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:
    G0
    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.
    & 2013 Elsevier B.V. All rights reserved.
    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
    0304-405X/$-see front matter & 2013 Elsevier B.V. All rights reserved.
    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
    1
    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
    t
    and
    has x n s shares outstanding. 3 Each time period t, young
    2
    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).
    3
    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
    2
    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 ∑
    s
    x s P s
    t rW
    i
    t
    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:
    i
    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
    1
    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
    1
    γ
    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
    M
    tþ1 Þ=var t ðr
    M
    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
    s
    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
    s
    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
    t
    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
    t
    and β H
    t , where β
    L
    t oβ
    H
    t . We then construct a
    betting against beta (BAB) factor as
    r BAB
    tþ1 ¼
    1
    β L
    t
    ðr L
    tþ1 ?r
    f Þ?
    1
    β H
    t
    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
    t
    ?β L
    t
    β L
    t β
    H
    t
    ψ t Z0 ð10Þ
    and increasing in the ex ante beta spread ðβ H
    t
    ?β L
    t Þ=ðβ
    L
    t β
    H
    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
    t
    for some of k,
    leads to a contemporaneous loss for the BAB factor
    ∂r BAB
    t
    ∂m k
    t
    r0 ð11Þ
    and an increase in its future required return:
    ∂E t ðr BAB
    tþ1 Þ
    ∂m k
    t
    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
    s
    t
    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
    4
    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.
    5
    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.
    6
    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
    M
    tþ1 ?E t ðP
    M
    tþ1 þδ
    M
    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
    7
    SMB, HML, and UMD are from Ken French's data library, and the
    liquidity risk factor is from Wharton Research Data Service (WRDS).
    8
    Our results are robust to the choice of benchmark (local versus
    global). We report these tests in Appendix B.
    9
    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
    i
    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
    i
    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
    i
    Þ
    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
    10
    The data can be downloaded at https://live.barcap.com.
    11
    The distress index was provided to us by Credit Suisse.
    12
    Daily returns are not available for our sample of US Treasury
    bonds, US corporate bonds, and US credit indices.
    13
    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
    i
    þð1?w i Þ ^ β
    XS
    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
    1
    β L
    t
    ðr L
    tþ1 ?r
    f Þ?
    1
    β H
    t
    ðr H
    tþ1 ?r
    f Þ;
    ð17Þ
    where r L
    tþ1 ¼ r
    tþ1 w L ; r
    H
    tþ1 ¼ r
    tþ1 w H ; β
    L
    t ¼ β
    t w L ; and β
    H
    t
    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
    14
    The Vasicek (1973) Bayesian shrinkage factor is given by
    w i ¼ 1?s 2
    i;TS =ðs
    2
    i;TS þs
    2
    XS Þ where s
    2
    i;TS is the variance of the estimated beta
    for security i and s 2
    XS
    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.
    15
    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.
    16
    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
    17
    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.
    18
    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
    t
    ?r f t ¼ ðr t ?r f t Þ? ^ θ t?1 ðr USGOV
    t
    ?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
    19
    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
    t
    ¼
    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ffi
    ∑ sAmonth t ðΔTED s ?ΔTED t Þ 2
    q
    . 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.
    20
    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
    1
    P s
    t
    e ′ s Ωx n
    ¼ r f þψ t þγ
    1
    P s
    t
    cov t ðP s
    tþ1 þδ
    s
    tþ1 ; P tþ1 þδ tþ1
    ? ? ′x n Þ
    ¼ r f þψ t þγcov t ðr s
    tþ1 ;r
    M
    tþ1 ÞP
    t x
    n
    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
    M
    tþ1 ÞP
    t x
    n
    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
     
    ¼
    1
    β L
    t
     
    t
    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
    t
    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
    1
    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
    t
    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
    1
    ariance of ψ t so that
    ^ β L
    t oβ
     
    ^ β H
    t
    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
    M
    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
    t
    covðP i tþ1 þδ i tþ1 ;P M
    tþ1 þδ
    M
    tþ1 Þ
    P i t varðP M
    tþ1 þδ
    M
    appendix.htm
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