Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 6831596, 15 pages

https://doi.org/10.1155/2017/6831596

## Effects of Common Factors on Dynamics of Stocks Traded by Investors with Limited Information Capacity

School of Economics and Management, Southeast University, Nanjing 211189, China

Correspondence should be addressed to Jianmin He

Received 4 June 2017; Revised 25 July 2017; Accepted 7 August 2017; Published 28 September 2017

Academic Editor: Ricardo López-Ruiz

Copyright © 2017 Songtao Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An artificial stock market with agent-based model is built to investigate effects of different information characteristics of common factors on the dynamics stock returns. Investors with limited information capacity update their beliefs based on the information they have obtained and processed and optimize portfolios based on beliefs. We find that with changing of concerned information characteristics the uncertainty of stock price returns rises and is higher than the uncertainty of intrinsic value returns. However, this increase is constrained by the limited information capacity of investors. At the same time, we also find that dependence between returns of stock prices also increased with the changing information environment. The uncertainty and dependency pertaining to prices show a positive relationship. However, the positive relationship is weakened when taking into account the features of intrinsic values, based on which prices are generated.

#### 1. Introduction

The concept of “information” is cornerstone of the EMH (Efficiency Market Hypothesis) [1, 2], which lays basis for development of other modern finance theories. Since information is an abstract concept, a large body of literature focuses on effects of information-bearing events on dynamics of the stock market [3–5]. Findings of these papers contradict the EMH and indicate that information in the stock market cannot be impounded into prices properly and immediately.

Ultimately, effects of information on stock dynamics or information efficiency of the market shall be fulfilled via activities of individual traders. For example, [6] argues that the proportion of informed traders affects information dissemination efficiency of the market. Before incorporating information into their activities or trading strategies, traders have to search and process information first. Suffering from limited information capacity, however, they cannot process and understand all information existing in the market [7]. Early studies concerning limited information capacity of decision makers are papers of Sims [8, 9]; then its implication is investigated in the field of stock market [10–12]. They find that limited information capacity of traders may cause excessive comovement [10], underdiversification [11], and home bias [12].

Factor analysis is one of the most popular practices for both financial practitioner and researchers, such as famous three-factor model [13, 14] and its numerous variants. It is also an information capacity-demanding task including analyzing lots of information, such as news, industry reports, and financial reports. Among all risk factors, common risk factors are a category of special ones. Not only dynamics of stocks but also their relations can be explained by common factors [15]. In addition, common factors are more specific than the market factor to be analyzed and have more extensive application potentiality than firm-specific factors. Thus, they are extensively exploited, like common economic factors [16], common company characteristics [17], common technical factors [18], and so forth. In fact, this emphasis or passion on common factors is supported by category learning behavior of traders due to limited information capacity [10].

Previous studies about common factors are almost empirical ones, and main purposes of these researches are identifying common factors with explanatory power for dynamics of stocks, such as momentum factor [19], short-term reversal factor [20], long-term reversal factor [3], and liquidity factor [21]. However, subject to sources of sample data, certain factors only apply to certain market scenarios. This has limited general explanation of roles played by common factors. Analytical approaches concerning limited information capacity [10–12] are usually accompanied with rigorous assumptions to keep the model tractable. The assumptions make analytical approaches not suitable for modeling adaptive behavior of traders in a complex market inundated with all kinds of information about risk factors.

In addition, stock market is a constantly evolving system. If we attempt to incorporate this market feature into our model, it is even harder to use traditional empirical researches or analytical approaches. Hopefully, the ABM (agent-based model) in finance field or ACF (Agent-based Computational Finance), pioneered by Sante Fe Institute, shows potentiality to cope with complex information environment of the market and the adaptive information searching and learning behavior of traders [22, 23].

The importance of studying effects of common factors on dynamics of stocks traded by investors with limited information capacity can be understood from perspective of EMH and behavioral finance and from practice of risk management and portfolio construction. Given limitation of empirical and analytical approaches above, we construct an artificial stock market using ABM to shed light on the effects. The rest of the paper is organized as follows. In Section 2, we introduce the structure of stock return, microstructure of the market, and the ABM. Simulation results and discussions are presented in Section 3. Robustness check is made in Section 4. In Section 5, conclusion remark is made.

#### 2. The Model

##### 2.1. Assets and Composition of Fundamental Return

Assuming that there are stocks in the market, their fundamental returns consisted of three types of risk factors: the market factor , common factors (), and each stock’s firm-specific factor (). Thus, the fundamental return of th stock at period is expressed as

In (1), , , and are drawn from mutually independent normal distributions , , and at each period [10]. The loading coefficient indicates the effect of the th common factor on the th stock; in the present we set (; ). With given initial value of the th stock , intrinsic values of the stock follow the dynamic . Thus, the variance of the th stock and the covariance of the th and th stocks (, ) are given in

In order to simulate the dynamic characteristics of the real stock market, we assume that the mean and variance of aforementioned normal distributions are, respectively, redrawn from uniform distributions , , , , and , at probabilities of , , and , which indicate intensities of dynamics of risk factors.

##### 2.2. Portfolio Optimization

There are investors with time window length in the market. They are endowed with a certain amount of initial wealth consisting of stocks and cash; that is, in which and are initial holding shares and price of th stock. In order to balance buying and selling power, the initial holding cash is set based on the initial holding shares and prices, such that .

At the trading day , a typical investor () either participates in portfolio trading of stocks at probability or holds current position waiting for better opportunities. The principle of investor for optimizing portfolio is maximizing the utility of wealth at trading day , namely, the statistical average timing at which investor participates in trading again. According to [24, 25], the investor ’s wealth at the trading day is jointly determined by the wealth at trading day and the return of portfolio over the time interval . Therefore, the evolution of wealth is given as where is the column vector of wealth where investor invests into each stock at trading day , and is the operator of transpose; is the corresponding return vector of stocks over the interval . Thus, investor ’s objective of portfolio optimization is calculating the optimal by maximizing utility of wealth .

According to the widely used utility function of CARA (Constant Absolute Risk Aversion) [22, 26–29], the optimization goal can be formally expressed as , where is the risk aversion attitude of investor . Under the assumption of normality of the expected return on trading day based on information gathered at trading day , it is well known that this optimization goal is equivalent to the following expression [30, 31]:

In the expression, () is variance-covariance matrix of expected returns over the interval . Following [32], the elements of can be written as (), where and are, respectively, the expected price and quote of the order, namely, the price at which investor hoped her orders had been traded. Therefore, according to the first-order partial derivative of expression (4) against , we obtain the optimal portfolio given in

##### 2.3. Heterogeneous Agents

From (5), we know that the optimal portfolio of investor is determined by three components: the risk aversion coefficient , the expected returns , and the variance-covariance matrix of expected returns. Investors are heterogeneous for all components. In respect of risk aversion coefficient, it is randomly and independently drawn from the uniform distribution for all investors.

Inspired by two distinctive beliefs held by investors in the real-world stock market, the present paper follows [32] and assumes that and are formed based on two same weighted beliefs: fundamentalist’s belief and chartist’s belief . Therefore, and . The superscript is substituted with and to indicate different beliefs, based on which expectations are formed. From analysis in previous subsections, we know that is determined by and which will be detailed in Section 2.5. Here, we try to focus on the formation of and when and .

With the belief of , investor holds the idea that the historical price trend will continue; that is, , where is settlement price of the last trading day; is the intensity of chartist’s belief responses to historical return . In terms of , investor forms her expectation based on calculation over historical prices of stocks.

When , investor believes that future prices will converge to future intrinsic values; that is, , in which is expected fundamental return based on information of future fundamental returns; is the reaction intensity of fundamentalist’s belief. As for , its th row and th column element is given as , if ; otherwise . After investor processed information and updated belief, she forms uncertainty perceptions about market and common and firm-specific factors, namely, , , and . Together with the formation of , the belief updating process will be described in detail in Section 2.4.

##### 2.4. Belief Updating with Constraint of Limited Capacity

In an ever-changing information environment of the stock market, investors have to constantly gather information and update their beliefs to adapt to the environment in order to construct a profitable portfolio. For , the information needed to update belief is derived from historical prices, which are freely available from the market. For , belief updating behavior needs analyzing information from all kinds of sources, like news, industry reports, and so on. This consumes some time and mental power, namely, information capacity in the present paper.

Sims [8, 9] first borrowed information entropy from the information theory [33] to describe limited information capacity of decision markers. Following these seminal papers, a large number of literatures [10–12] carry out in-depth studies concerning stock market with investors having limited information capacity. Take return of common factors in this paper, for example, the above-mentioned papers deem that uncertainty of posterior belief about must be smaller than that of prior belief after a signal about is received and that the amount of reduction in uncertainty equals consumed information capacity. According to [10], uncertainty of is the information entropy scaled in with base 2. The uncertainty of is determined by its variance-covariance matrix; that is, [11, 12], where is the number of stocks; is the determinant operator.

In fact, the above literatures modeled limited information capacity of decision makers as the limited transiting power of information channels. At the same time, they assume that the extraction of signal must reduce uncertainty perception of investors about returns. This is modeling investors’ behavior from perspective of learning, which does not apply to the belief updating case in this paper and is inconsistent with the fact that there is a lot of vague and even false information in the market. As the argument put forward by [7], understanding information is as much important as transiting or receiving information in communication. In the paper, limited information capacity of traders is measured as consumed information capacity in understanding the uncertainty. Since uncertainty is changing over time in a dynamic stock market, following [34] the present paper deems that the uncertainty perception of investors about fundamental returns may increase or decrease after acquiring and processing a piece of information. In either case, a certain amount of information capacity has to be consumed. We use absolute value of difference between uncertainty at trading days and ; that is, and to quantify consumed capacity of investors during their adaption from trading day to .

In the following, we will explain the process of information processing and belief updating when . For simplicity, notations with the same subscript, that is, (), successively denote variances of the market factor, firm-specific factors, and common factors. Since there are two distinctive phases including belief updating about risk factors and portfolio construction based on returns, we divide information processing and capacity consumption into and , to which investor allocates the fraction and of her total capacity . That is, and .

Once investor enters the market at trading day , she gets access to information about future fundamental returns of factors over the interval from various sources, such as industry reports and news, at probability indicating market transparency. First, investor processes information concerning returns at trading day for all risk factors; then she moves on to trading day , and so on. However, whether investor could process all information until depends on her information capacity. Therefore, the actual time window length over which investor can process all information must satisfy .

From (2), we know that is composed of ; thus evolves to after investor updates her uncertainty perception about risk factors. Therefore, the updating behavior of investor is constrained not only by the information capacity concerning risk factors, but also by with regard to return vector. In other words, two inequalities given in expression (6) must not be violated during belief updating.

In simulation, after belief updating investor applies the realizations generated from and according variances to expectation formation, namely, and , it can be seen that the magnitude and uncertainty of are positively correlated with and , which are referred to as information content and information amount in the present paper, respectively.

##### 2.5. Order Formation and Clearing Mechanism

Continuous double auction (CDA) is introduced as clearing and pricing mechanism. The operation of this mechanism requires investors to submit orders with elements of timing, quote, quantity, and trade direction. After investors enter the market sequentially and randomly, they form the quote and optimal portfolio under the constraint of resources, that is, no short selling and no short buying. Then order size and trade direction are determined given current shares of holding stocks. For the sake of brevity, we discard the subscript in the analysis that applies to every stock.

The formation of quote is affected by price range revealed by order book, expected price, and resources. After entering the market, investor observes the price range revealed by the book, where is the last transaction price at intraday time or if no transaction happens; is the tick size. After randomly drawing a price , the quote is preliminary determined as , from which we can see that is higher if is greater than , and vice versa. This is consistent with behavior of investors in real-life stock market. When it comes to only one stock, according to [32], we can obtain the only highest quote and the only lowest quote by taking resources constraint into account. Thus, the quote is finally given in

Let () be the fraction of wealth expected to be invested into each stock. At this point, () can be computed separately according to (5). However, considering resources constraint faced by entire portfolio, investor only distributes a fraction of total wealth to th stock with . Thus, actual wealth invested into th stock is (). Thus, the shares of th stock that investor would like to hold are , where is the operator of rounding down to the nearest integer. By comparing the demanding shares and the shares that investor already holds , order size and trade direction are given in (8), where 1, −1, and 0 represent buying, selling, and holding current position.

Orders are stored in the book with priority of price first and then with the priority of timing if the prices of orders are the same. The condition of transaction is met if the highest quote of orders on the buying side is greater than the lowest quote of orders on the selling side. The matched buying (selling) order is the one with highest (lowest) quote and with earlier submission time if quotes are the same for more than one order on the same side.

The traded volume of the transaction equals the smaller size of matched orders, and the transaction price is the quote of earlier submitted one of matched orders. After transaction, corresponding shares of stock and cash are added to or subtracted from positions of investors. At the same time, the matched order with smaller size is cleared from the book, while the remaining fraction of the bigger order is retained in the book. See [34, 35] for more technical details of the CDA mechanism. Settlement price of th stock at trading day , that is, , is the last transaction price at that day.

#### 3. Simulation Results and Discussion

##### 3.1. A Typical Simulation Result and Model Validation

Before simulation, we first assign values and ranges to parameters that will not change in the following. Those parameters in Table 1 denote traits of investors and economic environment of the artificial stock market.