turning alphas into betas: arbitrage and the cross-section
TRANSCRIPT
Turning Alphas into Betas:Arbitrage and the Cross-section of Risk
Thummim Cho∗London School of Economics
May 2018
Abstract
What determines the cross-section of asset betas with a risk factor? Arbitrageactivity plays an important role. I develop a model in which the cross-section of betasarises endogenously through the act of arbitrage. Testing the model’s predictions,I show empirically that arbitrage activity plays an important role in generating thecross-section of betas in multifactor and intermediary-based asset pricing models. Thearbitrage channel for the betas can cause a cross-sectional asset pricing regression tosuffer from endogeneity.
∗Department of Finance, London, UK. Email: [email protected]. I thank my advisors John Campbell, JeremyStein, Samuel Hanson, and Adi Sunderam for their outstanding guidance and support, and Andrei Shleifer for manyinsightful discussions. I also thank Tobias Adrian, Lauren Cohen, William Diamond, Erkko Etula, Wayne Ferson,Robin Greenwood, Valentin Haddad, Byoung-Hyoun Hwang, Christian Julliard, Yosub Jung, Dong Lou, Chris Malloy,Ian Martin, David McLean, Tyler Muir, Christopher Polk, Emil Siriwardane, Argyris Tsiaras, Dimitry Vayanos, YaoZeng, and seminar participants at the Adam Smith Asset Pricing Workshop, Boston College, Columbia BusinessSchool, Cornell University, Dartmouth College, Harvard Business School, London School of Economics, Universityof British Columbia, University of Southern California, Hanyang University, Korea University, and Seoul NationalUniversity for helpful discussions and comments. Robert Novy-Marx and Mihail Velikov generously allowed meto use their data on anomalies for preliminary empirical analyses. Jonathan Tan, Karamfil Todorov, and Yue Yuanprovided superb research assistance.
1 Introduction
What generates the cross-section of different riskiness of assets? That is, in the context of factor
models in asset pricing, what makes an asset have a higher beta than others with respect to sys-
tematic risk factors? Despite the obvious importance of these questions, most studies in empirical
asset pricing treat betas as the exogenous right-hand variable that explains expected returns, stay-
ing agnostic about where the betas come from. In contrast, this paper treats betas as the interesting
left-hand variable—the very thing we need to explain.
One obvious channel that can generate the cross-section of factor betas is the cash-flow funda-
mentals: the stocks of firms whose businesses are riskier have higher factor betas.1 But what are
the other important channels? And given those channels, is it valid to treat betas as an exogenous
regressor in an asset pricing regression that explains expected returns?
This paper explores a new channel for the cross-section of betas: the act of arbitrage that turns
“alphas” into “betas.”2 This channel arises naturally from reinterpreting limits to arbitrage (Shleifer
and Vishny, 1997) through the lens of a factor model: arbitrage with limited capital is endogenously
risky because it causes a mispriced asset with an “alpha” (abnormal return) to attain an endogenous
“beta” with the risk factors to which arbitrage capital is exposed. Applying this to the cross-section
of betas, assets may have higher betas because they had higher alphas prior to arbitrage and attract
correspondingly more arbitrage activity.
A second motivation for studying the “alphas-into-betas” channel is its implications for asset
pricing tests. A cross-sectional asset pricing regression attributes an asset’s expected return in
excess of the risk-free rate to its betas with respect to risk factors, with alpha being the residual:
E [rei ] = λ1βi,1 + ... + λJβi,J + αi. Hence, identifying the price of risk λj requires βj to be
“exogenous,” cross-sectionally uncorrelated with α. However, if the act of arbitrage on the test
assets turns their αs into βj during the sample period of the regression, the exogeneity assumption
fails due to simultaneity, biasing the price of risk estimate λj upward.
My key contribution is to flesh out this alphas-into-betas channel for the cross-section of betas
and to highlight its empirical relevance for factor models suggested in the literature.
1Campbell and Mei (1993) and Campbell, Polk, and Vuolteenaho (2010).2Andrew Lo first used this expression to refer to limits to arbitrage occurring in practice (Khandani and Lo, 2011).
1
To formalize my argument, I develop a model in which the cross-section of betas arises entirely
through the act of arbitrage. In my three-period model, a representative arbitrageur trades a contin-
uum of underpriced assets but faces exogenous wealth and funding shocks that generate variation
in the capital she can deploy. The assets have a known terminal value, but in the early (time 1)
and intermediate (time 2) trading periods, they are priced according to deterministic, downward-
sloping demand curves formed by behavioral investors. Crucially, the severity of the underpricing
in the absence of arbitrage capital—the “pre-arbitrage” alpha—differs across the assets.
In this model, the cross-section of betas with respect to arbitrage capital shocks—the sole risk
factor—arises entirely through the act of arbitrage. Furthermore, an asset with a higher pre-
arbitrage alpha attains a higher endogenous “post-arbitrage” beta with respect to arbitrage capital
shocks (Proposition 1). A higher pre-arbitrage alpha means that a larger fraction of the asset is held
by the arbitrageur, since the arbitrageur plays a larger price-correcting role in that asset in equilib-
rium. This, however, means that the asset’s price responds more to the variation in the arbitrage
capital. From this explanation follows the prediction that the cross-section of average arbitrage
positions in the assets should also explain their post-arbitrage betas (Proposition 2).
Furthermore, this endogenous post-arbitrage beta arises only when the arbitrageur is capital-
constrained (Proposition 3); an arbitrageur with does not generate endogenous betas in the assets
when she has “deep pockets.” An asset pricing test that does not account for this endogenous
generation of betas can lead to a false discovery of a pricing factor (Proposition 4).
Using these predictions, I study the extent to which the alphas-into-betas channel explains the
cross-section of betas in actual factor models. In particular, I study the beta exposures of 40
equity “anomalies”— the “long” and “short” portfolios (top and bottom deciles) of 20 anomaly
characteristics known to predict abnormal returns—to the five factors of Fama and French (FF)
(2015) and the funding-liquidity factor of Adrian, Etula, and Muir (2014). My main approach is
to study the anomalies’ betas before and after 1993, the approximate year when arbitrageurs such
as the long-short equity hedge funds began trading these anomalies according to short interest
data, but I also exploit different sample years used in the academic papers that first studied each
anomaly.3
3Chordia, Roll, and Subrahmanyam (2011) and Chordia, Subrahmanyam, and Tong (2014) were the first to suggestthat anomaly trading grew rapidly around 1993. McLean and Pontiff (2016) were the first to exploit the cross-sectionaldifferences in the publication sample years of anomalies.
2
R2adj = 0.76
-1-.5
0.5
Cha
nge
in R
MW
Bet
a
-15 -10 -5 0 5 10Pre-1993 FF 5-Factor Alpha
R2adj = 0.31
-1-.5
0.5
Cha
nge
in C
MA
Bet
a
-15 -10 -5 0 5 10Pre-1993 FF 5-Factor Alpha
R2adj = 0.59-4
-20
2C
hang
e in
Fun
ding
Bet
a
-15 -10 -5 0 5 10Pre-1993 FF 5-Factor Alpha
Figure 1: Turning “Alphas” into “Betas”The figures show that, in the cross-section of 40 equity anomalies, the pre- to post-1993 change in the factor beta ispredicted by the pre-1993 alpha. RMW and CMA are the profitability and investment factors of Fama and French(FF) (2015). The funding beta is with respect to the funding-liquidity factor of Adrian, Etula, and Muir (2014). Thealpha is with respect to the five-factor model of FF. The change in the beta is estimated as the post-1993 beta minusthe pre-1993 beta multiplied by the shrinkage factor of 0.6. Tables 5 and 9 show the regression counterparts that donot restrict the coefficient on the pre-1993 beta to be 0.6. The first two figures use monthly data over 1974-2016; thelast figure uses quarterly data over the same period.
The act of arbitrage plays a prominent role in determining the cross-section of factor betas. To
show this for the five-factor model of FF, I first ask which of the five factors represents risks borne
by arbitrageurs who trade the anomalies. I find that long-short equity hedge funds—the canonical
equity anomaly arbitrageurs—are positively exposed to profitability and investment factors (RMW
and CMA) but neutralize their exposures to market, size, and value factors (MKT, SMB, and HML).
Intuitively, in a more familiar three-factor context (Fama and French, 1993), these arbitrageurs
trade HML but not MKT and SMB. Since the new RMW and CMA factors are designed to replace
HML, the arbitrageurs have positive exposures to RMW and CMA but not to the other factors.4
Consistent with this observation, an anomaly’s pre-1993 five-factor alpha predicts an increase
in the post-1993 beta with respect to RMW and CMA (Figure 1; Proposition 1) but not with respect
to MKT, SMB, and HML. That is, anomalies with positive (negative) pre-1993 alphas see their
RMW and CMA betas increase (decrease) after arbitrage begins around 1993. For example, an
anomaly with a positive (negative) pre-arbitrage alpha but no fundamental RMW beta attains a
positive (negative) post-arbitrage endogenous RMW beta since a positive RMW shock increases
the level of arbitrage capital and exerts positive (negative) price pressure on the anomaly.
Although this alphas-into-betas effect around 1993 supports the arbitrage-based explanation for
the betas, the effect is also consistent with an alternative explanation: the betas of high-realized-
4Which includes the residual HML; see Fama and French (2015, 2016) for the redundancy of HML.
3
alpha anomalies in the pre-1993 period are measured with errors that subsequently disappear in the
post-1993 period. The concern is aggravated if anomaly “discovery” takes place when researchers
find strategies whose in-sample alpha is high because the in-sample beta is erroneously low, in
which case a high in-sample alpha can predict an increase in the beta out-of-sample. However, two
observations are at odds with this alternative explanation. First, the absence of the positive alphas-
into-betas effect in MKT, SMB, and HML suggests that measurement error is not a serious issue for
the FF betas. Second, the RMW and CMA betas do not increase from in-sample to out-of-sample
as they do from pre to post 1993, implying that “beta mining” does not explain the effect.
On the other hand, additional evidence points to the arbitrage-based explanation for the post-
1993 change in RMW and CMA betas. First, anomalies whose RMW and CMA betas increase
more post 1993 are the ones with more arbitrage activity in the same post-1993 period, consistent
with the intuition that an anomaly in which arbitrageurs play a larger price-correcting role has a
larger endogenous price sensitivity to the variation in arbitrage capital (Proposition 2). Second,
the predictability of the post-1993 betas from the pre-1993 alphas is mainly due to the part of the
post-1993 sample during which the VIX is high, consistent with limits to arbitrage arising only
when arbitrageurs are constrained (Proposition 3). Third, anomalies with high costs of arbitrage
show a weaker alphas-into-betas relation, consistent with the relation being driven by arbitrage.
The intermediary-based asset pricing model of Adrian, Etula, and Muir (AEM) (2014) offers
another interesting setting in which to test the model’s predictions. AEM find that anomalies tend
to have high betas with funding-liquidity shocks, but to what extent is this because arbitrageurs
with funding-liquidity risk transmit these exposures to the anomalies that they trade?
My tests again point to the arbitrage-based explanation for the cross-section of funding-liquidity
betas: (i) the anomalies have no significant exposures to funding liquidity in the pre-1993 period
but attain large exposures in the post-1993 period that line up with their pre-1993 FF five-factor
alphas (Figure 1); (ii) the cross-section of different funding-liquidity exposures are explained by
the differences in their average arbitrage position in the post-1993 period; and (iii) the funding-
liquidity exposures arise exclusively in the “constrained” part of the post-1993 period.
Finally, I use the funding-liquidity betas to illustrate the econometric implications of using
endogenous arbitrage-driven betas in cross-sectional asset pricing tests. If the alphas-into-betas
channel causes the teset assets’ betas in the post-arbitrage period (post 1993) to be cross-sectionally
4
correlated with their alphas in the pre-arbitrage period (pre 1993), the estimate for the price of risk
in a cross-sectional asset pricing test that pools the pre- and post-arbitrage periods would be biased
upwards and can lead to a false discovery of a pricing factor (Proposition 4).5 I discuss why the
intermediary-based asset pricing literature should be especially mindful of this issue.
This paper contributes to the vast cross-sectional asset pricing literature. To my knowledge,
I am the first to apply the limits-to-arbitrage insight (Shleifer and Vishny, 1997) to explain the
cross-section of asset betas and to point out that the arbitrage-based channel for betas can cause an
endogeneity problem in asset pricing regressions. I justify the validity of this concern by finding
evidence of this channel for the betas, but I do not deem my evidence definitive and more work
should investigate this potential endogeneity problem.
I also contribute to the literature on asset pricing in the presence of arbitrageurs (financial in-
termediaries) that trade against behavioral investors.6 Kozak, Nagel, and Santos (2017) show that
asset returns in such a “behavioral” model still follow a factor structure in the presence of ratio-
nal arbitrageurs and hence are not easily distinguishable from those in a fully rational model. I
show, however, that the factor model in this behavioral world should have its cross-section of betas
exhibit distinct patterns, including predictability using pre-arbitrage alphas.
Finally, this paper relates to limits to arbitrage. The cross-sectional alphas-into-betas prediction
of this paper can be used to test limits to arbitrage in other contexts (e.g., De Long, Shleifer, Sum-
mers, and Waldmann, 1990; Kyle and Xiong, 2001; Gromb and Vayanos 2002; Liu and Longstaff,
2004; Stein, 2009). My model also relates to recent models of multi-asset arbitrage (Brunnermeier
and Pedersen, 2009; Gromb and Vayanos, 2017; Kondor and Vayanos, 2018), but it differs in that
the key cross-sectional difference across the assets is their pre-arbitrage alphas. Moreover, this
paper adds to the empirical evidence that the act of arbitrage changes the riskiness of assets (e.g.,
Lou and Polk, 2013; Drechsler and Drechsler, 2016; McLean and Pontiff, 2016; Liu, Lu, Sun, and
Yan, 2015), but I focus on how it shapes the cross-section of betas in asset pricing models.7
5This problem is subtler than it may appear and cannot be easily avoided. See for Section 5 details.6Intermediary-based asset pricing falls into this category if investors with preferences different from those of finan-
cial intermediaries are labeled “behavioral.” The theoretical and empirical work in this literature includes Gertler andKiyotaki (2010), He and Krishnamurthy (2012, 2013), Adrian, Etula, and Muir (2014), Brunnermeier and Sannikov(2014), He, Kelly, and Manela (2017), Avdjiev, Du, Koch, and Shin (2017), and Haddad and Muir (2017).
7This cross-sectional focus is similar to that of Brunnermeier, Nagel, and Pedersen (2009), who find that the actof arbitrage on interest rate differentials (carry trade) generates the cross-section of different currency crash risks.
5
2 Asset Pricing Model with Endogenous Cross-section of Risks
2.1 Model setup
Time horizon, assets, and investors. The economy has three periods (t = 1, 2, 3) and two types
of security: a risk-free bond and a continuum of anomaly assets i ∈ [0, 1]. The risk-free bond is
supplied elastically at a zero interest rate. An anomaly asset (“asset”) is a claim to a stream of cash
flows δi,2, v + δi,3 over t ∈ 2, 3 and has a zero net supply. The dividends δi,t are i.i.d. across
assets and time with Et [δi,t+1] = 0, which makes the assets effectively risk-free for a diversified
investor.
There are two types of investor: the behavioral investor and the representative arbitrageur. Be-
havioral investors cause the assets to be mispriced and are modeled as a set of demand curves. An
arbitrageur with mass µ trades against these mispricings but may be capital-constrained. I analyze
the three-period equilibrium under two different assumptions about µ: the trivial “pre-arbitrage”
equilibrium with µ = 0 and the more interesting “post-arbitrage” equilibrium with µ = 1/2.
These two equilibria respectively capture the periods before and after the growth of arbitrage on
the assets.
Behavioral investor demand. Behavioral investors cause the assets to be underpriced, with the
severity of underpricing increasing in i.8 This is modeled using the following aggregate behavioral
investor demand for each asset i:
Bi,t =Et[rei,t+1
]rmax
− i ∀t, (1)
whereBi,t is the aggregate behavioral investor demand for anomaly i (in units of wealth),Et[rei,t+1
]is the objective conditional expected (excess) return, and rmax > 0 is a constant representing the
abnormal return on the most underpriced asset when the arbitrageur has a zero mass (µ = 0).9
Since Et[rei,t+1
]∝ p−1
i,t , eq. (1) implies a negative relationship between price pi,t and de-
mand Bi,t. Furthermore, as i increases, the demand curve shifts downward, implying that higher-
8This direction of mispricing is chosen for convenience and does not affect my theoretical predictions.9I.e., rmax = Et
[re1,t+1|µ = 0
].
6
i assets have lower prices for any given level of behavioral investor demand. Since the de-
mand curves are fixed over time and are the same in both the pre- and post-arbitrage economies,
this cross-sectional ordering of the force behind the underpricing is time-invariant, a key as-
sumption I maintain throughout the analysis.10 All assets have an equal “size” by construc-
tion: a marginal increase in the arbitrage position leads to the same constant fall in the equi-
librium expected return for all assets. That is, if xi,t denotes the arbitrage position on i at t,
∂Et[rei,t+1
]/∂xi,t = −µ
(∂Et
[rei,t+1
]/∂Bi,t
)= −µ rmax ∀i.
Specifying the demand curves rather than the preferences of the behavioral investors allows me
to proceed without specifying the reason for the apparent mispricing, which could be behavioral,
rational, or statistical. For instance, an asset may generate an “alpha” because it covaries with
consumption shocks that risk-averse households want to avoid but the arbitrageur does not care
about. In this case, the alpha reflects a rational compensation for risk but can still turn into “beta”
through the act of arbitrage.
Arbitrageur. A representative, risk-neutral arbitrageur can trade all assets to maximize the ex-
pected terminal wealth at time 3 but faces a capital constraint. Specifically, the arbitrageur can
borrow only up to a stochastic funding constraint ft ∈ [0,∞) and faces a mean-zero wealth shock
wt.11 The arbitrageur can take both long and short positions but faces the margin rate of 1, which
reflects that actual arbitrageurs, such as hedge funds, are not short-sale constrained but face a
non-zero margin requirement (e.g., Brunnermeier and Pedersen, 2009; Ang, Gorovyy, and van
Inwegen, 2010; Garleanu and Pedersen, 2011).12 If the arbitrageur wealth turns negative in any
period, the arbitrageur is forced to exit the market immediately and pay an interest cost c ≥ rmax
on the negative wealth in all future periods.13
10This can happen in practice despite the growth of institutional capital in the stock market if mutual fund managersexhibit behavioral patterns similar to those of retail investors, which seems to be the case (e.g., Frazzini, 2006; Frazziniand Lamont, 2008).
11Making the arbitrageur wealth and funding shocks “systematic” while shutting down shocks to the assets comingfrom dividends or behavioral investors is a conscious choice that elucidates how assets with only idiosyncratic riskscan attain systematic risks through the act of arbitrage.
12Keeping the margin constant rather than having it depend on the volatility of the asset, I emphasize that dif-ferent endogenous betas can arise without differences in the anomalies’ idiosyncratic volatilities, the key feature inBrunnermeier and Pedersen (2009) and Gromb and Vayanos (2017).
13This allows me to obtain the marginal value of wealth in the negative-wealth region.
7
Hence, the arbitrageur’s objective at time t is as follows:
maxxiEt [w3]
s.t. wt+1 =
wt +∫ 1
0ri,t+1xi,tdi+ wt+1
(1 + c)wt
if wt > 0
if wt ≤ 0∫ 1
0|xi,t| di ≤ 1 (wt > 0) kt
kt = wt + 1 (wt > 0) ft,
(2)
where xi is the unit arbitrageur’s sequence of dollar positions on anomaly i in all trading periods,
wt is the wealth of the arbitrageur, ri,t is the asset return, 1 (·) is an indicator function, and kt is the
arbitrageur’s deployable capital (“arbitrage capital”).
Equilibrium conditions. I look for a competitive equilibrium in which
1. The aggregate behavioral investor demand Bi,1 and Bi,2 satisfy equation (1) given
prices pi,1 and pi,2;
2. The arbitrageur’s chosen positions xi,1 and xi,2 solve problem (2), given prices pi,1and pi,2; and
3. All asset markets clear: µxi,t +Bi,t = 0 ∀i, t.
2.2 The pre-arbitrage equilibrium
Given this setup, in the “pre-arbitrage economy” with a negligible mass of arbitrageurs (µ = 0), the
assets have different alphas but no systematic risks (see Appendix B for derivations and proofs):
Lemma 1. (Asset returns in the pre-arbitrage economy). If µ = 0, the return on asset i follows
ri,t = αprei + εi,t, (3)
with εi,t denoting a mean-zero idiosyncratic return and the “pre-arbitrage alpha,”
αprei = rmaxi, (4)
increasing monotonically from asset i = 0 to asset i = 1.
8
Since the pre-arbitrage and post-arbitrage economies feature the same behavioral investors, the
“pre-arbitrage alpha” αprei measures the degree of latent mispricing that pushes the asset price away
from the correct level in both economies and is a key variable in my comparative statistics.
2.3 The post-arbitrage equilibrium
Now suppose the arbitrageur has a non-negligible mass of µ = 1/2. If the arbitrageur is always un-
constrained with sufficient capital (k1, k2 ≥ 1), all alphas are arbitraged away and no endogenous
risk arises:
Lemma 2. (Asset returns with unconstrained arbitrageurs). Suppose µ = 1/2 and k1, k2 ≥ 1
with certainty so that the arbitrageur is always unconstrained. Then, the return on asset i follows
ri,t = εi,t (5)
where εi,t is a mean-zero idiosyncratic return.
Hence, with frictionless “textbook” arbitrage, the assets that were differently mispriced in the pre-
arbitrage economy become effectively identical riskless assets in the post-arbitrage economy.
In contrast, suppose now that in the middle of arbitrage, the level of arbitrage capital may fall
below the value required to counteract all mispricings (Shleifer and Vishny, 1997). In the model,
this is when k2 can fall below 1. In this case, the asset returns and the level of arbitrage capital
comove endogenously at time 2, making the assets endogenously risky from the arbitrageur’s per-
spective. This implies that the asset return from time 1 to time 2 follows a beta pricing model:
Lemma 3. (Asset returns with constrained arbitrageurs). Suppose µ = 1/2 and k2 has full
support over [0, 1] so that the arbitrageur may be constrained. Then, the expected return on asset
i at time 2 follows
E1ri,2 = αposti,0 + λmβposti,m (6)
9
s.t. (i) βposti,m is the negative of the beta with respect to the arbitrageur’s time-2 stochastic discount
factor (SDF), which depends negatively on k2.
(ii) αposti,0 is the asset-specific zero-beta rate that is also the abnormal return by the zero-risk-
free-rate assumption.
(iii) λm > 0 and βposti,m > 0 for i > 0.
Lemma 3 restates the “limits of arbitrage” result of Shleifer and Vishny (1997) as a beta pricing
model. Arbitrage that requires capital is endogenously risky since its return comoves negatively
with the arbitrageur’s SDF. This limits a complete arbitrage, as the arbitrageur requires a premium
for bearing that risk. The zero-beta rate αposti,0 may differ across the assets, implying that the law of
one price may be violated. This happens if the arbitrageur’s capital constraint binds at time 1, as in
Garleanu and Pedersen (2011), Geanakoplos and Zame (2014), and Gromb and Vayanos (2017).
Since the arbitrageur’s SDF is determined by the level of arbitrage capital k2, eq. (6) can be
alternatively expressed as a factor model using k2 or its subcomponents as the factors:
Lemma 4. (A factor model of asset returns). Suppose µ = 1/2 and k2 has full support over [0, 1]
so that the arbitrageur may be constrained. Then, the expected return on asset i at time 2 follows
an approximate factor model:
E1ri,2 ≈ αposti,0 + λkβposti,k
= αposti,0 + λwβposti,w + λfβ
posti,f
(7)
where αposti,0 is the same as in Lemma 3 and βposti,z is the beta with respect to z ∈ k2, w2, f2 with
βposti,k > 0. By Stein’s lemma, these factor models are exact if k2 and ri,2 are jointly normally
distributed.
Lemma 4 elucidates the nature of the endogenous risk that arises through arbitrage. First,
mispriced assets become risky through the act of arbitrage as their returns comove endogenously
with shocks to arbitrage capital. Hence, the arbitrage-based explanation for the betas only applies
to factors that represent systematic shocks to arbitrage capital. Second, there are two kinds of
systematic shocks to arbitrage capital: (i) wealth shocks w2 coming from traded assets and investor
flows (see eq. (2)) and (ii) funding shocks f2. In the empirical sections, I proxy the systematic
arbitrageur wealth shocks using the RMW and CMA factors of Fama and French (2015) and proxy
10
the systematic arbitrageur funding shocks using the funding-liquidity factor of Adrian, Etula, and
Muir (2014).
2.4 The cross-sectional predictions of the model
The key theoretical contribution of this paper is in generating the cross-sectional predictions about
the betas that arise endogenously through arbitrage. The first prediction relates the degree of
mispricing prior to arbitrage to the size of the endogenous beta:14
Proposition 1. (Pre-arbitrage alpha determines the cross-section of endogenous post-arbitrage
betas). Suppose µ = 1/2 and k2 has full support over [0, 1] so that the arbitrageur may be
constrained.
(i) Then, an asset with a high pre-arbitrage alpha attains a higher arbitrage capital beta:
∂βposti,k
∂αprei
> 0. (8)
That is, “alphas turn into betas.”
(ii) The same goes for the beta with respect to the w2 and f2 that comprise k2.
A large pre-arbitrage alpha means that a larger fraction of the asset is held by the arbitrageur
since she plays a larger price-correcting role in the asset in equilibrium. This, however, also means
that the asset responds more to the variation in the arbitrage capital at time 2, making the asset
endogenously more exposed to each systematic shock to arbitrage capital. From this explanation,
it follows that the cross-section of average arbitrage positions in the assets should also explain the
cross-section of endogenous βs:
Proposition 2. (The cross-section of average arbitrage positions explains the cross-section of
endogenous post-arbitrage betas). The expected arbitrage position in an asset µE1 [xi,2] explains
the cross-section of betas βposti,k in the post-arbitrage equilibrium. This is also true for βposti,f and
βposti,w .
14I show this using βi,k instead of the exact measure of risk βi,m for the sake of a tighter link to the empiricalsection, but the proof in Appendix B holds analogously for βi,m.
11
Next, a useful restriction on the the alphas-into-betas relation in Proposition 1 is that the cross-
sectional “predictability” of the betas comes from the constrained states of time 2. Put differently,
an arbitrageur does not generate endogenous βs in the assets when she has “deep pockets”:
Proposition 3. (The endogenous post-arbitrage beta arises when the arbitrageur is constrained).
The endogenous post-arbitrage betas arise only when the arbitrageur is constrained. That is,
βposti,k | (k2 ≥ 1) = 0
βposti,k | (k2 < 1) > 0(9)
for all i ∈ (0, 1]. For this reason, if kt follows a process k∗t such that k∗1, k∗2 ≥ 1 almost surely, then
neither beta nor abnormal return arises:
βposti,k = 0 and E1 [ri,2] = 0 for all i ∈ [0, 1] (10)
Finally, asset pricing tests may generate biased results if the econometrician cannot distinguish
between pre- and post-arbitrage economies. The presence of the alphas-into-betas relation means
that part of the ability of beta to explain the cross-section of returns can come erroneously from
the fact that assets with high pre-arbitrage returns due to αprei endogenously attain high betas in the
post-arbitrage equilibrium:
Proposition 4. (An upward-biased price of risk in a naive asset pricing test). Suppose that the
arbitrageur enters time 1 to face a pre-arbitrage economy (µ = 0) with probability φ and a post-
arbitrage economy (µ = 1/2) with probability 1 − φ. If this arbitrageur’s capital kt has positive
support over [0, 1], an econometrician who does not condition on µ observes an upward-biased
price of risk associated with βi,k:
E [ri,2] ≈ αi,0 +
λk + φb−1︸︷︷︸bias
βi,k (11)
where αi,0 and βi,k are the unconditional zero-beta rate and arbitrage-capital beta over the two
economies, and b is the coefficient on the linearized alphas-into-betas effect, βposti,k = b αprei .
12
3 Application to Multifactor Models
3.1 The anomalies
Does the act of arbitrage help determine the cross-section of betas in mutlifactor models? To
answer this question, I study the cross-section of betas of equity anomalies, trading strategies
known to generate abnormal returns. Equity anomalies are well-suited for this purpose for two
reasons. First, they have been actively traded by arbitrageurs, such as the quantitative long-short
equity hedge funds, since the early 1990s (e.g., Hanson and Sunderam, 2014), suggesting that
some of these anomalies’ risks may be an endogenous outcome of the arbitrage itself. Second,
they offer a reasonably rich cross-section, as noted in Green, Hand, and Zhang (2016), allowing
me to test the model’s cross-sectional predictions.
In particular, I use a set of 40 equity anomalies which are the “long” and “short” portfolios
(top and bottom deciles) of 20 anomaly characteristics (see the list in Table 1).15 The return on
an anomaly is the monthly value-weighted average return of all domestic common stocks from the
three major exchanges (NYSE, AMEX, and NASDAQ) that belong to the extreme decile portfolios
determined each month.16 I use data from 1974m1 to 2016m12. Since some of the anomalies are
related, I use standard errors that account for the cross-anomaly correlations (see Appendix C).
3.2 The five Fama-French (2015) factors
I study the cross-section of betas with respect to the five factors of Fama and French (FF) (2015).
FF suggest that the market, size, value, profitability, and investment factors (labeled MKT, SMB,
HML, RMW, and CMA) summarize a stock’s systematic risks. Consistent with this claim, Fama and
French (2016) find that the five-factor model explains the high returns earned by beta arbitrage, net
15This list of 20 characteristics represents a standard set of low-turnover anomaly characteristics. One can arriveat this set by taking the 32 characteristics surveyed by Novy-Marx and Velikov (2016) and excluding the 5 redundant(e.g., “high-frequency combo”) and 7 highest-turnover (e.g., short-term reversal) characteristics. I thank Mihail Ve-likov for several correspondences that helped ensure my replication of their anomalies. I exclude the high-turnoveranomalies since endogenous beta should not arise in anomalies with a short mispricing horizon (i.e., if the asset priceachieves the fundamental value v at time 2, in the context of my model) (for a formal treatment of this point, seeGromb and Vayanos, 2017).
16See the online data appendix to this paper as well as Novy-Marx and Velikov (2016) for more information on theanomaly construction.
13
issuance, and volatility anomalies through their exposures to RMW and CMA in addition to the the
size, value, investment, and profitability anomalies that the model is intended to explain. Table 2
summarizes the cross-section of 40 anomalies’ betas with the five factors in two sample periods
based on the 1993 cutoff explained next.17
3.3 The 1993 cutoff
I use 1993 as the approximate year of the shift from the “pre-arbitrage” equilibrium to the “post-
arbitrage” equilibrium for equity anomalies. There are several reasons. First, the amount of hedge
fund capital grew rapidly in the 1990s, with the total assets under management growing from
$39 billion in 1990 to $1.73 trillion in 2008 (Stein, 2009). Second, some of the most influential
papers on equity anomalies were published around 1993, including Fama and French (1993) and
Jegadeesh and Titman (1993), igniting the practitioner interest in trading stocks based on anomaly
signals. Third, stock market liquidity improved significantly around 1993 (Chordia, Roll, and
Subrahmanyam, 2011), lowering the cost of arbitraging the anomalies (Chordia, Subrahmanyam,
and Tong, 2014).
I back up this anecdotal evidence using actual arbitrage activity inferred from short interests;
that is, the number of shares of a stock being shorted. Since shorting is primarily done by arbi-
trageurs such as long-short equity hedge funds, abnormally high or low shorting of a stock serves
as a proxy for arbitrage activity on the anomalies (Ben-David, Frazoni, and Moussawi, 2012;
Boehmer, Jones, and Zhang, 2013; Hanson and Sunderam, 2014; Hwang, Liu, and Xu, 2018).18 In
particular, I measure arbitrage activity as the negative (×−1) of the abnormal short interest, which
I define to be the value-weighted average of short interest ratio (shares shorted / shares outstand-
ing) minus the cross-sectional average short interest ratio of stocks that belong to the same NYSE
size decile, where the average is taken over all stocks that belong to the anomaly portfolio.
Arbitrage activity inferred from short interests strongly supports the case that arbitrage activity
on the anomalies began around 1993. Table 3 shows that the post-1993 dummy has the single
largest effect on the growth of arbitrage activity on the anomalies, controlling for both the post-
publication and post-sample effects (read the coefficients on variables interacted with the long17I thank Kenneth French for providing the factor data through his data library.18Abnormally low shorting signals that arbitrageurs prefer to take long positions on the stock and avoid shorting it.
14
anomaly dummy or pre-1993 alpha).19 Academic publication triggers arbitrage activity only in the
post-1993 period, when hedge funds had sufficient capital to generate an observable effect on short
interests in response to an academic publication. Comparing the R2s in columns 7-10 shows that
the pre-1993 multifactor alphas do a better job of capturing the latent mispricing perceived by the
arbitrageurs than the pre-1993 CAPM alpha. Columns 7, 11, and 12 suggest that the exact horizon
in which the pre-arbitrage alpha is measured matters less.
3.4 The evidence on the cross-section of multifactor betas
To what extent is the cross-section of five-factor betas endogenous? (Note: Throughout this paper,
“endogenous beta” refers to the part of an asset’s beta that arises from arbitrage on past returns,
which makes the beta an endogenous variable in an asset pricing regression with realized past
returns on the left-hand side.)
The first task is to infer which of the five factors represent systematic risks borne by the long-
short arbitrageurs of anomalies. Table 4 shows that in the pre-1993 period, the equal-weighted
long-short portfolio of anomalies that takes an equal positive (negative) position on long-side
(short-side) anomalies is positively exposed to RMW and CMA but not strongly exposed to MKT,
SMB, and HML (column 1). That is, long (short) anomalies tend to load positively (negatively) on
RMW and CMA, making it difficult for arbitrageurs to neutralize their RMW and CMA exposures
through a long-short strategy. Consistent with this, long-short arbitrageurs of anomalies proxied
by the long-short equity hedge funds are systematically exposed to RMW and CMA but not to the
other factors in the post-1993 period (column 3).20 This suggests that RMW and CMA represent
systematic shocks to arbitrage capital, implying that the anomalies’ post-1993 betas with these
factors may be endogenous.
19My finding on the 1993 cutoff is somewhat at odds with the finding that no return decay is observed in theanomalies following 1993 (McLean and Pontiff, 2016). The main reason for this difference is that short interestmeasures the arbitrage activity by a group of sophisticated arbitrageurs, whereas return decay reflects investment byall types of investor. Another contributing factor is that I use the year in which the anomaly was first published, notwhen it was first well publicized (e.g., the academic publication of the value anomaly is Rosenberg, Reid, and Lanstein(1985) in my data but Fama and French (1992) in McLean and Pontiff).
20That the long-short arbitrageurs are MKT- and SMB-neutral is not surprising, but it may surprise the reader thatthey are HML-neutral, given the importance of value strategy for quantitative hedge fund managers. Intuitively, inthe context of the Fama-French (1993) three-factor model, these arbitrageurs are exposed to HML. However, sincethe new RMW and CMA factors are designed to subsume HML, the arbitrageurs have positive exposures to RMW andCMA but not to the residual HML.
15
I therefore test, in the cross-section of 40 anomalies, if the pre-1993 five-factor alpha predicts
the post-1993 betas with respect to RMW and CMA but not with respect to MKT, SMB, and HML
(Proposition 1). Table 5 shows that a high pre-1993 alpha predicts high post-1993 RMW and CMA
betas, controlling for pre-1993 beta. Comparing the R2s suggests that as much as 38% (RMW) and
21% (CMA) of the cross-sectional variation in the post-1993 betas is endogenous. The coefficients
on the pre-1993 beta are around .6 for both RMW and CMA (and for SMB and HML as well),
suggesting that these factor betas shrink by a factor of 0.6 around the cross-sectional mean of zero
between the two sample periods, much as the market beta shrinks around 1 over time (e.g., Vasicek,
1973). The alphas-into-betas effect is stronger for RMW betas than for CMA betas, which reflects
that long-short arbitrageurs are more strongly exposed to the RMW factor (Table 4).
The alphas-into-betas prediction in Section 2 is derived under the assumption that the anomaly
assets differ only in their pre-arbitrage alpha. The anomalies used in my tests, however, may also
differ in their volatility. Columns 3 and 6 show that adding volatility to the regression does not
meaningfully affect the ability of pre-1993 alpha to explain the post-1993 RMW and CMA betas,
although the negative coefficient on αpre × V olatility suggests that, as expected, the arbitrage
channel for the betas is weaker for more volatile anomalies. The related issue of differences in the
anomalies’ trading cost is explored at the end of the section.
There is no alphas-into-betas effect for SMB and HML, consistent with the earlier observation
that a long-short strategy neutralizes exposure to these factors. With MKT, a high pre-1993 alpha
predicts a lower post-1993 beta, consistent with the simple long-short strategy on the anomalies
having a slightly negative exposure to MKT but inconsistent with long-short equity hedge funds
having a slightly positive market exposure (Table 4). This effect on MKT beta disappears, however,
once volatility has been controlled for.21
A. The measurement-error explanation
The results above are consistent with the arbitrage-based explanation for αs-into-βs, but what other
explanations exist? To answer this, I take a step back and consider the αs-into-βs regression. The
21I also find that the alphas-into-betas effect on MKT does not reappear when long and short anomalies are consid-ered separately.
16
assumed data-generating process is
βposti = b0 + b1αprei + b2β
prei + ui, (12)
where α and β respectively denote the realized alpha and true beta.22 Rewriting eq. (12),
βposti = b0 + b1αprei + b2β
prei + ui − b2e
prei , (13)
where eprei ≡ βprei − βprei denotes the measurement error in βprei .
The problem is that αprei and βprei are the intercept and slope coefficients from the same time-
series regression, which makes αprei and eprei positively correlated if the factor associated with the β
has a positive mean, as is the case with the FF five factors. This implies that if b2 > 0, the estimated
b1 can be positive even if b1 = 0. Intuitively, a negative measurement error in the beta leads to a
higher realized alpha, and this high realized alpha can appear to predict a subsequent increase in the
estimated beta. The problem is aggravated if anomaly “discovery” takes place when a researcher
finds a strategy whose in-sample alpha is high because its in-sample beta is erroneously low, in
which case the alphas-into-betas effect around 1993 may be driven by anomalies with a high in-
sample alpha experiencing an increase in the beta out-of-sample.
However, two observations are at odds with this measurement-error explanation. The first is the
absence of a positive αs-into-βs effect in MKT, SMB, and HML. All five factors have a positive
realized mean in the pre-1993 period, implying that the measurement error issue, if exists, should
apply to all five. Instead, the absence of a positive αs-into-βs effect in those three factors suggests
that betas are measured with a sufficient precision over the 240 months in the pre-1993 period.
Second, I use panel regressions to test if the anomalies’ RMW and CMA betas change predictably
from in-sample to out-of-sample, as they do around 1993. Table A1 shows that they do not, once
the post-1993 effect has been controlled for.23 The null finding on “beta mining” is also true for
the MKT, SMB, and HML betas.
22I use realized alpha as the determinant of βpost in the data-generating process because the magnitude of thearbitrage depends on the realized alpha that the arbitrageur observes rather than an unobserved true alpha.
23The result is similar if I add an interaction with publication year to the regression.
17
B. Further evidence for the arbitrage channel
On the other hand, two additional tests point to the arbitrage-based explanation for the alphas-
into-betas effect in the RMW and CMA betas. The first is the ability to explain the cross-section
of post-1993 betas using anomaly-specific arbitrage activity inferred from short interests used in
section 3.3. Table 6 (also see Figure 4) shows that anomalies that attain high post-1993 RMW and
CMA betas are the ones with larger average arbitrage positions in the same period, consistent with
the intuition that an anomaly in which arbitrageurs play a larger price-correcting role endogenously
has a larger price sensitivity to the variation in arbitrage capital (Proposition 2).
It is interesting to consider this result jointly with the alphas-into-betas result in Table 5. Con-
ceptually, both the pre-1993 alpha and the post-1993 arbitrage activity are valid predictors of en-
dogenous beta in the post-1993 period. The advantage of using the contemporaneous arbitrage
activity is that it is a more direct determinant of the endogenous beta. However, since the true arbi-
trage activity is unobserved and has to be measured with an error, the pre-1993 alpha can act like
an instrumental variable that provides a cleaner estimate of the alphas-into-betas effect if arbitrage
capital is allocated to different anomalies based on their past alphas.24
Second, the predictability of post-1993 betas using pre-1993 alphas comes from the constrained
times of the post-1993 period, consistent with the notion that an arbitrageur does not generate en-
dogenous betas in the anomalies when she has “deep pockets” (Proposition 3). To test this, I proxy
the constrained times within the post-1993 period as the months in which the three-month moving
average of the VIX was above the median, which implies that 1997m1-2003m8 and 2007m7-
2012m6 were the periods in which arbitrageurs’ capital constraint was binding (Figure 2). The
first period begins in the aftermath of the peso crisis and includes the LTCM, 9/11, and the Iraq
War, while the second period coincides with the Great Recession. I then ask if the alphas-into-
betas result earlier comes from these constrained times of the post-1993 period in which shocks
to arbitrage capital translate into a larger price response in the anomalies, since arbitrageurs have
difficulty replenishing their capital.
Comparing the R2s in Table 7 shows that for both RMW and CMA, the ability of pre-1993
alpha to predict post-1993 beta is stronger for betas during constrained times, consistent with price
24I thank Yao Zeng for this point.
18
pressure from arbitrageurs arising when they are constrained.25 There remains some predictability
of the unconstrained-time betas, suggesting that the constrained vs. unconstrained classification is
imperfect. I report the five-factor alpha and arbitrage activity in the two subperiods of post 1993 as
a sanity check. The larger alpha and lower arbitrage activity suggest that the high-VIX times are
indeed those when arbitrageurs are more capital-constrained.
An alternative explanation for this result is that pre-1993 alpha represents compensation for
conditional risk with respect to these two factors. However, I find that the differential alphas-into-
betas result does not arise in MKT, SMB, and HML (controlling, in the case of MKT, for volatility),
suggesting that the pre-1993 alpha is not a compensation for conditional risks with respect to the
five factors (Table A2).
C. Costs of arbitrage
Costs of arbitrage, such as idiosyncratic volatility and liquidity, may matter in arbitraging the
anomalies, especially for those involving short selling (Knez and Ready, 1996; Korajczyk and
Sadka, 2004; Lesmond, Schill, and Zhou, 2004; McLean and Pontiff, 2014). Hence, if the alphas-
into-betas effect arises through arbitrage, the strength of the effect may differ across anomalies
depending on their costs of arbitrage. I therefore include arbitrage cost and its interaction with pre-
1993 alpha in my main alphas-into-betas regression. I use size, idiosyncratic volatility, illiquidity
(Amihud, 2002), and bid-ask spread (Corwin and Schultz, 2012) to measure the cost of arbitrage.
Arbitrage costs affect the alphas-into-betas relation in the direction one would expect if the rela-
tion were driven by arbitrage (Table 8; each cost variable is signed so that a high value means high
arbitrage cost). Anomalies whose underlying stocks are small, volatile, and have a large bid-ask
spread see their alphas turn into smaller betas, although the effect—albeit weak—has an opposite
direction for the Amihud measure of illiquidity. Among different costs of arbitrage, idiosyncratic
volatility and bid-ask spread have the largest effects and their effects are statistically significant
for CMA but not for RMW. This finding that anomalies with high costs of arbitrage show a weaker
alphas-into-betas relation again suggest that the relation is driven by the act of arbitrage.
25The predictive regression for the constrained-time CMA betas, however, has a marginally insignificant (at the10% level) coefficient on αpre due to the larger residual variance during high-VIX times.
19
3.5 Other multifactor models
The literature suggests other factor models of equity anomalies, including—but are not limited
to—the models motivated by the arbitrage pricing theory (APT) (e.g., Fama and French, 1992,
1993; Carhart, 1997), intertemporal CAPM (e.g., Campbell and Vuolteenaho, 2002; Campbell,
Giglio, Polk, and Turley, 2017), q-theory (e.g., Hou, Xue, and Zhang, 2015), consumption and
macroeconomic risks (e.g., Parker and Julliard, 2005; Yogo, 2006; Lettau, Ludvigson, and Ma,
2017), and behavioral factors (e.g., Stambaugh and Yuan, 2016; Daniel, Hirschleifer, and Sun,
2017).
The arbitrage-based explanation for the cross-section of betas explored in this paper applies to
any factors that are systematic arbitrage capital shocks. This makes the Clarke’s (2016) “level,
slope, and curve” factor analysis useful, since a long-short arbitrage is likely to be exposed only
to “slope” factors, making those factors likely suspects for the alphas-into-betas effect and hence
also the endogenous beta problem in an asset pricing test discussed in Section 5.
4 Application to Intermediary-based Asset Pricing
Intermediary-based asset pricing models offer another interesting opportunity to test the arbitrage-
based explanation for the cross-section of betas. These models have enjoyed some empirical suc-
cess in pricing the cross-section of assets by measuring risks from the perspective of financial
intermediaries (e.g., Adrian, Etula, and Muir, 2014; He, Kelly, and Manela, 2017; Avdjiev, Du,
Koch, and Shin, 2017). These models, however, seem incomplete without an adequate explanation
of where the cross-section of different intermediary βs come from:
. . . ultimately we want to understand not only the pricing of assets’ covariances with
risk factors, but also the determinants of these covariances themselves. In the context
of [Adrian, Etula, and Muir (2014)]’s empirical analysis specifically, a full account of
margin risk should explain why some assets covary more strongly than others with the
leverage of security broker-dealers (Campbell, 2017).
Why do some assets covary more strongly than others do with systematic shocks to financial in-
20
termediaries? To what extent are the intermediary asset pricing betas an endogenous outcome of
arbitrage?
4.1 The funding-liquidity factor of Adrian, Etula, and Muir (2014)
I answer these questions using the funding-liquidity factor of Adrian, Etula, and Muir (AEM)
(2014). Motivated by the funding-liquidity model of Brunnermeier and Pedersen (2009), AEM
focus on aggregate funding-liquidity shocks to financial intermediaries, which the authors proxy
using shocks to the leverage of security broker-dealers (plotted in Figure 3). Using these shocks,
they find that high returns to value and momentum anomalies are explained by their positive expo-
sures to aggregate funding-liquidity shocks.
However, these funding-liquidity exposures of anomalies may be endogenous given that long-
short arbitrageurs such as hedge funds are strongly exposed to these shocks (Brunnermeier and
Pedersen, 2009; Aragon and Strahan, 2012; Mitchell and Pulvino, 2012). This seems especially
plausible for the AEM factor, since most security broker-dealers provide funding to hedge funds
as their prime brokers, meaning that the hedge funds may be transmitting these funding shocks to
the anomalies they trade.
4.2 The evidence on the cross-section of funding-liquidity betas
I therefore test if the cross-section of funding-liquidity betas of 40 anomalies are consistent with the
arbitrage-based explanation (Table 9). In the pre-1993 period, the anomalies did not have strong
exposures to this measure of funding liquidity (column 1). It is only in the post-1993 period that
the anomalies attain large beta exposures, consistent with anomalies on average having no inherent
exposure to funding liquidity (column 3).
Furthermore, the cross-section of funding-liquidity betas in the post-1993 period appear to be
driven by the act of arbitrage. The pre-arbitrage alpha proxied by the FF five-factor alpha ex-
plains the cross-section of post-1993 funding-liquidity betas (column 6), although the effect is
only marginally significant, due to large measurement errors in the quarterly data (Proposition 1).
The cross-section of betas are also explained by the cross-section of average arbitrage positions in
21
the post-1993 period, consistent with anomalies with more arbitrageur involvement being endoge-
nously more exposed to arbitrage capital shocks (columns 10-11; Proposition 2). The endogenous
part of the beta comes from parts of the post-1993 period in which arbitrageurs are likely to be
capital-constrained, proxied by quarters in which the VIX is above the post-1993 median (col-
umn 15). This is again consistent with the alphas-into-betas effect arising when arbitrageurs are
constrained but not when they are unconstrained (Proposition 3).
5 Implications for Asset Pricing Tests
Here, I explain why the alphas-into-betas channel for the cross-section of betas can be problematic
for asset pricing tests. This is because the alphas-into-betas effect is a source of endogeneity for
the βs in asset pricing regressions. To see this, consider a cross-sectional asset pricing regression
that attributes the expected excess return of an asset to its betas with respect to risk factors and the
residual alpha:
E [rei ] = λ1βi,1 + ...+ λJβi,J + αi. (14)
In this regression, identifying λj , which tells us whether or not factor j is a risk factor (i.e., λj >
0), requires βj to be “exogenous” in that it is cross-sectionally uncorrelated with the residual α.
However, if the act of arbitrage causes test assets with high αs to attain high βj during the sample
period of the regression, βj becomes endogenous due to simultaneity.
Since this potential endogeneity issue pertains to cross-sectional asset pricing tests, it is less of
a concern for factors that are traded portfolios (e.g., FF five factors), which can be tested using
the time-series approach. I therefore use the non-traded Adrian, Etula, and Muir (AEM) (2014)
funding-liquidity factor to illustrate the issue. In particular, I study how the alphas-into-betas
channel makes cross-sectional asset pricing of 40 anomalies using the funding-liquidity factor
sensitive to the choice of the sample period.
Table 10 shows the outcome of cross-sectional asset pricing tests using funding liquidity as the
sole factor to be consistent with AEM. In the pre-1993 period (1974-1993) with little arbitrage on
the anomalies, the estimated price of risk is close to zero, suggesting that the anomalies are not
priced by the funding-liquidity-exposed arbitrageurs during this period. In the post-1993 period
22
(1994-2016) with more arbitrage, however, the factor attains a larger price of risk and explains a fair
amount (34%) of cross-sectional variation in the expected return on 40 anomalies. The estimated
price of risk λ = 1.40 in this period is likely to reflect the correct price of risk associated with
funding liquidity in the post-arbitrage equilibrium. The large standard error makes the price of risk
coefficient statistically insignificant, but this does not necessarily preclude that funding liquidity is
priced in the cross-section, given the large estimator variance arising from measurement errors in
quarterly data and the fact that pricing 40 anomalies with a single factor is a very high hurdle.
The issue, however, is that the price of risk estimate becomes much larger (λ = 3.11) (and
statistically significant) in the sample period of 1974-2016, which pools the the pre-1993 and
post-1993 periods (column 3). This impressive performance of the factor in the pooled sample
period can be rationalized by the argument made earlier: including the pre-arbitrage period in a
cross-sectional asset pricing regression with an endogenous beta biases the price of risk upward
since the regression treats the high realized mean return due to the pre-arbitrage alpha to be risk
premium explained by the post-arbitrage beta. Consistent with this, reducing the exposure to the
pre-arbitrage period reduces the price of risk estimate (columns 4-6; Proposition 4).
To be fair, this illustration using the funding-liquidity factor and 40 anomaly portfolios does not
necessarily undermine AEM, who show that the same funding-liquidity factor (leverage factor)
explains the cross-section of 25 equity portfolios sorted by size and value, 10 equity portfolios
sorted by momentum, and 6 bond portfolios sorted by maturity. The appendix of their paper
conducts a subsample analysis to find that the funding-liquidity factor has a significant price of risk
in both the first half (1968-1988) and second half (1989-2009) of their sample period, although
the price of risk estimated in the pooled sample period is still larger. One way to reconcile the
subsample-robustness of their result with the illustration I give here using 40 different anomalies is
the segmented-market story: the basic anomaly strategies like value and momentum were priced by
arbitrageurs even in the pre-1993 period, but the other anomalies were priced by the arbitrageurs
only after both practioner interest in anomaly-characteristics-based strategies and the amount of
hedge fund capital grew rapidly around 1993.
Future work in empirical intermediary-based asset pricing should nonetheless be mindful of the
endogenous beta problem illustrated here. Although this problem can arise in any factor model,
intermediary asset pricing uses aggregate financial shocks as the pricing factors, so arbitrage capital
23
is likely to be exposed to those factors as well, in which case the alphas-into-betas effect would
arise with respect to those factors. Intermediary asset pricing also tends to use factors that are
not traded portfolios and hence rely exclusively on the cross-sectional asset pricing test approach,
which also makes the endogeneous beta issue more relevant.
6 Conclusion
This paper is the first attempt to endogenize the cross-section of asset betas using the limits-to-
arbitrage insight. I provide theoretical and empirical evidence on the arbitrage channel for the betas
and show that betas arising in this manner can cause cross-sectional asset pricing tests to suffer
from endogeneity and generate biased price of risk estimates. Future work could usefully provide
additional empirical evidence on the arbitrage channel for the cross-section of betas in equity and
other markets and examine the severity of the endogeneity problem caused by endogenous betas
in popular asset pricing models.
24
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Table 1: List of Anomaly Characteristics
Academic Publication Market Cap Share
No Anomaly Label Authors Year Sample Long Short
1 Beta arbitrage beta Fama and MacBeth 1973 1926-1968 0.09 0.092 Return on market equity rome Basu 1977 1956-1971 0.05 0.033 Ohlson’s O-score ohlson Ohlson 1980 1970-1976 0.29 0.014 Size size Banz 1981 1926-1975 0.02 0.585 Long-run reversals rev60m DeBondt and Thaler 1985 1926-1982 0.03 0.136 Value value Rosenberg, Reid, and Lanstein 1985 1980-1990 0.04 0.27 Momentum mom12m Jegadeesh 1990 1964-1987 0.1 0.048 Net issuance netissue Ikenberry, Lakonishok, and Vermaelen 1995 1980-1990 0.11 0.099 Net issuance monthly netissue_m Ikenberry, Lakonishok, and Vermaelen 1995 1980-1990 0.1 0.09
10 Accruals acc Sloan 1996 1962-1991 0.06 0.0511 Return on assets roa Haugen and Baker 1996 1979-1993 0.17 0.0312 Return on book equity roe Haugen and Baker 1996 1979-1993 0.14 0.0413 Failure probability failprob Dichev 1998 1981-1996 0.16 0.0214 Piotroski’s f-score piotroski Piotroski 2000 1976-1997 0.21 0.0915 Investment invest Titman, Wei, and Xie 2004 1973-1996 0.03 0.0716 Idiosyncratic volatility idiovol Ang et al. 2006 1986-2000 0.25 0.0417 Asset growth atgrowth Cooper, Gulen, and Schill 2008 1968-2003 0.03 0.118 Asset turnover ato Soliman 2008 1984-2002 0.05 0.0919 Gross margins gm Soliman 2008 1984-2002 0.2 0.0420 Gross profitability profit Balakrishnan, Bartov, and Faurel 2010 1976-2005 0.1 0.07
29
Table 2: Beta Exposures to Fama-French (2015) Five Factors
The pre-arbitrage period is from 1981m1 for the failure probability anomaly to account for its sensitivity to sample period em-phasized in Dichev (1998). Significance at the 5% level based on heteroskedasticity-robust OLS standard errors is expressedin boldface. Returns are in annualized percentages.
Pre-1993 Period (1974m1-1993m12) Post-1993 Period (1994m1-2016m12)
β β
No Anomaly re α MKT SMB HML RMW CMA re α MKT SMB HML RMW CMA
1 beta(L) 7.92 2.58 0.67 0.06 0.40 -0.28 -0.19 7.75 0.8 0.54 -0.01 0.09 0.28 0.432 rome(L) 16.39 5.43 1.10 0.30 0.58 0 -0.22 13.74 2.93 1.06 0.20 0.30 0.30 0.083 ohlson(L) 5.96 0.7 0.97 -0.22 -0.21 0.17 0.14 8.45 2.14 0.97 -0.15 -0.28 0.03 -0.014 size(L) 9.48 -2.21 0.89 1.19 0.01 -0.20 0.15 9.73 2.53 0.81 0.97 0.07 -0.40 0.125 rev60m(L) 11.23 -4.63 1.18 0.89 0.36 -0.08 0.4 12.61 2.12 1.16 0.59 0.33 -0.44 0.396 value(L) 13.38 -2.43 1.15 0.49 0.82 0.08 0.17 8.93 0 0.93 0.15 0.65 -0.12 0.067 mom12m(L) 14.06 6.29 1.11 0.38 -0.48 -0.03 0.32 8.74 1.62 1.02 0.36 -0.42 0.01 -0.078 netissue(L) 10.92 1.05 1.02 0.08 0.16 0.24 0.23 12.48 1.85 1.02 0.11 0.12 0.41 0.209 netissue_m(L) 11.04 1.18 1.05 0.10 0.27 0.33 -0.02 10.72 0.29 1.04 -0.04 0.12 0.33 0.2910 acc(L) 7.88 -0.37 1.02 0.21 -0.18 0.15 0.24 7.65 0.69 1.16 0.04 -0.18 -0.27 -0.0911 roa(L) 7.05 2.68 0.98 -0.07 -0.46 0.20 0.08 9.80 2.74 0.98 -0.02 -0.42 0.30 -0.112 roe(L) 8.72 3.58 1.04 0.03 -0.45 0.22 0.04 9.76 0.97 1.01 -0.04 -0.25 0.40 0.113 failprob(L) 7.75 2.48 0.93 0 -0.41 0.08 0.23 10.02 3.03 0.87 0.16 -0.31 0.14 0.1114 piotroski(L) 7.16 1.69 0.96 -0.03 -0.10 0.14 -0.1 7.70 -0.31 1.02 0.06 -0.1 0.19 -0.0915 invest(L) 12.03 0.58 1.14 0.37 -0.06 0 0.53 9.84 0.51 1.02 0.24 -0.11 -0.09 0.5316 idiovol(L) 6.31 1.09 0.83 -0.30 0.20 0.03 -0.02 8.04 0.33 0.79 -0.17 0.06 0.24 0.2717 atgrowth(L) 10.70 -1.82 1.14 0.62 -0.23 -0.15 0.80 10.17 0.4 1.08 0.11 -0.06 -0.15 0.6318 ato(L) 10.56 0.82 1.03 0.37 -0.04 0.37 0.07 8.84 0.34 0.95 0.16 -0.06 0.44 -0.219 gm(L) 4.7 0.45 0.93 -0.11 -0.37 0.11 0.1 9.27 3.71 0.97 -0.14 -0.35 -0.06 -0.0620 profit(L) 7.05 1.56 0.94 -0.02 -0.31 0.28 0.07 10.48 2.17 0.94 0.08 -0.27 0.29 0.1721 beta(S) 4.29 -3.71 1.31 0.55 0.02 -0.26 -0.51 5.73 -4.07 1.53 0.16 0.52 -0.54 -0.4222 rome(S) -1.13 -11.53 1.16 0.30 -0.1 -0.2 0.52 4.31 -3.59 1.38 0.34 0.25 -0.89 -0.123 ohlson(S) 3.08 -6.49 1.10 0.81 -0.13 -0.45 0.13 5.07 -2.14 1.18 0.66 0.1 -0.57 -0.3324 size(S) 5.06 -0.73 0.98 -0.29 -0.01 0.15 0.07 7.10 0.19 0.99 -0.26 0 0.02 -0.0225 rev60m(S) 4.86 -0.72 1.10 0.1 -0.52 0.20 0.08 9.13 2.12 1.14 0.03 -0.18 0.13 -0.5026 value(S) 3 -0.34 0.99 -0.02 -0.55 0.12 -0.03 7.45 1.18 1.03 -0.12 -0.35 0.07 -0.1627 mom12m(S) -4.5 -11.32 1.07 0.51 0.23 -0.48 -0.51 2.72 -5.35 1.44 0.19 0.62 -0.64 -0.7028 netissue(S) 3.16 -0.6 0.97 0.19 -0.04 -0.41 -0.42 3.44 -3.78 1.10 0.04 0.24 -0.14 -0.3929 netissue_m(S) 5.31 0.48 0.99 0.22 -0.01 -0.34 -0.34 3.92 -3.09 1.12 0.12 0.26 -0.28 -0.4130 acc(S) 3.81 -2.73 1.09 0.42 -0.27 0.09 -0.27 5.58 -1.83 1.08 0.44 -0.18 0.09 -0.4731 roa(S) -0.29 -11.89 1.09 0.66 0.05 -0.33 0.41 3.27 -1.83 1.22 0.37 -0.20 -0.88 -0.2132 roe(S) 0.51 -11.45 1.11 0.63 0.18 -0.30 0.3 1.57 -4.33 1.29 0.32 -0.11 -0.91 -0.1433 failprob(S) -2.39 -11.46 1.29 1.09 0.31 -0.47 -0.11 0.67 -8.94 1.69 0.32 0.46 -0.78 -0.5934 piotroski(S) 3.36 -3.51 1.04 0 -0.04 -0.33 0.26 7.09 -1.07 1.02 0.14 -0.04 -0.21 0.3435 invest(S) 3.77 -2.2 1.12 0.25 -0.18 0.11 -0.38 5.2 -3.1 1.06 0.30 0.11 0.22 -0.4836 idiovol(S) -3.6 -14.52 1.09 1.16 -0.02 -0.37 -0.1 3.67 -2.23 1.37 0.64 -0.19 -0.99 -0.3637 atgrowth(S) 4.19 -1 1.09 0.31 -0.29 0.02 -0.36 6.07 -0.33 1.10 0.19 -0.24 0.08 -0.5838 ato(S) 5.53 3.15 0.77 -0.19 0.33 -0.60 -0.39 3.69 -2.53 1.02 -0.12 0 -0.27 -0.0439 gm(S) 7.46 -1.58 1.08 0.27 -0.02 -0.20 0.28 6.43 -0.83 0.94 0.39 -0.02 -0.20 0.0540 profit(S) 4.88 1.8 0.90 -0.01 0.29 -0.92 -0.36 6.12 1.27 0.84 -0.10 -0.06 -0.33 0.07
30
Table 3: Arbitrage Activity on Anomalies Inferred from Short Interests
Baseline: ArbActivityi,t = b0 + (αprei ×Xi,t)
′b1 +X ′i,tb2 + ui + εi,t
The table studies the determinants of arbitrage activity on an anomaly using panel data (40 anomalies × 1974m1-2016m12) of short interests. The dependent variablemeasures arbitrage activity on anomaly i in month t using the negative of (−1 × 104) the “abnormal” short interest, defined as the value-weighted average of shortinterest ratio (shares shorted / shares outstanding) minus the cross-sectional average short interest ratio of stocks that belong to the same NYSE size decile, where theaverage is taken over all stocks that belong to the anomaly portfolio. I use short interests reported in mid-month and shares outstanding on the same day (if available)or the previous trading day. The post-1993 dummy is 0 for the pre-1993 period (1974m1-1993m12) and 1 for the post-1993 period (1994m1-2016m12). For columns(6)-(12), an anomaly’s “pre-arbitrage” alpha, denoted αpre, is measured by its alpha with respect to the multifactor model specified in the column head in the pre-1993period. Columns (11) and (12) use αpre computed in the last 15 years (1979m1-1993m12) and 10 years (1984m1-1993m12) leading to 1993m12, respectively. Forfailure probability, αpre is computed from 1981m1 onward. Post-Publication, Post-Sample, Post-1993, Post-1993×Post-Pub, and constant terms are included in theregression (whenever appropriate) but not reported in the table. In the parentheses are standard errors adjusted for cross-anomaly covariances. ***, **, and * indicate1%, 5%, and 10% significance levels.
Long vs. Short Model: FF5 CAPM FF3 Carhart FF5(’79-) FF5(’84-)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Long× Post-Publication 66.09∗∗∗ 5.42 -15.26(18.74) (14.19) (18.77)
Long× Post-Sample 71.70∗∗∗ 11.20 16.59(18.53) (18.69) (16.76)
Long× Post-1993 79.08∗∗∗ 69.12∗∗∗ 55.31∗∗∗
(17.11) (18.90) (14.61)
Long× Post-1993× Post-Pub 29.04(24.14)
αpre × Post-Pub 3.40∗∗ -0.25 -0.49 -1.48 -0.58 -1.22 -1.58(1.50) (1.57) (1.44) (1.33) (1.99) (1.54) (1.61)
αpre × Post-Sample 0.77 2.11 3.36∗∗ 1.80 0.66 2.24∗ 2.51(1.72) (1.35) (1.54) (1.18) (1.39) (1.25) (1.63)
αpre × Post-1993 8.86∗∗∗ 6.14∗∗∗ 5.71∗∗∗ 5.21∗∗∗ 7.62∗∗∗ 6.50∗∗∗ 6.77∗∗∗
(1.43) (1.06) (1.19) (1.02) (0.91) (1.08) (1.40)
αpre × Post-1993× Post-Pub 5.04∗∗∗ 2.69 5.41∗∗∗ 7.35∗∗∗ 6.32∗∗∗ 6.24∗∗∗
(1.64) (1.76) (1.44) (2.26) (1.36) (1.71)
Anomaly FE Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes YesObservations 20640 20640 20640 20640 20640 20640 20640 20640 20640 20640 20640 20640AdjustedR2 0.11 0.11 0.16 0.17 0.18 0.28 0.28 0.23 0.29 0.28 0.30 0.28
31
Table 4: To Which Fama-French Factors Are the Long-Short Arbitrageurs of AnomaliesExposed?
Baseline: rei,t = β0 + βMKT rMKT,t + βSMBrSMB,t + βHMLrHML,t + βRMW rRMW,t + βCMArCMA,t + εi,t
The table uses time-series regressions to show that long-short arbitrageurs of equty anomalies are positively exposed toRMW and CMA, but not strongly exposed to the other Fama-French (2015) factors. The “EW long-short portfolio ofanomalies” is the portfolio that gives equal positive weights to the 20 long-side portfolios and equal negative weightsto the 20 short-side portfolios. The “long-short equity hedge fund” returns are computed as a 90/10 mix of equity-market-neutral hedge fund returns and equity short-bias hedge fund indices from Hedge Fund Research (HFR). Thehedge fund returns are analyzed only in the post-1993 period. The standard errors are heteroskedasticity-robust OLSstandard errors. ***, **, and * indicate 1%, 5%, and 10% significance levels.
EW Long-Short Long-Short Equity Hedge Fund Return (Post-1993)
Portfolio of Anomalies Baseline Alternative Mix of Equity Market Neutral + Short Bias Hedge Funds
Pre-1993 Post-1993 (90/10) 100/0 80/20 60/40 40/60 20/80 0/100
(1) (2) (3) (4) (5) (6) (7) (8) (9)
MKT 0.015 -0.617∗∗∗ 0.007 0.085∗∗∗ -0.070∗∗∗ -0.225∗∗∗ -0.380∗∗∗ -0.534∗∗∗ -0.689∗∗∗
(0.074) (0.141) (0.015) (0.016) (0.016) (0.021) (0.029) (0.038) (0.047)
SMB 0.154 0.125 0.005 0.045∗∗ -0.035∗ -0.114∗∗∗ -0.194∗∗∗ -0.273∗∗∗ -0.352∗∗∗
(0.132) (0.192) (0.018) (0.019) (0.018) (0.025) (0.034) (0.046) (0.057)
HML 0.278∗ -0.070 0.008 -0.020 0.036 0.091∗∗∗ 0.147∗∗∗ 0.202∗∗∗ 0.258∗∗∗
(0.159) (0.216) (0.023) (0.024) (0.025) (0.032) (0.042) (0.053) (0.065)
RMW 0.960∗∗∗ 1.481∗∗∗ 0.087∗∗∗ 0.068∗∗ 0.105∗∗∗ 0.142∗∗∗ 0.178∗∗∗ 0.215∗∗∗ 0.252∗∗∗
(0.203) (0.260) (0.023) (0.028) (0.022) (0.031) (0.046) (0.063) (0.081)
CMA 1.364∗∗∗ 1.688∗∗∗ 0.076∗∗ 0.058 0.093∗∗∗ 0.127∗∗∗ 0.162∗∗∗ 0.196∗∗∗ 0.231∗∗
(0.264) (0.297) (0.032) (0.036) (0.031) (0.038) (0.054) (0.072) (0.092)
Constant 2.970∗∗∗ 1.653∗∗∗ 0.136∗∗∗ 0.146∗∗∗ 0.126∗∗∗ 0.107∗ 0.087 0.067 0.048(0.302) (0.410) (0.047) (0.050) (0.048) (0.064) (0.089) (0.117) (0.146)
Observations 240 276 276 276 276 276 276 276 276Adjusted R2 0.27 0.62 0.16 0.15 0.48 0.72 0.77 0.79 0.80
32
Table 5: Alpha Turns into Endogenous RMW and CMA Betas
Baseline: βposti = b0 + b1α
prei + b2β
prei + ui
The table uses cross-sectional regressions to show that pre-1993 alpha explains the cross-section of post-1993 betas. I do this separately for factors that are likelyto be systematic shocks to arbitrage capital (RMW and CMA) and for factors that are not (MKT, SMB, and HML), in light of the result in Table 4. Both alphas andbetas are with respect to the Fama-French (2015) five-factor model. Volatility is the cross-sectionally standardized standard deviation of the anomaly’s pre-1993excess return. In the parentheses are standard errors adjusted for cross-anomaly covariances in measurement errors. ***, **, and * indicate 1%, 5%, and 10%significance levels.
Factors That Proxy Arbitrage Capital Shocks Other Fama-French Factors
RMW βpost CMA βpost MKT βpost SMB βpost HML βpost
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
αpre 0.054∗∗∗ 0.053∗∗∗ 0.030∗∗ 0.027∗∗ -0.018∗∗∗ -0.008 0.005 -0.011(0.011) (0.010) (0.014) (0.013) (0.006) (0.005) (0.009) (0.012)
βpre 0.579∗∗∗ 0.981∗∗∗ 0.645∗∗∗ 0.676∗∗∗ 0.627∗∗∗ 0.753∗∗∗ 0.904∗∗∗ 1.265∗∗∗ 0.895∗∗∗ 0.628∗∗∗ 0.582∗∗∗ 0.625∗∗∗ 0.658∗∗∗
(0.115) (0.243) (0.118) (0.139) (0.157) (0.142) (0.244) (0.270) (0.333) (0.099) (0.100) (0.083) (0.103)
Volatility -0.055 -0.139∗∗∗ 0.007(0.041) (0.053) (0.038)
αpre × Volatility -0.009 -0.016∗ -0.012∗∗
(0.007) (0.009) (0.005)
Constant 0.012 -0.048 -0.009 -0.036 -0.086 -0.090∗ 0.104 -0.239 0.097 -0.000 0.004 -0.002 0.018(0.034) (0.045) (0.037) (0.048) (0.056) (0.047) (0.251) (0.269) (0.348) (0.025) (0.024) (0.034) (0.047)
Observations 40 40 40 40 40 40 40 40 40 40 40 40 40Adjusted R2 0.86 0.48 0.87 0.55 0.34 0.68 0.72 0.57 0.77 0.75 0.75 0.58 0.55
33
Table 6: Arbitrage Activity Explains the Endogenous Betas
Baseline: βposti = b0 + b1ArbActivity
posti + b2β
prei + ui
The table uses cross-sectional regressions to show that anomaly-specific arbitrage activity explains the cross-sectionof post-1993 RMW and CMA betas of 40 equity anomalies, controlling for the pre-1993 beta. Arbitrage activity ismeasured as the negative (× − 1) of the “abnormal” short interest, defined as the value-weighted average of shortinterest ratio (shares shorted / shares outstanding) minus the cross-sectional average short interest ratio of stocks thatbelong to the same NYSE size decile, where the average is taken over all stocks that belong to the anomaly portfolioand then over the post-1993 period. I use short interests reported in mid-month and shares outstanding on the same day(if available) or the previous trading day. In the parentheses are standard errors adjusted for cross-anomaly covariancesin measurement errors. ***, **, and * indicate 1%, 5%, and 10% significance levels.
RMW βpost CMA βpost
(1) (2) (3) (4)
Post-1993 Arbitrage Activity 0.306∗∗∗ 0.216∗∗∗
(0.073) (0.083)
Pre to Post-1993 Change in Arbitrage Activity 0.392∗∗∗ 0.259∗∗
(0.088) (0.101)
βpre 0.574∗∗∗ 0.554∗∗∗ 0.592∗∗∗ 0.614∗∗∗
(0.118) (0.118) (0.120) (0.122)
Constant 0.057 0.053 0.013 0.004(0.038) (0.039) (0.046) (0.046)
Observations 40 40 40 40Adjusted R2 0.78 0.80 0.64 0.63
34
Table 7: Endogenous Betas Are Generated during Constrained Times
Baseline: βpost, constrainedi = b0 + b1α
prei + b2β
prei + ui
The table uses cross-sectional regressions to show that the ability of pre-1993 alpha to explain the cross-section ofpost-1993 betas for RMW and CMA comes from its ability to predict betas in the constrained times of the post-1993period. The constrained vs. unconstrained post-1993 periods are proxied by months in which the 3-month movingaverage of the VIX is above vs. below the median (see Figure 2). Arbitrage activity on an anomaly is defined as thenegative (−1×102) of the time-series average of “abnormal” short interest on the anomaly as defined in Table 6. Bothalphas and betas are with respect to the Fama-French (2015) five-factor model. In the parentheses are standard errorsadjusted for cross-anomaly covariances in measurement errors. ***, **, and * indicate 1%, 5%, and 10% significancelevels.
Unconstrained Post-1993 Period (Low-VIX) Constrained Post-1993 Period (High-VIX)
RMWβ CMAβ α ArbActivity RMWβ CMAβ α ArbActivity
(1) (2) (3) (4) (5) (6) (7) (8)
αpre 0.036∗∗ 0.008 0.058∗∗∗ 0.031(0.017) (0.016) (0.013) (0.020)
βpre 0.439∗∗ 0.578∗∗∗ 0.580∗∗∗ 0.673∗∗∗
(0.183) (0.220) (0.126) (0.167)
Long 2.483∗ 1.067∗∗∗ 3.478 0.871∗∗∗
(1.485) (0.214) (2.301) (0.215)
Constant -0.024 -0.002 -1.535 -1.127∗∗∗ 0.001 -0.051 -1.455 -0.751∗∗∗
(0.046) (0.069) (1.420) (0.190) (0.038) (0.054) (2.036) (0.187)
Observations 40 40 40 40 40 40 40 40Adjusted R2 0.62 0.22 0.18 0.38 0.88 0.56 0.29 0.28
35
Table 8: Cost of Arbitrage Affects the Alphas-into-Betas Relation
Baseline: βposti = b0 + b1α
prei × Costi + b2α
prei + b3Costi + b4β
prei + ui
The table uses cross-sectional regressions to show that the alphas-into-betas effect is weaker for anomalies with highercosts of arbitrage, consistent with the effect being driven by the act of arbitrage. The dependent variable is the post-1993 beta with respect to RMW or CMA, a proxy for systematic shocks to arbitrage capital. Both alphas and betasare with respect to the Fama-French (2015) five-factor model. The cost of arbitrage of an anomaly is calculated asvalue-weighted decile ranks (based on NYSE stocks) of the underlying stocks’ costs of arbitrage as measured by size(expressed in × − 1 so that smaller size means higher arbitrage cost), idiosyncratic volatility, illiquidity (Amihud,2002), or bid-ask spread (Corwin and Schultz, 2012). This measure is averaged over time within the pre-1993 periodwith little arbitrage disturbance and then cross-sectionally standardized for the ease of interpretation. In the paranthesesare standard errors adjusted for cross-anomaly covariances in measurement errors. ***, **, and * indicate 1%, 5%,and 10% significance levels.
RMW βpost CMA βpost
(1) (2) (3) (4) (5) (6) (7) (8)
αpre × Cost -0.001 -0.005 0.001 -0.005 -0.002 -0.012∗∗ 0.000 -0.016∗∗
(0.005) (0.004) (0.005) (0.005) (0.007) (0.006) (0.008) (0.007)
αpre 0.052∗∗∗ 0.049∗∗∗ 0.051∗∗∗ 0.048∗∗∗ 0.033∗∗∗ 0.028∗∗ 0.033∗∗∗ 0.031∗∗
(0.009) (0.010) (0.009) (0.010) (0.012) (0.013) (0.012) (0.014)
Cost -0.024 -0.070∗ -0.020 -0.073∗∗ 0.022 -0.104∗ 0.036 -0.100∗
(0.030) (0.038) (0.029) (0.034) (0.042) (0.063) (0.041) (0.060)
βpre 0.559∗∗∗ 0.575∗∗∗ 0.563∗∗∗ 0.595∗∗∗ 0.675∗∗∗ 0.723∗∗∗ 0.665∗∗∗ 0.747∗∗∗
(0.094) (0.092) (0.092) (0.091) (0.119) (0.121) (0.120) (0.125)
Constant 0.006 -0.014 0.009 -0.014 -0.035 -0.084∗ -0.030 -0.092∗∗
(0.035) (0.034) (0.034) (0.033) (0.050) (0.048) (0.048) (0.045)
Cost of Arbitrage Size Idivol Amihud Spread Size Idivol Amihud SpreadObservations 40 40 40 40 40 40 40 40AdjustedR2 0.85 0.86 0.85 0.87 0.53 0.59 0.54 0.61
36
Table 9: Explaining the Cross-section of Funding-liquidity Betas
The table uses cross-sectional regressions to show that the cross-section of funding-liquidity betas of 40 equity anomalies are consistent with being determinedthrough the act of arbitrage. To do so, it applies the key cross-sectional tests to understand the cross-section of betas with respect to the funding-liquidity factor(also called “leverage factor”) of Adrian, Etula, and Muir (2014), controlling for the Fama-French (FF) (2015) five factors. The alphas and betas are computed intime-series regressions that include the FF five factors. The anomaly data used in this analysis are quarterly (1974Q1-2016Q4) to match the quarterly frequencyof the funding-liquidity factor. Arbitrage activity on an anomaly is defined as the negative (×− 1) of the time-series average of “abnormal” short interest on theanomaly as defined in Table 6. The constrained vs. unconstrained post-1993 periods are proxied by quarters in which the VIX is above vs. below the median. Inthe parentheses are standard errors adjusted for cross-anomaly covariances in measurement errors. ***, **, and * indicate 1%, 5%, and 10% significance levels.
βpref βpost
f Unconstrained βpostf Constrained βpost
f
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Long 0.02 0.75 -0.42 1.40(0.29) (0.57) (0.41) (0.88)
αpre -0.01 0.14∗ 0.14∗∗ -0.00 0.23∗∗
(0.04) (0.07) (0.07) (0.06) (0.11)
βpref 0.51 0.70 0.35 0.44 0.32 0.30 0.70 1.07
(0.67) (0.49) (0.44) (0.45) (0.62) (0.62) (1.01) (0.88)
Post-1993 Arbitrage Activity 0.99∗
(0.51)
Pre to Post-1993 Change in Arb Activity 1.21∗
(0.62)
Constant 0.07 0.07 -0.75 -0.08 -0.42 -0.13 0.05 0.01 0.20 -0.01 -1.37 -0.20(0.24) (0.09) (0.57) (0.20) (0.34) (0.22) (0.18) (0.18) (0.41) (0.18) (0.98) (0.39)
Observations 40 40 40 40 40 40 40 40 40 40 40 40Adjusted R2 -0.03 -0.02 0.10 0.53 0.01 0.60 0.58 0.57 0.11 -0.01 0.15 0.60
37
Table 10: Cross-sectional Asset Pricing Test Using Endogenous Beta
The table illustrates that cross-sectional asset pricing tests using “endogenous” betas that arise as an outcome ofarbitrage activity on alphas can generate upward-biased price of risk estimate. Specifically, it uses the beta withrespect to the funding-liquidity factor (also called “leverage factor”) of Adrian, Etula, and Muir (2014) to fit the cross-section of realized excess returns on 40 equity anomalies in a sample period that includes the “pre-arbitrage” periodwith little arbitrage activity on the test assets. By doing so, the table illustrates that mixing the pre-arbitrage period inthe regression can bias the price of risk estimate upward by attributing the relationship between pre-arbitrage alpha andpost-arbitrage beta due to arbitrage to risk premium associated with the beta. In the parentheses are Shanken (1992)standard errors. ***, **, and * indicate 1%, 5%, and 10% significance levels.
Dependent Variable: Realized Mean Excess Return (40 Equity Anomalies)
Pre-1993 Period Pre-1993 Period Different Mix of "Pre-Arbitrage" (Pre-1993) Period
(1974-1993) (1994-2016) 1974-2016 1979-2016 1984-2016 1989-2016
(1) (2) (3) (4) (5) (6)
Funding-liquidity Beta 0.25 1.40 3.11∗∗ 2.88∗ 2.53∗ 1.79(1.48) (1.44) (1.39) (1.55) (1.46) (1.41)
Constant 6.27 8.77∗∗ 5.54 7.45∗ 7.23∗ 9.15∗∗∗
(4.96) (3.66) (4.50) (3.90) (4.07) (3.29)
Observations 40 40 40 40 40 40Adjusted R2 -0.02 0.34 0.39 0.41 0.49 0.39
1020
3040
5060
VIX
1995m12 2000m12 2005m12 2010m12 2015m12
Month-end VIX MedianConstrained Unconstrained
Figure 2: Constrained vs. Unconstrained Post-1993 Periods Inferred from the VIXThe figure reports the constrained (unconstrained) months within the post-1993 period, defined as the months in whichthe 3-month (±1.5 months) moving average of the VIX was above (below) the post-1993 median.
38
-4-2
02
4Fu
ndin
g-Li
quid
ity S
hock
s
12
34
5Lo
g B
roke
r-D
eale
r Lev
erag
e
1975q1 1985q1 1995q1 2005q1 2015q1Quarter
Log BD Leverage Funding-Liquidity Shocks
Figure 3: Funding-liquidity Shocks Measured by Broker-dealer Leverage ShocksThe figure plots the funding-liquidity shocks used in Adrian, Etula, and Muir (2014) for 1974Q1-2016Q4. Thefunding-liquidity shocks are computed as the quarterly change in the log of aggregate broker-dealer leverage pub-lished in the Federal Reserve Board’s flow of funds data. Change in beta is estimated as the post-1993 beta minus thepre-1993 beta multiplied by the shrinkage factor of 0.6.
Figure 4a. RMW Beta Figure 4b. CMA Beta
R2adj = 0.64
-1-.5
0.5
Cha
nge
in R
MW
Bet
a
-3 -2 -1 0 1Arbitrage Activity
R2adj = 0.46-.5
0.5
Cha
nge
in C
MA
Bet
a
-3 -2 -1 0 1Arbitrage Activity
Figure 4: Arbitrage Activity Explains the Pre- to Post-1993 Change in RMW and CMA BetasThe figures illustrate the ability of post-1993 arbitrage activity to explain the cross-section of pre- to post-1993 changein the RMW and CMA betas. Change in beta is estimated as the post-1993 beta minus the pre-1993 beta multipliedby the shrinkage factor of 0.6. Arbitrage activity on an anomaly is defined as the negative (× − 1) of the time-seriesaverage of “abnormal” short interest on the anomaly as defined in Table 6, which also shows the regression counterpartsthat do not restrict the coefficient on the pre-1993 beta to be 0.6.
39
A Additional Tables
Table A1: In-Sample to Out-of-Sample Change in the Estimated RMW and CMA Betas
Baseline: βi,t = b0 + b1
(αinsamplei × 1 (t /∈ sample)
)+ b21 (t /∈ sample) + ui + εi,t
The table uses the panel data of anomaly factor betas (40 anomalies × 43 years) to show that, controlling for the post-1993 effect, the RMW and CMA betas do notchange in a predictable manner from in-sample to out-of-sample years used in the anomaly’s original academic publication. For comparison, I report results for the MKT,SMB, and HML betas. The dependent variable is the beta estimated over one year using daily returns on the anomaly and the factor. The in-sample alpha αinsample isthe five-factor alpha within the sample period used in the original publication. αpre is the pre-1993 five-factor alpha. I use the pre-1993 alpha as the in-sample alpha forthe four anomalies whose sample years fall before 1974, the beginning of the sample period I use in this study. In the parentheses are standard errors not adjusted forcross-anomaly covariances in betas but adjusted for serial correlations in the betas within an anomaly. ***, **, and * indicate 1%, 5%, and 10% significance levels.
Factors That Proxy Arbitrage Capital Shocks Other Fama-French Factors
RMW βpost CMA βpost MKT βpost SMB βpost HML βpost
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
Long× Out-of-Sample -0.006 -0.069 0.057 0.018 -0.002 0.098∗∗ 0.005(0.048) (0.051) (0.067) (0.066) (0.027) (0.047) (0.054)
Long× Post-1993 0.104∗ 0.065 0.009 -0.092 -0.005(0.059) (0.070) (0.035) (0.056) (0.056)
αinsample × Out-of-Sample 0.012∗∗ -0.006 0.012 0.003 -0.005 0.000 0.002(0.005) (0.005) (0.009) (0.009) (0.003) (0.006) (0.006)
αpre × Post-1993 0.024∗∗∗ 0.012∗ 0.009∗∗∗ -0.003 -0.004(0.005) (0.006) (0.003) (0.006) (0.007)
Out-of-Sample -0.016 0.065∗ -0.005 0.025 -0.078∗ -0.013 -0.036 -0.001 0.009 0.003 -0.025 0.024 0.079∗ 0.083∗∗∗
(0.032) (0.034) (0.023) (0.025) (0.039) (0.036) (0.039) (0.038) (0.021) (0.013) (0.033) (0.023) (0.043) (0.026)
Post-1993 -0.133∗∗ -0.040 -0.108∗∗ -0.055 -0.059∗∗ -0.039∗∗ 0.025 -0.027 -0.023 -0.032(0.050) (0.028) (0.047) (0.040) (0.026) (0.017) (0.036) (0.030) (0.047) (0.026)
Constant -0.128∗∗∗ -0.117∗∗∗ -0.125∗∗∗ -0.119∗∗∗ -0.012 -0.003 -0.010 -0.002 1.052∗∗∗ 1.050∗∗∗ 0.155∗∗∗ 0.155∗∗∗ -0.088∗∗∗ -0.088∗∗∗
(0.016) (0.017) (0.015) (0.015) (0.022) (0.023) (0.021) (0.023) (0.006) (0.006) (0.012) (0.013) (0.018) (0.018)
Anomaly FE Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes YesObservations 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720 1,720AdjustedR2 -0.00 0.03 0.01 0.07 0.00 0.01 0.01 0.02 0.04 0.07 0.01 0.00 0.01 0.01
40
Table A2: Constrained vs. Unconstrained Post-1993 Period Betas: MKT, SMB, and HML
Baseline: βpost, constrainedi = b0 + b1α
prei + b2β
prei + ui
The table uses cross-sectional regressions to study the ability of pre-1993 alpha to explain the cross-section of con-strained vs. unconstrained post-1993 period betas for MKT, SMB, and HML. The constrained vs. unconstrainedpost-1993 periods are indicated in Figure 2. Both alphas and betas are with respect to the Fama-French (2015) five-factor model. In the parentheses are standard errors adjusted for cross-anomaly covariances in measurement errors.***, **, and * indicate 1%, 5%, and 10% significance levels.
Unconstrained Post-1993 Period (Low-VIX) Constrained Post-1993 Period (High-VIX)
MKTβ SMBβ HMLβ MKTβ SMBβ HMLβ
(1) (2) (3) (4) (5) (6) (7) (8)
αpre -0.009 -0.001 -0.009 -0.002 -0.020∗∗ -0.008 0.009 -0.015(0.006) (0.006) (0.011) (0.012) (0.008) (0.006) (0.012) (0.018)
βpre 0.940∗∗∗ 0.770∗∗ 0.589∗∗∗ 0.575∗∗∗ 0.914∗∗∗ 0.984∗∗∗ 0.643∗∗∗ 0.640∗∗∗
(0.208) (0.319) (0.109) (0.153) (0.281) (0.363) (0.119) (0.118)
Volatility 0.031 -0.006(0.042) (0.042)
αpre × Volatility -0.007 -0.013∗∗
(0.005) (0.006)
Constant 0.067 0.236 0.058∗ -0.051 0.088 -0.004 -0.026 0.016(0.205) (0.335) (0.030) (0.045) (0.290) (0.380) (0.029) (0.049)
Observations 40 40 40 40 40 40 40 40Adjusted R2 0.72 0.76 0.74 0.45 0.68 0.74 0.66 0.52
41
B Theory Appendix
B.1 Solving the pre-arbitrage equilibrium
Proof of Lemma 1 (Asset returns in the pre-arbitrage economy). Since the behavioral in-
vestors alone clear the market, equation (1) implies that the price at time t is simply the price
at time t+ 1 discounted by the behavioral investors’ asset-specific discount factor 1 + rmaxi:
pi,t = (1 + rmaxi)−1 pi,t+1. (Note that δi,t+1 drops out since Et [δi,t+1] = 0.) It follows that
the return on asset i at time t + 1 is ri,t+1 = rmaxi + (1 + rmaxi)3−t δi,t+1
v≡ αprei + εi,t+1,
with δi,3 = 0 and ∂αprei /∂i = rmax > 0.
B.2 Solving the post-arbitrage equilibrium
The equilibrium in the post-arbitrage economy with µ = 1/2 is solved backward, starting from
time 2, which represents the period immediately before mispricing disappears and asset prices
converge to their fundamental value. Hence, an arbitrageur at time 2 invests all available capital
in the mispriced assets without worrying about asset returns covarying with the level of arbitrage
capital in the future. Time 1 represents the earlier periods of arbitrage in which an arbitrageur does
worry about asset returns covarying endogenously with the level of arbitrage capital before the
assets realize their fundamental value. The asset prices at time 1 therefore take these endogenous
risks into account.
To find the equilibrium in each period, note first that the arbitrageur’s objective function in (2)
implies the following value function at t ∈ 1, 2:
Vt (wt, ft) = maxxi,t Et [Vt+1 (wt+1, ft+1)]
s.t.∫ 1
0|xi,t| di ≤ (wt + ft)
wt+1 = wt +∫ 1
0
(pi,t+1+δi,t+1
pi,t− 1)xi,tdi+ wt+1
V3 = w3
(15)
42
in the non-default state (wt > 0), and
Vt = (1 + c)3−twt (16)
in the default state (wt ≤ 0).
Then, the equilibrium price at time 2 is given by the following lemma:
Lemma 5. (Time-2 equilibrium prices). The equilibrium price of asset i at time 2 is
pi,2 = mi,3v (17)
s.t. (i) mi,3 = 1/ (1 + rmaxi∗2) for the “exploited” assets i ∈ (i∗2, 1].
(ii) mi,3 = 1/ (1 + rmaxi) for the “unexploited” assets i ∈ [0, i∗2].(iii) i∗2 is the marginal asset s.t. i∗2 = 1, 1−
√k2, and 0 for k2 ∈ (−∞, 0], (0, 1), and [1,∞),
respectively.(iv) For completeness, the equilibrium arbitrage position is
xi,2 =
i− i∗2
0
if i ≥ i∗2if i < i∗2
.
Proof. The arbitrageur’s value function at time 2 in the non-default state (w2 > 0) is
V2 = w2 + maxxi,2
∫ 1
0
E2 [ri,3]xi,2di+ ψ2
[w2 + f2 −
∫ 1
0
|xi,2| di]
(18)
where ψ2 = 0 if the arbitrageur is unconstrained. Since the arbitrageur does not take negative
positions in equilibrium (doing so would generate a negative expected return due to the
behavioral investor demand), the first order condition with respect to xi,2 within the value
function at time 2 implies
E2 [ri,3] ≤ ψ2
Now, suppose that there exists a marginal asset i∗2 such that the condition binds for i ≥ i∗2
and is slack for i < i∗2. The unexploited assets (i < i∗2) would be priced by the behavioral
investors using eq. (1) so that mi,3 = 1/ (1 + rmaxi). The exploited assets would share the
same discount factor 1/ (1 + rmaxi∗2) that coincides with the behavioral investors’ discount
factor for the marginal asset. Finally, market clearing would require 12xi,1 = i − i∗2. Hence,
43
the marginal asset is given by
k2 =
∫ 1
i∗2
xi,1di = 2
∫ 1
i∗2
(i− i∗2) di = (1− i∗2)2 ⇐⇒ i∗2 = 1−√k2
in the constrained case. Given these discount factors, E2 [ri,3] = rmaxi∗2 for any i ≥ i∗2 and
E2 [ri,3] = rmaxi < rmaxi∗2 for i < i∗2, so that i∗2 is indeed the marginal asset. If k2 ≥ 1,
1−√k2 ≤ 0 and all assets are exploited, so that i∗2 = 0. If k2 ≤ 0, no asset is exploited, so
that i∗2 = 1.
These time-2 prices offer a glimpse into why high-i assets become endogenously riskier in
this post-arbitrage equilibrium. The prices of high-i assets respond more to the variation in k2:
as k2 ranges from 0 to 1, the price of asset i rises from v/ (1 + rmaxi) to v, implying a rmaxi-
percent increase in its price. The intuition is that the an initially more-mispriced asset relies more
heavily on the price-correcting role of arbitrage capital, which makes its price more sensitive to the
variation in the level of arbitrage capital. From Lemma 5 follows the arbitrageur’s marginal value
of wealth at time 2:
Lemma 6. (Time-2 marginal value of wealth). The arbitrageur’s value function at time 2 is
V2 = Λ2w2 (19)
where the marginal value of wealth in the non-default state (w2 > 0) is Λ2 = 1 + rmaxi∗2 and that
in the default state (w2 ≤ 0) is Λ2 = 1 + c.
Proof. First, consider w2 > 0. The derivative of the value function (18) with respect to w2 gives
Λ2 = 1 + ψ2. For ψ2 , the derivative with respect to any exploited asset’s xi,2 within the
bracket implies ψ2 = E2 [ri,3] = rmaxi∗2, where the second equality follows from equation
(17). Next, Λ2 for w2 ≤ 0 follows from equation (16). Finally, V2 = Λ2w2 since Lemma 5
implies that the marginal value of wealth Λ2 = 1+ψ2 = 1+rmaxi∗2 is also the average return
on wealth in the non-default state and w3 = (1 + c)w2 in the non-default case.
Lemma 6 implies that a low-k2 state is a “bad” state in which the arbitrageur’s marginal value
of wealth is high: Λ2 rises from 1 to 1 + rmax and to 1 + c as k2 decreases from∞ to 0+ and to
−∞. This inverse relationship between Λ2 and k2 is a natural result that is expected regardless
of the preference for risk or intertemporal substitution, much as the decreasing marginal utility of
44
consumption does not rely on the curvature of the utility function. With risk-neutrality in particular,
this happens because arbitrage capital k2 falls precisely when the investment opportunity rmaxi∗2improves. Given this, the equilibrium price at time 1 is as follows:
Lemma 7. (Time-1 equilibrium prices). The equilibrium price of asset i at time 1 is
pi,1 = E1 [mi,2 (pi,2 + δi,2)] (20)
s.t. (i) mi,2 = mA2 ≡ Λ2/Λ1 for the exploited assets i ∈ I∗1 where I∗1 is the set of exploited
assets.(ii) mi,2 = mB
i,2 ≡ 1/ (1 + rmaxi) for the unexploited assets i ∈ I∗1.(iii) Λ1 is the time-1 marginal value of wealth s.t. Λ1 = E1 [Λ2] + ψ1 where ψ1 > 0 if thearbitrageur is constrained and ψ1 = 0 if the arbitrageur is unconstrained.(iv) The arbitrageur is unconstrained if k1 is above a threshold k∗1 ≤ 1.
Proof. I first prove that the arbitrageur does not take a negative position at time 1. The proof is by
contradiction. Suppose xi,1 < 0. Then, by Lemma 6 and eq. (15), E1 [Λ2ri,2] = −ψ1 < 0,
where ψ1 is the Lagrangian multiplier on the capital constraint∫ 1
0|xi,1| di ≤ (w1 + f1).
However, market clearing in eq. (1) implies E1 [Λ2ri,2] ≥ 0. To see this, note that it implies
E1 [ri,2] = rmax (i− xi,1/2) > rmaxi =⇒ pi,2pi,1
> (1 + rmaxi)pi,2
E1[pi,2]≥ 1 for any possible
realization of pi,2, since pi,2 ∈ [v, (1 + rmaxi) v] by Lemma 7. Since pi,2/pi,1 ≥ 1 and Λ2 >
0, E1 [Λ2ri,2] ≥ 0. Now I am ready to prove the main lemma. Since the arbitrageur does
not take negative positions in the assets in equilibrium, in the unconstrained case in which
ψ1 = 0, taking the first order condition with respect to xi,1 within the maximization bracket
implies the fundamental theorem of asset pricing with Λ1 = E1 [Λ2]. In the constrained case,
taking the first order condition with respect to xi,1 within the maximization bracket implies
(E1 [Λ2] + ψ1) pi,1 ≥ E1 [Λ2 (pi,2 + δi,2)], which holds with equality if xi,1 > 0. Then, Λ1 =
dV1/dw1 = E1 [Λ2]+ψ1 pins down the prices of the exploited assets. The unexploited assets
are priced by the behavioral investors so that pi,1 = E1
[(1 + rmaxi)
−1 (pi,2 + δi,2)]. Given
these prices, the equilibrium arbitrage positions are given by behavioral investor demand (1).
Finally, to obtain k∗1 , assume that all assets are exploited and combine (1) and (20) to obtain
E1
[Λ2
E1 [Λ2](pi,2 + δi,2)
]=
E1 [pi,2 + δi,2]
1 + rmax (i− µxi,1),
45
which gives
xi,1 = 2
(i− 1
rmax
[(1 + Cov1
(Λ2
E1 [Λ2],pi,2 + δi,2E1 [pi,2]
))−1
− 1
]).
Rearranging and setting k∗1 =∫ 1
0xi,1di gives
k∗1 = 1− 2
rmax
∫ 1
0
(1 + Cov1
(Λ2
E1 [Λ2],pi,2 + δi,2E1 [pi,2]
))−1
− 1
,
which is less than or equal to 1 since Cov (Λ2, pi,2 + δi,2) = Cov (1 + rmaxi∗2, pi,2 + δ2) ≤ 0
∀i since pi,2 = v/ (1 + rmaxi∗1) or pi,2 = v/ (1 + rmaxi) and i∗2 = 1 −
√k2 where k2 =
w1 +∫ 1
0(pi,2 + δ2)xi,1di.
B.3 Proof of the main lemmas
Proof of Lemma 2. (Equilibrium with unconstrained arbitrageurs). Since k2 ≥ 1 with cer-
tainty, Λ2 = 1 and i∗2 = 1 with certainty. It follows from equation (17) that the price of an
asset at time 2 is pi,2 = v and ri,3 = εi,3 ≡ δi,3/v. It follows from this and the assumption
k1 ≥ k∗1 = 1 (note k∗1 = 1 in this special case with no endogenous risk) that, by Lemma 7,
pi,1 = E1 [pi,2 + δi,2] = v, so that ri,2 = εi,2 ≡ δi,2/v.
Proof of Lemma 3. (Asset returns with constrained arbitrageurs). The expected return for-
mula follows from an algebraic manipulation of Lemma 7. βposti,m > 0 is true because
Cov1
(ri,2,m
A2
)= Cov1
(pi,2 + δi,2,m
A2
)= Cov1
(v
mi,2
,mA2
),
where mi,2 = mA2 when i > i∗2 and mi,2 = (1 + rmaxi)
−1 when i ≤ i∗2. Finally, λm > 0 by
assumption, since k2 has full support over [0, 1].
Proof of Lemma 4. (A factor model of asset returns). Since mA2 =
(1 + rmax
(1−√k2
))/Λ1
at k ∈ (0, 1), a first-order approximation around k2 ≡ (1− r−1max (E1 [Λ2]− 1))
2 is mA2 ≈
46
E1
[mA
2
]− rmax
(2Λ1
√k2
)−1 (k2 − k2
). Thus,
E [ri,2] = αposti,0 + λmβposti,m ≈ αposti,0 +
rmaxV ar1 (k2)
2Λ1E1 [mA2 ]√k2︸ ︷︷ ︸
≡λk
Cov1 (ri,2, k2)
V ar1 (k2)︸ ︷︷ ︸≡βpost
i,k
.
Since k2 = w2 + f2, this also means
E [ri,2] ≈ αi,0 +rmaxV ar1 (w2)
2Λ1E1
[mA
2
]√k2︸ ︷︷ ︸
≡λw
βposti,w +rmaxV ar1 (f2)
2Λ1E1
[mA
2
]√k2︸ ︷︷ ︸
≡λf
βposti,f .
To see that βposti,k > 0 for i > 0, note that Cov1 (ri,2, k2) = p−1i,1Cov1 (pi,2 + δi,2, k2) =
p−1i,1Cov1
(v
mi,2, k2
), where we know ∂mi,2
∂k2≤ 0 for i > 0. Also, for any random variable x,
we know
Cov (x, f (x)) = E [(x− E [x]) (f (x)− E [f (x)])]
= E [(x− E [x]) (f (x)− f (E [x]))]︸ ︷︷ ︸≥0
+ E [(x− E [x]) (f (E [x])− E [f (x)])]︸ ︷︷ ︸=0
≥ 0
if f ′ (x) ≥ 0, which is the case when x is k2 and f (x) is mi,2 (k2).
B.4 Proof of the propositions
Proof of Proposition 1. (Pre-arbitrage alpha determines the cross-section of endogenous post-arbitrage betas). (i) The proof has two steps: first prove that the prices of high-i assets
respond more strongly to the variation in arbitrage capital and then prove that this implies
that those assets have higher arbitrage capital betas. For the first step, since Cov1 (pi,2, k2) =
E1 [pi,2k2]− E1 [pi,2]E1 [k2],
Cov1 (pi,2, k2) = v
∫ k2(i)
−∞
k2
1 + rmaxidF (k2) + v
∫ ∞k2(i)
k2
1 + rmaxi∗2dF (k2)
−vE1 [k2]
(∫ k2(i)
−∞
1
1 + rmaxidF (k2) +
∫ ∞k2(i)
1
1 + rmaxi∗2dF (k2)
),
47
where k2 (i) denotes the value of k2 that makes i the marginal asset and F is the conditional
cumulative density function of k2. The derivative of the covariance with respect to i gives
∂Cov1 (pi,2, k2)
∂i= −v
∫ k2(i)
−∞
k2
(1 + rmaxi)2dF (k2) + vE1 [k2]
∫ k2(i)
−∞
1
(1 + rmaxi)2dF (k2) ,
where the Leibniz terms cancel out by the fact that i∗2 (k2 (i)) = i. Rearranging the terms
gives
∂Cov1 (pi,2, k2)
∂i=
v
(1 + rmaxi)2 (E1 [k2]− E1 [k2| k2 ≤ k2 (i)])F (k2 (i)) > 0.
Next, to show how this monotonicity of the price covariance implies ∂Cov1 (ri,2, k2) /∂i >
0, it suffices to show that the equilibrium time-1 prices are non-increasing in i:
∂pi,1∂i≤ 0.
To see this, suppose for a contradiction that i < j but pi,1 < pj,1. Suppose also that j is
priced by the arbitrageur so that pj,1 = E0
[Λ2
Λ1pj,2
]. Since pi,2 ≥ pj,2 in all states of t = 2, it
must be that
pi,1 ≥ E1
[Λ1
Λ0
pi,2
]≥ E1
[Λ1
Λ0
pj,2
],
which is a contradiction. Now suppose that j is priced by the behavioral investors so that
pj,1 = 11+rmaxj
E1 [pj,2]. Again, since pi,2 ≥ pj,2 in all states of t = 2, it must be that
pi,1 ≥1
1 + rmaxiE1 [pi,2] ≥ 1
1 + rmaxjE1 [pj,2] ,
which is also a contradiction. Hence, pi,1 is non-increasing in i. Putting these together, we
see that Cov1 (ri,2, k2) is non-decreasing in i:
∂Cov1 (ri,2, k2)
∂i> 0.
It follows that∂βposti,k
∂αprei
=1
rmaxV ar1 (k2)× ∂Cov1 (ri,2, k2)
∂i> 0.
(ii) This follows from a similar application of the method above.
48
Proof of Proposition 2. (The cross-section of average arbitrage positions explains the cross-
section of endogenous post-arbitrage betas). (i) First, I show that a higher expected arbi-
trage position on an asset means that the capital required to exploit the asset is lower. Let
k2 (i) continue to denote the level of k2 that makes i the marginal asset. Then, E1 [µxi,2] =∫ 1
k2(i)
(√k2 −
√k2 (i)
)dF (k2)+
∫∞1
(1−
√k2 (i)
)dF (k2) =
∫∞k2(i)
[min
√k2, 1
−√k2 (i)
]·dF (k2). Hence, E1 [µxi,2] and k2 (i) are negatively related:
∂k2 (i)
∂E1 [µxi,2]< 0.
(ii) On the other hand, a lower k2 (i) means higher beta. To show this, I prove
∂Cov1 (ri,2, k2)
∂k2 (i)< 0.
This is because ∂pi,1∂k2(i)
=∂pi,1∂i︸ ︷︷ ︸≤0
· ∂i
∂k2 (i)︸ ︷︷ ︸<0
≥ 0 and ∂Cov1(pi,2,k2)
∂k2(i)= −v
∫ k2(i)
−∞k2
(1+rmaxi)2 ·
∂i∂k2(i)
dF (k2) + vE1 [k2]∫ k2(i)
−∞1
(1+rmaxi)2 · ∂i
∂k2(i)dF (k2) =
∂Cov1 (pi,2, k1)
∂i︸ ︷︷ ︸>0
· ∂i
∂k2 (i)︸ ︷︷ ︸<0
< 0.
Combining (i) and (ii) implies∂βpost
i,k
∂E1[µxi,2]> 0.
Proof of Proposition 3. (The endogenous post-arbitrage beta arises when the arbitrageur is
constrained). This follows trivially from the analysis in Lemma 2.
Proof of Proposition 4. (An upward-biased price of risk in a naive asset pricing test). The
data-generating process considered by the econometrician is
ri,2 =
rprei,2 = αprei + εprei,2
rposti,2 ≈ αposti,0 + λkβposti,k + εposti,2
with probability φ
with probability 1− φ
where βposti,k ≈ b αprei . Hence, E [ri,2] ≈ φαprei +(1− φ)(αposti,0 + λkβ
posti,k
)≈ (1− φ)αposti,0︸ ︷︷ ︸
≡αi,0
+
λk + φb−1︸︷︷︸bias
βi,k since βi,k ≈ (1− φ) βposti,k where βi,k is the unconditional beta.
49
C Empirical Appendix: Standard Error Correction
Studying the endogeneity of betas requires putting betas on the left-hand side of a regression.
What is the econometric consequence of such a regression and how should the standard errors be
adjusted?
The model of excess returns on anomaly i in the post-arbitrage economy is
rei,t = µi + g′tβposti,g + βposti,f ft + εi,t, (21)
where ft is the factor representing arbitrage-capital shocks, gt is a vector of the other k−1 factors,
t = 1, ..., T are time periods within the post-arbitrage equilibrium, and the assumptions made in
Shanken (1992) hold, including that Σε ≡ V ar (ε1,t, ..., εn,t | ft, gt) is not necessarily diagonal but
that (ε1,t, ..., εn,t) is i.i.d. over time. The model of the endogenous determination of βposti,f is
βposti,f = x′ib+ ui, (22)
where xi is anomaly i’s pre-arbitrage characteristics and the error term ui is heteroskedastic but
uncorrelated. For instance, xi = (1, αprei ) in the simple alphas-into-betas regression.26
Since βposti,f is unobserved, (22) cannot be estimated directly. Instead, I use βposti,f estimated from
(21) as the dependent variable of the equation, which implies the relation
βposti,f = x′ib+ ui + ei, (23)
where ei denotes the measurement error arising from estimating βi,f . When T is finite, estimating
(23) instead of (22) raises the concern that “similar” anomalies end up with correlated measurement
errors e, increasing the likelihood of obtaining a spuriously large b.
To derive the standard error correction, rewrite (21) as
rei = µi1n +Gβposti + εi (24)
26In this regression, the measurement error of αprei is not a concern since it is the observed alpha, not the true alpha,
that determines the strength of arbitrage.
50
where rei is a T -vector of excess returns, G is a T × J matrix of factor realizations, and εi is a
T -vector of unexplained returns. Then, the measurement error induced by the OLS estimation of
βposti,f is
ei ≡ βposti,f − βposti,f = (0, 0, ..., 1)︸ ︷︷ ︸
(1×J)
(F ′dF d)−1F ′dεi, (25)
where F d is a matrix of factor realizations demeaned by the realized time-series sample means.
Hence, (23) can be explicitly written as
βposti,f = x′ib+ ui + (0, 0, ..., 1) (F ′dF d)−1F ′dεi. (26)
It follows from (26) that the finite-sample conditional variance-covariance matrix of b is
V b = (X ′X)−1X ′
Ωu + V ar(ε′F d (F ′dF d)
−1(0, 0, ..., 1)′
)︸ ︷︷ ︸
≡Ωe
X (X ′X)−1, (27)
where X ≡ (x1, ...,xn), Ωu ≡ V ar ((u1, ..., un)), and ε is a T × n matrix of unexplained re-
turns. This shows that the correct standard errors that account for the measurement errors are (i)
higher due the second component Ωe within the parantheses and (ii) affected by the cross-sectional
covariances Cov(βposti,f , βpostj,f |X
)6= 0 between correlated anomalies.
The matrix Ωe, however, does not simplify to be a function of Σε and ΣG ≡ V ar ((g′t, ft))
only, so I estimate it through bootstrapping. To do this, I draw the cross-sectional vector (ε1,t, ..., εn,t, g′t, ft)
from a randomly chosen time period within the sample (t = 1, ..., T ) T times with replacement to
obtain a vector of boostrapped measurement errors es ≡ (e1,s, ..., en,s). Then, I repeat this 10, 000
times to obtain e1, ..., e10,000 and use this to estimate Ωe. The estimated Ωe is used to obtain the
correct standard errors for the regression model (23). As is well known, the unbiasedness and
consistency of the estimator b are not affected by the measurement errors.
51