Maximally Diverse Grouping Problem

In the previous chapter, we describe the grouping problem. The goal of the grouping problem is to divide n elements into K groups, with numbers of objects in groups as similar as possible, and each element belonging to a group. There is a research discussing the effect of diversity in grouping problem [29]. Therefore, the maximally diverse grouping problem (MDGP) starts to be discussed. The MDGP considers the diversity in the traditional grouping

problem. MDGP is to get maximal diversity among the elements in each group. Formally, the formulas of MDGP is as follows:

, between each pair of objects.

There is a research used to prove that MDGP is the NP-hard problem [4]. Therefore, there are many heuristic approaches proposed to solve MDGP [3] [4] [9] [18] [29] [35] [40] like the hybrid genetic algorithm [18], the variable neighborhood search [3], and the artificial bee colony algorithm [35]. Then, Chen et al. proposed an algorithm for mining stock portfolio for solving MDGP [9].

2.4 The Proposed Method for Stock Portfolio

In this section, we explore the literature on the stock portfolio. In 1952, Markowitz proposed the M-V model to lay the foundation for the theory of stock portfolio [28]. After the M-V model has been proposed, there are many studies carried out on the selection of portfolio.

Although this issue has been raised for a long time, there are still many studies to explore it until now. The following describes the recent discussion of the stock portfolio of literature.

The theory of stock portfolio is roughly divided into two categories depending on whether a model is used or not. Some research with a stock portfolio model is first introduced below [12] [21] [22] [37] [38]. Escobar et al. proposed a model for closed-form solutions that was for optimal allocation and value functions [12]. With the model, they pointed out that the stock return predictability significantly affects the optimal bond portfolio and the hedge components were larger in bond portfolio than the respective hedge components in stock portfolio. Sun and Liu established the empirical cross-correlation matrices to improve portfolio optimization by combining the Pearson’s correlation coefficients (PCC) method and the detrended cross-correlation analysis (DCCA) method. In the experiment, the stability analysis of portfolio weights demonstrates that the PCC method matrix has a greater effect than the DCCA method matrix on the portfolio weight stability [37]. Yu et al. developed two CVaR-based robust portfolio models. The first one is worst-case conditional value-at-risk (WCVaR) model, and the second one is relatively robust conditional value-at-risk (RRCVaR) model. They find that when required return is fixed, the RRCVaR model brings higher returns, lower trading costs, and higher portfolio diversity than the WCVaR model [38]. Then, some research without a stock portfolio model is introduced below [1] [5] [7] [13] [24] [42]. Baralis et al. presented an itemset-based approach to automatically identify promising sets of high-yield but diversified stocks to investors. They investigated the usage of itemsets to generate appropriate stock portfolios and recommend them to investors from historical stock data [1]. Thakur et al. used the fuzzy Delphi method to identify the critical factors in input data. They then used critical factors and historical data to rank the stocks by the Dempster– Shafer evidence theory [24].

There are still many approaches in progress.

2.5 Investor Sentiment Index

In 1990, DeLong et al. proposed the noise trader model [10]. The model considers that in the market there are some noise traders who give not only excessive response but the low response to information, which directly affects the stock price equilibrium and result to market risk rise. Investor sentiment is one of the irrational factors that cause the transaction of the noise. It represents the subjective judgment of the investor based on his attitude to the future stock price rising or falling. Hence, many studies have begun to look for emotional factors that can reflect the future changes in the stock market for traders. Schmeling used a new data set on investor sentiment to show that institutional and individual sentiment proxy for smart money

and noise trader risk.

In 2001, Brown et al. indicated that if investors believe that the actual value of the stock is higher than (below) the current price, their mentality will tend to be optimistic (pessimistic) [2]. The authors also found that investor sentiment will be affected by past stock returns and the degree of emotional volatility is also highly correlated with the stock returns in the same period. Schmeling used a new data set on investor sentiment to show that institutional and individual sentiment proxy for smart money and noise trader risk [36]. Lee et al. found that there is a close relationship between investor sentiment, market volatility and excess return [27].

The following describes the research related to investor sentiment indices. In 1998, Neal et al. researched the three investor sentiment indices including the degree of closed-end funds discount, odd-lot ratio, and net redemptions for forecasting stock returns [31]. They found that the degree of closed-end funds discount and net redemption are predictable for small companies, and the odd-lot ratio has no predictability for either large or small company. In 2000, Barber et al. considered that there is a strong negative relationship between the IPO rate and the following year's market returns [6]. This relationship can provide a stronger market return prediction. In 2012, Wang et al. pointed out that there is a high correlation between the three retail sentiment variables including the buy-sell imbalance, the individual investor turnover and the proportion of day-trades [41]. This result indicated that these three variables are likely to capture similar retail investment behaviors.

CHAPTER 3

Proposed Approach

In this section, we will introduce our proposed approach, which uses two investor sentiment indices in the grouping genetic algorithm to obtain a diverse group stock portfolio. We first design some rules based on two sentiment indices to decide the buying and selling points of all the input stocks. These stock price data with buying and selling points are then sent to the proposed based approach for obtaining a best group stock portfolio. The proposed GGA-based approach will use the sentiment indices and the other factors in its fitness to evaluate the chromosomes (possible group stock portfolios). After a predefined number of generations, the chromosome with the best fitness value at the last generation is output as the diverse group stock portfolio. The flowchart of the proposed approach is given in Fig. 1.

Fig. 1. The flowchart of the proposed approach.

Fig. 1 shows that the input data is preprocessed by trading strategy generated from investor sentiment index and trading signal. And, it puts the preprocess data into the grouping genetic algorithm to produce the best derived diverse GSPs with investor sentiment index. The components of the proposed approach are described as follows.

3.1 Data Preprocessing: Trading Strategy

For considering the sentiment factor in the proposed GGA-base approach for finding a good group stock portfolio, we first transform the input stock price sequences to the ones with sentiment considered. We design some rules based on two sentiment indices to decide the buying and selling points of all the input stocks.

Most of the portfolio research topics focus on how to find a high-risk and low-risk portfolio; however, little research is proposed to provide users with trading time points of the stock portfolio. In this study, we provide users with trading timings for each trading period. For instance, we assign the trading period from 2014 to 2016. With this research, we accurately informed consumers to buy the stock on February 3, 2014, and sell it on December 24, 2016.

What is more, this study still retains the advantages of high-risk and low-risk of traditional stock portfolios.

In this study, the decision rules for determining the timing of buying or selling the stocks are based on the investor sentiment index. In the previous chapter, there are many indices that can reflect the investor's sentiment, called investor sentiment index. After filtering, we choose the research [41] to provide investor sentiment indicators as the basis for this study. There are three reasons for selecting the investor sentiment indices. First, the sentiment indices provided by the paper apply to the Taiwan stock market and it consists with the data used in this paper.

Due to the different rules of the transaction, the floating price of the stock in different regions

will be different, and the impact of investor sentiment indices will also be affected. Therefore, selecting the same range of research as our paper is important. Second, the formulas for investor sentiment indices are clear and easy to understand. Too complicated formulas not only difficult to directly express its meaning but also cause the overlong operation time. Third, the indices are designed based on the behavior of retail investors. In Taiwan, retail investors' trading volume accounts for 70% of Taiwan's overall trading volume for the stock market. Therefore, studying Taiwan's retail investors' behavior is more meaningful and persuasive. Next, we introduce the two investor sentiment indices used in this paper to reflect the investor's sentiment.

The first index is called Buy-Sell Imbalance (BSIt), which is defined as follows:

,

The parameters in this formula are as follows: BMLt is the balance of margin loan of the stock on the day t; BSLt is the balance stock loan of the stock on the day t; Vt is the trading volume of the stock on the day t; BIt is the buying volume of institutional investors of the stock on the day t; and SIt is the selling volume of institutional investors of the stock on the day t.

With these defined parameters, BSIt means the value of buy-sell imbalance of the stock on the day t, VBt means the retail investors' buying volume of the stock on the day t, and VSt means the retail investors' selling volume of the stock on the day t.

Then we explain the proper terms used. The margin loan is an investor’s behavior which means investor borrows money from securities dealer to buy the stock. In Taiwan, the securities

dealer will borrow 60% of the value of the stock to investors with good credit to purchase the stock. For example, we assign that the price of the stock is one hundred, and the value of the stock is one hundred thousand. It means that investors can only use forty thousand to hold the stock by margin loan. In general, investors expect that stock price will rise in the future, but the hands of the funds is not enough; then they pay a part of the margin to securities dealer to borrow money for buying stock, and then wait for an opportunity to sell the stock at high prices and earn low selling low spreads. In contrast to a margin loan, the stock loan means investors borrow stock from securities dealer to sell it. If investors expect that stock price will fall in the future, they can borrow stock from securities dealer to sell by the stock loan. The trading volume represents the total number of transactions including institutional investors and retail investors, where institutional investors include foreign investment institutions, investment trusts, and dealers. The retail investors represent the individual investor. Retailers are usually composed of low-income individual investors and characterized by less investment per person.

Next is an analysis the formula of BSIt. Above all, we focus on the fractions. On a stock at the time t, the higher of the BMLt means the higher volume of investor borrowing money to buy the stock. This phenomenon can be interpreted as the degree of investors expecting the stock to turn high is increasing. On the other hand, the higher the BSLt means the higher volume of investor borrowing stock to sell the stock. Therefore, if the difference between BMIt and BSLt is high, it means the degree of investors expecting the stock to rise is high too; in other words, investors are optimistic about the stock. If the difference between BMIt and BSLt is low, it means investors are pessimistic about the stock. Then, we focus on the denominator. The sum of VBt and VSt means the overall retail investors' trading volume of the stock at the time t. To sum up, at time t, higher BSIt value means that the retail investors are more optimistic about the stock.

The second investor sentiment index is called proportion of Day-Trades (DTt), which is

defined as follows:

,

t t t Shares

DT  NDT ₍₄₎

where NDTt is the number of day-trades and Sharest is the outstanding shares. In the stock market, the number of day-trades (NDTt) means investors’ twice trading behaviors (including buy after sell or sell after buy) by margin loan and stock loan on the same day t on the same stock. For example, if a man buys (sells) a share of TSMC on December 12, 2016 by margin (stock) loan and sells (buys) it by stock (margin) loan on the same day, the number of day-trades of TSMC on December 12 adds one. The Sharest means the amount of stock that the company offers to investors on the day t. Generally, the magnitude of the Sharest change is extremely small. Only when the company makes a big change, the Sharest will be different.

Therefore, the Sharest is used to normalize NDTt.

Next is an analysis of the formula of DTt. When the DTt of the stock turns high, it means more and more investors day-trade the stock on the same day t. Generally, most of the investors do not day-trade, only those who are particularly interested in the stock will do so. Thus it can be seen that there is a positive relation between DTt and the degree of investors’ interest in the stock. However, the value of DTt does not represent the investor expecting the stock price to rise or fall. For instance, if investors are expecting the stock price to rise, they may buy a stock by margin loan and sell it by stock loan on the same day, and that causes the DTt of the stock to rise. Then, if investors are expecting the stock price to fall, they may sell a stock by stock loan and buy it by margin loan on the same day, and this behavior still causes the DTt of the stock to turn high.

With these two investor sentiment indices, we divide the investor sentiments on a stock into six cases as follows:

Table 1. Six cases for the two investor sentiment indices.

In Table 1, the investor sentiments are classified into six cases by the status of BSIt and the change of DTt. We use the six cases in Table 1 to judge the stocks on every trading day. For example, Case II reflects the active attitude of the investor on a stock. However, it does not give the rising or falling expectation of investors. Cases IV, V and IV reflect the investors holding the passive attitude to a stock. However, the passive attitude doesn’t affect the stock price. Therefore, Cases II, IV, V and VI can be ignored. On the contrary, the active attitude of investors to a stock will affect the stock price. Thus, we use Cases I and III to develop the

For distinguishing the status of BSIt, the BSI thresholds are defined. The BSI thresholds include Low BSI threshold and High BSI threshold as follows:

Low BSI threshold = PR10(BSIt),

High BSI threshold = PR90(BSIt),

where PR10(BSIt) is the value which is at the PR10 from all 𝐵𝑆𝐼t value, and PR90(BSIt) is the value which is at the PR90 from all 𝐵𝑆𝐼t value. The PR (percentile rank) value is a percentage of scores in its frequency distribution that are equal to or lower than it. It is mostly used in the examination results above the rankings. For example, a score that is greater than or equal to

50% of the scores of people is said to be at the 50th percentile, where 50 is the percentile rank.

To practice the PR value, two steps should be implemented. They are assigned a string of numbers S as follows:

S = {12, 8, 2, 1, 17, 15, 4, 18, 9, 3, 11, 16, 10, 19, 5, 20, 14, 7, 13, 6}.

First, the string S is sorted from small to large. Then the new string S' is as follows:

S' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}.

Then, we use the formula to calculate the PR value of every element in the string S'. The formula is as follows:



 



 N

PRvalue_i 100Rⁱ 50

100 , (5)

where Ri is the rank of element i, and N is the number of elements in the string S. Finally, we get the PR value of every element in the string S. In this case, N is 20. The PR value of the element ‘2’ is 7, called PR7(S), because the rank of ‘2’ is 19. Similarly, proving that ‘3’ is PR12(S), ‘4’ is PR17(S), …, ‘20’ is PR97(S). PR10(S) cannot be calculated due to the small number of elements in the string S. Therefore, we choose the element which is larger than PR10(S) and the closest one to PR10(S) to be the Low BSI threshold. The element ‘3’ is the Low BSI threshold. Similarly, we choose the element ‘19’ to be the High BSI threshold. With the Low BSI threshold and the High BSI threshold, it is defined that the High BSIt is the BSIt

which islarger than High BSI threshold, and the Low BSIt is the BSIt which is smaller than Low BSI threshold.

Following that, we introduce the trading strategy in this proposed approach. Before defining the trading strategy, we combine Case I and Case III with buying and selling 46 company shares for finding the rules which maximize the stock return. The combination1 is, if Case I is true, buy the stock and if Case III is true, sell it. The combination2 is, if Case III is true, buy the stock and if Case I is true, sell it. Compare combination1 with combination2, find combination2, and make the overall stock more profitable than combination1. The reason for this phenomenon is likely to be that the investor sentiment experienced the lowest point is likely to rise and then drive the growth of stock prices. After the consolidation of the above information, the trading strategy is set up of this study. The trading strategy is composed of two rules as follows:

Rule 1: On a day t, if a stock’s BSIt is Low and DTt is rising, then buy the stock.

Rule 2: On a day t, if a stock’s BSIt is High and DTt is rising, then sell it.

Using the defined rules, the trading points are added to the original data. In this study, we define that the number of transaction on every stock is one. Therefore, we choose the first buying point and the last selling point to be the trading timing on this stock. By following this investor sentiment method, we get the Buy price and Sell price on every stock in the input data.

3.2 Components of the Proposed Approach

3.2.1 Encoding Scheme

We use a set S to represent n stocks, and denote them as {s1, s2, …, sn}. The number of the groups is K. Then, the encoding scheme of the chromosome is defined as follows in Fig. 2.

Fig. 2. The encoding scheme of chromosome.

In Fig. 2, the chromosome consists of three parts including the grouping part, stock part, and stock portfolio. The number of groups in the grouping part is K. The results of the stock

In Fig. 2, the chromosome consists of three parts including the grouping part, stock part, and stock portfolio. The number of groups in the grouping part is K. The results of the stock