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Information Content of Option Prices: Comparing Analyst

Forecasts to Option-Based Forecasts

Anthony Sanford∗

March 31, 2021

Abstract

Finance theory dictates that public information is incorporated in asset pricing expectations.

Empirical research suggests that not all return forecasts are equal. Do diﬀerent forecasts weigh

information diﬀerently? This paper decomposes the information content of option and analyst

forecasts. The results show that analyst forecasts are constructed using a wide-spectrum of mar-

ket and ﬁrm-level data while option-based forecasts capture measures of uncertainty. Further,

we revisit the question of whether analyst forecast dispersion is a proxy for uncertainty. We

ﬁnd a negative relationship between analyst disagreement and option-based forecasts, indicating

that option traders view analyst disagreement as a source of uncertainty.

1 Introduction

What information is captured by analyst forecasts and how does it compare to forecasts obtained

from the options market? What information can be gleaned from analyst disagreement? Using an

option-based forecast derived from the Recovery Theorem (RT) (Ross, 2015), I assess the informa-

tional content of options on any given day for a certain time horizon. Does a forecast obtained

from a model like the RT provide additional information (or diﬀerent information) from a forecast

∗University of Maryland, College Park. Postdoctoral Fellow. Address: 4113AA Van Munching Hall, College Park,

MD 20742. Tel: 1.301.405.6300 Email: sanfoan@umd.edu

1

obtained from an analyst? In other words, if presented with a forecast for a stock widely analyzed

by analysts, would there be any value to using a model like the RT to obtain a separate forecast?

In this paper, I hypothesize that stock market forecasts from analysts will be diﬀerent from fore-

casts constructed using option prices, because the information sets1on which these forecasts are

based (and the incentives of the individuals creating these forecasts) are simply weighted diﬀer-

ently. Broadly speaking, I test this hypothesis in two stages. First, we determine if the two types of

forecasts actually produce diﬀerent results. Using percentile-based hypothesis tests (Wilcox, 2011),

I show that option-based forecasts are statistically signiﬁcantly diﬀerent from analyst forecasts at

every level of the expected return distribution.

Second, given that analyst forecasts are diﬀerent from a forecast derived from the RT, we deter-

mine whether the two sets of actors2weigh their information set diﬀerently when formulating their

expectations. To do so, I estimate cross-sectional regressions to determine the factor loadings of spe-

ciﬁc information used to construct the two types of forecasts. I analyze about seven hundred factors

known to characterize market returns, such as, for example, bond spreads and consumer sentiments.

The analyzed factors fall into one of three categories: macroeconomic factors (such as consumption

growth, inﬂation, and unemployment), stock market factors (such as book-to-market ratios, and

dividend-price ratios), and probability factors (such as overall market crash probabilities, reces-

sion probabilities, and sentiment indices). Cross-sectional regression results indicate that analyst

forecasts weigh information related speciﬁcally to the ﬁrm and macroeconomic fundamentals (e.g.

exchange rates, interest rates, and investments) more heavily while option-based forecasts tend to

weigh more heavily information related to the probability of extreme events (e.g. VIX and economic

1One could argue that the models being used by analysts is diﬀerent from that of the RT and therefore we would

expect to ﬁnd that the results are diﬀerent. Even if that is true, the model will be based on information captured by

that model and can be thought of as a multi-factor model based on various publicly available pieces of information.

In other words, if analysts were using the RT, then the cross-sectional regressions (and the hypothesis test) in

this paper would ﬁnd that the forecasts are based on the same information. The results in this paper thus apply

regardless of whether or not we think that analysts are using diﬀerent information sets or diﬀerent models.

2We deﬁne the two sets of actors as 1) analysts and 2) option traders.

2

uncertainty). Why is the fact that analyst forecasts and option-based forecasts diﬀerent important?

Recent research in ﬁnance has argued that option-based models incorporate the probability of rare

disaster events (Barro and Liao, 2021) while analyst forecasts tend to be overly optimistic and delay

incorporating negative news into their forecasts (Bryan and Tiras, 2007). As such, these ﬁndings,

which shows that analyst and option-based forecasts are diﬀerent, are indications that diﬀerent

forecasts should be viewed as complements rather than substitutes.

What might explain this discrepancy between analyst-based and option-based forecasts? I argue

that the incentives facing professional analysts and option traders play a major role in explaining

forecast discrepancies. On average, analysts are more optimistic, but also more conservative, about

the future prospects of the stock market than are option market participants, who generally use

options as a hedging instrument. On the one hand, analysts’ forecasts tend to capture slight

variations from the status quo in the market because analysts are penalized (in the most extreme

case, by losing their jobs) if their forecasts are “too out there.” Since ﬁrms that hire analysts are

paid for their forecasts, it stands to reason that bad forecasters become unreliable which makes them

dispensable. On the other hand, options market participants are using options to protect themselves

against the possibility of future adverse movements. As such, an option-based forecast that uses

the natural probability distribution will capture expected extreme movements in the market, and

not the slight variations from the status quo captured in an analyst-based forecast.

Scholars have examined the value of forecast models compared to analyst forecasts in the past.

For example, a strand of the literature compares time-series model forecasts with analyst forecasts

(Brown and Rozeﬀ, 1978; Brown et al., 1987; Clement, 1999), ﬁnding that, on average, analyst

forecasts are superior because analysts are capable of incorporating larger amounts of information

into their models. A recent literature has shown that option prices capture underlying rare disaster

probabilities (Barro and Liao, 2021; Backus et al., 2011; Seo and Wachter, 2019). These rare disaster

3

probabilities do not seem to be incorporated into analyst forecasts – further reinforcing the idea

that the two types of forecasts should be viewed in tandem rather than as competing forecasts.

Further, the RT is too recent to have been the object of an analysis comparing diﬀerent forecasts.

This paper further contributes to the growing literature focusing on the Recovery Theorem

speciﬁcally. So far, researchers have been somewhat mixed on the RT’s forecasting ability. Several

researchers have extended and/or tested the RT empirically and have found positive forecasting

results (Sanford, 2019a; Jensen et al., 2019; Audrino et al., 2019; Bakshi et al., 2018; Van Appel

and Maré, 2018). Yet, others have questioned the legitimacy of the model, claiming that it does not

recover what the model claims it recovers (Borovička et al., 2016). This particular study approaches

the legitimacy question from a diﬀerent angle: if the RT does not provide us with information

beyond what is already available (in analyst forecasts, for example), then there would be little

beneﬁt in using models such as the RT or in extending research on the extraction of the natural

probability distribution in the future. Ultimately, however, this article ﬁnds that the information

content used in the RT is signiﬁcantly diﬀerent from that used in analyst forecasts.

Another relevant strand of the literature focuses on the drivers of analyst forecasts (such as the

analyst’s career status, the aggressiveness of a forecast, and so on) (Clement and Tse, 2005; Givoly

et al., 2009; Bryan and Tiras, 2007). This literature, which focuses on characteristics of analysts

themselves, does not identify what information analysts use in their forecasts. Instead, it tries

to: 1) determine how good analysts are at forecasting stock prices/returns, and 2) what motives

or characteristics drive these analysts to forecast the way that they do. This article advances

this literature by analyzing the informational content driving the forecasts, and, by extension, by

positing the underlying motivations/incentives of the actors producing them.

4

2 Models

In this section, we ﬁrst start by mathematically deﬁning the RT and analyst forecasts. The RT will

be derived brieﬂy mostly for notational purposes. More details are available in Ross (2015); Sanford

(2019a). As for the analyst forecasts, we take them as given since the data is readily available

from the Institutional Brokers’ Estimate System (IBES) database. Instead, we derive a model that

outlines the decision making process of an analyst. In particular, we outline the idea that, although

rational, analysts do not always immediately include all available information into their forecast.

Instead, analysts’ incorporate information into their forecast once it has been veriﬁed and/or when

they are certain that it will have an impact. In other words, the analyst, who is incentivized to

keep his job, may not be willing to make outrageous calls on a stock unless she is certain that

the information will have an impact (Mikhail et al., 1999; Lim, 2001; Gu and Wu, 2003; Ramnath

et al., 2008). Finally, we derive a measure of uncertainty based on the analyst forecast. More

speciﬁcally, we deﬁne the standard deviation of analyst forecasts, the range of analyst dispersions,

and a measure of the distribution of the analyst forecasts proxied by the diﬀerence between the

mean and the median forecasts.

2.1 Recovery Theorem

The Recovery Theorem (RT) by Ross (2015) is a methodology that allows us to extract a forecast

for an asset for which options are traded. The methodology allows us to disentangle state prices

into their individual components (the discount rate, the pricing kernel, and the natural probability

distribution). To accomplish this, we must ﬁrst deﬁne state prices. In continuous time theory,

state prices are deﬁned as the second derivative of the option prices with respect to the strike price

5

(Breeden and Litzenberger, 1978):

s(K, T ) = ∂2C all(K, T )

∂K 2(1)

where s(K, T )is the state price, C all(K, T )is the call option price, Tis the time to maturity of the

option, and Kis the strike price. These prices can be estimated empirically in various ways, but

the method used in this paper is the method described in Sanford (2019b). Intuitively, state prices

are the price of an asset that pays you one dollar at some future date, T, if the underlying asset

reaches a speciﬁc state given a speciﬁc initial state.3State prices are used because they eﬀectively

standardize the future payout from our asset. In essence, we remove part of the uncertainty of the

future asset’s price, which then allows us to focus on extracting the stochastic component in the

pricing equation. Mathematically, we can write the state price as follows:

st

i,j =δu0(ct+1,j )

u0(ct,i)ft,t+1

i,j (2)

where sis a vector of state prices, iand jare states (e.g. S&P 500 levels), δis a discount rate, u0(·)

is a marginal utility, and fis the natural probability measure. To derive a forecast from equation

2, we need to extract the natural probability distribution of returns, f.

In order to have enough equations to solve this system, we ﬁrst derive and deﬁne contingent

3For example, a state price might be the price of an asset that guarantees its investor a payout of one dollar in one

month if the current level of an underlying index is at 1,000 today and ends at 1,500 in one month.

6

state prices.4Contingent state prices are obtained for the RT as follows:

st+1 =stP, t = 1, ..., m −1

P≥0

(3)

where st+1 is the next period’s state price, stis the current period’s state price, and Pis the con-

tingent state price. Writing equation 2 in terms of contingent state prices and rearranging so that

we are solving for the natural probability distribution, f, we have:

fi,j =1

δpi,j

u0(ci)

u0(cj)

Separating the marginal utilities (and re-arranging) gives us:

pi,j

1

u0(cj)=δ1

u0(ci)fi,j

Deﬁning the marginal utilities in terms of z and then multiplying both sides by the respective z’s,

we obtain:

pi,j zi=δzjfi,j

Noting that pi,j and fi,j are entries to a matrix, we can re-write the equation in matrix form as:

P*

zi=δ*

zjF

4Contingent state prices are deﬁned in the same way as state prices except that, for contingent prices, we generalize

the initial state and the ﬁnal state transitions so that they are not solely dependent on the current state of the

world. For example, if the current state of the S&P 500 is 1,000, state prices will be the prices for all transitions

from an initial state of 1,000 to some future state, whereas contingent state prices will be all possible pairs from

any current state to any future state.

7

Since Fis a stochastic matrix we can write the previous equations as:

P*

zi=δ*

zj

which is nothing more than an eigenvalue/eigenvector problem that can be solved using the Perron-

Frobenius theorem (Meyer, 2000). At this point in the RT, we have all of the components to solve

for the natural probability distribution as follows:

fi,j =1

δpi,j

zi

zj

(4)

Intuitively, equation 4 can be thought of as the risk-netural distribution which has been scaled for

the risk-aversion parameter, zi

zj, and the discount factor, δ. The natural probability distribution

is the option market’s best estimate of the risk-adjusted expected distribution of returns. Letting

return be deﬁned as a continuous random variable, we can deﬁne expected return obtained from

the natural probability distribution of the RT as follows:

E[r] = Z∞

−∞

f(r)rdr (5)

where f(r)is the natural probability distribution of returns obtained from the RT and ris the

random variable for returns.

2.2 Analyst Forecast

For this paper, the main driver for ﬁnancial analysts is not, as one might expect, to get the most

accurate forecast possible. Rather, the analyst wants to be as close as possible within a certain

margin of error (Mikhail et al., 1999; Lim, 2001; Gu and Wu, 2003; Ramnath et al., 2008). If the

8

analyst is beyond that margin of error, they run the risk of losing their jobs over the long run.5As

such, the analyst, on average, uses the information that allows them to construct a forecast that

is accurate enough, but never so far oﬀ that the analyst runs the risk of losing their livelihood. In

essence, the analyst is not irrational and choosing not to use all available information. Rather, the

analyst has a bias (conscious or unconscious) that prevents her from using the information that

might result in a forecast that is too far from the current level of the asset being analyzed.

A large literature (Mikhail et al., 1999; Lim, 2001; Gu and Wu, 2003; Ramnath et al., 2008)

shows that analysts are not remunerated based on how well they can forecast earnings, for example.

Instead, analysts are ﬁred when their performance is deemed to be inadequate. In order to keep the

mathematical problem simple, let us assume that analysts’ life earnings depend solely on their ability

to become “superstar” analysts. This means that analysts are motivated by the potential to become

a recognized analyst, a valuable commodity in the eyes of the analysts’ employer. We will assume

that an analyst has two potential outcomes: either they can continue their work, and earn 1$, or they

are terminated by their employer and they now earn 0$ in perpetuity. It is not necessary to become

a superstar as an analyst, but this certainly motivates the analyst to perform better. To keep the

problem even simpler, I assume that there is no probability of becoming a superstar. Instead, the

analyst is faced with the prospect of either continuing to do her job (motivated by the prospect

of becoming a “superstar”) or to get terminated (the model would simply include a third option in

the more complicated model – to become a “superstar” – with an associated probability dependent

on past performance). The probability of continuing to be an analyst (not getting terminated) is

dependent on the analyst’s ability to forecast earnings. We model this “ability to forecast” as the

average distance between the analyst’s forecast and realized earnings. For tractability purposes

with the RT, we will assume that the proxy for earnings is the stock’s price. In other words, we can

5Getting ﬁred might sounds extreme. However, one can think of this as the consequence that an analyst must face

when she is unreliable in her forecast. If the analyst is unreliable, since ﬁrms must pay for this analysts’ report, her

ﬁrm is more likely to ﬁre her than if she is consistent.

9

write the analyst’s performance as follows:

χ(P|It) =

n

X

t=1

Et[Pt+1|It]−Pr

t+1

n(6)

where Etis the expectation operator at time t,Pt+1 is the expected price for the next period

(the one being estimated by the analyst), Pr

t+1 is the next period’s realized price and Itis the

information set available to the analyst at time tover nperiods. Substituting a basic price equation,

pt=Et( ˜mt+1xt+1 ), we can rewrite equation 6 as follows:

χ(P|It) =

n

X

t=1

Et[ ˜mt+1 ˜xt+1|It]−Pr

t+1

n(7)

where ˜mt+1 is the stochastic discount factor in the pricing equation and ˜xt+1 is the future payoﬀ of

the asset. Normalizing the future payoﬀ xt+1 like we did for the Recovery Theorem in the previous

section, we obtain the following equation:

χ(P|It) =

n

X

t=1

Et[ ˜mt+1|It]−Pr

t+1

n(8)

The implication of the model is that the analyst will formulate her forecast such that the average

diﬀerence between her forecast and the realized price of the asset is minimized.6This will, in

essence, ensure that the probability of getting ﬁred is smallest. Looking at equation 8, one should

notice that the only thing that the analyst controls is how she formulates her expectation. Since the

stochastic discount factor is unknown, she will use the information set available to her at time tto

estimate it. As such, the only piece of information that the analyst controls is the information set

6In reality, the problem would be more complicated than this because the analyst would want to minimize her chance

of getting ﬁred (being below the threshold α) all while maximizing her probability of becoming a superstar. So,

the more complicated problem would be a minimax optimization problem. For simplicity, I am only looking at the

minimization problem.

10

used to formulate the expectation. Thus, the analyst’s problem is an inﬁnite horizon optimization

problem:

max

It

V(χ(P|It)) (9)

where we can deﬁne the value function, V(·), in equation 9 as a piece-wise function as follows:

V(χ(P|It)) =

1,if χ(P|It)≤αt

0,if χ(P|It)> αt

(10)

which can be understood as the analyst being allowed to continue in her current position if her

overall (average) forecast (χ(P|It)) is less than some threshold (αt). The analyst will choose the

information that she uses to formulate her forecast so as to minimize the possibility of getting ﬁred.

This formulation then allows us to formulate our ﬁrst formal hypothesis to be tested in this paper

as:

Hypothesis 1. Analyst forecasts will have a negative skew as a result of their incentive structure.

In an economy where markets are fully rational (Muth, 1961; Blanchard and Watson, 1982) and

all available information is used to price assets, an option-based forecast should, ultimately, be the

same as an analyst-based forecast. The model above aims to illustrate why the two forecasts may be

diﬀerent from one another even if option market participants and analysts are acting on the same

information. It suggests that it is possible to obtain two diﬀerent forecasts because of diﬀerent

incentives despite all parties acting on the same information set. This now allows us to formulate

the second hypothesis tested in this paper as:

Hypothesis 2. At any given time t, regardless of the information set available, an analyst forecast

will be diﬀerent from an option-based forecast.

11

Intuitively, this hypothesis implies that because of various analyst biases, the analyst will have a

forecast that is, itself, bias (see Lim (2001); Kothari et al. (2016) for an overview of the analyst bias

literature) which implies that we should expect option-based forecasts to be diﬀerent from analyst

forecasts. Given the fact that both the option-based and analyst forecasts use the same information

set and that the stock market is, in general, a random walk, we can formulate our third hypothesis

to be tested in this paper as:

Hypothesis 3. Since the information set available by option-based and analyst forecasts is the

same, we would expect the forecast residuals to be very similar.

This third hypothesis stems from the idea that since stock returns are random and therefore

unpredictable in the short-term, then both option-based and analyst forecasts should be equally

“inaccurate.” Of the information that is available to investors, is the information used by option

traders diﬀerent from that used by analysts? The fourth hypothesis tested in this paper can thus

be formulated as follows:

Hypothesis 4. The information set used at any time period tin an analyst forecast is the same

as that used in an option-based forecast.

This fourth hypothesis does not state that analysts do not incorporate all information into their

forecasts. Rather, the hypothesis posits that analysts, because of their biases, may downplay, choose

to ignore, or delay incorporating certain pieces of information when constructing their forecasts. As

an example of the information dissemination mechanism described in this section and the idea

that analysts might delay incorporating information into their forecasts, we can use the COVID-19

pandemic. The COVID-19 pandemic traces its roots to an outbreak in China in late 2019. The

virus started to spread throughout the world and the ﬁrst cases started to appear in the United

States in January 2020. Washington state was the ﬁrst to declare a state of emergency in February

12

2020 and the ﬁrst stay at home orders were issued by the state of California on March 19, 2020.

Despite all of the signals, the very ﬁrst analysts to start revising their forecasts downwards for

American companies did not come until late February/early March (information obtained from the

Bloomberg terminal). This is in line with the model described in this section – analysts incorporate

information available to them but do so when they are certain of the impact on the ﬁrms that they

are analyzing. In other words, they are not willing to put their livelihood (Lim, 2001) on the line

with every piece of information that they receive. Rather, they wait and analyze the situation until

they are more certain about their model predictions. In contrast, the VIX, which is the uncertainty

index derived from option prices, had already increased by almost 50% by January 27th, 2020 –

more than a month prior to the very ﬁrst analyst updating its forecast. Again, this is an indication

that the information contained in option prices is somewhat diﬀerent, or at least incorporated more

quickly, from the information used and disseminated by analysts.7

2.3 Analyst forecast dispersion

As a corollary to the previous hypothesis, we ask what information is contained in analyst forecasts

dispersion? A small body of work (see Kothari et al. (2016) for a literature review on the topic)

has shown that analyst disagreement is a proxy for stock return uncertainty. Miller (1977); Diether

et al. (2002) have argued that there is a negative relationship between analyst coverage and prices.

In a market with little to no short selling, Miller posits that analyst disagreement causes asset prices

to increase because of the increased number of positive forecasts from analysts. This causes prices

to become overvalued which in turn causes a correction in asset prices. As such, we should expect

to observe a negative relationship between analyst dispersion and realized return. This negative

relationship occurs because company information is revealed and corrections to the overvaluations

7Or at least, that there is a lag in the information dissemination of analysts’.

13

occur.

In contrast, Merton (1987) has argued that uncertainty should be positively associated with

returns since increased idiosyncratic risk should be priced positively. Intuitively, if we accept the

premise that increased disagreement among analysts is a measure uncertainty, then we should expect

a positive relationship between analyst dispersion and expected returns since investors will want to

be compensated for taking on risk.

We deﬁne measures of dispersion intuitively as the amount of disagreement amongst analysts.

Mathematically, we deﬁne them in three ways (Barron et al., 2009): the ﬁrst is standard deviation

of the analyst forecasts, the second is the absolute range of the analyst forecasts, and the third is

the absolute diﬀerence between the mean and the median of the analyst forecasts. The standard

deviation is deﬁned as:

DeviationS D =DSD =sPn

i=1(F Ci−¯

F C )2

n−1(11)

where F Ciis an individual analysts’ forecast. Intuitively, the standard deviation of the analyst

forecasts measures the level of disagreement between all of the analysts. A larger SD value is an

indication that there is increased uncertainty around a speciﬁc companies expected future cash ﬂows

and vice versa. This uncertainty, is precisely the information that we are trying to capture with

these measures of dispersion. The range, R, is deﬁned as the diﬀerence between the largest analyst

forecast and the lowest analyst forecast as follows:

DeviationR=DR=|Range(F C )|(12)

Intuitively, this represents the diﬀerence between the most optimistic and the most pessimistic

analysts. This measure should be viewed as an extreme measure of dispersion since it is calculated

14

using the two most extreme analyst forecasts. Finally, the mean minus median, M−M, is deﬁned

as the diﬀerence between the mean forecast and the median forecast as follows:

DeviationM−M=DM−M=|¯

F C −Median(F C )|(13)

Intuitively, the M-M dispersion variable can be thought of as the amount of skew in the distribution

of analyst forecasts. If the mean is larger than the median, then the distribution of analyst forecast

is positively skewed, and vice versa. The larger the diﬀerence in the M-M measure, the larger the

skew of the distribution. Hence, the DeviationM−Mvariable can be thought of as a summary of

the higher moments of the dispersion distribution.

Using these measures of dispersion and the idea that there should be information contained in

the amount of disagreement between analysts, we can now formulate our ﬁnal hypothesis.

Hypothesis 5. There is a positive relationship between expected returns and analyst forecast

dispersion.

The question posited by this paper is that if we assume that analyst dispersion is indeed a proxy

for uncertainty, then we would expect a positive relationship between the dispersion measures and

the option-forecast derived from the RT, as suggested by Merton (1987).

3 Methodology

The questions asked in this paper fall into one of three categories: 1) are analyst forecasts and

option-based forecasts equal to one another? 2) is the information content used to forecast the two

diﬀerent forecasts the same? and 3) what is the relationship between expected returns and analyst

disagreement? To answer these questions, I use option data to construct the forecast for the RT and

I use aggregated analyst data to construct the analyst forecast as outlined in the previous section.

15

These then allow us to test the hypothesis about whether or not the two forecasts are statistically

equivalent. From there, we will be able to determine whether or not the information in the analyst

and the option-based forecasts are the same. To test this, I conduct cross-sectional regressions.

The cross-sectional regressions will tell us what information is used at the time that the forecasts

are constructed. Finally, we want to determine the relationship between analyst dispersion and

expected returns. For this paper, we deﬁne return between time tand time t+ 1 as:

rt,t+1 =Pt+1 −Pt

Pt

(14)

where Ptand Pt+1 are the price of a stock at time tand t+ 1, respectively.

3.1 Expected Return Analysis

The ﬁrst hypothesis asks whether the analyst forecasts exhibit a positive skew. We will analyze

this using a simple descriptive statistics about the distributions skew. In order to test hypothesis 2

from the previous section, we conduct hypothesis tests. The hypothesis test answers the question

of whether or not returns calculated from analyst forecasts are equal to returns calculated from the

RT:

H0:Et[rt+1,RT ] = Et[rt+1,analyst ]

Ha:Et[rt+1,RT ]6=Et[rt+1,analyst ]

where Et[rt+1,RT ]is the expected return from the RT at time tand Et[rt+1,analyst]is the expected

return from the analyst forecast at time t. Hypothesis three compares the forecast residuals of the

16

option-based and the analyst-based forecasts, deﬁned mathematically as:

F Er rorRT,t+1 =Et[rt+1,RT ]−rt+1

F Er roranalyst,t+1 =Et[rt+1,analyst ]−rt+1

To compare these forecast residuals, we conduct the following regression:

F Er roranalyst,t+1 =α+βF E rrorRT ,t+1 +(15)

where F Er roranalyst and F Er rorRT are the analyst forecast errors and RT forecast errors, respec-

tively.

3.2 Cross-sectional regressions

To test hypothesis four, we conduct cross-sectional regressions in a way that has now become

ubiquitous in the ﬁnance literature. To explain the cross-sectional variation from the expected

returns, we regress the expected returns on various factors that are known to aﬀect the formulation

of expected returns (see, for example, Welch and Goyal (2007); Cochrane (2009); Fama and French

(2012)). For example, we might expect consumer conﬁdence to aﬀect our formulation of future

expected returns. If that information is truly used in formulating that speciﬁc expected return,

we would expect to see a statistically and economically signiﬁcant coeﬃcient associated with that

speciﬁc variable. This then allows us to compare what information is used in diﬀerent forecast

formulations. If, for example, an option-based forecast were to include variables that capture more

uncertainty (such as the VIX, the probability of a recession, or consumer sentiment, to name a few),

then we would conclude that the option-based forecasts reﬂects the expected uncertainty in the

17

market. We formulate the cross-sectional regression for the expected return of the RT as follows:

Et[rt+1,RT ] = α+

l

X

z=1

βzXz,t +F Ei+(16)

which is a multivariate cross-sectional linear regression where E[rt+1,RT ]is the expected return from

the RT, βzis the regression coeﬃcient for factor Xzwith a possibility of lfactors, and F Eiis a ﬁxed

eﬀect for ﬁrm i. Intuitively, this model represents the breakdown of information used to construct

the option-based forecast. The idea is to determine what, if any, information is being included to

construct the option-based forecast at time t. This is similar to breaking down the cross-section of

ﬁrm returns into various factors. The only diﬀerence here is that instead of using a cross-section of

returns to various factors, we are examining the factor composition of expected return models.

The cross-sectional regression for the analyst forecast is formulated in the same way with the

exception that the dependent variable is the expected return from the analyst rather than the RT

as follows:

Et[rt+1,analyst] = α+

l

X

z=1

βzXz,t +F Ei+(17)

which is a multivariate cross-sectional linear regression where Et[rt+1,analyst]is the expected return

from the analysts, βzis the regression coeﬃcient for the factor Xz.

3.3 Analyst forecast dispersion

As a corollary to the ﬁrst set of hypothesis in this paper, a natural question arises when we think

about the dispersion of analyst forecasts. The dispersion of analyst forecasts has been considered

to be a good proxy for the level of uncertainty for a speciﬁc ﬁrm’s stock at any point in time. The

idea is that when analysts’ forecasts are widely spread out, there is less certainty about the future

cash ﬂows of the ﬁrm and therefore more uncertainty. For this article, we deﬁned three measures of

18

dispersion: 1) the standard deviation, SD, of the analyst forecasts, 2) the range, R, of the analysts

forecast, and 3) the absolute diﬀerence between the mean and the median analyst forecast. Using

three measures of dispersion allows us to conclude more deﬁnitely (or more robustly) the relationship

between returns and uncertainty as previously discussed (Kothari et al., 2016). We proceed to test

hypothesis ﬁve using the following regression (Diether et al., 2002; Park, 2005):

DSD,t =α+ ΓiEt[rt+1,RT ] +

l

X

z=1

βzXz,t +F Ei+(18)

where DSD,t is the analyst forecast dispersion deﬁned in equation 11, Γiis the coeﬃcient of interest

on the expected return variable obtained from the RT, Et[rt+1,RT ], and Pl

z=1 βzXz,t represents

factors added to our regression as controls for robustness. For robustness, we also run the same

regressions but change our deﬁnition of the dispersion of analyst forecasts (see equations 12 and 13

for the dispersion deﬁnitions).

DR,t =α+ ΓiEt[rt+1,RT ] +

l

X

z=1

βzXz,t +F Ei+(19)

where DR,t is the dispersion range.

DM−M,t =α+ ΓiEt[rt+1,RT ] +

l

X

z=1

βzXz,t +F Ei+(20)

where DM−M,t is the dispersion measure obtained from the diﬀerence between the mean and the

median of analyst forecasts.

19

4 Data

The RT relies heavily on the estimation of state price densities (SPDs) which, in turn, rely on

interpolation techniques. As such, because of the limited amount of option data, the RT sometimes

suﬀers from unreliable results (Bakshi et al., 2018; Jensen et al., 2019; Audrino et al., 2019). In his

paper on the RT, Ross opted to use over-the-counter data to circumvent the interpolations issues

of SPDs. However, this data is private and is therefore not readily available. To resolve this issue,

I selected a sample that limits the amount of option price interpolation that is necessary, making

the results less dependent on SPD interpolation techniques which will improve the reliability of the

results presented in this paper. As such, sample selection was done using the following conditions:

ﬁrms needed to have a minimum of about 250,000 options during the period from January 1st

2010 to June 30th, 2019, ﬁrms needed to have options that expired in at a minimum of one year,

ﬁrms needed to have a minimum of six traded options based on time-to-maturity, ﬁrms needed to

have at least six options based on strike price (call and put), ﬁrms needed to have data during

the entire sample period, and ﬁrms needed to have analyst coverage for at least half of the time

period analyzed. Once all of these conditions were met, we were left with a sample of 58 ﬁrms. This

sample represents approximately 30 million option prices. A complete list of the ﬁrms included in the

analysis of this paper can be found in the appendix. Once we apply the RT, we have approximately

2.1 million probabilities which are used to construct the approximately 140,000 forecasts used in

this paper.

4.1 Option price data

The data needed to construct the forecasts used in this article come from the Wharton Research Data

Services. More speciﬁcally, the options data needed to apply the Recovery Theorem were obtained

from the OptionMetrics database. Using these option prices, we apply the SPD interpolation

20

technique outlined in Sanford (2019b) where we use a b-spline to interpolate the density based

on the strike price dimension and a linear interpolation with a hazard rate representing the ﬁrm’s

default probability8for the TTM dimension. Once we have the SPD, we take the second derivative

of the option prices with respect to the strike price (Breeden and Litzenberger, 1978) to get the

state prices needed to apply the RT. To convert the implied volatility to option prices, we used the

Black-Scholes model (Black and Scholes, 1973). The stock prices used as an input for the B-S model

were obtained from the CRSP dataset. The risk-free rate used as an input for the B-S model was

obtained from the Fama-French database.

4.2 Analyst forecast data

The analyst forecast data were obtained from the Institutional Brokers’ Estimate System (I/B/E/S)

database. The I/B/E/S database reports all of the data from analyst reports for a speciﬁc stock.

Using the individual forecasts from the analysts, I can then aggregate the information by looking

at the median and/or the mean forecast for all analysts. The mean and median forecasts from all

analysts’ therefore represents the analyst forecast and the disagreement among analysts’ represents

the dispersion measures previously described.

4.3 Firm level data

The return data for the ﬁrms analyzed in this paper were obtained from the CRSP database. In order

to determine what information about the ﬁrm, if any, was being used in the forecast models analyzed

in this paper, we gathered data related to the ﬁrm. This data was obtained from the Compustat

database. We used data such as the earnings per share, stock price, shares outstanding, dividend

yield, leverage, bond ratings, among others. All of this data is available from the Compustat

database at the quarterly level. Certain variables are not readily available and therefore need to

8The default probabilities were obtained from the Bloomberg terminal.

21

computed. Variables that need to be standardized for ﬁrm size are divided by total assets (Eberly

et al., 2008; İmrohoroğlu and Tüzel, 2014). Firm cash ﬂow is deﬁned as operating income after

depreciation (oiadpq) plus depreciation (dpq) divided by total assets (atq). Operating proﬁt is

computed as total revenue (revtq) plus research and development expenses (xrdq) net of cost of

goods sold (cogsq) and selling, general and administrative expenses (xsgaq) all divided by total

assets (atq). Inventory is calculated as inventory (invtq) divided by total assets (atq). Revenue

is calculated as revenue (revtq) divided by total assets (atq). Leverage is calculated as the total

long-term debt (dlttq) plus current liabilities (dlcq) divided by total assets. Capital investment is

calculated as capital expenditures (capxy) plus research and development expenses (xrdq) divided

by total assets. Finally, total investment is calculated as capital expenditures (capxy) plus research

and development expenses (xrdq) plus acquisitions (aqcy) divided by total assets. Firm-level data

was obtained quarterly.

4.4 Other data

The last portion of my statistical analysis examines which of almost 300 factors best explain the

analyst and option-based forecasts. I will only discuss major ones here. All of the macroeconomic

variables, such as economic growth, unemployment, and the NBER recession indicator, are publicly

available from the FRED database. The consumer sentiment index was obtained from the University

of Michigan’s Surveys of Consumers website. It is based on a consumer survey conducted by the

University of Michigan, and gives a pulse of the average consumer sentiment in the United States.

22

5 Results

5.1 Descriptive Statistics

We start the results by ﬁrst looking at the descriptive statistics for the returns and forecasts used in

this paper. Monthly return is the percent change in prices from time period tto t+ 1.RT expected

return is the expected return variable derived from the Recovery Theorem at time t.Analyst

expected return is the average of the analyst forecasts at time t.Analyst expected return median is

the median analyst forecast at time t.Analyst dispersion range,analyst dispersion sd, and analyst

dispersion M-M are, respectively: the diﬀerence between the most optimistic and least optimistic

analysts, the standard deviation of all analyst forecasts on a given day, and the diﬀerence between

the mean and median analyst forecasts on any given day.

Table 1: Summary statistics of main variables

N Mean SD 25th Percentile 75th Percentile

Monthly return 138,334 0.0125 0.129 -0.0334 0.0562

RT expected return 138,334 0.0217 0.0270 0.0044 0.0335

Analyst expected return 74,785 0.05768 0.08057 0.0115 0.0985

Analyst expected return median 74,785 0.05024 0.08585 0.00471 0.09836

Analyst dispersion range 74,785 0.06886 0.14003 0 0.10631

Analyst dispersion sd 74,785 0.02619 0.06590 0 0.03793

Analyst dispersion M-M 74,785 0.00744 0.02425 0 0.01136

Notes: Summary statistics for all main variables in our analysis. Monthly return is the percent change in

prices over the upcoming calendar month. RT expected return is the expected return calculated using the

Recovery Theorem. Analyst expected return is the average expected return from all analyst reports. Analyst

expected return median is the median of all analyst forecasts. Analyst dispersion range is the diﬀerence

between the highest and lowest analyst forecasts. Analyst dispersion sd is the standard deviation of all of the

analyst forecasts. Analyst dispersion M-M is the diﬀerence between the mean and median analyst forecasts.

The sample size for this paper is 138,334. This corresponds to the sample size of the returns

and the sample size of the RT. In total, we analyze 58 ﬁrms9between January 1st, 2010 and June

9The complete list of ﬁrms used in this paper is available in the appendix.

23

30th, 2019. The sample ends on June 30th, 2019 because that is when the OptionMetrics database

ends as of the date of this paper. All ﬁrms analyzed in this paper have a complete dataset so we

have data for all ﬁrms for the entire duration of the sample. The average monthly return for this

sample is about 1.25%. The average return forecasted from the RT is about 2.1%. The average

forecast return obtained from the analysts is about 5%. The analysts, on average, are much more

optimistic about the market than the RT. The volatility of the forecasts, however, is closer to the

market volatility for the analysts than for the RT. The RT has a standard deviation of forecasts of

about 2.6% compared to that of the analyst which is about 8%.

Table 2: Correlation of forecast variables

RT Forecast Mean Analyst Forecast Median Analyst Forecast

RT Forecast 1

Mean Analyst Forecast 0.0472 1

Median Analyst Forecast 0.044 0.9585 1

Notes: Pearson correlation between the RT forecast, mean analyst forecast, and median analyst forecast.

Table 2 shows the correlation matrix for the analyst forecasts and the RT. Already, there is an

indication that the forecasts are vastly diﬀerent. The correlation between the RT forecast and the

analyst forecast is approximately 0.05, indicating a very weak correlation between the two variables.

This weak correlation persists even after changing the leads and lags of the forecasts.

5.2 Forecast distributions

Hypothesis one asks whether or not analyst forecasts will have a negative skew as a result of their

incentive structure? To answer this question, we ﬁrst look at the distribution of forecasts (left panel

of ﬁgure 1). For comparison purposes, we also include the distribution of forecasts obtained from

the RT (right panel of ﬁgure 1).

24

0 5 10 15

Density

-1 -.5 0 .5 1

Mean Analyst Forecast

0 5 10 15 20 25

Density

-0.10 0.00 0.10 0.20 0.30

RT Forecast

Figure 1: The left panel is the histogram of analyst forecasts. The right panel is the histogram of

RT forecasts.

From the left panel in ﬁgure 1, we can clearly see that the majority of forecasts are positive. In

fact, we can conﬁrm that the distribution has a skewness equal to -10.7701. Intuitively, this is in line

with what we would expect: on average, analysts forecast small positive gains and they occasionally

forecast large negative gains. In this sample, there are 406 analyst forecasts that are less than zero

with an average forecast of about -18%. In contrast, there are 74,360 forecasts that are larger than

zero with an average forecast of about 6%. Again, smaller but more frequent positive forecasts

and larger but less frequent negative forecasts. In general, it seems to be the case that analysts

only forecast negative returns when very bad negative news becomes available. The RT forecast is

much more symmetric around zero (although positively skewed and slightly to the right of zero).

The skewness coeﬃcient for the RT forecast is 1.6346. Overall, the RT forecasted 19,010 negative

returns with an average of about -1% and 119,324 positive returns with an average of about 2.7%.

Again, the RT is much more symmetric around zero than the analyst forecasts. Comparing these

to the realized return, we ﬁnd that there were 61,611 negative returns with an average of about

-5.9% and 84,135 positive returns with an average of about 6.5%. Clearly, the analyst forecasts are

biased in comparison to realized return – on average, analysts forecast positive returns and only

25

very rarely forecast negative returns despite the fact that realized return is almost a 50-50 split

between positive and negative returns. Table 3 summarizes these ﬁndings.

Table 3: Distribution Summary Statistics

N Mean

Realized return positive 61,611 -0.05942

Realized return negative 84,135 0.06513

RT forecast positive 19,010 -0.00992

RT forecast negative 119,324 0.02672

Analyst forecast positive 406 -0.17686

Analyst forecast negative 74,360 0.05897

Notes: Summary statistics for returns broken down

into either positive or negative returns. The ﬁrst col-

umn reports the number of observations and the sec-

ond column reports the mean of the subsetted data.

5.3 Hypothesis Tests

Hypothesis two asks the following question: is the forecast obtained from the RT statistically

signiﬁcantly diﬀerent from the forecast obtained from analysts? To test this, we will employ a

simply two-sided hypotehsis test as follows:

H0:µRTi=µanalysti

Ha:µRTi6=µanalysti

(21)

where µRT is the expected return for the RTiand µanalystiis the expected return obtained from

the analyst forecast. The null hypothesis from equation 21 can be rejected at the 0.1% level of

signiﬁcance (p-value < 0.0001) with 74,785 daily observations. In order to be certain that the

above hypothesis test is not driven by certain outliers, we conduct percentile hypothesis test to

determine whether or not there is a statistically signiﬁcant diﬀerence between the two samples at

each percentile. Using a method outlined in Wilcox (2011), we are able to reject (p-value ≤0.05)

the null hypothesis that the two samples are the same regardless of which part of the distribution

26

we are looking at.

5.4 Forecast errors

Hypothesis three asks whether or not forecasts residuals from analysts and the RT are equal? To

answer this question, we conduct a cross-sectional regression of the forecast residuals. The results

are shown in table 5. We ﬁnd that the coeﬃcient is very close to one and that the R-square is 0.7568.

As such, it would seem as though, although not identical, the two processes are quite similar.

Table 4: Recovery Theorem cross-sectional regressions

(1)

VARIABLES Analyst Forecast Error

RT Forecast Error 0.971***

(0.0190)

Constant 0.0344***

(0.000223)

Observations 70,708

R-squared 0.7568

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: Cross-sectional regressions of the forecast residuals. In

this regression, the dependent variable is the analyst forecast error

and the independent variable is the RT forecast error. We deﬁne

forecast error as the diﬀerence between the forecast and the realized

return.

5.5 Cross-Sectional Factor Regressions

We now turn to decomposing the expected returns derived from analyst forecasts and those derived

from option prices via the RT. This will then allows us to provide an answer to hypothesis four which

asked whether or not the information set used by both option-based and analyst forecasts in the

same? To answer this question, we decompose the cross-sectional forecasts in order to understand

what information makes up the various forecasts. In the previous section, we concluded that, at

every quantile of the forecast distribution, the forecasts derived from analysts were statistically

27

signiﬁcantly diﬀerent from the forecasts obtained from the RT. The next question that we must

answer is what information actually makes up the forecasts and is that information diﬀerent between

the two forecasts of interest. For this section, we analyzed about 700 macroeconomic, ﬁrm-speciﬁc,

and market variables. Table 5 summarizes the important results for the RT and table 6 summarizes

the important results for the analysts.

28

Table 5: Recovery Theorem cross-sectional regressions

(1) (2) (3) (4) (5) (6) (7) (8)

VARIABLES RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast

Economic Policy Uncertainty 4.07e-06 3.82e-06 1.31e-05*** 1.72e-06 1.82e-06 5.68e-06***

(2.50e-06) (2.52e-06) (2.73e-06) (1.51e-06) (1.52e-06) (1.52e-06)

Eﬀective federal funds rate 0.000374 0.000361 8.06e-05 4.63e-05

(0.00337) (0.00336) (0.00179) (0.00179)

TED Spread -0.00331 -0.00327 -0.00221 -0.00218

(0.00375) (0.00375) (0.00176) (0.00177)

U.S. / Euro FX Rate -0.00437 -0.00468 -0.00125 -0.00126

(0.0159) (0.0159) (0.00679) (0.00678)

Trade Weighted U.S. Dollar Index 0.000140 0.000131 -0.000376*** 2.86e-05 2.49e-05 -0.000182***

(0.000549) (0.000549) (6.06e-05) (0.000238) (0.000237) (2.77e-05)

Chinese / U.S. FX Rate 0.00312 0.00320 0.00136 0.00139

(0.00365) (0.00365) (0.00164) (0.00163)

U.S. / Japan FX Rate -0.000162 -0.000162 -7.62e-05 -7.52e-05

(0.000119) (0.000119) (5.41e-05) (5.41e-05)

U.S. / Australia FX Rate 0.0249 0.0248 0.00846 0.00842

(0.0153) (0.0154) (0.00685) (0.00688)

CBOE S&P 500 3-Month Vol Index 0.000284 0.000259 0.000759** 9.99e-05 0.000111 0.000314**

(0.000294) (0.000296) (0.000310) (0.000144) (0.000146) (0.000148)

CBOE S&P 100 Vol Index 0.000354 0.000385* 0.000166* 0.000169*

(0.000213) (0.000219) (9.61e-05) (9.93e-05)

90-day AA commerical paper -0.000248 -0.000195 0.000560 0.000557

(0.00281) (0.00281) (0.00157) (0.00156)

1-month AA commerical paper -0.00171 -0.00177 -0.00139 -0.00135

(0.00261) (0.00261) (0.00185) (0.00186)

Yield 10 year zero coupon bond 0.00164 0.00164 0.000606 0.000611

(0.00108) (0.00108) (0.000453) (0.000454)

Russell 2000 Vol Index -0.000326* -0.000332* -0.000146* -0.000158*

(0.000185) (0.000187) (8.56e-05) (8.49e-05)

CBOE Vol Index -0.000406* -0.000167

(0.000222) (0.000108)

Volume 3.74e-11*** 1.60e-11***

(5.59e-12) (2.95e-12)

Market Excess Return -0.00779 0.0198** -0.00364 -0.00856 0.00397 -0.00757

(0.00748) (0.00946) (0.00905) (0.00868) (0.00974) (0.00947)

Small-minus-big 0.0217 -0.0188 0.0114 -0.00731 -0.0268** -0.0143

(0.0147) (0.0145) (0.0149) (0.0126) (0.0128) (0.0128)

High-minus-low 0.0102 -0.00639 0.00496 0.0187 0.0116 0.0101

(0.0152) (0.0137) (0.0155) (0.0148) (0.0166) (0.0151)

Momentum -0.00187 -0.00662 -0.00318 -0.00338 -0.00495 -0.00215

(0.00934) (0.0109) (0.0100) (0.00993) (0.0121) (0.0105)

Lag RT forecast 0.565*** 0.560*** 0.560*** 0.545***

(0.0168) (0.0172) (0.0172) (0.0169)

Constant 0.0214*** -0.0170 -0.0163 0.0431*** 0.00930*** -0.00210 -0.00205 0.0208***

(4.06e-06) (0.0565) (0.0566) (0.00602) (0.000358) (0.0249) (0.0251) (0.00280)

Firm Fixed Eﬀect YES YES YES YES YES YES YES YES

Observations 138,334 114,968 114,968 133,029 108,382 92,378 92,378 104,810

R-squared 0.050 0.092 0.092 0.089 0.354 0.379 0.379 0.361

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: For these cross-sectional regressions, the dependent variable is the RT forecast and the

independent variables are the various macro, ﬁrm-speciﬁc, and market factors included in this paper.

Standard errors are clustered at the ﬁrm-level. Data frequency for these regressions is daily. Model

speciﬁcations were selected using the RMSE criterion.

Table 5 shows the cross-sectional regressions with the RT forecast as the dependent variable and

the various factors analyzed as the independent variables. All of the regressions include clustered

standard errors at the ﬁrm-level. The variables chosen to be included in the regressions were the

ones that provided the lowest root-mean-squared deviation (RMSE). The ﬁrst four columns and the

29

last four columns are the same regressions with the exception that the last four columns include a

lag of the forecast. The results are fairly consistent whether or not we include the lagged forecast

variable in the regressions. Columns one and ﬁve are the baseline regressions which only include

Fama-French factors as baseline controls for size, value, momentum, and market risk exposures.

We ﬁnd that none of these four factors alone explain the RT forecast. In general, for models two,

three, six, and seven, the results are all fairly similar. We notice that the RT incorporates into its

forecast, information about various measures of volatility risk. In particular, model two indicates

that there is a negative relationship between the RT forecast and the Russell 2000 volatility index

(RVX). This may, at ﬁrst, seem counterintuitive since we should expect to see a positive relationship

between market volatility and expected return of individual stock since stockholders will want to be

compensated for increased market volatility by way of higher returns. However, the stocks included

in this paper are, generally speaking, large cap stocks while the RVX is a proxy for expected small

cap volatility. Given the propensity for investors to look for safe havens during uncertain times, it

should not be surprising that there would be a negative relationship between expected small cap

volatility and large cap expected return. As uncertainty for small cap stocks increases, investors ﬂee

these stocks and invest in “safer” large cap assets (Adrian et al., 2019). The increase in expected

return given higher expected uncertainty is more accurately depicted in model three where we have

a positive relationship between CBOE S&P 100 volatility index and the expected return from the

RT. Here, as one would expect, we observe that as volatility for large cap increases, so does the

expected return of our RT forecast. The fourth model includes another model that had a low RMSE

but also had a lower r-squared compared to the other models. Here, we see that, again, uncertainty

plays a central role in determining the expected return using the RT. The interesting part hers is

that the trade weighted index is negatively related to the RT forecast. This is likely an indicator

that ﬁrm speciﬁc variables are included in the forecast from options because the exchange rate

30

is likely to have a signiﬁcant adverse eﬀect on the bottom line of large caps stocks, like the ones

included in this analysis. Finally, volume is also included in the cross-sectional information that

explains returns. In other words, there is a strong relationship between the demand for a given

stock and its expected return. Next, we turn to our analysis of the cross-sectional regressions of the

mean analyst forecasts found in table 6.

Table 6: Analyst cross-sectional regressions

(1) (2) (3) (4) (5) (6) (7) (8)

VARIABLES Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast

Economic Policy Uncertainty 1.06e-05* 9.18e-06 3.29e-05*** 1.84e-06 1.06e-06 9.80e-06

(5.40e-06) (5.56e-06) (1.08e-05) (5.28e-06) (5.55e-06) (7.04e-06)

Eﬀective federal funds rate -0.0341*** -0.0304** -0.0316*** -0.0290***

(0.0114) (0.0123) (0.00779) (0.00949)

TED Spread -0.0448** -0.0400* -0.0428*** -0.0400**

(0.0203) (0.0220) (0.0157) (0.0175)

U.S. / Euro FX Rate -0.0797*** -0.0837*** -0.0313 -0.0332

(0.0283) (0.0273) (0.0263) (0.0248)

Trade Weighted U.S. Dollar Index -0.00266** -0.00316** 0.000260 -0.000871 -0.00110 0.000238

(0.00113) (0.00129) (0.000267) (0.000986) (0.00109) (0.000198)

Chinese / U.S. FX Rate 0.0344*** 0.0384*** 0.0189*** 0.0201***

(0.00613) (0.00829) (0.00535) (0.00739)

U.S. / Japan FX Rate -0.000310 -0.000239 -0.000460 -0.000444

(0.000435) (0.000466) (0.000344) (0.000356)

U.S. / Australia FX Rate -0.144*** -0.162*** -0.104*** -0.116***

(0.0188) (0.0241) (0.0205) (0.0260)

CBOE S&P 500 3-Month Vol Index 0.00286*** 0.00294*** 0.00530*** 0.00172*** 0.00191*** 0.00386***

(0.000587) (0.000614) (0.000628) (0.000430) (0.000468) (0.000447)

CBOE S&P 100 Vol Index -0.00109** -0.00112** -0.000760** -0.000994***

(0.000419) (0.000448) (0.000332) (0.000351)

90-day AA commerical paper 0.0172** 0.0105 0.0154** 0.0109

(0.00713) (0.00747) (0.00727) (0.00825)

1-month AA commerical paper 0.0130 0.0168* 0.0123* 0.0149**

(0.00868) (0.00853) (0.00679) (0.00734)

Yield 10 year zero-coupon bond 0.00585** 0.00451 0.00458*** 0.00398*

(0.00261) (0.00323) (0.00164) (0.00217)

Russell 2000 Vol Index -0.000809** -0.000905** -0.000414 -0.000405

(0.000344) (0.000350) (0.000262) (0.000262)

Market Excess Return -0.0940*** -0.0910*** -0.105*** -0.162*** -0.124*** -0.178***

(0.0230) (0.0226) (0.0207) (0.0401) (0.0379) (0.0351)

Small-minus-big 0.246*** 0.177** 0.247*** 0.280* 0.247* 0.238*

(0.0639) (0.0703) (0.0613) (0.141) (0.135) (0.130)

High-minus-low -0.0330 -0.0324 0.0174 -0.00704 -0.0191 -0.0220

(0.0526) (0.0518) (0.0510) (0.0470) (0.0481) (0.0451)

Momentum 0.0858** 0.0718* 0.108*** 0.0759 0.0346 0.114*

(0.0359) (0.0385) (0.0370) (0.0567) (0.0724) (0.0631)

CBOE Volatility Index -0.00329*** -0.00252***

(0.000499) (0.000311)

Volume 4.83e-11*** 2.76e-11***

(1.02e-11) (8.84e-12)

Lag analyst forecast 0.387*** 0.355*** 0.354*** 0.376***

(0.0509) (0.0272) (0.0276) (0.0441)

Constant 0.0571*** 0.283*** 0.315*** -0.0134 0.0398*** 0.152* 0.175** -0.0114

(1.18e-05) (0.0913) (0.0892) (0.0225) (0.00330) (0.0801) (0.0780) (0.0169)

Firm Fixed Eﬀect YES YES YES YES YES YES YES YES

Observations 70,720 62,416 59,600 69,468 39,964 36,482 34,706 39,476

R-squared 0.168 0.217 0.209 0.193 0.309 0.341 0.332 0.334

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: For these cross-sectional regressions, the dependent variable is the analyst forecast and the

independent variables are the various macro, ﬁrm-speciﬁc, and market factors included in this paper.

Standard errors are clustered at the ﬁrm-level. Data frequency for these regressions is daily. Model

speciﬁcations were selected using the RMSE criterion.

Table 6 shows the cross-sectional regressions with the mean analyst forecast as the dependent

variable and the various factors analyzed as the independent variables. All of the regressions include

31

clustered standard errors at the ﬁrm-level. The variables chosen to be included in the regressions

were the ones that provided the lowest RMSE. The ﬁrst four columns and the last four columns are

the same regressions with the exception that the last four columns include a lag of the mean analyst

forecast. The variables that make up the forecasts of analysts are related to 1) interest rates in the

macroeconomy, 2) exchange rates, 3) ﬁrm risk factors, 4) volatility, and 5) economic uncertainty.

In other words, analyst forecasts are constructed using a little of everything.

What can be concluded from the results presented in table 5 is that the forecast derived from

options by way of the RT, incorporate data related to expected volatility and information about the

uncertainty in the macroeconomy. In comparison, table 6 shows that analyst forecast incorporate

“ a little of everything” in their forecasts. The analyst forecasts also includes information about

uncertainty but also about ﬁrm risk characteristics, the macroeconomics indicators, and uncertainty

measures for the economy and the stock market. In other words, analyst forecasts are inﬂuenced

by a little bit of everything whereas option-based forecasts are more concerned with the onset of

volatility in ﬁnancial markets rather than uncertainty about the macroeconomy. One observation

that is worth noting, however, is that the analyst forecasts have an r-squared that is much larger

in the case of the ﬁrst four models than that of the RT forecast (about 20% versus 10%). As will

be seen in the subsequent cross-sectional regressions (tables 7 and 8), this is likely due to the fact

that the option-based forecasts are incorporating more ﬁrm-speciﬁc information than the analyst

forecasts. Since the data for tables 5 and 6 is based on the daily data and Compustat data is not

available daily, we are not able to include these variables at the daily time interval.

The appendix (table 12) also includes a cross-sectional regression where the dependent variable

is the median analyst forecast instead of the mean analyst forecast presented in table 6. These

regressions are included as a robustness check. The results are largely consistent both in terms of

statistical signiﬁcance and magnitude with the results presented in table 6.

32

Tables 7 and 8 present the results for the cross-sectional regressions at the quarterly interval.

Naturally, our sample sizes in this section are signiﬁcantly smaller. However, these regressions allow

us to include variables that are not available at the daily interval such as ﬁrm-speciﬁc accounting

data. Comparing the two sets of results, we note that, for the RT forecast, variables such as leverage,

the dividend-price ratio, total assets, and income are statistically and economically signiﬁcant while

for the analyst forecasts, for the most part, none of these variables seem to impact the forecast. As

such, it would seem to be the case that for the RT, ﬁrm-level information is used but for analyst

forecasts, only macro-level variables are used.

33

Table 7: Recovery Theorem (RT) cross-sectional regressions (quarterly)

(1) (2) (3) (4) (5) (6)

VARIABLES RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast RT Forecast

Leverage -0.0304** -0.0291** -0.0303** -0.0228 -0.0182 -0.0523**

(0.0134) (0.0137) (0.0137) (0.0139) (0.0166) (0.0226)

Capital Investment -0.109* -0.133** -0.116* -0.109* -0.0346 0.0544

(0.0618) (0.0637) (0.0641) (0.0638) (0.0768) (0.0947)

Total Investment 0.0389** 0.0397* 0.0351* 0.0374* 0.0141 -0.0181

(0.0194) (0.0203) (0.0203) (0.0202) (0.0276) (0.0325)

Dividend Price Ratio 0.790* 0.854* 0.861* 0.866* 1.457** 1.301*

(0.474) (0.485) (0.484) (0.481) (0.572) (0.692)

Total Assets -3.09e-07*** -2.87e-07*** -2.02e-07** -2.48e-07*** -4.51e-07*** -2.87e-07*

(7.40e-08) (7.64e-08) (8.85e-08) (8.98e-08) (1.22e-07) (1.68e-07)

Change in Investments 5.13e-07** 5.33e-07** 4.22e-07* 1.64e-07 5.13e-07

(2.53e-07) (2.52e-07) (2.54e-07) (2.96e-07) (3.44e-07)

Income -3.15e-06* -2.96e-06* -2.20e-06 -4.62e-06*

(1.67e-06) (1.66e-06) (2.05e-06) (2.46e-06)

Common Shates Outstanding 1.32e-05** 1.33e-05** 2.91e-06

(5.15e-06) (6.16e-06) (7.58e-06)

Market Excess Return 0.106 -0.0210

(0.165) (0.279)

Small-minus-big -0.0155 0.0281

(0.253) (0.478)

High-minus-low 0.259 0.256

(0.329) (0.804)

Momentum 0.325 0.352

(0.245) (0.546)

Economic Policy Uncertainty Index for United States 6.23e-05

(7.63e-05)

Eﬀective federal funds rate 0.0364

(0.0617)

TED Spread 0.0132

(0.0477)

U.S. / Euro Foreign Exchange Rate 0.172

(0.144)

Trade Weighted U.S. Dollar Index 0.00582

(0.00487)

Chinese / U.S. Foreign Exchange Rate -0.0142

(0.0394)

U.S. / Japan Foreign Exchange Rate -0.00156

(0.00128)

U.S. / Australia Foreign Exchange Rate -0.0137

(0.0903)

CBOE S&P 500 3-Month Volatility Index -0.000793

(0.00482)

CBOE S&P 100 Volatility Index 0.000537

(0.00419)

90-day AA commerical paper -0.0641

(0.0580)

1-month AA commerical paper 0.0223

(0.0957)

Yield 10 year zero-coupon bond 0.00968

(0.00916)

Russell 2000 Volatility Index -3.80e-05

(0.00196)

Constant 0.0387*** 0.0375*** 0.0375*** 0.0184** 0.0224** -0.407

(0.00474) (0.00505) (0.00504) (0.00898) (0.0105) (0.414)

Observations 547 502 502 502 380 278

R-squared 0.059 0.067 0.075 0.088 0.094 0.154

Number of statenum 31 29 29 29 29 29

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: For these cross-sectional regressions, the dependent variable is the RT forecast and the

independent variables are the various macro, ﬁrm-speciﬁc, and market factors included in this paper.

Standard errors are clustered at the ﬁrm-level. Data frequency for these regressions is quarterly. Model

speciﬁcations were selected using the RMSE criterion.

34

Table 8: Analyst cross-sectional regressions (quarterly)

(1) (2) (3) (4) (5) (6)

VARIABLES Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast

Leverage 0.0456 0.0339 0.0331 0.0467 -0.0185 -0.0575

(0.0524) (0.0536) (0.0537) (0.0557) (0.0778) (0.112)

Capital Investment 0.191 0.243 0.214 0.229 0.213 -0.642

(0.214) (0.222) (0.225) (0.226) (0.285) (0.473)

Total Investment -0.0360 -0.0714 -0.0668 -0.0614 -0.0495 0.0251

(0.0684) (0.0714) (0.0718) (0.0721) (0.103) (0.145)

Dividend Price Ratio -0.937 -1.545 -1.511 -1.576 -0.921 -1.538

(1.674) (1.706) (1.709) (1.711) (2.003) (2.756)

Total Assets -6.47e-08 -4.75e-08 -8.98e-08 -1.22e-07 -4.50e-07 1.10e-08

(2.23e-07) (2.29e-07) (2.36e-07) (2.39e-07) (4.29e-07) (6.13e-07)

Change in Investments -1.31e-06* -1.28e-06* -1.41e-06* -1.57e-06* -2.15e-06

(7.64e-07) (7.66e-07) (7.78e-07) (9.11e-07) (1.30e-06)

Income 2.60e-06 3.43e-06 -1.43e-05* -6.25e-06

(3.63e-06) (3.75e-06) (7.50e-06) (9.00e-06)

Common Shates Outstanding 1.24e-05 2.81e-05 1.26e-05

(1.37e-05) (1.74e-05) (2.52e-05)

Market Excess Return 0.922* -1.569

(0.538) (1.680)

Small-minus-big 0.494 1.365

(0.878) (2.623)

High-minus-low 0.131 -1.446

(1.182) (3.861)

Momentum -1.029 -5.400**

(0.814) (2.583)

Economic Policy Uncertainty Index for United States 0.000309

(0.000425)

Eﬀective federal funds rate 0.00492

(0.300)

TED Spread -0.0613

(0.222)

U.S. / Euro Foreign Exchange Rate 0.771

(0.651)

Trade Weighted U.S. Dollar Index 0.0548*

(0.0289)

Chinese / U.S. Foreign Exchange Rate -0.259

(0.239)

U.S. / Japan Foreign Exchange Rate -0.0174**

(0.00726)

U.S. / Australia Foreign Exchange Rate 0.609

(0.589)

CBOE S&P 500 3-Month Volatility Index -0.0309

(0.0234)

CBOE S&P 100 Volatility Index -0.000405

(0.0227)

90-day AA commerical paper -0.00861

(0.290)

1-month AA commerical paper -0.0409

(0.541)

Yield 10 year zero-coupon bond 0.0881

(0.0535)

Russell 2000 Volatility Index 0.0234**

(0.0111)

Constant 0.0606*** 0.0706*** 0.0694*** 0.0440 0.0671 -2.590

(0.0179) (0.0192) (0.0193) (0.0340) (0.0410) (2.176)

Observations 266 244 244 244 165 120

R-squared 0.008 0.024 0.026 0.030 0.110 0.458

Number of statenum 30 29 29 29 29 28

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: For these cross-sectional regressions, the dependent variable is the analyst forecast and the

independent variables are the various macro, ﬁrm-speciﬁc, and market factors included in this paper.

Standard errors are clustered at the ﬁrm-level. Data frequency for these regressions is quarterly. Model

speciﬁcations were selected using the RMSE criterion.

35

5.6 Dispersion

Hypothesis ﬁve asks whether or not there is a positive relationship between analyst disagreement

(analyst forecast dispersion) and expected returns. To serve as a baseline we ﬁrst present the results

for realized return instead of the expected return obtained from the RT. We do this because the

literature has already established that the relationship between the return and dispersion measures

is negative (Miller, 1977; Kothari et al., 2016; Diether et al., 2002).

Table 9: Analyst dispersion vs. return regressions

(1) (2) (3) (4) (5) (6)

VARIABLES Analyst Range Analyst SD Analyst Distribution Analyst Range Analyst SD Analyst Distribution

Realized Return -0.0116*** -0.00364** -0.00161** -0.0110*** -0.00341* -0.00152**

(0.00355) (0.00172) (0.000629) (0.00364) (0.00177) (0.000647)

Constant 0.0690*** 0.0262*** 0.00889*** 0.0697*** 0.0265*** 0.00900***

(0.000467) (0.000226) (8.28e-05) (0.000487) (0.000237) (8.67e-05)

F-F Controls NO NO NO YES YES YES

Firm Fixed Eﬀect YES YES YES YES YES YES

Observations 74,773 74,773 74,773 70,708 70,708 70,708

R-squared 0.176 0.131 0.101 0.174 0.129 0.099

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: The dispersion variables are deﬁned as (1) the range of analyst forecasts, (2) the standard deviation

(SD) of the analyst forecast, and (3) the distribution of the analyst forecasts proxied by the diﬀerence

between the mean and the median forecasts. The F-F controls are the standard Fama-French factors.

Standard errors are clustered at the ﬁrm-level. Data frequency for these regressions is daily.

Table 9 shows the results comparing the realized returns and analyst dispersion. The ﬁrst three

columns are linear regressions comparing the return to the three diﬀerent dispersion methods: 1)

the range of analyst forecasts, 2) the standard deviation (SD) of the analyst forecasts, and 3) the

distribution deﬁned as the diﬀerence between the mean and the median of the analyst forecasts.

Columns 4 through 6 are the same regressions but we added Fama–French controls. What we

ﬁnd is that, regardless of how we deﬁne analyst disagreement and regardless of whether or not we

include controls, there is a negative relationship between analyst forecast dispersion and returns.

This result (both in signiﬁcance and in magnitude) is in line with other studies that have examined

this relationship (Diether et al., 2002; Ang et al., 2006; Stambaugh et al., 2015). The magnitude of

the relationship ranges from about 0.3% to about 1%, depending on the measure of dispersion and

36

controls.

Next, and perhaps more interestingly, we examine the relationship between expected returns

obtained from the RT and the dispersion of analyst forecasts. We are interested in determining the

sign of this relationship so as to ascertain the validity of hypothesis 5: is the relationship between

expected returns and analyst forecast dispersion positive? In other words, is analyst dispersion a

proxy for uncertainty? The intuition here is that if analysts themselves, who are sophisticated con-

sumers of market information, do not agree on the valuation of a ﬁrm then this itself is information

about how uncertain future cash ﬂows for that speciﬁc ﬁrm are.

Table 10: Analyst dispersion vs. Recovery Theorem

(1) (2) (3) (4) (5) (6)

VARIABLES Analyst Range Analyst SD Analyst Distribution Analyst Range Analyst SD Analyst Distribution

RT Forecast 0.0683*** 0.0425*** 0.0127*** 0.0682*** 0.0424*** 0.0126***

(0.0190) (0.00925) (0.00339) (0.0190) (0.00925) (0.00339)

Constant 0.0680*** 0.0255*** 0.00869*** 0.0681*** 0.0255*** 0.00870***

(0.000648) (0.000315) (0.000115) (0.000649) (0.000315) (0.000115)

F-F Controls NO NO NO YES YES YES

Firm Fixed Eﬀect YES YES YES YES YES YES

Observations 70,720 70,720 70,720 70,720 70,720 70,720

R-squared 0.174 0.129 0.099 0.174 0.129 0.099

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: The dispersion variables are deﬁned as (1) the range of analyst forecasts, (2) the standard deviation

(SD) of the analyst forecast, and (3) the distribution of the analyst forecasts proxied by the diﬀerence between

the mean and the median forecasts. The F-F controls are the standard Fama-French factors. Standard errors

are clustered at the ﬁrm-level. Data frequency for these regressions is daily.

Based on the above, at least for an option-based forecast like the RT, there is a positive relation-

ship that is both statistically and economically signiﬁcant between expected returns and analyst

forecast dispersion. This relationship persists even after controlling for the F-F factors (columns

4 through 6). As such, analyst disagreement is priced positively in an option-based forecast. We

therefore conclude that analyst disagreement is a proxy for uncertainty that is positively priced by

a model like the RT.

37

6 Conclusion

Is there a beneﬁt to using a stock forecast obtained from the RT compared to a forecast created by a

ﬁnancial analyst? Do we need option-based forecasts when we already have perfectly good forecasts

from professional analysts? First, using simple two-sided hypothesis tests and using quantile re-

gressions, I determined that the option-based and analyst-based forecasts were statistically diﬀerent

from one another. Our ﬁndings indicate that the RT tends to forecast much larger movements in

the tail than analysts and that the distribution of analyst forecasts tend to be more optimistic and

very rarely “overly” negative.

But what information are these forecasts based on? Using cross-sectional regressions, I was able

to ascertain that the RT forecast more heavily weighs information related to expected uncertainty.

Analysts forecasts, on the other hand, weigh more heavily information related to macroeconomics

and the ﬁrm. Analysts use information that is known to aﬀect stock prices. The models used by

these analysts tend to be simple and use information that is easy to understand, easy to calculate,

and readily available. This results in predictions that are fairly intuitive and do not stray far from

where we would expect stock prices to go. In other words, the distribution of expected returns

obtained from analysts is far narrower and negatively skewed compared to that of other models.

Analysts have an inherit bias not to obtain a forecast that is far “out there,” so that they do not lose

their job. This was made even more apparent when looking at the distribution of forecasts from

analysts, which was negatively skewed and had very few negative forecasts. Ultimately, this result

conﬁrmed the idea that, on average, analysts tend to be biased towards positive returns.

The results from the RT are also in line with the idea that option market participants or traders

are likely those that are interested in mitigating their exposure to large movements in the market.

As such, measures such as probability of uncertainty indices/forecasts and changes in capital risk

make sense as drivers because they predict extreme events (such a collapse of the credit market

38

or of a currency). Ultimately, this paper shows that the forecast obtained from analysts is very

diﬀerent from the forecasts obtained from more complicated asset pricing models like the RT. We

conclude that, analyst forecasts and option-based forecasts should be viewed as complements rather

than substitutes since they weigh stock price information diﬀerently.

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A Appendix

Table 11 shows the tickers for the ﬁrms used in the analysis of this paper. In total, 58 ﬁrms were

selected at random.

Table 11: Tickers

ABC ABT ACN ADBE AKAM

ALGN ALXN AMD AMT AMZN

ATVI BA BAC BAX BLK

BMY C CCL CMCSA CRM

CSCO DAL DHI DISH DLTR

EBAY F FCX GILD GOOGL

GS HUM ILMN INTC JNJ

JPM M MCK MET MGM

MMM MO MRK MRO MSFT

MU NEM NSC PFE PM

SCHW SLB T TPR USB

V WFC WHR

Notes: List of tickers included in the analysis

of this paper.s

42

Table 12 shows the results using the same analysis as in table 6 with the exception that here, the

dependent variable used in the cross-sectional regression is the median analyst forecast. Given the

correlations between the mean and the median analyst forecast, it is not surprising that the results

are very similar. These results are included in the appendix as a robustness check.

Table 12: Median Analyst cross-sectional regressions

(1) (2) (3) (4) (5) (6) (7) (8)

VARIABLES Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast Analyst Forecast

Economic Policy Uncertainty 1.13e-05* 9.77e-06 3.39e-05*** 5.47e-06 4.26e-06 1.58e-05**

(5.94e-06) (6.16e-06) (1.06e-05) (6.43e-06) (6.74e-06) (7.75e-06)

Eﬀective federal funds rate -0.0406*** -0.0373*** -0.0463*** -0.0437***

(0.0115) (0.0123) (0.00822) (0.00988)

TED Spread -0.0519*** -0.0470** -0.0589*** -0.0557***

(0.0187) (0.0201) (0.0139) (0.0157)

U.S. / Euro FX Rate -0.0911*** -0.0955*** -0.0497 -0.0518*

(0.0289) (0.0282) (0.0313) (0.0302)

Trade Weighted U.S. Dollar Index -0.00285** -0.00333*** 0.000271 -0.00122 -0.00145 0.000273

(0.00111) (0.00123) (0.000258) (0.00119) (0.00128) (0.000230)

Chinese / U.S. FX Rate 0.0342*** 0.0380*** 0.0223*** 0.0236**

(0.00575) (0.00751) (0.00697) (0.00886)

U.S. / Japan FX Rate -0.000374 -0.000302 -0.000613 -0.000598

(0.000407) (0.000430) (0.000373) (0.000379)

U.S. / Australia FX Rate -0.158*** -0.176*** -0.135*** -0.147***

(0.0186) (0.0231) (0.0272) (0.0324)

CBOE S&P 500 3-Month Vol Index 0.00296*** 0.00305*** 0.00551*** 0.00196*** 0.00216*** 0.00467***

(0.000628) (0.000650) (0.000605) (0.000554) (0.000586) (0.000603)

CBOE S&P 100 Vol Index -0.00126*** -0.00131*** -0.00103*** -0.00129***

(0.000423) (0.000446) (0.000382) (0.000389)

90-day AA commerical paper 0.0179** 0.0117 0.0211*** 0.0164*

(0.00787) (0.00797) (0.00750) (0.00820)

1-month AA commerical paper 0.0183* 0.0220** 0.0207** 0.0235**

(0.00945) (0.00971) (0.00818) (0.00896)

Yield 10 year zero-coupon bond 0.00678*** 0.00558** 0.00606*** 0.00558***

(0.00228) (0.00278) (0.00164) (0.00208)

Russell 2000 Vol Index -0.000779** -0.000867** -0.000334 -0.000310

(0.000336) (0.000342) (0.000317) (0.000321)

Market Excess Return -0.0881*** -0.0921*** -0.103*** -0.179*** -0.130*** -0.195***

(0.0226) (0.0237) (0.0221) (0.0472) (0.0395) (0.0415)

Small-minus-big 0.243*** 0.178** 0.252*** 0.335** 0.298* 0.292**

(0.0623) (0.0743) (0.0631) (0.153) (0.151) (0.145)

High-minus-low -0.0149 -0.0153 0.0337 0.0194 0.00384 0.000555

(0.0594) (0.0635) (0.0594) (0.0610) (0.0673) (0.0589)

Momentum 0.125*** 0.112** 0.148*** 0.137* 0.0800 0.181**

(0.0458) (0.0516) (0.0483) (0.0752) (0.0912) (0.0843)

CBOE Volatility Index -0.00347*** -0.00303***

(0.000475) (0.000372)

Volume 0*** 0**

(0) (0)

Lag analyst forecast 0.257*** 0.218*** 0.215*** 0.245***

(0.0865) (0.0645) (0.0643) (0.0800)

Constant 0.0496*** 0.325*** 0.356*** -0.0229 0.0429*** 0.224** 0.246** -0.0189

(1.21e-05) (0.0917) (0.0900) (0.0223) (0.00493) (0.0967) (0.0955) (0.0199)

Observations 70,720 62,416 59,600 69,468 39,964 36,482 34,706 39,476

R-squared 0.139 0.185 0.177 0.160 0.214 0.242 0.233 0.233

Robust standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Notes: For these cross-sectional regressions, the dependent variable is the median analyst forecast and the

independent variables are the various macro, ﬁrm-speciﬁc, and market factors included in this paper. The

independent variables are deﬁned in tables 12 and 13. Standard errors are clustered at the ﬁrm-level.

Data frequency for these regressions is daily. Model speciﬁcations were selected using the BIC criterion.

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