Defining, Modeling, and Measuring Investor Sentiment

Transcription

1 Defining, Modeling, and Measuring Investor Sentiment Cathy Zhang University of California, Berkeley Department of Economics April Abstract This thesis attempts to come closer at resolving three highly contested issues involving investor sentiment. First are the theoretical issues of defining and modeling sentiment in an intuitive and parsimonious way. To date, there has been no commonly accepted definition, let alone theory, of investor sentiment since the term may be used in different ways depending on the context. In this thesis, I narrow the scope by focusing on a definition of sentiment expressed in terms of investors erroneous beliefs relative to fundamentals. Using this definition, I derive a model of individual sentiment in which individual biases or systematic tendencies to overvalue private pseudo-information signals may lead to the formation of erroneous beliefs. I argue intuitively that so long as beliefs are correlated among investors and limited arbitrage is present, sentiment can affect asset prices in equilibrium. Furthermore, defining and modeling sentiment in this way also paves the way for resolving the third main issue regarding investor sentiment: how to measure it. Evaluated within the context of my definition and model, I assess two general approaches to measuring sentiment: an indirect approach using financial proxies and a direct approach using survey results. I argue that at best, only the latter approach captures investor sentiment as I have defined it. I conclude by suggesting a second-best measure of sentiment consistent with the initial definition I set forth, while also taking into account empirical and methodological constraints. This thesis would not be possible without the helpful discussions, thoughtful comments, and continued support of my advisor, Professor Botond Kőszegi. Thanks, Botond, for facilitating this endeavor and for making the thesis-writing process an enjoyable experience. 0

2 Introduction As a whole, this thesis makes a contribution to understanding a topic that has been of substantial interest to economists and the public alike. Investor sentiment has been traditionally regarded as a myth by classical financial theories and has received little attention by researchers prior to The standard argument was that in the highly competitive financial market, suboptimal trading behaviors such as paying attention to sentiment signals unrelated to fundamental value will be quickly eliminated by aggressive arbitrageurs. Yet this implication has not been consistent with actual historical events, such as the dot-com bubble of the late 1990 s. In reaction to this period of so-called irrational exuberance, 1 many researchers appealed in hindsight to behavioral explanations such as investor sentiment. Thus, from a historical perspective investor sentiment is important since it is a plausible explanation for the speculative episodes of the 1990 s: extreme bullish sentiments pushed prices far above fundamentals, leading to an inevitable return and crash in Although many researchers now agree that investor sentiment can be economically significant, the concept itself is still largely regarded as cryptic and abstract. At the crux of the problem is the fact that the term sentiment is used in different ways among economic researchers, professional traders, and the media. To help resolve the confusion, this thesis takes three fundamental issues related to investor sentiment and proposes some resolutions to each. Discussed in turn, the three issues are: (i) how to define sentiment clearly and concisely, (ii) a way to model sentiment that is consistent with actual investor behavior, and (iii) suggestions for measuring sentiment in light of methodological constraints. The first part of the thesis deals with the theoretical problem of coming up with a general definition and theory of investor sentiment. In this thesis, I focus on a definition of sentiment that captures why sentiment is important in the first place. Namely, I regard sentiment as pertaining to erroneous beliefs across investors. In the context of financial markets, this can be thought of as a stock price that is objectively correct, but which individual investors only posses subjective beliefs (Shefrin 2007). Sentiment is thus the beliefs and expectations of market participants relative to fundamental value. By restricting our attention to this particular notion of sentiment, it is then possible to draw implications from 1 The term irrational exuberance was first coined in 1996 by former Chairman of the Federal Reserve Alan Greenspan in reaction to the stock market boom of the 1990 s. Later on, the term was commonly used by financial pundits and the media in reference to market over-valuation (Shiller 2003). 1

3 it and come closer to some kind of cohesive model, relying of course on specific assumptions and preexisting theories of asset pricing and investor behavior. To that end, Part I of this thesis first provides a definition of investor sentiment and then uses this definition to suggest a possible model of investor sentiment. Section 1 gives a brief background on two contrasting approaches to thinking about investor sentiment, namely the classical market efficiency approach and the behavioral noise trader approach. Traditionally, classical finance makes no room for the presence of investor sentiment. For sentiment to be economically important, we need the assumptions of the latter behavioral framework in order to conclude that equilibrium asset prices and market outcomes can be substantially impacted by the presence of irrational agents. Next, Section 2 presents an intuitive definition of investor sentiment that I use throughout the thesis. I provide a simple mathematical expression for individual investor sentiment and argue how it captures potentially erroneous beliefs across investors. With this definition, I derive a model of individual sentiment in Section 3. In the model, I assume that all investors hold common priors but introduce the possibility that investors do not put correct weights on incoming information signals. I show that investor sentiment may result because of individual biases or systematic tendencies to overvalue private pseudo-information signals. I also explore some applications of the model to current topics in behavioral corporate finance, namely market timing of equity issues and managerial over-confidence. Finally, I summarize a set of necessary conditions required for investor sentiment to matter in the aggregate economy. Thus as a whole, Part I forms the theoretical groundwork for the thesis, providing the basis for answering the second main issue with investor sentiment: how to measure it. In general, measuring sentiment is controversial since doing so requires a preliminary judgment as to what the ideal measure should capture. Further complicating things is the fact that researchers typically have a wide range of priors regarding what sentiment really is, and thus how to best measure it. With that in mind, I apply the theoretical implications from Part I to assess the methodology and underlying assumptions of commonly used sentiment measures. Part II of this thesis is thus a comparative analysis of existing measures of sentiment. Namely, I use my definition of investor sentiment to assess the merits of two general approaches to measuring sentiment: indirect measures based on financial proxies, and direct measures based on surveys. While certain existing proxies are fundamentally inconsistent with the initial notion of sentiment I set forth, there do exist promising measures 2

4 that appear to capture the precise idea of sentiment I have in mind. I end by proposing a way to measure sentiment based on the arguments and considerations presented throughout the thesis. 3

5 Part I Defining and Modeling Investor Sentiment In this part of the thesis, my main objective is to define and model investor sentiment in a way that is both intuitively appealing and consistent with actual investor behavior. In Section 1, I provide a mapping of how sentiment is typically regarded in the literature: first, the market efficiency approach, and second, the noise trader approach. Throughout the thesis, it will be useful to keep this mapping in mind. After defining sentiment in Section 2, I introduce a simple model in Section 3 that presents a likely possibility for why erroneous beliefs arise. I also provide a behavioral interpretation of the model and assess the model in light of common stylized facts from corporate finance. Finally, I conclude this part of the thesis by summarizing a set of necessary conditions in order for investor sentiment to matter in the aggregate. 1 Theory and Background Before formally defining sentiment, I first summarize two existing approaches to thinking about investor sentiment. For now, investor sentiment can be thought of as potentially erroneous beliefs that investors have about an aggregate economic variable, such as stock price. Later on in the paper, I will clarify this definition in more detail. Before doing so however, it is useful to have in mind a mapping of how economists have traditionally viewed sentiment. The first approach is based on the traditional asset-pricing theories of classical finance, which argue that asset prices are rational assessments of expected future payoffs. Traditionally, this view makes no room for investor sentiment, since price changes only reflect the arrival of external news about future cash-flows and interest rates. An alternative approach, behavioral finance, suggests instead that that investor sentiment may significantly distort market outcomes thereby affecting asset prices in equilibrium. Specifically, the noise trader model posits that if there are limits to arbitrage and investor beliefs are correlated, then noise unrelated to fundamentals, such as sentiment, may lead asset prices to deviate 4

6 from what is expected from the benchmark of market efficiency. 1.1 Classical Finance and Investor Sentiment In classical finance, there is typically no room for the presence of investor sentiment. Such theories have mostly ignored or assumed away investor sentiment, arguing that in the highly competitive financial market, suboptimal trading behaviors such as paying attention to signals unrelated to fundamental value will be quickly eliminated. In short, classical finance revolves around two basic premises, that when taken together implies the lack of prolonged arbitrage opportunities: 1. Financial markets are informationally efficient. 2. Market participants are rational. First, the cornerstone of modern financial economics, the Efficient Markets Hypothesis, maintains that asset prices should reflect all available information about the fundamental value of the underlying security. Assuming no frictions, the price of a security should equal its fundamental value, defined as the discounted sum of future cash flows. Mathematically, this means that the price P t of a particular stock or portfolio equals the expected forecast P t+1 of subsequent cash flows and investment risks, conditional on all information available at the current time period. This can be stated concisely as: P t = E t [P t+1 I t ] (1) Hence, the Efficient Markets Hypothesis says price equals the optimal forecast of it. This implies that any surprising movements in the stock market must originate with new information about the fundamental value P t+1 (Fama 1965). From this, it then follows that fundamental value is comprised of a predicable component and an unpredictable component: P t+1 = P t + u t (2) Here, u t represents the forecast error and must be uncorrelated with any information available at time t, otherwise it would not be taking into account all available information (Shiller 5

7 2003). Since the price P t is also information, P t and u t must also be uncorrelated with each other. Consistent with the market efficiency paradigm is the presumption that individuals behave rationally and fully take into account all available information in the decision-making process. Therefore, when there is new information about a security, rational investors will quickly respond, leaving no room for excess risk-adjusted returns based on the information signal. Through motivations of self-interest and the forces of arbitrage, modern finance has traditionally assumed that irrational investors will be quickly eliminated from the market, along with risk-free profit opportunities. In real life financial markets however, there are limits to arbitrage. Trading costs, including transaction costs, information costs, and financing costs may prevent rational arbitrageurs from taking advantage of market mis-pricings. Since real life financial markets are far from perfect, these frictions may make it difficult to find and take advantage of a perfectly substitutable asset (Shleifer 2000). However, even after taking into account fundamental risk and transaction costs, standard financial theories still have a hard time explaining prolonged mispricings and unexploited arbitrage opportunities (Shleifer, Vishny 1997). For example, financial puzzles such as the closed-end fund discount and IPO underpricing are empirical observations that provide evidence that markets may not always be informationally efficient. To explain these anomalies, one approach has been to appeal to behavioral explanations that relax the strict rationality requirement of standard theories. In particular, behavioral finance has been an increasingly fruitful branch of research that, in short, takes account of deviations from perfect rationality and explores the ways this may affect market outcomes, asset prices, and even the behavior of other investors. With regards to investor sentiment, behavioral finance offers models that are much more flexible about investor behavior and in doing so, can explain financial anomalies such as limited arbitrage. I provide a brief summary of behavioral finance in the next sub-section, commenting especially on its relevance to investor sentiment. 1.2 Behavioral Finance and Investor Sentiment Although it is often observed that most individuals are not the hyper-rational agents postulated in classic economic models, standard theory suggests that in competitive Arrow-Debreu 6

8 financial markets, cognitive biases and misguided beliefs that lead to suboptimal trading decisions will immediately be arbitraged away by aggressive arbitrageurs. Recent empirical findings show however that this is not always the case. In particular, not only are individual investors just as prone to biases as the population at large, but in some situations they may be even more likely to display over-confidence, herding behavior, and speculation (Barberis, Thaler 2003). Psychological studies and recent research show that these findings are robust and systematic (Barber, Odean, Zhu 2003). This implies that even in a highly incentivized financial market with a large number of investors interacting with one another, it might still be the case that investors with suboptimal biases are not completely eliminated from the market. In other words, there are limits to arbitrage, and behavioral finance formalizes and posits ways this might happen. One way behavioral finance formalizes the possibility of limited arbitrage is through the noise trader model, which is arguably one of the most cited alternatives to the Efficient Markets paradigm. The model claims that because investors are risk-averse and have short horizons, real-life arbitrage must take account of the fact that arbitrageurs may not want to expose themselves too much undiversifiable risk (DeLong, Shleifer, Summers, Waldmann 1990). In particular, an important consideration for rational arbitrageurs is the behavior of other investors who may be prone to exogenous sentiment. These so-called noise traders are not fully rational in the sense that they may trade on the basis of noisy sentiment rather than information. Although noise traders have no access to insider information, they trade on noisy sentiment as if it were valuable information that would give them an edge on the trading floor (Black 1986). As a result, noise traders expectations about asset returns are sensitive to fluctuations in sentiment that is, they overestimate expected returns in some periods and underestimate them in other periods such that their trades are not randomly distributed across assets. Because sentiment is correlated across these noise traders, this risk cannot be diversified away. This implies that limits to arbitrage may persist due to noise-trader risk, defined as the risk incurred by rational arbitrageurs from the unpredictability of noise traders. Specifically, noise traders add in the risk that their beliefs may not revert back to the mean and become even more extreme over time. This risk is borne by market participants, and is thus a potential explanation for the existence of unexploited arbitrage opportunities (DeLong, Shleifer, Summers, Waldmann 1990). 7

9 As opposed to the traditional view that stock return co-movements can only be explained by either changes in fundamental value or the discount rate, an immediate implication of the noise trader model is that the correlated trading activities of unpredictable noise traders can also induce return co-movements. This means that stock prices are determined by the interaction of sophisticated arbitrageurs and unpredictable noise traders, in addition to standard risk factors and macroeconomic variables. In this way, the noise trader theory makes room for the presence of investor sentiment. However, despite the noise trader theory s implication on how market participants may respond to irrational noise traders, the model does not define explicitly what sentiment is, or how sentiment in particular as opposed to noise in general, can affect market outcomes. To this end, I provide a definition of investor sentiment in the next section and explain why the particular definition I present leads to implications consistent with the noise trader model. 2 Defining Investor Sentiment In this section, I present a simple and intuitive definition of investor sentiment that will be used throughout this thesis. There has been no single commonly accepted definition of investor sentiment to date. Existing definitions of sentiment in the literature range from vague statements about investors mistakes to specific psychological biases that are model-specific (Shefrin 2007). Furthermore, the term itself is subject to a wide spectrum of classifications and is used in different ways by academic researchers, financial analysts, and the media (Barberis, Shleifer, Vishny 1998; Daniel, Hirshleifer, and Subrahmanyan 1998; Welch and Qiu 2004; Cliff and Brown 2004; Shefrin 2007; Baker and Wurgler 2007). For example, while some researchers may refer to investor sentiment as a propensity to trade on noise rather than information, the same term is used colloquially to refer to investor optimism or pessimism. The term sentiment also has connotations with emotions, so the media may refer to it as investor fear or risk-aversion. Despite the family resemblance, all these notions lack a set of necessary and sufficient conditions that makes clear exactly what we mean when we refer to investor sentiment. This is not to say however that previous research has been incorrect or mistaken in their approach; in fact the exact opposite is true: it is only because of the significance of previous works that we are any closer to a single, coherent theory or at the very least a definition of investor 8

10 sentiment. One approach to thinking about individual investor sentiment, and the approach that I subscribe to, is to think about it in terms of beliefs. The classical notion of a rational agent is one who has well-defined preferences and forms correct beliefs through Bayesian updating. In the present context, I assume that the former always holds and focus instead on the latter. That is, throughout this thesis I will assume that investors are susceptible to erroneous beliefs but are otherwise rational in the sense that their preferences satisfy standard preference axioms. This simplification allows for a simple definition of investor sentiment defined in terms of erroneous beliefs. More precisely, I define investor sentiment as the following: Investor sentiment represents market participants beliefs about future cash flows relative to some objective norm, namely the true fundamental value of the underlying asset. As defined above, sentiment corresponds to erroneous beliefs that investors have against some kind of objective benchmark. One possibility for this benchmark is the true fundamental value of the underlying asset, defined as the discounted sum of future cash flows and investment risks. Accordingly, there are two possibilities for why erroneous beliefs occur: individuals correctly use wrong information, or that they wrongly use correct information. In other words, sentimental investors may update their beliefs through news about fundamentals in addition to noisy signals unrelated to fundamentals, and may do so in a way that is statistically incorrect. In this way, sentiment can be expressed as the component of expectations about asset returns not warranted by fundamentals and Bayesian updating alone. For example, this could be because some investors are overconfident in their stock-picking abilities and act on their beliefs in a way different from a Bayesian updater who bases trading decisions on news about fundamental value alone. In my model of sentiment presented in Section 3, I assume that erroneous beliefs only reflect the first possibility that individuals correctly use wrong information. In the literature, modeling sentiment in terms of the latter possibility has been taken by previous work, namely by Barberis, Shleifer, and Vishny (1998). In this framework then, it is useful to think about investor sentiment as the discrepancy between two forecasts: (i) an individual s subjective assessment based on all available 9

11 information in addition to potentially biased private information, and (ii) the objectively correct forecast based only on relevant information alone. Defining sentiment in this way is attractive since it allows for formal expressions consistent with intuitions. I claim that individual investor sentiment can be formalized by the following expression: S i,t E i,t [P t+1 I i,t] E t [P t+1 I t ] (3) In this formulation, S i,t denotes individual sentiment of investor i. P represents the stock price, a random variable whose true value is unknown at time t but revealed to all at time t + 1. I t denotes all public information available about fundamentals at time t, and I i,t denotes the information actually used by individual investors to arrive at their forecasts. I i,t may be different from I t if individuals have access to and use biased or misguided private information, which is a possibility that I model more formally in Section 3 of this thesis. Thus from (3), we can think of sentiment as the difference between the means of two probability density functions, one representing an objective distribution and the other representing an individual investor s subjective probability distribution. The above definition of sentiment also allows for the possibility that beliefs are correct, in which case an individual is rational, narrowly defined. Note that whenever the two terms are exactly equal, we have E i,t [P t+1 I i,t] = E t [P t+1 I t ] and S i,t = 0. In this case, beliefs are realistic and only reflect fundamental value. Thus, I refer to zero-sentiment as a necessary but not sufficient condition for a rational investor. In other words, S i,t = 0 is necessary to infer that beliefs are formed according to standard probability axioms and only reflect relevant pricing information. In this formulation, an investor prone to sentiment therefore has S i,t 0. I define a bullish investor as one who is unrealistically optimistic about next period s stock price. That is, E i,t [P t+1 I i,t] > E t [P t+1 I t ] and S i,t > 0. On the other hand, a bearish investor is unrealistically pessimistic so that E i,t [P t+1 I i,t] < E t [P t+1 I t ]. Note however that the way I use the terms bullish and bearish may differ from the colloquial usage. 10

12 3 Modeling Investor Sentiment To explore the intuitions and implications of investor sentiment more closely, I now present a simple model of sentiment based on the definition presented in the previous section. In this model, investor sentiment arises because individuals incorporate potentially wrong information in updating their beliefs. I assume that all investors hold common priors but nonetheless arrive at different posterior beliefs because of some private pseudo-signal that varies across investors. Hence, I introduce the possibility that investors do not place correct weights on incoming information, which then results in sentiment. I present a behavioral interpretation of the model in which sentiment results because of individual biases or systematic tendencies to pay more attention to private information that is irrelevant from an ex ante perspective. So long as beliefs are correlated and there is empirical evidence that supports this then sentiment can affect market outcomes, as suggested by the noise trader model. 3.1 Assumptions In order to preface for my model of investor sentiment to be presented in Section 3.2, I first review some key assumptions regarding investor beliefs and how differing forecasts of the future may arise. Although some of these assumptions were mentioned briefly in the previous section, I include them all here for easy reference. 1. Investor sentiment is defined to only reflect beliefs, and not preferences or risk appetites. 2. Rational investors update beliefs in a Bayesian way that only reflect news about fundamental value. Sentimental investors form erroneous beliefs because they update their beliefs through the same news about fundamentals in addition to noisy signals unrelated to fundamentals, but otherwise use Bayes Rule correctly. 3. All investors maximize their utility given their beliefs. In other words, both rational and sentimental investors act in a way that is consistent with their personal forecasts and subjective beliefs. The first assumption that sentiment pertains to beliefs, rather than preferences or risk appetites, is both a simplifying assumption and one that is motivated by how I defined sentiment 11

13 in the first place. Although others may define sentiment to also include violations of preference axioms or propensity for a particular risk profile, for simplicity and parsimony, I only allow for the possibility that sentiment reflects erroneous beliefs. Hence in the analysis that follows, I hold constant investor preferences and risk profiles, focusing mainly on differences in beliefs. Next, I assume that erroneous beliefs only reflect the possibility that individuals correctly use wrong information. Another possibility for the formation of erroneous beliefs is that investors incorrectly use Bayes Rule and hence arrive at forecasts that are statistically incorrect. 2 In my model, I assume instead that all investors use Bayes Rule correctly. The possibility of erroneous beliefs thus occurs because they may incorporate irrelevant information in their forecasts. In the model below, I call information that is irrelevant to fundamentals from an ex ante perspective, a pseudo-signal. For example, an investor might trade based on their mood, which has no bearing on actual stock price behavior. My final assumption is that although investors may form incorrect beliefs, they still maximize their expected utility given their potentially wrong beliefs. Hence, individual forecasts are always consistent with beliefs. 3.2 A Model of Investor Sentiment with Private Pseudo-Signals In this section, I present a model of investor sentiment in which potentially erroneous beliefs arise because investors use common public information in addition to private pseudo-signals to arrive at their forecasts of fundamental value. Investors are thus faced with a signal extraction problem. In this framework, equilibrium asset prices reflect the average belief across market participants so that information that may be irrelevant to fundamental value from an ex ante perspective, might turn out to affect asset prices in equilibrium. I model this possibility directly by introducing a private pseudo-signal that each investor believes to reflect the unobserved fundamental value, but is just pure noise in reality. Clearly, if investors believe that their private signal reflects fundamental value with some positive probability, they will arrive at posterior beliefs different than what is warranted by the fact of the matter. I argue that if the private pseudo-signals are correlated across investors so that their effect does not wash out in equilibrium, sentiment can result from investors placing positive weight 2 For a model that seeks this approach, please see Barberis, Shleifer, Vishny (1998). 12

14 on their private information. To begin the analysis, first recall that the basic conclusion of the Efficient Markets Hypothesis is that the current price of an asset reflects the discounted expected value of the asset s payoff stream and investment risks, conditional on all current information available to investors. If all market participants use the exact same information to forecast asset prices and there is no private information, then asset prices should just reflect the average beliefs of all market participants. This is just the law of iterated expectations, which says that the marginal investor s expectation today of tomorrow s expectation of future payoffs should be exactly equal to his expectation today of future payoffs. However, if investors hold different information, then surely this will be reflected in the average expectation about future prices. It has been shown that in general, average expectations fail to satisfy the law of iterated expectations in the presence of differential information between investors (Allen, Morris, Shin 2004). Therefore, in the presence of informational asymmetries across individual investors, it is not generally the case that the average expectation today of the average expectation tomorrow is equal to the average expectation of future payoffs. This possibility thus gives rise to sentiment. Instead of assuming that all market participants arrive at their forecasts using the exact same information, I introduce in my model the possibility that individuals have access to two information sources: public information that is common knowledge to all, and private information. Each investor therefore faces a signal extraction problem and must deduce how to weight incoming information. I define a rational investor as one who places correct weights on incoming signals, whereas an investor prone to sentiment uses incorrect weights. To illustrate with an example, suppose that the two information signals have equal value in forecasting the future payoff. In that case, investors should simply place equal weight on the two signals when forming their posterior beliefs. Hence, a rational investor will correctly deduce an equal weighting, and a sentimental investor will either tend to overweight public information or her private signal. To formalize this idea more precisely and show how this illustrates investor sentiment, I now present a simple model in which a large population of investors makes predictions about the stock market. Modeling concepts are inspired by Harrison and Kreps (1979), Allen, Morris, and Shin (2003), and Kurz (2006). In the model economy, investors try to forecast the unobserved fundamental value of an aggregate asset, which I denote by P. For example, P can be thought of as an index of 13

15 the S& P 500. At time t, investors try to predict the value of P at time t + 1. Interest rates are normalized to zero so that there is no time discounting. I assume that all investors hold the same prior belief that the future unknown value P is normally distributed with mean µ p and variance 1 α : P t+1 N(µ p, 1 α ) (4) Since investors do not observe P t+1 directly, they can only make probabilistic inferences from noisy incoming information. At time t, each investor receives informational signals about P. Suppose for now that investors only observe a common signal, which I denote by x t. Further assume that the signal x t reflects all information available at time t and that all investors observe the exact same signal, which reflects fundamental value P with some noise: x t = P t+1 + ε t (5) ε t N(0, 1 β ) (6) Since x t reflects all information available at time t, each investor should then arrive at the exact same posterior beliefs. Using standard Bayesian inference, each individual investor has the following belief at time t, regarding the value of P at time t + 1: E i,t [P t+1 x t ] = αµ p + βx t α + β (7) = α α + β µ p + β α + β x t (8) To model how individual sentiment may arise, consider the case where in addition to the specifications made above, investors also receive a private signal at time t, which I denoted by s i,t. Unlike the common signal from above, the private signal varies across investors and hence affects their individual posterior belief about P. Intuitively, if these private signals are correlated with one another so that the effect does not just average out in the end, then equilibrium market beliefs will clearly be affected. I call the private signal, s i,t, a pseudo-information signal because ex ante, it reveals no information about fundamental value. That is, s i,t is just pure noise. When arriving at their respective posterior beliefs, each investor should put zero weight on this private signal since it is not relevant information for forecasting P. Crucially however, each investor falsely believes that her own private signal reflects fundamental value. I denote by s i,t an individual 14

16 investor s perception about the private signal: s i,t = P t+1 + η i,t (9) η i,t N(0, 1 γ i ) (10) The dynamics of the pseudo-signal, s i,t, are similar to the common signal, x t, except for the fact that each investor potentially interprets her own private signal differently. To summarize the model so far, I have assumed that all investors hold a common prior belief that the unknown value of P t+1 is drawn from a normal distribution with mean µ p and precision α. At time t, each investor receives two signals. The first is a public signal, x t, which is known by all to reflect fundamental value, albeit with some noise. For example, this can be thought of as a public earnings announcement or some economy-wide news shock. The error term on this public signal is normally distributed with zero mean and precision β. At time t, each investor also receives another signal. This is a private signal that in actuality is just pure noise, but each investor believes is relevant information for forecasting P t+1. Investors believe that the error term on this private pseudo-signal, s i,t is normally distributed with mean 0 and precision γ i. Conditional on all this information, each investor arrives at their subjective belief regarding P t+1. Each deduces the following: = E i,t [P t+1 x t, s i,t] = αµ p + βx t + γ i s i,t α + β + γ i (11) α β γ i µ p + x t + s i,t α + β + γ i α + β + γ i α + β + γ i (12) As before, investors place positive weight on the common prior that P t+1 is Gaussian with mean µ p and on the common signal x t. Note however, that each investor also places positive weight on the private pseudo-signal s i,t. From an ex ante and perfect information perspective, s i,t is just pure noise and is irrelevant in forecasting P t+1. However, since each investor believes that the private signal contains relevant information about fundamentals, they will clearly extract information from it to arrive at their personal forecasts. We now have all the necessary ingredients for arriving at individual sentiment. Recall from the previous section that I defined individual investor sentiment as the difference in two forecasts: (i) an individual s subjective assessment based on all available information including private information, and (ii) the objectively correct forecast based 15

17 only on relevant information alone: S i,t E i,t [P t+1 I i,t] E t [P t+1 I t ] (13) Individual investor sentiment reflects erroneous beliefs, which is the difference in an investor s subjective forecast, E i,t [P t+1 I i,t], and the objectively correct forecast E t [P t+1 I t ]. In (13) recall that I denote by I t all information relevant to fundamentals at time t. I t on the other hand reflects the information actually used by investors, which if different from I t gives rise to individual sentiment. In this particular model, false beliefs arise because investors pay undue attention to incoming pseudo-signals. Therefore, investors treat noise as if it were information and so arrive at beliefs different than what is objectively warranted by relevant information alone. From the model and set-up from above, we arrive at the following formulation for individual sentiment: S i,t = E i,t [P t+1 x t, s i,t] E t [P t+1 x t ] (14) Notice that in the model, x t is a common signal assumed to reflect all information relevant to fundamentals at time t. Hence, x t is tantamount to I t from (13). In addition to relying on x t, sentimental investors also rely on their perceived private signal s i,t. Plugging everything in, we get: α = ( µ p + α + β + γ i β α + β + γ i x t + S i,t = E i,t [P t+1 x t, s i,t] E t [P t+1 x t ] (15) γ i s α α + β + γ i,t) ( i α + β µ p + β α + β x t) (16) The above expression represents the individual sentiment of an investor in the way I defined it in Section 2. Sentiment is thus the difference between two forecasts: an individual s subjective forecast that uses relevant information in addition to noise about fundamentals, and the objectively correct forecast. An investor subject to sentiment places incorrect weights on incoming signals and so arrives at a forecast different than what is warranted by the objective facts at hand. A rational investor on the other hand is defined to have zero sentiment so that the two forecasts are precisely equal. In this way, the magnitude of the expression above quantifies the amount of discount from full rationality, in the narrow sense that I defined it. 16

18 Intuitively, there are many reasons why investors may arrive at false beliefs. In the next sub-section, I review some possibilities. 3.3 Rational vs. Behavioral Interpretation A rational interpretation for why investors possess incorrect beliefs is simply that they make random mistakes. Note that nowhere in my model did I assume that investor beliefs are correlated with one another. Hence, if mistaken beliefs are not systematic so that it just cancels out in the aggregate, then there will be no room for sentiment to affect equilibrium asset prices. However, the possibility that mistakes are systematic gives rise to a behavioral interpretation. My model of investor sentiment above is consistent with a behavior interpretation that views individuals as not fully rational in the sense that they are susceptible to holding systematically false beliefs. So far, I have assumed that for whatever reason, investors believe that s i,t is relevant forecasting information. From the model, recall that the private pseudosignal should not enter into an investor s forecast since it is just pure noise and contains no relevant information about P from an ex ante perspective. However, the investor falsely believes that s i,t is relevant, and hence places positive weight on the private signal when forming beliefs: E i,t [P t+1 x t, s i,t] = α α + β + γ i µ p + β α + β + γ i x t + γ i α + β + γ i s i,t (17) There are a number of behavioral reasons for why investors may hold this false belief. For example, individuals may be biased in their ability to discern between relevant forecasting information and so may weigh private signals more than public signals because of overconfidence in their trading ability, which some research shows is more likely for men (Barber, Odean 2001). Alternatively, suppose that the private signal is actually information on an individual s past forecasting successes and that these past successes were just lucky guesses. If that individual believes that she has a hot hand, then perhaps she will place undue weight on the private signal that tells her to just guess her lucky number, say. Nonetheless, erroneous beliefs will only matter in the aggregate if such beliefs are correlated across investors. In order for the behavioral interpretation to carry weight then, 17

19 there needs to be convincing reason why this may be. In terms of the above model, correlated beliefs would thus require that individuals private pseudo-signals are also correlated. For example, if the model is augmented to include a variety of stocks rather than just a single aggregate asset, then correlated beliefs could mean that there are shared preferences among individual investors for buying certain kinds of stocks. Recent empirical research has shown that small retail investors do indeed have coordinated trading activities, in the sense that they tend to buy and sell certain stocks in concert (Barber, Odean, Zhu 2003). The more difficult question is why this may be. One possible explanation for correlated pseudo-signals and thus coordinated beliefs is that individual investors possess superior private information or superior ability to interpret public information, in which case they will tend to buy undervalued stocks and sell overvalued stocks as a group (Barber, Odean, Zhu 2003). If this is the case, trades are correlated. But note that this interpretation does not follow from my model since I assume that an individual s private signal is just pure noise. There is also limited empirical evidence that this is actually the case, since intuitively, there is no reason that small investors have access to insider information and that this information is available to a large enough group of small investors. As it is supported by a large amount of empirical evidence, a more plausible explanation for correlated trading activity is that individual investors as a whole exhibit certain psychological biases that affect how they form beliefs in the first place. There is a large body of evidence that suggests that as a whole, individuals make decisions using simple rules of thumb or heuristics. One that is commonly used is the representativeness heuristic, in which people expect small samples and short time series of data to be representative of the underlying population or distribution (Tversky, Kahneman 1974). For example, investors may extrapolate past performance to the future so that they overweight a stock s recent history. This is consistent with psychological studies in which subjects forecast higher future prices after observing past price increases (DeBondt 1993). Hence, as a whole, investor beliefs may be correlated in a way that results in a systematic over-valuation of stocks with strong past returns and under-valuation of stocks with poor past returns. Investors reliance on the representative heuristic may also lead stock prices to mean revert (Barberis, Shleifer, Vishny 1998). Another behavioral explanation for correlated beliefs is that investors may have a 18

20 tendency to over-value certain stocks that, for whatever reason, grab their attention. Indeed, there is empirical evidence that suggests that small investors disproportionately buy attention-grabbing glamor stocks (Barber, Odean 2002). Finally, another possibility is that if we think of sentiment as related to mood, then correlated beliefs might imply that investors are on average happy together or sad together. Hence, that beliefs are correlated in this way requires that there be some exogenous common factor that affects individual mood. One controversial example of this is the weather. Although the overall consensus is mixed, some research points to evidence that stock returns are related to weather due to its purported effects on mood (Hirshleifer, Shumway 2003). 3.4 Applications to Behavioral Corporate Finance In this sub-section, I show that my definition and model of investor sentiment can be applied to current topics in behavioral corporate finance. 3 Though a relatively new field, behavioral corporate finance has become an increasingly prolific area of research. Broadly speaking, two general categories dominate the field: 1. Investor irrationality: inefficient markets with rational managers who exploit investor biases (Application: market timing of IPOs) 2. Managerial irrationality: efficient markets with managerial biases and rational investors (Application: CEO over-confidence) In the remainder of this sub-section, I summarize some applications of the above two categories, noting how they relate to the way I defined and modeled sentiment earlier in this thesis. 3 Much of this section has been motivated by Professor Ulrike Malmendier s lectures from the course Economics 138 Behavioral and Corporate Finance (U.C. Berkeley, Spring 2008) 19

21 Investor Biases in Inefficient Markets: Applications In the first category, rational managers take advantage of the irrationalities of small outside investors to increase the value of their companies. That managers can systematically profit from others misvaluations of course assumes that markets are inefficient so that prices may be too high or too low relative to fundamentals in which case managerial decisions may either respond to or encourage mispricings. Assuming that for whatever reason, managers know when their company is being over-valued or under-valued relative to fundamentals presumably because they have more information than small investors about the true value of the company it might be the case that managers exploit the prevailing market sentiment to raise shareholder value (Tirole 1993). This so-called market timing hypothesis has been used to explain common empirical observations such as IPO waves and merger misvaluations (Shleifer, Vishny 2003). I first provide a brief summary of what market timing is and then explain how it fits in with the way I defined sentiment. In the corporate finance literature, market timing refers to the tendency for firms to issue equity when stock prices are high, relative to book and past market values, and repurchase when prices are low, thus implying that equity issues may occur in waves. Many possible reasons for this phenomenon have been proposed, 4 but of particular relevance to this thesis is the possibility that market timing is due to firms exploiting periods of market over-valuation, or high sentiment. 5 In particular, market timing has been suspected to have implications on corporate decision-making with regards to new equity offerings. As it relates to IPOs, the basic assumption of the market timing hypothesis is that there is asymmetric information between a shareholder-value maximizing manager and outside investors subject to sentiment. When managers observe that the market is overvaluing the company, it then becomes more favorable to issue equity. Thus, the basic decision problem that managers face is trading off maximizing the short-run value of their firm from exploiting sentiment 4 A popular alternative explanation for market timing based on agency considerations is that during economic booms, the problem of adverse selection becomes less relevant, thus enabling issuers to raise equity. Adverse selection in capital markets occurs when firms of low type are pooled with firms of high type, because their types are unobservable by investors. During economic booms, the intrinsic value of an investment project may be large enough to dwarf the costs of the adverse selection problem. For a more comprehensive review and other explanations of market timing, please see Jean Tirole (1993): The Theory of Corporate Finance, Section 2.5 and Section Although never defined explicitly in the corporate finance literature, it seems to be implicit that sentiment refers to under- or over-valuation relative to fundamentals. Though this is precisely the definition of sentiment that I subscribe to, I find it nonetheless puzzling that it was never more explicitly or formally spelled out. 20

22 against the potentially lower long-run value as prices eventually correct and revert to fundamentals. Indeed analysis of earnings forecasts and realizations around equity issues confirm the suspicion that firms are more likely to issue equity when investors are too enthusiastic about the firm s earning prospects (Loughran, Ritter 1997). Perhaps even more convincing is the fact that managers admit to market timing; for example, in an anonymous survey of chief financial officers of public corporations, two-thirds admit that the amount by which our stock is undervalued or overvalued was an important or very important consideration in issuing equity (Graham, Harvey 2001). I now claim that my definition of sentiment is consistent with the underlying notion of investor exuberance in the market timing hypothesis, which the literature usually takes as given. First recall that value-creating managers know precisely when the market is over- or under-valuing their company, presumably because they know underlying fundamental value. Investor over- or under-valuation then is precisely the notion of sentiment I presented earlier in which individual beliefs are assessed relative to fundamentals: investor over-valuation thus corresponds to S i,t > 0, and investor under-valuation corresponds to S i,t < 0. In other words, what market timing models have always taken as given, is what I formalize and clarify in this thesis. Note however that although the way I defined sentiment fits in with the underlying idea of biased investor enthusiasm in the corporate finance literature, my particular definition of sentiment is bottom-up, rather than top-down, as is with the case of market timing models. In other words, my definition and model of sentiment is expressed in terms of individual beliefs, which then may translate to broader market outcomes under certain conditions. On the other hand, market timing models are macroeconomic, in which case investor sentiment, though relevant, is just taken as given. Nonetheless, I believe that my definition of sentiment, specified in terms of individual beliefs, makes the underlying idea of the market timing hypothesis more clear. Managerial Biases and Efficient Markets: Applications With regard to the second main category of behavioral corporate finance, instead of investor biases and inefficient markets, we now have managerial biases and, for simplicity, efficient markets. One important application in this category of research has been to model and formalize managerial over-confidence, in which case the company leader exhibits misguided and potentially false beliefs. Recent research has shown the empirical effects of corporate 21