Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 8, 142 - 150, 2016
doi:10.21311/001.39.8.19 Generalized Entropy Optimization Model Applied to Investment Decision Based on Investor Sentiment
Hua Li1, 2 1 School of Business Administration, University of Science and Technology Liaoning, Anshan, 114051, China 2 Institute of Quantitative &Technical Economics, Chinese Academy of Social Sciences, Beijing, 100732, China
Shuying Zhao School of Business Administration, University of Science and Technology Liaoning, Anshan, 114051, China
Qiubai Sun* School of Business Administration, University of Science and Technology Liaoning, Anshan, 114051, China *Corresponding author(E-mail: [email protected])
Tao Zhang Institute of Quantitative &Technical Economics, Chinese Academy of Social Sciences, Beijing, 100732, China
Abstract How the investors to get the expected return with the great opportunities and challenges in financial markets, is becoming one of the problems that investors need to solve. The generalized entropy optimization model studied in this paper can better measure the investment risk and make up the insufficiency of the traditional investment portfolio model. On the optimization theory, this paper studies generalized entropy model based on investor sentiment to help investors to make decisions. Firstly, the investor sentiment is quantified and processed to reflect the fluctuation of investor sentiment by market data, and then reflect the investors attitude to the market situation; Secondly, combined with the investor sentiment evaluation indexes, the generalized entropy investment decision model is constructed to reflect the emotional changes of the investors, selecting 24 stocks from the industries to portfolio invest, determining the optimal investment ratio of each stock and ensuring that the minimum investment risk to get the maximum investment income. The theory of entropy optimization with investor sentiment is more close to the investors' investment preferences, and it is more satisfying to the investors' intention.
Key words: Optimization Model Investor Sentiment, Principal Component Analysis, Generalized Entropy, Investment Decision.
1. INTRODUCTION
Unlike modern portfolio theory hypothesis, behavioral finance theory holds that investors may make the irrational investment, because of being affected by the external and own psychological effects. Kahneman and Tversky (1979) proposed the "prospect theory"; Debondt and Thaler (1985) pointed out that most people would violate the Bias rule to unexpected events and appeared "overreaction" phenomenon, which sparked a renaissance of behavioral finance; Shefrin (1994) and Statman (2000) put forward respectively the single account portfolio theories and the multiple accounts portfolio theories; De long, Shieifer, Summers and Waldmann (1990); Barberis, Shieifer and Vishny (1998); Hong and Stein (1999) proposed respectively the DSSW model, BSV model, HS model ,which are the important behavioral finance theories; M Baker,J Wurgler (2006) studied the impact of investor sentiment on the cross section of the stock, which is based on the study of small stocks, young, high volatility, non-payment of dividends, loss, extreme growth stocks and bad stocks; Hu Changsheng, Chi Yangchun(2013) built the rational and irrational sentiment indexes and did a research. Research showed that rational investors and irrational investors were makers of the market volatility and rational investors didn’t have the effect of stabilizing the market; Yi Zhigao, MaoNing(2009)constructed a composite index measuring stock market investor sentiment (CICSI) by six individual sentiment indicators, such as closed- end Fund discounts, IPO quantity, First-day gains, Consumer Confidence Index, New Investors Accounts and did a preliminary inspection that CICSI had the effectiveness to measure the investors' irrational behavior; Li Fengyu (2014) explained the ETF premium phenomenon of the A-share market based on the perspective of investor sentiment; Liu Weiqi, Liu Xinxin (2014) comparatively studied of the effects of individual and institutional investor sentiment on stock returns, which showed that the institutional investors emotions can help predict individual investors, and vice versa not established; Zhang Liping, Wang Lin (2014) found that the investor emotional index had a strong explanatory power on the stock return through the empirical analysis of EGARCH model; Wang Bo (2014) constructed the asset pricing model based on investor sentiment and the 142
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 8, 142 - 150, 2016 study found that stock equilibrium price can be decomposed into the rational and emotional part, which investor sentiment lead to the price deviate from the intrinsic value and the higher volatility and price bubble. Because of the stock data processing complex and the large calculation, how to quantify stock data became a hot topic in mathematical finance field. As a gradually mature mathematical analysis tool, entropy has been gradually applied to securities investment and recognized by scholars; AM Mathai, HJ Haubold (2007) introduced the characteristics of generalized entropy index through the analysis of Mathai path mode; E Maasoumi,M Trede (2001) derived a generalized flow model and had the comparison of United States and Germany’s earnings mobility by using panel data; As C.E.Shannon proposed the information entropy, the entropy theory has been used in machine learning algorithms and get the promotion, Karayiannis (1994) made up the defect of the fuzzy clustering by the entropy theory and he proposed the maximum entropy clustering algorithm; Li (1995) proposed the maximum entropy clustering algorithm to solve the problem of uncertainty; Wang Yijiao, Feng Qi, Chai Lihe (2009) used the generalized entropy to analyze the evolution process of the stock market; For the comprehensive evaluation system of virtual enterprise partner selection, Wang shuo, Tang Xiaowo (2003) put forward the generalized entropy model. According to the analysis of the investors’ research results, the investor sentiment evaluation index system is constructed and combined the generalized entropy model; this paper constructs the generalized entropy investment decision model reflecting investor sentiment. The model is used to invest and to determine the proportion of stock investments, which can maximize investment returns and provides a theoretical basis for investors to make investment decisions. With the advancement in networking and multimedia technologies enables the distribution and sharing of multimedia content widely. In the meantime, piracy becomes increasingly rampant as the customers can easily duplicate and redistribute the received multimedia content to a large audience. Insuring the copyrighted multimedia content is appropriately used has become increasingly critical.
2. MODEL CONSTRUCTION
2.1. Investor Sentiment Comprehensive Evaluation Index System Construction Baker and Wurgler(2006) pointed out that the various proxy variables of investor sentiment may have some lead-lag effect, making these variables at different times can influence the investor sentiment. Based on this method, considering the influence of past data on investor sentiment, in the process of selection indicators, the paper selects a lag phase of indicators as explanatory variables to illustrate the investor sentiment changes. Therefore, in view of the selection frequency of various index and the difficulty of obtaining stock data, this article choose the Consumer Confidence Index (CCI), Closed-End Fund Discount rate (CEFD), Turnover rate (TURN), New Investor Account growth rate (NIA), Price Earnings Ratio (PER), The Shanghai Composite Index Returns rate (SCIR) and lag a period of Price Earnings Ratio (PER-1) to construct the comprehensive index of investor sentiment. Data is selected from 2008.01 to 2014.12, which are a total of 84 months. The empirical samples are selected from Shanghai A-share market in China. The types and selection reason of quantitative indicators are shown in Table 1. Table 1. Investor Sentiment Indexes Index Name Paraphrasing The response of the strength of stock liquidity and the stock's speculative demand to Turnover Rate investors. Source: CSMAR Solution The indicators of assessing whether the stock price level is reasonable, representing the Price Earnings stock is overvalued or undervalued. Ratio Source: CSMAR Solution The optimistic degree of the future economy, the more optimistic the market is, the higher Consumer the index. Confidence Index Source: Eastmoney New Investor On behalf of investors’ the degree of demand and participation to securities, the higher Account Growth the index, the higher the degree of participation, the higher the mood. Iate Source: Eastmoney Closed end fund discount rate can be used as a proxy variable of investor sentiment, and Closed-End Fund it has a negative correlation with emotion. Discount Rate Source: CSMAR Solution The Shanghai Significant influence factors of investor sentiment, with the market in a rising phase, the Composite Index higher the index, the higher the mood. Returns Rate Source: CSMAR Solution
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The calculation formula of CEFD , NIR are:
m CEFD NAV / NAV t i 1 wpit it it it (1)
w ， p ， NAV it represents the market proportion of i fund at t time it is the price of i fund at t time it is the unit net value of i fund at t time. New Investors Number NIR (2) Total Investors Number last month The principal component analysis (PCA) was used to construct comprehensive investor sentiment index, the following results can be got including descriptive statistical analysis of the variables, the KMO and Bartlett test output results, the cumulative contribution of factors and the Rotating Factor Load Matrix. Descriptive statistical analysis of the variables is shown in table 2; the KMO and Bartlett test output results is shown in table 3;, the cumulative contribution of factors is shown as table 4 treated by SPSS 21 which reflects the cumulative contribution rate of factors and the choice of the factors number; the Rotating Factor Load Matrix is shown as table 5.
Table 2. Variable Descriptive Statistical Analysis
N Minimum Maximum Average standard deviation PER 84 9.77 49.4 18.4035 8.37766 TURN 84 5.81 64.63 19.8733 14.54636 CCI 84 92.4 108.5 102.044 4.0591 NIR 85 0.000387937 0.010690319 0.001788621 0.001460129 CEFD 84 -32.3810625 -8.230097917 -17.60000026 6.529449011 SCIR 84 -0.246313945 0.205692475 -0.002196407 0.083273873 PER_1 84 9.77 59.24 18.9183 9.48359 Effective N (list) 84
Table 3. The KMO and Bartlett Test Output Results
KMO and Bartlett test Kaiser-Meyer-Olkin measures of samples adequate 0.661 Approximate chi-square 427.848 Bartlett's sphericity test df 21 Sig. 0
From table 3, the KMO value is 0.661, which is generally considered to be suitable for doing the factor analysis if the KMO value is more than 0.6. So the sentiment variable is suitable for the factor analysis. From table 4, we can easily see that the eigenvalues of the first three factors is more than 1, the fourth factor eigenvalues is 0.547, the variance accounted for 7.811% of the total variance, compared with the third factor, the gap is large. So, this article selects the first three factors, the cumulative contribution is rate more than 85%, meeting the cumulative contribution rate requirements for the factor analysis. The three factors contains variables most of the information.
Table 4. Factor Contribution Rate Total variance explained Ingredients Initial eigenvalues Extraction of square sum loads Total Variance % Cumulative % Total Variance % Cumulative % 1 3.573 51.038 51.038 3.573 51.038 51.038 2 1.313 18.759 69.797 1.313 18.759 69.797 3 1.138 16.253 86.049 1.138 16.253 86.049 4 0.547 7.811 93.86 5 0.268 3.824 97.684 6 0.104 1.492 99.176 7 0.058 0.824 100 Extraction method: principal component analysis.
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Table 5. Rotating Factor Load Matrix
Component matrix a Ingredients
1 2 3 PER 0.874 -0.015 0.384 PER_1 0.867 -0.22 0.378 TURN 0.775 0.481 -0.259 CCI -0.459 0.475 0.548 NIR 0.894 0.175 0.227 CEFD -0.624 -0.156 0.654 SCIR -0.236 0.868 0.018 Extraction method: principal component analysis. Three ingredients have been extracted.
According to table 5, the investor sentiment comprehensive evaluation index is:
IS 0.34PBR 0.30PBR-1 0.29TURN 0.04CCI 0.35NIR-0.11CEFD 0.09SCIY (3) Note that the index value should be the standard data of the original index. Then the comprehensive index of investor sentiment has been finished by the principal component analysis (PCA). The IS’ s plus and minus represent the emotional state of the investors. If the IS>0, indicating that market investors sentiment is more optimistic and belongs to positive emotional states; If the IS<0, illustrating the more pessimistic investor sentiment changes, which belongs to the negative emotional States. By formula 3, IS index and the sentiment indicators have shown a certain relationship, which is consistent with the economic significance. The higher PER, TURN, CCI, SCIR, NIR, PER-1index and the lower the CEFD, the higher the market investor enthusiasm and the more active market transactions; it also can be seen that NIR has the greatest emotional impact, the value is 0.35, that is, controlling the other variables to remain unchanged, NIR increased by one percent, mood will increase by 35%.
2.2. General Entropy Optimization Model The word "entropy" comes from the Greek “(change)," which means the capacity of the change, the first introduced by German physicist Clausius (Clausius) in thermodynamics in 1856. In the evolution of the entropy theory, the most representative theories is the information theory founded by American mathematician wiener (Wiener) and Shannon (Shannon), having the epoch-making significance. In the process of the communication, they called the uncertainty of information sources as information entropy, the role of information as the elimination of the things uncertainty. The information entropy is defined as equation 4:
H piiln p (4)
In the formula, H represents the system entropy, pi represents a probability. In 1972, Csiszer defined such a measure: n p D(:)() p q q i i (5) i 1 qi In the formula, is the twice differentiable convex function, and (1) 0 , making: xx (x ) , 0, 1 (6) 1 So, 11nnpp D( p : q ) q (ii ) ( p q 1- 1) (7) 2 i i i -1ii11qqii -1 This is Havrada-Charvat directional divergence measurements, which is the basis of constructing generalized entropy model in this paper.
According to the formula (6), let maxp D ( p : u ) represent the maximum value, then:
1 -1 maxp D( p : u ) ( u 1) (8) -1 Defining Hp() as:
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1 n Hp() maxDpu (:) Dpu (:) u-1 (1 p ) (9) i p -1 i 1 The model is based on the minimum cross entropy model, a "distance" formula, representing the difference between investment income and expected returns. If the value is smaller, indicating that when the investors make investments, the real return is closer to the expected returns, the risk is lower under the uncertainty relation. In the formula (9), u -1 can be any multiple as entropy measure form, simplifying the model: 1 n H( p ) (1 p ) (10) i -1 i 1
2.3 General Entropy Optimization Model with Investor Sentiment In making investment decisions, investors often tend to have a psychological bias, which cause they not be completely rational investment according to rational man hypothesis, so they were affected by many factors such as gender, psychological quality, and cultural degree and so on when they choose the stocks, namely investor sentiment. Based on past stock data and the sentiment comprehensive index, the market investor sentiment can be obtained. Based on the formula (10), giving the indicators of the formula the specific economic meaning, investor sentiment factors can be taken into account, thereby it can be more consistent with the actual situation of the investors when distribute the proportion of investment and ensure that the investors get their expected return. If we treat α as investor sentiment, we can get the generalized entropy investment decision model in the investment portfolio. Considering n stocks in all T time periods of stock return rate, πt represents the t time income accounting for the proportion of total revenue, its value can be obtained in the past stock data; rit represents the i stock in t time of stock return rate, so the expected return rate of the i stock is:
T r r, i 1,2, , n (11) it1 t it The variance and covariance can be expressed as:
T 22（r- r ) , i 1,2, , n t 1 t it i (12) T （r-) r（ r -), r i , k 1,2,,; n i k ikik t 1 titi kti
Assume that the proportion of the total investment will be allocated for each stock xi ( i = 1,2, ..., n n), then: n xx10， (13) i1 ii According to Markowitz mean - variance model, the portfolio's expected return and variance can be expressed as the formula (14), (15): n E x r i1 ii (14) nn V x2 22 x x x 2 2 i11i i i i k i k ik i k i i (12), (13) into (14), available: T n n T V（ r x-)() r x22 R R (15) t1t i 1 it i i 1 i i t 1 t t Among them, n R r x， ti1 it i (16) n n T T n T R r x x r r x R i1i i i 1 i t 1 t it t 1 t i 1 it i t 1 t t
Rt represents the portfolio investment income at the t stage, indicates the average earnings, namely
investors expected return. Meanwhile, Pt is defined as the probability of the stock returns for the t time period:
tRR t t t Pt T (17) R R k 1 kk Combining formula (10), (15), (17) can be:
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11mmp H( p ) ( p 11 1) [ (jj ) 1] 11jj11j jR j (18) 1m R 1 1 ( j 1) (ER ( ) 1) 11 j1 j RR
T In formula (18), RR is on behalf of the investors expected return. In general condition, we t1 tt hope that this value remains the same, and therefore to simplified calculation, it can be omitted. Thus according to formula (15) and (11), (16), the formula (18) can reduce to equation (19):
1 n minH ( p ) [1 ( x r ) ] 1- i1 t i i Tn s.. t x r c ti11t i i n x 1 (19) i1 i
xi 0, i 1,2, , n 1
Wherein, α represents investor sentiment index, its value by formula (3) decided; t represents the t time income accounting for the proportion of total revenue, ri is on behalf of stock returns within a certain period of time, c and is treated as investors expected return. In this chapter, c =0.012. This is the generalized entropy investment decision-making model of the portfolio analysis in this chapter.
3. EMPIRICAL ANALYSIS
3.1 Symbol Description (1) The Calculation of Stock Returns There are two common ways calculating the stock return rate: the percentage and the logarithm method. If the pt indicates the closing price of a stock at t time, pt-1 is the closing price of a stock at t-1 time; rt indicates return of a stock at the t moment, then the calculation formula of the percentage method and logarithmic method as the following formula (20):
rt()/ p t p t1 p t
pt (20) rt ln()()() ln p t ln p t 1 pt1 But in percentage yield formula, the range of the next phase yield is (-1, + ∞), does not meet the hypothesis of the real return rate obeying normal distribution. However, logarithmic method can remedy this defect, at the same time using the logarithmic method can make operation process simple, more suitable for modeling requirements of financial assets, so this paper uses the logarithmic method to calculate each stock return and this This paper selects a total of 84 month stock yields from 2008.01 to 2014.12 as the analysis object, πt represents the t mouth income accounting for the proportion of 84 months average yield, α is on behalf of the investors sentiment index for the market, and α≠1.
3.2 Data Selection Based on the generalized Entropy investment decision model established in this chapter, given that ordinary investors are risk-neutral or risk averse, so in the choice of stock, investors tend to invest in more stable stocks that can provide reliable support for the investors to make long-term investment. The ups and downs of the stock in China stock market, although in the short term it may receive higher returns, but it does not guarantee long-term continuous gains, and the stochastic volatility of this kind of stocks is too large. It is difficult to satisfy the premise of model building, so give up such stocks on the data selection. According to the nature of the portfolio, this paper selects the 24 stocks of 14 industry in the Shanghai stock market, electricity, real-estate, construction, finance, resident services and other services, mining, Agriculture, forestry and fisheries, wholesale and retail trade, software, water conservancy, culture, sports and entertainment, transportation, accommodation catering industry and leasing business service business and so on. There are Qianjiang Water Resources(600283), Guidong Electric Power Industry(600310), Hongcheng Water Industry(600461), Baoli Real Estate(600048), Nanjing High-Tech(600064), Shanghai Construction(600170), Minsheng Bank(600016), Merchants Bank(600036), Medium Body Industry(600158), Sinopec(600028),
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Yangquan Coal Mining(600348), China Petroleum(601857), Yasheng Group(600108), Zhejiang Orient(600120), Chinese Medicine Shares (600511), Gehua Wired(600037), China Unicom(600050), Entrepreneurial Environment(600874), Broadcast Television Network(600831), Zhongyuan High- Speed(600020), Chongqing Road Bridge(600106), Jinling Hotel Corporation (601007), China International Trade(600007), CYTS (600138). Each stock earnings data of 2008.01 - 2014.12 is downloaded by RESSET, using lingo software to optimize formula(19).
3.3 Model Solution According to the formula (19), (20) and the data obtained, running the lingo software, which can obtain the final results getting the investment ratio of the investors in the stock, the result as shown in table 6:
Table 6. The Proportion of Stock Investment
General Entropy Model Selected stocks α=-0.4 α=-0.2 α=0.3 α=0.5 α=0.6 China International Trade 0.0663 0.0635 0.0455 0.0410 0.0423 Minsheng Bank 0.0429 0.0396 0.0487 0.0489 0.0532 Zhongyuan High- Speed 0.0348 0.0429 0.0376 0.0350 0.0310 Sinopec 0.0595 0.0616 0.0524 0.0510 0.0485 Merchants Bank 0.0489 0.0455 0.0572 0.0590 0.0612 Gehua Wired 0.0075 0.0098 0.0218 0.0310 0.0417 Baoli Real Estate 0.0343 0.0279 0.0367 0.0299 0.0431 China Unicom 0.0753 0.0699 0.0421 0.0450 0.0398 Nanjing High-Tech 0.0369 0.0311 0.0511 0.0491 0.0452 Chongqing Road Bridge 0.0117 0.0152 0.0285 0.0398 0.0361 Yasheng Group 0.0333 0.0354 0.0368 0.0382 0.0384 Zhejiang Orient 0.0574 0.0591 0.0437 0.0410 0.0424 CYTS 0.0308 0.0341 0.0401 0.0397 0.0402 Medium Body Industry 0.0563 0.0547 0.0490 0.0473 0.0413 Shanghai Construction 0.0376 0.0388 0.0381 0.0432 0.0456 Qianjiang Water Resources 0.0584 0.0472 0.0500 0.0525 0.0517 Guidong Electric Power Industry 0.0200 0.0233 0.0382 0.0312 0.0299 Yangquan Coal Mining 0.0291 0.0353 0.0392 0.0310 0.0287 Hongcheng Water Industry 0.0097 0.0110 0.0310 0.0285 0.0184 Chinese Medicine Shares 0.0465 0.0483 0.0322 0.0364 0.0308 Broadcast Television Network 0.0590 0.0621 0.0499 0.0473 0.0379 Entrepreneurial Environment 0.0595 0.0512 0.0384 0.0421 0.0457 Jinling Hotel Corporation 0.0242 0.0301 0.0397 0.0401 0.0473 China Petroleum 0.0601 0.0624 0.0521 0.0518 0.0596 Objective Function Value 0.8921 0.9548 1.1432 1.5637 1.8492
From table 6, it can be concluded that in the generalized entropy model, when α=-0.4, the investors can choose China Petroleum(601857), China Unicom(600050), China International Trade(600007), Sinopec(600028) according to the proportion of investment, the objective function value is 0.8921; α=-0.2, the investors can choose China Unicom(600050), China International Trade(600007), China Petroleum (601857), Sinopec(600028), the objective function value is 0.9548; α=0.3, the investors can choose Merchants Bank(600036), Sinopec(600028), China Petroleum (601857), Nanjing High-Tech (600064), the objective function value is 1.1432; α=0.5, the investors can choose Merchants Bank(600036), Sinopec(600028), China Petroleum(601857), Qianjiang Water Resources(600283), the value is 1.5637; α=0.6, the investors can choose Merchants Bank(600036), China Petroleum(601857), Minsheng Bank(600016), Qianjiang Water Resources (600283), the value is 1.8492. According to equation (19), α is on behalf of investors sentiment, when the value of α is -0.4, -0.2, 0.3, 0.5, 0.6, namely investor sentiment index is -0.4, -0.2, 0.3, 0.5, 0.6. Minus sign is on behalf of the investors in a negative attitude toward the market, meaning the market in the doldrums; plus sign represents the investors holding a positive attitude, the market is bullish, and the objective function value is 0.8921, 0.9548, 1.1432, 1.5637 and 1.8492 respectively. It finds that when investor sentiment is minus, the negative attitude of the market, the investors choose a more robust stock to invest to avoid the risks, so the objective function value is small, and the less risk, mainly to guarantee the principal. However, when investors are optimistic about the market, they tend to the higher interest stocks, showing the "profit" nature. In the pursuit of high returns is accompanied by a high risk, so the objective function value is high, meaning the risk degree is higher. Construction of generalized entropy investment decision model is basically consistent with economic theory.
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Under the same investment interests situation, investors choose a lower risk of stock to invest, which have the certain reference significance to investors’ stocks selection. 3.4 Model Evaluation Making investment decisions, because of the influence of market external and internal factors, the investors obtain the return meanwhile they must take on different degrees of risk, which reflects the uncertainty of financial market. In addition to the influence of the above factors, the investors are also affected by their own psychological factors such as the degree of bearing risk, so it is different for every investor in the choice of individual stocks investment share proportion. The modern investment portfolio model is based on the analysis of the investors, under the uncertainty situation it can reasonably distributed stock share, so as to obtain a better income. The generalized entropy investment decision-making model constructed above covers the advantages both the maximum entropy criterion and the minimum cross entropy criterion, and takes into account the non- exponential problem in the portfolio problem. The paper also quantifies the investor sentiment, overcoming the shortcomings of the previous investment decision models. Considering the psychological behavior factors, the article optimizes the investment risk analysis, making investment decisions of investors are more in line with investors' mental bearing capacity. Passing verification, the model has some applicability and provides a new way of thinking for avoiding the system risk of the portfolio. But the model (19) also exists many defects, such as: on the selection of the data, because of ignoring a share dividend, dividends, suspension the stock market appeared and other factors, there may be some bias in the stock daily return process; Meanwhile, this paper does not give a more rigorous mathematical proof and reasoning process, which is the focus of future research.
4. CONCLUSIONS
Firstly, based on the study of sentiment indicators, this paper has selected six indicators to reflect the emotional state, which can be effectively quantify the investor sentiment, then building a comprehensive index system of investor sentiment. Secondly, based on entropy optimization theory, the risk of securities portfolio has been effectively processed and the generalized entropy model has been constructed. Combined with sentiment index system, the article has established the mood of generalized entropy investment decision model, thus the investors' psychology and behavior factors have been considered in the model, meeting the different investment trends and making up the inadequacy of previous quantitative models, more in line with the needs of investors. The model has not only extended the entropy optimization theory, but also taken into account the psychological factors of investors' investment. It can giving a more objective reference for investors to minimum the investment risk under the preconditions of a certain expect returns level and to maximize the investment revenue in certain level risk. Therefore the entropy optimization theory has been better applied to the securities investment field, which providing the reference value and practical significance for theory and practice.
Acknowledgements This work was supported by Natural Science Foundation of China (71371092, 71301015) and China Postdoctoral Science Foundation(2014M561134).
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