment of order 2. FTR v u u E(max( R,0) ) v E(max( R,0) ) where E(max(-R, 0) v ) and E(max(R, 0) u ) are the lower artial moment of order v and the uer artial moment of order u, resectively. The selection of u and v are associated to investors styles or references. e can consider the following cases for u and v according to: u 0.5, v 2 for a defensive investor; u 1.5, v 2 for a conservative investor; u 1, v 1 for a moderate investor. Additionally, it is known that if u 1, v 1, the FTR reduces to the Omega ratio Results of the Emirical Study For comaring the erformance of this model with other models, we select randomly two datasets of 40 stocks from Shanghai Stock Exchange in China, and the eriod of datasets san from January 4, 2010 to ovember 2, 2010 and from May 20, 2011 to March 21, 2012, resectively. The catalog of 40 stocks codes is shown in Table 1. Table 1 The catalog of the 40 stocks codes Serial Stock code Serial Stock code Serial Stock code Serial Stock code number number number number First, we comute the ortfolio weights according to the MVM, MVSM and EESM based on the samle estimates of the first dataset. Then, utilize these ortfolio weight vectors q k, the out-of-samle return of ortfolio in the second dataset, denoted by R, is calculated by R = q k R ik, where R ik denotes the return vector in the second dataset. Thus, the out-of-samle returns generated by each of the considered ortfolio models can be obtained. Based on the out-of-samle returns, the SR, ASR, MADR, SSR and FTR measures are calculated to evaluate the erformance of the EESM relative to MVM and MVSM. The values of mean, standard deviation, skewness, SR, ASR, MADR, SSR, FTR(0.5,2), FTR(1.5,2) and FTR(1,1) for each ortfolio obtained from MVM, MVSM and EESM are resectively lotted in from Fig.1 to Fig. 10 in Aendix. As seen from Fig. 1 to Fig. 10, the MVM rovide oor results in terms of all erformance measures (21) 859

8 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness excet FTR relative to the MVSM and EESM. Moreover, by contrast between indexes obtained from the EESM and the MVSM, it should be noted about Fig. 1, Fig. 3, Fig. 5, and Fig. 7 that EESM shows better erformances in mean, skewness, ASR, and SSR than the MVSM. Additionally, it can be seen from the Fig.2 that the fluctuation of standard deviation comuted by the EESM is more stable as well excet fewer extreme values. The results of SR and MADR in Fig. 4 and Fig. 6 resectively show that the values of both two erformance measures above obtained by the EESM and MVSM are aroximately similar, while EESM dislays more steadily than the MVSM. On the other hand, in terms of the FTR (0.5, 2) and FTR (0.5, 2) and FTR (1, 1), we can find that there is no obvious differences among the EESM, MVSM, and MVM from Fig Overall, we can say that ortfolios obtained from the EESM erform better in terms of variety ortfolio erformance measures than the MVM and MVSM. 5. Conclusions e resent a multi-objective model which includes the information entroy, the incremental entroy and the skewness. e use the incremental entroy to reflect the incremental seed of caital, and entroy to measure the risk of ortfolio, and also consider effects of skewness. The fuzzy rogramming technique is alied to address the ortfolio selection model. And then, by comaring their erformance with the two classical models based on a series of advanced erformance measures and a multitude of data sets from the Shanghai Stock Exchange in China, we find that the erformance of EESM is better than the considered other models. The transaction costs in the EESM should be considered and alied into the financial market in the future. 6. Acknowledgments The authors thank the anonymous reviewers for roviding constructive feedback. This work is suorted by ational atural Science Foundation of China Grant o and o , and the training lan of science research of undergraduate of Ministry of Education of the Peole s Reublic of China Grant o , and disciline construction roject of Beijing University of Chemical Technology Grant o All errors are the authors own. 7. Aendix Fig. 1. Mean obtained from MVM, MVSM and EESM. 860

9 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness Fig. 2. Standard deviation obtained from MVM, MVSM and EESM. Fig. 3. Skewness obtained from MVM, MVSM and EESM. Fig. 4. SR obtained from MVM, MVSM and EESM. 861

10 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness Fig. 5. ASR obtained from MVM, MVSM and EESM. Fig. 6. MADR obtained from MVM, MVSM and EESM Fig. 7. SSR obtained from MVM, MVSM and EESM 862

11 Portfolio Selection Model with the Measures of Information Entroy-Incremental Entroy-Skewness Fig. 8. FTR (0.5, 2) obtained from MVM, MVSM and EESM Fig. 9. FTR (1.5, 2) obtained from MVM, MVSM and EESM 8. References Fig. 10. FTR (1, 1) obtained from MVM, MVSM and EESM [1] U.Krzysztof, J.A. Hołyst, Investment strategy due to the minimization of ortfolio noise level by observations of coarse-grained entroy, Physica A, Statistical Mechanics and its Alications, no. 344, , [2] P. Jana, T.K. Roy, S.K. Mazumder, Multi-objective mean-variance-skewness model for ortfolio otimization, Advanced Modeling and Otimization, no. 9, ,

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