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Information Content of Option Prices: Comparing
Analyst Forecasts to Option-Based Forecasts
Anthony Sanford∗
February 8, 2024
Abstract
The asset pricing literature has been producing increasingly complex and computationally
intensive models of stock returns. Separately, professional analysts’ forecast stock returns. Are
the sophisticated methods found in the asset pricing literature achieving different forecasts
to those of analysts’? Do the two forecasts’ even capture the same information? In this
paper, I hypothesize that analyst forecasts and forecasts constructed using option prices will
be different because they place different weights on available information. Using hypothesis
tests and quantile regressions, I find that option-based forecasts are statistically significantly
different from analyst forecasts at every level of the forecast distribution. Using cross-sectional
regressions, I find that the difference originates in the weighting structure of the information
sets used to create the forecasts: option-based forecasts incorporate information about the
probability of extreme events more heavily while analyst forecasts focus on information about
firm and macroeconomic fundamentals.
Keywords: Recovery theorem, analyst forecasts, forecasting, derivatives.
JEL Code:G12, G17, C53
∗HEC Montréal, Department of Finance. Assistant Professor. Address: 3000 chemin de la Côte-Ste-
Catherine, Montréal, QC H3T 2A7. Email: anthony.sanford@hec.ca. Tel: (1) 514 340-6421
1
1 Introduction
The asset pricing literature is marked by the development of increasingly complex and com-
putationally intensive models for predicting stock returns. Simultaneously, professional ana-
lysts continue to provide forecasts based on their assessments of firms’ future performances.
This raises a critical question: do the advanced models found in the literature, such as the
Recovery Theorem (RT) (Ross, 2015), offer forecasts that significantly differ from or add
value to those provided by analysts? This paper hypothesizes that the differences between
analyst forecasts and forecasts constructed using option prices through the RT arise from
their reliance on distinct information sets and objectives.
The analysis is structured into two main phases. Initially, the study seeks to establish
whether there is a statistical difference in the forecasts produced by analysts and those gen-
erated through the RT. By applying hypothesis testing and quantile regression techniques,
we show that option-based forecasts show statistically significant differences from analyst
forecasts at every quantile of the expected return distribution. This finding is significant
as it suggests divergences not only at the median but also, and especially, at the tails of
the distribution, indicating variations in the treatment of extreme market conditions. Fol-
lowing the demonstration of these differences, the paper explores whether these divergences
stem from the reliance on distinct sets of information by analysts and the RT in formulat-
ing their forecasts. An examination through cross-sectional regressions of a broad array of
variables known to influence market returns—including but not limited to macroeconomic
factors, stock market factors, and probability factors—reveals the underlying reasons for the
differences in forecasts. It is found that while analyst forecasts are largely predicated on
2
firm-specific and macroeconomic fundamentals, option-based forecasts incorporate a wider
array of information, with a particular emphasis on the probability of extreme market events.
This exploration leads to the consideration of the practical implications of employing
sophisticated models like the RT. If these models are not substantively enhancing forecasts
beyond the insights already available through analyst forecasts, their usefulness might be
questioned. However, this study underscores the unique contribution of the RT and similar
models in capturing aspects of market dynamics, such as tail risks, that are not typically
addressed in analyst forecasts. Our findings suggest that there may be valuable synergies in
integrating insights from both analyst and option-based forecasts into a comprehensive fore-
casting framework. Such an approach could potentially offer a more nuanced and complete
picture of expected stock returns, leveraging the strengths of both types of forecasts.
What might account for the noted differences between analyst-based and option-based
forecasts? A key factor lies in the contrasting incentives of professional analysts and option
traders. Typically, analysts exhibit a more optimistic yet conservative stance regarding future
market prospects compared to option market participants, who primarily view options as
hedging tools. Analysts’ forecasts often reflect minor deviations from current market trends,
a reflection of the high stakes involved, including potential job loss for markedly divergent
predictions. Given that analysts’ accuracy directly impacts their employability, there is a
natural tendency towards caution.
Conversely, option traders leverage forecasts to mitigate risks of future market downturns,
thereby focusing on potential extreme movements rather than incremental changes. This
fundamental difference in focus and incentive structures between the two groups results
in distinct forecast characteristics: option-based forecasts inherently account for potential
3
market extremes (see, for example, Bakshi et al. (2023)), while analyst forecasts typically
center on more immediate, less volatile trends.
This paper also situates its contribution within the broader debate on forecast mod-
els’ efficacy, particularly in comparison to analyst forecasts. Previous studies (Brown and
Rozeff, 1978; Brown et al., 1987; Clement, 1999) have often highlighted analysts’ ability to
integrate extensive information into their forecasts, suggesting their superiority over time-
series models. However, the RT introduces a novel dimension to this comparison, offering an
opportunity to evaluate how option-based forecasts using the RT compare with traditional
analyst predictions in terms of information utilization and forecasting accuracy.
Moreover, this study enriches the discourse surrounding the RT, a model with mixed
reviews regarding its predictive capabilities. While some have found it to offer valuable
forecasting insights (Jensen et al., 2018; Audrino et al., 2021; Bakshi et al., 2017; Sanford,
2021; Sanford and Yang, 2022; Sanford, 2022), others remain skeptical of its ability to de-
liver on its theoretical promises (Borovička et al., 2016; Dillschneider and Maurer, 2019;
Jackwerth and Menner, 2020). By examining the unique information content captured by
the RT in comparison to analyst forecasts, this paper contributes a fresh perspective to the
debate, ultimately finding distinct differences in the information utilized by each forecasting
approach.
Lastly, this analysis extends the literature on what drives analyst forecasts by not only as-
sessing their forecasting prowess but also exploring the specific motivations and information
sets that underpin their predictions. Unlike previous research focused on analysts’ character-
istics and motivations (Clement and Tse, 2005; Givoly et al., 2009; Bryan and Tiras, 2007),
this study delves into the actual informational content of forecasts, offering insights into the
4
underlying reasons for the observed differences in forecast models.
2 Models
Market participants in the options market typically seek to hedge against potential future
adverse scenarios, using options as instruments to mitigate risks associated with uncertain-
ties like foreign exchange fluctuations (Nance et al., 1993; Bodnar et al., 1998; Graham and
Smith, 1999). In contrast, financial analysts aim to synthesize a broad spectrum of infor-
mation to forecast a firm’s future earnings trajectory. Their primary objective is to provide
forecasts that are sufficiently accurate to maintain their employment and credibility, as their
professional stability depends on the reliability of their reports (Ramnath et al., 2008; Mozes,
2003; Groysberg et al., 2011; Mikhail et al., 1999; Hong et al., 2000).
This fundamental divergence in motivations suggests that forecasts derived from options
market activities (via the RT) and those produced by analysts are based on different infor-
mation sets, or different weighting schemes on the information sets. While option market
participants focus on hedging against downside risks, analysts strive for forecasts that bal-
ance accuracy with conservatism to avoid jeopardizing their careers. The analysis of forecasts
hinges on the concept of the stochastic discount factor (SDF), which discounts future cash
flows to their present value, incorporating the expected risk and the investor’s risk aversion.
Both analysts and option-based forecasts aim to estimate this factor, yet they diverge in the
specifics of the information set used and the representative investor’s risk aversion profile.
5
The expected future price of an asset, reflective of the SDF, is given by:
pt=Et( ˜mt+1xt+1 |It)(1)
where ptis the price at time t,˜mt+1 the SDF, xt+1 the future cash flow, and Itthe information
set available at time t. This equation highlights the critical role of the information set in
shaping forecasts, indicating that the divergence in forecasting methods may stem from either
differing information sets or the weight assigned to the same information.
The subsequent sections delve deeper into how these forecasts are derived, examining the
specific methodologies employed by both the RT and financial analysts. This exploration
sheds light on the theoretical and practical aspects of each approach, offering insights into
the nuances that drive the divergence in their forecasts.
2.1 Recovery Theorem (RT)
The Recovery Theorem (RT) by Ross (2015) introduces a method for extracting the natural
probability distribution of returns from options markets, allowing us to disentangle state
prices into their constituent elements: the discount rate, the pricing kernel, and the natural
probability distribution. Central to this methodology is the concept of state prices, repre-
sented as the second derivative of option prices with respect to the strike price. These prices
standardize future payouts by reducing the uncertainty surrounding the asset’s future price.
State prices for transitions between states iand jat time tare mathematically defined as:
st
i,j =δu′(ct+1,j )
u′(ct,i)ft,t+1
i,j (2)
6
where δis the discount rate, u′() denotes the marginal utility, and fis the natural probability
measure. Our goal is to obtain the natural probability distribution ffor the S&P 500, which
necessitates the empirical determination of each component in the equation. Recall that the
derivation of state prices is defined as the second derivative of option prices with respect to
the strike price as follows:
s(K, T ) = ∂2Call(K, T )
∂K 2(3)
This equation, based on the work of Breeden and Litzenberger (1978), empirically estimates
state prices from call option prices and the strike price K. Next, we turn to contingent state
prices, which generalizes the concept of state prices to encompass transitions between any
two states and thus eliminates the need for a time subscript. This generalization is crucial
for solving the RT system of equations:
st+1 =stP, t = 1, ..., m −1P≥0(4)
Here, Prepresents the matrix of contingent state prices, enabling the model to account for
a broad spectrum of state transitions. Further development in the RT’s application involves
isolating the natural probability distribution, f, through manipulation of the previously
established equations:
fi,j =1
δpi,j
u′(ci)
u′(cj)
This step hinges on the contingent state prices derived from equation 4, leading to a reeval-
uation of the relationship between marginal utilities and the pricing kernel. Separating the
7
marginal utilities (and re-arranging) gives us:
pi,j
1
u′(cj)=δ1
u′(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 it as:
P⇀
zi=δ⇀
zjF
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
(5)
The application of the RT inherently involves transitioning from the risk-neutral probability
measure, or Q-measure (the Pprices in equation 4), to the physical probability measure,
8
known as the P-measure (the natural probability distribution, f, in equation 5). This tran-
sition is crucial for extracting the natural probability distribution from options markets.
The RT facilitates this by adjusting the Q-measure, which abstracts risk preferences and
market premiums, to reflect real-world probabilities that incorporate risk aversion and time
preferences.
This adjustment process is central to the RT’s methodology, effectively rescaling risk-
neutral valuations to align with the natural probability distribution under the P-measure.
This ensures that the derived forecasts and analyses account for the actual market dynamics
and investor behaviors, bridging the gap between theoretical risk neutrality and empirical
market expectations.
2.2 Analyst Forecast
Contrary to the straightforward goal of maximizing forecast accuracy, financial analysts nav-
igate a more nuanced landscape, aiming to produce forecasts that stay within an acceptable
margin of error. This balancing act is driven by the dual objectives of maintaining credibility
and ensuring job security, as excessively deviating forecasts can jeopardize their positions
(Mikhail et al., 1999; Lim, 2001; Gu and Wu, 2003; Ramnath et al., 2008). Analysts, there-
fore, operate within a framework where accuracy is weighed against the risk of significant
error, leading them to prefer forecasts that are conservative yet sufficiently informed.
The literature reveals that analysts’ compensation and job stability are not directly tied
to the precision of their forecasts but rather to their ability to avoid poor performance
evaluations. This dynamic incentivizes a forecasting approach that is cautious, utilizing
9
information in a way that mitigates the risk of job loss without necessarily striving for perfect
accuracy (Mikhail et al., 1999; Lim, 2001; Gu and Wu, 2003; Ramnath et al., 2008). The core
of the analysts’ forecasting dilemma is captured by the following performance evaluation:
χ(P|It) =
n
X
t=1
Et[Pt+1|It]−Pr
t+1
n(6)
Here, Etrepresents the expectation based on the information set Itavailable at time t,
with Pt+1 and Pr
t+1 denoting the forecasted and realized prices, respectively. This equation
reflects the analysts’ effort to minimize the average difference between their forecasts and
actual outcomes, leveraging their assessment of future cash flows (˜xt+1) and the stochastic
discount factor ( ˜mt+1) within their information constraints.
To model the analysts’ strategic decision-making, we consider their ultimate aim: to
optimize their forecasting strategy to minimize the risk of termination while aligning with
their professional incentives:
max
It
V(χ(P|It)) (7)
The value function V(χ(P|It)) is thus defined as a binary outcome reflecting the analysts’
employment status based on their forecasting accuracy relative to a threshold αt:
V(χ(P|It)) =
1,if χ(P|It)≤αt
0,if χ(P|It)> αt
(8)
This formulation illustrates the analysts’ predicament in selecting information sets that en-
able them to stay within the safety margin of forecasting accuracy, thereby securing their
10
employment. The underlying assumption of rational markets, where all information is fully
used in asset pricing (Muth, 1961; Blanchard and Watson, 1982), contrasts with the reality
observed in analysts’ forecasts. Despite access to similar information as option market par-
ticipants, analysts’ unique incentives and risk aversion lead to different forecasting outcomes.
This divergence is rooted in their strategic approach to information utilization, highlighting
the complexity of financial forecasting where identical information can yield varied predic-
tions due to different prioritizations of accuracy and risk.
3 Methodology
Recall that the our goal is to address two main hypothesis: 1) whether forecasts derived
from analysts and those based on option pricing (option-based forecasts) are statistically
equivalent, and 2) whether the underlying information content guiding these two forecasting
approaches is identical. To tackle these questions, our analysis proceeds in three method-
ologically distinct steps. Initially, we make use of two-sided hypothesis tests to evaluate the
equivalence of analyst-based and option-based forecasts. This step is crucial for establishing
a baseline comparison between the two forecasting methods.
Subsequently, we employ cross-sectional quantile regressions to investigate the differential
behavior of these forecasts across the distribution spectrum. Specifically, we aim to discern
if the forecasts diverge significantly at the distribution tails compared to the median. This
approach allows us to determine whether the informational basis for the forecasts remains
consistent across varying market conditions, particularly focusing on the extremes versus the
median of the distribution. The quantile regressions facilitate a detailed comparison, first
11
between each forecast type and the realized returns, and subsequently, directly between the
two forecasting methodologies themselves.
Finally, to pinpoint the specific information used in generating these forecasts, we con-
duct cross-sectional regression analyses. This phase involves an extensive examination of
nearly three hundred factors recognized for their influence on market returns, categorized into
macroeconomic factors, stock market factors, and probability factors. Macroeconomic factors
encompass variables derived from broader economic data, such as consumption growth, in-
flation, and unemployment rates. Stock market factors pertain to variables directly affecting
stock performance, including book-to-market ratios and dividend-price ratios. Probability
factors, on the other hand, incorporate variables indicative of the likelihood of significant
market events, such as market crash probabilities, recession probabilities, and sentiment in-
dices. This detailed investigation seeks to clarify the distinct informational components that
underpin the construction of the two types of forecasts. The analysis within this section is
based on linear cross-sectional regressions. Each of these methodological steps are detailed
in the subsequent subsections.
3.1 Hypothesis tests
A pivotal question this paper seeks to address is the comparative value of different fore-
casting methods: specifically, the value of sophisticated models like the RT, which derives
stochastic discount factors from options markets, against traditional forecasts provided by
financial analysts. This inquiry is not merely academic; it has practical implications for
investment strategies, risk management, and market analysis. The core of this investigation
12
lies in understanding whether forecasts based on advanced financial models offer a distinct
advantage, in terms of predictive accuracy and information content, over those generated by
human experts analyzing market fundamentals.
To systematically examine this question, we initiate our analysis with hypothesis testing,
aiming to statistically evaluate the equality of expected returns as forecasted by the RT and
financial analysts. The formulation of the hypothesis test is as follows:
H0:µUV RT =µanalyst
Ha:µUV RT =µanalyst
(9)
Here, µUV RT represents the mean forecasted return derived from the RT, while µanalyst de-
notes the mean forecasted return from financial analysts. The null hypothesis (H0) posits
that there is no significant difference in the expected returns as predicted by the two methods,
suggesting that the sophisticated mathematical underpinnings of the RT do not necessar-
ily translate into different forecast performance relative to analyst predictions. Conversely,
the alternative hypothesis (Ha) asserts that the expected returns differ, implying that the
methodologies lead to fundamentally different forecasts.
This test is critical for several reasons. First, it quantitatively assesses the divergence
or convergence in forecasts from theoretical models versus human judgment. Secondly, it
sets the stage for further analysis into the nature of these differences, should they exist,
by exploring the distributional characteristics and the informational underpinnings of each
forecasting method. Understanding whether the advanced computational techniques embed-
ded in the RT provide additional value is crucial for both academic research and practical
13
financial decision-making.
3.2 Quantile regressions
Quantile regressions, as developed by Koenker and Hallock (2001) and further explored by
Yu et al. (2003) and Koenker and Xiao (2002), offer a robust methodological framework
for examining differences across the entire distribution of forecasted returns, not just at the
mean or median. This approach is particularly valuable in our analysis for identifying where,
within the distribution, the forecasts from the RT and financial analysts diverge.
The application of quantile regressions allows us to assess the conditional distribution
of the forecasted returns given a set of explanatory variables. Unlike traditional regression
models that only estimate the mean of the dependent variable, quantile regressions enable
us to estimate any quantile, providing a more comprehensive view of the distribution. This
is crucial for our analysis as it helps to uncover whether the differences between RT and
analyst forecasts are more pronounced in the tails of the distribution — potentially indicating
different risk assessments or information processing between the two forecasting approaches.
To implement the quantile regressions, we follow the specifications laid out by Koenker
and Hallock (2001), adapting the model to our specific context of comparing forecast dis-
tributions. The quantile regression model is specified as follows for each quantile τ(where
0< τ < 1):
Qτ(Y|X) = Xβτ(10)
Here, Qτ(Y|X)represents the τ-th quantile of the dependent variable Y(either RT or analyst
forecasts, or realized returns) conditioned on the independent variables X, which in our case,
14
consists of the other forecasts or the market returns.
By estimating this model across different quantiles, we can systematically compare the
behavior of RT-derived forecasts and analyst forecasts across the distribution, particularly
focusing on the tails. This enables us to not only determine if there are significant differences
between the two types of forecasts but also to understand the distributional characteristics of
these differences. For instance, finding that discrepancies are more significant in the upper or
lower tails could suggest that the RT model and analysts weigh extreme outcomes differently,
reflecting distinct risk considerations or information assimilation processes.
The insights gained from quantile regressions are instrumental in deepening our under-
standing of the comparative dynamics between theoretical model-based forecasts and expert
judgment-based forecasts. By delineating where along the distribution these forecasts di-
verge, we can better assess the implications for risk management, portfolio optimization,
and strategic decision-making in financial contexts.
3.3 Cross-sectional regressions
To decompose the underlying information used in constructing forecasts, this study employs
cross-sectional regressions. This approach is pivotal for unraveling the specific factors that
potentially inform the different forecasting methodologies under analyzed—those based on
analyst insights and those based off of sophisticated asset pricing models like those of the
RT. The core of our regression analysis is encapsulated in the following specification:
Et[rt+1] = α+
n
X
i=1
βixi,t +ϵ(11)
15
Here, Et[rt+1]represents the expected return as forecasted for the subsequent period, serving
as our dependent variable. The independent variables, xi,t, encompass a variety of factors
at time tthat are posited to influence these forecasts. Each factor iis weighted by its
corresponding coefficient βi, with αdenoting the intercept, and ϵcapturing the regression’s
error term.
The selection of factors (xi,t) is informed by a comprehensive review of the literature on
factor models, drawing from seminal works by Fama and French (1993), Fama and French
(1996), Fama and French (2012), Welch and Goyal (2007), and Martin (2017). These factors
are not only recognized for their impact on stock prices but are also acknowledged for their
predictive power concerning macroeconomic conditions (Estrella and Mishkin, 1998; Chau-
vet and Piger, 2008; Manela and Moreira, 2017). Such an inclusive approach ensures that
our analysis covers a broad spectrum of influences, acknowledging the intricate relationship
between individual stock performances and broader economic indicators.
Given the interconnectivity of broad market indexes like the S&P 500 with the overarching
U.S. economy (Cochrane, 2009), incorporating macroeconomic factors into our regressions is
not just methodological rigor; it is a necessity. This inclusion allows us to gauge the extent
to which macroeconomic dynamics versus market-specific information drive the forecasting
processes of analysts and RT-based models.
By analyzing the coefficients βi, this study aims to highlight which factors significantly
contribute to the forecasting models, thereby shedding light on the informational content
underpinning each method. Are analysts and RT models leveraging common information,
or does each approach draw on distinct data sources or interpret shared data differently?
Answering these questions not only enriches our understanding of forecasting methodolo-
16
gies but also offers insights into how different market participants might use or prioritize
information.
4 Data
The empirical analysis presented in this paper relies on data sourced from Wharton Research
Data Services. Specifically, the return data for the S&P 500 were extracted from the CRSP
database, while the options data necessary for applying the Recovery Theorem were obtained
from the OptionMetrics database. Analyst forecast data were sourced from the Institutional
Brokers’ Estimate System (I/B/E/S) database, which compiles individual analyst reports for
specific stocks. By aggregating these individual forecasts, we derive a consolidated forecast
metric, considering both the median and mean forecasts provided by analysts. This aggre-
gated forecast facilitates a direct comparison with the forecasts generated by the Recovery
Theorem, matched by date and forecast horizon to ensure consistency. The most detailed
level of data granularity used in this study corresponds to monthly forecasts. Due to some
data points missing in the I/B/E/S database, the dataset spans from September 2003 to
July 2013, yielding 103 monthly data points. When analyzing forecasts derived solely from
the RT, the dataset extends from February 1996 to August 2015, encompassing 235 data
points.
In order to apply the RT, we first must construct the state price density, which makes
use of the risk-neutral density. To derive the risk-neutral density, we apply the methodology
outlined in Sanford (2021) and Figlewski (2008). We construct the risk-neutral density
(RND) by selectively including at-the-money (ATM) and out-of-the-money (OTM) options,
17
while excluding in-the-money (ITM) options and options that are deeply OTM. As is common
in the literature, we extract the RNDs using a mixture of OTM calls and puts (Figlewski,
2008, 2018). This approach is designed to leverage the liquidity and market representation of
ATM and slightly OTM options, which are most indicative of prevailing market sentiments,
avoiding the erratic pricing behavior associated with deeply OTM options. In lieu of ITM
options, we use a mixture of put and call options, which we combine using what Figlewski
(2008) calls a weighting function using the midpoint of the bid-ask spread of option prices.
For the conversion of option prices into implied volatilities, the Black-Scholes equation is
employed. This step enables the calculation of state price densities (SPDs) by interpolating
across the available option prices. The interpolation employs a b-spline for options around the
ATM strike price, supported by linear interpolation techniques that incorporate firm survival
probabilities, effectively constructing a detailed implied volatility surface across different
strike prices and maturities. The final estimation of SPDs, essential for the subsequent
application of the recovery theorem, involves taking the second derivative of these option
prices with respect to their strike prices.
The final phase of our statistical analysis examines the efficacy of nearly 300 factors in
explaining the variance in the two types of forecasts. While the first 16 variables listed in Ta-
ble 1 are extensively documented in the literature, particularly by Welch and Goyal (2007),
additional variables warrant mention due to their potential obscurity but significant rele-
vance. For example, the Smooth Rec Prob variable, defined by Chauvet and Piger (2008),
represents the smoothed probability of a recession occurring in the United States, sourced
from the FRED database. The Money Growth variable indicates the monthly percentage
change in the M3 money supply, also from FRED. Recession Indicator, another FRED-
18
sourced variable, reflects the NBER’s recession indicator for the U.S. The Macro Leading
Index predicts the growth rate of the American economy over the subsequent six months.
UMICH Sentiment, derived from the University of Michigan’s Surveys of Consumers, gauges
consumer sentiment in the U.S. The variables Unclassified and News VIX, discussed in
Manela and Moreira (2017), relate to news-implied volatility, with Unclassified capturing
the volatility from indeterminate news categories. Int Capital Risk, as conceptualized
by He et al. (2017), quantifies shocks to the equity capital ratio of financial intermediaries.
Lastly, Consumption Growth measures the growth rate of consumption in the U.S., again
sourced from FRED. These variables, spanning macroeconomic indicators and stock mar-
ket metrics, are instrumental in elucidating the factors that underpin the forecasts under
analysis.
Notably, the VIX, often referred to as the market’s "fear gauge," exhibits a wide range
from 10.42 to 59.89, highlighting periods of relative calm and spikes in market volatility.
This variability is crucial for understanding market sentiment and its impact on forecasting
accuracy. Similarly, the S&P 500 Volatility index ranges significantly, underscoring the
fluctuating nature of market conditions over the dataset’s timeframe.
Another critical variable, the Smooth Recession Probability, has a spread from 0 to
100%, indicating the model’s sensitivity to changing economic conditions and its potential
predictive power regarding economic downturns. This contrasts sharply with the more sta-
ble metrics, such as the AAA and BAA bond rates, which move within a narrower band,
reflecting the relative stability of credit markets compared to the equity markets represented
by the VIX and S&P 500 volatility. The Earnings Yield and Dividends Yield variables offer
insights into the market’s valuation and income-generating capabilities, with their ranges
19
suggesting periods of varying attractiveness to investors. The Book-to-Market ratio further
complements this picture by providing a measure of the market’s valuation of companies
relative to their book value, a key factor in asset pricing models.
Macroeconomic indicators like Money Growth and the Recession Indicator provide a
broader economic context, with Money Growth’s negative minimum highlighting periods of
contraction in the money supply. The Macro Leading Index and UMICH Sentiment, reflect-
ing expectations for economic growth and consumer confidence, respectively, underscore the
interconnectedness of market forecasts with broader economic sentiment.
Variable nMin q1
e
x ¯x q3Max #NA
VIX 103 10.42000 13.48500 16.44000 19.80874 23.17000 59.89000 0
Stock Index 103 735.09000 1123.44500 1257.64000 1244.42699 1378.63000 1685.73000 0
Dividends Yield 103 16.58600 21.84350 24.46533 24.56178 27.83650 33.64553 0
Earnings Yield 103 6.86000 52.69167 69.83000 64.16929 84.61858 92.09000 0
Book-to-Market 103 0.21605 0.27083 0.30923 0.30364 0.34309 0.44110 0
Treasury Bills 103 0.00010 0.00090 0.00920 0.01596 0.02905 0.05030 0
AAA 103 0.03400 0.04960 0.05330 0.05072 0.05540 0.06280 0
BAA 103 0.04510 0.05770 0.06270 0.06240 0.06670 0.09210 0
Long Term Yield 103 0.02160 0.03635 0.04430 0.04172 0.04860 0.05390 0
Net Equity Exp 103 -0.05768 -0.01475 -0.00236 -0.00598 0.00909 0.01629 0
Risk-free Rate 103 0.00001 0.00007 0.00077 0.00133 0.00242 0.00419 0
Inflation 103 -0.01915 -0.00103 0.00240 0.00205 0.00518 0.01222 0
Long Term Return 103 -0.11240 -0.01760 0.00830 0.00474 0.02400 0.14430 0
Corporate Bond Rate 103 -0.09490 -0.01255 0.00640 0.00555 0.02360 0.15600 0
Stock Volatility 103 0.00036 0.00075 0.00124 0.00354 0.00232 0.05809 0
S&P 500 Volatility 103 -0.16698 -0.01246 0.01299 0.00786 0.03252 0.10901 0
Smooth Rec Prob 103 0.00000 0.10000 0.10000 14.19320 0.45000 100.00000 0
Money Growth 103 -0.45823 0.33081 0.48193 0.51209 0.63670 2.29745 0
Recession Indicator 103 0.00000 0.00000 0.00000 0.11064 0.00000 1.00000 0
Macro Leading Index 103 -2.68000 0.69000 1.33000 0.91427 1.53500 1.94000 0
UMICH Sentiment 103 55.30000 70.45000 78.60000 79.26408 90.30000 103.80000 0
20
Unclassified 103 -1.59383 5.47829 9.34211 9.15198 12.38244 35.96375 0
News VIX 103 13.62252 21.05267 25.23244 25.40327 29.11102 57.89771 0
Int Capital Risk 103 -0.24670 -0.03215 0.00637 0.00301 0.03955 0.39650 0
Consumption Growth 103 -0.01401 0.00200 0.00383 0.00381 0.00594 0.02789 0
Table 1: This table presents the descriptive statistics for key independent variables used in
this paper’s analysis, including a mix of macroeconomic and stock market factors. These
variables are instrumental in examining the factors influencing the forecasts. The columns
detail the sample size (n), minimum, first quartile (q1), mean (¯x), median (ex), third quartile
(q3), maximum values, and the count of missing values (#NA), respectively.
Table 2 shows the descriptive statistics for the analyst forecast, the univariate RT (UVRT),
and the realized returns, respectively. The columns are as follows: nis the number of ob-
servations in the sample, Min is the minimum value, q1is the first quantile value, e
xis the
median, ¯x is the mean, q3is the third quantile, Max is the maximum value, and #NA is
the number of missing values in the sample. The descriptive statistics presented in table 2
give a good indication of just how different the forecasts are in comparison to the realized
return. First, focusing on the tail values by looking at the min and max values, we notice
that the closest value to the realized return is the one from the UVRT (with a minimum
at -0.05 compared to -0.24 for the realized return). The analyst and UVRT forecasts are
both quite far from the value for the realized return (-0.05701 and -0.01513, respectively),
but the UVRT is certainly closer. This is the first indication that the forecast obtained from
the UVRT accounts for a lot more of the variation in the returns than the analyst forecast.
Granted, at this point we do not yet know the specific timing of the values obtained so it is
important, since this is a time-series analysis, that we look into that next. Figures 1 through
3 compare the timings and differences between the CDFs of the realized returns and various
forecasts.
Variable nMin q1e
x ¯x q3Max #NA
Analyst 103 -0.01513 -0.00212 0.00244 0.00472 0.00694 0.07449 0
UVRT 235 -0.05701 -0.00409 0.00320 0.00342 0.00964 0.04832 0
21
Return 235 -0.23884 -0.01754 0.01769 0.00693 0.03542 0.13022 0
Table 2: This table provides the descriptive statistics for two forecasts and the empirical
returns used in the analysis of this paper. The first row presents the summary statistics for
the analysts. The second row represents the summary statistics for the RT. The third row
represents the summary statistics for the realized returns. The columns, in their respective
order, represent the number of observations, the minimum value, the value for the first
quartile, the average, the median, the value for the third quartile, the maximum value, and
the number of missing values in the sample.
The time series plots in figure 1 are ordered as follows: top left is the plot for the realized
return, top right is the plot for the UVRT, and bottom left is the plot for the analyst forecasts.
The analysts forecasts (bottom left graph) seem to have a lot less variation than the other
time series graphs. This is in line with the quantile–quantile plots in figure 2. Ultimately, the
time series shows that the analyst forecasts have fairly consistent results (positive) and that
their negative forecasts have a tendency to be small negative forecasts. This is in line with
our proposition that analysts tend to be much more conservative than option traders. The
largest positive forecast (from table 2) is about 7.5% while the largest negative expected
return is about -1.5%. This is certainly not the case for the UVRT (bottom left graph)
which has substantially larger negative and positive expected returns. The next questions
that arise are: 1) how far are these distributions from a normal distribution?, and 2) how
far are these distributions from their empirical counterpart?
22
Month
Realized Return
0 20 40 60 80 100
−0.2 −0.1 0.0 0.1
Month
UVRT
0 20 40 60 80 100
−0.06 −0.04 −0.02 0.00 0.02 0.04
Month
Analyst Forecast
0 20 40 60 80 100
0.00 0.02 0.04 0.06
Figure 1: This figure presents the QQ-plots of the variables used in the analysis of this paper.
The top left figure is the QQ-plot for the realized return. The top right figure is the QQ-plot
for the RT. The bottom left figure is the QQ-plot for the analyst forecast.
Figure 2 presents the quantile–quantile (QQ) plots for the realized return, UVRT, and
analyst forecasts. The QQ plot shows how far the quantiles for an empirical distribution
(the dots) are from a theoretical normal distribution (diagonal line) (Hyndman and Fan,
1996). As the asset pricing literature shows (Akgiray, 1989; Cenesizoglu and Timmermann,
2008; Cochrane, 2009; Maheu and McCurdy, 2011), most of the forecasts are fairly normally
distributed. The only exception is at the tails, which clearly deviate from the normal distri-
bution for all of the empirical distributions. Ultimately, all of these empirical distributions
exhibit fat tails (Rachev et al., 2005; Cont, 2001). The most notable fat tail is on the right
side of the distribution in the QQ plot for the analyst forecasts. This indicates that, on aver-
age, the analysts are very optimistic (high expected returns) whenever forecasts are positive.
Analysts seem to produce very large positive forecasts, but negative forecasts that are in line
23
with the normal distribution. The UVRT is much closer to the realized return. This is an
indication that the UVRT is much more realistic in its expectations of future movements of
stock prices.
−2 −1 0 1 2
−0.2 −0.1 0.0 0.1
Realized Return
Theoretical Quantiles
Sample Quantiles
−2 −1 0 1 2
−0.06 −0.04 −0.02 0.00 0.02 0.04
UVRT
Theoretical Quantiles
Sample Quantiles
−2 −1 0 1 2
0.00 0.02 0.04 0.06
Analyst Forecast
Theoretical Quantiles
Sample Quantiles
Figure 2: This figure presents the time-series plots of the time-series for the variables used
in the analysis of this paper. The top left figure is the time-series for the realized return.
The top right figure is the time-series for the RT. The bottom left figure is the time-series
for the analyst forecast.
Figure 3 shows the empirical cumulative density function (ecdf) for the UVRT (black line)
compared to the expected return from the analysts (red line) and the realized return (blue
line). Clearly, the UVRT forecasts a fatter tail than both the realized return (blue line) and
the expected return from the analysts (red line). It is important to note that the empirical
CDFs are compared in “real-time.” This means that the graphs show the distributions of
the different forecasts moved forward by one month compared to the actual realized return.
This is so that the realized return and the forecasts are compared at the same “time.”1
1For example, the one-month forecast for February would have been obtained in January. So the comparison
24
−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06
0.0 0.2 0.4 0.6 0.8 1.0
Return
Fn(x)
Figure 3: This figure presents the empirical cumulative density functions for the variables
used in the analysis of this paper. The blue line is the empirical cumulative density function
for the realized return. The black line is the empirical cumulative density function for the
RT. The red line is the empirical cumulative density function for the analyst forecast.
5 Results
5.1 Hypothesis Testing
We want to answer the following question: why do we need a model like the Recovery
Theorem when we have forecasts available from financial analysts? In other words, is there
value in forecasts from models that obtain the stochastic discount factors from options like
the RT, or are we better off using the forecast from analysts? The first test in this paper is
would be between the the January forecasts for February and the realized return for February so that we
are comparing apples to apples.
25
a hypothesis test as follows:
H0:µUV RT =µanalyst
Ha:µUV RT =µanalyst
(12)
where µUV RT is the expected return for the univariate RT and µanalyst is the expected return
obtained from the analyst forecast. The null hypothesis from equation 12 can be rejected at
the 1% level of significance (p-value = 0.00348) with 103 monthly observations. This result
indicates that the information set used for the UVRT forecast is likely different than that
used by analysts.
5.2 Cross-Sectional Quantile Regressions
Figures 4 through 7 are the results for the different forecasts compared to the realized return.
Figures 8 and 9 are the results for the RT forecasts compared to the analyst forecast. The
quantile regressions in this section unveil a nuanced landscape of forecast behaviors across
the distribution spectrum. The core of the distribution, as captured in Figures 4 to 9, reveals
a striking congruence between analyst and UVRT forecasts with the realized market returns,
suggesting a shared median expectation. This alignment, however, diverges as we explore the
distributional extremities, where the UVRT forecasts’ expansive uncertainty encapsulation
starkly contrasts with the pronounced conservatism of analyst forecasts. This conservative
bias, most evident in the left tail, could curtail the predictive power of analyst forecasts in
anticipating significant market downturns.
The UVRT’s broader uncertainty integration not only corroborates our hypothesis about
26
the conservative nature of analyst forecasts but also underscores the UVRT’s superior ca-
pability in capturing potential market volatilities. Such insights are important, offering
a richer understanding of market dynamics and informing more robust risk management
strategies. Moreover, the divergence in the tails hints at underlying differences in risk per-
ception and information processing between analysts and the market mechanisms that the
UVRT captures. This differentiation is critical for portfolio managers and investors seek-
ing to navigate the complexities of market predictions and underscores the importance of
incorporating a diverse set of forecasting tools in financial analysis. The quantile regression
analysis, therefore, does not merely highlight discrepancies in forecasting behaviors but also
invites a deeper interrogation of the fundamental assumptions driving market expectations.
We turn to cross-sectional regressions next in an effort to capture the information contained
in each of the forecasts.
0.00 0.02 0.04 0.06
−0.2 −0.1 0.0 0.1
Return
Figure 4: This figure represents the scatterplot along with the fitted line for the quantile
regression between the analyst forecast and the empirical returns.
27
0.0 0.2 0.4 0.6 0.8 1.0
−4 −2 0 2 4
Intercept
0.0 0.2 0.4 0.6 0.8 1.0
−4 −2 0 2 4
Analyst
Figure 5: This figure is a visual representation of the results from the quantile regression
between the analyst and the empirical returns. The top panel shows the result for the
constant while the bottom panel shows the result for the slope of the estimated quantile
regression.
−0.06 −0.04 −0.02 0.00 0.02 0.04
−0.2 −0.1 0.0 0.1
Return
Figure 6: This figure represents the scatterplot along with the fitted line for the quantile
regression between the UVRT and the empirical returns.
28
0.0 0.2 0.4 0.6 0.8 1.0
−1 0 1 2 3 4 5
Intercept
0.0 0.2 0.4 0.6 0.8 1.0
−1 0 1 2 3 4 5
UVRT
Figure 7: This figure is a visual representation of the results from the quantile regression
between the UVRT and the empirical returns. The top panel shows the result for the constant
while the bottom panel shows the result for the slope of the estimated quantile regression.
0.00 0.02 0.04 0.06
−0.06 −0.04 −0.02 0.00 0.02 0.04
UVRT
Figure 8: This figure represents the scatterplot along with the fitted line for the quantile
regression between the UVRT and the analyst forecast.
29
0.0 0.2 0.4 0.6 0.8 1.0
−3 −2 −1 0 1 2
Intercept
0.0 0.2 0.4 0.6 0.8 1.0
−3 −2 −1 0 1 2
Analyst vs UVRT
Figure 9: This figure is a visual representation of the results from the quantile regression
between the UVRT and the analyst. The top panel shows the result for the constant while
the bottom panel shows the result for the slope of the estimated quantile regression.
5.3 Cross-Sectional Factor Regressions
In this section of the paper, we explore the informational underpinnings of forecast models
through cross-sectional regressions. Specifically, we aim to identify which variables signifi-
cantly influence the forecasts produced by analysts and the Recovery Theorem-based model.
This analytical approach allows us to determine the components of these forecasts, high-
lighting the types of information or factors that are most important in shaping market
expectations. By focusing on key factors identified through statistical analysis, we can offer
insights into the differential reliance on various types of information by different forecasting
methodologies, thus shedding light on the mechanisms driving market predictions.
30
In the analysis of cross-sectional regressions, we focus on identifying the primary fac-
tors influencing the UVRT forecasts. Our examination in Table 3 specifically targets the
predictive power of the previous period’s UVRT forecasts, recession indicators, and money
growth on the current UVRT expectations, as the most prominent or important variables in
forming the expectations for the RT. This analytical approach aims to distill the essence of
the information sets used by market participants in forming these forecasts.
The regression outcomes highlight the significant role of recession indicators, affirming
the hypothesis that forecasts from the UVRT deeply integrate the potential for extreme mar-
ket movements. This is a critical insight, highlighting the UVRT’s sensitivity to economic
downturn signals, in stark contrast to the findings from analyst forecasts, which we will turn
to next. Such differentiation not only emphasizes the UVRT’s broader scope in uncertainty
capture but also signals its use in anticipating significant market adjustments. The positive
coefficient for Recession suggests that an increase in the likelihood of a recession is associ-
ated with higher expected returns as forecasted by the UVRT, indicating that the UVRT
model is sensitive to economic downturn signals and incorporates this risk into its forecast.
The positive coefficient for Money Growth implies that higher money supply growth is also
associated with higher expected returns, suggesting that liquidity or expansionary monetary
policy conditions are viewed positively by the UVRT model in terms of future market re-
turns. More specifically, the positive coefficient on Money Growth indicates that the UVRT
correctly predicts that an increase in the money supply should lead to lower interest rates,
which ultimately will lead to economic growth, a positive outcome for the stock market. In
essence, these coefficients make sense as they align with economic intuition: recession indica-
tors signal economic risk which the market may price in, and money growth could be related
31
to economic expansion or inflation expectations, both of which can affect market returns.
This analysis underscores the UVRT’s ability to capture broader economic conditions and
risk perceptions in its forecasts.
Model 1 Model 2 Model 3
U V RTt−1-0.0565 -0.0966 -0.0946
(0.0658) (0.0646) (0.0632)
Recessions 0.0111∗∗∗ 0.0097∗∗∗
(0.0028) (0.0028)
Money Growth 0.0080∗∗∗
(0.0024)
Constant 0.0030∗∗ 0.0018†-0.0021
(0.0009) (0.0009) (0.0015)
R20.0032 0.0669 0.1106
Num. obs. 235 235 235
∗∗∗p < 0.001,∗∗ p < 0.01,∗p < 0.05,†p < 0.1
Table 3: This table presents the results for three different regression specifications using
the UVRT as the dependent variable. Model 1 includes only a lagged forecast. Model 2
includes both the lagged forecast along with the recession indicator. Model 3 includes all of
the variables in model 2 along with the money growth variable.
Table 4 shows the results for the current expectation obtained from the analyst forecasts
as the dependent variable. Here, the smooth recession probability variable, the VIX, the
book to market variable of the companies in the Dow Jones Industrial Average, the t-bill
rate for the 3-month treasury bills, the default rate for firms rated AAA, the default rate
for firms rated BAA, the inflation rate, the macro leading index, and the University of
Michigan’s consumer sentiment index are the key explanatory variables. Note that the high
R2values in this table are not surprising since analysts are using the same information to
formulate their expectations as other investors. All of this data is made available to the
public, which indicates that analysts, on average, use unsurprising data in formulating their
expectations. These variables are commonly known to be good (general) indicators of where
32
the economy is heading. If analysts are trying to stay conservative, one would expect them
to use data that is conservative in nature. These variables will not produce something that
goes far beyond the current trajectory of the market, especially at the monthly interval. For
example, the VIX is a measure of the fear index for the economy. One would expect that as
fear increases, investors would expect higher returns to compensate them for the increased
level of risk in the economy. This result is apparent in the positive coefficient of the VIX
variable. The book-to-market ratio is a proxy for the relative value of the firm. This is a
well-known risk factor in the asset pricing literature and it has been well documented that
value firms command a higher return (Fama and French, 1993). These variables are used
in many of the most mainstream economic models of asset pricing. They are considered to
be aggregate measures of risk for the economy, and as such, should have an impact on an
aggregate stock market index like the S&P 500.
33
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9
Smoothed Recession 0.0003∗∗∗ 0.0001∗∗∗ 0.0002∗∗∗ 0.0001∗∗∗ 0.0001∗∗∗ -0.0000 0.0000 0.0000 0.0000
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
VIX 0.0008∗∗∗ 0.0006∗∗∗ 0.0008∗∗∗ 0.0005∗∗∗ 0.0002†0.0002 0.0002 0.0003∗
(0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001)
Book-to-Market 0.0865∗∗∗ 0.0910∗∗∗ 0.1678∗∗∗ 0.1676∗∗∗ 0.1685∗∗∗ 0.1619∗∗∗ 0.1664∗∗∗
(0.0167) (0.0165) (0.0167) (0.0141) (0.0146) (0.0154) (0.0155)
T-Bill 0.1005∗−0.1136∗−0.1397∗∗ −0.1471∗∗∗ −0.1372∗∗ −0.1505∗∗∗
(0.0475) (0.0475) (0.0418) (0.0420) (0.0425) (0.0427)
AAA 0.9865∗∗∗ 0.1162 -0.0481 -0.0090
(0.1314) (0.2249) (0.2552) (0.2536)
BAA 1.1610∗∗∗ 1.0436∗∗∗ 1.2463∗∗∗ 1.1348∗∗∗
(0.1185) (0.2278) (0.2725) (0.2774)
Inflation −0.2892∗−0.2798∗−0.2495∗
(0.1229) (0.1226) (0.1226)
Macro Leading Index 0.0021 0.0014
(0.0015) (0.0016)
UMICH Sentiment 0.0001†
(0.0001)
Constant 0.0005 −0.0131∗∗∗ −0.0366∗∗∗ −0.0416∗∗∗ −0.1067∗∗∗ −0.1203∗∗∗ −0.1178∗∗∗ −0.1232∗∗∗ −0.1308∗∗∗
(0.0011) (0.0023) (0.0050) (0.0054) (0.0097) (0.0089) (0.0090) (0.0098) (0.0107)
R20.4776 0.6317 0.7103 0.7229 0.8248 0.8608 0.8685 0.8709 0.8749
Num. obs. 103 103 103 103 103 103 103 103 103
∗∗∗p < 0.001,∗∗p < 0.01,∗p < 0.05,†p < 0.1
Table 4: This table outlines the results of nine cross-sectional regression models analyzing
analyst forecasts as the dependent variable. Each model progressively incorporates differ-
ent independent variables such as Smoothed Recession Probability, VIX, Book-to-Market,
Treasury Bills, AAA and BAA bond rates, Inflation, Macro Leading Index, and UMICH
Sentiment, to explore their predictive power on analyst forecasts. Model 1 starts with the
basic relationship, and subsequent models add more variables to capture various market and
economic factors’ effects. The table shows the regression coefficients with their significance
levels.
One conclusion that can be drawn from this set of tables (tables 3 and 4) is that the
information used by analysts to create their forecasts is quite different from the information
used to formulate the RT forecast. The RT is characterized by variables that are related
to expected risks (such as capital risk) or the probability of a recession. These are the
probabilities of large movements that would affect a firm substantively if the event occurred.
Despite the very small probability of these events occurring in the upcoming month, they
are still used as part of the information set in the RT’s forecast. This is one of the biggest
differences between the forecast of the RT and the forecast obtained from the analysts. For
34
the RT, the information used is focused on extreme movements, no matter how likely these
movements are, whereas for the analyst, the focus is on information that is much more likely
to have an impact of the stock, albeit by not as much. This is in line with the “conservative”
model for analyst forecasts presented in this paper. Although the information to construct
forecasts that are quite volatile is out there, analysts prefer to construct their forecasts based
on models and information that is more of a “sure” bet.
6 Conclusion
This study conducted a comparative analysis between stock forecasts derived from the Re-
covery Theorem and those produced by financial analysts, focusing on whether option-based
models provide unique advantages over traditional analyst forecasts. Through the appli-
cation of hypothesis tests and quantile regression analyses, significant differences were un-
covered between option-based and analyst-based forecasts, particularly in their approach to
predicting extreme market movements. The Recovery Theorem forecasts are notably more
inclined to anticipate significant deviations in the market, highlighting a key distinction from
analysts’ more conservative predictions that generally align with current market trends.
Further investigation into the informational foundations of these forecasts confirmed that
RT-based forecasts utilize a distinct set of data, including uncertainty indices, capital risk,
and volatility, indicative of a focus on macroeconomic uncertainties and potential market
extremes. Conversely, analyst forecasts rely on traditional economic indicators such as t-bill
rates and book-to-market values, which align with their aim to provide stable and intuitive
market predictions. The differentiation in the informational and motivational frameworks
35
guiding these forecasts emphasizes their distinct uses. Analysts provide market insights that
adhere closely to conventional expectations, while option-based forecasts, like those obtained
using the RT, address the needs of market participants concerned with hedging against or
exploiting extreme market variations.
Acknowledging the limitations of this research, including its dataset constraints and the
focus on the S&P 500, this paper suggests avenues for future studies to expand upon, such as
exploring a wider range of data sources, examining individual assets beyond market indices,
and delving deeper into option pricing dynamics. In sum, the study highlights the comple-
mentary nature of analyst and option-based forecasts in market analysis. The divergences
in their predictive focus and informational reliance highlights the potential for integrating
diverse forecasting models to enhance overall market prediction accuracy. Future research
in this direction may reveal ways to leverage the strengths of both approaches for more
comprehensive and nuanced financial analyses.
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