Content uploaded by Anthony Sanford
Author content
All content in this area was uploaded by Anthony Sanford on Mar 31, 2021
Content may be subject to copyright.
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.
References
Adrian, T., Crump, R. K., and Vogt, E. (2019). Nonlinearity and flight-to-safety in the risk-return trade-off
for stocks and bonds. The Journal of Finance, 74(4):1931–1973.
Ang, A., Hodrick, R. J., Xing, Y., and Zhang, X. (2006). The cross-section of volatility and expected returns.
The Journal of Finance, 61(1):259–299.
Audrino, F., Huitema, R., and Ludwig, M. (2019). An empirical implementation of the ross recovery theorem
as a prediction device. Journal of Financial Econometrics.
Backus, D., Chernov, M., and Martin, I. (2011). Disasters implied by equity index options. The journal of
finance, 66(6):1969–2012.
Bakshi, G., Chabi-Yo, F., and Gao, X. (2018). A recovery that we can trust? deducing and testing the
restrictions of the recovery theorem. The Review of Financial Studies, 31(2):532–555.
Barro, R. J. and Liao, G. Y. (2021). Rare disaster probability and options pricing. Journal of Financial
Economics, 139(3):750–769.
Barron, O. E., Stanford, M. H., and Yu, Y. (2009). Further evidence on the relation between analysts’
forecast dispersion and stock returns. Contemporary Accounting Research, 26(2):329–357.
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political
Economy, pages 637–654.
Blanchard, O. J. and Watson, M. W. (1982). Bubbles, rational expectations and financial markets. NBER
Working Paper No. 945.
39
Borovička, J., Hansen, L. P., and Scheinkman, J. A. (2016). Misspecified recovery. Journal of Finance.
Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices.
Journal of Business, pages 621–651.
Brown, L. D., Hagerman, R. L., Griffin, P. A., and Zmijewski, M. E. (1987). Security analyst superiority
relative to univariate time-series models in forecasting quarterly earnings. Journal of Accounting and
Economics, 9(1):61–87.
Brown, L. D. and Rozeff, M. S. (1978). The superiority of analyst forecasts as measures of expectations:
Evidence from earnings. The Journal of Finance, 33(1):1–16.
Bryan, D. M. and Tiras, S. L. (2007). The influence of forecast dispersion on the incremental explanatory
power of earnings, book value, and analyst forecasts on market prices. The Accounting Review, 82(3):651–
677.
Clement, M. B. (1999). Analyst forecast accuracy: Do ability, resources, and portfolio complexity matter?
Journal of Accounting and Economics, 27(3):285–303.
Clement, M. B. and Tse, S. Y. (2005). Financial analyst characteristics and herding behavior in forecasting.
The Journal of finance, 60(1):307–341.
Cochrane, J. H. (2009). Asset Pricing:(Revised Edition). Princeton university press.
Diether, K. B., Malloy, C. J., and Scherbina, A. (2002). Differences of opinion and the cross section of stock
returns. The Journal of Finance, 57(5):2113–2141.
Eberly, J., Rebelo, S., and Vincent, N. (2008). Investment and value: A neoclassical benchmark. Technical
report, National Bureau of Economic Research.
Fama, E. F. and French, K. R. (2012). Size, value, and momentum in international stock returns. Journal
of financial economics, 105(3):457–472.
Givoly, D., Hayn, C., and Lehavy, R. (2009). The quality of analysts’ cash flow forecasts. The Accounting
Review, 84(6):1877–1911.
40
Gu, Z. and Wu, J. S. (2003). Earnings skewness and analyst forecast bias. Journal of Accounting and
Economics, 35(1):5–29.
İmrohoroğlu, A. and Tüzel, Ş. (2014). Firm-level productivity, risk, and return. Management Science,
60(8):2073–2090.
Jensen, C. S., Lando, D., and Pedersen, L. H. (2019). Generalized recovery. Journal of Financial Economics,
133(1):154–174.
Kothari, S. P., So, E., and Verdi, R. (2016). Analysts’ forecasts and asset pricing: A survey. Annual Review
of Financial Economics, 8:197–219.
Lim, T. (2001). Rationality and analysts’ forecast bias. The Journal of Finance, 56(1):369–385.
Merton, R. C. (1987). A simple model of capital market equilibrium with incomplete information. The
journal of finance, 42(3):483–510.
Meyer, C. D. (2000). Matrix analysis and applied linear algebra, volume 2. SIAM: Society for Industrial and
Applied Mathematics, Philadelphia, PA.
Mikhail, M. B., Walther, B. R., and Willis, R. H. (1999). Does forecast accuracy matter to security analysts?
The Accounting Review, 74(2):185–200.
Miller, E. M. (1977). Risk, uncertainty, and divergence of opinion. The Journal of finance, 32(4):1151–1168.
Muth, J. F. (1961). Rational expectations and the theory of price movements. Econometrica: Journal of the
Econometric Society, pages 315–335.
Park, C. (2005). Stock return predictability and the dispersion in earnings forecasts. The Journal of Business,
78(6):2351–2376.
Ramnath, S., Rock, S., and Shane, P. (2008). The financial analyst forecasting literature: A taxonomy with
suggestions for further research. International Journal of Forecasting, 24(1):34–75.
Ross, S. (2015). The recovery theorem. The Journal of Finance, 70(2):615–648.
41
Sanford, A. (2019a). Recovery theorem with a multivariate markov chain. Working paper.
Sanford, A. (2019b). State price density estimation with an application to the recovery theorem. Working
Paper.
Seo, S. B. and Wachter, J. A. (2019). Option prices in a model with stochastic disaster risk. Management
Science, 65(8):3449–3469.
Stambaugh, R. F., Yu, J., and Yuan, Y. (2015). Arbitrage asymmetry and the idiosyncratic volatility puzzle.
The Journal of Finance, 70(5):1903–1948.
Van Appel, V. and Maré, E. (2018). The ross recovery theorem with a regularised multivariate markov chain.
ORiON, 34(2):133–155.
Welch, I. and Goyal, A. (2007). A comprehensive look at the empirical performance of equity premium
prediction. The Review of Financial Studies, 21(4):1455–1508.
Wilcox, R. R. (2011). Introduction to robust estimation and hypothesis testing. Academic press.
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.
43
