Speculative Betas∗
Harrison Hong†
David Sraer‡
Preliminary – Please do not circulate this draft.
First Draft: September 29, 2011
This Draft: May 4, 2012
Abstract
We provide a theory for why high beta assets are more subject to speculative overpricing
than low beta ones. When investors disagree about the common factor of cash-flows,
high beta assets are more sensitive to this aggregate disagreement and hence experience
a greater divergence-of-opinion about their payoffs. Short-sales constraints for some
investors such as retail mutual funds result in high beta assets being over-priced. When
aggregate disagreement is low, expected return increases with beta due to risk-sharing.
But when it is large, expected return initially increases but then decreases with beta.
High beta assets also have greater shorting from unconstrained arbitrageurs and greater
share turnover. Using security analysts’ forecasts about firm earnings, we find evidence
consistent with these predictions.
∗
Hong acknowledges support from the National Science Foundation through grant SES-0850404. Sraer
gratefully acknowledges support from the European Research Council (Grant No. FP7/2007-2013 - 249429)
as well as the hospitality of the Toulouse School of Economics. For insightful comments, we thank Andrea
Frazzini, Augustin Landier, Ailsa Roll and seminar participants at the Toulouse School of Economics, Norges
Bank-Stavanger Microstructure Conference, Norges Bank, Brazilian Finance Society Meetings, Indiana University, Arizona State University, the 2nd Miami behavioral finance conference and CNMV Securities Market
Conference, University of Bocconi, University of Lugano, University of Colorado, Brandeis University, and
Temple University.
†
Princeton University and NBER (e-mail: [email protected])
‡
Princeton University and CEPR (e-mail: [email protected])
1.
Introduction
There is compelling evidence that the risk and return trade-off, the cornerstone of modern
asset pricing theory, is often of the wrong sign. This literature, which dates back to the
works of Black (1972) and Black, Jensen, and Scholes (1972), shows that low risk stocks,
as measured by a stock’s co-movement with the stock market or Sharpe (1964)’s Capital
Asset Pricing Model (CAPM) beta, have significantly outperformed high risk stocks over
the last thirty years.1 For instance, Baker, Bradley, and Wurgler (2011) point out that a
dollar invested in a portfolio of the lowest decile of beta stocks on January of 1968 would
have yielded $11 in real terms at the end of December 2008. A dollar invested in the highest
decile of beta stocks would have yielded around 64 cents. This under-performance is as
economically significant as famous excess stock return predictability patterns such as the
value-glamour or price momentum effects.2
We provide a theory for this high-risk and low-return puzzle by allowing investors to disagree about the market or common factor of firm cash flows and prohibiting some investors
from short-selling. To begin with, there is substantial evidence of disagreement among professional forecasters’ and households’ expectations about many macroeconomic state variables such as market earnings, industrial production growth and inflation (Cukierman and
Wachtel (1979), Zarnowitz and Lambros (1987), Kandel and Pearson (1995), Mankiw, Reis,
and Wolfers (2004)). In particular, dispersion in professional forecasts about stock market
earnings varies over time and is correlated with aggregate economic uncertainty indicators
(Bloom (2009), Lamont (2002)). Disagreement might emanate from many sources such as
heterogeneous priors or cognitive biases such as overconfidence that result in agents over1
A non-exhaustive list of studies include Haugen and Heins (1975), Blitz and Vliet (2007), Cohen, Polk,
and Vuolteenaho (2005)), Frazzini and Pedersen (2010), Ang, Hodrick, Xing, and Zhang (2006).
2
The value-growth effect (Fama and French (1992), Lakonishok, Shleifer, and Vishny (1994)), buying
stocks with low price-to-fundamental ratios and shorting those with high ones, generates a reward-to-risk or
Sharpe (1964) ratio that is two-thirds of a zero-beta adjusted strategy of buying low beta stocks and shorting
high beta stocks. The corresponding figure for the momentum effect (Jegadeesh and Titman (1993)), buying
past year winning stocks and shorting past year losing ones, is roughly three-fourths of the long low beta,
short high beta strategy.
1
weighing private signals.3
Moreover, most investors in the stock market such as retail mutual funds, which in 2010
have 20 trillion dollars under management, are prohibited from shorting by charter (Almazan,
Brown, Carlson, and Chapman (2004)) or through the use of derivatives (Koski and Pontiff
(1999)). Only a small subset of investors, such as hedge funds with 1.8 trillion dollars in
asset management, tend to short. Short-sales constraints that bind for some investors are
due to institutional rationales as opposed to the literal physical cost of shorting.4
We incorporate these two assumptions into a CAPM framework, in which firms’ cash
flows follow a one factor model. Investors only disagree about the mean of the macro-factor
or common component of cash flows. There are two groups of investors: buyers such as
retail mutual funds and arbitrageurs such as hedge funds who can also short. Our model is
the multi-asset extension of Chen, Hong, and Stein (2002)’s rendition of Miller (1977) who
originally considered how disagreement and short-sales constraints affects the pricing of a
single stock. Large divergence of opinion leads to over-pricing because price reflects only
the views of the optimists as pessimists are sidelined due to short-sales constraints. The
consideration of a general disagreement structure about both means and covariances of asset
returns with short-sales restrictions in a CAPM setting is developed in Jarrow (1980), who
shows that short-sales restrictions in one asset might increase the prices of others. It turns
out that a focus on a simpler one-factor disagreement structure about common cash-flows
yields closed form solutions and a host of testable implications for the cross-section of asset
prices that would otherwise not be possible.
Our main result is that high beta assets are over-priced compared to low beta ones when
disagreement about the common factor or the aggregate market is high. High beta stocks
are more subject to investor disagreement about the market factor of cash flows because
3
See Hong and Stein (2007) for a discussion of the various rationales. A large literature starting with
Odean (1999) and Barber and Odean (2001) argues that retail investors engage in excessive trading due to
overconfidence.
4
See Lamont (2004) for a discussion of the many rationales why there is a tremendous bias against
shorting in financial markets, including historical events such as the Great Depression in which short-sellers
were blamed for the Crash of 1929.
2
they load more on this common factor than low beta stocks. If investors disagree about the
macro-economy, their views about the cash flows of high beta stocks like retail companies
will diverge much more than their views about low beta stocks like utilities. Because of
short-sales constraints, there is over-pricing of high beta stocks as their prices only partially
reflect the pessimists’ beliefs. During times of high uncertainty and disagreement, high beta
stocks are only held by optimists. Arbitrageurs attempt to correct this over-pricing but their
limited risk aversion results only in limited shorting leading to over-pricing in equilibrium.5
The reason high beta assets get short-sold first is that a larger fraction of their variance
comes from systematic as opposed to idiosyncratic variance, which turns out to be true
empirically as we discuss below. Our theory more generally predicts that stocks that have
a higher fraction of systematic to total variance hit the short-sales constraints first and are
thus more likely to be overpriced.
Our simple model delivers the following novel testable implications. When disagreement
is low, all investors are long and short-sales constraints do not bind in that no investor takes
a short position. The traditional risk-sharing motive leads high beta assets to attract a
lower price or higher expected return. The relationship between risk and return is kinkshaped for higher levels of aggregate disagreement. For assets with a beta below a certain
cut-off, expected returns are increasing in beta as there is little disagreement about these
stock’s cash flows and therefore no short-selling in equilibrium. For assets with a beta above
an equilibrium cut-off, disagreement about the dividend becomes sufficiently large that the
pessimist investors are sidelined. As disagreement increases, the cut-off level for beta below
which there is no desire to short falls and the expected return initially rises with beta but
then falls with beta above this cut-off.6
5
High beta stocks might also be more difficult to arbitrage because of incentives for benchmarking and
other agency issues (Brennan (1993), Baker, Bradley, and Wurgler (2011), Karceski (2002).
6
When aggregate disagreement is so large that pessimists short-sell all assets, the relationship between
risk and return can be entirely inverted and the entire market becomes overpriced, leading to a negative
market return. We assume that all assets in our model have a strictly positive loading on the aggregate
factor. Thus, it is always possible that pessimists want to be short an asset, provided disagreement is large
enough.
3
Our model also delivers the following three auxiliary predictions. First, there is more
investor disagreement about the cash flows of high beta assets during times when there is
more uncertainty or disagreement about the market. Second, high beta stocks are more
likely to be shorted by arbitrageurs and especially so when disagreement is high. And third,
in a dynamic extension of our static model, we show that as disagreement increases, the
share turnover gap between high and low beta assets also increases. Investors anticipate
that high beta assets are more likely to experience binding short-sales constraints and hence
have a higher resale option than low beta ones (Harrison and Kreps (1978), Morris (1996),
Scheinkman and Xiong (2003) and Hong, Scheinkman, and Xiong (2006)). Since a high
resale option leads to high turnover, high beta assets end up having more share turnover
than low beta assets. Our insight that high beta assets are more speculative and have higher
turnover builds on Hong and Sraer (2011)’s analysis of credit bubbles. They point out how
debt, with a bounded upside, is less disagreement sensitive than equity and hence less prone
to speculative over-pricing and over-trading.
We test these predictions using a monthly time series of aggregate disagreement. Disagreement for a stock’s cash-flow is simply measured by the standard deviation of its analysts
forecasts as in Diether, Malloy, and Scherbina (2002). The aggregate disagreement measure
is typically a beta-weighted average of analyst earnings forecast dispersion for all stocks, similar in spirit to Yu (2010). These two papers find that dispersion of these forecasts predicts
low returns in the cross-section and for the market index in the time series respectively, consistent with the standard predictions of models with disagreement and costly short-selling.
Our theory suggests that we weigh each stock’s disagreement measure by that stock’s beta.
After all, stocks with very low betas have by definition almost no sensitivity to aggregate
disagreement, and their disagreement should mostly reflect idiosyncratic disagreement.
Our empirical analysis delivers the following results. First, aggregate disagreement magnifies the increasing relationship between stock-level short-interest ratio, disagreement, share
turnover and beta. Second, in months with low aggregate disagreement, expected returns
4
are increasing with beta. Third, in months with high aggregate disagreement, there is an
inverted-U shape to the risk-return relationship. This is shown in the context of a standard
Fama-MacBeth analysis where the concavity of the excess return/β relationship is shown to
be strictly increasing with aggregate disagreement. Finally, the average slope of the Security
Market Line is strictly decreasing with our measure of aggregate disagreement.
Our model provides an alternative to Black (1972)’s model for the low-return and high
risk puzzle as emanating from leverage constraints. The unique inverted-U shape between
risk and return, predicted by our model and confirmed in the data, is not predicted in his
model. Our model also naturally generates market segmentation in the sense that during
high uncertainty periods only optimist own high beta stocks. Hence, we deliver an analog
to Merton (1987)’s segmented CAPM due to clientele effects, except that volatility attracts
lower returns in our setting due to speculation as opposed to higher returns in his setting due
to risk absoption. More generally, our model generates radically different predictions from
a risk-sharing, liquidity motive or even many behavioral approaches for the cross-sectional
pricing of assets. In Delong, Shleifer, Summers, and Waldmann (1990), high noise trader risk
yields high return. In Campbell, Grossman, and Wang (1993), high liquidity risk yields high
expected return. In Barberis and Huang (2001), mental accounting by investors still leads to
a positive relationship between risk and return. The exception is the model of overconfident
investors and the cross-section of stock returns in Daniel, Hirshleifer, and Subrahmanyam
(2001) that might yield a negative relationship as well.
Our paper proceeds as follows. We present the model in Section 2. We describe the
empirical findings in Section 3. We conclude in Section 4. All proofs are in the Appendix
Section 5.
5
2.
Model
2.1.
Static Setting
We consider an economy populated with a continuum of investors of mass 1. There are two
periods, t = 0, 1. There are N risky assets and the risk-free rate is r and is exogenous.
Risky asset i delivers a dividend d˜i at date 1. We decompose the dividend processes into
systematic and idiosyncratic components:
∀i ∈ {1, . . . , N },
d˜i = wi z̃ + ˜i ,
where z̃ ∼ N (z̄, σz2 ), ˜i ∼ N (0, σ2 ), and Cov (z̃, ˜i ) = 0 ∀i ∈ {1, . . . , N }. wi is the cash-flow
beta of asset i and is assumed to be non-negative. Our model assumes homoskedasticity of the
dividend process. If dividends are heteroskedastic, what matters for equilibrium asset prices
is no longer the cash-flow beta wi but the ratio of the cash-flow beta wi to the idiosyncratic
volatility of stock i. In the data, however, there is a strong monotonic relationship between
betas and idiosyncratic volatility. For instance, figure A shows the ratio of aggregate to total
volatility for 20 β portfolios from 1981 to 2010.7 This assumption is thus innocuous from an
empirical perspective.8
Each asset i ∈ {1, . . . , N } is in supply
19
N
and we assume w.l.o.g. that:
w1 < w2 < · · · < wN .
That is, we index assets in the economy by their cash-flow betas, with these betas increasing
in i. Asset N has the highest cash-flow beta while asset 1 has the lowest. We assume that
7
For the construction of our 20 β portfolios, see Section 3.2.
The general model with heteroskedastic dividends is easily derived and available from the authors upon
request.
9
This normalization is without loss of generality. If asset i is in supply si , then what matters is the
ranking of assets along the wsii dimension. The rest of the analysis is then left unchanged
8
6
the value-weighted average w in the economy is 1 (
PN
wi
i=1 N
= 1).10
The population of investors is divided into two groups. A fraction α of investors hold
heterogenous beliefs and cannot short. For concreteness, one might interpret these buyers
as mutual funds, who are generally prohibited from going short. These investors are divided
in two groups of mass 21 , A and B, which hold heterogenous beliefs about the mean value of
the aggregate shock z̃. Agents in group A believe that EA [z̃] = z̄ + λ while agents in group B
believe that EB [z̃] = z̄ − λ. We assume w.l.o.g. that λ > 0 so that group A is the optimistic
group and group B the pessimistic one. Investors maximize their date-1 wealth and have
mean-variance preferences:11
U (W̃j,1 ) = Ej [W̃j,1 ] −
1
V ar(W̃j,1 )
2γ
where j ∈ {A, B} and γ is the investors’ risk tolerance.
A fraction of 1 − α of ”a” investors (the “arbitrageurs”) have homogeneous and correct
beliefs and are not subject to the short-sales constraint.12 They also have mean-variance
preferences given by:
U (W̃a,1 ) = Ej [W̃a,1 ] −
1
V ar(W̃a,1 ).
2γ
The following theorem characterizes the equilibrium.
α
Theorem 1. Let θ = 1−2 α and let (u)i∈[0,N +1] be a sequence such that uN +1 = 0,
P 2 P
wj
1
2
2
2
2
ui = N wi σ + σz
w
+
σ
for i ∈ [1, N ] and u0 = ∞. u is a strictly
z
j≥i N
j<i j
decreasing sequence.
10
This is just to ensure that as markets become complete (N → ∞), the economy remains bounded.
In our static model, investors could also simply be endowed with CARA preferences.
12
That these investors have homogenous beliefs is simply assume for expositional convenience. Heterogeneous priors for unconstrained investors wash out in the aggregate and have thus no impact on equilibrium
asset prices.
11
7
Suppose that disagreement λ is such that for some ī ∈ [1, N ]:
uī−1 > λγ ≥ uī
(1)
Group A agents are long all assets. Group B agents are long assets i < ī − 1. Equilibrium
asset prices are given by:

1


wi σz2 +
z̄wi −


γ




1
Pi (1 + r) =
 z̄wi −
wi σz2 +


γ





σ2
N
σ2
N
for i < ī


σz2
N
P


1
wi 
 −  for i ≥ ī
P
+
2
γ
N
σ2 + σz2
i<ī wi
|
{z
}
θσ2
λγ −
i≥ī wi
(2)
π i =speculative premium
Assets with high cash-flow betas, i.e. i ≥ ī, are over-priced (relative to the benchmark
with no short-sales constraints or when α = 0) and the degree of over-pricing is increasing
with disagreement λ and with the fraction of short-sales constrained agents α. Assets with
high cash-flow betas also receive greater short interest than low beta assets and this difference
is increasing with disagreement λ.
Proof. See Appendix.
Holding fixed the other parameters, high wi assets are more sensitive to aggregate disagreement (λ) and hence experience a greater divergence of opinion than low wi assets. When
disagreement grows, agents in the pessimistic group become more pessimistic on high wi
stocks first and then on low wi stocks. Thus, their desired holdings of risky securities decreases and especially more so for high wi stocks, until it becomes negative. At this point,
agents in group B becomes short-sales constrained on high wi stocks. This creates a speculative premium or over-pricing on these stocks. The larger the disagreement, the more high
wi stocks experience short-sales constraints and thus the greater the over-valuation.
When there are only arbitrageurs, α = 0, there is no disagreement left in the model and
the standard CAPM formula applies. When α > 0, θ > 0 and the equilibrium prices of high
wi assets, i.e. those assets such that wi ≥ wī , depend on λ and there is then over-pricing
8
since group B investors are prevented to short. The amount of mispricing is also strictly
increasing with λ. Thus, arbitrageurs will try to short high β assets first, but especially so
when aggregate disagreement λ is large.
When short-sales constraints are binding, the difference between the unconstrained price
and the constrained price is given by:
θσ 2
π = γ
i
wi
λγ −
σ2 +
σz2
N
σz2
P
i≥ī
P
wi
2
i<ī wi
!
1
−
N
!
,
This term, which we call the speculative premium, captures the degree of over-pricing due to
the short-sales constraints. This speculative premium is strictly increasing with θ, i.e. with
the fraction of short-sales constrained agents α. For a fixed wi , the larger is the divergence
of opinion λ, the greater the over-pricing. The greater is the risk absorption capacity
σ2
,
γ
the
greater is the over-pricing. When there is less risk absorption capacity, even in the presence
of disagreement, investors require a lower price to hold the asset and everyone, even the
pessimists, are long the asset.
It is interesting to consider more deeply the intuition for why the high beta assets hit
short-sales constraints first. As we alluded to in the Introduction, the reason is that the
fraction of systematic variance to total variance is higher for high beta assets since we
assumed that the idiosyncratic variance is the same across all assets. It turns out that this
is a good assumption as there is little correlation between beta and idiosyncratic volatility.
If high beta assets also have higher idiosyncratic volatility, then our findings need not hold.
In these disagreement models, greater idiosyncratic volatility makes prices low due to a
positive net supply of the assets and hence requires greater disagreement over cash flows for
short-sales constraints to bind. Our theory more generally predicts that stocks that have a
higher fraction of systematic to total variance should get shorted first and experience more
overpricing.
We can also restate the equilibrium prices as expected returns and relate them to the
9
familiar βi from the CAPM. Because stocks with large w’s naturally have larger β’s, we
show here that higher disagreement will flatten the Security Market Line when there is
costly short-selling.
Corollary 1. Let βi =
Cov(R̃i ,R˜M )
˜ )
V ar(RM
and ī is such that Group B investors are long assets i < ī
and short assets i ≥ ī. Define κ(λ) =
2 P
σz
σ2 λγ− N ( i≥ī wi )
P
2
σz2 σ2 +σz2 (
i<ī wi )
> 0. The expected returns are given
by:


σz2 +


 βi
γ
E[R̃i ] =
2

 σz +

 βi
γ
σ2
N
σ2
N
for i < ī
(1 − θκ(λ)) +
σ2
θ (1 + κ(λ)) for i ≥ ī
γN
The intuition for this corollary is the following. Notice that for α = 0 (θ = 0), agents have
homogenous beliefs so that λ does not affect the expected returns of the assets and the
standard CAPM formula holds. Precisely, the expected returns depend on the covariance of
the asset with the market return, and the risk premium is simply determined by the ratio of
the variance of the market return (which is close to the variance of the aggregate factor σz2
when N is large) to the risk tolerance of investors γ.
However, when a fraction α > 0 of agents are short-sales constrained and disagreement
is large enough, the expected returns for assets i ≥ ī depend on the disagreement parameter
λ. Higher beta stocks are subject to more disagreement as to their expected cash flows and
will experience more binding short-sales constraints, higher prices and hence lower expected
returns. The CAPM does not hold and the Security Market Line is kink-shaped. For assets
with a beta above some cut-off (ī is determined endogenously and depends itself on λ), the
expected return is increasing with beta but at a lower pace than for assets with a beta below
this cut-off (this is the −θκ(λ) < 0 term above). This is due to high beta assets being
over-priced and having lower expected returns due to costly short-sales constraints.
The return/β relationship is not linear in the model. However, it is easily seen that
provided N is large enough, the relation between expected excess returns and β will be close
to linear. As an illustration, Figure 2 shows the actual relationship between excess returns for
10
the following parameterization, which represents a simple calibration of our model: N = 100,
σz2 = .0022, σ2 = .029, γ = .6 and θ = 1.75 (i.e. α ≈ .63.). The calibration for σz2 and
σe psilon2 correspond to the empirical aggregate and idiosyncratic variance of monthly stock
returns over the 1970-2010 period.13 α ≈ .63 corresponds to the fraction of the stock market
held by mutual funds and retail investors, for which the cost of shorting is presumably nontrivial. γ is set to .6 so as to match the average in-sample monthly stock excess returns
observed in the data of .5% monthly. Three cases are displayed on this figure: λ = 0 (no
disagreement and no shorting at equilibrium), λ = .005 (51 assets are shorted in equilibrium)
– this corresponds to 1/10 of the standard deviation of stock return and λ = .0075 (77 assets
shorted at equilibrium). When λ is equal to 0, the standard CAPM analysis applies and
the Security Market Line is an upward slopping line. As λ rises, the security market line
rises with beta up to some point and then its slope flattens (Case λ = .005). For large λ,
the security market line is initially upward slopping and then strictly decreasing with beta
(Case λ = .0075).
In our empirical analysis below, we look for a concavity in the expected excess return /
β relationship, as suggested in our model and in Figure 2. In addition, rather than relying
fully on the structure of the model, we take a simpler approach and also estimate directly
the slope of the security market line, i.e. the coefficient estimate of an OLS regression of
realized excess returns on β. We show in the next corollary that that our model predicts
that this coefficient is strictly decreasing with idiosyncratic disagreement.
Corollary 2. Let µ̂ be the coefficient estimate of a cross-sectional regression of realized
returns R̃i on βi (and assuming there is a constant term in the regression). The coefficient
µ̂ decreases with λ the aggregate disagreement. This effect is larger when there are less
arbitrageurs in the economy (i.e. when α increases).
In the absence of disagreement and short-sales constraints (α = 0 – i.e. all investors are
13
Aggregate variance is the variance of the market return. Idiosyncratic variance is the variance of the
residual of a CAPM regression of monthly returns on the contemporaneous market return.
11
arbitrageurs), the slope of the security market line is simply µ̂ =
P
σz2 +( N
i=1
γ
1
N2
)σ2
. When at
least one asset is being constrained (i.e. when λγ ≥ uN ), µ̂, the slope of the security market
line (as estimated from a regression of excess returns on β), is strictly decreasing with λ, the
aggregate disagreement parameter. In particular, it is direct to show that µ̂ will be strictly
negative, provided that λ is large enough relative to γ (i.e. that the speculation motive for
trading is large relative to the risk-sharing motive for trading). Furthermore, the role of
aggregate disagreement is magnified by the presence of short-sales constrained agents: in an
economy with a low fraction of arbitrageurs (i.e. high short-sales constraints), an increase
in λ leads to a much larger decrease in the estimated slope of the security market line than
in an economy with many arbitrageurs.
Thus our proposition states that: (1) the slope of the SML (µ̂) is strictly lower when
short-sales constraints are binding (λγ > uN ) than in the absence of binding short-sales
constraints (λγ < uN ) and (2) the slope of the security market line is strictly decreasing
with λ as soon as λγ > uN . In particular, provided γλ is high enough, the estimated slope
of the security market line µ̂ becomes negative.
2.2.
Limiting Cases
In this section, we consider the role disagreement may play in a market with an infinite
number of assets but where shorting is costly.
To simplify the exposition, consider the following parameterization:14 for all i > 0,
PN
wi = N2i+1 . Then, wi ∈]0, 2[ and
i=1 wi = 1. The next proposition characterizes the
security market line in this limiting case:
Proposition 1. Three cases arise:
14
This is without loss of generality. All results carry through in the general case but are harder to expose.
12
1. When λγ < σz2 , expected returns follow the standard CAPM formula i > 0:
E[R̃i ] = βi
2. When λγ > σz2 +
σ2
,
2
σz2
γ
the speculative premium applies on all assets’ prices so that
expected returns can be written as:
σz2
E[R̃i ] = βi (1 + θ) − λθ
γ
3. Finally, when σz2 < λγ < σz2 + σ2 , there is a finite number of assets ī − 1 < ∞ that are
not shorted at equilibrium and expected returns have the following expression:

σz2


if i < ī
 βi
γ
E[R̃i ] =

σz2

 βi (1 + θ) − λθ
γ
(3)
if i ≥ ī
ī increases weakly with λ.
Proof. See Appendix.
As the number of assets goes to infinity, four cases happen. When disagreement is high
(λγ > σz2 +
σ2
),
2
all assets experience binding short-sales constraints. There is a speculative
premium in the price of each asset and conversely a speculative discount on their returns.
When disagreement is low, (λγ <
2 2
σ ),
3 z
there are no assets shorted at equilibrium and
therefore prices are set according to the standard CAPM formula with N → ∞. For low
intermediate disagreement, (σz2 > λγ > 23 σz2 ), there is a strictly positive fraction y of assets
that are not shorted at equilibrium, i.e. the number of assets not being shorted (ī = yN ) goes
P
to infinity and the speculative premium thus goes down to 0 (as j<ī wj2 → ∞). Therefore,
the CAPM pricing formula eventually holds for all λγ < σz2 despite the fact that investors
have 0 holdings on an infinite number of assets. Finally, for high intermediate disagreement,
13
(σz2 +
σ2
2
> λγ > σz2 ), there is a finite number of assets that are not shorted at equilibrium.
For the assets without binding short-sales constraints, the standard CAPM formula applies.
For the assets with binding short-sales constraints, the speculative premium applies, and in
particular the expression for the speculative premium is the same as the one obtained in the
case where all assets are shorted at equilibrium.
2.3.
Dynamic Setting
TO BE COMPLETED
2.4.
Discussions
In our framework, we only allow investors to disagree on the expectation of the aggregate
factor, z̃. A more general framework would allow investors to also disagree on the idiosyncratic component of stocks dividend ˜i . However, introducing this idiosyncratic disagreement
would not significantly modify our analysis. The natural case is where this idiosyncratic disagreement is independent of the β of the stock (more precisely independent of wi ). Then, the
potential over-pricing generated by the idiosyncratic disagreement would also be independent of β so that our conclusions in terms of the slope of the security market line would be
left unaffected. If idiosyncratic disagreement and the β of the stock are positively correlated,
then this would reinforce our effect.
Another restriction in the model is that investors only disagree on the first moment of
the aggregate factor z̃ and not on the second moment σz2 . From a theoretical viewpoint,
this is not very different. In the same way that β scales disagreement regarding z̄, β would
scale disagreement about σz2 . In other words, label the group that underestimates σz2 as
the optimists and the group that overestimate σz2 as the pessimists. Optimists are more
optimistic about the utility derived from holding a high β asset than a low β asset and
symmetrically, the pessimists are more pessimistic about the utility derived from holding a
high β asset than a low β asset. Again, high β assets are more sensitive to disagreement
14
about the variance of the aggregate factor σz2 than low β assets. As in our model, this
would naturally lead to high β stocks being overpriced when this disagreement about σz2 is
large. However, while proxies for disagreement about the mean of the aggregate factor can
be constructed somewhat easily, it is not clear how one would proxy for disagreement about
its variance.
Our results yield the following testable implications, which we take to the data in the
next section. First, short-interest ratios, stock-level disagreement and turnover all increase
with β and this relationship is steeper for larger levels of aggregate disagreement. Second,
the Security Market Line is upward sloping when aggregate disagreement is sufficiently low.
In periods in which aggregate disagreement is high enough, the SML takes an an inverted-U
shape: initially increasing for low beta stocks and then flat to downward sloping for high beta
stocks. Finally, the slope of the Security Market Line decreases with aggregate disagreement.
3.
Data
In this section, we present the data used in this paper. Table 1 presents descriptive statistics
of the variables we construct in this section.
3.1.
Data Source
The data in this study are collected from two main sources. U.S. stock return data are from
the CRSP tape and include all available common stocks on CRSP between January 1970 and
December 20010. We exclude penny stocks defined as stocks with a share price below $5.
βs are computed with respect to the value-weighted market returns provided on Ken French
website. Excess returns are above the US Treasury bill rate, which we also download from
Ken French website. Monthly turnover is defined as monthly volume normalized by number
of shares outstanding.15 This paper also uses analyst forecasts of the earnings-per- share
15
For NASDAQ stocks, we only take 1/2 the volume to share outstanding ratio as is standard using CRSP
data.
15
(EPS) long-term growth rate (LTG) as the main proxy for investors beliefs regarding the
future prospects of individual stocks. The data are provided by the I/B/E/S database. As
explained in detail in Yu (Yu (2010)), the long-term forecast has several advantages. First,
it features prominently in valuation models. Second, it is less affected by a firms earnings
guidance relative to short-term forecasts. Because the long-term forecast is an expected
growth rate, it is directly comparable across firms or across time. Analyst forecasts from
December 1981 through December 2010 are used in this study. Finally, we obtain shortinterest ratio data from Bloomberg. These data cover 17,716 stocks over the 1988-2009
period.
3.2.
Constructing β portfolios
We follow the literature (see for instance Frazzini and Pedersen (2010)) in constructing
beta portfolios in the following manner. Each month, we use the past twelve months of
daily returns to estimate the market beta of each stock in that cross-section. This is done
by regressing, at the stock-level, the stock’s excess return on the contemporaneous excess
market return as well as five lags of the market return to account for the illiquidity of small
stocks (Dimson (1979)). Our measure of β is then the sum of these six OLS coefficients. To
limit the influence of large absolute returns on the estimation of these βs, we windsorize the
daily returns using as thresholds the median return +/- five times the interquartile range of
the daily return distribution.
We then sort stocks into 20 beta portfolios based on these pre-ranking betas. We compute
the daily equal-weighted returns on these portfolios. We then compute the post-ranking βs
by regressing each portfolio daily returns on the excess market returns, as well as five lags
of the market return and adding these six estimates. This post-ranking βs are computed
using the entire sample period (Fama and French (1992)). Using a one-year trailing window
to compute post-ranking time-varying βs give quantitatively similar but noisier estimates.
16
3.3.
Measuring Aggregate Disagreement
Our measure of aggregate disagreement is similar in spirit to Yu (2010). Stock-level disagreement is measured as the dispersion in analyst forecasts of the earnings-per-share (EPS)
long-term growth rate (LTG). We then aggregate this stock-level disagreement measure,
weighting each stock by its post-ranking β. Intuitively, our model suggests that there are
two components to aggregate disagreement: (1) a first component coming from the disagreement about the aggregate factor z̃ – the λ in our model (2) a second component coming
from disagreement about the idiosyncratic factor ˜i . We are interested in constructing an
empirical proxy for the first component only (see our discussion in Section 2.4). To that end,
disagreement about low β stocks should only play a minor role since disagreement about a
low β stock has to come mostly from idiosyncratic disagreement (in the limit, disagreement
about a β = 0 stock can only come from idiosyncratic disagreement). Thus, we weight each
stock-level disagreement by the stock’s post-ranking β.
Figure A provides the time-series of our aggregate disagreement measure. It peaks during
the 1981/82 recession, the dot-com bubble of the late 90s and the recent recession.
4.
4.1.
Empirical Analysis
Auxiliary Predictions on Disagreement, Shorting and Share
Turnover
We turn to the auxiliary implications of our model. Figure 4 highlights the role played
by aggregate disagreement on the relationship between turnover, shorting and stock-level
disagreement and β. For each of our 20 β portfolios, we compute the average of the stocklevel short ratio (top panel), average stock-level dispersion in analyst earnings forecasts
and average share turnover. These averages are calculated for high aggregate disagreement
months (red dots) and low aggregate disagreement months (blue dots), where high (resp.
17
low) aggregate disagreement months are defined as being in the top (resp. bottom) quartiles
of in-sample aggregate disagreement. As shown in Figure 4, there is a strictly increasing
relation between the short-ratio, stock-level disagreement, share turnover and β; this relation
is steeper in months with high aggregate disagreement relative to month with low aggregate
disagreement.
Those results are confirmed by a simple regression analysis, where we estimate the following equations:
∀t ∈ [1981 : 12, 2010 : 12], ∀i ∈ [1, 20] :
yit = αt + µβi + νSizei + δβi × Agg. Dis.t−1 + ρSizei × Agg. Dis.t−1 + it
where yit is the equal-weighted average short-interest ratio of stocks in portfolio i at date t
(column 1 and 2), the equal-weighted average disagreement of stocks in portfolio i at date t
(column 3 and 4) and the equal-weighted average turnover of stocks in portfolio i at date t
(column 5 and 6).
Column 1, 3 and 5 estimate the previous equation using a simple pooled cross-sectional
regressions where standard errors are clustered at the date level. Column 2, 4 and 6 use a
Fama-MacBeth approach. This is a two-stage procedure. In the first-stage, we estimate the
following cross-sectional regression every month:
∀i ∈ [1, 20] : yit = α + δt βi + ρt Sizei + it
The second-stage projects the time-series of coefficients obtained in the first-stage (δt , ρt )
against the time-series of aggregate disagreement, using a simple OLS estimation with robust
standard errors.
∀t ∈ [1981 : 12, 2010 : 12],


 ρt = ν̄ + ρ̄Agg. Dis.t−1 + ωt

 δt = µ̄ + δ̄Agg. Dis. + ηt
t−1
18
Column 2, 4 and 6 reports the coefficients (δ̄, µ̄, ρ̄, ν̄) estimated using this two-steps
procedure. The resulting δ̄ coefficients are very similar to the δ coefficients obtained using
the simple pooled cross-sectional approach in column 1, 3 and 5.
The findings are broadly consistent with our model. Consider first the result regarding
stock-level disagreement. We notice that even for the lowest possible value of aggregate
disagreement (i.e. aggregate disagreement of 3.26), there is a strictly increasing relationship
between stock-level disagreement and β (1.1 × 3.26 − 2.5 > 0 in column 3). This validates an
important modeling choice in our theory, which is to neglect idiosyncratic disagreement. Indeed, even in low aggregate disagreement months, high β stocks experience more stock-level
disagreement than low β stocks. Thus, idiosyncratic disagreement appears to be positively
correlated with β in the data. As we discussed in Section 2.4, this positive correlation magnifies the speculativeness of high β stocks. Moreover, we see that the relationship between
stock-level disagreement and β becomes significantly steeper as aggregate disagreement rises.
This is consistent with the basic premise of our analysis that β scales aggregate disagreement.
In column 1 and 2, we report that while the short-interest ratio typically increases with
aggregate disagreement, this relation is significantly stronger in high aggregate disagreement
months. This is consistent with our theory, since, in the model, mispricing is larger for high
β stocks, so that arbitrageurs are on average more likely to short those stocks. However,
mispricing is also increasing with λ, the aggregate disagreement parameter – so that there
will be more shorting by the arbitrageurs for high disagreement months. Note that these
effects are quite sizable. Consider a one s.d. increase in aggregate disagreement (.90). It
will lead to a relative increase in the short interest ratio of a β = 1 stock relative to a β = 0
stock of .004 × .90 = .0036, which represents 20% of the s.d. of the short ratio.
Finally, column 5 and 6 shows that the same patterns hold for turnover: turnover in
general increases with β, but much more so in months of high aggregate disagreement. This is
consistent with our simple dynamic extension whereby an increase in aggregate disagreement
increases the resale option of high β stocks relative to low β stocks.
19
To conclude, note that the regressions in Table 2 controls for size as a way to account
for the heterogeneity in our 20 β portfolio.
4.2.
Aggregate Disagreement and the Security Market Line
The concavity of the Security Market Line
The second part of our empirical analysis examines how the Security Market Line is affected
by our aggregate disagreement measure. To this end, we first present in Figure 5 the empirical relationship between β and excess returns. For each of the 20 β portfolio in our sample,
we compute the average excess forward return for high (red dots) and low (blue dots) disagreement months (defined as top vs. bottom quartile of aggregate disagreement). We do
this using various horizons: 1-month (top left panel), 3-months (top-right panel), 6 months
(bottom-left panel) and 12 months (bottom-right panel). While the relationship between
excess forward returns and β is quite noisy at the 1 and 3 months horizon, two striking facts
emerge from at the 6 and 12 months horizons. First, the excess return/β relationship seems
to be increasing for months with low aggregate disagreement. This is consistent with our
theory whereby low aggregate disagreement means low or even no mispricing and hence a
strictly upward sloping security market line. Second, in months of high aggregate disagreement the excess returns/β relationship appears to exhibit the inverted-U shape predicted by
the theory.
To evaluate the importance of this inverted-U shaped relationship in the data, we consider
the following Fama-MacBeth exercise. In the first stage, we estimate, for each month in our
sample the following cross-sectional regression:
e
∀i ∈ [1, 20] : ri,t→t+k
= αtk + πtk βi + φkt βi2 + kit
e
where ri,t→t+k
is the k-months forward excess return of portfolio i at date t and βi is
the post-ranking β of portfolio i. This gives us a time-series estimate of the coefficient
20
(αt , πt , φt )t∈[1981:12,2010:12] . We are especially interested in how the concavity in the excess
return/β relationship evolves with aggregate disagreement. In fact, our theory predicts that
the relationship should be more concave as aggregate disagreement increases. To formally
test this, we then simply projects these time-series coefficients onto our aggregate disagreement measure. Table 3 reports our findings. The first line corresponds to the ψ k coefficient
in the estimation of the following equation:
φkt = ck + ψ k × Agg. Dis.t + ωtk
where k ∈ {1, 3, 6, 12} corresponds to the various horizons used in the first-stage crosssectional regressions. In this context, ψ k is naturally interpreted as how the concavity in the
excess returns/β relationship evolves with aggregate disagreement. In column 2, 4, 6 and 8,
we control for other variables potentially affecting the shape of the security market line as
well, i.e. we estimate an augmented version of the previous equation:
φkt = ck + ψ k × Agg. Dis.t + τ k Xt + ωtk
where Xt includes the returns on HML, SMB, the PE and DP ratio, the TED spread, the
one year inflation rate as well as the k year forward market return. When k is strictly larger
than 1, we use Newey-West standard errors with k − 1 lags to account for the overlapping
returns in our dependent variable. When k = 1, we simply report Huber-White standard
error.
The results in Table 3 indicates that, at the 3, 6 and 12 months horizon, the Security
Market Line is significantly more concave for months with high aggregate disagreement relative to months with low aggregate disagreement, i.e. ψ k is negative and significantly different
from 0 at the 5 or 1% level depending on the horizon and whether or not the control predictors are included in the regression. To better grasp the implication of this regression analysis
on the shape of the security market line, we present, in figure A, the security market line as
21
predicted from the regression in Table 3 column 5 for two levels of aggregate disagreement:
low (corresponding to the 10th percentile of the aggregate disagreement distribution) and
high (corresponding to the 90th percentile of the aggregate disagreement distribution). As
we see from this figure, the 6-months forward security market line is predicted to be upward
slopping for the low disagreement months and inverted-U shaped for the high disagreement.
While the excess return/β relationship is similar for low beta stocks in both high and low aggregate disagreement months, excess returns become significantly lower for high beta stocks
in months of high aggregate disagreement relative to the low aggregate disagreement months:
high β stocks are predicted to have 0 excess returns in high aggregate disagreement months
at the 6-months horizon, as opposed to low disagreement months where excess returns are
around 10%. These results are overall consistent with the predictions from our speculative
asset pricing model.
The Slope of the Security Market Line
In Table 4, we present the results of a similar analysis omitting the quadratic term in the
cross-sectional regressions. We are thus interested in the average slope of the security market
line and in how it relates with aggregate disagreement. To this end, we again use a two-stages
procedure, where we first regress in the cross-section:
e
∀i ∈ [1, 20] : ri,t→t+k
= akt + bkt βi + kit
and then project bkt onto aggregate disagreement at date t, as well as the control predictors
Xt . The results in Table 4 show that at the 6 and 12 months horizon, the slope of the
Security Market Line is strictly decreasing with aggregate disagreement. Indeed, a simple
corollary of our theoretical analysis is that, as aggregate disagreement increases, the slope of
the security market line should decrease. This is because high β stocks are speculative when
aggregate disagreement is high, which makes their expected returns go down. The results in
22
Table 4 are again consistent with this particular prediction of our model.
Alternative Measure of disagreement
As a robustness check, we repeat our empirical analysis using an alternative measure of
disagreement. To this end, we use a measure of uncertainty provide in Bloom et al. (2012).
This measure consists in the cross-sectional dispersion of plant-level sales growth and is discussed extensively in Bloom et al (2012). This measure captures the uncertainty underlying
the economic fundamentals of the economy. When these fundamentals are more uncertain,
there is naturally more scope for disagreement among investors. An advantage of using this
alternative measure is that it is available since 1970, which provides us with more statistical
power to identify the role played by disagreement on asset prices.
In Figure A, we simply represent the time-series of this measure of sales growth uncertainty. As we see, it is strongly correlated with our measure of aggregate disagreement. In
Table 5, we repeat the analysis in Table 3 using this uncertainty measure as our proxy for
disagreement. At the 6 and 12 months horizons, we see again that the Security Market Line
is significantly more concave for months with higher disagreement.
5.
Conclusion
We show that incorporating the speculative motive for trade into asset pricing models yields
strikingly different results from the risk-sharing or liquidity motives. High beta assets are
more speculative since they are more sensitive to disagreement about common cash-flows.
Hence they experience greater divergence of opinion and in the presence of costly shortselling, they end up being over-priced relative to low beta assets. When the disagreement is
low, the risk-return relationship is upward sloping. As opinions diverge, the slope is initially
positive and then negative for high betas. Empirical tests using security analyst disagreement
measures confirm these predictions. We also verify the premise of our speculative mecha23
nism by finding that high beta stocks have higher individual disagreement about cash-flows,
higher shorting and higher share turnover than low beta ones and that these gaps grow with
aggregate disagreement. Hence our model provides a plausible explanation for the high-risk,
low-return puzzle.
24
6.
Appendix
Proof of Theorem 1:
Proof. Consider an equilibrium where group B investors are long on assets i < ī and hold a short position
for assets i ≥ ī. The first order conditions for group A agents are simply:
N
X
1
∀i ∈ [1, N ] : (z̄ + λ)wi − Pi (1 + r) =
γ
!
wk µA
k
!
wi σz2
+
2
µA
i σ
k=1
The first order condition for agents in group B however depends on the assets:


1


(z̄ − λ)wi − Pi (1 + r) =



γ
ī−1
X
!
wk µB
k
!
wi σz2
+
2
µB
i σ
for i < ī
k=1


1

B


 µi = 0 ⇔ (z̄ − λ)wi − Pi (1 + r) − γ
ī−1
X
!
wk µB
k
!
wi σz2
< 0 for i ≥ ī
k=1
Finally, for arbitrageurs, the first order condition is simply:
1
∀i ∈ [1, N ] : z̄wi − Pi (1 + r) =
γ
We use the market clearing condition (α
B
µA
i +µi
2
N
X
!
wk µak
!
wi σz2 + µai σ2
k=1
+ (1 − α)µai =
1
N)
and sum the first-order conditions of
all agents for asset i to obtain:

1
σ2

2

z̄w − Pi (1 + r) =
wi σz +
for i < ī


 i
γ
N
!
ī−1

1
σ2
α 2 X
α
α

2
B

wk µk
for i ≥ ī

 1 − 2 (z̄wi − Pi (1 + r)) + 2 λwi = γ wi σz + N − 2 σz wi
(4)
k=1
Call S =
Pī−1
i=1
wi µB
i . We look for an expression for S. We start by using the first order conditions of group
B agents and plug in the equilibrium price of assets i < ī:
−λγwi + wi σz2 +
σ2
2
= Swi σz2 + µB
i σ for i < ī
N
We can simply sum up the previous relation for all i < ī:








2
X
X
X
X
σ
Sσ2 = −λγ 
wk2  − Sσz2 
wk2  + σz2 
wk2  + 
wk  for i < ī
N
k<ī
k<ī
k<ī
25
k≥ī
(5)
From the previous expression, we can compute S:
S =1−
σ2
P
wi
i≥ī N
σ2 + σz2
+ λγ
P
P
i<ī
i<ī
wi2
wi2
We can now plug the previous equation into equation 4 for assets i ≥ ī and obtain the equilibrium price of
assets i ≥ ī:
1
Pi (1 + r) = z̄wi −
γ
σ2
wi σz2 + N

+ θ wi
σ2
P
2
σ2 + σz
|
α
2
where θ =
1− α
2



2 X
2
σ
σ
λ − z
wk  − 
2
γN
γN
i<ī wi
k≥ī
{z
}
(6)
π i =speculative premium
is a strictly increasing function of α. For assets i < ī, the price is simply given by:
1
γ
Pi (1 + r) = z̄wi −
σ2
wi σz2 + N
(7)
We can now use the equilibrium prices to derive the equilibrium holdings of B agents:

 P

wi
2

σ
−
λγ

z
i≥ī N
1


 for i < ī

P
+ wi  2
2
2
N
B
σ
+
σ
z
i<ī wi
µi =




 0 for i ≥ ī
We now need to derive the conditions under which this is indeed an equilibrium. The marginal asset is asset
ī if and only if
∂U B
(µB ∗)
∂µB
ī
< 0 and µB
> 0, where µB ∗ is the holdings of group B agents in the proposed
ī−1
equilibrium. These conditions are easily shown to be equivalent to:
σz2
γN
Call uk =

X
k≥ī
1
γN wk
wk +

1
σ2 + σz2
γN wī−1
σ2 + σz2
P
i<k
wi2
+
X
wk2  > λ >
k<ī
σz2
γN
P
wi
i≥k N
σz2
γN

X
k≥ī
1  2
σ + σz2
γN wī
wk2 
k<ī
. Clearly, uk is a strictly decreasing sequence as :
uk − uk−1
wk +

X
1
1
=
−
σ2 + σz2
N wk
N wk−1
|
{z
}
!!
X
wi2
<0
i<k
<0
Define u0 = +∞ and uN +1 = 0. Then the sequence (ui )i∈[0,N +1] spans R+ and the marginal asset is the
unique asset ī that verifies:
uī ≤ λγ < uī−1
26
It is direct to show that provided this is verified, group B agents have strictly positive holdings for assets
i < ī and 0 holdings for assets i ≥ ī.
Overpricing for assets i ≥ ī is simply defined as the difference between the equilibrium price and the price
that would prevail in the absence of heterogenous beliefs and short sales costs. It is equal to the speculative
premium:

π i = θ wi



2 X
2
σ
σ
λ − z
wk  − 
2
γN
γN
w
i<ī i
σ2
P
2
σ2 + σz
for i ≥ ī
k≥ī
Note that λ ≥ uī ⇔ π i ≥ 0 so that assets i ≥ ī are overpriced. That mispricing is increasing with the
fraction of short-sales constrained agents α is direct as θ is a strictly increasing function of θ.
We now derive the equilibrium holdings of arbitrageurs. Their first-order condition writes:
∀i ∈ [1, N ],
Define S a =
PN
k=1
1
z̄wi − Pi (1 + r) =
γ
N
X
wi σz2
!
µak wk
!
+ µai σ2
k=1
µak wk . Using the equilibrium pricing equation, this first-order condition can be
rewritten as:
wi σz2 +
∀i ∈ [1, N ],
σ2
− γπ i 1i≥ī = wi σz2 S a + µai σ2
N
By multiplying each of these equations by wi and summing up all the equations, we get:
P
wk π k
P
k≥ī
a
S =1−γ
σ2 + σz2
N
k=1
wk2
Using this expression for S a in the previous first-order condition yields the following expression for
arbitrageurs’ holdings of assets i:
∀i ∈ [1, N ],


P
k
w
π
γ
1
k
k≥ī
P

µai =
− 2 π i 1{i≥ī} − wi σz2
N
N
σ
2
2
σ +σ
w2
z
k=1
k
First remark that if i < ī, µai > 0, so that arbitrageurs are long. Since both group B and group A agents
are also long, it is immediate that: V i = 1 for all i < ī.
Now consider the case i ≥ ī. First notice from the expression of the speculative premium that:
∀k, i ≥ ī,
θσ 2
wk
πk + =
γN
wi
27
θσ 2
π + γN
i
Thus, multiplying the previous expression by wk and summing over all k ≥ ī:
X
w k πk +
k≥ī
θσ2
γN




θσ2
i
X
π
+
γN 
wk  = 
wk2  
wi

X

k≥ī
k≥ī
Thus:
k
k≥ī wk π
P
N
2
P
π i − wi σz2
σ2 + σz
k=1
wk2
=
=
=
=





θσ2
i
2
X
X
π
+
θσ
γN 

π i − wi
− 
wk2  
wk 
wi
γN
2
w
σ2 + σz
k≥ī
k≥ī
k=1 k


P
2
2
2
2
2
X
X
w
σ
+
σ
σz
θσ
z
P k<ī k −
P

πi wk2 − wi
wk 
N
N
γN
2
2
2
2
2
2
σ + σz
w
σ
+
σ
w
k≥
ī
k≥
ī
z
k=1 k
k=1 k

 
σz2 P
P
2
i≥ī wi
1 σ2 + σz2
θσ2   λγ − N
k<ī wk

−
P
wi
N
γ
N σ 2 + σ 2 PN w2
2
σ2 + σz2
z
i=1 wi
k=1 k


X
X
1
σz2

−
wk2 − wi
wk 
N σ 2 + σ 2 PN w2
k≥
ī
k≥
ī
z
k=1 k
P


N
2
2
2
+
σ
w
σ
2
z
i=1 i
θσ
1

P
wi λγ −
N
γ σ2 + σ2
N
w2
σz2
P
N
2
z
i
i=1
We can now derive the actual holding of arbitrageurs on assets i ≥ ī:

∀i ≥ ī,
µai =

1
wi λγ
wi λγ
1
1+θ
P
 =
P
+θ −
−θ
N
N
N
N
N
2
2
2
2
2
2
σ + σz
σ
+
σ
w
w
z
i=1 i
i=1 i
First, notice that arbitrageurs holdings are decreasing with i. There is at least one asset shorted by
P
2
σ2 +σz2 ( N
i=1 wi )
1
arbitrageurs provided that µaN < 0 which is equivalent to λ > λ̄ = 1+θ
θ
N
γwN . If this is verified,
there exists a unique î ∈ [1, N − 1] such that µai < 0 ⇔ i > î.
Proof of Corollary 1:
Proof. Note that βi =
2
σ
N
1
i=1 N 2
wi σz2 +
PN
, so that wi = βi
σz2 +(
)σ2
equations in terms of expected returns:
P
σz2 +( N
i=1
σz2


wi σz2 +


 E[Ri ] =
γ
2

wσ +


 E[Ri ] = i z
γ
28
σ2
N
1
N2
)σ2
for i < ī
1 2
N σ
− πi
−
2
1 σ
N σz2 .
We can rewrite the pricing

σ2

wi σz2 + N


for i < ī
E[R
]
=

i


γ
P


wi
2
2
2 λγ − σz

i≥
ī
N

σ
σ2
w
σ
i z 



P
+
E[R
]
=
1
−
θ
(1 + θ) for i ≥ ī
i

2

γ
σz2 σ2 + σz2
γN
i<ī wi
wi
2 P
σ2 λγ−σz ( i≥ī N )
P
2
2 .
σz σ2 +σz2 (
i<ī wi )
Define κ(λ) =
Using the definition of βi , this can be rewritten as:

σ2 +


 E[Ri ] = βi z

γ

2


 E[Ri ] = βi σz +
γ
σ2
N
σ2
N
for i < ī
(1 − κ(λ)θ) +
σ2
θ(1 + κ(λ)) for i ≥ ī
γN
Proof of Corollary 2:
Proof. We can write the actual returns as:

σ2
σz2 + N



+ η̃ i for i < ī
 βi
γ
R̃i =
σ2

2
2


 E[Ri ] = βi σz + N (1 − κ(λ)θ) + σ θ(1 + κ(λ)) + η̃ i for i ≥ ī
γ
γN
where η˜i = wi ũ + ˜i and ũ = z̃ − z̄.
Using the fact that by definition,
P
i
wi =
P
i
βi = N , a cross-sectional regression of realized returns R̃i
on βi and a constant would deliver the following coefficient estimate:
PN
µ̂ =
=
Let
uī−1
γ
PN
βi R̃i − i=1 R̃i
PN 2
i=1 βi − N
i=1
σz2 +
γ
σ2
N
γ
1 + 2 ũ −
σz
> λ1 > λ2 >
uī
γ .
2
i≥ī βi
PN
i=1
P
−
P
i≥ī
βi
!
βi2 − N
!
θκ(λ)
+
1
N
N −ī+1
i≥ī βi −
N
PN 2
β
−
N
i=1 i
P
!
σ2
θ (1 + κ(λ))
γ
Call ī1 (ī2 ) the threshold associated with disagreement λ1 (resp. λ2 ). We have
that ī1 = i¯2 = ī. Then:

µ̂(λ1 ) − µ̂(λ2 )

N
X
θ (κ(λ1 ) − κ(λ2 ))  2 
σz
= − PN
βi2 −
2
i=1 βi − N
i≥ī
29
N
X
i≥ī

βi  +
σ2
N
N
X
i≥ī

2
(βi − 1) 
P
P
P
PN 2
2
We show that
yi = 0. Then:
i≥ī βi ≥
i≥ī βi . Call βi = 1 + yi with yi such that
i=1 βi =
N
X
PN
PN
N +2
yi + i=1 y 2 > N = i=1 βi . Thus, the relationship is true for ī = 0. Now assume it is true for
i=1
| {z }
=0
P
P
P
P
2
2
2
ī = k. We have:
i≥k+1 βi −
i≥k+1 βi =
i≥k βi −
i≥k βi + βk − βk . Either βk > 1 in which case it
P
P
is evident that i≥k+1 βi2 − i≥k+1 βi > 0 as βi > 1 for i ≥ k. Or βk ≤ 1 in which case βk − βk2 > 0 and
P
P
using the recurrence assumption, i≥k+1 βi2 − i≥k+1 βi > 0. Thus, µ̂(λ1 ) − µ̂(λ2 ) < 0.
Moreover, we show now that µ̂(λ) is continuous at ui , for all i ∈ [1, N ]. When λ = u+
i , we have ī = i.
When λ = u−
i , we have ī = i + 1. First, notice that κ(λ) is continuous at ui . Indeed:
+
κ(u−
i ) = κ(ui ) =
σ2 1
σz2 N γwi
Thus:
µ̂
u+
i
− µ̂
u−
i
σ2
σ2
2
−
(1 + κ(ui ))
βi κ(ui ) σz +
N
N
θ
βī − 1
σ2
wi σz2 κ(ui ) − = 0
PN 2
γ i=1 βi − N
N
βī − 1
θ
P
2
γ N
i=1 βi − N
=
=
Thus µ̂ is continuous and strictly decreasing over ]ui+1 , ui [, so that it is overall strictly decreasing with
aggregate disagreement λ.
We can also easily show that the slope of the security market line, µ̂, is strictly decreasing with θ, the
fraction of short-sales constrained agents in the model. Indeed (noting that the thresholds ui are independent
of θ):



2 X
X
X
∂ µ̂
κ(λ)
σ
2
 = − P
(βi − 1) + σz2 
βi2 −
βi  < 0
N
∂θ
N
2
γ
β −N
i=1
i≥ī
i
As we have already shown that:
P
i≥ī
βi2 −
P
i≥ī
i≥ī
i≥ī
βi .
Finally, Going back to the expression for µ̂(λ1 ) − µ̂(λ2 ), we see that this difference can be expressed as
−Cθ with C > 0, so that it is clearly increasing with θ. Thus, when α, the fraction of short-sales constrained
agents increases, θ increases, and the difference between µ̂(λ1 ) and µ̂(λ2 ) decreases. Because this difference
is strictly negative, this means that the gap between the two slopes becomes wider.
Proof of Proposition 1:
30
2i
N +1 :
Proof. First, using the new parameterization of the model wi =
∀i > 0,
1 i2 − 1
N +1 1 2
ui = σz2 1 −
+
σ
3 N (N + 1)
N 2i As in the general case, ui is clearly a strictly decreasing sequence and uN →N →∞
σz2 +
σ2
2 .
Trivially, if λγ <
2 2
3 σz
2 2
3 σz
and u1 →N →∞
then there are no assets shorted at equilibrium when N → ∞ and asset
prices can be written using the usual CAPM formula. Symmetrically, if λγ > σz2 +
σ2
2 ,
then when N → ∞
all assets are shorted at equilibrium and the speculative premium applies to all assets. In particular, in this
limiting case, βi = w + i, si = 0 and j̄ = 1 so that the speculative premium has the following expression:
πi = θ
βi
λγ − σz2
γ
σ2
This leads to the following expression for expected returns: E[R̃i ] = βi (1 + θ) γz − λθ
Consider now one intermediate case where σz2 < λγ < σz2 +
σ2
2 .
Then there exists a finite ī > 0 such
that limN →∞ ui < λγ < limN →∞ ui−1 , i.e. there are a finite number of assets that are not shorted at
equilibrium. But then, at the limit limN →∞
ī
N
= 0, so that the speculative premium on asset i ≥ ī has the
same expression as in the case where all assets are shorted. The equilibrium expected returns are thus again:
 σz2


β
(1
+
θ)
−
λθ
 i
γ
E[R̃i ] =
2


 βi σz for i < ī
γ
for i ≥ ī
The final intermediate case is when 32 σz2 < λγ < σz2 . In this case, it needs to be that ī → ∞ as N → ∞
when N → ∞, otherwise we know that for all i ui > σz2 at the limit so that λγ would have to be > σz2 ,
2ī
which would be a contradiction. Call x = limN →∞ wī = N
the β of the marginal stock. It is easily seen
q
that x = 3 1 − λγ
σ 2 . More importantly, the speculative premium now goes down to zero. This is because
z
the speculative premium can be written in the limit as:
i
π ≈n→∞
σ2
θ βi
γ
λγ − σz2 1 − x2
σ2 + 34 σz2 x3 N
31
!
→N →∞ 0
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34
A.
Tables and Figures
Table 1: Summary Statistics for Time Series Analysis
SML slope
3-months ∆ ret.
6-months ∆ ret.
12-months ∆ ret.
Agg. Dis
Agg. For
Price/Earnings
Div./Price
SMB
HML
TED
Inflation
V IX
Mean
1.99
1.37
2.07
4.14
3.73
14.78
22.12
0.03
0.14
0.37
0.56
3.07
20.40
Median
2.91
1.90
3.44
6.43
3.52
14.04
19.00
0.03
-0.03
0.34
0.40
3.00
19.49
Std. Dev.
13.72
12.87
18.28
27.31
0.79
3.06
15.44
0.01
3.19
3.13
0.56
1.42
7.87
25th p 75th p.
-7.46
10.00
-5.35
8.27
-7.89
13.40
-12.27 20.51
3.11
4.14
13.26 15.54
14.69 25.59
0.02
0.04
-1.61
1.84
-1.38
1.91
0.22
0.68
2.23
3.92
14.12 24.44
Min
Max
-40.25 47.11
-37.78 52.06
-60.92 70.70
-82.02 81.12
2.68
6.14
9.77
27.10
7.48 123.79
0.01
0.06
-16.62 22.06
-12.87 13.88
0.03
4.62
-2.10
8.39
10.42 59.89
Obs.
337
346
343
337
349
349
349
349
349
349
348
348
252
Note: SML slope is the coefficient estimate of a monthly OLS regression of the 10 betaportfolios returns on each portfolio beta. 3-months ∆ ret. (resp. 6 and 12) is the 3 months
(resp. 6 and 12) forward returns of a value-weighted portfolio long in stocks in the top decile
of betas and short in stocks in the bottom decile of beta. Agg. Dis. is the value/β-weighted
sum of stock level dispersion measured as the standard deviation of analyst forecasts on the
stock. Agg. For. is the value/β-weighted sum of analyst forecast on the stock. TED is the
TED spread and Inflation is the yearly inflation rate.
35
Table 2: β portfolios characteristics and disagreement
Short
Interest
OLS
FAMA
β× Agg. Dis.t−1
β
Size × Agg. Dis.t−1
Size
Observations
Adj. R2
.004***
(7.6)
.0066***
(2.9)
.0016***
(7.8)
-.0068***
(-8.3)
5,239
.91
.0044***
(7.7)
.0054**
(2.2)
.00083***
( 5.08 )
-.0033***
(-4.85)
262
Stock
Disagreement
OLS
FAMA
1.1***
(19)
-2.5***
(-10)
.057**
(2.4)
-.6***
(-6.2)
6,947
.72
1.2***
(18)
-2.6***
(-9.6)
-.052**
(-2.39)
-.1652*
(-1.83)
348
Turnover
OLS
FAMA
.2***
(12)
-.14**
(-2.1)
.02***
(3.7)
-.081***
(-3.7)
6,960
.9
.2***
(12)
-.15**
(-2)
-.0069
(-1.20)
.024
(1.0)
348
Notes: OLS and Fama-MacBeth estimation of monthly β portfolio characteristics on β, size and β and
size interacted with aggregate disagreement. Aggregate disagreement is the β-weighted sum of stock level
disagreement. The dependent variables are: the equal-weighted average monthly short interest ratio (column
1), the equal-weighted average stock-level disagreement (column 2), the equal-weighted turnover. β is the
post-ranking portfolio β of the 20 β portfolio. Size is the logarithm of the average market capitalization of
stocksin each 20-β portfolio. ∗ , ∗∗ , and ∗∗∗ means statistically different from zero at 10, 5 and 1% level of
significance.
36
37
-3.113
(-1.25)
β
Yes
348
-12.89***
(-3.31)
2.046***
(2.73)
6.749**
(2.53)
No
348
-9.211
(-1.36)
2.877*
(1.79)
8.68*
(1.96)
Yes
348
-32.50***
(3.38)
5.26***
(2.74)
17.6***
(2.90)
No
348
-14.25
(-1.32)
4.54*
(1.76)
16.09**
(2.27)
Yes
348
-63.98***
(-3.70)
9.73**
(3.21)
36.44***
(3.70)
Portfolio forward excess return
3-months
6-months
(3)
(4)
(5)
(6)
-2.30** -3.39***
-4.21**
-6.60***
(-2.12)
(-2.81)
(-2.45)
(-3.37)
No
348
-19.45
(-1.06)
6.11
(1.35)
26.47**
(2.50)
Yes
348
-56.01**
(-2.22)
6.32
(1.51)
49.66***
(3.82)
12 months
(7)
(8)
-6.67**
-8.51
(2.57)
(-3.65)
Notes: Fama-MacBeth estimation with Newey-West adjusted standard-errors allowing for 2 (column 3 and 4), 5 (column 5 and 6) and 11 lags (column
7 and 8). T-statistics are in parenthesis. The dependent variables are the 1/3/6/12 months forward equal-weighted return of 20 β portfolios. Aggregate
disagreement is the β weighted average of stock-level disagreement measured as the dispersion of analysts forecasts on the long-run growth of EPS.
Controls include the 1/3/6/12 months forward market return, HML, SMB, D/P, P/E, TED spread, past 12 months inflation. ∗ , ∗∗ , and ∗∗∗ means
statistically different from zero at 10, 5 and 1% level of significance.
No
348
1.00*
(1.68)
β× Aggregate Disagreement
Controls
Observations
2.645
(1.39)
β2
β 2 × Aggregate Disagreement
1-month
(1)
(2)
-.728
-1.238**
(-1.56)
(-2.37)
Table 3: Disagreement and concavity of the security market line.
38
2.3
(1.3)
347
4.4
(1.6)
347
(2)
-.84*
(-1.8)
.031
(.97)
3
(.071)
-.2*
(-1.7)
-.035
(-.3)
-.54
(-1.5)
.75
(1)
(3)
-.49*
(-1.7)
.05**
(2.5)
-75***
(-2.8)
-.12
(-1.6)
-.12*
(-1.7)
.35
(1.5)
1**
(2.2)
1.1***
(23)
.92
(.52)
347
8.6
(1.6)
345
(4)
-1.8
(-1.4)
Slope of Security Market Line
(5)
(6)
(7)
(8)
-2.6**
-1.7**
-4.1* -6.4***
(-2)
(-2)
(-1.9)
(-3.2)
.17**
.16***
.35***
(2.4)
(4.1)
(3.4)
-3.2
-232***
-98
(-.032)
(-3.4)
(-.52)
-.84**
-.53**
-1***
(-2.6)
(-2.5)
(-3.1)
-.66***
-.4**
-1***
(-2.9)
(-2.6)
(-3.6)
-.47
1.3**
-.46
(-.49)
(2.5)
(-.31)
1.4
2.5***
4
(.61)
(2.9)
(1.4)
1***
(13)
9.4
3.6
19**
23**
(1.3)
(.76)
(2.1)
(2)
345
345
342
342
(9)
-3.8**
(-2.5)
.29***
(4.2)
-467***
(-2.9)
-.56***
(-3.1)
-.72***
(-3.9)
2.1**
(2.2)
4.9***
(3.3)
.96***
(10)
11
(1.2)
342
35**
(2.2)
336
(10)
-7.5*
(-1.9)
56***
(2.7)
336
(11)
-13***
(-4.4)
.59***
(4)
-434
(-1.3)
-1.2**
(-2.1)
-1.5**
(-2.6)
-.41
(-.17)
7.8*
(1.7)
(12)
-7.4***
(-3.1)
.44***
(3.1)
-873***
(-2.9)
-1**
(-2.4)
-1.1***
(-2.8)
1.8
(.99)
8.2***
(3.1)
.84***
(5.7)
31**
(2)
336
Notes: OLS estimation with Newey-West adjusted standard-errors allowing for 2/5/11 lags in column 4 to 12. Columns 1 to 3 use HuberWhite
standard errors. T-statistics are in parenthesis. The dependent variable is the coefficient estimate of a monthly cross-sectional regression of the 20
beta-portfolio 1 month forward excess returns (columns 1 to 3), 3 months forward excess returns (columns 4 to 6), 6 months forward excess returns
(columns 7 to 9) and 12 months forward excess returns (columns 10 to 12) on the portfolio beta. Aggregate disagreement is the β-weighted average of
stock-level disagreement. Inflation rate is the past year CPI growth rate. D/P is the aggregate dividend to price ratio and P/E is the aggregate price
to earnings ratio, both from Robert Shiller’s website. HML and SMB are the returns on the HML and SMB portfolios from Ken French’s website. ∗ ,
∗∗
, and ∗∗∗ means statistically different from zero at 10, 5 and 1% level of significance.
Observations
Adj. R2
Constant
Forward Market Return
TED Spread
Inflation
SMB
HML
D/P
P/E
Aggregate Disagreement
(1)
-.49
(-1.3)
Table 4: Disagreement and the average slope of the security market line
39
.424
(.91)
-2.64
(-.74)
No
476
β× Aggregate Disagreement
β
Controls
Observations
Yes
342
-9.817**
(-2.19)
.688
(1.23)
4.03
(1.42)
No
476
-10.17
(-1.23)
1.48
(1.37)
6.026
(1.57)
Yes
342
-22.32**
(2.31)
1.576
(1.41)
9.249
(1.60)
No
476
-29.38*
(1.93)
3.99**
(2.01)
15.41**
(2.14)
Yes
342
-47.63***
(2.98)
3.826**
(2.32)
24.11**
(2.57)
Portfolio forward excess return
3-months
6-months
(3)
(4)
(5)
(6)
-.839
-.916
-2.058** -2.513**
(-1.64)
(-1.34)
(-2.17)
(-2.40)
No
476
-79.85***
(4.00)
10.194***
(4.35)
37.96***
(3.26)
Yes
342
-76.749***
(3.16)
6.492***
(2.83)
48.53***
(3.22)
12 months
(7)
(8)
-4.80***
-4.896***
(-3.23)
(-3.41)
Notes: Fama-MacBeth estimation with Newey-West adjusted standard-errors allowing for 2 (column 3 and 4), 5 (column 5 and 6) and 11 lags (column
7 and 8). T-statistics are in parenthesis. The dependent variables are the 1/3/6/12 months forward equal-weighted return of 20 β portfolios. Aggregate
disagreement is the β weighted average of stock-level disagreement measured as the dispersion of analysts forecasts on the long-run growth of EPS.
Controls include the 1/3/6/12 months forward market return, HML, SMB, D/P, P/E, TED spread, past 12 months inflation. ∗ , ∗∗ , and ∗∗∗ means
statistically different from zero at 10, 5 and 1% level of significance.
1.84
(1.0)
β2
β 2 × Aggregate Disagreement
1-month
(1)
(2)
-.270
-.359
(1.10)
(-.98)
Table 5: Aggregate Uncertainty (Bloom (2012)) and concavity of the security market line.
.5
fraction of aggregate vol. to total vol
.6
.7
.8
.9
Figure 1: Ratio of aggregate to total volatility, by β.
0
.5
1
post-ranking beta
40
1.5
2
.01
Expected excess returns
0
.005
-.005
0
.5
1
beta
lambda=0
lambda=.0075
1.5
2
lambda=.005
Figure 2: Model simulation.
Parameterization: N = 100, σz2 = 4.66, σ2 = 298.1, γ = 15 and θ = 1.75
3
4
aggregate disagreement
5
6
7
Figure 3: Time series of aggregate disagreement
01jan1980
01jan1990
01jan2000
date
41
01jan2010
0
.01
short-interest ratio
.02
.03
.04
.05
Figure 4: Equal-weighted average of analyst forecasts’ dispersion, short interest ratio and
share turnover for stocks by β (20 β portfolios) during low and high aggregate disagreement
months.
0
.5
1
post-ranking beta
Low disagreement
1.5
2
High disagreement
2
stock-level disagreement
4
6
8
10
(a) Short interest ratio
0
.5
1
post-ranking beta
Low disagreement
1.5
2
High disagreement
0
.5
turnover ratio
1
1.5
2
(b) Dispersion of analyst earnings forecasts
0
.5
1
post-ranking beta
Low disagreement
High disagreement
(c) Share turnover
42
1.5
2
3.5
.2
1.5
.4
2
.6
2.5
.8
3
1
1.2
Figure 5: Average 1-month, 3-months, 6-months, and 12-months excess of risk-free returns
for equal-weighted beta decile portfolios during low disagreement and high aggregate disagreement months.
0
.5
1
post-ranking beta
Low disagreement
1.5
2
0
High disagreement
.5
1
post-ranking beta
Low disagreement
2
High disagreement
(b) 3 months equal-weighted return
5
0
2
10
4
15
6
8
20
(a) 1 month equal-weighted return
1.5
0
.5
1
post-ranking beta
Low disagreement
1.5
2
0
High disagreement
.5
1
post-ranking beta
Low disagreement
(c) 6 months equal-weighted return
1.5
High disagreement
(d) 12 months equal-weighted return
43
2
-5
0
5
10
15
Figure 6: Predicted Security Market Line for 10% and 90% aggregate disagreement.
0
.5
1
beta_post_average
10% disagreement
44
1.5
90% disagreement
2
3
6
8
aggregate disagreement
4
5
6
10
12
Sales uncertainty
7
14
Figure 7: Time-series of Aggregate Disagreement and Aggregate Uncertainty from Bloom
(2012)
01jan1970
01jan1980
01jan1990
date
aggregate disagreement
45
01jan2000
01jan2010
Sales uncertainty