Advances in Decision Sciences

Volume 2015 (2015), Article ID 971269, 18 pages

http://dx.doi.org/10.1155/2015/971269

## Loss Aversion, Adaptive Beliefs, and Asset Pricing Dynamics

Department of Social Science Computing, Faculty of Economics and Political Science, Cairo University, Cairo 11431, Egypt

Received 17 March 2015; Revised 30 June 2015; Accepted 10 September 2015

Academic Editor: Wing K. Wong

Copyright © 2015 Kamal Samy Selim 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

We study asset pricing dynamics in artificial financial markets model. The financial market is populated with agents following two heterogeneous trading beliefs, the technical and the fundamental prediction rules. Agents switch between trading rules with respect to their past performance. The agents are loss averse over asset price fluctuations. Loss aversion behaviour depends on the past performance of the trading strategies in terms of an evolutionary fitness measure. We propose a novel application of the prospect theory to agent-based modelling, and by simulation, the effect of evolutionary fitness measure on adaptive belief system is investigated. For comparison, we study pricing dynamics of a financial market populated with chartists perceive losses and gains symmetrically. One of our contributions is validating the agent-based models using real financial data of the Egyptian Stock Exchange. We find that our framework can explain important stylized facts in financial time series, such as random walk price behaviour, bubbles and crashes, fat-tailed return distributions, power-law tails in the distribution of returns, excess volatility, volatility clustering, the absence of autocorrelation in raw returns, and the power-law autocorrelations in absolute returns. In addition to this, we find that loss aversion improves market quality and market stability.

#### 1. Introduction

In 1987, the Wall Street Stock Market faced a severe financial crisis. That crisis provoked economists to realize that traditional economic theories such as the* theory of rational expectations* and* efficient market hypothesis* cannot explain the emergent aggregate behaviour of real markets. In addition, classical economic theories suggest that financial markets are populated with rational agents and rationality is common knowledge to all agents. However, these assumptions lead to* no trade* theorems [1], which contradict the excessive trading volume observed in real financial markets. Moreover, the rational agent cannot survive in a heterogeneous world^{1}. This observation motivated the researchers to use more advanced computational technique to better understand the behaviour of financial markets.

Therefore, the high trading activities in financial markets present evidence for the presence of heterogeneous predictions for asset prices. Heterogeneous agent models aim to relax the classical hypothesis of a representative agent and the rational expectations towards heterogeneous bounded rational agents [3]. These models study the individual-based behaviour in markets populated with bounded rational, heterogeneous agents using simple heuristics and simple extrapolation methods. This approach seems to provide plausible results in generating a near-realistic financial time series [4] and replicating the so-called stylized facts of financial markets [5–7]. Stylized facts represent a set of statistical properties common across many markets and time periods, such as bubbles and crashes, fat-tailed return distributions, uncorrelated returns, and volatility clustering.

Frankel and Froot [8], Taylor and Allen [9], and Menkhoff [10] conducted different questionnaire surveys to investigate the traders’ main heuristics in order to model their behaviours. The studies revealed that traders rely on two trading philosophies, the technical and the fundamental analysis, to determine their trading strategies. According to chartists, the ones who believe that price trend will continue and follow technical analysis will try to maximize their profits by taking advantages of asset price fluctuations [11]. Chartists compare the current price with the previous one; they buy (sell) when the asset price increases (decreases). On the other hand, fundamentalists, the ones who follow fundamental analysis, predict that the asset price will revert to its fundamental value [12]. Therefore, fundamentals buy (sell) when the asset price decreases (increases) as compared to its fundamental value.

Financial markets are comprised of many traders with heterogeneous beliefs, attributes, and level of rationality [13]. Traders face high uncertainty since they have to make expectations of future prices to submit their current orders. This uncertainty is due not only to the limited traders’ ability to collect and process information, but also to the algorithmic complexity of the problem they face. Consequently, traders are continuously enforced to learn and adapt to highly dynamic environments. This adaptation causes the whole financial system to coevolve. Studying financial markets as adaptive evolutionary systems of heterogeneous agents is a completely different approach from that used in the traditional economic models [14]. This encourages the use of agent-based modelling as the most suitable approach, as it provides more flexible tools to simulate the real world [15–18]. This approach implies new challenges and opportunities for making policy as well as managing economic crisis.

Artificial financial markets are models developed using the agent-based modelling approach. The main aim of artificial financial markets is to understand the endogenous variables that cause aggregate behaviours and patterns to emerge at the macro level [4, 19–23]. These artificial markets serve as test-beds for policy makers to explore the effect of different regulatory policies which improves the decision making process. Many of the market crashes can be limited by identifying the sensitive parameters that affect the financial market either directly or indirectly [24]. In this paper, we propose an artificial financial market which is capable of generating realistic stock market dynamics.

Meanwhile, behavioural finance is a relatively new paradigm seeking to link behavioural and cognitive psychological theories with finance to understand the bounded rational decisions of financial traders. Since 1979, Kahneman and Tversky provoked the idea of the choice under uncertainty. They spent many years to study this concept by conducting surveys and collecting data about the traders’ behaviour under uncertainty [25–28]. Kahneman and Tversky propose that the outcomes of risky prospects are estimated by a value function. This function is mainly characterized by loss aversion; that is, the function is steeper in the negative than in the positive domain. This characteristic describes asymmetric S-shaped value function, which is concave above a reference point and convex below it.

Although the prospect theory has been developed since 1979; yet there is no clear definition of gains and losses and how to measure them. Also, there is no clear identification of the reference point. Accordingly, its application into financial markets framework is very challenging. The model proposed in this paper provides a novel application of the prospect theory, where agents recognize their gains and losses in terms of an evolutionary fitness measure.

Many studies have been developed to model the switching dynamics between fundamental analysis and technical analysis, such as [4, 29, 30]. It is worth noting that deterministic agent-based financial market models, such as [31, 32], are able to explain boom-bust cycles while stochastic models, such as [4, 22, 33–35], can replicate more detailed stylized facts of financial markets. Unfortunately, few authors studied behavioural biases in their agent-based financial framework, such as [36–39].

In this paper, we explore the agent-based modelling as a tool for studying loss aversion behavioural bias introduced by the prospect theory. Our model contributes to behavioural finance research by linking the macro and the micro behaviours. This link is ignored in the classical models studied behavioural finance. To our knowledge, no research has been conducted to study the impact of loss aversion behavioural bias on the adaptive belief system and asset pricing dynamics, which is considered as our main contribution in the current work.

The rest of this paper is organized as follows. In Section 2, we introduce an agent-based financial market model in which the chartist traders are loss averse, along with the basic parameter settings and the model implementation. In Section 3, we investigate the extent to which our agent-based model is able to replicate the stylized facts of the Egyptian Stock Exchange. In addition, the results of a large Monte Carlo simulation we performed are presented. Finally, in Section 4, we summarize our main results and conclusions.

#### 2. An Agent-Based Model under Loss Aversion

In this section we introduce an agent-based financial model populated with heterogeneous agents with loss aversion behavioural bias. At the beginning we discuss the model definition and assumptions. In Section 2.2, the detailed model is provided. Finally, the parameter settings are depicted in Section 2.3.

##### 2.1. Model Definition and Assumptions

The main assumptions of the proposed artificial financial market can be summarized as follows.(i)There is only one risky asset to be traded.(ii)There are two types of agents, the market maker and the traders.(iii)In each time step , each trader decides on taking one of two alternative actions, either to submit orders or to abstain from the market.(iv)If a trader chooses to submit an order, she/he can either follow technical or fundamental trading rule. It is assumed that, at time , the orders are submitted without knowing the asset price. It is also assumed that fundamental traders can calculate the fundamental values.(v)Beliefs adaptation rule: the agents are bounded rational as they tend to choose the strategy performed well in the recent past and therefore display some kind of learning behaviour. It is assumed that the fitness of each trading strategy is available and publicly known by all agents.(vi)The chartist agents are loss aversion so that they recognize losses more than twice recognizing their gains. Consequently, they consider a value function proposed by the prospect theory to evaluate the fitness of each trading strategy.(vii)The fraction of traders following each strategy is determined via a discrete choice model.(viii)The market maker correlates the orders and adjusts the asset price according to the net submitted orders. It is assumed that the market maker is a risk neutral and settles the asset prices without intervention.(ix)Agents in our market interact indirectly through their impact on price adjustment which affects the performance of the trading rules which in turn affects the agent decision to select trading strategy and so on.

##### 2.2. Model Formulation

The behaviour of the market maker is described as in Farmer and Joshi [34], where the price settlement is formulated as a log-linear price impact function. This function measures the relation between the quantity ordered (demand/supply) and the price of the asset. Thus, the log-price of the asset in period is given bywhere is a positive price settlement parameter, and are the orders submitted by chartists and fundamentalists, respectively, at time , and and are the weights of technical strategy and fundamental strategy, respectively, at time . In order to make our assumptions close to the real market, the noise terms are added to catch any random factors affecting the price settlement process. It is assumed that are IID normally distributed random variables with mean zero and constant standard deviation .

The goal of the technical analysis used by the chartists is to exploit the price changes [40]. Orders exploiting technical trading rules at time can be written aswhere is a positive reaction parameter (also called extrapolating parameter) that capture the strength of agents’ sensitivity to the price signals. The first term at the right-hand side of (2) represents the difference between current and last price, which indicates the exploitation of price changes. The second term captures additional random orders of technical trading rules. And are IID normally distributed random variables with mean zero and constant standard deviation .

Fundamental analysis assumes that prices will revert to their fundamental values in the short run [12]. Orders generated by fundamental trading rules at time can be formalized as where is a reaction parameter (also called a reverting parameter) for the sensitivity of fundamentalists’ excess demand to deviations of the price from the underlying fundamental value. ^{2} are log-fundamental values (or simply fundamental values) [31]. is introduced to capture additional random orders of fundamental trading rules. are IID normally distributed random variables with mean zero and constant standard deviation .

The evolutionary part of the model, inspired by Brock and Hommes [32], depicts how beliefs are evolved over time. That is, how agents adapt their beliefs and switching between strategies. This part is mirrored in the fractions , where represents the fraction of inactive agents and are as indicated in (1), and the strategy weights add up to one. Fractions are updated according to evolutionary fitness measure (or attractiveness of the trading rules) which can be presented as follows:where , , and are the fitness measures of using chartist strategy, fundamental strategy, and no-trade strategy, respectively. The inactive traders submit zero orders, so they got zero attractiveness of taking such an action. The fitness measure of the other two trading rules, the technical and the fundamental analysis, depends on two components. The first term of the right-hand sides of (4) and (5) is the performance of the strategy rule in most recent time. Notice that orders submitted in period are executed at the price declared in period . The gains or losses depend on the price declared in period . The second term of the right-hand sides of (4) and (5) represents agents’ memory, where is the memory parameter that measures the speed of recognizing current myopic profits. For , agent has no memory, while for they compute the fitness of the rule as a sum of all observed myopic profits.

While, in Westerhoff [22] model, agents symmetrically perceive gains and losses in terms of fitness, in our model we propose a realistic behavioural bias, so that chartists evaluate their strategy fitness in terms of a value function of gains and losses. The proposed value function implies that chartists recognize losses more than twice their recognition of gains. As our focus is to study loss aversion, we adopt the Tversky and Kahneman [27] and Benartzi and Thaler [41] piecewise linear value function proposed by the prospect theory. Accordingly, the value of the fitness of technical strategy is given bywhere is the parameter of loss aversion that measures the relative sensitivity to gains and losses. However, setting reduces the value function to ; we call this case loss-neutral chartists.

Following Manski and McFadden [42], the market share of each strategy can be obtained by the discrete choice model^{3} as follows:

The higher attractive strategy will be chosen by the agents. The parameter , in (8), is called the intensity of choice and measures the sensitivity of mass of agents selecting the trading strategy with higher fitness measure. In such adaptive beliefs, financial market prices and fractions of trading strategies coevolve over time.

##### 2.3. Basic Parameter Settings

Model parameter settings are determined following Tversky and Kahneman [27], Winker and Gilli [43], Farmer and Joshi [34], and Westerhoff [22]. The values of model parameters are chosen so that the model can mimic the dynamics of real financial markets.

The main idea behind choosing specific values of the parameters can be summarized as follows. The reaction parameters of technical and fundamental trading rules (multiplied by the price settlement parameter) are between zero and 0.1 for daily data.

To keep the autocorrelation^{4} of raw returns^{5} close to zero, parameters and are chosen as follows. The population of the chartists is matched with the population of the fundamentalists, so that the positive short-term autocorrelations induced by the chartists are cancelled by the negative short-term autocorrelation of the fundamentalists. Therefore, the reaction parameters of technical and fundamental trading rules are set to be the same.

The value of is assumed to be higher than to reflect the level of noise associated with technical trading rule. The value of is assumed to be near one, so the agents have good memory. Also, the value of reflects the bounded rationality in choosing the trading rule with highest fitness measure. Finally, many experiments estimate loss aversion parameter to be in the neighbourhood of 2; that is, the utility of losses is twice as great as the utility of gains [27, 41]. Experimental estimation of has been estimated by Tversky and Kahneman [28], such as . The values of model parameters are summarised in Table 1. In the following section we study the evolutionary dynamics of our proposed model.