Structural Uncertainty and Stock Market Volatility∗
Daniel Andrei†
Bruce Carlin‡
Michael Hasler§
November 19, 2015
Abstract
When the economy is subject to model uncertainty, agents continuously calibrate
the parameters of their models to rationalize what they publicly observe. This causes
structural uncertainty and disagreement, which affects asset prices in substantial ways.
Structural uncertainty is associated with persistent stock-return volatility, consistent
with GARCH-type processes. It magnifies stock-return volatility both during recessions
and expansions, consistent with empirical evidence. Our model can explain why we
observe occasional large changes in asset prices without a correspondingly large change
in underlying fundamentals. Finally, disagreement earns a risk premium and is the
primary channel through which uncertainty is priced.
Keywords: Asset Pricing, Learning, Uncertainty, Disagreement, Volatility, Risk Premium
JEL Classification. D51, D83, G12, G14
∗
This paper was circulated previously under the title “Model Disagreement and Economic Outlook.” We
would like to thank Tony Berrada, Mike Chernov, Julien Cujean, Alexander David, Jerome Detemple (EFA
discussant), Bernard Dumas, Andrea Eisfeldt, Barney Hartman-Glaser, Burton Hollifield, Julien Hugonnier,
Arvind Krishnamurthy, Lars-Alexander Kuehn, Ali Lazrak (NFA discussant), Francis Longstaff, Hanno Lustig,
Emilio Osambela, Monika Piazzesi, Nick Roussanov (Cavalcade discussant), Martin Schneider, Ken Singleton,
Pascal St.-Amour, Wei Xiong, and Hongjun Yan for their useful advice. We would also like to acknowledge
comments from conference and seminar participants at the SFI meeting in Gerzensee, 4th International Forum
on Long-Term Risks, UCLA Anderson, 2014 Mathematical Finance Days, 2014 SFS Finance Cavalcade, 2014
WFC, 2014 SITE Summer Workshop, 2014 EFA meeting, 2014 NFA meeting, and CMU Tepper. Financial
support from the Swiss Finance Institute, NCCR FINRISK of the Swiss National Science Foundation, UCLA,
and the University of Toronto is gratefully acknowledged.
†
UCLA, Anderson School of Management, 110 Westwood Plaza, Suite C420, Los Angeles, CA 90095, USA,
[email protected], danielandrei.info
‡
UCLA, Anderson School of Management, 110 Westwood Plaza, Suite C413, Los Angeles, CA 90095, USA,
[email protected], www.anderson.ucla.edu/finance/faculty/carlin
§
University of Toronto, Rotman School of Management, 105 St. George Street, Suite 431, Toronto, ON,
M5S 3E6, Canada, [email protected], www.rotman.utoronto.ca/faculty/hasler
1
1
Introduction
Explaining what drives the volatility of asset returns is a long-standing theoretical challenge.
Asset volatility is known to be excessively high relative to the volatility of fundamentals
(Shiller, 1981), to be time-varying and predictable (Engle, 1982), to increase during recessions
(Schwert, 1989) but also during periods of rapid technological progress (Pastor and Veronesi,
2006). So far, no consensus has been reached as to which theoretical framework can simultaneously explain all of these empirical observations. In this paper, we propose an equilibrium
asset pricing model that endogenously generates all of the aforementioned properties of asset
return volatility. Our approach is based on the view that agents do not know all the elements
of the structural model of the economy.
When economic agents use a forecasting model to describe the outcomes of various processes, they are typically confronted with two types of uncertainty. The first type pertains to
the inherent randomness of nature and dictates the ability of the model to predict tomorrow’s
(or next year’s) outcomes. The second type arises when economic agents lack knowledge about
the model itself—they do not perfectly trust its functional form and are uncertain whether
the model adequately reflects reality. Commonly denoted model uncertainty (Hansen, 2007),
it can arise from a variety of sources, such as calibration accuracy, parameter uncertainty, or
structural changes in model’s parameters over time.
To solve the problem of model uncertainty, economic agents perform a learning exercise
in which they continuously update their model or re-calibrate its parameters as new data
becomes available, in the hope of getting as close as possible to the “true” model of the
world. Nonetheless, in the presence of model uncertainty, this updating exercise is inherently
an under-identified problem that generates disagreement as a side-effect (Acemoglu, Chernozhukov, and Yildiz, 2015). Indeed, agents are uncertain about how exactly to interpret and
use publicly available information and therefore end up almost surely in disagreement.
As we show in this paper, model uncertainty and disagreement jointly affect asset prices in
substantial ways. To demonstrate this formally, we build on two strands of literature. First, we
draw on research that analyses incomplete-information and learning in economics and finance,
with one key distinction. Instead of assuming that agents adopt a “top-down” approach
and learn about the economy without changing the economic model, we propose a “bottomup” approach whereby agents adjust their economic models as new data becomes available,
attempting in this way to rationalize what they publicly observe (think, for instance, of an
econometrician who revises the regression coefficients after observing new data points). The
key implication of bottom-up learning is time-varying uncertainty, which stands in contrast
2
with learning models with similar structures.1 This implication is backed up by a considerable
amount of empirical evidence which clearly shows that uncertainty fluctuates. For instance,
Jurado, Ludvigson, and Ng (2015) find significant variation in their econometric estimates of
macroeconomic uncertainty, which calls for a theoretical understanding of the origin of these
fluctuations.2
The second related strand of research explores the impact of heterogeneous beliefs on asset
prices. Typically, this literature assumes that agents agree to disagree about the fundamental that governs the drift of the consumption/dividend stream in a continuous-time economy
(e.g., Scheinkman and Xiong, 2003; Dumas, Kurshev, and Uppal, 2009).3 We depart from this
particular form of disagreement and assume instead that the fundamental is publicly observed
and agreed upon at all times, but agents disagree about the parameters of the model that govern its dynamics. Our setup is motivated by clear empirical evidence of model disagreement.
For instance, Carlin, Longstaff, and Matoba (2014) document substantial disagreement among
Wall Street mortgage dealers about prepayment speed forecasts. Because all of the dealers in
the survey are large financial institutions having access to all publicly available information
and only very little private information, disagreement must necessarily stem from differences
in the assumptions or the structure of the models that dealers use.4
We build a continuous-time pure exchange economy populated by two agents. There exists
one risky asset which pays a continuous stream of dividends. Both the dividend process and its
expected growth rate (i.e., the fundamental ) are publicly observed. Model uncertainty arises
because agents do not observe the persistence of the expected growth rate, or (equivalently)
its mean-reverting parameter, which itself is time-varying. Variations in this parameter can be
interpreted as structural changes in the economy, such as long-lasting technological revolutions
or periods of “secular stagnation”. Agents learn via Kalman filtering and have discretion over
how to rationalize the observable fundamental. The discretion parameter that we use in the
model is meant to capture the number of levers that the agents have when calibrating their
1
Uncertainty is constant in most learning models, a common result in the broad literature of learning (see
Detemple, 1986; Gennotte, 1986; Dothan and Feldman, 1986; Brennan and Xia, 2001, among many others).
This arises because priors are Gaussian and all variables are normally distributed. In this case, the conditional
variance of the unobservable variable—the Bayesian uncertainty—follows a deterministic path and converges
to a steady-state. Stochastic uncertainty arises with non-Gaussian distributions (see, for instance, Detemple,
1991; David, 1997; Veronesi, 1999).
2
See also Bloom (2009), Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2014) and Bachmann,
Elstner, and Sims (2013).
3
See also Harris and Raviv (1993), Detemple and Murthy (1994), Kandel and Pearson (1995), Zapatero
(1998), Basak (2000, 2005), Berrada (2006, 2009), Buraschi and Jiltsov (2006), Li (2007), David (2008), Yan
(2008), Xiong and Yan (2010), Cvitanic and Malamud (2011), Napp, Malamud, Jouini, and Cvitanic (2012),
Bhamra and Uppal (2013), Buraschi, Trojani, and Vedolin (2014), Buraschi and Whelan (2013), Ehling,
Gallmeyer, Heyerdahl-Larsen, and Illeditsch (2013), Osambela (2015), and Baker, Hollifield, and Osambela
(2015).
4
See also Patton and Timmermann (2010), Andrade, Crump, Eusepi, and Moench (2014).
3
model (problem dimensions) or the liberty that they may take when choosing input values
(input constraints).
The model yields novel asset pricing implications. We show that learning about persistence
leads to time-varying Bayesian uncertainty. This introduces the main object of interest in our
model, which we call structural uncertainty. It is defined as the product of two quantities: the
Bayesian uncertainty about growth persistence and the difference between the fundamental
and its long-term mean. Intuitively, structural uncertainty (which would be zero without
model uncertainty) is a function of how much uncertainty is perceived by agents when they
calibrate their model and the degree to which the economy is in an expansion or contraction.
This implies that the way in which structural uncertainty impacts the price of risk, risk premia,
and the volatility of asset returns depends on the nature of the economy. It has little impact
during periods of normal balanced growth and strong impact during recessions and (rapid)
expansions.
We find that structural uncertainty directly drives stock return volatility. This result does
not require heterogeneous agents in the economy—even in a single-agent setup, structural
uncertainty generates time-varying and persistent stock market volatility. Furthermore, when
the economy is going through recessions and expansions, agents are facing large structural
uncertainty, which significantly increases stock return volatility. Intuitively, if the economy is
away from the balanced growth path, its convergence back to normal times is highly sensitive
to the mean-reverting parameter—for instance, if the economy is going through a recession
today, a slight update of this parameter is what differentiates a “mild recession” from a “secular
stagnation.” As such, our model can explain why we observe occasional large changes in stock
prices without a correspondingly large change in underlying fundamentals and implies a Ushaped relation between economic growth and structural uncertainty. This differentiates our
paper from other models of learning (e.g., Veronesi, 1999, 2000), in which uncertainty is high
when agents perceive the economy to be “in-between” a discrete set of growth states.5
An additional key variable arising from learning in our model is the structural disagreement
between agents, which is defined as the product of two quantities: the degree to which agents’
estimates of the mean-reversion parameter diverge from each other and the difference between
the fundamental and its long-term mean. Structural disagreement impacts the market price
of risk by increasing each agent’s consumption share risk. It is associated with a positive
risk premium that grows both with model disagreement and as the economy goes through
expansions or recessions. When agents have more discretion over which information to use
when rationalizing what they publicly observe, the risk premium becomes more volatile, and
5
See also David (1997), Cagetti, Hansen, Sargent, and Williams (2002), David and Veronesi (2002), and
David (2008).
4
the agent with the least favorable outlook requires a large risk premium for holding the risky
asset.
Structural uncertainty and disagreement interact with each other to generate additional
implications. When uncertainty is high and there is sufficient disagreement among agents,
the agent with the least favorable economic outlook requires an even larger risk premium for
holding the risky asset. However, without structural disagreement, uncertainty does not affect
the risk premium, which leads us to conclude that disagreement is a primary channel through
which uncertainty commands a risk premium.6
Our paper satisfies several empirical regularities. First, our model delivers an amplification
mechanism through which the impact of disagreement and uncertainty on the risk premium
and volatility is stronger during recessions or expansions. In these periods, model disagreement
leads to a higher risk premium and more uncertainty causes large and persistent fluctuations in
volatility. These features rationalize the high levels of volatility and large risk premia observed
not only during recessions (Schwert, 1989; Patton and Timmermann, 2010; Barinov, 2014),
but also during expansions such as the Nasdaq bubble in the late 1990’s (Pastor and Veronesi,
2006).
Second, our paper provides a theoretical foundation for GARCH. The persistence of volatility has been described extensively in the empirical literature, but there is a paucity of theoretical explanations.7 The explanation that we provide here is based on structural uncertainty,
which arises endogenously from learning about growth persistence. Agents’ learning generates
slow-moving Bayesian uncertainty, which itself is the main driver of stock return volatility,
and thus endogenously generates volatility clustering.8
Finally, our work rationalizes the empirical findings in Carlin, Longstaff, and Matoba
(2014) who analyze how model disagreement in the MBS market affects asset prices. The
authors show that there is a risk premium associated with disagreement and that disagreement varies over time. It appears that disagreement rises during periods of large market
6
This result arises although agents have CRRA utility. In typical models with CRRA utility, uncertainty
does not appear directly in the risk premium. See Collin-Dufresne, Johannes, and Lochstoer (2015) for a
model where uncertainty is priced in a model with Epstein and Zin (1989) preferences.
7
A few preference-based foundations for volatility clustering are provided by Campbell and Cochrane (1999),
Barberis, Huang, and Santos (2001), and McQueen and Vorkink (2004). See also Osambela (2015).
8
Timmermann (2001) shows that existence of structural breaks in the fundamental process induces volatility
clustering. David (1997) and Veronesi (1999) show that, when the fundamental follows a Markov-switching
process, learning implies volatility clustering. A commonality between our work and theirs is that agents
have imperfect information about the economic model, although our model does not feature structural breaks
or a finite number of states. See also Collin-Dufresne, Johannes, and Lochstoer (2015), who show that
parameter learning generates long-lasting risks when a representative agent has a preference for early resolution
of uncertainty. In our model, we allow the persistence parameter to change over time (and interpret these
variations as structural changes of the economy). Thus, learning is continuously re-generated and the parameter
never converges to a constant.
5
movements, which is consistent with our idea of structural disagreement. Further, the authors
show that disagreement is the primary channel through which uncertainty leads to trading
volume. Ostensibly, this is consistent with our finding that uncertainty is only incorporated
into risk premia when sufficient structural disagreement is present.
The rest of the paper proceeds as follows. Section 2 defines our model and the learning
processes that the agents use. Section 3 characterizes the market equilibrium and shows
how uncertainty and disagreement affect asset prices. Section 4 present numerical results
and didactic examples, with parameters resulting from an empirical calibration of the model
using the Simulated Method of Moments. Section 5 concludes. The Appendix contains all
proofs, investigates the accuracy of our numerical approximation, and describes our calibration
exercise.
2
Structural Uncertainty and Disagreement
Consider a pure exchange economy defined over a continuous-time horizon [0, ∞), in which a
single consumption good serves as the numéraire. There is a single risky asset (the stock) in
positive unit supply, which is the claim to the aggregate consumption stream. There is also
a risk-free asset available in zero net supply. The aggregate consumption/dividend stream δ
follows the process
dδt
= f˜t dt + σδ dWtδ
δt
p
df˜t = λt f¯ − f˜t dt + σf ρdWtδ + 1 − ρ2 dWtf
dλt = κ λ̄ − λt dt + ΦdWtλ ,
(1)
(2)
(3)
where W δ , W f , and W λ are three independent Brownian motions under the physical (objective) probability measure P. The expected consumption growth rate f˜ in (3) is referred
to as the fundamental, which mean-reverts to its long-term mean f¯ at speed λ. In turn, λ
also follows a mean-reverting process.9 The parameters σδ and σf are the volatilities of the
dividend growth and of the fundamental, and the parameters κ and Φ govern the dynamics
of the mean-reverting speed λ.
Discussions regarding the mean-reversion parameter λ are ubiquitous both among researchers and practitioners. This parameter basically tells how quickly the economy is likely
9
The mean-reversion speed can become negative because it is modeled with an Ornstein-Uhlenbeck process.
The parameter estimation performed in Appendix A.4, however, implies that the probability of the meanreversion speed turning negative is small, and therefore insures that the fundamental is stationary. The
stationarity properties of an Ornstein-Uhlenbeck process with stochastic mean-reversion speed are discussed
by Benth and Khedher (2013).
6
to grow, i.e., it captures the pace of the economy. For instance, “secular stagnation” (i.e., a
period of a lower trend growth in the economy) is a popular research topic during the recent
years (Summers, 2014; Hamilton, Harris, Hatzius, and West, 2015). The same parameter λ is
also an important input for the Fed in assessing how quick the recovery is, as it is generally accepted that the pace of recovery after the 2007 financial crisis has been unusually sluggish when
compared to the average pace of previous recoveries in the postwar period (Bernanke, 2013).
Other discussions regarding the same parameter λ are about different types of recoveries and
recessions. In particular, it is debated whether recoveries from banking and financial crises
are different (Reinhart and Rogoff, 2009; Howard, Martin, and Wilson, 2011). Researchers
also debate about the existence of a long-run component in the consumption growth (Beeler
and Campbell, 2012; Bansal, Kiku, and Yaron, 2012). Therefore, both the uncertainty about
the mean-reversion parameter and its time-varying feature seem to be uniformly accepted.10
The economy is populated by two agents: A and B. Agents trade with each other and derive utility from consumption. Each agent chooses a consumption-trading policy to maximize
her expected lifetime utility
i
Z
Ui = E
∞
e
0
−βt
c1−α
it
dt ,
1−α
(4)
where β > 0 is the time discount rate, α > 0 is the relative risk aversion coefficient, and cit
denotes the consumption of agent i ∈ {A, B} at time t. The expectation in (4) depends on
agent i’s perception of future economic conditions and her preferences.
At all times, the agents both observe δ and the fundamental f˜.11 The parameters f¯, σδ ,
σf , ρ, κ, λ̄, and Φ are commonly known. Given this, both agents are able to compute and
observe df˜t and dWtδ . However, they are unable to observe the mean-reversion speed λ and
are therefore tasked with estimating it to rationalize the fundamentals that they publicly
observe. This is the key feature that distinguishes our model from the previous literature
(e.g., Scheinkman and Xiong, 2003; Dumas, Kurshev, and Uppal, 2009). Indeed, in other
models, it is assumed that agents have heterogeneous beliefs about fundamentals. Here, both
agents publicly observe fundamentals and their evolution with no disagreement, but try to
rationalize how those fundamentals evolve in order to predict future economic conditions.
As we will show below, this alternative way of modeling learning and disagreement leads to
10
Besides long-run behavior, there are other dimensions of parameter uncertainty analyzed in the literature,
such as tail events (Liu, Pan, and Wang, 2005) or regime changes (Ju and Miao, 2012). See also Collin-Dufresne,
Johannes, and Lochstoer (2015) and their Online Appendix for a good survey of the literature.
11
We interpret the fundamental f˜ as the average/median forecast of the growth rate among a large survey of
professional forecasters (such as the future GDP growth estimated by the Survey of Professional Forecasters).
Agents in this economy observe this forecast and agree that it is the best available estimate of the expected
growth rate.
7
theoretical predictions distinct from the rest of the literature.
Each agent privately studies the market and uses a proprietary model to characterize
λ. This may involve using different information resources, parameter estimates, or model
structures. Thus, there is discretion over how agents choose to rationalize fundamentals. To
capture this, we assume that each agent uses a different process (model) to derive λ:
p
1 − φ2 dWtA
p
λ
dmB
1 − φ2 dWtB ,
t = φdWt +
λ
dmA
t = φdWt +
(5)
(6)
where W A and W B are independent Brownian motions, also independent from W δ , W λ , and
W f . The parameter φ affects the amount of discretion that the agents have and the difference
in how they rationalize the fundamental that is publicly observed. If φ < 1, the models will
never be exactly the same, even though they both yield the same f˜t as an output. Note that,
as we discussed in the introduction, it is not necessary for the mit to differ structurally from
each other, only that the choice of inputs be different. The fact that dWtA 6= dWtB almost
surely is meant to capture this, and we interpret a low φ as the presence of more levers that
agents have when they match what they observe in the market.
Going forward, we assume that φ < 1 so that the agents will arrive at different estimates
B
of λ. Notwithstanding, both agents do indeed observe dmA
t and dmt and acknowledge the
discretion that they have each used, but adhere to their own analysis when maximizing (4).
This implies that while estimates are made publicly, both agents trade based on their own
analysis. This is common in financial markets. For example, consider the mortgage-backed
security (MBS) market as studied in Carlin, Longstaff, and Matoba (2014). There, the MBS
dealers periodically announce their prepayment speed forecasts publicly, which are frequently
different, but trade on the results of their own analysis.
b and its posterior variance γ are such that λt is normally
The filtered mean-reversion speed λ
bt and variance γt . Based on (1)-(6), the Kalman filter implies that
distributed with mean λ
agent i ∈ {A, B} has the following system of state variables in mind:12
dζt
df˜t
bi
dλ
t
1
2
cδ
= f˜t − σδ dt + σδ dW
t
2
p
bi f¯ − f˜t dt + σf ρdW
ctf i
c δ + σf 1 − ρ2 dW
=λ
t
t
bi dt + pγt
¯ − f˜t dW
ctf i + φΦdW
ctmi
= κ λ̄ − λ
f
t
2
σf 1 − ρ
(7)
(8)
(9)
cδ , W
c f i , and W
c mi are independent Brownian motions under the probwhere ζ ≡ log δ and W
12
See Theorem 12.7 in Liptser and Shiryaev (2001) and Appendix A.1 for computational details.
8
ability measure of agent i, as defined in Appenidx A.1. Note that the discretion that each
agent uses to estimate λ in mA and mB leads to different results because of the last term in
(9).
b are particular to this learning exercise. First, its instantaneous variance
The dynamics of λ
is directly driven by the fundamental and it is higher when the expected growth rate is away
from its long-term mean. This is because agents learn more about λ when f¯ − f˜ is large,
b can exhibit regimes of positive or negative correlation
as can be seen from (8).13 Second, λ
b the same happens with
c f i shocks lower λ;
with the fundamental. When f˜ is high, positive dW
negative shocks when f˜ is low. In other words, agents form “extrapolative expectations”: they
regard unusual good or bad past performance of the economy as indicators of a slowly moving
economy, or as the economy’s “new normal.”
The dynamics of the Bayesian uncertainty γ are the same for both agents
(f¯ − f˜t )2 2
dγt
= (1 − φ2 )Φ − 2κγt − 2
γ .
dt
σf (1 − ρ2 ) t
(10)
The first and last terms in (10) distinguish our set-up from previous models. First, when
t
φ is low and the agents have more discretion, dγ
is higher. Setting the fundamental to its
dt
long-term mean in Equation (10) and solving for the “steady-state” uncertainty γss yields
γss =
(1 − φ2 )Φ2
.
2κ
(11)
By inspection, dγdφss < 0, so that more discretion leads to higher uncertainty.
Second, the level of the fundamental and its deviation from its long-term mean affect the
amount of uncertainty the agents have about λ. For example, if today the economy is in good
times (i.e., f˜t > f¯)14 and agents observe a positive df˜t , then λ is likely to be small (i.e., the
fundamental is persistent).15 The same intuition holds if the economy is in bad times (i.e.,
f˜t < f¯) and the agents observe a negative df˜t . On the other hand, if f˜t − f¯ = 0, fluctuations
in the fundamental are pure noise and do not provide any valuable information to agents.
At the extreme, when f˜t − f¯ = 0, fluctuations in the fundamental are pure noise and therefore do not
provide any valuable information to agents. A similar learning mechanism is described in Xia (2001).
14
Good and bad times are well-defined in our model because agents agree on the value of the fundamental.
If, instead, agents were to disagree on the value of the fundamental, then good and bad times would be defined
according to one of the agent’s beliefs or to an average belief.
15
This particular feature distinguishes learning about the mean-reversion speed λ from learning about the
fundamental f . Uncertainty fluctuates in our model as learning about the speed of mean-reversion makes it
dependent on the fundamental. In contrast, learning about the fundamental implies that uncertainty converges
rapidly to a constant steady-state, and therefore does not influence the dynamics of asset prices. Fluctuating
uncertainty also arises when agents learn about a Markov chain or about a regression coefficient (e.g., David,
1997; Veronesi, 2000; Xia, 2001).
13
9
The observable fundamental provides the link between the probability measure of the
agents, PA and PB . As such,
ctf A
dW
B
A
b
b
¯
˜
λt − λt
f − ft
ctf B +
p
= dW
dt,
σf 1 − ρ2
(12)
bB − λ
bA is the difference in the way in which agents rationalize the fundamental. Since
where λ
each agent perceives the economy under a different probability measure, any random economic
variable X has two expectations that are related to each other by
EA [X] = EB [ηX] .
(13)
Under PB , the change of measure ηt satisfies
A
dP ηt ≡
dPB =e
− 21
Rt
0
(λbBt −λb√At )(f¯−f˜t )
σf
1−ρ2
!2
ds−
Rt
0
(λbBt −λb√At )(f¯−f˜t ) dW
cf B
σf
1−ρ2
s
,
(14)
Ot
where Ot is the observation filtration at time t and η has the following dynamics
bB − λ
bA f¯ − f˜t
λ
t
t
dηt
ctf B .
p
=−
dW
2
ηt
σf 1 − ρ
(15)
Going forward, we define two objects. The first is the structural uncertainty
U ≡ (f¯ − f˜)γ,
(16)
which appears in the dynamics of uncertainty (10) and is a function of the Bayesian uncertainty
and the state of the economy. The second is the structural disagreement
bB − λ
bA ),
D ≡ (f¯ − f˜)(λ
(17)
which drives the diffusion of the change of measure (15). It is a function of the state of the
economy and the difference in the way in which agents rationalize the fundamental. Both U
and D will play a key role in understanding the patterns of stock-return volatility and risk
premia observed in financial markets.
10
3
3.1
Equilibrium Quantities
Asset Prices
Without loss of generality, in what follows we work under agent B’s probability measure PB .
We assume that markets are complete and solve for the equilibrium using the martingale
approach of Karatzas, Lehoczky, and Shreve (1987) and Cox and Huang (1989).16
The state-price density perceived by agent B, ξ B , satisfies
ξtB = e−βt δt−α
=
"
ηt
ΛA
1/α
+
1
ΛB
1/α #α
(18)
1 −βt −α −α
e δt ωBt ,
ΛB
(19)
where ΛA and ΛB are the Lagrange multipliers associated with the static budget constraints
of agents A and B, and ωit the consumption share of agent i ∈ {A, B} at time t.17 The
consumption shares are functions of ηt and their sum equals one.
According to (19), agent B’s stochastic discount factor depends on the aggregate consumption δ but also on her consumption share ωBt . Intuitively, the agent cares not only about the
aggregate level of consumption (which would be the case in a representative agent economy),
but also on how much of δ she consumes, because now she shares consumption with agent A.
The following proposition provides the equilibrium risk-free rate, the market price of risk
vector perceived by agent B, and the price-dividend ratio for the risky asset.
Proposition 1. (Equilibrium) The risk-free rate, r, and the market price of risk vector
perceived by agent B, θB , are:
1
1α−1
rt = β + αf˜t − α(α + 1)σδ2 +
2
2 α
>
Dt
ω
0
0
At
θtB = ασδ σ √1−ρ
.
2
D
p t
σf 1 − ρ2
!2
ωAt ωBt
(20)
(21)
f
Assuming that the coefficient of relative risk aversion α is an integer,18 the equilibrium price16
The martingale approach transforms the dynamic consumption and portfolio choice problem into a consumption choice problem subject to a static lifetime budget constraint.
17
See Appendix A.2.
18
This assumption simplifies the calculus. To the best of our knowledge, it has been first pointed out by
Yan (2008) and Dumas, Kurshev, and Uppal (2009). If the coefficient of relative risk aversion is real, the
computations can still be performed using Newton’s generalized binomial theorem. Bhamra and Uppal (2013)
offer a comprehensive analysis for all possible values of the risk aversion.
11
dividend ratio of the risky asset satisfies
α St X α j α−j
=
ω ω Fj (Zt ),
δt
j At Bt
j=0
(22)
where
Fj (Zt ) ≡ EB
t
"Z
∞
e−β(u−t)
t
ηu
ηt
αj δu
δt
1−α
#
du .
(23)
>
bA , λ
bB , γ .19
The 4-dimensional vector of state variables, Z, is defined by Z ≡ f˜, λ
By inspection of (20) and (21), structural disagreement is a key driver of both the risk-free
rate and the market price of risk. The price-dividend ratio expressed in Equation (22) consists
in a weighted sum of expectations, with weights characterized by the consumption shares.
The first three components in (20) form the usual risk-free rate in a representative agent
economy. First, the risk-free rate increases with the discount factor β. Second, the risk-free
rate increases with the fundamental f˜. In this case, agents expect higher future consumption
and hence lower future marginal utility. Future payments due to saving have lower value,
which decreases the demand for the risk-free asset and increases the equilibrium risk-free rate.
Third, the risk-free rate decreases with the uncertainty about aggregate consumption σδ . The
greater the uncertainty, the more agents demand risk-free payments and hence a lower risk-free
rate is necessary to clear the market for borrowing and lending.
The last term in (20) comprises additional effects on the risk-free rate due to fluctuations
in the consumption shares. From the perspective of agent B (and without loss of generality)
if she expects a larger consumption share in the future20 and thus has lower future marginal
utility, she saves less and consumes more today. This increases the equilibrium risk-free rate.
Second, the greater the uncertainty about consumption shares, the more agents save today,
which decreases the equilibrium risk-free rate. These two effects are combined in the last term
in (20). If the risk aversion in the economy is larger than one, then the first effect dominates
and thus divergence of the agents’ models increases the risk-free rate in the economy.
19
See Appendix A.2 and A.3 for computational details. The model is not affine-quadratic because of 1) the
bf˜ governing the drift of f˜ (see Cheng and Scaillet, 2007). Nonetheless,
dynamics of γ and 2) the product λ
all the state variables in Fj (Zt ) have a long-term mean and thus the setup is particularly suitable to a Taylor
expansion.
20
It is worth nothing that each agent expects at all points in time a higher consumption share in the future
(i.e., the drift of each agent consumption share is positive under her own probability measure). This might
seem puzzling at first, but it is a natural effect arising from the difference in beliefs. Each agent believes that
her model is the true model of the world and that the other agent’s model is (partially) innacurate. Because
of this, each agent believes that she will be right in the future and thus her consumption share will increase.
12
Given this, model disagreement affects the risk-free rate (i) when α 6= 1, (i.e., the agents
do not have logarithmic utility), (ii) when the fundamental deviates from its long-term mean,
f˜t = f¯, and (iii) when there is a reasonable amount of parity among the agents in the economy
(i.e., it is not the case that ωit >> ωjt ).
The market price of risk perceived by agent B, shown in Equation (21), comprises two
terms. First, agent B requires a positive price for bearing the risk of fluctuations in aggregate
consumption δ, as determined by the first element in the vector (market price of aggregate
consumption risk ). The second element of the vector is the price required by agent B for
bearing the risk of fluctuations in her own consumption share ωB (market price of consumption
share risk ).
To understand the sign of the market price of consumption share risk, it is instructive to
consider two situations. First, if Dt > 0, agent B’s model has a more favorable economic
outlook.21 In this case, whenever there is a positive shock to the fundamental, agent B’s
consumption share increases. This induces a positive correlation between her consumption
share and the fundamental, and thus agent B requires a positive premium to bear fundamental
risk. Alternatively, if Dt < 0, agent B’s model has a less favorable outlook. In this case, her
consumption share is negatively correlated with fundamental risk, and agent B is willing to
pay a price to take this risk.
In short, any risk that is positively correlated with agent B’s consumption share earns a
positive price because taking this risk pays off when the agent’s marginal utility of consumption
is low. Furthermore, the magnitude of the market price of consumption share risk depends on
how much agent A consumes, ωA . Naturally, agent B requires a higher market price of risk
when the proportion of agent A in the economy is large. Finally, consumption shares do not
bA = λ
bB (in these cases agents have the same forecasts) and
fluctuate when f˜t = f¯ or when λ
t
t
thus there is no market price of consumption share risk.
3.2
Return Volatility and Risk Premia
Application of Itô’s lemma on the stock price provides the stock-return volatility, the risk
premium, and the Sharpe ratio in this economy.
Proposition 2. (Stock-Return Volatility, Risk Premium, and Sharpe Ratio) The
diffusion vector of stock returns, Σ, the risk premium according to agent B, RPB , and the
This happens when (i) the economy is going through good times (f¯ − f˜t < 0) and agent B believes the
bB − λ
bA < 0), or (ii) when the economy is going through
fundamental to be more persistent than agent A (λ
bad times and agent B believes the fundamental to be less persistent than agent A.
21
13
Sharpe ratio according to agent B, SRB , satisfy

S
σδ + hSf σf ρ
 Sf p
SλA
Sµ
 S σf 1 − ρ2 + √1
D
+
+
−
S
S
2

σf 1−ρ
Σ=
S
S

φΦ λSA φ2 + λSB

p
SλA
φΦ 1 − φ4
S
>
i
SλB
U 
S




RPB ≡ ΣθB
SλA SλB
Sf
Sf
ωAt
Sµ 2
= ασδ σδ + σf ρ + ωAt D + 2
+
UD − D
S
S
σf (1 − ρ2 )
S
S
S
RPB
SRB ≡
,
||Σ||
(24)
(25)
(26)
(27)
where ||·|| is the norm operator, Sy denotes the partial derivative of the stock price with respect
to the state variable y, and U and D are defined in (17) and (16). The economy is driven by
c δ , fundamental risk W
c f B , agent A’s model
four sources of risk: aggregate consumption risk W
c mA∗ , and agent B’s model risk W
c mB , which are defined in Appendix A.2, Equation
risk W
(42).
Because the economy is driven by four sources of risk, the stock-return diffusion Σ, expressed in Equation (24), has four components. The local volatility of the stock is computed
as the norm of its diffusion vector, σ ≡ ||Σ||. However, as we describe more later, only the
second component of Σ significantly affects the volatility.22 The state variables that directly
impact this component are the structural disagreement D and the structural uncertainty U.
We analyze the impact of these objects in Sections 4.1 and 4.2.
The risk premium perceived by agent B in (26) is computed as the vector product between
the market price of risk θB and the stock return diffusion Σ. Because the market price of
risk is driven solely by structural disagreement, it follows that uncertainty affects the risk
bB 6= λ
bA . If we set D = 0 in (26), then uncertainty has no direct impact
premium only when λ
on the risk premium. This implies that model disagreement is the main channel through which
uncertainty impacts the risk premium. We will explore this more in Section 4.2.
Learning about the mean-reversion speed λ connects both the uncertainty and model
disagreement with the observable fundamental f˜, as shown in (16) and (17). This feature is
unique to our model and implies that uncertainty and model disagreement have an impact
only when the economy is away from “normal times” (i.e., when f˜ 6= f¯). As the fundamental
is closer to its long-run mean, the structural uncertainty and structural disagreement both get
22
As we show numerically, the partial derivatives of the stock price with respect to the state variables exhibit
little variation and therefore do not significantly drive the volatility.
14
Parameter
Volatility of dividend growth
Symbol Estimate
σδ
0.0132∗∗∗
Long-term mean of the fundamental
Volatility of fundamental
f¯
σf
Correlation between dividend and fundamental
Persistence of mean-reversion speed
Long-term mean of mean-reversion speed
Volatility of mean-reversion speed
ρ
κ
λ̄
Φ
Discretion parameter
φ
(2.95 × 10−4 )
∗∗∗
0.0261
(6.59 × 10−4 )
∗∗∗
0.0195
(1.41 × 10−3 )
∗∗∗
0.1501
(2.16 × 10−2 )
∗∗∗
0.4989
(0.1101)
∗∗∗
1.1503
(6.62 × 10−2 )
∗∗∗
0.6495
(6.23 × 10−2 )
∗∗
0.0955
(3.93 × 10−2 )
Table 1: Calibration to the U.S. economy (SMM estimation)
Parameter values resulting from a SMM estimation with 13 moment conditions. Standard
errors are in brackets and statistical significance at the 10%, 5%, and 1% levels is labeled ∗ ,
∗∗ , and ∗∗∗ , respectively.
close to zero. Conversely, a fundamental far from its long-run mean enhances the effects of
uncertainty and model disagreement on asset returns.
Given the nature of the expressions in (24)-(26), characterizing the effects of uncertainty,
discretion, and model disagreement further necessitates a numerical implementation. In the
analysis that follows, we set α = 3, β = 0.01, and the ratio of Lagrange multipliers equal
to one. This last choice assures that both agents are endowed with the same initial share
of consumption. However, before assigning values to the other parameters, we calibrate the
model to the real U.S. GDP growth rate and its 1-quarter-ahead median analyst forecast using
the Simulated Method of Moments. Our parameter estimates are summarized in Table 1.23
4
Results
4.1
Stock-return volatility
Equation (24) of Proposition 2 suggests a non-linear relation between stock-return volatility
σ and the fundamental f˜ ceteris paribus. To see this, we plot in the left panel of Figure 1 the
23
See Appendix A.4 for details on the estimation method and a discussion of the estimated parameters.
15
stock-return volatility versus the fundamental. When γ = 0, the agents learn perfectly about
λ and volatility does not depend on the fundamental. In this case, the structural uncertainty
U is also zero. When uncertainty increases, volatility is higher when the fundamental is far
away from its long-term mean. In this case, the effect of uncertainty on volatility is amplified
by the state of the economy.24
With a few exceptions,25 models of learning do not generate fluctuations in uncertainty. In
our setup, these fluctuations arise endogenously from the learning exercise: when agents try to
rationalize a particular level of the fundamental, uncertainty depends on it and thus fluctuates
with the state of the economy. This, in equilibrium, generates fluctuations in volatility.
bB − λ
bA . However,
Based on (24), it appears that stock-return volatility also depends on λ
it turns out that this relationship is significantly weaker. In the right panel of Figure 1, we
plot the stock-return volatility versus the fundamental for different values of disagreement.26
The plot shows that disagreement has only a marginal effect on volatility.
We confirm the results from Figure 1 by simulating the economy at a 1-week frequency
over 50 years. We perform 350 such simulations. For each simulated point, we compute the
structural uncertainty U and plot it against the volatility of stock returns. We then perform a
quadratic fit of volatility on structural uncertainty. The quadratic fit line is shown in Figure
2 (solid line), where we also plot the 95% confidence intervals across simulations. The narrow
confidence intervals suggest that structural uncertainty generates most of the fluctuations in
volatility, whereas disagreement has only a minor effect, in line with our interpretation of
Figure 1.
According to Figure 2, fluctuations in the structural uncertainty U are the main driver
of fluctuations in volatility. This implies that the persistence of stock-return volatility is
directly related to the persistence of the structural uncertainty. Given that uncertainty is
endogenously generated by the learning of agents, it is a strongly persistent process27 and
24
A common empirical observation is that bad economic times tend to be characterized by higher uncertainty.
Although this is not a feature of our model, the results from Figure 1 suggest that we should observe countercyclical volatility, i.e., higher volatility during times of higher uncertainty. Veronesi (1999) obtains a similar
result in a single-agent economy in which the set of possible drifts for the aggregate dividend is finite. This
discreteness of the state space generates fluctuating uncertainty.
25
Veronesi (1999, 2000), Xia (2001), David (1997, 2008).
26
For this plot, γ = γss , so that the solid lines in the two panels of Figure 1 are identical.
27
It is straightforward to understand why learning implies persistent uncertainty. If uncertainty is high,
agents need not one but a succession of high quality signals for uncertainty to decrease significantly. Conversely, a succession of low quality signals are needed for uncertainty to increase significantly. Collin-Dufresne,
Johannes, and Lochstoer (2015) show that, when investors need to learn about the constant expected consumption growth rate of the economy, uncertainty converges to a constant steady-state and is therefore persistent.
In their equilibrium model, such persistence implies a large risk premium when investors have Epstein and Zin
(1989) preferences with a sufficiently large elasticity of intertemporal substitution. In our model, uncertainty
is persistent but never converges to a constant steady-state because its dynamics depend on the fundamental,
as shown in (10).
16
0.12
0.12
γ = 0.42
γ = 0.2
γ=0
Volatility
0.1
b = −0.4
∆λ
b=0
∆λ
0.1
0.08
0.08
0.06
0.06
0.04
0.04
−0.02 0
0.02 0.04 0.06 0.08
Fundamental f˜
b = 0.4
∆λ
−0.02 0
0.02 0.04 0.06 0.08
Fundamental f˜
Figure 1: Volatility vs. fundamental
The left panel depicts the stock-return volatility, σ, against the fundamental f˜ for different
values of the uncertainty γ. The solid blue curve is for γ = γss ≈ 0.42, the dashed red curve
for γ = 0.2, and the dotted black curve for γ = 0; for this plot, model disagreement is fixed to
zero. The right panel depicts stock-return volatility, σ, against the fundamental f˜ for different
b≡λ
bB − λ
bA . The dashed red curve is for ∆λ
b = 0.4, the solid
values of model disagreement ∆λ
b
b
blue curve for ∆λ = 0, and the dotted black curve for ∆λ = −0.4; for this plot, uncertainty
is fixed at its steady state value. The dashed vertical lines mark the steady state level of the
fundamental, f¯. Parameter values are provided in Table 1.
consequently the structural uncertainty itself becomes persistent.28 Stock-return volatility is
thus persistent not because some exogenous variable is assumed to be persistent, but because
economic agents try to rationalize the observed fundamental and learn about its underlying
model. This particular type of learning implies a persistent uncertainty process and a strong
relationship between volatility and uncertainty—particularly when the economy is away from
“normal times.” Structural uncertainty offers thus a theoretical foundation of the GARCH
behavior commonly observed in financial markets (Engle, 1982; Bollerslev, 1986).
4.2
Equity Risk Premium and Sharpe Ratio
Proposition 2 shows that the structural uncertainty U and the structural disagreement D have
a direct impact on the risk premium perceived by agent B. The equity risk premium is defined
as the dot product between the market price of risk and the stock return diffusion. This has
an important implication: because structural disagreement is the sole driver of the market
28
The 1-week autocorrelation of uncertainty in our simulated sample is 0.999. The 1-week autocorrelation of
stock-return volatility resulted from our simulated sample is 0.981, showing that volatility is indeed a persistent
process.
17
Quadratic fit
95% CI
0.12
Volatility
0.1
0.08
0.06
0.04
−0.01
0
0.01
Structural uncertainty U
Figure 2: Volatility vs. structural uncertainty: quadratic fit on simulated data
The plot depicts the stock-return volatility, σ, against the structural uncertainty U defined by
(f¯ − f˜)γ. The solid line represents a quadratic fit on the model-generated data resulted from
350 50-year simulations at weekly frequency. The dashed lines represent the 95% confidence
bA , λ
bB are their
bands. Parameter values are provided in Table 1. The initial values for f˜, λ
long-term means, γ at its steady state value, and δ0 = η0 = 1.
price of risk, uncertainty affects the risk premium only when disagreement is different from
zero. In other words, disagreement is the primary channel through which uncertainty affects
the risk premium in the economy.
To illustrate this effect, we plot in panel (a) of Figure 3 the risk premium as a function
of the fundamental f˜, for different values of disagreement. For this plot, we keep uncertainty
fixed at its steady-state level γss . The plot shows that agent B requires a high premium to
hold the risky asset when her model forecasts a less favorable economic outlook, i.e., when
eB − λ
eA = −0.4 and vary the amount of uncertainty
D < 0. In panel (b), we consider the case λ
(the dotted lines in the two panels are identical). The figure shows that uncertainty has an
impact on the risk premium only when disagreement is different from zero. Specifically, higher
uncertainty amplifies the risk premium required by agent B to hold the asset (when agent B’s
model forecasts a less favorable economic outlook).
In panel (c) of Figure 3 we plot the Sharpe ratio versus the fundamental, for different
values of disagreement. The price of risk follows a similar pattern as the risk premium. The
effect of uncertainty on the Sharpe ratio is negligible.
According to the above discussion, the risk premium and the Sharpe ratio are mainly
driven by the structural disagreement D. Furthermore, uncertainty magnifies the risk premium only when disagreement is different from zero. We confirm these effects again through
simulations. We perform 350 simulations of the economy at a 1-week frequency over 50 years.
18
(a) Risk premium
b = 0.4
∆λ
b=0
∆λ
0.05
b = −0.4
∆λ
(c) Sharpe ratio
(b) Risk premium
γ=0
γ = 0.2
γ = 0.42
0
1
0.5
b = 0.4
∆λ
b=0
∆λ
b = −0.4
∆λ
0
−0.05
−0.5
−0.02 0 0.02 0.04 0.06 −0.02 0 0.02 0.04 0.06
Fundamental f˜
Fundamental f˜
−0.02 0 0.02 0.04 0.06
Fundamental f˜
Figure 3: Risk premium and Sharpe ratio vs. fundamental f˜
The left panel plots the equity risk premium perceived by agent B, RPB , against the fundab≡λ
bB − λ
bA . Uncertainty is assumed
mental f˜ for different values of model disagreement ∆λ
to be at its steady state. The middle panel plots the equity risk premium for different values
b = −0.4. The right panel plots the
of the uncertainty. Disagreement is assumed to be ∆λ
Sharpe ratio perceived by agent B, SRB against the fundamental f˜ for different values of
b For both panels, uncertainty is fixed at its steady state value. The
model disagreement ∆λ.
dashed vertical lines mark the steady state level of the fundamental, f¯. Parameter values are
provided in Table 1.
For each simulated point, we compute the structural disagreement D and plot it against the
risk premium. We then perform a cubic fit of the risk premium on the structural disagreement.
The cubic fit line is shown in Figure 4, where we also plot the 95% confidence intervals across
simulations. The narrow confidence intervals suggest that indeed the relationship is non-linear
and uncertainty affects the risk premium only when disagreement is away from zero.29
Disagreement about the speed of mean-reversion implies that the risk premium and the
Sharpe ratio are persistent in equilibrium. Indeed, the 1-week autocorrelations of the risk
premium and Sharpe ratio computed from our simulated sample are 0.948 and 0.959, respectively. Persistence arises because disagreement, which is persistent in our model (1-week
autocorrelation is 0.992), is a key driver of the risk premium and Sharpe ratio.
4.3
Illustration: volatility and risk premium dynamics after a growth
shock
We conclude this section with an illustration of the dynamics of the volatility and risk premium
after the economy experiences a recessionary shock. Suppose that professional forecasters
29
When the structural disagreement D is negative (resp., positive), agent B’s model has a less (resp., more)
favorable economic outlook. Thus, we expect to obtain lines with similar shapes as those in panel (b) of Figure
3.
19
Non-linear fit
95% CI
Risk premium
0.01
0.005
0
−0.005
−0.01
−0.015
−0.003
0
0.003 0.006
Structural disagreement D
Figure 4: Volatility vs. structural disagreement: cubic fit on simulated data
bB −
The graph depicts the risk premium against the structural disagreement D defined by (λ
bA )(f¯ − f˜). The solid line represents the cubic fit function on the model-generated data
λ
resulted from 350 50-year simulations at weekly frequency. The dashed lines represent the
95% confidence bands. Parameter values are provided in Table 1. The initial values for f˜,
bA , λ
bB are their long-term means, γ at its steady-state, and ζ = µ = 0.
λ
estimate the growth rate of the economy at -2.5%. Assume further that the two agents
bA = 1.15
rationalize this level of the fundamental with two different mean-reversion speeds, λ
bB = 0.9 (agent B’s model has a less favorable economic outlook—it predicts a relatively
and λ
longer recession).
Figure 5 illustrates the evolution of the volatility, risk premium and model disagreement
B
b
bA for three years at weekly frequency. The solid lines represent averages across 10,000
λ −λ
simulations.30
Panel (a) shows that volatility increases above 10% (in yearly terms) and then goes down
slowly over the next year (solid line)31 . We plot with dashed lines the 5th and 95th percentiles
computed across 10,000 simulations. The take-away message of this exercise is that volatility
is persistent and features a lot of variability. We have tried several specifications, including an
initial situation without model disagreement, and the results are similar. This illustration thus
shows that learning about the mean-reversion speed of the fundamental induces GARCH-like
30
The average volatility across simulations and time is 4.5%, the average risk premium is 0.7%, and the
average risk-free rate is 8.7%. These averages would better match their empirical counterparts if agents had
habit formation, as in Chan and Kogan (2002), Xiouros and Zapatero (2010), Bhamra and Uppal (2013), and
Ehling, Gallmeyer, Heyerdahl-Larsen, and Illeditsch (2013) among others. As this would not add any insights
to the main predictions of our model, we decide to keep the setup simple and focus on the dynamic properties
of asset prices.
31
Given our parameter values, this increase in volatility is substantial: the volatility of the dividend growth
is 1.3%, the volatility of the fundamental is 1.9% (see Table 1), and the risk aversion is α = 3.
20
(c) Model disagreement
(b) Risk premium
(a) Volatility
0.04
0.1
0.1
0
0.02
0.05
−0.1
−0.2
0
0
0
50
100
Weeks
150
0
50
100
Weeks
150
0
50
100
Weeks
150
Figure 5: Evolution of volatility, risk premium, and model disagreement after a
recessionary shock.
The solid lines show the average values for volatility (left panel), risk premium (middle panel)
and model disagreement (right panel), averaged across 10,000 simulated paths of length 3 years
at weekly frequency. The dashed lines plot the 5th and 95th percentiles computed across the
simulated paths. The starting value of the fundamental is assumed to be f˜ = −2.5% and the
bA = 1.15, λ
bB = 0.9. Parameter values are provided in Table 1. The
mean-reverting speeds λ
initial value for γ is its steady-state, and ζ = µ = 0.
variation in the volatility of stock returns.
Turning to panel (b), the risk premium required by agent B for holding the asset is large and
positive (recall that agent B’s model has a less favorable economic outlook). One important
consideration here is that if we start with an initial situation without disagreement, then
the risk premium is close to zero (although it still experiences fluctuations). Disagreement is
therefore the main driver of risk premia, whereas uncertainty mostly explains the persistent
fluctuations in volatility.
Finally, panel (c) shows the evolution of model disagreement over the 3-year period of
our simulated sample (averaged across 10,000 simulations). Model disagreement takes a long
time to converge back to zero.32 Furthermore, the 5th and 95th percentile lines show that
disagreement is very volatile. The discretion parameter φ is an important determinant of the
variation in disagreement and thus it also drives the variation in the risk premium. Note also
that, although agent B believes initially that the fundamental is more persistent than agent
A, there is a high chance than in the near future these beliefs will reverse and agent B’s model
features a less persistent fundamental.
32
The stochastic process for disagreement is given in Equation (52) in the Appendix. Disagreement meanreverts towards zero with a mean-reverting speed higher or equal to κ. It has a conditional variance of
2Φ2 φ2 (1 − φ2 ) (Equation (53) in the Appendix).
21
5
Conclusion
Understanding how uncertainty affects security prices in financial markets remains one of the
most important issues in finance. Ever since Mehra and Prescott (1985) raised the equity
premium puzzle, many authors have rationalized it by considering that investors may have
heterogeneous beliefs or differences of opinion (e.g., Varian, 1985; Abel, 1989; David, 2008).
But the origins of this disagreement among investors has been assumed to stem from either
a staunch difference of opinion about the best model or about the right outputs. In many
instances, uncertainty is constant and does not play any role.
In this paper, we take a different stance. We show that in the presence of model uncertainty,
disagreement may arise as market participants calibrate their models differently, not because
they stiffly support one output prediction over another, but because they are human beings
facing an under-identified problem. In this light, investors may disagree over time even if they
are sitting side-by-side entering the same inputs and observing what each other do, because
they rationalize differently what they publicly observe.
Because of this calibration process, uncertainty fluctuates and models deviate to varying degrees over time, leading to asset pricing implications. We characterize these effects in
the paper and reach three primary conclusions. First, rationalizing fundamentals differently
implies persistent stock-return volatility, which provides a theoretical explanation for GARCHtype processes. Second, the state of the economy governs the “structural uncertainty”, which
fluctuates and magnifies stock-market volatility in recessions and expansions. Third, model
disagreement is associated with a risk premium and is the primary channel through which uncertainty commands a risk premium. To our knowledge, these are novel implications that add
to the established literatures on learning and on heterogeneous beliefs and provide plausible
explanations for known empirical observations.
22
A
A.1
Appendix
Filtering Problem
Following the notations of Liptser and Shiryaev (2001), agent i observes the vector



dWtδ
dζt
 df˜t  = (A0 + A1 λt )dt + B1 dWtλ + B2 dW f 

(28)
t
dWti
dmit
 
 
 




σδ
0
0
dWtδ
0
0
f˜t − 12 σδ2
p
 + f¯ − f˜t  λt  dt +  0  dWtλ + σf ρ σf 1 − ρ2
 dW f  ,
= 
0
t
p 0
φ
0
0
dWti
0
0
1 − φ2
(29)
where i ∈ {A, B}. The unobservable process λ satisfies

dWtδ
dλt = (a0 + a1 λt )dt + b1 dWtλ + b2 dWtf 
dWti

(30)

dWtδ
= κλ̄ + (−κ)λt dt + ΦdWtλ + 0 0 0 dWtf  .
dWti

(31)
Therefore,
bob = b1 b01 + b2 b02 = Φ2
(32)

σδ2

ρσf σδ 0
0
σf2
BoB =
+
0
1
0
0
boB = b1 B1 + b2 B2 = 0 0 φΦ .
B1 B10
B2 B20
= ρσf σδ
0
bi = EPi (λt |Ot ) has dynamics
The estimated process defined by λ
t



dζt
bi = a0 + a1 λ
bi dt + (boB + γt A0 )(BoB)−1  df˜t  − A0 + A1 λ
bi dt ,
dλ
t
t
1
t
i
dmt
(33)
(34)
(35)
where the uncertainty γ solves the following Ordinary Differential Equation
dγt
= a1 γt + γt a01 + bob − (boB + γt A01 )(BoB)−1 (boB + γt A01 )0 .
dt
(36)
Consequently,

bi = κ λ̄ − λ
bi
dλ
t
t
dt + 0
(f¯−f˜t )γt
√
σf 1−ρ2
23

cδ
dW
t
cf i 
φΦ 
 dW
t ,
c
dWtmi
(37)
where the three Brownians are independent and are defined as follows:
ctδ = dWtδ
dW
cf i =
dW
t
(38)
1
σf
p
1 − ρ2
h
i
bi (f¯ − f˜t )dt − σf ρdW δ
df˜t − λ
t
t
ctmi = dmit
dW
(39)
(40)
The dynamics of uncertainty are
dγt
(f¯ − f˜t )2 2
γ .
= 1 − φ2 Φ2 − 2κγt − 2
dt
σf (1 − ρ2 ) t
A.2
(41)
Proof of Proposition 1
Consider the following 4-dimensional Brownian motion defined under the probability measure of
agent B:


cδ
W
 cf B 
W

c=
dW
(42)
 c mB 

W
c mA∗
W
where the first three Brownians are defined in (38), (39) and (40). The last Brownian is defined such
c mA and dW
c mB is φ2 ):
that (this ensures that the correlation between dW
p
c mA = φ2 dW
c mB + 1 − φ4 dW
c mA∗
dW
(43)
Write the dynamics of the vector of state variables under the probability measure of agent B:
1
2
ct
dζt = f˜t − σδ dt + σδ 0 0 0 dW
2
p
bB f¯ − f˜t dt + σf ρ σf 1 − ρ2 0 0 dW
ct
df˜t = λ
t


bB − λ
bA γt λ
2
t
t
bA +
bA = κ λ̄ − λ
dλ
f¯ − f˜t  dt
t
t
2
2
σf (1 − ρ )
p
γt
3
4
¯
˜
ct
√
f − ft
φ Φ φΦ 1 − φ dW
+ 0
σf 1−ρ2
γt
¯ − f˜t
bB dt + 0
bB = κ λ̄ − λ
ct
√
f
φΦ
0
dλ
dW
t
t
σf 1−ρ2
!
˜t − f¯)2
(
f
dγt = (1 − φ2 )Φ − 2κγt − 2
γ 2 dt
σf (1 − ρ2 ) t
2 2
bB − λ
bA
¯ − f˜t
λ
f
bA ¯ ˜
t
1 t
(λbB
t −λt )(f −ft )
ct ,
√
0 0 dW
dµt = −
dt + 0 −
σf 1−ρ2
2
σf2 (1 − ρ2 )
24
(44)
(45)
(46)
(47)
(48)
(49)
(50)
The dynamics of disagreement are given by:

2 
¯
˜
γ
f
−
f
t
t
p
 bB bA
2
bB − λ
bA = − 
ct ,
4 dW
d λ
κ
+
λ
−
λ
dt
+
0
0
φΦ
1
−
φ
φΦ
1
−
φ


t
t
t
t
σf2 (1 − ρ2 )
(51)
(52)
and thus its conditional variance is
2φ2 Φ2 1 − φ2 .
The optimization problem of agent B is
"Z
#
1−α
c
max E
e−βt Bt dt
cBt
1−α
0
Z ∞
B
s.t. E
ξt cBt dt ≤ xB0 ,
(53)
∞
(54)
(55)
0
where ξ B denotes the state-price density perceived by agent B and xB0 is her initial wealth. Under
the probability measure PB , the problem of agent A is
"Z
#
1−α
∞
c
max E
ηt e−βt At dt
(56)
cAt
1−α
0
Z ∞
B
s.t. E
ξt cAt dt ≤ xA0 .
(57)
0
Note that the change of measure enters directly the objective function of agent A but not its budget
constraint (57). The reason is that the budget constraint depends on the state-price density perceived
by agent B.33
The first-order conditions are
− 1
α
cBt = ΛB eβt ξtB
(58)
1
ΛA βt B − α
cAt =
e ξt
,
(59)
ηt
where ΛA and ΛB are the Lagrange multipliers associated with the budget constraints of agents A and
B, respectively. Summing up agents’ optimal consumption policies and imposing market clearing,
i.e., cAt + cBt = δt , yields the state-price density perceived by agent B
"
#α
1 1/α
ηt 1/α
B
−βt −α
ξt = e δt
+
.
(60)
ΛA
ΛB
Substituting the state-price density ξ B in the optimal consumption policies yields the following
33
Alternatively,
by agent A, ξ A . Then, we would
A weB could
A have
defined
B the
state-price density perceived
A
B
B
have E ξ 1x = E ηξ 1x = E ξ 1x for any event x. That is, ξ = ηξ A .
25
consumption sharing rules
cAt = ωAt δt
(61)
cBt = ωBt δt = (1 − ωAt )δt ,
(62)
where ωit denotes agent i’s share of consumption at time t for i ∈ {A, B}. Agent A’s share of
consumption satisfies
ωAt = ηt
ΛA
ηt
ΛA
1/α
1/α
+
1
ΛB
1/α .
(63)
As in Yan (2008) and Dumas, Kurshev, and Uppal (2009), we assume that the coefficient of
relative risk aversion α is an integer. In this case, the state-price density at time u satisfies
"
#α
ηu 1/α
1 1/α
B
−βu −α
(64)
+
ξu = e
δu
ΛB
ΛA
j
α X
α
ηu ΛB α
−βu −α 1
=e
δu
(65)
ΛB
ΛA
j
j=0
=
e−βu δu−α
j
α j 1 X α
1 α ηt ΛB α αj
ηu
ΛB
ηt
ΛA
j
(66)
j
j α 1 X α
ηu α
ωAt
,
ΛB
ηt
1 − ωAt
j
(67)
j=0
=
e−βu δu−α
j=0
where the last equality comes from the fact that
ωAt = 1 − ωAt = ηt
ΛA
1/α
(68)
1/α
1/α
+ ΛηAt
1/α
1
ΛB
1
ΛB
1/α
1
ΛB
+
ηt
ΛA
1/α ,
(69)
and consequently
ηt ΛB
ΛA
1
α
=
ωAt
.
1 − ωAt
(70)
Rewriting Equation (69) yields
1
ΛB
1/α
+
ηt
ΛA
1/α
=
26
1
1 − ωAt
1
ΛB
1
α
,
(71)
and thus
ξtB
=
e−βt δt−α
1
1 − ωAt
α
1
.
ΛB
(72)
Therefore, the price-dividend ratio satisfies
Z ∞ B
ξu δ u
St
= Et
du
δt
ξtB δt
t


j j
P
α
α
η
α
ω
1
−βu δ −α
u
Z
At
 ∞e
u ΛB
j=0 j
ηt
1−ωAt
δu 
(67) (71) (72)
=
Et 
du
α

1 −α
1
δt 
t
δ
e−βt 1−ω
ΛB t
At


j j
P
α
α
η
α
ω
1−α
u
Z
At
 ∞ −β(u−t) j=0 j

ηt
1−ωAt
δu

= Et 
e
du
α


1
δt
t
(73)
(74)
(75)
1−ωAt
α X
α j
=
ωAt (1 − ωAt )α−j Et
j
j=0
Z
∞
−β(u−t)
e
t
ηu
ηt
j α
δu
δt
!
1−α
du .
Let us define the function Fj (Z) as follows
Z ∞
χ δu
−β(u−t) ηu
Fj (Zt ) ≡ Et
du ,
e
ηt
δt
t
(76)
(77)
>
bA , λ
bB , γ
where = αj , j = 0, . . . , α, χ = 1 − α, and Z ≡ f˜, λ
is a vector of state-variables that
does not comprise ζ = log δ and µ = log η.
Using these notations, the price-dividend ratio satisfies
α St X α j
=
ωAt (1 − ωAt )α−j Fj (Zt )
δt
j
j=0
α X
α j α−j
ωAt ωBt Fj (Zt ),
=
j
(78)
(79)
j=0
which is Equation (22) in Proposition 1.
A.3
Exponential-Quadratic Approximation
We have
− 21
Ru
bA
¯ ˜
(λbB
s −λs )(f −fs )
√
!2
R
ds− tu
bA ¯ ˜
(λbB
s −λs )(f −fs )
√
0 t ηu
σf 1−ρ2
=e
ηt
χ
Ru
R u
˜ 1 2
ct
δu
0 0 0 dW
t χ(fs − 2 σδ )ds+ t χσδ
=e
,
δt
where and χ are some constants.
27
σf
1−ρ2
!
0 0
ct
dW
(80)
(81)
Therefore,

ηu
ηt
δu
δt
Ru
χ
t
χ(f˜s − 1 σ 2 )− 1 2
δ
2
¯ ˜
bA
(λbB
s −λs )(f −fs )
√
σf
=e

− 21
Ru
t
χ2 σ 2 +2
δ
!2
1−ρ2
¯ ˜
bA
(λbB
s −λs )(f −fs )
√
σf
×e

+ 12 χ2 σδ2 +2
1−ρ2
!2 
R
ds− u
t
¯ ˜
bA
(λbB
s −λs )(f −fs )
√
σf
1−ρ2
−χσδ !2 
ds
(82)
bA ¯ ˜
(λbB
s −λs )(f −fs )
√
σf
1−ρ2
!
0 0
ct
dW
.
(83)
Note that last term of the first row cancels the first term of the second row. Importantly, the second
row defines a change of measure. The change of measure is

!
!2 
b B −λ
bA )(f¯−f˜s )
b A )(f¯−f˜s )
b B −λ
λ
R
Ru
λ
(
(
s
s
s
s
1 u 2 2
2
ct

√
√
− 2 t χ σδ +
ds− t −χσδ 0 0 dW
σf 1−ρ2
dP̄ σf 1−ρ2
≡ νt = e
,
dPB Ot
(84)
where the P̄-Brownian motion W̄ is defined as
ct + yt dt
dW̄t = dW
>
bA ¯ ˜
(λbB
t −λt )(f −ft )
√
yt = −χσδ 0
0
.
2
σf
1−ρ
Rewriting the problem under the probability measure P̄ yields
Z ∞ R
u
Xs ds
t
F (Zt ) ≡ Ēt
e
du ,
(85)
(86)
(87)
t
where Xt = −β +
bB bA 2 ¯ ˜ 2
1 (λt −λt ) (f −ft )
2
σf2 (1−ρ2 )
2 − + χ f˜t − 12 σδ2 + 21 χ2 σδ2 . For notational ease, we drop
the index j when defining the function F (.) by keeping in mind that = αj and χ = 1 − α in our
setup. We now transform this expression to obtain a P̄-martingale. We have
Z ∞ R
Z t R
Rt
u
u
X
ds
X
ds
X
ds
s
s
F (Zt )e 0 S +
e0
du = Ēt
e0
du ≡ M̄t ,
(88)
0
0
where M̄ is a P̄-martingale.
Applying Itô’s lemma to the martingale M̄ and setting its drift to zero yields the following Partial
Differential Equation for the function F (Z)
L Z F (Z) + F (Z)X(Z) + 1 = 0,
(89)
where L Z is the P̄-infinitesimal generator with respect to the vector of state variables Z. The
dynamics of the vector of state variables are defined as follows:
ct .
dZt = µ(Zt )dt + σ(Zt )dW
(90)
Since the function F (Z) is a standard transform, we rewrite PDE (89) by setting F (Z) = eL(Z) .
28
That is, the function L(Z) solves the following PDE
i
1 h
∇L(Z)> µ(Z) + tr ∇L(Z)∇L(Z)> + Hess(L(Z)) σ(Z)σ(Z)> + X(Z) + e−L(Z) = 0,
2
(91)
(92)
2
∂
∂
where ∇f (Z) = ∂Z
f (Z) is the gradient of the function f (.), Hess(f (Z)) = ∂Z
2 f (Z) is the Hessian
matrix of the function f (.), and tr(.) is the trace operator.
Because each function can be characterized by a Taylor expansion around a reference point Z0 ,
the solution to PDE (92) can be expressed in the following Polynomial representation
L(Z) =
∞
X
ai (Z0 )(Z − Z0 )i
(93)
|i|=0
=
X
ai (Z0 )(Z − Z0 )i +
X
ak (Z+ )(Z − Z0 )k
(94)
|k|=m+1
|i|≤m
≡ L̄(Z) + R (Z − Z0 )m+1 ,
(95)
where L : Rn → R; the multi-indices i, k are such that |i| = i1 + · · · + in and |k| = k1 + · · · + kn ;
the ai (Z0 ) coefficients characterize the partial derivatives of degree |i| of the function L(Z) at the
reference point Z0 ; the ak (Z+ ) coefficients characterize the partial derivatives of degree |k| = m + 1
of the function L(Z) at a point Z+ ∈ ΛZ,Z0 , where ΛZ,Z0 is the line segment connecting Z and
Z0 ; Y l = Y1l1 · · · Ynln for any n-dimensional
vector Y ; L̄ : Rn → R is an approximation of L(.) of
degree m; and R (Z − Z0 )m+1 is the remainder of degree m + 1. As m converges to infinity, the
approximation L̄(X) converges to the true function L(Z) because the remainder R (Z − Z0 )m+1
converges to zero.
The procedure to solve for the coefficients ai (Z0 ), |i| ∈ N in (93) consists in (1) substituting (93)
in PDE (92), (2) performing a Taylor expansion of the resulting expression at the reference point Z0 ,
and (3) setting the loadings on Y 0 ≡ (Z − Z0 )0 , Y 1 ≡ (Z − Z0 )1 , Y 2 ≡ (Z − Z0 )2 , . . . to zero. Since
this problem involves an infinite number of equations with an infinite number of unknowns, the idea
is to truncate the Taylor expansion of L(Z) and that described in (2) at a degree m and solve for
the coefficients ai (Z0 ), |i| ≤ m. That is, the latter procedure characterizes the approximation L̄(Z)
defined in (95).
As noted by Benzoni, Collin-Dufresne, and Goldstein (2011), PDE (92) would admit a closed
form solution of the form
L(Z) = A + B > Z
(96)
if the last term were absent (e−L(Z) ≈ 0) and both the vector of state variables Z and X(Z) were
affine in Z. Assuming again that the last term were absent, PDE (92) would admit a closed form
solution of the form
L(Z) = A + B > Z + Z > CZ
(97)
if the vector of state variables Z were affine-quadratic and X(Z) were quadratic in Z (Cheng and
Scaillet, 2007). It is worth noting that, even in a single-agent economy where the fundamental f˜ and
29
·10−9
·10−3
·10−11
PDE Residuals
2
0
5
−2
0
0
−4
−5
−2
0
0.02
0.04 0.06
Fundamental f˜
−0.4 −0.2 0
0.2 0.4
b≡λ
bB − λ
bA
Disagreement ∆λ
−6
0
0.2
0.4
Uncertainty γ
Figure 6: PDE Residuals vs. State Variables
The left, middle, and right panels depict the PDE residuals against the fundamental f˜, model
b≡λ
bB − λ
bA , and uncertainty γ, respectively. If not mentioned otherwise, the
disagreement ∆λ
bA = λ
bB = λ̄, and γ = γss . Parameter values are provided in Table
state variables are f˜ = f¯, λ
1.
b have constant diffusions
the mean-reversion speed λ
bt f¯ − f˜t dt + σ ˜dW
ct
df˜t = λ
f
bt = κ λ̄ − λ
bt dt + σλ dW
ct ,
dλ
(98)
(99)
the vector of state variables belongs to a class of processes that is more complex that the affineb2 , and f˜λ
b and computing
quadratic class. Indeed, augmenting the vector of state variables with f˜2 , λ
the drift and variance-covariance matrix of the augmented vector involves terms of order larger than
two. This shows that, even in a single-agent economy, time-varying mean-reversion implies a solution
to PDE (92) that is more complex than the quadratic form in (97).
To solve our problem, we do however choose to truncate the Taylor expansion of L(Z) around
the reference point Z0> = (f¯, λ̄, λ̄, γss )> at a degree m = 2, and therefore consider a quadratic
approximation of the form
L̄(Z) = A + B > Z + Z > CZ,
(100)
>
bA , λ
bB , γ
because adding terms of higher order does not significantly affect (1)
where Z > = f˜, λ
the residuals of the PDE34 and (2) the value of the approximation. Figure 6 depicts the residuals
of the PDE when = 0 and χ = 1 − α = −2 for different values of the state variables. Note that
we choose to illustrate the PDE residuals when = 0, as opposed to ∈ {1/3, 2/3, 1}, because this
parametrization yields the largest residuals. This figure shows that our approximation is accurate
for a large range of values of the state variables.
34
The residuals of the PDE are simply obtained by plugging the approximation (100) in PDE (92). By definition, a closed form solution to PDE (92) yields residuals that are equal to zero, whereas a good approximation
yields residuals that are close to zero.
30
A.4
Calibration to the U.S. Economy
In the model, each agent models the dynamics of the dividend growth and the expected dividend
growth using three sources of risk: a dividend growth shock, an expected dividend growth shock,
and an information shock. Because agents consider three shocks but we only observe two time-series,
the model cannot be estimated by Maximum-Likelihood. Moreover, the fact that the fundamental
features a stochastic mean-reversion speed implies that only few moments can be computed in closed
form and thus we cannot apply the Generalized Method of Moments either.
We choose therefore to estimate the parameters of the model by the Simulated Method of Moments (SMM). The data that we used spans Q1:1969-Q3:2014 and was obtained from the Federal
bB , and
Reserve Bank of Philadelphia. First, we simulate 20,000 paths of the state variables ζ, f˜, λ
γ over a 45-year horizon at weekly frequency. The horizon is chosen to match its empirical counterpart, and the relatively high frequency is chosen to mitigate simulation errors generated by the Euler
discretization scheme. Second, we record the log-dividend growth rate and the expected dividend
growth rate at a quarterly frequency for each simulation. Third, we compute the following moments
and their average across simulations:
• the mean of the log-dividend growth rate,
• the volatility of the log-dividend growth rate,
• the 1-quarter autocorrelation of the log-dividend growth rate,
• the mean of the expected dividend growth rate,
• the volatility of the expected dividend growth rate,
• the 1-quarter autocorrelation of the expected dividend growth rate,
• the correlation between the log-dividend growth rate and the expected dividend growth rate,
• the 2-, 3-, and 4-quarter-ahead variance ratios of the log-dividend growth rate,
• the 2-, 3-, and 4-quarter-ahead variance ratios of the expected dividend growth rate for each
simulation.
The vector of parameters, Θ = (σδ , f¯, σf , ρ, κ, λ̄, Φ, φ)> , is chosen to minimize to following criterion
Θ = argmin[M e − M (Θ)]> W [M e − M (Θ)],
(101)
where M (Θ) is the 13-dimensional vector of simulated moments, M e is its empirical counterpart,
W = Ω−1 is the optimal weighting matrix, and Ω is the variance-covariance matrix (computed across
simulations) of the vector of moments M (Θ).
Table 1 presents the estimated parameters and their statistical significance. All but the precision
of the signal φ are statistically significant at the 99% confidence level; the precision φ is statistically
significant at the 95% confidence level. The volatility of dividend growth σδ , the long-term mean
of the fundamental f¯, and the volatility of the fundamental σf are consistent with the values used
in the asset-pricing literature (e.g., Bansal and Yaron, 2004; Brennan and Xia, 2001; Bansal, Kiku,
and Yaron, 2010; Croce, Lettau, and Ludvigson, 2014). The positive correlation ρ between the
fundamental and the dividend growth rate reflects the fact that analysts use the observation of the
GDP to forecast its future growth. Indeed, analysts adjust their forecasts upward when they observe
a positive growth surprise, and downward when they observe a negative growth surprise. Moreover,
a correlation coefficient smaller than one shows that, on top of the observation of the GDP, analysts
31
use other sources of information to infer the expected growth. The long-term mean-reversion speed
of the fundamental implies a half-life of approximately seven months, which is significantly shorter
than what is assumed in the long-run risk literature. The mean-reversion speed itself is relatively
persistent as its half-life is approximately 17 months. It is also highly volatile (Φ is large), which
lends support to our assumption of time-varying mean-reversion speed. The discretion parameter φ
is significantly higher than zero.
32
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