The Behavior of Financial Markets under Rational Expectations

The Behavior of Financial

Markets under Rational

Expectations

HAN Yan

Bridge21 | Alhambra, CA

The Behavior of Financial Markets under Rational Expectations

By HAN Yan

Copyright ©2022 HAN Yan

Published by Bridge21 Publications, LLC

Distributed by Casemate Group

Casemate Academic (Havertown, Pennsylvania): USA and North America

Oxbow Books (Oxford): UK, Europe, and the Rest of the World

This research was sponsored by the National Natural Science Foundation of China under grant number 71772013.

No reproduction and distribution without permission.

Published in the United States of America

ISBN 978-1-62643-087-7 Hardcover

ISBN 978-1-62643-088-4 ePub | ebook pdf | Kindle

Contents

Chapter 1 Information

1.1 Rational expectations

1.2 Different opinions

1.3 Information aggregation and learning

1.3.1 Information percolation and learning

1.3.2 Hidden information

1.3.3 Price informativeness

1.4 How do agents learn new information

1.5 Information and pricing

1.5.1 Information asymmetry and asset pricing

1.5.2 Liquidity and asset pricing

1.6 Further issues of learning and reacting

1.7 Information providers

Chapter 2 How are prices formed?

2.1 The perspective of asset pricing literature

2.2 Inventory costs based microstructure models

2.3 Information based microstructure models

2.4 What is risk?

2.5 Trading mechanism

2.6 Short selling

2.7 Other topics in microstructure

Chapter 3 Liquidity

3.1 Measuring liquidity

3.2 Volume

3.3 Determinants of liquidity

3.4 Markets’ and liquidity providers’ conditions and liquidity

3.5 The effect of liquidity on the firm’s well-being

3.6 Who is providing liquidity

Chapter 4 Limit orders

4.1 The models of limit orders

4.2 Investor’s choice of limit versus market orders

4.3 Limit order revisions and aggressiveness

4.4 Limit order patterns

Chapter 5 Depicting investors

5.1 Theory based investor taxonomy

5.1.1 Informed investors

5.1.2 Liquidity investors, noise investors, and speculators

5.2 Identity based investor taxonomy

5.2.1 Individual investors

5.2.2 Institutional investors

5.2.3 Foreign investors

5.3 Investor sophistication

5.3.1 What does sophistication mean?

5.3.2 The effect of investor sophistication

5.4 Herding and correlated trading

5.4.1 Empirical evidence

5.4.2 Theoretical explanations

5.5 Trading behavior

5.5.1 Trading and contemporaneous returns

5.5.2 Trading horizon

5.5.3 Value investing

Chapter 6 Mutual funds

6.1 Fund performance and fund manager skills

6.1.1 Overview of fund manager skills

6.1.2 Factors affecting fund performance

6.2 Funds herding

6.3 Incentives and the risk shifting

6.4 Fund flow and investor’s preferences

6.4.1 Flow-performance relationship

6.4.2 Explanations for asymmetric performance-flow relationship

6.4.3 Investor’s purchasing process

6.4.4 Advertising

6.4.5 Smart or dumb money effect

6.5 Fund fees

6.6 Governance of funds

Chapter 7 Prices of IPO and SEO

7.1 Why and when to issue new equities

7.2 Choice of issue types

7.3 New issuance pricing

7.4 Trading around IPO and SEO

7.5 Market reaction and long term performance

Chapter 8 Behavioral explanation

8.1 General discussion on the behavioral explanation

8.2 Disposition effect and prospect theory

8.3 Overconfidence and over-reaction

8.4 Awareness, familiarity, and attention

8.5 Law of small numbers

8.6 Rational structural uncertainty models

Chapter 9 Modeling economic behavior

9.1 Nash equilibrium

9.1.1 Dominant and dominated strategy

9.1.2 Definition of Nash equilibrium

9.1.3 Evolutionary stability

9.1.4 Continuous strategy space

9.2 Bayesian games

9.3 Utility maximization models

Chapter 10 Mathematical techniques for economic modeling

10.1 Difference equations

10.2 Differential equations

10.2.1 Existence and uniqueness of the solutions

10.2.2 Solving one-variable first-order differential equation

10.2.3 General method solving high order linear differential equations

10.3 Fixed point

10.4 Static optimization

10.5 Duality

Chapter 11 Empirical methodology

11.1 Regressions

11.1.1 Simple regressions

11.1.2 Regression discontinuity

11.1.3 Ordered probit regression

11.1.4 Vector autoregression

11.1.5 Panel data regressions

11.2 Non-regression methods

11.3 Endogeneity

11.3.1 Natural experiment

11.3.2 Difference approach

11.3.3 Instrumental variables

11.3.4 Simultaneous equations

11.3.5 Self-selection and Heckman model

11.3.6 Smart experimental designs getting around endogeneity

11.3.7 When should we be concerned about endogeneity

11.3.8 Other research design issues

11.4 Measurements

11.4.1 Sample and data

11.4.2 Information and risk

11.4.3 Trading

11.4.4 Firm characteristics

11.4.5 Asset pricing related measures

Bibliography

Chapter 1

Information

1.1 Rational expectations

Rational expectation models are arguably the most important starting point for understanding the economic behaviors in financial markets. The notable seminal works along this line of research are: Milgrom (1981), Grossman (1981), and Milgrom and Stokey (1982).

Hellwig (1980, p. 479) shows that for an agent with constant absolute risk aversion (CARA), her demand for the risky asset is independent of her initial wealth. Admati (1985) extends Hellwig (1980) into multi assets. The following is from Admati (1985) except for some change in notation.

There is a continuum of agents, indexed by i , in the interval[0,1].¹ They trade in two periods. In period 0, each agent allocates her initial wealth w0i between one riskless asset, and n risky assets. Let p be the price vector of the risky assets.² In period 1, the riskless asset pays r units, and the risky assets pay v units of the single consumption good. Therefore the terminal wealth for agent i is given by

where xi is agent i’s optimal holding of the risky assets, which maximizes her expected utility .

Further suppose before allocating her wealth in period 0, agent i receives a private signal si concerning about v, i.e., . And there is a per capita random supply shock z for the risky assets. Let V, U, and Si be the variance-covariance matrices of v, z, and .

The equilibrium satisfies the following conditions:

1. Optimal holding condition: argmax .

2. Market clearing condition: .

Admati (1985, p. 637) shows that there is a unique linear equilibrium price function of the form

where and .

Agent i’s conditional expectation of v is given by (Admati 1985, p. 639)

where is not necessarily equal to the unconditional variance covariance matrix V.

And the equilibrium demand function of agent i is given by (Admati, 1985, pp. 639–640):

1.2 Different opinions

It should be noted that rational expectation models assume that agents have heterogeneous beliefs. This forms an important distinct from representative agent paradigm in the traditional economic models. The traditional economic models typically assume that there is a representative agent, which represent all the agents (investors) in the game. By assuming a representative agent, the traditional economic models implicitly assume that all agents (investors) share the same belief, i.e., homogeneous beliefs. Although one can argue that the distinction between heterogeneous versus homogeneous beliefs is nothing more than different modeling techniques, with this distinction becoming vital for understanding the economic behaviors in financial markets, because the dynamics of financial markets are essentially driven by the interactions among investors who hold different beliefs. Put differently, a market in which all investors share the same belief is a dead market because no one has the incentive to buy or sell the financial instruments. That is why we shall pay close attentions to the models’ aspects of different opinions.

Lamont (2012, p. 5) provides an excellent explanation for different opinions:

Harrison and Kreps (1978) construct a model with rational investors where differences of opinion, together with short sale constraints, create a speculative premium in which stock prices are higher than even the most optimistic investor’s assessment of their value (see also Duffie, Garleanu, and Pedersen 2002). These differences of opinion can be interpreted as arising from different prior beliefs, which rationally converge as information arrives (Morris, 1996), or as irrational overconfidence (Scheinkman and Xiong, 2003).

A key feature of rational expectation model is that investors are trying to infer other investors’ private information, and update their information according to the inference. If we assume investors agree to disagree, or put it alternatively, investors hold dogmatic belief, that is, they simply deem wrong what the others believe. There’s a term, agree to disagree, to summarize this situation.³

Diether et al. (2002) use the analyst dispersion to proxy divergence of opinion, and find that higher divergence of opinion is followed by a lower future stock return. Chen et al. (2002) use the mutual fund ownership as the divergence of opinion proxy, and find that higher divergence of opinion is associated with higher stock price. Chatterjee et al. (2012) argue that these two pieces of evidence support Miller’s (1977) prediction that the greater the divergence of opinion among investors about the security’s value, the higher the security’s market price.

Moeller et al. (2007) show that the higher diversity of opinion decreases the acquirer’s return in stock payment acquisitions, with cash payment acquisitions unaffected.

Kim (2013) shows that there are pricing errors at the opening sessions, and the errors are more pronounced among the stocks that trade earlier. The evidence supports the idea that investors learn from each other and this learning process inevitably incurs errors, especially at its early stage.

1.3 Information aggregation and learning

1.3.1 Information percolation and learning

Some market indicators, among which is surely the stock price, serve as information aggregators. Stock price aggregate all the information on the market. This is the fundamental belief behind market efficiency. The inquiry on information aggregation can be categorized into two lines. One line of research focuses on the equilibrium result of information aggregation, e.g., Grossman (1976). The other line of research focuses on the micro-level process of information aggregation, e.g., Kyle (1985). Besides stock price, trading volume is also viewed as containing information, e.g., Schneider (2009).

Why can these indicators serve as information aggregator? The most important reason is that every investor tries to infer other agent’s private information by observing these indicators. It’s this process that aggregates all the agent’s private information. Rational expectation models depicts these processes. The competition among insiders drive stock price to its true value more quickly. See Holden and Subrahmanyam (1994, 1992), Foster and Viswanathan (1996), Back et al. (2000), and Baruch (2002).

In Kyle’s (1985) model, information is gradually incorporated into price, because the informed trader with monopoly power splits her order. Baruch (2002), Back et al. (2000), Foster and Viswanathan (1996, 1994, 1993), Holden and Subrahmanyam (1994, 1992), and Admati and Pfleiderer (1988) extend Kyle’s (1985) model, allowing for multiple informed traders.

A key role of market is to aggregate the private information held by various agents (Hayek, 1945). For this process to occur, agents must infer others information and update theirs. This inferring-updating process is also studied under terms like information percolation and learning. Here learning means that investors interpret their own and infer other investors’ signals. Another similar concept of learning is learning by doing. That is, investors improve their investment skills through the accumulation of experiences. Experiences, of course, can help investors better understand and use the signals. But it seems that the learning by doing literature emphasizes more on long term effect of the improved skills, rather than the signal interpretation in individual decisions. So in this subsection, the learning refers to the interpretation of signals.

Duffie et al. (2010) define two channels of information percolation. One is the public channel, and the other is the private channel: Information aggregation occurs through the public observation of variables that reflect other agents’ actions (such as prices or public bids for an asset) or through the private observation of other agents’ actions (such as bilateral bargaining in a decentralized market). … Private information sharing is typical in functioning over-the-counter markets for many types of financial assets, including bonds and derivatives. In these markets, trades occur at private meetings in which counterparties offer prices that reveal information to each other, but not to other market participants. In addition to this form of private information sharing, many over-the-counter markets also have public releases of a selection of price quotations or executed trades (pp. 1575–1576). The main conclusion is summarized in the abstract: [i]f the private learning channel is present, convergence of the distribution of beliefs to the perfect-information limit is exponential at a rate equal to the sum of the mean arrival rate of public information and the mean rate at which individual agents are randomly matched with other agents. If, however, there is no private information sharing, then convergence is exponential at a rate strictly lower than the mean arrival rate of public information.

Nicolosi et al. (2009, p. 318) make and prove the following argument:

[I] nvestors’ demand for risky assets will increase with their confidence in their private signal precision. Specially, if investors privately receive signals (e.g., ranging from negative one to positive one) regarding the quality of potential purchases’ future returns, then investors who believe their signals to be perfectly accurate (based on their previous trades’ subsequent returns) would be willing to trade whenever the signals were, even marginally, positive. However, if investors believe their signals to be less accurate, then they would be less willing to trade after receiving only slightly positive signals. Clearly, investors’ willingness to trade would increase with their perceived signal precision.

Nicolosi et al. (2009) provide strong evidence that [trading] ability measures significantly affect investors’ future investment performance and trade intensity. [And] trading experience helps future investment performance.

It’s now a too simple assumption that informed traders are just informed due to exogenous reasons. Whether or not to acquire costly information is a decision for a trader to make. Incorporating this endogenous information acquisition makes the model better. The following models assume endogenous information acquisition: Goettler et al. (2009), Mendelson and Tunca (2004).

1.3.2 Hidden information

Hong and Stein (2003) argue that short-sale-constrained investors have to sit out of the market when the market price is higher than their deemed value. Therefore, these investors’ private information does not reflect the market price. Their private information becomes incorporated when the market price is low so that they can begin to buy the stock. This hidden information leads to changes in price, volatility, and volume even without new information coming to the market. Romer (1993) also discusses hidden information. But his hidden information is symmetric, whereas Hong and Stein’s (2003) is asymmetric.

Chen et al. (2002) empirically test the idea of hidden information. They use the breadth of ownership as a proxy of the private information excluded from the market price. That is, if the stock ownership becomes concentrated in the hands of some investors, then more investors just sit out of the market if they cannot short sell. Therefore the contemporaneous price of concentrated owned stocks are inherently optimistic, because only the optimist’s information is incorporated in the price. Hence the subsequent stock price will be lower, i.e., the following stock return is low. They refer to the hidden information as pent-up information.

1.3.3 Price informativeness

Price reflects information. My fundamental idea is that the only channel of price’s incorporating information is informed trading. Consider the following players in this game: maker makers, informed and liquidity traders, analysts, intermediaries, affiliated investment banks⁴ (Ellis et al., 2000; Schultz, 2003), etc.

Anand et al. (2011) argue that location