代写Journal of Financial Economics

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代写Journal of Financial Economics
Betting against beta $

Andrea Frazzini

a , Lasse Heje Pedersen a,b,c,d,e, n

a AQR Capital Management, CT 06830, USA

b Stern School of Business, New York University, 44 West Fourth Street, Suite 9-190, NY 10012, USA

c Copenhagen Business School, 2000 Frederiksberg, Denmark

d Center for Economic Policy Research (CEPR), London, UK

e National Bureau of Economic Research (NBER), MA, USA

a r t i c l e i n f o

Article history:

Received 16 December 2010

Received in revised form

10 April 2013

Accepted 19 April 2013

Available online 17 October 2013

JEL classification:

Keywords:

Asset prices

Leverage constraints

Margin requirements

Liquidity

Beta

CAPM

a b s t r a c t

We present a model with leverage and margin constraints that vary across investors and

time. We find evidence consistent with each of the model's five central predictions:

(1) Because constrained investors bid up high-beta assets, high beta is associated with low

alpha, as we find empirically for US equities, 20 international equity markets, Treasury

bonds, corporate bonds, and futures. (2) A betting against beta (BAB) factor, which is long

leveraged low-beta assets and short high-beta assets, produces significant positive risk-

adjusted returns. (3) When funding constraints tighten, the return of the BAB factor is low.

(4) Increased funding liquidity risk compresses betas toward one. (5) More constrained

investors hold riskier assets.

1. Introduction

A basic premise of the capital asset pricing model

(CAPM) is that all agents invest in the portfolio with the

highest expected excess return per unit of risk (Sharpe

ratio) and leverage or de-leverage this portfolio to suit

their risk preferences. However, many investors, such as

individuals, pension funds, and mutual funds, are con-

strained in the leverage that they can take, and they

therefore overweight risky securities instead of using

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jfec

Journal of Financial Economics

http://dx.doi.org/10.1016/j.jfineco.2013.10.005

☆ We thank Cliff Asness, Aaron Brown, John Campbell, Josh Coval (discussant), Kent Daniel, Gene Fama, Nicolae Garleanu, John Heaton (discussant),

Michael Katz, Owen Lamont, Juhani Linnainmaa (discussant), Michael Mendelson, Mark Mitchell, Lubos Pastor (discussant), Matt Richardson, William

Schwert (editor), Tuomo Vuolteenaho, Robert Whitelaw and two anonymous referees for helpful comments and discussions as well as seminar participants

at AQR Capital Management, Columbia University, New York University, Yale University, Emory University, University of Chicago Booth School of Business,

Northwestern University Kellogg School of Management, Harvard University, Boston University, Vienna University of Economics and Business, University of

Mannheim, Goethe University Frankfurt, the American Finance Association meeting, NBER conference, Utah Winter Finance Conference, Annual

Management Conference at University of Chicago Booth School of Business, Bank of America and Merrill Lynch Quant Conference, and Nomura Global

Quantitative Investment Strategies Conference. Lasse Heje Pedersen gratefully acknowledges support from the European Research Council (ERC Grant

no. 312417) and the FRIC Center for Financial Frictions (Grant no. DNRF102).

n Corresponding author at: Stern School of Business, New York University, 44 West Fourth Street, Suite 9-190, NY 10012, USA.

E-mail address: lpederse@stern.nyu.edu (L.H. Pedersen).

Journal of Financial Economics 111 (2014) 1–25

leverage. For instance, many mutual fund families offer

balanced funds in which the “normal” fund may invest

around 40% in long-term bonds and 60% in stocks, whereas

the “aggressive” fund invests 10% in bonds and 90% in

stocks. If the “normal” fund is efficient, then an investor

could leverage it and achieve a better trade-off between

risk and expected return than the aggressive portfolio with

a large tilt toward stocks. The demand for exchange-traded

funds (ETFs) with embedded leverage provides further

evidence that many investors cannot use leverage directly.

This behavior of tilting toward high-beta assets sug-

gests that risky high-beta assets require lower risk-

adjusted returns than low-beta assets, which require

leverage. Indeed, the security market line for US stocks is

too flat relative to the CAPM (Black, Jensen, and Scholes,

1972) and is better explained by the CAPM with restricted

borrowing than the standard CAPM [see Black (1972,

1993), Brennan (1971), and Mehrling (2005) for an excel-

lent historical perspective].
代写Journal of Financial Economics

Several questions arise: How can an unconstrained

arbitrageur exploit this effect, i.e., how do you bet against

beta? What is the magnitude of this anomaly relative to

the size, value, and momentum effects? Is betting against

beta rewarded in other countries and asset classes? How

does the return premium vary over time and in the cross

section? Who bets against beta?

We address these questions by considering a dynamic

model of leverage constraints and by presenting consistent

empirical evidence from 20 international stock markets,

Treasury bond markets, credit markets, and futures

markets.

Our model features several types of agents. Some

agents cannot use leverage and, therefore, overweight

high-beta assets, causing those assets to offer lower

returns. Other agents can use leverage but face margin

constraints. Unconstrained agents underweight (or short-

sell) high-beta assets and buy low-beta assets that they

lever up. The model implies a flatter security market line

(as in Black (1972)), where the slope depends on the

tightness (i.e., Lagrange multiplier) of the funding con-

straints on average across agents (Proposition 1).

One way to illustrate the asset pricing effect of the

funding friction is to consider the returns on market-

neutral betting against beta (BAB) factors. A BAB factor is

a portfolio that holds low-beta assets, leveraged to a beta

of one, and that shorts high-beta assets, de-leveraged to a

beta of one. For instance, the BAB factor for US stocks

achieves a zero beta by holding $1.4 of low-beta stocks and

shortselling $0.7 of high-beta stocks, with offsetting posi-

tions in the risk-free asset to make it self-financing. 1 Our

model predicts that BAB factors have a positive average

return and that the return is increasing in the ex ante

tightness of constraints and in the spread in betas between

high- and low-beta securities (Proposition 2).

When the leveraged agents hit their margin constraint,

they must de-leverage. Therefore, the model predicts that,

during times of tightening funding liquidity constraints,

the BAB factor realizes negative returns as its expected

future return rises (Proposition 3). Furthermore, the model

predicts that the betas of securities in the cross section are

compressed toward one when funding liquidity risk is high

(Proposition 4). Finally, the model implies that more-

constrained investors overweight high-beta assets in their

portfolios and less-constrained investors overweight low-

beta assets and possibly apply leverage (Proposition 5).

Our model thus extends the Black (1972) insight by

considering a broader set of constraints and deriving the

dynamic time series and cross-sectional properties arising

from the equilibrium interaction between agents with

different constraints.

We find consistent evidence for each of the model's

central predictions. To test Proposition 1, we first consider

portfolios sorted by beta within each asset class. We find

that alphas and Sharpe ratios are almost monotonically

declining in beta in each asset class. This finding provides

broad evidence that the relative flatness of the security

market line is not isolated to the US stock market but that

it is a pervasive global phenomenon. Hence, this pattern of

required returns is likely driven by a common economic

cause, and our funding constraint model provides one such

unified explanation.

To test Proposition 2, we construct BAB factors within

the US stock market and within each of the 19 other

developed MSCI stock markets. The US BAB factor realizes

a Sharpe ratio of 0.78 between 1926 and March 2012.

To put this BAB factor return in perspective, note that its

Sharpe ratio is about twice that of the value effect and 40%

higher than that of momentum over the same time period.

The BAB factor has highly significant risk-adjusted returns,

accounting for its realized exposure to market, value, size,

momentum, and liquidity factors (i.e., significant one-,

three-, four-, and five-factor alphas), and it realizes a

significant positive return in each of the four 20-year

subperiods between 1926 and 2012.

We find similar results in our sample of international

equities. Combining stocks in each of the non-US countries

produces a BAB factor with returns about as strong as the

US BAB factor.

We show that BAB returns are consistent across coun-

tries, time, within deciles sorted by size, and within deciles

sorted by idiosyncratic risk and are robust to a number of

specifications. These consistent results suggest that coin-

cidence or data mining are unlikely explanations. How-

ever, if leverage constraints are the underlying drivers as in

our model, then the effect should also exist in other

markets.

Hence, we examine BAB factors in other major asset

classes. For US Treasuries, the BAB factor is a portfolio that

holds leveraged low-beta (i.e., short-maturity) bonds and

shortsells de-leveraged high-beta (i.e., long-term) bonds.

This portfolio produces highly significant risk-adjusted

returns with a Sharpe ratio of 0.81. This profitability of

shortselling long-term bonds could seem to contradict the

well-known “term premium” in fixed income markets.

There is no paradox, however. The term premium means

While we consider a variety of BAB factors within a number of

markets, one notable example is the zero-covariance portfolio introduced

by Black (1972) and studied for US stocks by Black, Jensen, and Scholes

(1972), Kandel (1984), Shanken (1985), Polk, Thompson, and Vuolteenaho

(2006), and others.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 2

that investors are compensated on average for holding

long-term bonds instead of T-bills because of the need for

maturity transformation. The term premium exists at all

horizons, however. Just as investors are compensated for

holding ten-year bonds over T-bills, they are also compen-

sated for holding one-year bonds. Our finding is that the

compensation per unit of risk is in fact larger for the one-

year bond than for the ten-year bond. Hence, a portfolio

that has a leveraged long position in one-year (and other

short-term) bonds and a short position in long-term bonds

produces positive returns. This result is consistent with

our model in which some investors are leverage-

constrained in their bond exposure and, therefore, require

lower risk-adjusted returns for long-term bonds that give

more “bang for the buck”. Indeed, short-term bonds

require tremendous leverage to achieve similar risk or

return as long-term bonds. These results complement

those of Fama (1984, 1986) and Duffee (2010), who also

consider Sharpe ratios across maturities implied by stan-

dard term structure models.

We find similar evidence in credit markets: A leveraged

portfolio of highly rated corporate bonds outperforms a

de-leveraged portfolio of low-rated bonds. Similarly, using

a BAB factor based on corporate bond indices by maturity

produces high risk-adjusted returns.

We test the time series predictions of Proposition 3

using the TED spread as a measure of funding conditions.

Consistent with the model, a higher TED spread is asso-

ciated with low contemporaneous BAB returns. The lagged

TED spread predicts returns negatively, which is incon-

sistent with the model if a high TED spread means a high

tightness of investors' funding constraints. This result

could be explained if higher TED spreads meant that

investors' funding constraints would be tightening as their

banks reduce credit availability over time, though this is

speculation.

To test the prediction of Proposition 4, we use the

volatility of the TED spread as an empirical proxy for

funding liquidity risk. Consistent with the model's beta-

compression prediction, we find that the dispersion of

betas is significantly lower when funding liquidity risk

is high.

Lastly, we find evidence consistent with the model's

portfolio prediction that more-constrained investors hold

higher-beta securities than less-constrained investors

(Proposition 5). We study the equity portfolios of mutual

funds and individual investors, which are likely to be

constrained. Consistent with the model, we find that these

investors hold portfolios with average betas above one.

On the other side of the market, we find that leveraged

buyout (LBO) funds acquire firms with average betas

below 1 and apply leverage. Similarly, looking at the

holdings of Warren Buffett's firm Berkshire Hathaway,

we see that Buffett bets against beta by buying low-beta

stocks and applying leverage (analyzed further in Frazzini,

Kabiller, and Pedersen (2012)).

Our results shed new light on the relation between risk

and expected returns. This central issue in financial eco-

nomics has naturally received much attention. The stan-

dard CAPM beta cannot explain the cross section of

unconditional stock returns (Fama and French, 1992) or

conditional stock returns (Lewellen and Nagel, 2006).

Stocks with high beta have been found to deliver low

risk-adjusted returns (Black, Jensen, and Scholes, 1972;

Baker, Bradley, and Wurgler, 2011); thus, the constrained-

borrowing CAPM has a better fit (Gibbons, 1982; Kandel,

1984; Shanken, 1985). Stocks with high idiosyncratic

volatility have realized low returns (Falkenstein, 1994;

Ang, Hodrick, Xing, Zhang, 2006, 2009), but we find that

the beta effect holds even when controlling for idiosyn-

cratic risk. 2 Theoretically, asset pricing models with bench-

marked managers (Brennan, 1993) or constraints imply

more general CAPM-like relations (Hindy, 1995; Cuoco,

1997). In particular, the margin-CAPM implies that high-

margin assets have higher required returns, especially

during times of funding illiquidity (Garleanu and

Pedersen, 2011; Ashcraft, Garleanu, and Pedersen, 2010).

Garleanu and Pedersen (2011) show empirically that

deviations of the law of one price arises when high-

margin assets become cheaper than low-margin assets,

and Ashcraft, Garleanu, and Pedersen (2010) find that

prices increase when central bank lending facilities reduce

margins. Furthermore, funding liquidity risk is linked to

market liquidity risk (Gromb and Vayanos, 2002;

Brunnermeier and Pedersen, 2009), which also affects

required returns (Acharya and Pedersen, 2005). We com-

plement the literature by deriving new cross-sectional and

time series predictions in a simple dynamic model that

captures leverage and margin constraints and by testing its

implications across a broad cross section of securities

across all the major asset classes. Finally, Asness, Frazzini,

and Pedersen (2012) report evidence of a low-beta effect

across asset classes consistent with our theory.

The rest of the paper is organized as follows. Section 2

lays out the theory, Section 3 describes our data and

empirical methodology, Sections 4–7 test Propositions 1–5,

and Section 8 concludes. Appendix A contains all proofs,

Appendix B provides a number of additional empirical

results and robustness tests, and Appendix C provides a

calibration of the model. The calibration shows that, to

match the strong BAB performance in the data, a large

fraction of agents must face severe constraints. An interest-

ing topic for future research is to empirically estimate

agents' leverage constraints and risk preferences and study

whether the magnitude of the BAB returns is consistent

with the model or should be viewed as a puzzle.

2. Theory

We consider an overlapping-generations (OLG) econ-

omy in which agents i¼1,…,I are born each time period t

with wealth W i t and live for two periods. Agents trade

securities s¼1,…,S, where security s pays dividends δ s

and

has x n s shares outstanding. 3 Each time period t, young

This effect disappears when controlling for the maximum daily

return over the past month (Bali, Cakici, and Whitelaw, 2011) and when

using other measures of idiosyncratic volatility (Fu, 2009).

The dividends and shares outstanding are taken as exogenous. Our

modified CAPM has implications for a corporation's optimal capital

structure, which suggests an interesting avenue of future research

beyond the scope of this paper.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 3

agents choose a portfolio of shares x¼(x 1 ,…,x S )′, investing

the rest of their wealth at the risk-free return r f , to

maximize their utility:

max x′ðE t ðP tþ1 þδ tþ1 Þ?ð1þr f ÞP t Þ?

γ i

x′Ω t x; ð1Þ

where P t is the vector of prices at time t, Ω t is the variance–

covariance matrix of P tþ1 þδ tþ1 , and γ i is agent i's risk

aversion. Agent i is subject to the portfolio constraint

m i t ∑

x s P s

t rW

ð2Þ

This constraint requires that some multiple m i t of the total

dollars invested, the sum of the number of shares x s times

their prices P s , must be less than the agent's wealth.

The investment constraint depends on the agent i. For

instance, some agents simply cannot use leverage, which is

captured by m i ¼1 [as Black (1972) assumes]. Other agents

not only could be precluded from using leverage but also

must have some of their wealth in cash, which is captured

by m i greater than one. For instance, m i ¼1/(1?0.20)¼1.25

represents an agent who must hold 20% of her wealth in

cash. For instance, a mutual fund could need some ready

cash to be able to meet daily redemptions, an insurance

company needs to pay claims, and individual investors

may need cash for unforeseen expenses.

Other agents could be able to use leverage but could

face margin constraints. For instance, if an agent faces a

margin requirement of 50%, then his m i is 0.50. With this

margin requirement, the agent can invest in assets worth

twice his wealth at most. A smaller margin requirement m i

naturally means that the agent can take greater positions.

Our formulation assumes for simplicity that all securities

have the same margin requirement, which may be true

when comparing securities within the same asset class

(e.g., stocks), as we do empirically. Garleanu and Pedersen

(2011) and Ashcraft, Garleanu, and Pedersen (2010) con-

sider assets with different margin requirements and show

theoretically and empirically that higher margin require-

ments are associated with higher required returns

(Margin CAPM).

We are interested in the properties of the competitive

equilibrium in which the total demand equals the supply:

∑

x i ¼ x n ð3Þ

To derive equilibrium, consider the first order condition for

agent i:

0 ¼ E t ðP tþ1 þδ tþ1 Þ?ð1þr f ÞP t ?γ i Ωx i ?ψ i t P t ; ð4Þ

where ψ i is the Lagrange multiplier of the portfolio con-

straint. Solving for x i gives the optimal position:

x i ¼

γ i

Ω ?1 ðE t ðP tþ1 þδ tþ1 Þ?ð1þr f þψ i t ÞP t Þ: ð5Þ

The equilibrium condition now follows from summing

over these positions:

x n ¼

Ω ?1 ðE t ðP tþ1 þδ tþ1 Þ?ð1þr f þψ t ÞP t Þ; ð6Þ

where the aggregate risk aversion γ is defined by 1/γ¼

Σ i 1/γ i and ψ t ¼ ∑ i ðγ=γ i Þψ i t is the weighted average Lagrange

multiplier. (The coefficients γ/γ i sum to one by definition of

the aggregate risk aversion γ.) The equilibrium price can

then be computed:

P t ¼

E t ðP tþ1 þδ tþ1 Þ?γΩx n

1þr f þψ t

; ð7Þ

Translating this into the return of any security r i tþ1 ¼

ðP i tþ1 þδ i tþ1 Þ=P i t ?1, the return on the market r M

tþ1 ,

and using the usual expression for beta, β s

t ¼ cov t

ðr s

tþ1 ;r

tþ1 Þ=var t ðr

tþ1 Þ, we obtain the following results.

(All proofs are in Appendix A, which also illustrates the

portfolio choice with leverage constraints in a mean-

standard deviation diagram.)

Proposition 1 (high beta is low alpha).

(i) The equilibrium required return for any security s is

E t ðr s

tþ1 Þ ¼ r

f þψ

t þβ

t λ t

ð8Þ

where the risk premium is λ t ¼ E t ðr M

tþ1 Þ?r

f ?ψ t

and ψ t

is the average Lagrange multiplier, measuring the tight-

ness of funding constraints.

(ii) A security's alpha with respect to the market is

α s

t ¼ ψ t ð1?β

t Þ. The alpha decreases in the beta, β

代写Journal of Financial Economics

t .

(iii) For an efficient portfolio, the Sharpe ratio is highest for

an efficient portfolio with a beta less than one and

decreases in β s

for higher betas and increases for lower

betas.

As in Black's CAPM with restricted borrowing (in which

m i ¼1 for all agents), the required return is a constant plus

beta times a risk premium. Our expression shows expli-

citly how risk premia are affected by the tightness of

agents' portfolio constraints, as measured by the average

Lagrange multiplier ψ t . Tighter portfolio constraints (i.e., a

larger ψ t ) flatten the security market line by increasing the

intercept and decreasing the slope λ t .

Whereas the standard CAPM implies that the intercept

of the security market line is r f , the intercept here is

increased by binding funding constraints (through the

weighted average of the agents' Lagrange multipliers).

One could wonder why zero-beta assets require returns

in excess of the risk-free rate. The answer has two parts.

First, constrained agents prefer to invest their limited

capital in riskier assets with higher expected return.

Second, unconstrained agents do invest considerable

amounts in zero-beta assets so, from their perspective,

the risk of these assets is not idiosyncratic, as additional

exposure to such assets would increase the risk of their

portfolio. Hence, in equilibrium, zero-beta risky assets

must offer higher returns than the risk-free rate.

Assets that have zero covariance to the Tobin (1958)

“tangency portfolio” held by an unconstrained agent do

earn the risk-free rate, but the tangency portfolio is not the

market portfolio in our equilibrium. The market portfolio

is the weighted average of all investors' portfolios, i.e., an

average of the tangency portfolio held by unconstrained

investors and riskier portfolios held by constrained inves-

tors. Hence, the market portfolio has higher risk and

expected return than the tangency portfolio, but a lower

Sharpe ratio.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 4

The portfolio constraints further imply a lower slope λ t

of the security market line, i.e., a lower compensation for a

marginal increase in systematic risk. The slope is lower

because constrained agents need high unleveraged returns

and are, therefore, willing to accept less compensation for

higher risk. 4

We next consider the properties of a factor that goes

long low-beta assets and shortsells high-beta assets.

To construct such a factor, let w L be the relative portfolio

weights for a portfolio of low-beta assets with return

r L

tþ1 ¼ w

′

L r tþ1

and consider similarly a portfolio of high-

beta assets with return r H

tþ1 . The betas of these portfolios

are denoted β L

and β H

t , where β

t oβ

t . We then construct a

betting against beta (BAB) factor as

r BAB

tþ1 ¼

β L

ðr L

tþ1 ?r

f Þ?

β H

ðr H

tþ1 ?r

f Þ

ð9Þ

this portfolio is market-neutral; that is, it has a beta of

zero. The long side has been leveraged to a beta of one, and

the short side has been de-leveraged to a beta of one.

Furthermore, the BAB factor provides the excess return on

a self-financing portfolio, such as HML (high minus low)

and SMB (small minus big), because it is a difference

between excess returns. The difference is that BAB is not

dollar-neutral in terms of only the risky securities because

this would not produce a beta of zero. 5 The model has

several predictions regarding the BAB factor.

Proposition 2 (positive expected return of BAB). The expected

excess return of the self-financing BAB factor is positive

E t ðr BAB

tþ1 Þ¼

β H

?β L

β L

t β

ψ t Z0 ð10Þ

and increasing in the ex ante beta spread ðβ H

?β L

t Þ=ðβ

t β

t Þ and

funding tightness ψ t .

Proposition 2 shows that a market-neutral BAB portfolio

that is long leveraged low-beta securities and short higher-

beta securities earns a positive expected return on average.

The size of the expected return depends on the spread in

the betas and how binding the portfolio constraints are in

the market, as captured by the average of the Lagrange

multipliers ψ t .

Proposition 3 considers the effect of a shock to the

portfolio constraints (or margin requirements), m k , which

can be interpreted as a worsening of funding liquidity,

a credit crisis in the extreme. Such a funding liquidity

shock results in losses for the BAB factor as its required

return increases. This happens because agents may need to

de-leverage their bets against beta or stretch even further

to buy the high-beta assets. Thus, the BAB factor is

exposed to funding liquidity risk, as it loses when portfolio

constraints become more binding.

Proposition 3 (funding shocks and BAB returns). A tighter

portfolio constraint, that is, an increase in m k

for some of k,

leads to a contemporaneous loss for the BAB factor

∂r BAB

∂m k

r0 ð11Þ

and an increase in its future required return:

∂E t ðr BAB

tþ1 Þ

∂m k

Z0 ð12Þ

Funding shocks have further implications for the cross

section of asset returns and the BAB portfolio. Specifically,

a funding shock makes all security prices drop together

(that is, ð∂P s

t =∂ψ t Þ=P

is the same for all securities s).

Therefore, an increased funding risk compresses betas

toward one. 6 If the BAB portfolio construction is based

on an information set that does not account for this

increased funding risk, then the BAB portfolio's conditional

market beta is affected.

Proposition 4 (beta compression). Suppose that all random

variables are identically and independently distributed (i.i.d.)

over time and δ t is independent of the other random

variables. Further, at time t?1 after the BAB portfolio is

formed and prices are set, the conditional variance of the

discount factor 1/(1þr f þψ t ) rises (falls) due to new informa-

tion about m t and W t . Then,

(i) The conditional return betas β i t?1 of all securities are

compressed toward one (more dispersed), and

(ii) The conditional beta of the BAB portfolio becomes

positive (negative), even though it is market neutral

relative to the information set used for portfolio

formation.

In addition to the asset-pricing predictions that we

derive, funding constraints naturally affect agents' portfo-

lio choices. In particular, more-constrained investors tilt

toward riskier securities in equilibrium and less-

constrained agents tilt toward safer securities with higher

reward per unit of risk. To state this result, we write next

While the risk premium implied by our theory is lower than the

one implied by the CAPM, it is still positive. It is difficult to empirically

estimate a low risk premium and its positivity is not a focus of our

empirical tests as it does not distinguish our theory from the standard

CAPM. However, the data are generally not inconsistent with our

prediction as the estimated risk premium is positive and insignificant

for US stocks, negative and insignificant for international stocks, positive

and insignificant for Treasuries, positive and significant for credits across

maturities, and positive and significant across asset classes.

A natural BAB factor is the zero-covariance portfolio of Black (1972)

and Black, Jensen, and Scholes (1972). We consider a broader class of BAB

portfolios because we empirically consider a variety of BAB portfolios

within various asset classes that are subsets of all securities (e.g., stocks in

a particular size group). Therefore, our construction achieves market

neutrality by leveraging (and de-leveraging) the long and short sides

instead of adding the market itself as Black, Jensen, and Scholes

(1972) do.

Garleanu and Pedersen (2011) find a complementary result, study-

ing securities with identical fundamental risk but different margin

requirements. They find theoretically and empirically that such assets

have similar betas when liquidity is good, but when funding liquidity risk

rises the high-margin securities have larger betas, as their high margins

make them more funding sensitive. Here, we study securities with

different fundamental risk, but the same margin requirements. In this

case, higher funding liquidity risk means that betas are compressed

toward one.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 5

period's security payoffs as

P tþ1 þδ tþ1 ¼ E t ðP tþ1 þδ tþ1 Þþb P M

tþ1 þδ

tþ1 ?E t ðP

tþ1 þδ

tþ1 Þ

? ?

þe

ð13Þ

where b is a vector of market exposures, and e is a vector of

noise that is uncorrelated with the market. We have the

following natural result for the agents' positions.

Proposition 5 (constrained investors hold high betas). Uncon-

strained agents hold a portfolio of risky securities that has a

beta less than one; constrained agents hold portfolios of risky

securities with higher betas. If securities s and k are identical

except that s has a larger market exposure than k, b s 4b k ,

then any constrained agent j with greater than average

Lagrange multiplier, ψ j t 4ψ t , holds more shares of s than k.

The reverse is true for any agent with ψ j t oψ t .

We next provide empirical evidence for Propositions

1–5. Beyond matching the data qualitatively, Appendix C

illustrates how well a calibrated model can quantitatively

match the magnitude of the estimated BAB returns.

3. Data and methodology

The data in this study are collected from several

sources. The sample of US and international equities has

55,600 stocks covering 20 countries, and the summary

statistics for stocks are reported in Table 1. Stock return

data are from the union of the Center for Research in

Security Prices (CRSP) tape and the Xpressfeed Global

database. Our US equity data include all available common

stocks on CRSP between January 1926 and March 2012,

and betas are computed with respect to the CRSP value-

weighted market index. Excess returns are above the US

Treasury bill rate. We consider alphas with respect to the

market factor and factor returns based on size (SMB),

book-to-market (HML), momentum (up minus down,

UMD), and (when available) liquidity risk. 7

The international equity data include all available

common stocks on the Xpressfeed Global daily security

file for 19 markets belonging to the MSCI developed

universe between January 1989 and March 2012. We

assign each stock to its corresponding market based on

the location of the primary exchange. Betas are computed

with respect to the corresponding MSCI local market

index. 8

All returns are in US dollars, and excess returns are

above the US Treasury bill rate. We compute alphas with

respect to the international market and factor returns

based on size (SMB), book-to-market (HML), and momen-

tum (UMD) from Asness and Frazzini (2013) and (when

available) liquidity risk. 9

We also consider a variety of other assets. Table 2

contains the list of instruments and the corresponding

ranges of available data. We obtain US Treasury bond data

from the CRSP US Treasury Database, using monthly

returns (in excess of the one-month Treasury bill) on the

Table 1

Summary statistics: equities.

This table shows summary statistics as of June of each year. The sample includes all commons stocks on the Center for Research in Security Prices daily

stock files (shrcd equal to 10 or 11) and Xpressfeed Global security files (tcpi equal to zero). Mean ME is the average market value of equity, in billions of US

dollars. Means are pooled averages as of June of each year.

Country Local market index Number of

stocks, total

Number of

stocks, mean

Mean ME (firm,

billion of US dollars)

Mean ME (market,

billion of US dollars)

Start year End year

Australia MSCI Australia 3,047 894 0.57 501 1989 2012

Austria MSCI Austria 211 81 0.75 59 1989 2012

Belgium MSCI Belgium 425 138 1.79 240 1989 2012

Canada MSCI Canada 5,703 1,180 0.89 520 1984 2012

Denmark MSCI Denmark 413 146 0.83 119 1989 2012

Finland MSCI Finland 293 109 1.39 143 1989 2012

France MSCI France 1,815 589 2.12 1,222 1989 2012

Germany MSCI Germany 2,165 724 2.48 1,785 1989 2012

Hong Kong MSCI Hong Kong 1,793 674 1.22 799 1989 2012

Italy MSCI Italy 610 224 2.12 470 1989 2012

Japan MSCI Japan 5,009 2,907 1.19 3,488 1989 2012

Netherlands MSCI Netherlands 413 168 3.33 557 1989 2012

New Zealand MSCI New Zealand 318 97 0.87 81 1989 2012

Norway MSCI Norway 661 164 0.76 121 1989 2012

Singapore MSCI Singapore 1,058 375 0.63 240 1989 2012

Spain MSCI Spain 376 138 3.00 398 1989 2012

Sweden MSCI Sweden 1,060 264 1.30 334 1989 2012

Switzerland MSCI Switzerland 566 210 3.06 633 1989 2012

United Kingdom MSCI UK 6,126 1,766 1.22 2,243 1989 2012

United States CRSP value-weighted index 23,538 3,182 0.99 3,215 1926 2012

SMB, HML, and UMD are from Ken French's data library, and the

liquidity risk factor is from Wharton Research Data Service (WRDS).

Our results are robust to the choice of benchmark (local versus

global). We report these tests in Appendix B.

These factors mimic their U.S counterparts and follow Fama and

French (1992, 1993, 1996). See Asness and Frazzini (2013) for a detailed

description of their construction. The data can be downloaded at http://

www.econ.yale.edu/?af227/data_library.htm.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 6

Table 2

Summary statistics: other asset classes.

This table reports the securities included in our data sets and the corresponding date range.

Asset class Instrument Frequency Start year End year

Equity indices Australia Daily 1977 2012

Germany Daily 1975 2012

Canada Daily 1975 2012

Spain Daily 1980 2012

France Daily 1975 2012

Hong Kong Daily 1980 2012

Italy Daily 1978 2012

Japan Daily 1976 2012

Netherlands Daily 1975 2012

Sweden Daily 1980 2012

Switzerland Daily 1975 2012

United Kingdom Daily 1975 2012

United States Daily 1965 2012

Country bonds Australia Daily 1986 2012

Germany Daily 1980 2012

Canada Daily 1985 2012

Japan Daily 1982 2012

Norway Daily 1989 2012

Sweden Daily 1987 2012

Switzerland Daily 1981 2012

United Kingdom Daily 1980 2012

United States Daily 1965 2012

Foreign exchange Australia Daily 1977 2012

Germany Daily 1975 2012

Canada Daily 1975 2012

Japan Daily 1976 2012

Norway Daily 1989 2012

New Zealand Daily 1986 2012

Sweden Daily 1987 2012

Switzerland Daily 1975 2012

United Kingdom Daily 1975 2012

US Treasury bonds Zero to one year Monthly 1952 2012

One to two years Monthly 1952 2012

Two to three years Monthly 1952 2012

Three to four years Monthly 1952 2012

Four to five years Monthly 1952 2012

Four to ten years Monthly 1952 2012

More than ten years Monthly 1952 2012

Credit indices One to three years Monthly 1976 2012

Three to five year Monthly 1976 2012

Five to ten years Monthly 1991 2012

Seven to ten years Monthly 1988 2012

Corporate bonds Aaa Monthly 1973 2012

Aa Monthly 1973 2012

A Monthly 1973 2012

Baa Monthly 1973 2012

Ba Monthly 1983 2012

B Monthly 1983 2012

Caa Monthly 1983 2012

Ca-D Monthly 1993 2012

Distressed Monthly 1986 2012

Commodities Aluminum Daily 1989 2012

Brent oil Daily 1989 2012

Cattle Daily 1989 2012

Cocoa Daily 1984 2012

Coffee Daily 1989 2012

Copper Daily 1989 2012

Corn Daily 1989 2012

Cotton Daily 1989 2012

Crude Daily 1989 2012

Gasoil Daily 1989 2012

Gold Daily 1989 2012

Heat oil Daily 1989 2012

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 7

Fama Bond portfolios for maturities ranging from one to

ten years between January 1952 and March 2012. Each

portfolio return is an equal-weighted average of the

unadjusted holding period return for each bond in the

portfolio. Only non-callable, non-flower notes and bonds

are included in the portfolios. Betas are computed with

respect to an equally weighted portfolio of all bonds in the

database.

We collect aggregate corporate bond index returns

from Barclays Capital's Bond.Hub database. 10 Our analysis

focuses on the monthly returns (in excess of the one-

month Treasury bill) of four aggregate US credit indices

with maturity ranging from one to ten years and nine

investment-grade and high-yield corporate bond portfo-

lios with credit risk ranging from AAA to Ca-D and

Distressed. 11 The data cover the period between January

1973 and March 2012, although the data availability varies

depending on the individual bond series. Betas are com-

puted with respect to an equally weighted portfolio of all

bonds in the database.

We also study futures and forwards on country equity

indexes, country bond indexes, foreign exchange, and

commodities. Return data are drawn from the internal

pricing data maintained by AQR Capital Management LLC.

The data are collected from a variety of sources and

contain daily return on futures, forwards, or swap con-

tracts in excess of the relevant financing rate. The type of

contract for each asset depends on availability or the

relative liquidity of different instruments. Prior to expira-

tion, positions are rolled over into the next most-liquid

contract. The rolling date's convention differs across con-

tracts and depends on the relative liquidity of different

maturities. The data cover the period between January

1963 and March 2012, with varying data availability

depending on the asset class. For more details on the

computationce between the

three-month Eurodollar LIBOR and the three-month US

Treasuries rate. Our TED data run from December 1984 to

March 2012.

3.1. Estimating ex ante betas

We estimate pre-ranking betas from rolling regressions

of excess returns on market excess returns. Whenever

possible, we use daily data, rather than monthly data, as

the accuracy of covariance estimation improves with the

sample frequency (Merton, 1980). 12 Our estimated beta for

security i is given by

^ β ts

¼ ^ ρ

^ s i

^ s m

; ð14Þ

where ^ s i and ^ s m are the estimated volatilities for the stock

and the market and ^ ρ is their correlation. We estimate

volatilities and correlations separately for two reasons.

First, we use a one-year rolling standard deviation for

volatilities and a five-year horizon for the correlation to

account for the fact that correlations appear to move more

slowly than volatilities. 13 Second, we use one-day log

returns to estimate volatilities and overlapping three-day

log returns, r 3d

i;t

¼ ∑ 2

k ¼ 0 lnð1þr

tþk Þ, for correlation to con-

trol for nonsynchronous trading (which affects only corre-

lations). We require at least six months (120 trading days)

of non-missing data to estimate volatilities and at least

three years (750 trading days) of non-missing return data

for correlations. If we have access only to monthly data, we

use rolling one and five-year windows and require at least

12 and 36 observations.

Finally, to reduce the influence of outliers, we follow

Vasicek (1973) and Elton, Gruber, Brown, and Goetzmann

(2003) and shrink the time series estimate of beta ðβ TS

Table 2 (continued)

Asset class Instrument Frequency Start year End year

Hogs Daily 1989 2012

Lead Daily 1989 2012

Nat gas Daily 1989 2012

Nickel Daily 1984 2012

Platinum Daily 1989 2012

Silver Daily 1989 2012

Soymeal Daily 1989 2012

Soy oil Daily 1989 2012

Sugar Daily 1989 2012

Tin Daily 1989 2012

Unleaded Daily 1989 2012

Wheat Daily 1989 2012

Zinc Daily 1989 2012

The data can be downloaded at https://live.barcap.com.

The distress index was provided to us by Credit Suisse.

Daily returns are not available for our sample of US Treasury

bonds, US corporate bonds, and US credit indices.

See, for example, De Santis and Gerard (1997).

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 8

toward the cross-sectional mean ðβ XS Þ:

^ β i ¼ w i ^ β TS

þð1?w i Þ ^ β

ð15Þ

for simplicity, instead of having asset-specific and time-

varying shrinkage factors as in Vasicek (1973), we set

w¼0.6 and β XS ¼1 for all periods and across all assets.

However, our results are very similar either way. 14

Our choice of the shrinkage factor does not affect how

securities are sorted into portfolios because the common

shrinkage does not change the ranks of the security betas.

However, the amount of shrinkage affects the construction of

the BAB portfolios because the estimated betas are used to

scale the long and short sides of portfolio as seen in Eq. (9).

To account for the fact that noise in the ex ante betas

affects the construction of the BAB factors, our inference is

focused on realized abnormal returns so that any mis-

match between ex ante and (ex post) realized betas is

picked up by the realized loadings in the factor regression.

When we regress our portfolios on standard risk factors,

the realized factor loadings are not shrunk as above

because only the ex ante betas are subject to selection

bias. Our results are robust to alternative beta estimation

procedures as we report in Appendix B.

We compute betas with respect to a market portfolio,

which is either specific to an asset class or the overall

world market portfolio of all assets. While our results hold

both ways, we focus on betas with respect to asset class-

specific market portfolios because these betas are less

noisy for several reasons. First, this approach allows us to

use daily data over a long time period for most asset

classes, as opposed to using the most diversified market

portfolio for which we only have monthly data and only

over a limited time period. Second, this approach is

applicable even if markets are segmented.

As a robustness test, Table B8 in Appendix B reports

results when we compute betas with respect to a proxy for a

world market portfolio consisting of many asset classes. We

use the world market portfolio from Asness, Frazzini, and

Pedersen (2012). 15 The results are consistent with our main

tests as the BAB factors earn large and significant abnormal

returns in each of the asset classes in our sample.

3.2. Constructing betting against beta factors

We construct simple portfolios that are long low-beta

securities and that shortsell high-beta securities (BAB factors).

To construct each BAB factor, all securities in an asset class are

ranked in ascending order on the basis of their estimated

beta. The ranked securities are assigned to one of two

portfolios: low-beta and high-beta. The low- (high-) beta

portfolio is composed of all stocks with a beta below (above)

its asset class median (or country median for international

equities). In each portfolio, securities are weighted by the

ranked betas (i.e., lower-beta securities have larger weights in

the low-beta portfolio and higher-beta securities have larger

weights in the high-beta portfolio). The portfolios are reba-

lanced every calendar month.

More formally, let z be the n?1 vector of beta ranks

z i ¼rank(β it ) at portfolio formation, and let z ¼ 1 ′ n z=n be the

average rank, where n is the number of securities and 1 n is

an n?1 vector of ones. The portfolio weights of the low-

beta and high-beta portfolios are given by

w H ¼ kðz?zÞ þ

w L ¼ kðz?zÞ ?

ð16Þ

where k is a normalizing constant k ¼ 2=1 ′ n jz?zj and x þ

and x ? indicate the positive and negative elements of a

vector x. By construction, we have 1 ′ n w H ¼ 1 and 1 ′ n w L ¼ 1.

To construct the BAB factor, both portfolios are rescaled to

have a beta of one at portfolio formation. The BAB is the

self-financing zero-beta portfolio (8) that is long the low-

beta portfolio and that shortsells the high-beta portfolio.

r BAB

tþ1 ¼

β L

ðr L

tþ1 ?r

f Þ?

β H

ðr H

tþ1 ?r

f Þ;

ð17Þ

where r L

tþ1 ¼ r

′

tþ1 w L ; r

tþ1 ¼ r

′

tþ1 w H ; β

t ¼ β

′

t w L ; and β

¼ β ′ t w H .

For example, on average, the US stock BAB factor is long

$1.4 of low-beta stocks (financed by shortselling $1.4 of

risk-free securities) and shortsells $0.7 of high-beta stocks

(with $0.7 earning the risk-free rate).

3.3. Data used to test the theory's portfolio predictions

We collect mutual fund holdings from the union of the

CRSP Mutual Fund Database and Thomson Financial CDA/

Spectrum holdings database, which includes all registered

domestic mutual funds filing with the Securities and Exchange

Commission. The holdings data run from March 1980 to

March 2012. We focus our analysis on open-end, actively

managed, domestic equity mutual funds. Our sample selection

procedure follows that of Kacperczyk, Sialm, and Zheng

(2008), and we refer to their Appendix for details about the

screens that were used and summary statistics of the data.

Our individual investors' holdings data are collected

from a nationwide discount brokerage house and contain

trades made by about 78 thousand households in the

period from January 1991 to November 1996. This data

set has been used extensively in the existing literature on

individual investors. For a detailed description of the

brokerage data set, see Barber and Odean (2000).

Our sample of buyouts is drawn from the mergers and

acquisitions and corporate events database maintained by

AQR/CNH Partners. 16 The data contain various items,

including initial and subsequent announcement dates,

and (if applicable) completion or termination date for all

takeover deals in which the target is a US publicly traded

The Vasicek (1973) Bayesian shrinkage factor is given by

w i ¼ 1?s 2

i;TS =ðs

i;TS þs

XS Þ where s

i;TS is the variance of the estimated beta

for security i and s 2

is the cross-sectional variance of betas. This

estimator places more weight on the historical times series estimate

when the estimate has a lower variance or when there is large dispersion

of betas in the cross section. Pooling across all stocks in our US equity

data, the shrinkage factor w has a mean of 0.61.

See Asness, Frazzini, and Pedersen (2012) for a detailed description

of this market portfolio. The market series is monthly and ranges from

1973 to 2009.

We would like to thank Mark Mitchell for providing us with

these data.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 9

firm and where the acquirer is a private company. For

some (but not all) deals, the acquirer descriptor also

contains information on whether the deal is a leveraged

buyout (LBO) or management buyout (MBO). The data run

from January 1963 to March 2012.

Finally, we download holdings data for Berkshire Hath-

away from Thomson-Reuters Financial Institutional (13f)

Holding Database. The data run from March 1980 to

March 2012.

4. Betting against beta in each asset class

We now test how the required return varies in the

cross-section of beta-sorted securities (Proposition 1) and

the hypothesis that the BAB factors have positive average

returns (Proposition 2). As an overview of these results,

the alphas of all the beta-sorted portfolios considered in

this paper are plotted in Fig. 1. We see that declining

alphas across beta-sorted portfolios are general phenom-

ena across asset classes. (Fig. B1 in Appendix B plots the

Sharpe ratios of beta-sorted portfolios and also shows a

consistently declining pattern.)

Fig. 2 plots the annualized Sharpe ratios of the BAB

portfolios in the various asset classes. All the BAB portfo-

lios deliver positive returns, except for a small insignif-

icantly negative return in Austrian stocks. The BAB

portfolios based on large numbers of securities (US stocks,

international stocks, Treasuries, credits) deliver high risk-

adjusted returns relative to the standard risk factors

considered in the literature.

4.1. Stocks

Table 3 reports our tests for US stocks. We consider ten

beta-sorted portfolios and report their average returns,

alphas, market betas, volatilities, and Sharpe ratios. The

average returns of the different beta portfolios are similar,

which is the well-known relatively flat security market

line. Hence, consistent with Proposition 1 and with Black

(1972), the alphas decline almost monotonically from the

low-beta to high-beta portfolios. The alphas decline when

estimated relative to a one-, three-, four-, and five-factor

model. Moreover, Sharpe ratios decline monotonically

from low-beta to high-beta portfolios.

The rightmost column of Table 3 reports returns of the

betting against beta factor, i.e., a portfolio that is long

leveraged low-beta stocks and that shortsells de-leveraged

high-beta stocks, thus maintaining a beta-neutral portfo-

lio. Consistent with Proposition 2, the BAB factor delivers a

high average return and a high alpha. Specifically, the BAB

factor has Fama and French (1993) abnormal returns of

0.73% per month (t-statistic¼7.39). Further adjusting

returns for the Carhart (1997) momentum factor, the BAB

portfolio earns abnormal returns of 0.55% per month

(t-statistic¼5.59). Last, we adjust returns using a five-

factor model by adding the traded liquidity factor by

Pastor and Stambaugh (2003), yielding an abnormal BAB

return of 0.55% per month (t-statistic¼4.09, which is

lower in part because the liquidity factor is available

during only half of our sample). While the alpha of the

long-short portfolio is consistent across regressions, the

choice of risk adjustment influences the relative alpha

contribution of the long and short sides of the portfolio.

Our results for US equities show how the security

market line has continued to be too flat for another four

decades after Black, Jensen, and Scholes (1972). Further,

our results extend internationally. We consider beta-

sorted portfolios for international equities and later turn

to altogether different asset classes. We use all 19 MSCI

developed countries except the US (to keep the results

separate from the US results above), and we do this in two

ways: We consider international portfolios in which all

international stocks are pooled together (Table 4), and we

consider results separately for each country (Table 5). The

international portfolio is country-neutral, i.e., the low-

(high-) beta portfolio is composed of all stocks with a beta

below (above) its country median. 17

The results for our pooled sample of international

equities in Table 4 mimic the US results. The alpha and

Sharpe ratios of the beta-sorted portfolios decline

(although not perfectly monotonically) with the betas,

and the BAB factor earns risk-adjusted returns between

0.28% and 0.64% per month depending on the choice of

risk adjustment, with t-statistics ranging from 2.09 to 4.81.

Table 5 shows the performance of the BAB factor within

each individual country. The BAB delivers positive Sharpe

ratios in 18 of the 19 MSCI developed countries and

positive four-factor alphas in 13 out of 19, displaying a

strikingly consistent pattern across equity markets. The

BAB returns are statistically significantly positive in six

countries, while none of the negative alphas is significant.

Of course, the small number of stocks in our sample in

many of the countries makes it difficult to reject the null

hypothesis of zero return in each individual country.

Table B1 in Appendix B reports factor loadings. On

average, the US BAB factor goes long $1.40 ($1.40 for

international BAB) and shortsells $0.70 ($0.89 for interna-

tional BAB). The larger long investment is meant to make

the BAB factor market-neutral because the stocks that are

held long have lower betas. The BAB factor's realized

market loading is not exactly zero, reflecting the fact that

our ex ante betas are measured with noise. The other

factor loadings indicate that, relative to high-beta stocks,

low-beta stocks are likely to be larger, have higher book-

to-market ratios, and have higher return over the prior 12

months, although none of the loadings can explain the

large and significant abnormal returns. The BAB portfolio's

positive HML loading is natural since our theory predicts

that low-beta stocks are cheap and high-beta stocks are

expensive.

Appendix B reports further tests and additional robust-

ness checks. In Table B2, we report results using different

window lengths to estimate betas and different bench-

marks (local, global). We split the sample by size (Table B3)

and time periods (Table B4), we control for idiosyncratic

volatility (Table B5), and we report results for alternative

We keep the international portfolio country neutral because we

report the result of betting against beta across equity indices BAB

separately in Table 8.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 10

definitions of the risk-free rate (Table B6). Finally, in Table

B7 and Fig. B2 we report an out-of-sample test. We collect

pricing data from DataStream and for each country in

Table 1 we compute a BAB portfolio over sample period

not covered by the Xpressfeed Global database. 18 All of the

results are consistent: Equity portfolios that bet against

betas earn significant risk-adjusted returns.

4.2. Treasury bonds

Table 6 reports results for US Treasury bonds. As before,

we report average excess returns of bond portfolios

formed by sorting on beta in the previous month. In the

cross section of Treasury bonds, ranking on betas with

respect to an aggregate Treasury bond index is empirically

equivalent to ranking on duration or maturity. Therefore,

in Table 6, one can think of the term “beta,” “duration,” or

“maturity” in an interchangeable fashion. The right-most

column reports returns of the BAB factor. Abnormal

returns are computed with respect to a one-factor model

in which alpha is the intercept in a regression of monthly

excess return on an equally weighted Treasury bond

excess market return.

The results show that the phenomenon of a flatter security

market line than predicted by the standard CAPM is not

limited to the cross section of stock returns. Consistent with

Proposition 1, the alphas decline monotonically with beta.

Likewise, Sharpe ratios decline monotonically from 0.73 for

low-beta (short-maturity) bonds to 0.31 for high-beta (long-

maturity) bonds. Furthermore, the bond BAB portfolio deli-

vers abnormal returns of 0.17% per month (t-statistic¼6.26)

with a large annual Sharpe ratio of 0.81.

Fig. 1. Alphas of beta-sorted portfolios. This figure shows monthly alphas. The test assets are beta-sorted portfolios. At the beginning of each calendar

month, securities are ranked in ascending order on the basis of their estimated beta at the end of the previous month. The ranked securities are assigned to

beta-sorted portfolios. This figure plots alphas from low beta (left) to high beta (right). Alpha is the intercept in a regression of monthly excess return. For

equity portfolios, the explanatory variables are the monthly returns from Fama and French (1993), Asness and Frazzini (2013), and Carhart (1997)

portfolios. For all other portfolios, the explanatory variables are the monthly returns of the market factor. Alphas are in monthly percent.

DataStream international pricing data start in 1969, and Xpress-

feed Global coverage starts in 1984.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 11

Because the idea that funding constraints have a sig-

nificant effect on the term structure of interest could be

surprising, let us illustrate the economic mechanism that

could be at work. Suppose an agent, e.g., a pension fund,

has $1 to allocate to Treasuries with a target excess return

of 2.9% per year. One way to achieve this return target is to

invest $1 in a portfolio of Treasuries with maturity above

ten years as seen in Table 6, P7. If the agent invests in one-

year Treasuries (P1) instead, then he would need to invest

$11 if all maturities had the same Sharpe ratio. This higher

leverage is needed because the long-term Treasures are 11

times more volatile than the short-term Treasuries. Hence,

the agent would need to borrow an additional $10 to lever

his investment in one-year bonds. If the agent has leverage

limits (or prefers lower leverage), then he would strictly

prefer the ten-year Treasuries in this case.

According to our theory, the one-year Treasuries there-

fore must offer higher returns and higher Sharpe ratios,

flattening the security market line for bonds. Empirically,

short-term Treasuries do offer higher risk-adjusted returns

so the return target can be achieved by investing about $5

in one-year bonds. While a constrained investor could still

prefer an un-leveraged investment in ten-year bonds,

unconstrained investors now prefer the leveraged low-

beta bonds, and the market can clear.

While the severity of leverage constraints varies across

market participants, it appears plausible that a five-to-one

leverage (on this part of the portfolio) makes a difference

for some large investors such as pension funds.

4.3. Credit

We next test our model using several credit portfolios and

report results in Table 7. In Panel A, columns 1 to 5, the test

assets are monthly excess returns of corporate bond indexes

by maturity. We see that the credit BAB portfolio delivers

abnormal returns of 0.11% per month (t-statistic¼5.14) with a

large annual Sharpe ratio of 0.82. Furthermore, alphas and

Sharpe ratios decline monotonically.

In columns 6 to 10, we attempt to isolate the credit

component by hedging away the interest rate risk. Given

the results on Treasuries in Table 6, we are interested in

testing a pure credit version of the BAB portfolio. Each

calendar month, we run one-year rolling regressions of

excess bond returns on the excess return on Barclay's US

government bond index. We construct test assets by going

long the corporate bond index and hedging this position

by shortselling the appropriate amount of the government

bond index: r CDS

?r f t ¼ ðr t ?r f t Þ? ^ θ t?1 ðr USGOV

?r f t Þ, where

^ θ t?1

is the slope coefficient estimated in an expanding

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

US equities

Australia

Austria

Belgium

Canada

Switzerland

Germany

Denmark

Spain

Finland

France

United Kingdom

Hong Kong

Italy

Japan

Netherlands

Norway

New Zealand

Singapore

Sweden

International equities

Credit indices

Corporate bonds

Credit, credit default swaps

Treasuries

Equity indices

Country bonds

Foreign exchange

Commodities

Sharpe ratio

Fig. 2. Betting against beta (BAB) Sharpe ratios by asset class. This figures shows annualized Sharpe ratios of BAB factors across asset classes. To construct

the BAB factor, all securities are assigned to one of two portfolios: low beta and high beta. Securities are weighted by the ranked betas and the portfolios are

rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The BAB factor is a self-financing portfolio that is

long the low-beta portfolio and shorts the high-beta portfolio. Sharpe ratios are annualized.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 12

regression using data from the beginning of the sample

and up to month t?1. One interpretation of this returns

series is that it approximates the returns on a credit

default swap (CDS). We compute market returns by taking

the equally weighted average of these hedged returns, and

we compute betas and BAB portfolios as before. Abnormal

Table 3

US equities: returns, 1926–2012.

This table shows beta-sorted calendar-time portfolio returns. At the beginning of each calendar month, stocks are ranked in ascending order on the basis

of their estimated beta at the end of the previous month. The ranked stocks are assigned to one of ten deciles portfolios based on NYSE breakpoints. All

stocks are equally weighted within a given portfolio, and the portfolios are rebalanced every month to maintain equal weights. The right-most column

reports returns of the zero-beta betting against beta (BAB) factor. To construct the BAB factor, all stocks are assigned to one of two portfolios: low beta and

high beta. Stocks are weighted by the ranked betas (lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger

weights in the high-beta portfolio), and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio

formation. The betting against beta factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. This table

includes all available common stocks on the Center for Research in Security Prices database between January 1926 and March 2012. Alpha is the intercept

in a regression of monthly excess return. The explanatory variables are the monthly returns from Fama and French (1993) mimicking portfolios, Carhart

(1997) momentum factor and Pastor and Stambaugh (2003) liquidity factor. CAPM¼Capital Asset Pricing Model. Regarding the five-factor alphas the Pastor

and Stambaugh (2003) liquidity factor is available only between 1968 and 2011. Returns and alphas are in monthly percent, t-statistics are shown below the

coefficient estimates, and 5% statistical significance is indicated in bold. Beta (ex ante) is the average estimated beta at portfolio formation. Beta (realized) is

the realized loading on the market portfolio. Volatilities and Sharpe ratios are annualized.

Portfolio P1

(low beta)

P2 P3 P4 P5 P6 P7 P8 P9 P10

(high beta)

BAB

Excess return 0.91 0.98 1.00 1.03 1.05 1.10 1.05 1.08 1.06 0.97 0.70

(6.37) (5.73) (5.16) (4.88) (4.49) (4.37) (3.84) (3.74) (3.27) (2.55) (7.12)

CAPM alpha 0.52 0.48 0.42 0.39 0.34 0.34 0.22 0.21 0.10 ?0.10 0.73

(6.30) (5.99) (4.91) (4.43) (3.51) (3.20) (1.94) (1.72) (0.67) (?0.48) (7.44)

Three-factor alpha 0.40 0.35 0.26 0.21 0.13 0.11 ?0.03 ?0.06 ?0.22 ?0.49 0.73

(6.25) (5.95) (4.76) (4.13) (2.49) (1.94) (?0.59) (?1.02) (?2.81) (?3.68) (7.39)

Four-factor alpha 0.40 0.37 0.30 0.25 0.18 0.20 0.09 0.11 0.01 ?0.13 0.55

(6.05) (6.13) (5.36) (4.92) (3.27) (3.63) (1.63) (1.94) (0.12) (?1.01) (5.59)

Five-factor alpha 0.37 0.37 0.33 0.30 0.17 0.20 0.11 0.14 0.02 0.00 0.55

(4.54) (4.66) (4.50) (4.40) (2.44) (2.71) (1.40) (1.65) (0.21) (?0.01) (4.09)

Beta (ex ante) 0.64 0.79 0.88 0.97 1.05 1.12 1.21 1.31 1.44 1.70 0.00

Beta (realized) 0.67 0.87 1.00 1.10 1.22 1.32 1.42 1.51 1.66 1.85 ?0.06

Volatility 15.70 18.70 21.11 23.10 25.56 27.58 29.81 31.58 35.52 41.68 10.75

Sharpe ratio 0.70 0.63 0.57 0.54 0.49 0.48 0.42 0.41 0.36 0.28 0.78

Table 4

International equities: returns, 1984–2012.

This table shows beta-sorted calendar-time portfolio returns. At the beginning of each calendar month, stocks are ranked in ascending order on the basis

of their estimated beta at the end of the previous month. The ranked stocks are assigned to one of ten deciles portfolios. All stocks are equally weighted

within a given portfolio, and the portfolios are rebalanced every month to maintain equal weights. The rightmost column reports returns of the zero-beta

betting against beta (BAB) factor. To construct the BAB factor, all stocks are assigned to one of two portfolios: low beta and high beta. The low- (high-) beta

portfolio is composed of all stocks with a beta below (above) its country median. Stocks are weighted by the ranked betas (lower beta security have larger

weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio), and the portfolios are rebalanced every calendar

month. Both portfolios are rescaled to have a beta of one at portfolio formation. The betting against beta factor is a self-financing portfolio that is long the

low-beta portfolio and short the high-beta portfolio. This table includes all available common stocks on the Xpressfeed Global database for the 19 markets

listed in Table 1. The sample period runs from January 1984 to March 2012. Alpha is the intercept in a regression of monthly excess return. The explanatory

variables are the monthly returns of Asness and Frazzini (2013) mimicking portfolios and Pastor and Stambaugh (2003) liquidity factor. CAPM¼Capital

Asset Pricing Model. Regarding the five-factor alphas the Pastor and Stambaugh (2003) liquidity factor is available only between 1968 and 2011. Returns are

in US dollars and do not include any currency hedging. Returns and alphas are in monthly percent, t-statistics are shown below the coefficient estimates,

and 5% statistical significance is indicated in bold. Beta (ex-ante) is the average estimated beta at portfolio formation. Beta (realized) is the realized loading

on the market portfolio. Volatilities and Sharpe ratios are annualized.

Portfolio P1

(low beta)

P2 P3 P4 P5 P6 P7 P8 P9 P10

(high beta)

BAB

Excess return 0.63 0.67 0.69 0.58 0.67 0.63 0.54 0.59 0.44 0.30 0.64

(2.48) (2.44) (2.39) (1.96) (2.19) (1.93) (1.57) (1.58) (1.10) (0.66) (4.66)

CAPM alpha 0.45 0.47 0.48 0.36 0.44 0.39 0.28 0.32 0.15 0.00 0.64

(2.91) (3.03) (2.96) (2.38) (2.86) (2.26) (1.60) (1.55) (0.67) (?0.01) (4.68)

Three-factor alpha 0.28 0.30 0.29 0.16 0.22 0.11 0.01 ?0.03 ?0.23 ?0.50 0.65

(2.19) (2.22) (2.15) (1.29) (1.71) (0.78) (0.06) (?0.17) (?1.20) (?1.94) (4.81)

Four-factor alpha 0.20 0.24 0.20 0.10 0.19 0.08 0.04 0.06 ?0.16 ?0.16 0.30

(1.42) (1.64) (1.39) (0.74) (1.36) (0.53) (0.27) (0.35) (?0.79) (?0.59) (2.20)

Five-factor alpha 0.19 0.23 0.19 0.09 0.20 0.07 0.05 0.05 ?0.19 ?0.18 0.28

(1.38) (1.59) (1.30) (0.65) (1.40) (0.42) (0.33) (0.30) (?0.92) (?0.65) (2.09)

Beta (ex ante) 0.61 0.70 0.77 0.83 0.88 0.93 0.99 1.06 1.15 1.35 0.00

Beta (realized) 0.66 0.75 0.78 0.85 0.87 0.92 0.98 1.03 1.09 1.16 ?0.02

Volatility 14.97 16.27 17.04 17.57 18.08 19.42 20.42 22.05 23.91 27.12 8.07

Sharpe ratio 0.50 0.50 0.48 0.40 0.44 0.39 0.32 0.32 0.22 0.13 0.95

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 13

returns are computed with respect to a two-factor model

in which alpha is the intercept in a regression of monthly

excess return on the equally weighted average pseudo-

CDS excess return and the monthly return on the Treasury

BAB factor. The addition of the Treasury BAB factor on the

right-hand side is an extra check to test a pure credit

version of the BAB portfolio.

The results in Panel A of Table 7 columns 6 to 10 tell the

same story as columns 1 to 5: The BAB portfolio delivers

significant abnormal returns of 0.17% per month (t-

statistics¼4.44) and Sharpe ratios decline monotonically

from low-beta to high-beta assets.

Last, in Panel B of Table 7, we report results in which the

test assets are credit indexes sorted by rating, ranging from

AAA to Ca-D and Distressed. Consistent with all our previous

results, we find large abnormal returns of the BAB portfolios

(0.57% per month with a t-statistics¼3.72) and declining

alphas and Sharpe ratios across beta-sorted portfolios.

Table 5

International equities: returns by country, 1984–2012.

This table shows calendar-time portfolio returns. At the beginning of each calendar month, all stocks are assigned to one of two portfolios: low beta and

high beta. The low- (high-) beta portfolio is composed of all stocks with a beta below (above) its country median. Stocks are weighted by the ranked betas,

and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The zero-beta betting

against beta (BAB) factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. This table includes all available

common stocks on the Xpressfeed Global database for the 19 markets listed in Table 1. The sample period runs from January 1984 to March 2012. Alpha is

the intercept in a regression of monthly excess return. The explanatory variables are the monthly returns of Asness and Frazzini (2013) mimicking

portfolios. Returns are in US dollars and do not include any currency hedging. Returns and alphas are in monthly percent, and 5% statistical significance is

indicated in bold. $Short (Long) is the average dollar value of the short (long) position. Volatilities and Sharpe ratios are annualized.

Country Excess

return

t-Statistics

Excess return

Four-factor

alpha

t-Statistics

alpha

$Short $Long Volatility Sharpe ratio

Australia 0.11 0.36 0.03 0.10 0.80 1.26 16.7 0.08

Austria ?0.03 ?0.09 ?0.28 ?0.72 0.90 1.44 19.9 ?0.02

Belgium 0.71 2.39 0.72 2.28 0.94 1.46 16.9 0.51

Canada 1.23 5.17 0.67 2.71 0.85 1.45 14.1 1.05

Switzerland 0.75 2.91 0.54 2.07 0.93 1.47 14.6 0.61

Germany 0.40 1.30 ?0.07 ?0.22 0.94 1.58 17.3 0.27

Denmark 0.41 1.47 ?0.02 ?0.07 0.91 1.40 15.7 0.31

Spain 0.59 2.12 0.23 0.80 0.92 1.44 15.6 0.45

Finland 0.65 1.51 ?0.10 ?0.22 1.08 1.64 24.0 0.33

France 0.26 0.63 ?0.37 ?0.82 0.92 1.57 23.7 0.13

United Kingdom 0.49 1.99 ?0.01 ?0.05 0.91 1.53 13.9 0.42

Hong Kong 0.85 2.50 1.01 2.79 0.83 1.38 19.1 0.54

Italy 0.29 1.41 0.04 0.17 0.91 1.35 11.8 0.30

Japan 0.21 0.90 0.01 0.06 0.87 1.39 13.3 0.19

Netherlands 0.98 3.62 0.79 2.75 0.91 1.45 15.4 0.77

Norway 0.44 1.15 0.34 0.81 0.85 1.33 21.3 0.25

New Zealand 0.74 2.28 0.62 1.72 0.94 1.36 18.1 0.49

Singapore 0.66 3.37 0.52 2.36 0.79 1.24 11.0 0.72

Sweden 0.77 2.29 0.22 0.64 0.89 1.34 19.0 0.48

Table 6

US Treasury bonds: returns, 1952–2012.

This table shows calendar-time portfolio returns. The test assets are the Center for Research in Security Prices Treasury Fama bond portfolios. Only non

callable, non flower notes and bonds are included in the portfolios. The portfolio returns are an equal-weighted average of the unadjusted holding period

return for each bond in the portfolios in excess of the risk-free rate. To construct the zero-beta betting against beta (BAB) factor, all bonds are assigned to

one of two portfolios: low beta and high beta. Bonds are weighted by the ranked betas (lower beta bonds have larger weight in the low-beta portfolio and

higher beta bonds have larger weights in the high-beta portfolio) and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to

have a beta of one at portfolio formation. The BAB factor is a self-financing portfolio that is long the low-beta portfolio and shorts the high-beta portfolio.

Alpha is the intercept in a regression of monthly excess return. The explanatory variable is the monthly return of an equally weighted bond market

portfolio. The sample period runs from January 1952 to March 2012. Returns and alphas are in monthly percent, t-statistics are shown below the coefficient

estimates, and 5% statistical significance is indicated in bold. Volatilities and Sharpe ratios are annualized. For P7, returns are missing from August 1962 to

December 1971.

Portfolio P1

(low beta)

P2 P3 P4 P5 P6 P7

(high beta)

BAB

Maturity (months) one to 12 13–24 25–36 37–48 49–60 61–120 4120

Excess return

0.05 0.09 0.11 0.13 0.13 0.16 0.24 0.17

(5.66) (3.91) (3.37) (3.09) (2.62) (2.52) (2.20) (6.26)

Alpha 0.03 0.03 0.02 0.01 ?0.01 ?0.02 ?0.07 0.16

(5.50) (3.00) (1.87) (0.99) (?1.35) (?2.28) (?1.85) (6.18)

Beta (ex ante) 0.14 0.45 0.74 0.98 1.21 1.44 2.24 0.00

Beta (realized) 0.16 0.48 0.76 0.98 1.17 1.44 2.10 0.01

Volatility 0.81 2.07 3.18 3.99 4.72 5.80 9.26 2.43

Sharpe ratio 0.73 0.50 0.43 0.40 0.34 0.32 0.31 0.81

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 14

4.4. Equity indexes, country bond indexes, currencies,

and commodities

Table 8 reports results for equity indexes, country bond

indexes, foreign exchange, and commodities. The BAB port-

folio delivers positive returns in each of the four asset classes,

with an annualized Sharpe ratio ranging from 0.11 to 0.51. We

are able to reject the null hypothesis of zero average return

only for equity indexes, but we can reject the null hypothesis

of zero returns for combination portfolios that include all or

some combination of the four asset classes, taking advantage

of diversification. We construct a simple equally weighted

BAB portfolio. To account for different volatility across the

four asset classes, in month t we rescale each return series to

10% annualized volatility using rolling three-year estimates

up to month t?1 and then we equally weight the return

series and their respective market benchmark. This portfolio

construction generates a simple implementable portfolio that

targets 10% BAB volatility in each of the asset classes. We

report results for an all futures combo including all four asset

classes and a country selection combo including only equity

indices, country bonds and foreign exchange. The BAB all

futures and country selection deliver abnormal return of

0.25% and 0.26% per month (t-statistics¼2.53 and 2.42).

4.5. Betting against all of the betas

To summarize, the results in Tables 3–8 strongly sup-

port the predictions that alphas decline with beta and BAB

factors earn positive excess returns in each asset class.

Fig. 1 illustrates the remarkably consistent pattern of

declining alphas in each asset class, and Fig. 2 shows the

consistent return to the BAB factors. Clearly, the relatively

flat security market line, shown by Black, Jensen, and

Scholes (1972) for US stocks, is a pervasive phenomenon

that we find across markets and asset classes. Averaging all

of the BAB factors produces a diversified BAB factor with a

large and significant abnormal return of 0.54% per month

(t-statistics of 6.98) as seen in Table 8, Panel B.

5. Time series tests

In this section, we test Proposition 3's predictions for

the time series of BAB returns: When funding constraints

Table 7

US credit: returns, 1973–2012.

This table shows calendar-time portfolio returns. Panel A shows results for US credit indices by maturity. The test assets are monthly returns on corporate

bond indices with maturity ranging from one to ten years, in excess of the risk-free rate. The sample period runs from January 1976–March 2012. Unhedged

indicates excess returns and Hedged indicates excess returns after hedging the index's interest rate exposure. To construct hedged excess returns, each

calendar month we run one-year rolling regressions of excess bond returns on the excess return on Barclay's US government bond index. We construct test

assets by going long the corporate bond index and hedging this position by shorting the appropriate amount of the government bond index. We compute

market excess returns by taking an equal weighted average of the hedged excess returns. Panel B shows results for US corporate bond index returns by

rating. The sample period runs from January 1973 to March 2012. To construct the zero-beta betting against beta (BAB) factor, all bonds are assigned to one

of two portfolios: low beta and high beta. Bonds are weighted by the ranked betas (lower beta security have larger weight in the low-beta portfolio and

higher beta securities have larger weights in the high-beta portfolio) and the portfolios are rebalanced every calendar month. Both portfolios are rescaled

to have a beta of 1 at portfolio formation. The zero-beta BAB factor is a self-financing portfolio that is long the low-beta portfolio and short the high-beta

portfolio. Alpha is the intercept in a regression of monthly excess return. The explanatory variable is the monthly excess return of the corresponding

market portfolio and, for the hedged portfolios in Panel A, the Treasury BAB factor. Distressed in Panel B indicates the Credit Suisse First Boston distressed

index. Returns and alphas are in monthly percent, t-statistics are shown below the coefficient estimates, and 5% statistical significance is indicated in bold.

Volatilities and Sharpe ratios are annualized.

Panel A: Credit indices, 1976–2012

Unhedged Hedged

Portfolios One to

three years

Three to

five years

Five to

ten years

Seven to

ten years

BAB One to

three years

Three to

five years

Five to

ten years

Seven to

ten years

BAB

Excess return 0.18 0.22 0.36 0.36 0.10 0.11 0.10 0.11 0.10 0.16

(4.97) (4.35) (3.35) (3.51) (4.85) (3.39) (2.56) (1.55) (1.34) (4.35)

Alpha 0.03 0.01 ?0.04 ?0.07 0.11 0.05 0.03 ?0.03 ?0.05 0.17

(2.49) (0.69) (?3.80) (?4.28) (5.14) (3.89) (2.43) (?3.22) (?3.20) (4.44)

Beta (ex ante) 0.71 1.02 1.59 1.75 0.00 0.54 0.76 1.48 1.57 0.00

Beta (realized) 0.61 0.85 1.38 1.49 ?0.03 0.53 0.70 1.35 1.42 ?0.02

Volatility 2.67 3.59 5.82 6.06 1.45 1.68 2.11 3.90 4.15 1.87

Sharpe ratio 0.83 0.72 0.74 0.72 0.82 0.77 0.58 0.35 0.30 1.02

Panel B: Corporate bonds, 1973–2012

Portfolios Aaa Aa A Baa Ba B Caa Ca-D Distressed BAB

Excess return 0.28 0.31 0.32 0.37 0.47 0.38 0.35 0.77 ?0.41 0.44

(3.85) (3.87) (3.47) (3.93) (4.20) (2.56) (1.47) (1.42) (?1.06) (2.64)

Alpha 0.23 0.23 0.20 0.23 0.27 0.10 ?0.06 ?0.04 ?1.11 0.57

(3.31) (3.20) (2.70) (3.37) (4.39) (1.39) (?0.40) (?0.15) (?5.47) (3.72)

Beta (ex ante) 0.67 0.72 0.79 0.88 0.99 1.11 1.57 2.22 2.24 0.00

Beta (realized) 0.17 0.29 0.41 0.48 0.67 0.91 1.34 2.69 2.32 ?0.47

Volatility 4.50 4.99 5.63 5.78 6.84 9.04 14.48 28.58 23.50 9.98

Sharpe ratio 0.75 0.75 0.68 0.77 0.82 0.50 0.29 0.32 ?0.21 0.53

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 15

become more binding (e.g., because margin requirements

rise), the required future BAB premium increases, and the

contemporaneous realized BAB returns become negative.

We take this prediction to the data using the TED

spread as a proxy of funding conditions. The sample runs

from December 1984 (the first available date for the TED

spread) to March 2012.

Table 9 reports regression-based tests of our hypoth-

eses for the BAB factors across asset classes. The first

column simply regresses the US BAB factor on the lagged

level of the TED spread and the contemporaneous change

in the TED spread. 19 We see that both the lagged level and

the contemporaneous change in the TED spread are

negatively related to the BAB returns. If the TED spread

measures the tightness of funding constraints (given by ψ in

the model), then the model predicts a negative coefficient

for the contemporaneous change in TED [Eq. (11)] and a

positive coefficient for the lagged level [Eq. (12)]. Hence, the

coefficient for change is consistent with the model, but the

coefficient for the lagged level is not, under this interpreta-

tion of the TED spread. If, instead, a high TED spread

indicates that agents' funding constraints are worsening,

then the results would be easier to understand. Under this

interpretation, a high TED spread could indicate that banks

are credit-constrained and that banks tighten other inves-

tors' credit constraints over time, leading to a deterioration

of BAB returns over time (if investors do not foresee this).

However, the model's prediction as a partial derivative

assumes that the current funding conditions change while

everything else remains unchanged, but, empirically, other

things do change. Hence, our test relies on an assumption

that such variation of other variables does not lead to an

omitted variables bias. To partially address this issue,

column 2 provides a similar result when controlling for a

number of other variables. The control variables are the

market return (to account for possible noise in the ex ante

betas used for making the BAB portfolio market neutral),

the one-month lagged BAB return (to account for possible

momentum in BAB), the ex ante beta spread, the short

volatility returns, and the lagged inflation. The beta spread

is equal to (β S ?β L )/(β S β L ) and measures the ex ante beta

difference between the long and short side of the BAB

portfolios, which should positively predict the BAB return

as seen in Proposition 2. Consistent with the model,

Table 9 shows that the estimated coefficient for the beta

spread is positive in all specifications, but not statistically

significant. The short volatility returns is the return on a

portfolio that shortsells closest-to-the-money, next-to-

expire straddles on the S&P500 index, capturing potential

sensitivity to volatility risk. Lagged inflation is equal to the

one-year US CPI inflation rate, lagged one month, which is

included to account for potential effects of money illusion

as studied by Cohen, Polk, and Vuolteenaho (2005),

although we do not find evidence of this effect.

Columns 3–4 of Table 9 report panel regressions for

international stock BAB factors and columns 5–6 for all the

BAB factors. These regressions include fixed effects and

standard errors are clustered by date. We consistently find

a negative relation between BAB returns and the TED

spread.

Table 8

Equity indices, country bonds, foreign exchange and commodities: returns, 1965–2012.

This table shows calendar-time portfolio returns. The test assets are futures, forwards or swap returns in excess of the relevant financing rate.

To construct the betting against beta (BAB) factor, all securities are assigned to one of two portfolios: low beta and high beta. Securities are weighted by the

ranked betas (lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio),

and the portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The BAB factor is a self-

financing portfolio that is long the low-beta portfolio and short the high-beta portfolio. Alpha is the intercept in a regression of monthly excess return. The

explanatory variable is the monthly return of the relevant market portfolio. Panel A reports results for equity indices, country bonds, foreign exchange and

commodities. All futures and Country selection are combo portfolios with equal risk in each individual BAB and 10% ex ante volatility. To construct combo

portfolios, at the beginning of each calendar month, we rescale each return series to 10% annualized volatility using rolling three-year estimate up to moth

t?1 and then equally weight the return series and their respective market benchmark. Panel B reports results for all the assets listed in Tables 1 and 2. All

bonds and credit includes US Treasury bonds, US corporate bonds, US credit indices (hedged and unhedged) and country bonds indices. All equities

includes US equities, all individual BAB country portfolios, the international stock BAB, and the equity index BAB. All assets includes all the assets listed in

Tables 1 and 2. All portfolios in Panel B have equal risk in each individual BAB and 10% ex ante volatility. Returns and alphas are in monthly percent,

t-statistics are shown below the coefficient estimates, and 5% statistical significance is indicated in bold. $Short (Long) is the average dollar value of the

short (long) position. Volatilities and Sharpe ratios are annualized.

n Denotes equal risk, 10% ex ante volatility.

BAB portfolios Excess return t-Statistics

excess return

Alpha t-Statistics

alpha

$Short $Long Volatility Sharpe

ratio

Panel A: Equity indices, country bonds, foreign exchange and commodities

Equity indices (EI) 0.55 2.93 0.48 2.58 0.86 1.29 13.08 0.51

Country bonds (CB) 0.03 0.67 0.05 0.95 0.88 1.48 2.93 0.14

Foreign exchange (FX) 0.17 1.23 0.19 1.42 0.89 1.59 9.59 0.22

Commodities (COM) 0.18 0.72 0.21 0.83 0.71 1.48 19.67 0.11

All futures (EIþCBþFXþCOM) n 0.26 2.62 0.25 2.52 7.73 0.40

Country selection (EIþCBþFX) n 0.26 2.38 0.26 2.42 7.47 0.41

Panel B: All assets

All bonds and credit n 0.74 6.94 0.71 6.74 9.78 0.90

All equities n 0.63 6.68 0.64 6.73 10.36 0.73

All assets n 0.53 6.89 0.54 6.98 8.39 0.76

We are viewing the TED spread simply as a measure of credit

conditions, not as a return. Hence, the TED spread at the end of the return

period is a measure of the credit conditions at that time (even if the TED

spread is a difference in interest rates that would be earned over the

following time period).

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 16

6. Beta compression

We next test Proposition 4 that betas are compressed

toward one when funding liquidity risk is high. Table 10

presents tests of this prediction. We use the volatility of

the TED spread to proxy for the volatility of margin

requirements. Volatility in month t is defined as the

standard deviation of daily TED spread innovations,

s TED

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ffi

∑ sAmonth t ðΔTED s ?ΔTED t Þ 2

. Because we are com-

puting conditional moments, we use the monthly volatility

as of the prior calendar month, which ensures that the

conditioning variable is known at the beginning of the

measurement period. The sample runs from December

1984–March 2012.

Panel A of Table 10 shows the cross-sectional dispersion

in betas in different time periods sorted by the TED

volatility for US stocks, Panel B shows the same for inter-

national stocks, and Panel C shows this for all asset classes

in our sample. Each calendar month, we compute cross-

sectional standard deviation, mean absolute deviation, and

inter-quintile range of the betas for all assets in the

universe. We assign the TED spread volatility into three

groups (low, medium, and high) based on full sample

breakpoints (top and bottom third) and regress the times

series of the cross-sectional dispersion measure on the full

set of dummies (without intercept). In Panel C, we compute

the monthly dispersion measure in each asset class and

average across assets. All standard errors are adjusted for

heteroskedasticity and autocorrelation up to 60 months.

Table 10 shows that, consistent with Proposition 4, the

cross-sectional dispersion in betas is lower when credit

constraints are more volatile. The average cross-sectional

standard deviation of US equity betas in periods of low

spread volatility is 0.34, and the dispersion shrinks to 0.29

in volatile credit environment. The difference is statistically

significant (t-statistics¼?2.71). The tests based on the other

dispersion measures, the international equities, and the other

assets all confirm that the cross-sectional dispersion in beta

shrinks at times when credit constraints are more volatile.

Appendix B contains an additional robustness check.

Because we are looking at the cross-sectional dispersion of

estimated betas, one could worry that our results was

driven by higher beta estimation errors, instead of a higher

variance of the true betas. To investigate this possibility,

we run simulations under the null hypothesis of a constant

standard deviation of true betas and test whether the

measurement error in betas can account for the compres-

sion observed in the data. Fig. B3 shows that the compres-

sion observed in the data is much larger than what could

be generated by estimation error variance alone. Naturally,

while this bootstrap analysis does not indicate that the

Table 9

Regression results.

This table shows results from (pooled) time series regressions. The left-hand side is the month t return of the betting against beta (BAB) factors.

To construct the BAB portfolios, all securities are assigned to one of two portfolios: low beta and high beta. Securities are weighted by the ranked betas

(lower beta security have larger weight in the low-beta portfolio and higher beta securities have larger weights in the high-beta portfolio), and the

portfolios are rebalanced every calendar month. Both portfolios are rescaled to have a beta of one at portfolio formation. The BAB factor is a self-financing

portfolio that is long the low-beta portfolio and short the high-beta portfolio. The explanatory variables include the TED spread and a series of controls.

Lagged TED spread is the TED spread at the end of month t?1. Change in TED spread is equal to TED spread at the end of month t minus Ted spread at the

end of month t?1. Short volatility return is the month t return on a portfolio that shorts at-the-money straddles on the S&P 500 index. To construct the

short volatility portfolio, on index options expiration dates we write the next-to-expire closest-to-maturity straddle on the S&P 500 index and hold it to

maturity. Beta spread is defined as (HBeta?LBeta)/(HBeta n LBeta) where HBeta (LBeta) are the betas of the short (long) leg of the BAB portfolio at portfolio

formation. Market return is the monthly return of the relevant market portfolio. Lagged inflation is equal to the one-year US Consumer Price Index inflation

rate, lagged one month. The data run from December 1984 (first available date for the TED spread) to March 2012. Columns 1 and 2 report results for US

equities. Columns 3 and 4 report results for international equities. In these regressions we use each individual country BAB factors as well as an

international equity BAB factor. Columns 5 and 6 report results for all assets in our data. Asset fixed effects are included where indicated, t-statistics are

shown below the coefficient estimates and all standard errors are adjusted for heteroskedasticity (White, 1980). When multiple assets are included in the

regression, standard errors are clustered by date and 5% statistical significance is indicated in bold.

US equities International equities, pooled All assets, pooled

Left-hand side: BAB return

(1) (2) (3) (4) (5) (6)

Lagged TED spread ?0.025 ?0.038 ?0.009 ?0.015 ?0.013 ?0.018

(?5.24) (?4.78) (?3.87) (?4.07) (?4.87) (?4.65)

Change in TED spread ?0.019 ?0.035 ?0.006 ?0.010 ?0.007 ?0.011

(?2.58) (?4.28) (?2.24) (?2.73) (?2.42) (?2.64)

Beta spread 0.011 0.001 0.001

(0.76) (0.40) (0.69)

Lagged BAB return 0.011 0.035 0.044

(0.13) (1.10) (1.40)

Lagged inflation ?0.177 0.003 ?0.062

(?0.87) (0.03) (?0.58)

Short volatility return ?0.238 0.021 0.027

(?2.27) (0.44) (0.48)

Market return ?0.372 ?0.104 ?0.097

(?4.40) (?2.27) (?2.18)

Asset fixed effects No No Yes Yes Yes Yes

Number of observations 328 328 5,725 5,725 8,120 8,120

Adjusted R² 0.070 0.214 0.007 0.027 0.014 0.036

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 17

Table 10

Beta compression.

This table reports results of cross-sectional and time-series tests of beta compression. Panels A, B and C report cross-sectional dispersion of betas in US

equities, international equities, and all asset classes in our sample. The data run from December 1984 (first available date for the TED spread) to March

2012. Each calendar month we compute cross sectional standard deviation, mean absolute deviation, and inter quintile range of betas. In Panel C we

compute each dispersions measure for each asset class and average across asset classes. The row denoted all reports times series means of the dispersion

measures. P1 to P3 report coefficients on a regression of the dispersion measure on a series of TED spread volatility dummies. TED spread volatility is

defined as the standard deviation of daily changes in the TED spread in the prior calendar month. We assign the TED spread volatility into three groups

(low, neutral, and high) based on full sample breakpoints (top and bottom one third) and regress the times series of the cross-sectional dispersion measure

on the full set of dummies (without intercept). t-Statistics are shown below the coefficient estimates, and 5% statistical significance is indicated in bold.

Panels D, E and F report conditional market betas of the betting against beta (BAB) portfolio based on TED spread volatility as of the prior month. The

dependent variable is the monthly return of the BAB portfolios. The explanatory variables are the monthly returns of the market portfolio, Fama and French

(1993), Asness and Frazzini (2013), and Carhart (1997) mimicking portfolios, but only the alpha and the market betas are reported. CAPM indicates the

Capital Asset Pricing Model. Market betas are allowed to vary across TED spread volatility regimes (low, neutral, and high) using the full set of dummies.

Panels D, E and F report loading on the market factor corresponding to different TED spread volatility regimes. All assets report results for the aggregate

BAB portfolio of Table 9, Panel B. All standard errors are adjusted for heteroskedasticity and autocorrelation using a Bartlett kernel (Newey and West,1987)

with a lag length of sixty months.

Cross-sectional dispersion Standard deviation Mean absolute deviation Interquintile range

Panel A: US equities

All 0.32 0.25 0.43

P1 (low TED volatility) 0.34 0.27 0.45

P2 0.33 0.26 0.44

P3 (high TED volatility) 0.29 0.23 0.40

P3 minus P1 ?0.05 ?0.04 ?0.05

t-Statistics (?2.71) (?2.43) (?1.66)

Panel B: International equities

All 0.22 0.17 0.29

P1 (low TED volatility) 0.23 0.18 0.30

P2 0.22 0.17 0.29

P3 (high TED volatility) 0.20 0.16 0.27

P3 minus P1 ?0.04 ?0.03 ?0.03

t-Statistics (?2.50) (?2.10) (?1.46)

Panel C: All assets

All 0.45 0.35 0.61

P1 (low TED volatility) 0.47 0.37 0.63

P2 0.45 0.36 0.62

P3 (high TED volatility) 0.43 0.33 0.58

P3 minus P1 ?0.04 ?0.03 ?0.06

t-Statistics (?3.18) (?3.77) (?2.66)

Conditional market beta

Alpha P1 (low TED volatility) P2 P3 (high TED volatility) P3?P1

Panel D: US equities

CAPM 1.06 ?0.46 ?0.19 ?0.01 0.45

(3.61) (?2.65) (?1.29) (?0.11) (3.01)

Control for three factors 0.86 ?0.40 ?0.02 0.08 0.49

(4.13) (?3.95) (?0.19) (0.69) (3.06)

Control for four factors 0.66 ?0.28 0.00 0.13 0.40

(3.14) (?5.95) (0.02) (1.46) (4.56)

代写Journal of Financial Economics

measurement error.

Panels D, E, and F report conditional market betas of

the BAB portfolio returns based on the volatility of the

credit environment for US equities, international equities,

and the average BAB factor across all assets, respectively.

The dependent variable is the monthly return of the BAB

portfolio. The explanatory variables are the monthly

returns of the market portfolio, Fama and French (1993)

mimicking portfolios, and Carhart (1997) momentum

factor. Market betas are allowed to vary across TED

volatility regimes (low, neutral, and high) using the full

set of TED dummies.

We are interested in testing Proposition 4(ii), studying

how the BAB factor's conditional beta depends on the TED-

volatility environment. To understand this test, recall first

that the BAB factor is market neutral conditional on the

information set used in the estimation of ex ante betas

(which determine the ex ante relative position sizes of the

long and short sides of the portfolio). Hence, if the TED

spread volatility was used in the ex ante beta estimation,

then the BAB factor would be market-neutral conditional on

this information. However, the BAB factor was constructed

using historical betas that do not take into account the effect

of the TED spread and, therefore, a high TED spread volatility

means that the realized betas will be compressed relative to

the ex ante estimated betas used in portfolio construction.

Therefore, a high TED spread volatility should increase the

conditional market sensitivity of the BAB factor (because the

long side of the portfolio is leveraged too much and the short

side is deleveraged too much). Indeed, Table 10 shows that

when credit constraints are more volatile, the market beta of

the BAB factor rises. The right-most column shows that the

difference between low- and high-credit volatility environ-

ments is statistically significant (t-statistic of 3.01). Controlling

for three or four factors yields similar results. The results for

our sample of international equities (Panel E) and for the

average BAB across all assets (Panel F) are similar, but they are

weaker both in terms of magnitude and statistical significance.

Importantly, the alpha of the BAB factor remains large

and statistically significant even when we control for the

time-varying market exposure. This means that, if we

hedge the BAB factor to be market-neutral conditional on

the TED spread volatility environment, then this condi-

tionally market-neutral BAB factor continues to earn

positive excess returns.

7. Testing the model's portfolio predictions

The theory's last prediction (Proposition 5) is that

more-constrained investors hold higher-beta securities

than less-constrained investors. Consistent with this pre-

diction, Table 11 presents evidence that mutual funds and

individual investors hold high-beta stocks while LBO firms

and Berkshire Hathaway buy low-beta stocks.

Before we delve into the details, let us highlight a

challenge in testing Proposition 5. Whether an investor's

constraint is binding depends both on the investor's ability

to apply leverage (m i in the model) and on its unobser-

vable risk aversion. For example, while a hedge fund could

apply some leverage, its leverage constraint could never-

theless be binding if its desired volatility is high (especially

if its portfolio is very diversified and hedged).

Given that binding constraints are difficult to observe

directly, we seek to identify groups of investors that are

Table 11

Testing the model's portfolio predictions, 1963–2012.

This table shows average ex ante and realized portfolio betas for different groups of investors. Panel A reports results for our sample of open-end actively-

managed domestic equity mutual funds as well as results a sample of individual retail investors. Panel B reports results for a sample of leveraged buyouts

(private equity) and for Berkshire Hathaway. We compute both the ex ante beta of their holdings and the realized beta of the time series of their returns. To

compute the ex-ante beta, we aggregate all quarterly (monthly) holdings in the mutual fund (individual investor) sample and compute their

ex-ante betas (equally weighted and value weighted based on the value of their holdings). We report the time series averages of the portfolio betas.

To compute the realized betas, we compute monthly returns of an aggregate portfolio mimicking the holdings, under the assumption of constant weight

between reporting dates (quarterly for mutual funds, monthly for individual investors). We compute equally weighted and value-weighted returns based on the

value of their holdings. The realized betas are the regression coefficients in a time series regression of these excess returns on the excess returns of the Center

for Research in Security Prices value-weighted index. In Panel B we compute ex ante betas as of the month-end prior to the initial takeover announcements

date. t-Statistics are shown to right of the betas estimates and test the null hypothesis of beta¼1. All standard errors are adjusted for heteroskedasticity and

autocorrelation using a Bartlett kernel (Newey and West, 1987) with a lag length of 60 months. A 5% statistical significance is indicated in bold.

Ex ante beta of positions Realized beta of positions

Investor, method Sample period Beta t-Statistics

(H0: beta¼1)

Beta t-Statistics

(H0: beta¼1)

Panel A: Investors likely to be constrained

Mutual funds, value weighted 1980–2012 1.08 2.16 1.08 6.44

Mutual funds, equal weighted 1980–2012 1.06 1.84 1.12 3.29

Individual investors, value weighted 1991–1996 1.25 8.16 1.09 3.70

Individual investors, equal weighted 1991–1996 1.25 7.22 1.08 2.13

Panel B: Investors who use leverage

Private equity (all) 1963–2012 0.96 ?1.50

Private equity (all), equal weighted 1963–2012 0.94 ?2.30

Private equity (LBO, MBO), value weighted 1963–2012 0.83 ?3.15

Private equity (LBO, MBO), equal weighted 1963–2012 0.82 ?3.47

Berkshire Hathaway, value weighted 1980–2012 0.91 ?2.42 0.77 ?3.65

Berkshire Hathaway, equal weighted 1980–2012 0.90 ?3.81 0.83 ?2.44

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 19

plausibly constrained and unconstrained. One example of

an investor that could be constrained is a mutual fund. The

1940 Investment Company Act places some restriction on

mutual funds' use of leverage, and many mutual funds are

prohibited by charter from using leverage. A mutual funds'

need to hold cash to meet redemptions (m i 41 in the

model) creates a further incentive to overweight high-beta

securities. Overweighting high-beta stocks helps avoid

lagging their benchmark in a bull market because of the

cash holdings (some funds use futures contracts to “equi-

tize” the cash, but other funds are not allowed to use

derivative contracts).

A second class of investors that could face borrowing

constraints is individual retail investors. Although we do

not have direct evidence of their inability to employ

leverage (and some individuals certainly do), we think

that (at least in aggregate) it is plausible that they are

likely to face borrowing restrictions.

The flipside of this portfolio test is identifying relatively

unconstrained investors. Thus, one needs investors that

could be allowed to use leverage and are operating below

their leverage cap so that their leverage constraints are not

binding. We look at the holdings of two groups of

investors that could satisfy these criteria as they have

access to leverage and focus on long equity investments

(requiring less leverage than long/short strategies).

First, we look at the firms that are the target of bids by

leveraged buyout (LBO) funds and other forms of private

equity. These investors, as the name suggest, employ

leverage to acquire a public company. Admittedly, we do

not have direct evidence of the maximum leverage

available to these LBO firms relative to the leverage they

apply, but anecdotal evidence suggests that they achieve

a substantial amount of leverage.

Second, we examine the holdings of Berkshire Hath-

away, a publicly traded corporation run by Warren Buffett

that holds a diversified portfolio of equities and employs

leverage (by issuing debt, via insurance float, and other

means). The advantage of using the holdings of a public

corporation that holds equities such as Berkshire is that we

can directly observe its leverage. Over the period from

March 1980 to March 2012, its average book leverage,

defined as (book equityþtotal debt) / book equity, was

about 1.2, that is, 20% borrowing, and the market leverage

including other liabilities such insurance float was about

1.6 (Frazzini, Kabiller, and Pedersen, 2012). It is therefore

plausible to assume that Berkshire at the margin could

issue more debt but choose not to, making it a likely

candidate for an investor whose combination of risk

aversion and borrowing constraints made it relatively

unconstrained during our sample period.

Table 11 reports the results of our portfolio test.

We estimate both the ex ante beta of the various investors'

holdings and the realized beta of the time series of their

returns. We first aggregate all holdings for each investor

group, compute their ex-ante betas (equal and value

weighted, respectively), and take the time series average.

To compute the realized betas, we compute monthly

returns of an aggregate portfolio mimicking the holdings,

under the assumption of constant weight between report-

ing dates. The realized betas are the regression coefficients

in a time series regression of these excess returns on the

excess returns of the CRSP value-weighted index.

Panel A shows evidence consistent with the hypothesis

that constrained investors stretch for return by increasing

their betas. Mutual funds hold securities with betas above

one, and we are able to reject the null hypothesis of betas

being equal to one. These findings are consistent with

those of Karceski (2002), but our sample is much larger,

including all funds over 30-year period. Similar evidence is

presented for individual retail investors: Individual inves-

tors tend to hold securities with betas that are significantly

above one. 20

Panel B reports results for our sample of private equity.

For each target stock in our database, we focus on its

ex ante beta as of the month-end prior to the initial

announcements date. This focus is to avoid confounding

effects that result from changes in betas related to the

actual delisting event. We consider both the sample of all

private equity deals and the subsample that we are able to

positively identify as LBO/MBO events. Since we only have

partial information about whether each deal is a LBO/MBO,

the broad sample includes all types of deals where a

company is taken private. The results are consistent with

Proposition 5 in that investors executing leverage buyouts

tend to acquire (or attempt to acquire in case of a non-

successful bid) firms with low betas, and we are able to

reject the null hypothesis of a unit beta.

The results for Berkshire Hathaway show a similar

pattern: Warren Buffett bets against beta by buying stocks

with betas significantly below one and applying leverage.

8. Conclusion

All real-world investors face funding constraints such

as leverage constraints and margin requirements, and

these constraints influence investors' required returns

across securities and over time. We find empirically that

portfolios of high-beta assets have lower alphas and

Sharpe ratios than portfolios of low-beta assets. The

security market line is not only flatter than predicted by

the standard CAPM for US equities (as reported by Black,

Jensen, and Scholes (1972)), but we also find this relative

flatness in 18 of 19 international equity markets, in

Treasury markets, for corporate bonds sorted by maturity

and by rating, and in futures markets. We show how this

deviation from the standard CAPM can be captured using

betting against beta factors, which could also be useful as

control variables in future research (Proposition 2). The

return of the BAB factor rivals those of all the standard

asset pricing factors (e.g., value, momentum, and size) in

terms of economic magnitude, statistical significance, and

robustness across time periods, subsamples of stocks, and

global asset classes.

As further consistent evidence, younger people and people with

less financial wealth (who might be more constrained) tend to own

portfolios with higher betas (Calvet, Campbell, and Sodini, 2007, Table 5).

Further, consistent with the idea that leverage requires certain skills and

sophistication, Grinblatt, Keloharju, and Linnainmaa (2011) report that

individuals with low intelligence scores hold higher-beta portfolios than

individuals with high intelligence scores.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 20

Extending the Black (1972) model, we consider the

implications of funding constraints for cross-sectional and

time series asset returns. We show that worsening funding

liquidity should lead to losses for the BAB factor in the

time series (Proposition 3) and that increased funding

liquidity risk compresses betas in the cross section of

securities toward one (Proposition 4), and we find con-

sistent evidence empirically.

Our model also has implications for agents' portfolio

selection (Proposition 5). To test this, we identify investors

that are likely to be relatively constrained and uncon-

strained. We discuss why mutual funds and individual

investors could be leverage constrained, and, consistent

with the model's prediction that constrained investors go

for riskier assets, we find that these investor groups hold

portfolios with betas above one on average.

Conversely, we show that leveraged buyout funds and

Berkshire Hathaway, all of which have access to leverage,

buy stocks with betas below one on average, another

prediction of the model. Hence, these investors could be

taking advantage of the BAB effect by applying leverage to

safe assets and being compensated by investors facing

borrowing constraints who take the other side. Buffett bets

against beta as Fisher Black believed one should.

Appendix A. Analysis and proofs

Before we prove our propositions, we provide a basic

analysis of portfolio selection with constraints. This ana-

lysis is based on Fig. A1. The top panel shows the mean-

standard deviation frontier for an agent with mo1, that is,

an agent who can use leverage. We see that the agent can

leverage the tangency portfolio T to arrive at the portfolio

T. To achieve a higher expected return, the agent needs to

leverage riskier assets, which gives rise to the hyperbola

segment to the right of T. The agent in the graph is

assumed to have risk preferences giving rise to the optimal

portfolio C. Hence, the agent is leverage constrained so he

chooses to apply leverage to portfolio C instead of the

tangency portfolio.

The bottom panel of Fig. A1 similarly shows the mean-

standard deviation frontier for an agent with m41, that is,

an agent who must hold some cash. If the agent keeps the

minimum amount of money in cash and invests the rest in

the tangency portfolio, then he arrives at portfolio T′.

To achieve higher expected return, the agent must invest

in riskier assets and, in the depicted case, he invests in

cash and portfolio D, arriving at portfolio D′.

Unconstrained investors invest in the tangency portfo-

lio and cash. Hence, the market portfolio is a weighted

average of T and riskier portfolios such as C and D.

Therefore, the market portfolio is riskier than the tangency

portfolio.

A.1. Proof of Proposition 1

Rearranging the equilibrium-price Eq. (7) yields

E t ðr s

tþ1 Þ¼ r

f þψ

t þγ

P s

e ′ s Ωx n

¼ r f þψ t þγ

P s

cov t ðP s

tþ1 þδ

tþ1 ; P tþ1 þδ tþ1

? ? ′x n Þ

¼ r f þψ t þγcov t ðr s

tþ1 ;r

tþ1 ÞP

′

t x

ð18Þ

where e s is a vector with a one in row s and zeros

elsewhere. Multiplying this equation by the market port-

folio weights w s ¼ x n s P s

t =∑ j x

n j P j

t and summing over s gives

E t ðr M

tþ1 Þ ¼ r

f þψ

t þγvar t ðr

tþ1 ÞP

′

t x

ð19Þ

that is,

γP ′ t x n ¼

λ t

var t ðr M

tþ1 Þ

ð20Þ

Inserting this into Eq. (18) gives the first result in the

proposition. The second result follows from writing the

expected return as

rtfolio must have a lower

expected return and beta (strictly lower if and only if some

agents are constrained). □

A.2. Proof of Propositions 2–3

The expected return of the BAB factor is

β L

ψ t ð22Þ

Consider next a change in m k

t . Such a change in a time

t margin requirement does not change the time t betas for

two reasons. First, it does not affect the distribution of

prices in the following period tþ1. Second, prices at time

t are scaled (up or down) by the same proportion due to

the change in Lagrange multipliers as seen in Eq. (7).

Hence, all returns from t to tþ1 change by the same

multiplier, leading to time t betas staying the same.

Given Eq. (22), Eq. (12) in the proposition now follows

if we can show that ψ t increases in m k because this lead to

∂E t ðr BAB

40 ð23Þ

Further, because prices move opposite required returns,

Eq. (11) then follows. To see that an increase in m k

increases ψ t , note that the constrained agents' asset

expenditure decreases with a higher m k

t . Indeed, summing

the portfolio constraint across constrained agents [where

i constrained

m i

W i t ð24Þ

Because increasing m k decreases the right-hand side,

the left-hand side must also decrease. That is, the total

market value of shares owned by constrained agents

decreases.

Fig. A1. Portfolio selection with constraints. The top panel shows the mean-standard deviation frontier for an agent with mo1 who can use leverage, and

the bottom panel shows that of an agent with m41 who needs to hold cash.

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 22

Next, we show that the constrained agents' expendi-

ture is decreasing in ψ. Hence, because an increase in m k

decreases the constrained agents' expenditure, it must

increase ψ t as we wanted to show.

∂

∂ψ

? ?

o0 ð25Þ

to see the last inequality, note that clearly ð∂P t =∂ψÞ′x i o0

since all the prices decrease by the same proportion [seen

in Eq. (7)] and the initial expenditure is positive. The

second term is also negative because

∑

i constrained

P ′ t

∂

ψ i t Þ

E t ðP tþ1 þδ tþ1 Þ?γΩx n

1þr f þψ

¼ ?P ′ t

∂

∂ψ

Ω ?1

ariance of ψ t so that

^ β L

t oβ

^ β H

40 □ ð31Þ

A.4. Proof of Proposition 5

To see the first part of the proposition, note that an

unconstrained investor holds the tangency portfolio,

which has a beta less than one in the equilibrium with

funding constraints, and the constrained investors hold

riskier portfolios of risky assets, as discussed in the proof

32Þ

The first term shows that each agent holds some (positive)

weight in the market portfolio x* and the second term

shows how he tilts his portfolio away from the market. The

direction of the tilt depends on whether the agent's

Lagrange multiplier ψ i t is smaller or larger than the

weighted average of all the agents' Lagrange multipliers

ψ t . A less-constrained agent tilts toward the portfolio

Ω ?1 E t ðP tþ1 þδ tþ1 Þ (measured in shares), while a more-

constrained agent tilts away from this portfolio. Given the

A. Frazzini, L.H. Pedersen / Journal of Financial Economics 111 (2014) 1–25 23

expression (13), we can write the variance-covariance

matrix as

Ω ¼ s 2

M bb′þΣ

ð33Þ

where Σ¼var(e) and s 2

M ¼ varðP

tþ1 Þ. Using the Matrix

Inversion Lemma (the Sherman-Morrison-Woodbury for-

mula), the tilt portfolio can be written as

Ω ?1 E t ðP tþ1 þδ tþ1 Þ

¼ Σ

where y ¼ b′Σ ?1 E t ðP tþ1 þδ tþ1 Þ=ðs 2

M þb′Σ

?1 bÞ is a scalar.

It holds that ðΣ ?1 bÞ s 4ðΣ ?1 bÞ k because b s 4b k and because

s and k have the same variances and covariances in Σ,

implying that (Σ ?1 ) s,j ¼(Σ ?1 ) k,j for jas,k and (Σ ?1 ) s,s ¼

(Σ ?1 ) k,k Z(Σ ?1 ) s,k ¼(Σ ?1 ) k,s . Similarly, it holds that

[Σ ?1 E t (P tþ1 þδ tþ1 )] s o[Σ ?1 E t (P tþ1 þδ tþ1 )] k since a higher

market exposure leads to a lower price (as seen below).

So, everything else equal, a higher b leads to a lower weight

in the tilt portfolio.

Finally, security s also has a higher return beta than k

because

β i t ¼

P M

covðP i tþ1 þδ i tþ1 ;P M

tþ1 þδ

tþ1 Þ

P i t varðP M

tþ1 þδ

appendix.htm

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