How Important Is Mispricing?

July 2011

Aydoğan Altı and Paul C. Tetlock*

University of Texas at Austin and Columbia University

Abstract

There is much controversy over whether risk or mispricing explains observed differences in

firms’ average stock returns. We contribute to this debate by proposing and estimating a model

in which investors’ information processing biases cause differences in firms’ returns. The model

matches many features of empirical data on asset pricing and firm investment. We estimate the

extent of biases that would reproduce empirical asset pricing anomalies and explore the

implications of these biases. Our findings indicate that the magnitude of mispricing is several

times larger than observed anomalies suggest. The model also provides novel insights into when

and how information processing biases can cause substantial capital misallocation.

                                                            * We thank UT Austin and Columbia for research support. We appreciate helpful suggestions from Nick Barberis, Jonathan Berk, Alon Brav, Murray Carlson, Martin Lettau, Erica Li, Sheridan Titman, Toni Whited, Jeff Wurgler, and seminar participants at the Michigan SFS, the NBER Behavioral, and the WFA conferences, as well as Stanford, UC Berkeley, UNC Chapel Hill, and UT Austin. Please send all correspondence to aydogan.alti@mccombs.utexas.edu and paul.tetlock@columbia.edu.

  

Many studies document that certain firms have higher average stock returns than others.1

There is much controversy over whether firms’ returns differ because of differences in risk or

because some firms are mispriced, relative to their fundamental values. To investigate this

question, several researchers propose, calibrate, and estimate models of risk-based explanations.2

In contrast, most empirical research on mispricing focuses on the qualitative predictions of

behavioral models.3 It is difficult to test the risk and mispricing explanations on a level playing

field without knowing whether models of mispricing make quantitatively realistic predictions.

To address this question, we propose and estimate a model of mispricing in which

investors’ information processing biases cause differences in firms’ asset returns. The model

generates endogenous relationships among firm investment, profitability, valuation, and asset

returns from exogenous variation in firm productivity and investors’ information about

productivity. We estimate the model parameters by matching key features of firms’ asset return

and investment data. In particular, we identify the magnitude of information processing biases

from observed differences in firms’ asset returns—i.e., return “anomalies.” To our knowledge,

we are the first to use asset prices to structurally estimate behavioral biases. These estimates

allow us to quantify empirically unobservable aspects of mispricing, such as firms’ “true”

expected returns from the perspective of a rational agent who knows the economy’s parameters

and investors’ information. The analysis produces several interesting results, which we discuss

below after describing the model.

                                                            1 Respectively, Banz (1981), Bernard and Thomas (1989), and Fama and French (1992) show that stocks with small sizes, positive earnings news, and high ratios of book-to-market equity have positive risk-adjusted returns. 2 Examples of risk-based models include Berk, Naik, and Green (1999), Lettau and Wachter (2007), and Liu, Whited, and Zhang (2009). 3 Examples of theories in which errors in investors’ expectations lead to return predictability include Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Hong and Stein (1999). Models in which investors’ preferences lead to return predictability include Benartzi and Thaler (1995), Barberis and Huang (2001), Barberis, Huang, and Santos (2001), and Barberis and Huang (2008).

1

One general insight from our analysis is that observed return anomalies are the tip of the

iceberg: insofar as behavioral biases cause anomalies, unobservable asset mispricing is much

larger than its observable proxies, such as the value anomaly. The reason is that empirical

anomalies are poor proxies for the errors in investors’ beliefs that cause mispricing. For example,

although the ratio of a firm’s book value to its market value (B/M) is a common proxy for

mispricing, only some of the cross-sectional variation in B/M comes from errors in investors’

beliefs about firms’ future prospects. Much of the variation in B/M comes from rational variation

in expected growth rates across firms. Model simulations show that sorting by firms’ true

expected returns produces a difference in expected returns that is at least three times larger than

the difference produced by sorting by firms’ observable characteristics, such as B/M, investment,

and profitability.

Our model is designed to serve as a natural starting point for the structural estimation of

behavioral biases. As such, it features homogeneous investors who value firm assets and

managers who choose firm investment to maximize the firm’s current asset price. Firm

productivity varies over time, but adjusting the capital stock is costly. Thus, if a firm’s expected

productivity increases, its manager dynamically responds by gradually increasing the firm’s

capital stock, generating persistence in the firm’s growth opportunities and its B/M ratio.

Agents cannot observe firm productivity, so they must estimate it using two pieces of

information: realized profits and a soft information signal. Past profits are a noisy indicator of the

firm’s current productivity. The soft information signal summarizes all information about

productivity that is not reflected in past profits. In practice, some soft signals such as analyst and

management forecasts are easy for econometricians to observe, whereas others such as investors’

subjective interpretations of news are intangible and harder to observe.

2

Investors in the model may suffer from two biases in estimating firm productivity:

overconfidence and excessive extrapolation. Overconfident agents believe the precision of the

soft signal to be higher than it actually is, as in Odean (1998), Daniel, Hirshleifer, and

Subrahmanyam (1998), and Scheinkman and Xiong (2003). Agents who excessively extrapolate

believe the persistence of firm productivity to be higher than it actually is, as in Barberis,

Shleifer, and Vishny (1998).

Previous research proposes that these two biases could explain why value stocks earn

higher returns than growth stocks.4 They could also explain why firms with high investment

experience lower returns, as shown in Titman, Wei, and Xie (2004) and Cooper, Gulen, and

Schill (2008). This investment anomaly links managers’ decisions to investors’ valuations of

firms, which is a central feature of our model. We use the empirical value and investment

anomalies to help us identify the combined magnitude of the two biases in our model.

We empirically distinguish between the two biases using their opposing predictions for

return predictability based on profitability. In estimating firm productivity, overconfident agents

pay more attention to the soft information signal than to past profits, whereas extrapolative

agents place too much weight on past profits. As a result, when agents are overconfident

(excessively extrapolate), profitability positively (negatively) predicts asset returns.

We estimate model parameters using the simulated method of moments (SMM).

Simulations based on the SMM parameter estimates closely replicate the relevant features of the

empirical data, including asset return predictability, as well as firm investment, profitability, and

Tobin’s Q. Twelve of the 13 empirical moments that the eight-parameter model attempts to

                                                            4 Beyond the theories already cited, empirical evidence consistent with extrapolation appears in Lakonishok, Shleifer, and Vishny (1994), La Porta (1996), La Porta et al. (1997), and Benartzi (2001). Further evidence on investor overconfidence appears in Odean (1999) and Barber and Odean (2001).

3

match are within 1.2 standard errors of the model’s predicted moments. The SMM parameter

estimates suggest that both information processing biases contribute to return predictability.

Our main finding is that asset mispricing can be much larger than return anomalies based

on observable proxies would suggest. Model simulations allow us to measure empirical

quantities that econometricians cannot typically observe, such as mispricing and true (rational)

expectations of returns. Mispricing is the ratio of a firm’s asset price to its fundamental value,

which is the present value of future cash flows under rational expectations. True expected

returns, which measure the profitability of trading strategies under rational expectations, quantify

how quickly mispricing corrects over time. We find that no observable firm characteristic, such

as B/M, is able to explain even 11% (28%) of the variance in true expected returns (mispricing).

Importantly, significant differences in firms’ true expected returns persist after 10 years.

We investigate various model parameters and alternative specifications to understand the

mechanism behind these findings. Controlled experiments with parameters reveal that the

overconfidence bias, not the excessive extrapolation bias, explains why empirical anomalies are

poor proxies for investors’ beliefs. Extrapolative agents’ biases are highly correlated with

observable proxies for productivity. When only the extrapolation bias is present, firm B/M ratios

capture nearly all of the variation in mispricing and firm profitability explains expected returns

quite well. In contrast, when only the overconfidence bias is present, agents’ beliefs depend on

the unobservable soft signal. Firm characteristics, such as B/M and profitability, fail to explain

the cross section of mispricing and expected returns induced by overconfidence.

Next we compare the return predictability from the benchmark model in which managers

choose investment to maximize the firm’s current asset price to the predictability from an

alternative model in which managers have rational expectations and maximize the firm’s long-

4

term fundamental value. We refer to the benchmark model as the “biased representative agent”

model because managers act as if they have the same biased beliefs as investors. One

justification for this behavior is that managers often have incentives to maximize the firm’s

current asset price, regardless of whether they share investors’ biased beliefs. The alternative

model of managerial behavior, which we refer to as the “rational long-term manager” model,

could arise if managers are rational and have incentives to maximize long-term firm value

perhaps because they own restricted stock or options.

We find that the rational long-term manager model incorrectly predicts that firm

investment only weakly forecasts abnormal asset returns. The biased representative agent model,

however, matches this aspect of return predictability data well. It is also consistent with

empirical findings that not only equity issuers, but also debt issuers experience lower future

returns, as shown by Spiess and Affleck-Graves (1999) and Cooper, Gulen, and Schill (2008).

The benchmark model with the biased representative agent implies that capital

misallocation is large. There are two ways to quantify the cost of capital misallocation due to

biased beliefs. First, one can contrast economic outcomes, such as value added, in a biased

economy with those in a rational one by comparing the two economies’ steady states. Second,

one can evaluate the hypothetical impact of rational long-term managers seizing control of firms

with biased investment policies and determining investment from thereon. The second method

differs from the first because rational long-term managers incur adjustment costs in order to

reach the optimal capital stock after taking control of firms making biased investment decisions.

Using the first method, we find that the aggregate value added of invested capital in a

steady state economy with the biased representative agent is 16.9% lower than it is in a steady

state economy in which managers maximize long-term rational value. Using the second method,

5

the aggregate value of invested capital is only 3.7% lower under biased investment policies

because of adjustment costs incurred by the rational manager. Underlying these aggregate

welfare costs of biased management, there is a wide range of inefficiencies across firms. Firms

with high investment rates despite low profitability tend to overinvest the most. Intuitively,

managers overinvest when agents are too optimistic about firm productivity, which typically

occurs when the soft information signal is high relative to firm profitability.

Further analysis shows that the excessive extrapolation and overconfidence biases play

different roles in capital allocation. When the biased representative agent is overconfident, the

firm invests too much when the soft signal is high and too little when it is low. These mistakes

are costly, but they may not cause average investment to be too high or low. Consistent with this

argument, simulations show that overconfident agents choose more volatile investment rates than

rational long-term managers, but their average investment rates are similar.

In contrast, a biased representative agent that excessively extrapolates productivity trends

puts too much value on the firm’s future growth opportunities, believing that she can act on

trends before they disappear. Consequently, she increases the capital stock today to lower

expected adjustment costs. Consistent with this reasoning, simulations indicate that extrapolative

managers hold more capital than rational managers, causing firms to be inefficiently large. Both

biases contribute to welfare costs, but overconfidence plays a greater role.

This study’s quantitative structural approach differs from the reduced form approach in

several earlier studies that test the validity of their models’ qualitative predictions. Examples of

this approach include Baker, Stein, and Wurgler (2003), Gilchrist, Himmelberg, and Huberman

(2005), and Polk and Sapienza (2009) among many others. These papers focus on how

mispricing affects rational managers’ investment decisions in the presence of financing

6

constraints or incentives to cater to investors. Our model focuses on the simple benchmark case

in which financing is frictionless and managers maximize the firm’s current asset price.

Most studies using structural methods do not jointly model investment behavior and asset

prices and even fewer allow for information processing biases. For example, Hennessy and

Whited (2007) employ a structural model to estimate the magnitude of corporate financing

frictions, rather than asset pricing frictions. Nearly all of the studies that model firms’ asset

returns and investment decisions, such as Gomes, Yaron, and Zhang (2006) and Liu, Whited, and

Zhang (2009), do not consider behavioral biases. For example, Liu, Whited, and Zhang (2009)

show that the average stock returns on portfolios formed on size and value could be consistent

with the optimality conditions from rational firms’ investment choices. In their model, stocks are

priced efficiently based on differences in risk and investment is efficient. In our model, assets are

priced inefficiently based on errors in investors’ expectations and investment is inefficient.

Two of the only papers to model firms’ asset prices and investment in the presence of

mispricing are Chirinko and Schaller (2001) and Panageas (2005). Both use aggregate time series

data and do not use data on the cross section of firms to estimate parameters, as we do. Panageas

(2005) models mispricing using heterogeneous investors and short-sales constraints. In contrast,

our model features homogeneous investors with no constraints that cause both pricing and

investment inefficiencies. Thus, our insights and analyses are quite different.

An overview of our paper follows. Section I develops our model of firm investment

dynamics and asset prices. Appendix A provides the technical details of our solution method.

Section II explains the empirical identification strategy and methodology. Appendix B describes

the empirical and estimation methods in greater detail. Section III presents the estimation results

and evaluates the model predictions using empirical benchmarks. Section IV analyzes the extent

7

of mispricing and investment inefficiencies in simulated model economies under various

specifications, including one in which managers maximize the firm’s long-term rational value.

Section V discusses and interprets the implications of our results for future research.

I. A Model of Investment and Asset Prices

A. Modeling Framework

This section presents the model that we analyze empirically in later sections. The model

builds on the standard dynamic neoclassical framework by allowing for agents to exhibit

information processing biases. There are two groups of agents in the economy: investors who

price assets, and managers who choose firm investment. All agents are risk-neutral and have a

discount rate of r. All agents have homogeneous information. For most of the analysis, we

assume that managers maximize the firm’s current asset price. In this case, a representative

investor/manager effectively determines both asset prices and firms’ investment policies. This

representative agent case is a natural starting point for a structural model that attempts to infer

agents’ beliefs from asset prices.5

All firms in the model share the same production technology, but their productivity at any

given time may differ. Here we describe the model for an individual firm; later in our empirical

analysis we consider a cross-section of such firms with differing productivity and capital.

Because we do not model tax or financing frictions, capital structure is irrelevant. Without loss

of generality, one can assume that investors finance firms using only equity.

We denote the continuous passage of time by t. The firm pays cash flows dπt given by

                                                            5 Firms’ asset prices would not change if we introduced risk-averse arbitrageurs because prices depend only on biased investors’ expectations when these investors are risk neutral. Other models, such as Daniel, Hirshleifer, and Subrahmanyam (1998), include both risk-averse arbitrageurs and risk-neutral investors with biased expectations.

8

dt f tdt ad ta mt

1−Kt.

(1)

The firm’s capital stock is Kt, and (0,1]α ∈ is its returns-to-scale parameter. The capital stock

evolves according to

dKt Itdt − Ktdt,

(2)

where It denotes the firm’s instantaneous investment rate and δ is the depreciation rate. To allow

for variation in growth opportunities, we adopt the standard assumption that adjusting the capital

stock is costly. When the firm invests at rate It, it pays the cost of the newly installed capital and

a quadratic adjustment cost of

It,Kt 2

ItKt−

2Kt.

(3)

In Equation (3), the parameter φ determines the cost of adjusting the capital stock.

The variable mt in Equation (1) denotes the economy-wide component of the cash flow

process,6 which grows geometrically at a rate of g according to

.t

t

dm gdtm

= (4)

We define the firm-specific component of the cash flow process in Equation (1) as

dat ≡ f tdt ad ta .

(5)

The term ft in Equation (5) is the firm’s productivity, which reverts to its long-term mean of f

according to the law of motion

                                                            6 We design the m process so that the firm’s asset price includes a component that represents economy-wide growth.

9

df t −f t − fdt fd tf.

(6)

The parameter λ measures the extent of mean reversion in productivity. We normalize the long-

term mean of productivity f to be one. The second term in Equation (5), aa tdσ ω , represents

noise in cash flows that is uncorrelated with the firm’s true productivity.

Because agents in the economy do not observe productivity ft directly, they estimate it

using two available pieces of information. First, agents observe the realized cash flow rate dat in

Equation (5), which is informative about ft . Second, agents observe a soft information signal st

that is correlated with the innovations in ft.7 The signal st summarizes all productivity-related

information, such as subjective interpretations of news events, other than cash flow realizations.

We assume that the soft signal st evolves according to

dst d tf 1 − 2 d t

s.

(7)

The stdω term represents noise in the signal. Equation (7) ensures that the variance of the signal

is equal to one for any value of signal informativeness, [0,1]η ∈ . We define signal precision as

≡

1 − 2.

(8)

A value of one (zero) for θ represents a fully revealing (uninformative) signal. In summary, there

are three jointly independent sources of randomness in the model: ftdω (the firm’s productivity

innovation), atdω (the cash flow noise), and s

tdω (the signal noise).

                                                            7 The modeling of the soft information signal follows Scheinkman and Xiong (2003).

10

B. Information Processing

We model two information processing biases, overconfidence and excessive

extrapolation. Similar to prior models of overconfidence, we assume that agents believe the soft

signal informativeness to be ηB > η and hence the signal precision to be θB > θ. The difference

θB – θ then measures how overconfident the agents are. We model the productivity extrapolation

bias by assuming that agents believe the productivity mean reversion parameter in Equation (6)

to be λB < λ. The difference λ – λB measures agents’ excessive extrapolation of productivity.8

As agents observe cash flow and signal realizations, they revise their beliefs about

productivity ft using Bayes’ rule, given their possibly incorrect beliefs (λB and ηB) about the

productivity and signal processes. Let tf denote the conditional estimate of ft given information

available at time t. Then the law of motion of tf is

( )1

2

,t

t

t tBB f B tt t

a ad

d

da f dtd f f f dt dsω

ω

γλ σ ησ σ

≡≡

−= − − + +

(9)

where 1t td dω ≡ s and ( )2

/t t td da f dt aω σ≡ − are standard Brownian motions, and Bγ is the steady-

state variance of the estimation error ttf f− . The variance Bγ is the solution to

( )22

2

variance of estimation errorvariance of

variance of

1 ,2 2

f Bf B B

B B a

ff

σ γσ η γλ λ σ

⎡ ⎤⎛ ⎞⎢ ⎥= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(10)

which gives

( ) ( )( )2 2 21 .B a B a B a B fγ σ λ σ λ σ η σ⎛ ⎞= − + + −⎜ ⎟⎝ ⎠

(11)

                                                            8 In estimating the model, we allow for θB and λB to be higher or lower than θ and λ, respectively. As we show below, however, the empirical estimates indicate overconfidence (θB > θ) and excessive extrapolation (λB < λ).

11

The learning process for a rational agent satisfies equations identical to (9) through (11), except

that we replace Bη and Bλ with their rational counterparts, η and λ.9

C. The Firm’s Optimization Problem

In the biased representative agent model, a manager who maximizes the firm’s current

asset price given investors’ information selects the firm’s investment policy (I) according to

( )( )( ) 1( , , ) max , ,r u tt t t u u u u ut uu tI

V K f m E e f m K I I K duα α∞ − − −

=⎡ ⎤= − −Ψ⎣ ⎦∫ (12)

subject to the evolution of capital, estimated productivity, and economy-wide growth:

( ) 1 2

,

,

.

u u u

Bu uB f Bu u

a

u

u

dK I du K du

d f f f du d d

dm gdum

δγλ σ η ω ωσ

= −

= − − + +

=

(13)

The firm’s asset price is the expected discounted value of future net cash flows—i.e.,

cash flows minus investment and adjustment costs—according to agents’ beliefs and

information. The Hamilton-Jacobi-Bellman equation for this maximization is

( ) ( )

( ) ( )

1

22

,

( , , ) max 12

t t t t t Kt

t tt BIB f t m f Bt

a

f m K I I K I K V

rV K f mf f V gmV V

α α δ

γλ σ ησ

−⎧ ⎫− −Ψ + −⎪ ⎪⎪ ⎪⎛= ⎨ ⎬⎛ ⎞

⎜ ⎟− − + + +⎪ ⎪⎜ ⎟⎜ ⎟⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭

.ff

⎞

                                                           

(14)

Substituting adjustment costs from Equation (3) and maximizing with respect to I yields

 9 We assume that initial productivity f0 is normally distributed with mean f and variance ( )2 / 2fσ λ . There is also an

initial signal 0 0s f 0ε= + , where ε0 is normally distributed with mean zero and variance v ( )0 22 2/f fγσ σ= γλ− . To

ensure the model starts at the steady state, we choose the initial signal variance such that the biased agents perceived initial estimation error variance matches their perceived steady-state estimation error variance γB .

12

ItKt

VK − 1

.

(15)

Equation (15) is the standard neoclassical investment policy: the firm invests at a rate that

exceeds the replacement of depreciated capital if and only if the marginal Q, VK, exceeds one.

The rate of adjustment of the capital stock is inversely related to adjustment costs φ.

Substituting Equation (15) into Equation (14) and suppressing the t subscripts to simplify

notation, we obtain a partial differential equation that characterizes the firm’s asset price V:

( )

( ) ( )

21

22

1 12

1 .2

K

BB f m f B

a

rV f m K K V K

fff f V gmV V

α α δφ

γλ σ ησ

−= − + −

⎛ ⎞⎛ ⎞⎜ ⎟− − + + + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(16)

We solve for the firm’s asset price V in Equation (16) and the resulting investment policy I in

Equation (15) using the numerical methods described in Appendix A.

II. Empirical Estimation of the Model

We determine the values of the 11 model parameters using direct estimation and the

simulated method of moments (SMM). First, we directly estimate three parameters—g, r, and

δ—using the values of three empirical moments. The discount rate (r) is equal to the sum of the

annualized averages of the equal-weighted excess return on firm assets (5.91%) and the real risk-

free rate of interest (1.26%), which is the yield on one-month nominal Treasury Bills deflated by

the Consumer Price Index (CPI). We set the average growth rate of economy-wide dividends (g)

equal to the average of value-weighted growth in firm assets (1.40%). The depreciation rate of

firm assets (δ) is the equal-weighted average depreciation rate on firm assets (2.96%). Appendix

B provides detailed definitions of asset returns, depreciation rates, and other empirical quantities.

13

Next, we apply our SMM procedure to determine the values of the remaining eight model

parameters. We select 13 moments, ten based on firms’ production process and three based on

firms’ asset returns, as inputs for this procedure to ensure that they collectively provide sufficient

information to identify the eight parameters. Strictly speaking, each of the technology and

information processing parameters affects all 13 of these moments, implying that all moments

are informative about all parameters—but some are more informative than others.

In fact, if agents were rational, the model would consist of just six parameters, α, φ, σa,

σf, θ, and λ, and the (ten) moments based on firms’ production processes would be sufficient for

identifying all six parameters. To help us identify the magnitude of the two information

processing biases, which arise from the discrepancy between θ and θB and between λ and λB, we

use three additional moments based on differences in firms’ expected asset returns. Our point

estimates of the eight model parameters above are the values that produce model behavior that

most closely matches the 13 moments described below. We estimate each empirical moment

using the time series average of cross-sectional estimates, following Fama and MacBeth (1973).

Here we provide intuition for identifying each parameter. We use the averages of firms’

Tobin’s Q and profitability, as defined by return on assets, to identify the return-to-scale

parameter (α). As α decreases, firms’ average profitability (or cash flows) and Tobin’s Q

increase. We infer the adjustment cost parameter (φ) from the coefficients in a regression of

investment on Q and profitability. High adjustment costs cause firm investment to respond less

to growth opportunities. The volatility of firm asset returns, profitability, investment, and Q

jointly provide information about the volatility of productivity and cash flows (σf and σa). A high

response of firms’ returns to their profitability suggests that agents must rely on realized profits

to learn about productivity, implying that the soft signal of productivity has low precision (θB).

14

The persistence in firm profitability helps us infer the true and perceived rates of mean reversion

in firm productivity (λ and λB).

We use differences in firms’ average asset returns to identify the differences between the

true and perceived productivity and signal processes. Recall that the difference λ and λB measures

the excessive extrapolation of trends in productivity, while the difference between θB and θ

captures the magnitude of overconfidence in the precision of the soft productivity signal. An

increase in either of these biases produces an increase in the value anomaly and an increase in

the investment anomaly.

We measure the value (investment) anomaly using the difference between the one-year

average returns of firms in the top and bottom deciles ranked by the inverse of Tobin’s Q

(investment) in the previous year. Sorting on inverse Q, which is the ratio of book assets to

market assets (B/M), is the only way to measure the value anomaly in the model because there is

no distinction between assets financed by equity versus debt. When forming portfolios based on

B/M, investment, and other accounting variables, we use data from each firm’s most recently

ended fiscal year, allowing for a three-month reporting lag. We rebalance all portfolios monthly.

To distinguish which of the two biases produces the observed value and investment

anomalies, we exploit their conflicting predictions about return predictability coming from firm

profitability. If investors excessively extrapolate from recent productivity information, they will

be too optimistic about firms that have high profitability, implying such firms will have low

future returns. In contrast, if investors are overconfident in their soft productivity signal and

place too little weight on hard profitability information, they will be too pessimistic about firms

with high profitability, implying such firms will have high future returns. Thus, the extent to

which firms with high current profitability exhibit high future returns reveals the relative

15

importance of the extrapolation and overconfidence biases. To quantify the profitability

anomaly, we compute the difference between the one-year average returns of firms in the top and

bottom deciles of profitability, as measured by return on assets.

In Appendix B, we describe the empirical methodology for measuring each of the 13

empirical moments, including ten production-related moments and three return anomalies. The

empirical methods are standard, except that the return anomalies are measured using firms’ asset

returns—rather than firms’ equity returns—to conform to the model. Appendix B also explains

how we use SMM to estimate the model parameters. In brief, the estimated parameters are those

that minimize the distance between the 13 moments predicted by the model and the 13 empirical

moments, where distance is measured in units of the moments’ empirical standard errors.

Appendix B contains further details.

III. Estimation Results

This section analyzes the biased representative agent model’s ability to match the

empirical moments described in Section II. The first two columns in Panel A of Table I show the

estimates and standard errors of the 13 empirical moments. The average Tobin’s Q of 1.580 and

average profitability of 13.2% suggest that the average firm has diminishing returns to scale and

valuable growth opportunities. The cross-sectional standard deviation of Q of 0.937 may seem

high relative to average Q. More than 5% of firms have Q values exceeding 4.375, which

increases the dispersion in Q. The persistence of profitability is 0.790, suggesting that firm

profitability is highly persistent. The investment sensitivities to Q and profitability are 0.067 and

0.660, which are within the respective ranges of values reported in Kaplan and Zingales (1997).

[Insert Table I here.]

16

The three anomaly rows in Panel A of Table I report the differences in asset returns

between the top and bottom decile portfolios sorted on B/M, investment, and profitability. The

annualized return differences from sorts on B/M and investment are 6.38% and −6.71%,

respectively, one year after portfolio formation. The significantly positive value (B/M) anomaly

is consistent with the well-known value effect in equity returns found in Fama and French

(1992). The negative investment anomaly is consistent with the investment and asset growth

effects found in Titman, Wei, and Xie (2004), and Cooper, Gulen, and Schill (2008),

respectively. The positive but statistically insignificant profitability anomaly of 1.39% per year

shows that post-earnings announcement drift is not large in our sample. Nevertheless, this

anomaly is informative about the relative importance of the two information processing biases,

as discussed in Section II. Some anomaly estimates appear slightly smaller than estimates in the

literature because we measure anomalies using firms’ asset returns, not just their equity returns.

The third column in Table I presents the results from SMM estimation of the model. The

last two rows in this column summarize how the model fits the data, using χ2 statistics with five

degrees of freedom.10 The χ2(5) value of 12.71 indicates that the simulated model moments

match the 13 empirical moments reasonably closely. Formally, the p-value of 0.048 implies that

we can marginally reject the hypothesis that the five over-identifying restrictions are valid at the

5% level. However, as discussed below, this rejection is driven by a single moment.

Figure 1 graphically summarizes the fit of the model to the 13 empirical moments. The

height of each vertical bar represents the t-statistic of the model’s prediction error, as measured

by the difference between the model’s prediction and the empirical moment, divided by the

empirical standard error. The model matches 12 of the 13 moments within 1.2 standard errors,

                                                            10 There are five degrees of freedom because the SMM procedure uses 13 moments to estimate eight parameters.

17

which is quite close. Visually, the ten leftmost bars in Figure 1 represent the prediction errors for

technology moments, while the three rightmost bars represent the errors for return anomalies.

[Insert Figure 1 here.]

The economic magnitudes of the model’s prediction errors are small. One can measure

their economic significance in percentage terms, using the prediction error divided by the

empirical moment. For the ten technology moments, excluding the standard deviation of

investment, the largest percentage model error is 16%. The prediction error for the volatility of

firm investment is −44%, meaning that model predicts too little investment volatility. One likely

reason is that the model does not include fixed adjustment costs, which could produce lumpy and

hence more volatile firm investment. Such adjustment costs also imply that empirical investment

is a noisy reflection of agents’ beliefs about productivity.

The model matches the three empirical return anomalies reasonably well in both

statistical and economic terms. The three model prediction errors for the profitability, value, and

investment anomalies are all less than one empirical standard error.11 Economically, the model

predicts a profitability anomaly that is too big by 1.29% per year and a value anomaly that is too

small by 1.78% per year. The model fits the investment anomaly nearly perfectly, suggesting that

it predicts an investment anomaly that is slightly too large relative to the value anomaly. The

lumpy investment data discussed above could explain this minor shortcoming of the model.

Overall, 12 of the model’s 13 prediction errors, all except investment volatility, are not

statistically significantly different from zero at even the 10% level.

Panel B of Table I shows the estimates of model parameters that produce the best fit to

the data, along with parameters’ standard errors. Reassuringly, the standard errors of all eight

parameters are small relative to the parameter values, indicating that there is enough statistical                                                             11 In unreported tests, we find that the model also matches the magnitude of the value anomaly beyond one year.

18

power to identify each parameter. The estimates of the technology parameters are reasonable

given the empirical data. The returns to scale (α) estimate of 0.844 is significantly below one,

implying diminishing returns scale.12 The adjustment cost parameter (φ) of 3.46 is smaller than

the estimates of 4.81 to 6.55 in Gomes, Yaron, and Zhang (2006), which uses aggregated

investment data which tends to be less volatile. The estimate of mean reversion in productivity

(λ) of 0.165 is low because profitability is quite persistent. The last column in Panel A of Table I

shows that the model matches this feature of the data well.

Panel B also reports the information processing parameter estimates. The perceived

precision of the soft information signal (θB) is high at 0.839, but significantly less than 1.0. This

high precision in the soft signal enables the model to match the weak response of asset returns to

profitability. The estimated true precision of the soft signal (θ) is 0.470, which is significantly

lower than the perceived precision of 0.839, implying that agents overreact to the soft

information signal. The difference between the true and perceived signal precisions helps the

model match the return predictability from B/M and investment.

Another way for the model to reproduce the return predictability from B/M and

investment is to set the perceived mean reversion in productivity (λB) lower than the true mean

reversion (λ). Indeed, the parameter estimates do satisfy this excessive extrapolation condition.

The overconfidence bias is strong relative to the extrapolation bias so that the model can match

the positive empirical profitability anomaly, which it does within one standard error.

Next we gauge the combined economic magnitude and plausibility of the two biases. The

last three rows in Panel B of Table I report the volatility of the error in estimating productivity

for a rational agent and a biased agent, along with the standard deviation of true productivity.

                                                            12 The returns to scale estimate of 0.844 is somewhat higher than the estimate of 0.627 (standard error = 0.219) in Hennessy and Whited (2007) mainly because our model allows firm productivity to grow at a positive rate of g.

19

Based on the volatilities in the estimation errors, a typical perceived error for the biased agent is

0.130, which is about half of the rational agent’s typical perceived error of 0.289. This

calculation suggests the two information processing biases are large. However, the difference in

the biased and rational agents’ perceived errors of 0.159 is only 18% of the magnitude of the

volatility in true productivity (0.870). This comparison suggests such biases may be difficult to

detect and correct. We discuss learning further in the next section.

IV. Analysis of the Simulated Economy

This section analyzes the properties of simulated model economies that mimic real-world

firm behavior and asset return predictability, using parameters from the biased representative

agent model estimation. Using simulated data allows us to explicitly measure many features of

the economy that econometricians cannot typically observe. Our discussion focuses on the

estimates of model parameters in Table I.

We divide the analysis into four subsections. First, we analyze firms’ expected returns

and asset mispricing from the perspective of a rational investor. When this investor evaluates

firms’ expected cash flows, she knows that biased agents select firm investment and determine

asset prices. Second, we distinguish the individual impacts of the two behavioral biases,

overconfidence and excessive extrapolation, on expected returns and mispricing. Third, to assess

how return predictability depends on managerial behavior, we simulate an economy in which

managers choose firm investment to maximize long-term rational firm value but biased investors

still determine asset prices. Fourth, we analyze investment inefficiencies by contrasting

economic outcomes, such as firm value and capital allocation, when managers choose firm

investment to maximize its biased asset price versus its fundamental value.

20

A. Analysis of Mispricing in the Biased Representative Agent Model

Empirical asset pricing research often uses the predictability in average realized returns

as a proxy for mispricing. Instead, we use two direct measures with natural interpretations that

are unavailable to empiricists: firms’ true expected returns and values from the perspective of an

agent with rational expectations conditional on investors’ information.13 A firm’s true expected

return measures the rate of convergence to its true value, where true value is the rationally

expected present value of the firm’s cash flows discounted at the appropriate rate (r).14 We first

analyze the predictability in firms’ true (abnormal) expected returns (minus the risk-free rate).

Then we compare firms’ observed asset prices to their true firm values. We define mispricing as

the firm’s market value divided by its true value, so that a ratio above one implies price exceeds

value. This definition is featured in the market efficiency notion proposed in Black (1986) : “we

might define an efficient market as one in which price is within a factor of 2 of value” (p. 533).

In his view, efficiently priced firms are ones with mispricing values between 0.5 and 2.0.

Panel A in Table II reports summary statistics of true one-year abnormal expected returns

and mispricing. The summary statistics in Panel A reveal that the cross-sectional standard

deviations of expected returns and mispricing are high in the model (5.66% and 21.66%) relative

to typical empirical anomaly estimates, implying quite inefficient asset pricing. In the model,

firms’ average value-weighted returns of –0.84% are lower than their equal-weighted returns of

0.39%, implying that larger firms have lower expected returns, as explained below.

[Insert Table II here.]

                                                            13 We estimate firms’ expected returns for up to 10 years from the perspective of a rational agent by computing the firms’ average returns, including both price changes and dividends, based on 5,000 simulated sample paths of the economy. Although there is no aggregate risk in our model, we use abnormal returns in these computations to facilitate interpretation. We compute a firm’s abnormal return using its raw return minus the risk-free rate. 14 True firm value is a solution to a partial differential equation analogous to Equation (16), except that it includes an extra state variable representing the rational estimate of productivity. We solve this equation numerically using the same method described in Appendix A.

21

Panel A shows that the equal-weighted average mispricing (i.e., price-to-value) ratio is

1.238 in the model, meaning that the average firm is substantially overpriced. This occurs

because investors overestimate the value of firms’ growth options. Both information processing

biases cause agents to think their information about future productivity is more precise than it

actually is. Consequently, they expect firms to make better-informed investment decisions than

they actually will, which leads agents to overprice firms’ assets.

Panels B and C in Table II show the distribution of true one-year abnormal expected

returns and mispricing for portfolios sorted on firm characteristics. The first column in Panel B

shows that simulations of the 13-moment model produce large variation in expected returns: the

top-to-bottom decile difference in firms’ expected returns is 19.89% (i.e., 11.21% − (−8.68%)).

The distribution of expected returns is smooth and positively skewed, much like empirical data

on firms’ realized returns.

The key result shown in Panel B is that the majority of the differences in portfolios’ true

expected returns remain unexplained by observable variables, such as B/M, investment, and

profitability. The top-to-bottom decile spreads in expected returns are 4.46% for B/M, 6.99% for

investment, and 2.38% for profitability, all of which are small compared to the spread of 19.89%

in firms’ true expected returns. By this criterion, investment is the characteristic that is most able

to explain differences in expected returns. This implies that investment is the best proxy—though

it is quite noisy—for the rate at which agents correct their expectations of firms’ cash flows.

Notably, the model-generated return predictability from observable variables such as

B/M and investment is asymmetric, whereas the true expected returns are nearly symmetric. For

example, the (growth) firms ranked in the bottom B/M decile have expected abnormal returns of

−0.43%, which is close to zero and not close to the −8.68% expected abnormal return for firms

22

with the lowest one-year expected returns. In contrast, (value) firms ranked in the top B/M decile

exhibit highly positive returns of 4.03% per year, which captures a significant part of the 11.21%

expected return for firms ranked in the top decile of one-year expected returns. These facts

suggest that the overpricing in growth firms is slower to correct itself.

Panel C shows that market prices often deviate widely from firm values. The top-to-

bottom decile spread in mispricing is 74.6% (i.e., 1.666 − 0.920) in the model, even though the

model predicts a value anomaly that is slightly too small. If one compares these mispricing

estimates to Black’s (1986) efficiency criteria, the average firms in the bottom and top

mispricing deciles are not “inefficiently” priced because 0.5 < 0.920 and 1.666 < 2.0.

Interestingly, Panel C shows that B/M, rather than investment, does the best job in explaining

mispricing, implying that B/M is the best proxy for errors in agents’ expectations of firms’ cash

flows. Comparing the explanatory power of B/M in Panels B and C, one can deduce that B/M

captures a long-term component of mispricing, most of which is not eliminated in the next year.

Table III presents cross-sectional regressions of expected returns and mispricing on five

observable firm characteristics using simulated data. Panel A reports estimated regression

coefficients of one-year abnormal expected returns on firm B/M, investment, the log of market

capitalization (size), profitability, and last year’s asset returns. All five firm characteristics are

standardized to have a mean of zero and unit standard deviation. The bottom row in Panel A

reports the R2 statistics from these regressions.

[Insert Table III here.]

The low R2 statistics reveal that the ability of each of the individual firm characteristics to

explain true abnormal expected returns is quite low. The R2s from the univariate regressions in

Panel A range from 0.2% to 10.9%. The main reason for the low R2s is that the econometrician

23

cannot directly measure the soft signal to which investors overreact. This is why using data from

the simulated economy is necessary to measure the true extent of mispricing. Regression (8)

shows that the joint explanatory power of the five firm characteristics (81.7%) is much greater

than the sum of their univariate explanatory power (27.5% = 6.8% + ... + 0.2%). Mechanically,

this occurs because profitability and investment are negatively correlated and predict returns with

opposite signs. Intuitively, only the joint specification is able to separately identify the two

distinct information processing biases, which have different implications for the profitability

anomaly but similar implications for the value and investment anomalies. The success of this

joint model in our simulations may have an empirical counterpart: Chen, Novy-Marx, and Zhang

(2010) argue that an asset pricing model with factors representing firm investment, profitability,

and market risk outperforms the widely used Fama-French (1993) three-factor model.

In the simulated model, the return predictability coefficients on firm size in the univariate

Regression (4) and the multivariate Regression (8) are highly negative. Although the estimation

does not include moments based on the empirical size anomaly, the model predicts a negative

size premium. This is consistent with the empirical findings in Banz (1981) and Fama and

French (1992). In the model, large firms tend to be those in which investors are too optimistic,

either because of overweighting positive soft signals or extrapolating positive productivity

trends, which leads to a negative size premium. In the univariate regressions (Regressions (1) to

(3)), firm investment negatively predicts returns, while B/M and profitability positively predict

returns, consistent with the anomalies reported in Table I. The multivariate regressions also

reinforce the conclusion from Table II that investment is the best predictor of expected returns.

The contemporaneous coefficient on past one-year returns in predicting one-year true

expected returns is close to zero, showing that the model does not predict positive return

24

momentum of the sort identified empirically in Jegadeesh and Titman (1993). This is interesting

because, in the model, profitability positively predicts returns as overconfident agents underreact

to profitability signals. This effect alone would produce return momentum because profitability

is positively associated with returns. There is, however, an offsetting effect that comes from the

positive correlation between the soft information signal and returns. Agents overreact to the soft

signal, which could cause negative autocorrelation in returns. The two countervailing effects

roughly offset at the empirical parameter estimates in this specification.15

Panel B reports analogous cross-sectional regressions of mispricing on firm

characteristics, again using simulated data. As expected, the coefficients on the characteristics

usually have opposite signs in Panels A and B because overpriced firms tend to have low

expected returns. Some discrepancies between the panels can arise because of nonlinearities in

the relationships and differences across firms in the rate at which mispricing dissipates. In

particular, the R2 of the B/M regression is much higher in Panel B than Panel C, supporting the

notion that B/M does a better job in explaining mispricing than it does in predicting returns.

Figure 2 graphically compares the cross-section of firms’ unobservable true expected

returns to the cross-section of expected returns conditional on observable variables. In each year,

firms are sorted into deciles based on their true one-year expected returns, mispricing, B/M, and

investment. We then measure the average returns on the top and bottom decile portfolios for each

of the next 10 years. The eight lines in the figure show these estimated expected returns for firms

ranked in the top and bottom deciles based on their true one-year expected returns, mispricing,

B/M values, and investment rates. The top and bottom lines depict sorts on true expected returns.

The next two outermost lines represent sorts on mispricing, while the middle four lines show

                                                            15 Naturally, including moments that measure return momentum in the estimations could lead to model parameters that would produce positive momentum.

25

sorts on investment and B/M. Because investment and mispricing negatively predict returns,

firms in the top investment and mispricing deciles have the lowest estimated expected returns in

the investment and mispricing sorts.

[Insert Figure 2 here.]

Importantly, Figure 2 shows that sorts based on true one-year expected returns produce

much larger differences in future returns than the two sorts based on the observable B/M and

investment variables. The differences between firms’ one-year expected returns are the same as

in Table II: 19.89% using sorts based on one-year expected returns, but only 4.46% (6.99%)

using sorts based on B/M (investment). In this sense, the observable return anomalies vastly

understate the true differences in expected returns.

Figure 2 provides two additional insights. First, differences in firms’ expected returns of

more than 1% per year persist beyond the 10-year investment horizon. Empirical studies face

challenges in detecting return predictability at such long horizons. Second, sorting on true

mispricing does not predict future returns as well as sorting on true expected returns. This occurs

because mispricing in some firms dissipates more rapidly than in other firms, where the exact

rate of convergence to fundamentals depends on the history of each firm’s soft signal and profits.

We conclude this subsection by considering a question that arises naturally in a structural

behavioral model: can agents use empirical data on asset returns to learn about their mistaken

parameter beliefs?16 To explore this, we simulate 39 years of data from 1,000 hypothetical

economies in which agents’ beliefs about the model parameters in Table I are actually correct,

implying that asset prices are rational, too.17 We then estimate regression coefficients of realized

                                                            16 Brav and Heaton (2002) also consider the implications of rational learning about structural uncertainty. 17 In this analysis, we treat an entire decile of firms with similar characteristics, such as B/M, as a single firm. This assumption is intended to reflect the high empirical correlations in returns of firms with similar characteristics, along with the low correlations in returns of firms with different characteristics.

26

firm returns on firm characteristics in these 1,000 economies to construct sampling distributions

of coefficients under the null hypothesis of correct agent beliefs. By comparing the population

regression coefficient under biased pricing in Table III to these 1,000 sample regression

coefficients under rational pricing, we test whether the hypothesis of rational pricing can be

rejected at various significance levels.

We briefly describe two such tests, suppressing the others to conserve space. If they use

univariate regressions of realized returns on firm characteristics, as in Regressions (1) to (5) in

Panel A in Table III, agents in the model could not reject the null hypothesis that their parameter

beliefs are correct at the 10% level. In contrast, if they use multivariate regressions, as in

Regressions (6) to (8) in Panel A in Table III, agents could reject their parameter beliefs at the

5% level—mainly because of significant coefficients on the investment and profitability

anomalies. These results mirror those in Table III. Multivariate tests of biased pricing are more

powerful because they are able to disentangle the investment and profitability anomalies. Agents

using such sophisticated tests could learn about their biases using a 39-year sample of data.

In summary, this subsection presents structural estimates suggesting that firm mispricing

can be quite large and persistent. The estimates of agents’ behavioral biases, however, are not

implausibly large in the sense that agents would need to apply sophisticated techniques to a large

amount of data to become aware of their biases. Biases are difficult to detect because firms’

observable characteristics are poor proxies for errors in agents’ expectations of firms’ cash flows.

B. Distinguishing the Impact of Overconfidence and Extrapolation

This subsection analyzes the separate impacts of the overconfidence and excessive

extrapolation biases on mispricing. Using the parameter estimates from the model in Table I as a

27

starting point, we consider two parameter changes. First, we eliminate agents’ overconfidence by

setting perceived signal precision equal to true signal precision (θB = θ = 0.470), while retaining

all other parameter values, including the excessive extrapolation parameters (λB = 0.125 and

λ = 0.165). Second, we eliminate agents’ excessive extrapolation by setting λB = λ = 0.165, while

retaining all other parameter values, including the overconfidence parameters (θ = 0.470 and

θB = 0.839). We simulate the behavior of economies with these two sets of parameter estimates

to isolate the impact of the two individual biases.

Table IV reports the moments from the simulation in which agents excessively

extrapolate, the simulation in which agents are overconfident, and a fully rational economy

without both biases. For comparison, the last two columns in Table IV redisplay the simulated

model moments with both biases and the empirical moments. The bottom rows in Table IV show

that the extrapolation and overconfidence biases both contribute to the value and investment

anomalies, whereas the two biases have opposing impacts on the profitability anomaly.

[Insert Table IV here.]

Many of the production-related moments in Table IV are quite similar in the biased and

rational economies. As we will show in Section IV.D, this does not imply that biases have no

real effects, but it does mean that many production moments play a small role in identifying

biases. There are two notable exceptions: Tobin’s Q and the sensitivity of returns to profitability.

Both of these moments differ significantly in the biased and rational economies because they

depend on asset prices. In fact, the sensitivity of returns to profitability depends further on

whether agents are overconfident or extrapolative, and thus helps to distinguish the two biases in

the model. Overconfident agents overweight the soft signal and ignore profitability information,

whereas extrapolative agents put too much weight on trends in profitability.

28

Table V analyzes the pricing inefficiencies in both one-bias simulations in more detail.

The two panels show the equal-weighted expected returns and mispricing (market price / rational

value) for portfolios consisting of firms sorted into deciles using firm characteristics, such as

B/M and profitability. These panels are analogous to Panels B and C in Table II, which show the

results from the simulation with both biases. As before, the analysis focuses on top-to-bottom

decile spreads in expected returns and mispricing.

[Insert Table V here.]

One striking fact in Panel A in Table V is that the simulation with excessive extrapolation

produces a spread in true expected returns of only 5.76% and two different observable variables,

profitability and investment, capture nearly all of this spread (5.04% and 4.63%). In contrast, the

simulation with just overconfident agents produces a much larger spread in true expected returns

(17.79%) and none of the observable variables, such as B/M, can reproduce even one-third of the

spread in returns. The reason is that corrections in overconfident agents’ beliefs are driven by the

soft productivity signal, which is not directly observable. In contrast, many of the corrections in

extrapolative agents’ beliefs occur through profit signals, which are linked to productivity trends.

Because overconfidence causes most of the differences in firms’ expected returns, the cross-

section of expected returns cannot be explained by “tangible” information, such as past profits.

This prediction of the model is consistent with the empirical results in Daniel and Titman (2006).

Panel A in Table V also shows that profitability negatively predicts returns in the

extrapolative investor simulation and positively predicts returns in the overconfident investor

simulation. In the two-bias model in Table I, these two effects roughly offset, producing a small

net positive profitability anomaly. An overconfident investor ignores public profit signals in

29

favor of his soft information signal, whereas an extrapolative investor puts too much weight on

trends in past profits, thinking that such trends will continue.

Panel B in Table V shows the cross-section of mispricing in the two one-bias simulations.

The simulation with excessive extrapolation only produces a spread in true mispricing of 28.0%

(1.299 – 1.019), whereas the simulation with just overconfident agents produces a much larger

spread in mispricing of 51.6% (1.357 – 0.841). The B/M variable is the most effective proxy for

mispricing when the extrapolation bias is the source of mispricing. Apparently, one can

effectively infer extrapolative agents’ mistaken beliefs from B/M because it summarizes past

productivity information. The profitability variable is the most effective proxy for mispricing

when agents are overconfident. Its effectiveness is limited, however, because errors in agents’

beliefs are based on the difference between their soft signal and a firm’s true productivity, both

of which are latent and imperfectly correlated with profitability.

[Insert Figures 3 and 4 here.]

Figures 3 and 4 depict the expected returns over the next 10 years of portfolios of firms

ranked based on their characteristics. The only differences between Figures 2, 3, and 4 are that

investors exhibit both biases in Figure 2, whereas they exhibit only the extrapolation bias in

Figure 3 and only the overconfidence bias in Figure 4. Consistent with Table V, Figure 3 with

only extrapolation shows that ranking firms based on observable firm characteristics, such as

investment, captures the majority of cross-sectional differences in firms’ true expected returns. In

Figure 4, however, when investors are overconfident in the soft signal precision, cross-sectional

differences in firms’ true expected returns are much greater than observable return anomalies.

Comparing Figures 3 and 4, one sees the expected returns are more persistent under the

30

extrapolative bias, where about 20% of the initial spread in expected returns remains after 10

years. That is, mispricing caused by excessive extrapolation corrects especially slowly.

In summary, three key points emerge from the analysis of the one-bias simulations. First,

although both biases contribute to the spread in firms’ true expected returns, the overconfidence

bias is a much bigger contributor. Second, the spread in true expected returns and mispricing

coming from the overconfidence bias is much harder to detect based on observable variables.

Third, both biases produce abnormal expected returns and mispricing that persist for many years,

with the pricing errors from the extrapolative bias correcting especially slowly.

C. Analysis of Mispricing when Managers Maximize Long-term Rational Firm Value

The models analyzed so far feature managers who maximize firms’ asset prices, and thus

choose investment according to investors’ information and beliefs. In this subsection, we

investigate whether a model with managers who maximize long-term rational firm value can

match the empirical data. That is, the manager in this model selects investment to maximize true

firm value, rather than investors’ perception of firm value.18 Because the manager uses the true

rate of mean reversion in productivity and true signal precision to forecast productivity, her

forecast differs from investors’ forecasts. Biased investors know that the manager is not pursuing

their ideal policy and think that this reduces firm value.19

We analyze the steady state properties of the economy with these rational long-term

value maximizing managers in Table VI. Panel A compares the simulated empirical moments for                                                             18 In the rational manager model, we assume that the manager maximizes total firm value, and thus does not care about timing the market. This is not realistic, but it is a reasonable benchmark. The rich set of circumstances observed in practice—e.g., some firms are partly owned by managers, others are cash rich and are not expected to raise financing, and others need external financing—are beyond the scope of this paper. 19 The differing beliefs of investors and managers imply that the firm value function in Equation (16) depends on an additional state variable that represents the rational manager’s beliefs. We solve for the value function using the polynomial approximation described in Appendix A. The approximation now includes a third state variable with a polynomial approximation degree of 10.

31

the economy with managers who maximize rational long-term firm value to the simulated

moments in an otherwise identical economy where the managers maximize the firm’s biased

asset price. The third column in Panel A computes the percentage changes in each steady state

moment that occur when switching from the biased to the rational manager economy.

[Insert Table VI here.]

The main result in Panel A is that the value anomaly increases by over 50%, while the

investment anomaly completely disappears in the economy with rational managers. The value

anomaly arises when investors are too pessimistic about value firms’ productivity prospects,

leading to high B/M ratios and high future returns. When rational managers run such value firms,

they spend more on new capital than investors would like. This makes investors even more

pessimistic about value firms and thus exacerbates the value anomaly. That is, the realized cash

flows of rationally managed value firms positively surprise the biased investors even more than

the realized cash flows of value firms run by biased managers. The reverse logic applies to

rationally managed growth firms.

The investment anomaly, however, arises when high managerial investment coincides

with excessive investor optimism. These two events coincide the most in the biased

representative agent economy, where managers maximize the firm’s current asset price, which is

set by biased investors. In contrast, in the rational long-term manager model, the relationship

between managerial investment and investor optimism is ambiguous. Both investment and

optimism depend positively on productivity signals, but they are negatively related when

investors disagree with rational managers’ productivity estimates. These competing effects

explain why simulations of the rational manager model produce an investment anomaly that is

much smaller than the empirical anomaly (0.11% versus −6.71%) and can have the wrong sign.

32

Panel B in Table VI provides further insights into mispricing in the rational and biased

manager economies. Row four shows that the cross-sectional standard deviation of expected

returns is slightly higher in the rational manager economy at 6.39% versus 5.66% in the biased

manager economy. The higher spread in expected returns in the rational manager economy

occurs because mispricing corrects itself sooner when the manager does not invest simply to

indulge investors’ misperceptions. In addition, the equal-weighted mean of mispricing is much

closer to one in the economy where the rational manager runs the firm (1.144 vs. 1.238). This

reduction in overpricing occurs because biased investors think that the rational manager chooses

a suboptimal investment policy, whereas a biased manager selects the optimal policy. Overall,

the biased representative agent model seems to fit the empirical data better than the rational

manager model in that we observe value and investment anomalies of comparable size, which

justifies our focus on the biased representative agent model.

D. Analysis of Investment Inefficiencies

Finally, we examine the biased representative agent model’s implications for capital

allocation. The model enables us to quantify inefficiencies in investment using several measures

that are unavailable to empirical researchers. Specifically, we compare firm values, capital

stocks, and investment rates when a manager selects investment to maximize the firm’s current

asset price and when a rational manager selects investment to maximize long-term value. From

the earlier simulations, we know firm values, capital stock, and investment rates in the biased

economy. We simulate these quantities in the hypothetical rational economy by setting investors’

information processing biases equal to zero, while holding all other model parameters constant.

That is, we set the perceived mean reversion in productivity (λB) equal to the true mean reversion

33

(λ) and the perceived precision of the soft productivity signal (θB) equal to the true precision of

the signal (θ). We refer to the quantities, such as firm values, from these two simulations as

“biased” and “rational” quantities.

Table VII reports several comparisons between the biased and rational economies. To

compare firm values, we adopt the perspective of a rational agent who properly computes the

expected discounted value of future cash flows. Value added is the difference between a firm’s

value (V) and the cost of its capital stock (K). A firm’s value per unit of capital is V/K, which is

equal to average Q in the rational economy but differs from average Q in the biased economy

because market prices differ from rational firm value.

[Insert Table VII here.]

Panel A in Table VII shows that behavioral biases cause large capital misallocation in the

steady state. The steady state aggregate value added in the biased economy is only 83.1% as

large as value added in the rational economy, as shown in the top row. The biased economy has a

larger capital stock and lower value per unit of capital (V/K). The intuition is that extrapolative

investors perceive future investment opportunities to be more persistent than they are, so they

want managers to install more capital to minimize future adjustment costs.

The top row in Panel A shows that the aggregate value added in the economies with only

overconfidence and only extrapolation are 90.9% and 98.8%, respectively, of the rational

economy’s value added. The combined value added reduction of 10.7% (1 – 0.909 + 1 – 0.988)

from the individual biases is much less than the value added reduction of 16.9% from both biases

together (1 – 0.831). One explanation is that a second distortion in an economy often causes

greater harm than the first distortion because changes in investment policy have a small first-

order impact on firm value when investment policy is close to the value-maximizing policy. A

34

complementary explanation is that overconfidence exacerbates the tendency for extrapolative

agents to increase firm size, as shown in the last two rows of Panel A. An overconfident and

extrapolative investor is more certain than a simply extrapolative investor that the firm can act on

growth opportunities before they dissipate.

The analysis in Panel A of Table VII compares the steady states of rational versus biased

economies. An alternative experiment is to measure the impact on economic outcomes of

switching from biased to rational management—i.e., supposing that rational managers take

control of firms that were managed by biased managers until that point in time. The resulting

changes in firm investment and value will be different from the steady-state comparisons in

Panel A because adjustment costs prevent rational managers from instantly changing the capital

stock and reaching the rational steady state immediately. Because it accounts for adjustment

costs, the rational takeover experiment is informative from a policy perspective.

Panel B of Table VII reports the results of the rational takeover experiment. The positive

values in the top row show that rational value-maximizing managers would invest much less than

biased managers, on average, which is consistent earlier findings. The median overinvestment

rate is 1.0% of the capital stock per year. The standard deviation of overinvestment of 4.4% is

much larger than the median, implying that many firms do not invest enough at any given time.

Both overinvestment and underinvestment are common because biased managers react too much

to noise in the soft information signal, which could be positive or negative.

The last two rows in Panel B of Table VII show that an instant replacement of a biased

manager by a rational manager would create substantial value for the typical firm. The median

biased firm is worth 95.0% as much as the rationally managed firm. This 5.0% marginal value

reduction from irrational management is much less than the value added reductions in the steady

35

state because rational managers incur large costs in adjusting the firm’s capital stock to the

optimal level. The median value reduction from biased management of 5.0% is greater than the

aggregate reduction of 3.7% (1 – 0.963). Biases reduce the value of small firms by a larger

percentage because a greater proportion of a small firm’s value consists of growth opportunities,

which are the main source of investors’ biases.

The second and third columns in both panels in Table VII show the separate impacts of

each bias on firm value and real investment. The first row in Panel B reveals that, when investors

are extrapolative, price-maximizing managers exhibit median investment rates that are 0.9% per

year higher than rational value-maximizing managers’ rates. As argued earlier, an extrapolative

investor overprices the firm’s future growth opportunities, believing that firms can act on

productivity shocks before they dissipate. Thus, the price-maximizing manager increases

investment today to lower investors’ expectations of adjustment costs. In contrast, when

investors are overconfident, the median price-maximizing manager’s investment rate is much

less biased (0.2% versus 0.9%).

The second row in Panel B shows that firms’ investment rates are much more volatile in

the simulation with overconfident agents (3.7%) than in the simulation with extrapolative agents

(1.8%). In the overconfidence simulation, the volatility in the soft information signal must be

large to produce the observable value anomaly because noise in the signal is only partially

correlated with B/M. The extrapolation bias does not need to be as large because past

productivity trends are more directly related to B/M and thus the observable value anomaly.

Overall, both biases contribute significantly to capital misallocation, but overconfidence plays a

larger quantitative role.

36

Lastly, we analyze the cross section of managerial overinvestment, as measured by the

difference in marginal investment rates of biased and rational managers following a takeover by

rational managers. Table VIII displays the results from regressions of overinvestment on lagged

investment, B/M, size, profitability, and one-year returns, all of which are standardized. The best

predictor of overinvestment is a firm’s observable lagged investment rate, which accounts for

32.8% of the variation in overinvestment, as shown in Regression (2). The general intuition is

that biased agents overreact to productivity information, coming from both investor

overconfidence and excessive extrapolation, which makes investment decisions too volatile. This

is why lagged investment positively predicts overinvestment in all the regressions in Table VIII.

[Insert Table VIII here.]

Similarly, biased managers of firms with valuable growth opportunities overreact to

positive productivity information and invest too much relative to a manager who maximizes

rational long-term value. Excessive investor optimism about growth explains the positive size

and negative B/M coefficients in Regressions (1) and (4). In the multivariate Regressions (6) to

(8), the lagged investment coefficient discussed above subsumes the B/M coefficient, implying

that lagged investment is the better proxy for excessive optimism. Holding B/M and investment

constant, managers underreact to the profitability signal, which explains the negative profitability

coefficient in Regression (7). In summary, firms with high investment rates despite relatively

low profitability tend to invest too much.

V. Concluding Discussion

This is one of the first studies to quantify the asset pricing and investment distortions

implied by behavioral theories of information processing biases. Our first finding is that

37

observed asset return anomalies understate the inefficiencies in firms’ asset prices and

investment decisions. Intuitively, econometricians cannot accurately measure investors’

information sets, which makes the efficient market hypothesis difficult or impossible to test, as

argued in Summers (1986) and Hansen and Richard (1987). Conversely, any observed evidence

of asset return anomalies could represent the tip of an iceberg of true mispricing. We use a

dynamic structural model to quantify the size of this iceberg.

The analysis shows that the overconfidence bias causes unobserved mispricing to be

larger than observable return anomalies. The mechanism is that overconfident investors have

incorrect beliefs that depend on an unobservable signal, making it difficult to see the true extent

of mispricing. Another notable behavioral bias, excessive extrapolation, does not share this

feature.

We also find that a model with biased investors but managers who maximize rational

long-term firm value fails to match a prominent empirical feature of return predictability: the

large investment anomaly. This suggests that models in which managers act to maximize current

asset prices are promising to explore. Future work could test whether more nuanced models of

managerial behavior fit the empirical data significantly better than our benchmark model where

managers invest exactly as much as biased investors would like them to invest.

Our model generates return predictability from behavioral biases alone, whereas in fact

some return predictability may come from risk premiums. This paper does not test the relative

merits of these explanations. Rather, our goal is to understand the implications of the behavioral

explanation. Future models could combine both return predictability mechanisms. Although

many of the qualitative insights here would apply to such models, there could be novel and

unanticipated interactions between behavioral and risk-based channels.

38

One need not interpret the parameter estimates from our model as constant over time. It is

quite possible that the asset return predictability patterns in the next 40 years will be markedly

different from predictability patterns observed in the past 40 years. As the economy’s latent

structural parameters change, behavioral biases may dissipate or be replaced by new biases.

Furthermore, rational agents may learn only gradually about such changes, generating return

patterns that are predictable only in hindsight. Under these alternative interpretations, our model

still provides a useful ex-post description of the underlying mispricing and investment

inefficiencies that occurred over the past 40 years. This description of the past may or may not

apply to the future. Thus, our main contribution is not the specific parameter estimates but the

modeling framework and the general insights it provides.

39

Appendix A: Numerical Solution of the Model

In Appendix A, we solve for the firm’s value function V. The partial differential equation

in Equation (16) characterizes firm value as a function of three state variables: capital stock Kt,

productivity estimate tf , and state of the economy mt. Now we introduce a variable

transformation that reduces the number of state variables in Equation (16) to two. We define

X ≡ m−1K,

(A1)

so that Equation (16) simplifies to

( ) ( )

( ) ( )

2

22

1 12

1 ,2

X X

BB f f B

a

r g J f X X J X gXJ

fff f J J

α δφ

γλ σ ησ

− = − + − −

⎛ ⎞⎛ ⎞⎜ ⎟− − + + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(A2)

where

( ) ( )1, ,J X f m V K f m−= , . (A3)

Next, one can solve for the function J in Equation (A2) and use Equation (A3) to obtain the

value function V. By differentiating both sides of Equation (A3) with respect to K, one obtains

VK JX .

(A4)

Equation (A4) states that marginal Q, VK, equals the derivative of J with respect to X.

We use a dual numerical approximation strategy to solve for functions J and JX. First, we

approximate JX. Differentiating both sides of Equation (A2) with respect to X, we obtain

40

( )

( ) ( )

21

22

11 12

1 .2

XX X

BB fX f B ffX

a

JrJ f X J g XJ

f f J J

αα δφ φ

γλ σ ησ

− ⎡ ⎤−= − + − + −⎢ ⎥

⎣ ⎦⎛ ⎞⎛ ⎞⎜ ⎟− − + + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

XX

(A5)

We then approximate JX in Equation (A5) using a sum of products of Chebyshev polynomials:20

( ) ,0 0

, 2 1 2M N

L LX m n m n

m n H L H L

X X f fJ X f a T TX X f f= =

⎛ ⎞⎛ ⎞− −= − ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠∑∑ 1 .− (A6)

In Equation (A6), M and N are integers that specify approximation degrees, { },m na is a set of

polynomial coefficients, and { }mT and { }nT are Chebyshev polynomials.21 The approximation

applies over intervals [ ],L HX X X∈ and [ ],L Hf f f∈ , which we choose such that the realized state

variables X and f attain values outside the bounds with less than 1% probability. To solve for

the unknown polynomial coefficients { },m na , we write Equation (A5) in terms of the Chebyshev

approximation and its partial derivatives. Then we minimize the sum of squared differences

between the right- and left-hand sides of this equation over a grid of ( ),X f points using a

nonlinear least squares routine.

Next, we use a similar polynomial approximation for the function J:

J X,f ∑

m0

M

∑n0

N

bm ,nTm 2 X − XLXH − XL

− 1 Tn 2f − fL

fH − fL− 1 .

(A7)

Equation (A7) differs from Equation (A6) only in the polynomial coefficients { },m nb . The partial

differential equation that characterizes J in Equation (A2) is linear in J and its partial derivatives,

                                                            20 We provide only a brief overview of this procedure—see Judd (1998) for details. Chebyshev polynomials are a family of orthogonal functions that constitute a basis for the space of continuous functions. 21 We use approximation degrees M = 80 and N = 10.

41

except for the term that includes ( )21XJ − . Because we already know the numerical

approximation for JX, we can use this function to compute ( 21XJ )− . Finally, using the computed

value of JX and the Chebyshev approximations for J and its partial derivatives in Equation (A7),

we express Equation (A2) as a linear system of equations in coefficients { },m nb , which we can

easily solve by matrix inversion. The resulting solution for J allows us to compute the firm’s

value function and its optimal investment policy using Equations (A3) and (15), respectively.

 

42

Appendix B: Empirical Methodology and Estimation Procedure

Empirical Methodology for Estimating Moments

Our goal in constructing an empirical sample is to provide an appropriate testing ground

for the model in Section I. The basis for the sample is the universe of all public US firms from

1968 through 2006. We restrict the sample to firms that have common stocks trading on the

NYSE, Nasdaq, or AMEX and those with comprehensive accounting information.

Because the model considers a firm in its steady state, all sample firms must have at least

60 months of continuous stock price data. The accounting information necessary for the

identification of key model parameters includes data on market equity, book equity, book debt,

cash, property, plant, and equipment (PP&E), profits, investment, sales, and total assets for each

of the last two years. The sample excludes firms with negative values of market equity, book

debt, PP&E, sales, and non-cash assets. It also excludes public utilities, regulated firms, and

financial firms—those with Standard Industry Classification codes in the ranges 4900-4999,

9000-9999, or 6000-6999—because our model of the capital stock may not apply to these firms.

We also exclude firms with stock prices less than $1 and those without trading volume in the

most recent month because these firms’ stock returns may contain significant measurement error.

Computing the empirical moments requires two types of raw inputs: accounting and asset

market variables. Each firm’s accounting variables include book equity, book debt, total assets,

Tobin’s Q, investment, profits, and depreciation. We define book equity as in Davis, Fama, and

French (2000), book debt as in Kaplan and Zingales (1997), total assets using the non-cash asset

definition in Cooper, Gulen, and Schill (2008). We do not include cash in firm assets because

firm assets in the model do not include any cash. Thus, we also subtract cash from firm value

43

when computing Q, which is (market equity + book debt – cash) / (non-cash assets). Market

equity is shares outstanding times the stock price at the end of the firm’s fiscal year. Investment

is the annual growth rate of assets. Profitability is after-tax operating income before depreciation

and before research and development expenses, divided by assets. That is, we treat research and

development as an investment, not an expense. Depreciation is the annual change in the

difference between gross capital and PP&E, divided by assets.

We compute monthly asset returns for each firm as the weighted average of equity and

debt returns, using the firm’s capital structure weights. We use standard CRSP data to measure

the equity return. A lack of firm-specific bond return data presents a challenge for measuring

debt returns. For the years 1973 through 2006, we use the monthly return on the Lehman

Brothers investment grade bond index as a proxy for firms’ debt returns. For the years 1968 to

1972, when the Lehman index is unavailable, we use the equal-weighted monthly return on US

Treasuries with maturities between five and 10 years from the CRSP US Monthly Treasury

Database. From 1973 to 2006, a regression of the investment grade returns on the Treasury index

return yields a coefficient of 1.00 with a standard error of 0.03 and an R2 of 83%. The correlation

of 0.91 and regression coefficient near 1.0 suggest that the Treasury index return is a reasonable

proxy for investment grade bond returns in most economic climates.

We estimate each empirical moment using the time series average of cross-sectional

estimates following Fama and MacBeth (1973). For the three return anomaly moments, such as

the one-year value anomaly, we use the monthly difference between the equal-weighted returns

of the firms in the top and bottom deciles of the accounting variable. In forming portfolios, we

allow for a reporting lag of three months for accounting variables. We compute the averages and

standard deviations of variables such as Tobin’s Q, profitability, investment, and returns using

44

monthly cross-sectional estimates. We winsorize Q, profitability, and investment at the 5% level

to reduce the influence of outliers. We winsorize returns at the 5% level only for computing

return volatility, not for the other return moments.

The point estimate for each empirical moment is the time series average of the cross-

sectional moment. Let d be a 13 by 1 vector consisting of all 13 point estimates. We use the time

series autocorrelation and standard deviation of the annual moments as the basis for computing

standard errors. The annual estimates of return anomalies are 12 times the monthly anomaly

moments. We model the yearly time series of each annual moment as a first-order autoregressive

process. For moments with an autoregressive coefficient that is significantly different from zero,

the delta method provides the moment’s standard error. For other moments, we use the Newey-

West (1987) standard error with the Newey-West (1994) recommendation for the number of

yearly lags. The covariance matrix (W) of the 13 annual moments is the element-wise product of

the moments’ correlation matrix and the outer product of the standard error vector.

Estimation Procedure

The goal of the SMM estimation is to select model parameters that produce simulated

model moments that are as close as possible to the empirical moments above. We compute the

distance between the model and empirical moments (d) using an SMM criterion function that

assumes a quadratic form: (m – d)TW-1(m – d), where m is the 13 by 1 vector of simulated model

moments. We select model parameters to minimize this SMM criterion function, using a

nonlinear least squares routine. We use the inverted empirical covariance matrix (W-1) to weight

the model’s errors in predicting the 13 moments (m – d). This is the efficient method for

selecting model parameters under the conditions described in Cochrane (2001).

45

Section II summarizes the empirical computation of d and W. The first step in computing

the vector of simulated model moments (m) is to obtain numerical approximations of the firm

value and policy functions, using methods described in Appendix A. Next, we apply these

numerical approximations over discrete time intervals of 40 periods per year to generate

simulated sample paths of model firms. We generate simulated data for a cross section of 1,000

firms, using 20 samples of 50 years each. For each sample firm, we draw initial values of true

productivity and agents’ productivity estimate from their respective steady-state distributions.22

To allow the capital stock to reach its steady-state distribution, we discard the first 10 years of

data in each sample. We use the remaining 40 years of data in each sample to compute the model

counterparts of empirical data moments and the resulting SMM criterion function. The 40 years

of simulated data corresponds closely to the 39 years of empirical data that we attempt to match.

                                                            22 For convenience, we assume productivity shocks are independent across firms. Because there are many firms in this economy that exhibit predictable and independent asset returns, a rational arbitrageur could obtain an extremely high Sharpe ratio. Of course, there are no rational agents in our model. Moreover, introducing productivity shocks that are correlated across firms, even if they are not priced, would eliminate the possibility of nearly risk-free arbitrage and would not affect the qualitative conclusions below.

46

References Baker, Malcolm, Jeremy Stein, and Jeffrey Wurgler, 2003, When does the market matter? Stock

prices and the investment of equity-dependent firms, Quarterly Journal of Economics 118, 969−1005.

Banz, Rolf W., 1981, The relationship between return and market value of common stocks,

Journal of Financial Economics 9, 3−18. Barber, Brad, and Terrance Odean, 2001, Boys will be boys: Gender, overconfidence, and

common stock investment, Quarterly Journal of Economics 116, 261−292. Barberis, Nicholas, and Ming Huang, 2001, Mental accounting, loss aversion, and individual

stock returns, Journal of Finance 56, 1247−1292. Barberis, Nicholas, and Ming Huang, 2008, Stocks as lotteries: The implications of probability

weighting for security prices, American Economic Review 98, 2066−2100. Barberis, Nicholas, Ming Huang, and Tano Santos, 2001, Prospect theory and asset prices,

Quarterly Journal of Economics 116, 1−53. Barberis, Nicholas, Andrei Shleifer, and Robert Vishny, 1998, A model of investor sentiment,

Journal of Financial Economics 49, 307−343. Benartzi, Shlomo, 2001, Excessive extrapolation and the allocation of 401(k) accounts to

company stock, Journal of Finance 56, 1747−1764. Benartzi, Shlomo, and Richard H. Thaler, 1995, Myopic loss aversion and the equity premium

puzzle, Quarterly Journal of Economics 110, 73−92. Bernard, Victor L., and Jacob K. Thomas, 1989, Post-earnings-announcement drift: Delayed

price response or risk premium? Journal of Accounting Research 27, 1−36. Berk, Jonathan B., Richard C. Green, and Vasant Naik, Optimal investment, growth options, and

security returns, Journal of Finance 54, 1553−1607. Black, Fischer, 1986, Noise, Journal of Finance 41, 529−543. Brav, Alon, and J.B. Heaton, 2002, Competing theories of financial anomalies, Review of

Financial Studies 15, 575−606. Chen, Long, Robert Novy-Marx, and Lu Zhang, 2010, An alternative three-factor model, Ohio

State University working paper. Chirinko, Robert S., and Huntley Schaller, 2001, Business fixed investment and “bubbles”: The

Japanese case, American Economic Review 91, 663−680.

47

Cochrane, John H., 2001, Asset Pricing, Princeton University Press. Cooper, Michael J., Huseyin Gulen, and Michael J. Schill, 2008, Asset growth and the cross-

section of stock returns, Journal of Finance 63, 1609−1651. Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 1998, Investor psychology and

security market under- and overreactions, Journal of Finance 53, 1839−1886. Daniel, Kent, and Sheridan Titman, 2006, Market reactions to tangible and intangible

information, Journal of Finance 56, 1605−1643. Davis, James L., Eugene F. Fama, and Kenneth R. French, 2000, Characteristics, covariances,

and average returns: 1929 to 1997, Journal of Finance 55, 389−406. Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock returns,

Journal of Finance 47, 427−465. Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on stocks

and bonds, Journal of Financial Economics 33, 3−56. Fama, Eugene F., and James D. MacBeth, 1973, Risk, return, and equilibrium: Empirical tests,

Journal of Political Economy 81, 607−636. Gilchrist, Simon, Charles P. Himmelberg, and Gur Huberman, 2005, Do stock price bubbles

influence corporate investment? Journal of Monetary Economics 52, 805−827. Gomes, João, Amir Yaron, and Lu Zhang, 2006, Asset pricing implications of firms’ financing

constraints, Review of Financial Studies 19, 1321−1356. Hansen, Lars P., and Steven F. Richard, 1987, The role of conditioning information in deducing

testable restrictions implied by dynamic asset pricing models, Econometrica 55, 587−613.

Hennessy, Christopher A., and Toni M. Whited, 2007, How costly is external financing? Journal

of Finance 62, 1705−1745. Hong, Harrison, and Jeremy C. Stein, 1999, A unified theory of underreaction, momentum

trading, and overreaction in asset markets, Journal of Finance 54, 2143−2184. Jegadeesh, Narasimhan, and Sheridan Titman, 1993, Returns to buying winners and selling

losers: Implications for stock market efficiency, Journal of Finance 48, 65−91. Judd, Kenneth L., 1998, Numerical methods in economics, The MIT Press.

48

49

Kaplan, Steven N., and Luigi Zingales, 1997, Do investment-cash flow sensitivities provide useful measures of financing constraints? Quarterly Journal of Economics 112, 169−215.

La Porta, Rafael, Josef Lakonishok, Andrei Shleifer, and Robert W. Vishny, 1997, Good news

for value stocks: Further evidence on market efficiency, Journal of Finance 52, 859−874. Lakonishok, Josef, Andrei Shleifer, and Robert Vishny, 1994, Contrarian investment,

extrapolation, and risk, Journal of Finance 49, 1541−1578. Liu, Laura Xiaolei, Toni M. Whited, and Lu Zhang, 2009, Investment-based expected stock

returns, Journal of Political Economy 117, 1105−1139. Newey, Whitney K., and Kenneth D. West, 1987, A simple positive semi-definite,

heteroskedasticity and autocorrelation consistent covariance matrix estimator, Econometrica 55, 703−708.

Newey, Whitney K., and Kenneth D. West, 1994, Automatic lag selection in covariance matrix

estimation, Review of Economic Studies 61, 631−653. Odean, Terrance, 1998, Volume, volatility, and profit when all traders are above average,

Journal of Finance 53, 1887−1934. Odean, Terrance, 1999, Do investors trade too much? American Economic Review 89,

1279−1298. Panageas, Stavros, 2005, The neoclassical theory of investment in speculative markets,

University of Chicago working paper. Polk, Christopher, and Paola Sapienza, 2009, The stock market and corporate investment: A test

of catering theory, Review of Financial Studies 22, 187−217. Scheinkman, Jose A., and Wei Xiong, 2003, Overconfidence and speculative bubbles, Journal of

Political Economy 111, 1183−1219. Summers, Lawrence H., 1986, Does the stock market rationally reflect fundamental values?

Journal of Finance 41, 591−601. Titman, Sheridan, K.C. John Wei, and Feixue Xie, 2004, Capital investment and stock returns,

Journal of Financial and Quantitative Analysis 39, 677−700.

50

Table I: Estimation results The table summarizes the results of the estimation of the biased representative agent model. Panel A reports empirical estimates, empirical standard errors, and simulated model values of the 13 moments used in the SMM estimation, along with the χ2 statistic and the corresponding p-value of the estimation. Panel B reports the point estimates and standard errors of the eight model parameters. Also reported are the standard deviations of the rational and biased agents’ perceived errors in estimating productivity and the standard deviation of productivity. These quantities are computed using the parameter estimates. Panel A: Empirical and simulated moments

Moments Empirical estimate

Standard error Model

Mean Tobin’s Q 1.580 0.370 1.735

Mean Profitability 0.132 0.033 0.111

Std. Dev. of Tobin’s Q 0.937 0.299 1.071

Std. Dev. of Abnormal Return 0.211 0.025 0.216

Std. Dev. of Profitability 0.094 0.021 0.101

Std. Dev. of Investment 0.169 0.033 0.094

Persistence of Profitability 0.790 0.017 0.810

Abnormal Return Sensitivity to Profitability 1.757 0.149 1.833

Investment Sensitivity to Tobin’s Q 0.067 0.021 0.056

Investment Sensitivity to Profitability 0.660 0.054 0.634

Value Anomaly 6.38% 2.35% 4.60%

Investment Anomaly −6.71% 1.39% −6.79%

Profitability Anomaly 1.39% 1.43% 2.68%

χ2 statistic 12.71

χ2 p-value (0.048)

51

Table I: Estimation results (continued)

Panel B: Parameter estimates

Parameters Estimate Standard error

Returns-to-Scale α 0.844 0.029

Adjustment Cost φ 3.459 0.449

Productivity Shock σf 0.499 0.110

Noise in Cash Flow σa 0.250 0.037

True Mean Reversion λ 0.165 0.034

Biased Mean Reversion λB 0.125 0.024

True Signal Precision θ 0.470 0.053

Biased Signal Precision θB 0.839 0.045

Std. Dev. of estimation error, rational 0.289

Std. Dev. of estimation error, biased 0.130

Std. Dev. of productivity 0.870

52

Table II: Expected returns and mispricing The table reports summary statistics and the cross-sectional properties of firms’ one-year expected returns and mispricing generated by the estimated model. One-year expected return is the firm’s average return in excess of the risk-free rate from holding its assets for one year. Mispricing is the ratio of the firm’s asset price to the rationally expected present value of its future cash flow stream. Panel A reports the summary statistics of one-year expected returns and mispricing. Panel B reports the equal-weighted average of one-year expected returns for firms sorted into deciles based on the one-year expected return variable itself, B/M, investment, and profitability, where investment is the lagged value of annual capital expenditures normalized by capital stock and profitability is the lagged value of annual profits normalized by capital stock. Panel C replicates the analysis in Panel B for mispricing. Panel A: Summary statistics for expected returns and mispricing Summary statistic Value-weighted mean Equal-weighted mean Median Standard deviation

One-year expected returns −0.84% 0.39% 0.04% 5.66%

Mispricing 1.212 1.238 1.215 0.217 Panel B: Cross-section of one-year expected returns Sorting variable One-year expected return B/M Investment Profitability Lowest decile −8.68% −0.43% 5.00% −0.96% 2 −5.17% −0.86% 2.45% −0.13% 3 −3.40% −0.97% 1.42% 0.04% 4 −1.97% −0.55% 0.67% 0.22% 5 −0.63% −0.38% 0.15% 0.31% 6 0.70% −0.01% −0.34% 0.45% 7 2.12% 0.32% −0.69% 0.65% 8 3.76% 1.02% −1.17% 0.81% 9 5.96% 1.72% −1.65% 1.03% Highest decile 11.21% 4.03% −1.99% 1.42%

53

Table II: Expected returns and mispricing (continued) Panel C: Cross-section of mispricing Sorting variable Mispricing B/M Investment Profitability Lowest decile 0.920 1.508 1.153 1.396 2 1.034 1.363 1.152 1.259 3 1.094 1.311 1.173 1.233 4 1.144 1.265 1.193 1.219 5 1.192 1.236 1.211 1.219 6 1.239 1.205 1.234 1.208 7 1.292 1.182 1.254 1.205 8 1.356 1.145 1.286 1.208 9 1.445 1.113 1.326 1.207 Highest decile 1.666 1.055 1.403 1.230

54

Table III: Cross-sectional regressions of expected returns and mispricing on firm characteristics The table reports cross-sectional regressions of one-year expected returns and mispricing on observable firm characteristics using simulated data from the estimated model. The firm characteristics include B/M, investment, profitability, size (the log of firm’s market value), and past one-year raw return. Reported coefficients and R2 values are time-series averages of cross-sectional regression estimates. All right-hand side variables are standardized in each cross-sectional regression. Panel A: One-year expected returns Regressions (1) (2) (3) (4) (5) (6) (7) (8)

B/M 0.015 0.003 −0.004 0.013

Investment −0.019 −0.016 −0.069 −0.063

Profitability 0.007 0.059 0.087

Size −0.016 −0.041

Past one-year return −0.001 −0.004

R2 6.8% 10.9% 1.5% 8.2% 0.2% 11.3% 50.6% 81.7% Panel B: Mispricing

Regressions (1) (2) (3) (4) (5) (6) (7) (8)

B/M −0.113 −0.113 −0.079 −0.077

Investment 0.077 −0.001 0.240 0.250

Profitability −0.044 −0.272 −0.304

Size −0.047 0.029

Past one-year return 0.017 0.024

R2 27.9% 13.0% 4.2% 4.7% 0.8% 28.0% 84.8% 87.2%

55

Table IV: Simulated moments with different biases in the biased representative agent model The table reports simulated values of the 13 estimation moments under different bias specifications. The first four columns correspond to cases with the excessive extrapolation bias only, overconfidence bias only, no biases, and both biases, respectively. The bias magnitudes are set equal to the estimated bias magnitudes in Panel B of Table I. The last column replicates the empirical moment values from Panel A of Table I. Bias in the model Extrapolation Overconfidence No Biases Both Biases Empirical

Mean Tobin’s Q 1.694 1.527 1.534 1.735 1.580

Mean Profitability 0.116 0.115 0.116 0.111 0.132

Std. Dev. Of Tobin’s Q 0.951 0.463 0.502 1.071 0.937

Std. Dev. of Abnormal Return 0.219 1.189 0.192 0.216 0.211

Std. Dev. of Profitability 0.101 0.103 0.104 0.101 0.094

Std. Dev. of Investment 0.091 0.076 0.077 0.094 0.169

Persistence of Profitability 0.808 0.812 0.810 0.810 0.790

Abnormal Return Sensitivity to Profitability 2.913 1.631 2.542 1.833 1.757

Investment Sensitivity to Tobin’s Q 0.066 0.135 0.121 0.056 0.067

Investment Sensitivity to Profitability 0.705 0.362 0.498 0.634 0.660

Value Anomaly 2.89% 2.94% - 4.60% 6.38%

Investment Anomaly −4.59% −2.97% - −6.79% −6.71%

Profitability Anomaly −4.98% 6.34% - 2.68% 1.39%

56

Table V: Expected returns and mispricing with different biases in the biased representative agent model

The table shows the cross-sectional properties of one-year expected returns and mispricing in one-bias model specifications in which either the excessive extrapolation bias or the overconfidence bias applies. Panel A reports the equal-weighted average of one-year expected returns for firms sorted into deciles based on the one-year expected return variable itself, B/M, investment, and profitability, where investment is the lagged value of annual capital expenditures normalized by capital stock and profitability is the lagged value of annual profits normalized by capital stock. Panel B replicates the analysis in Panel A for mispricing. Panel A: Cross-section of one-year expected returns

Sorting variable

One-yr expected return

B/M

Investment

Profitability

Bias in the model

Extrapolat. Overconf.

Extrapolat. Overconf.

Extrapolat. Overconf.

Extrapolat. Overconf. Lowest decile −2.18% −8.38% −0.42% −0.75% 3.10% 2.24% 3.23% −3.15%

2 −1.30% −4.85% −0.64% −0.69% 1.54% 1.20% 1.72% −1.46%

3 −0.89% −3.11% −0.57% −0.37% 0.89% 0.73% 1.01% −0.76%

4 −0.54% −1.75% −0.43% −0.29% 0.43% 0.47% 0.53% −0.29%

5 −0.19% −0.53% −0.24% −0.22% 0.11% 0.12% 0.14% 0.08%

6 0.17% 0.68% −0.07% 0.18% −0.20% −0.03% −0.22% 0.69%

7 0.56% 1.97% 0.23% 0.34% −0.50% −0.30% −0.51% 1.05%

8 1.06% 3.42% 0.55% 0.69% −0.75% −0.61% −0.85% 1.49%

9 1.75% 5.30% 1.09% 1.09% −1.07% −0.68% −1.21% 1.81%

Highest decile 3.58% 9.41% 2.50% 2.19% −1.53% −0.98% −1.81% 2.69%

57

Table V: Expected returns and mispricing with different biases in the biased representative agent model (continued)

Panel B: Cross-section of mispricing

Sorting variable

Mispricing

B/M

Investment

Profitability

Bias in the model

Extrapolat. Overconf.

Extrapolat. Overconf.

Extrapolat. Overconf.

Extrapolat. Overconf.

Lowest decile 1.019 0.841 1.292 1.104 1.076 1.050 1.113 1.220

2 1.065 0.919 1.214 1.087 1.086 1.048 1.096 1.127

3 1.088 0.961 1.181 1.073 1.100 1.052 1.106 1.095

4 1.107 0.996 1.159 1.070 1.113 1.052 1.116 1.074

5 1.123 1.030 1.141 1.067 1.129 1.061 1.125 1.061

6 1.140 1.064 1.123 1.057 1.139 1.061 1.136 1.040

7 1.159 1.101 1.107 1.053 1.153 1.067 1.148 1.027

8 1.182 1.146 1.090 1.046 1.170 1.075 1.161 1.012

9 1.216 1.208 1.067 1.039 1.192 1.075 1.179 0.999

Highest decile 1.299 1.357 1.024 1.028 1.243 1.083 1.220 0.967

58

Table VI: Comparing the rational value-maximizing manager model to the biased manager model The table provides comparisons of the rational long-term value maximizing manager model and the biased manager model. Panel A reports the simulated values of the 13 estimation moments for the rational and biased manager model specifications, along with the percentage change that occurs when switching from the biased to the rational manager model. Panel B reports summary statistics of one-year expected returns and mispricing in the two model specifications. Panel A: Comparing simulated moments in the rational and biased manager models

Manager type Rational manager

Biased manager % Change

Mean Tobin’s Q 1.775 1.735 2.3%

Mean Profitability 0.114 0.111 2.3%

Std. Dev. of Tobin’s Q 0.741 1.071 −30.9%

Std. Dev. of Abnormal Return 0.234 0.216 8.6%

Std. Dev. of Profitability 0.103 0.101 2.3%

Std. Dev. of Investment 0.076 0.094 −18.5%

Persistence of Profitability 0.809 0.810 −0.1%

Abnormal Return Sensitivity to Profitability 1.878 1.833 2.5%

Investment Sensitivity to Tobin’s Q 0.072 0.056 29.0%

Investment Sensitivity to Profitability 0.544 0.634 −14.1%

Value Anomaly 6.94% 4.60% 50.9%

Investment Anomaly 0.11% −6.79% −101.6%

Profitability Anomaly 2.79% 2.68% 4.1%

59

Table VI: Summarizing the rational value-maximizing manager model’s fit to the data (continued)

Panel B: Comparing expected returns and mispricing in the rational and biased manager models

Manager type Rational manager Biased manager

Value-weighted mean one-year expected return −0.69% −0.84%

Equal-weighted mean one-year expected return 0.42% 0.39%

Median one-year expected return 0.10% 0.04%

Std. dev. of one-year expected return 6.39% 5.66%

Value-weighted mean mispricing 1.165 1.212

Equal-weighted mean mispricing 1.144 1.238

Median mispricing 1.139 1.215

Std. dev. of mispricing 0.175 0.217

60

Table VII: Comparing the real effects of different biases The table reports summary statistics on the real effects of information processing biases in model specifications with both excessive extrapolation and overconfidence biases, the excessive extrapolation bias only, and the overconfidence bias only. The summary statistics in Panel A compare the steady state outcomes in these specifications to the steady state in a no-bias specification. Panel B evaluates the changes in outcomes that occur when rational managers take control of firms previously run by biased managers. Panel A: Steady-state comparisons

Model Both biases Extrapolation only Overconfidence only

Aggregate value added V – K, biased / rational 0.831 0.988 0.909

Aggregate value per unit of capital V/K, biased 1.236 1.293 1.365

Aggregate value per unit of capital V/K, rational 1.410 1.410 1.410

Median firm value per unit of capital V/K, biased 1.303 1.356 1.409

Median firm value per unit of capital V/K, rational 1.473 1.473 1.473

Aggregate capital stock K, biased / rational 1.442 1.369 1.027

Median firm capital stock K, biased / rational 1.306 1.269 1.044 Panel B: Rational managers taking control of firms run by biased managers Model Both biases Extrapolation only Overconfidence only

Median marginal investment rate, biased – rational 0.010 0.009 0.002

Std. dev. marginal investment rate, biased – rational 0.044 0.018 0.037

Aggregate firm value, biased / rational 0.963 0.992 0.977

Median firm value, biased / rational 0.950 0.988 0.970

61

Table VIII: Cross-sectional regressions of the biased manager’s overinvestment on firm characteristics The table reports cross-sectional regressions of the biased manager’s overinvestment on B/M, investment, profitability, size (the log of a firm’s market value), and past one-year raw return using simulated data from the estimated model. The biased manager’s overinvestment is defined as the marginal investment rate of the biased manager minus the marginal investment rate that a rational manager would choose for the same firm. Reported coefficients and R2 values are time-series averages of cross-sectional regression estimates. All right-hand side variables are standardized to have a zero mean and unit standard deviation in each cross-sectional regression.

Regression (1) (2) (3) (4) (5) (6) (7) (8)

B/M −0.016 0.003 0.008 0.002

Investment 0.025 0.027 0.061 0.063

Profitability 0.007 −0.038 −0.061

Size 0.018 0.027

Past one-year return 0.007 0.010

R2 12.6% 32.8% 2.4% 17.5% 3.0% 33.2% 60.1% 89.7%

Figure 1: t-statistics on Modeling Errors = (Simulated – Empirical Moment) / Empirical Standard Error

The figure shows the statistical significance of the difference between the 13 simulated moments and the 13 empirical moments used in the estimation of the model.

‐2.50

‐2.00

‐1.50

‐1.00

‐0.50

0.00

0.50

1.00

1.50

Mean Tobin’s Q

Mean Profita

bility

Std. Dev. of Tob

in's Q

Std. Dev. of A

bnormal Return

Std. Dev. of P

rofitability

Std. Dev. of Investm

ent

Persistence of Profitability

Abn

. Ret. Sen

sitivity

 to Profitability

Investment S

ensitivity

 to Tob

in's Q 

Investment S

ensitivity

 to 

Profita

bility

Value Ano

maly

Investment A

nomaly

Profita

bility Ano

maly

t‐statistics on Mod

el Prediction Errors

62

Figure 2: Persistence of expected returns in the biased representative agent model

The figure plots the one-year expected returns of decile portfolios for ten years after portfolio formation. Firms are independently sorted into decile portfolios with respect to their one-year expected return, mispricing, B/M, and lagged one-year investment rate. The upper (lower) four lines in the figure show one-year expected returns for firms ranked in the top (bottom) deciles by the one-year expected return sort and the B/M sort, and for firms ranked in the bottom (top) deciles by the mispricing sort and the investment sort.

63

Figure 3: Persistence of expected returns with extrapolation bias only

The figure replicates the analysis in Figure 2 for the model specification in which only the excessive extrapolation bias is present.

   

64

65

Figure 4: Persistence of expected returns with overconfidence bias only

The figure replicates the analysis in Figure 2 for the model specification in which only the overconfidence bias is present.