Betting Against Beta

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1 Being Agains Bea Andrea Frazzini and Lasse Heje Pedersen * This draf: May 10, 2013 Absrac. We presen a model wih leverage and margin consrains ha vary across invesors and ime. We find evidence consisen wih each of he model s five cenral predicions: (1) Since consrained invesors bid up high-bea asses, high bea is associaed wih low alpha, as we find empirically for U.S. equiies, 20 inernaional equiy markes, Treasury bonds, corporae bonds, and fuures; (2) A being-agains-bea (BAB) facor, which is long leveraged lowbea asses and shor high-bea asses, produces significan posiive risk-adjused reurns; (3) When funding consrains ighen, he reurn of he BAB facor is low; (4) Increased funding liquidiy risk compresses beas oward one; (5) More consrained invesors hold riskier asses. * Andrea Frazzini is a AQR Capial Managemen, Two Greenwich Plaza, Greenwich, CT 06830, e- mail: andrea.frazzini@aqr.com; web: hp:// Lasse H. Pedersen is a New York Universiy, Copenhagen Business School (FRIC Cener for Financial Fricions), AQR Capial Managemen, CEPR, and NBER, 44 Wes Fourh Sree, NY ; lpederse@sern.nyu.edu; web: hp:// We hank Cliff Asness, Aaron Brown, John Campbell, Josh Coval (discussan), Ken Daniel, Gene Fama, Nicolae Garleanu, John Heaon (discussan), Michael Kaz, Owen Lamon, Juhani Linnainmaa (discussan), Michael Mendelson, Mark Michell, Lubos Pasor (discussan), Ma Richardson, William Schwer (edior), Tuomo Vuoleenaho, Rober Whielaw and wo anonymous referees for helpful commens and discussions as well as seminar paricipans a AFA, NBER, Columbia Universiy, New York Universiy, Yale Universiy, Emory Universiy, Universiy of Chicago Booh, Kellogg School of Managemen, Harvard Universiy, Boson Universiy, Vienna Universiy of Economics and Business, Universiy of Mannheim, Goehe Universiy Frankfur, Uah Winer Finance Conference, Annual Managemen Conference a Universiy of Chicago Booh School of Business, Bank of America/Merrill Lynch Quan Conference and Nomura Global Quaniaive Invesmen Sraegies Conference. Pedersen graefully acknowledges suppor from he European Research Council (ERC gran no ). Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 1

2 A basic premise of he capial asse pricing model (CAPM) is ha all agens inves in he porfolio wih he highes expeced excess reurn per uni of risk (Sharpe raio), and leverage or de-leverage his porfolio o sui heir risk preferences. However, many invesors such as individuals, pension funds, and muual funds are consrained in he leverage ha hey can ake, and hey herefore overweigh risky securiies insead of using leverage. For insance, many muual fund families offer balanced funds where he normal fund may inves 40% in long-erm bonds and 60% in socks, whereas he aggressive fund invess 10% in bonds and 90% in socks. If he normal fund is efficien, hen an invesor could leverage i and achieve a beer rade-off beween risk and expeced reurn han he aggressive porfolio wih a large il owards socks. The demand for exchange-raded funds (ETFs) wih embedded leverage provides furher evidence ha many invesors canno use leverage direcly. This behavior of iling oward high-bea asses suggess ha risky high-bea asses require lower risk-adjused reurns han low-bea asses, which require leverage. Indeed, he securiy marke line for U.S. socks is oo fla relaive o he CAPM (Black, Jensen, and Scholes (1972)) and is beer explained by he CAPM wih resriced borrowing han he sandard CAPM (Black (1972, 1993), Brennan (1971), see Mehrling (2005) for an excellen hisorical perspecive). Several quesions arise: How can an unconsrained arbirageur exploi his effec, i.e., how do you be agains bea? Wha is he magniude of his anomaly relaive o he size, value, and momenum effecs? Is being agains bea rewarded in oher counries and asse classes? How does he reurn premium vary over ime and in he cross secion? Who bes agains bea? We address hese quesions by considering a dynamic model of leverage consrains and by presening consisen empirical evidence from 20 inernaional sock markes, Treasury bond markes, credi markes, and fuures markes. Our model feaures several ypes of agens. Some agens canno use leverage and herefore overweigh high-bea asses, causing hose asses o offer lower reurns. Oher agens can use leverage bu face margin consrains. They underweigh (or shor-sell) high-bea asses and buy low-bea asses ha hey lever up. The model Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 2

3 implies a flaer securiy marke line (as in Black (1972)), where he slope depends on he ighness (i.e., Lagrange muliplier) of he funding consrains on average across agens (Proposiion 1). One way o illusrae he asse-pricing effec of he funding fricion is o consider he reurns on marke-neural being agains bea (BAB) facors. A BAB facor is a porfolio ha holds low-bea asses, leveraged o a bea of 1, and ha shors high-bea asses, de-leveraged o a bea of 1. For insance, he BAB facor for U.S. socks achieves a zero bea by holding $1.4 of low-bea socks and shor-selling $0.7 of high-bea socks, wih offseing posiions in he risk-free asse o make i self-financing. 1 Our model predics ha BAB facors have a posiive average reurn and ha he reurn is increasing in he ex-ane ighness of consrains and in he spread in beas beween high- and low-bea securiies (Proposiion 2). When he leveraged agens hi heir margin consrain, hey mus de-leverage. Therefore, he model predics ha, during imes of ighening funding liquidiy consrains, he BAB facor realizes negaive reurns as is expeced fuure reurn rises (Proposiion 3). Furhermore, he model predics ha he beas of securiies in he cross secion are compressed oward 1 when funding liquidiy risk is high (Proposiion 4). Finally, he model implies ha more-consrained invesors overweigh high-bea asses in heir porfolios while less-consrained invesors overweigh low-bea asses and possibly apply leverage (Proposiion 5). Our model hus exends Black s (1972) cenral insigh by considering a broader se of consrains and deriving he dynamic ime-series and cross-secional properies arising from he equilibrium ineracion beween agens wih differen consrains. We find consisen evidence for each of he model s cenral predicions. To es Proposiion 1, we firs consider porfolios sored by bea wihin each asse class. We find ha alphas and Sharpe raios are almos monoonically declining in bea in each asse class. This finding provides broad evidence ha he relaive flaness of he 1 While we consider a variey of BAB facors wihin a number of markes, one noable example is he zero-covariance porfolio inroduced by Black (1972) and sudied for U.S. socks by Black, Jensen, and Scholes (1972), Kandel (1984), Shanken (1985), Polk, Thompson, and Vuoleenaho (2006), and ohers. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 3

4 securiy marke line is no isolaed o he U.S. sock marke bu ha i is a pervasive global phenomenon. Hence, his paern of required reurns is likely driven by a common economic cause, and our funding consrain model provides one such unified explanaion. To es Proposiion 2, we consruc BAB facors wihin he U.S. sock marke, and wihin each of he 19 oher developed MSCI sock markes. The U.S. BAB facor realizes a Sharpe raio of 0.78 beween 1926 and March To pu his BAB facor reurn in perspecive, noe ha is Sharpe raio is abou wice ha of he value effec and 40% higher han ha of momenum over he same ime period. The BAB facor has highly significan risk-adjused reurns, accouning for is realized exposure o marke, value, size, momenum, and liquidiy facors (i.e., significan 1-, 3-, 4-, and 5-facor alphas), and realizes a significan posiive reurn in each of he four 20-year subperiods beween 1926 and We find similar resuls in our sample of inernaional equiies; indeed, combining socks in each of he non-u.s. counries produces a BAB facor wih reurns abou as srong as he U.S. BAB facor. We show ha BAB reurns are consisen across counries, ime, wihin deciles sored by size, wihin deciles sored by idiosyncraic risk, and robus o a number of specificaions. These consisen resuls sugges ha coincidence or daamining are unlikely explanaions. However, if leverage consrains are he underlying drivers as in our model, hen he effec should also exis in oher markes. Hence, we examine BAB facors in oher major asse classes. For U.S. Treasuries, he BAB facor is a porfolio ha holds leveraged low-bea (i.e., shormauriy) bonds and shor-sells de-leveraged high-bea (i.e., long-erm) bonds. This porfolio produces highly significan risk-adjused reurns wih a Sharpe raio of This profiabiliy of shor-selling long-erm bonds may seem o conradic he well-known erm premium in fixed income markes. There is no paradox, however. The erm premium means ha invesors are compensaed on average for holding long-erm bonds raher han T-bills because of he need for mauriy ransformaion. The erm premium exiss a all horizons, however: Jus as invesors are compensaed for holding 10-year bonds over T-bills, hey are also compensaed for holding 1-year Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 4

5 bonds. Our finding is ha he compensaion per uni of risk is in fac larger for he 1-year bond han for he 10-year bond. Hence, a porfolio ha has a leveraged long posiion in 1-year (and oher shor-erm) bonds and a shor posiion in long-erm bonds produces posiive reurns. This resul is consisen wih our model in which some invesors are leverage-consrained in heir bond exposure and, herefore, require lower risk-adjused reurns for long-erm bonds ha give more bang for he buck. Indeed, shor-erm bonds require remendous leverage o achieve similar risk or reurn as long-erm bonds. These resuls complemen hose of Fama (1986) and Duffee (2010), who also consider Sharpe raios across mauriies implied by sandard erm srucure models. We find similar evidence in credi markes: A leveraged porfolio of highly raed corporae bonds ouperforms a de-leveraged porfolio of low-raed bonds. Similarly, using a BAB facor based on corporae bond indices by mauriy produces high risk-adjused reurns. We es he ime-series predicions of Proposiion 3 using he TED spread as a measure of funding condiions. Consisen wih he model, a higher TED spread is associaed wih low conemporaneous BAB reurns. The lagged TED spread predics reurns negaively, which is inconsisen wih he model if a high TED spread means a high ighness of invesors funding consrains. This resul could be explained if higher TED spreads mean ha invesors funding consrains would be ighening as heir banks reduce credi availabiliy over ime, hough his is speculaion. To es he predicion of Proposiion 4, we use he volailiy of he TED spread as an empirical proxy for funding liquidiy risk. Consisen wih he model s bea-compression predicion, we find ha he dispersion of beas is significanly lower when funding liquidiy risk is high. Lasly, we find evidence consisen wih he model s porfolio predicion ha more-consrained invesors hold higher-bea securiies han less-consrained invesors (Proposiion 5). On he one hand, we sudy he equiy porfolios of muual funds and individual invesors, which are likely o be consrained. Consisen wih he model, we find ha hese invesors hold porfolios wih average beas above 1. On he oher side of he marke, we find ha leveraged buyou (LBO) funds acquire Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 5

6 firms wih average beas below 1 and apply leverage. Similarly, looking a he holdings of Berkshire Hahaway, we see ha Warren Buffe bes agains bea by buying low-bea socks and applying leverage. Our resuls shed new ligh on he relaionship beween risk and expeced reurns. This cenral issue in financial economics has naurally received much aenion. The sandard CAPM bea canno explain he cross-secion of uncondiional sock reurns (Fama and French (1992)) or condiional sock reurns (Lewellen and Nagel (2006)). Socks wih high bea have been found o deliver low risk-adjused reurns (Black, Jensen, and Scholes (1972), Baker, Bradley, and Wurgler (2010)); hus, he consrained-borrowing CAPM has a beer fi (Gibbons (1982), Kandel (1984), Shanken (1985)). Socks wih high idiosyncraic volailiy have realized low reurns (Falkensein (1994), Ang, Hodrick, Xing, Zhang (2006, 2009)), 2 bu we find ha he bea effec holds even when conrolling for idiosyncraic risk. Theoreically, asse pricing models wih benchmarked managers (Brennan (1993)) or consrains imply more general CAPM-like relaions (Hindy (1995), Cuoco (1997)), in paricular he margin-capm implies ha high-margin asses have higher required reurns, especially during imes of funding illiquidiy (Garleanu and Pedersen (2009), Ashcraf, Garleanu, and Pedersen (2010)). Garleanu and Pedersen (2009) show empirically ha deviaions of he Law of One Price arises when highmargin asses become cheaper han low-margin asses, and Ashcraf, Garleanu, and Pedersen (2010) find ha prices increase when cenral bank lending faciliies reduce margins. Furhermore, funding liquidiy risk is linked o marke liquidiy risk (Gromb and Vayanos (2002), Brunnermeier and Pedersen (2010)), which also affecs required reurns (Acharya and Pedersen (2005)). We complemen he lieraure by deriving new cross-secional and ime-series predicions in a simple dynamic model ha capures leverage and margin consrains and by esing is implicaions across a broad cross secion of securiies across all he major asse classes. Finally, Asness, Frazzini, and Pedersen (2011) repor evidence of a low-bea effec across asse classes consisen wih our heory. 2 This effec disappears when conrolling for he maximum daily reurn over he pas monh (Bali, Cakici, and Whielaw (2010)) and when using oher measures of idiosyncraic volailiy (Fu (2009)). Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 6

7 The res of he paper is organized as follows: Secion I lays ou he heory, Secion II describes our daa and empirical mehodology, Secions III-VI es Proposiions 1-5, and Secion VII concludes. Appendix A conains all proofs, Appendix B provides a number of addiional empirical resuls and robusness ess, and Appendix C provides a calibraion of he model. The calibraion shows ha, o mach he srong BAB performance in he daa, a large fracion of agens mus face severe consrains. An ineresing opic for fuure research is o empirically esimae agens leverage consrains and risk preferences and sudy wheher he magniude of he BAB reurns is consisen wih he model or should be viewed as a puzzle. I. Theory We consider an overlapping-generaions (OLG) economy in which agens i=1,...,i are born each ime period wih wealh rade securiies s=1,...,s, where securiy s pays dividends i W and live for wo periods. Agens s and has x * s shares ousanding. 3 Each ime period, young agens choose a porfolio of shares x=(x 1,...,x S ), invesing he res of heir wealh a he risk-free reurn r f, o maximize heir uiliy: i f max x'( E P 1 1 (1 r ) P ) x' x (1) 2 where P is he vecor of prices a ime, Ω is he variance-covariance marix of P 1 1 consrain:, and γ i is agen i s risk aversion. Agen i is subjec o he following porfolio (2) m x P W i s s i s 3 The dividends and shares ousanding are aken as exogenous. We noe ha our modified CAPM has implicaions for a corporaion s opimal capial srucure, which suggess an ineresing avenue of fuure research beyond he scope his paper. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 7

8 This consrain requires ha some muliple i m of he oal dollars invesed he sum of he number of shares x s imes heir prices P s mus be less han he agen s wealh. The invesmen consrain depends on he agen i. For insance, some agens simply canno use leverage, which is capured by m i =1 (as Black (1972) assumes). Oher agens no only may be precluded from using leverage bu also mus have some of heir wealh in cash, which is capured by m i greaer han 1. For insance, m i = 1/(1-0.20)=1.25 represens an agen who mus hold 20% of her wealh in cash. For insance, a muual fund may need some ready cash o be able o mee daily redempions, an insurance company needs o pay claims, and individual invesors may need cash for unforeseen expenses. Oher agens ye may be able o use leverage bu may face margin consrains. For insance, if an agen faces a margin requiremen of 50%, hen his m i is Wih his margin requiremen, he agen can inves in asses worh wice his wealh a mos. A smaller margin requiremen m i naurally means ha he agen can ake greaer posiions. We noe ha our formulaion assumes for simpliciy ha all securiies have he same margin requiremen, which may be rue when comparing securiies wihin he same asse class (e.g., socks), as we do empirically. Garleanu and Pedersen (2009) and Ashcraf, Garleanu, and Pedersen (2010) consider asses wih differen margin requiremens and show heoreically and empirically ha higher margin requiremens are associaed wih higher required reurns (Margin CAPM). We are ineresed in he properies of he compeiive equilibrium in which he oal demand equals he supply: i x x* (3) i To derive equilibrium, consider he firs order condiion for agen i: 0 E P (1 r ) P x P (4) f i i i 1 1 Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 8

9 where ψ i is he Lagrange muliplier of he porfolio consrain. Solving for x i gives he opimal posiion: i 1 1 f i x E P r i P (5) The equilibrium condiion now follows from summing over hese posiions: 1 1 f x* E P r P (6) where he aggregae risk aversion γ is defined by 1/ γ = Σ i 1/ γ i, and i i is i he weighed average Lagrange muliplier. (The coefficiens i sum o 1 by definiion of he aggregae risk aversion.) The equilibrium price can hen be compued: P E P x 1 1 * f 1r (7) Translaing his ino he reurn of any securiy M he marke 1 i i i i r 1 P 1 1 / P 1, he reurn on r, and using he usual expression for bea, s cov s 1, M 1 / var M r r r 1, we obain he following resuls. (All proofs are in Appendix A, which also illusraes he porfolio choice wih leverage consrains in a mean-sandard deviaion diagram.) Proposiion 1 (high bea is low alpha). (i) The equilibrium required reurn for any securiy s is: s f s E r 1 r (8) Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 9

10 where he risk premium is E r M 1 r and is he average Lagrange f muliplier, measuring he ighness of funding consrains. s s (ii) A securiy s alpha wih respec o he marke is (1 ). The alpha decreases in he bea, s. (iii) For an efficien porfolio, he Sharpe raio is highes for an efficien porfolio wih a bea less han 1 and decreases in beas. s for higher beas and increases for lower i As in Black s CAPM wih resriced borrowing (in which m 1 for all agens), he required reurn is a consan plus bea imes a risk premium. Our expression shows explicily how risk premia are affeced by he ighness of agens porfolio consrains, as measured by he average Lagrange muliplier. Indeed, igher porfolio consrains (i.e., a larger ) flaen he securiy marke line by increasing he inercep and decreasing he slope. Whereas he sandard CAPM implies ha he inercep of he securiy marke line is r f, he inercep here is increased by binding funding consrains (hrough he weighed average of he agens Lagrange mulipliers). One may wonder why zero-bea asses require reurns in excess of he risk-free rae. The answer has wo pars: Firs, consrained agens prefer o inves heir limied capial in riskier asses wih higher expeced reurn. Second, unconsrained agens do inves considerable amouns in zero-bea asses so, from heir perspecive, he risk of hese asses is no idiosyncraic, as addiional exposure o such asses would increase he risk of heir porfolio. Hence, in equilibrium, zero-bea risky asses mus offer higher reurns han he risk-free rae. Asses ha have zero covariance o Tobin s (1958) angency porfolio held by an unconsrained agen do earn he risk-free rae, bu he angency porfolio is no he marke porfolio in our equilibrium. Indeed, he marke porfolio is he weighed average of all invesors porfolios, i.e., an average of he angency porfolio Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 10

11 held by unconsrained invesors and riskier porfolios held by consrained invesors. Hence, he marke porfolio has higher risk and expeced reurn han he angency porfolio, bu a lower Sharpe raio. The porfolio consrains furher imply a lower slope of he securiy marke line, i.e., a lower compensaion for a marginal increase in sysemaic risk. The slope is lower because consrained agens need high unleveraged reurns and are herefore willing o accep less compensaion for higher risk. 4 We nex consider he properies of a facor ha goes long low-bea asses and shor-sells high-bea asses. To consruc such a facor, le w be he relaive L L porfolio weighs for a porfolio of low-bea asses wih reurn r 1 wl ' r and 1 consider similarly a porfolio of high-bea asses wih reurn r H 1. The beas of hese porfolios are denoed and L L H, where. We hen consruc a beingagains-bea (BAB) facor as: H 1 1 r r r r r (9) H BAB L f H f 1 L 1 1 This porfolio is marke neural, ha is, i has a bea of zero: he long side has been leveraged o a bea of 1, and he shor side has been de-leveraged o a bea of 1. Furhermore, he BAB facor provides he excess reurn on a self-financing porfolio, such as HML and SMB, since i is a difference beween excess reurns. The difference is ha BAB is no dollar-neural in erms of only he risky securiies since his would no produce a bea of zero. 5 The model has several predicions regarding he 4 While he risk premium implied by our heory is lower han he one implied by he CAPM, i is sill posiive. I is difficul o empirically esimae a low risk premium and is posiiviy is no a focus of our empirical ess as i does no disinguish our heory from he sandard CAPM. We noe, however, ha he daa is no inconsisen wih our predicion as he esimaed risk premium is posiive and insignifican for U.S. socks, negaive and insignifican for Inernaional socks, posiive and insignifican for Treasuries, posiive and significan for credis across mauriies, and posiive and significan across asse classes. 5 A naural BAB facor is he zero-covariance porfolio of Black (1972) and Black, Jensen, and Scholes (1972). We consider a broader class of BAB porfolios since we empirically consider a variey of BAB porfolios wihin various asse classes ha are subses of all securiies (e.g., socks in a paricular size group). Therefore, our consrucion achieves marke neuraliy by leveraging (and deleveraging) he long and shor sides raher han adding he marke iself as Black, Jensen, and Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 11

12 BAB facor: Proposiion 2 (posiive expeced reurn of BAB). The expeced excess reurn of he self-financing BAB facor is posiive E (10) H L BAB r 1 0 L H H L and increasing in he ex-ane bea spread and funding ighness L H. This proposiion shows ha a marke-neural BAB porfolio ha is long leveraged low-bea securiies and shor higher-bea securiies earns a posiive expeced reurn on average. The size of he expeced reurn depends on he spread in he beas and how binding he porfolio consrains are in he marke, as capured by he average of he Lagrange mulipliers. The nex proposiion considers he effec of a shock o he porfolio consrains (or margin requiremens), m k, which can be inerpreed as a worsening of funding liquidiy, a credi crisis in he exreme. Such a funding liquidiy shock resuls in losses for he BAB facor as is required reurn increases. This happens because agens may need o de-leverage heir bes agains bea or srech even furher o buy he high-bea asses. Thus, he BAB facor is exposed o funding liquidiy risk, as i loses when porfolio consrains become more binding. Proposiion 3 (funding shocks and BAB reurns). k A igher porfolio consrain, ha is, an increase in m for some of k, leads o a conemporaneous loss for he BAB facor r BAB k m 0 (11) Scholes (1972) do. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 12

13 and an increase in is fuure required reurn: E r m BAB 1 k 0 (12) Funding shocks have furher implicaions for he cross secion of asse reurns and he BAB porfolio. Specifically, a funding shock makes all securiy prices drop ogeher (ha is, is he same for all securiies s). Therefore, an increased funding risk compresses beas owards one. 6 If he BAB porfolio consrucion is based on an informaion se ha does no accoun for his increased funding risk, hen he BAB porfolio s condiional marke bea is affeced. Proposiion 4 (bea compression). Suppose ha all random variables are i.i.d. over ime and is independen of he oher random variables. Furher, a ime -1 afer he BAB porfolio is formed and prices are se, he condiional variance of he f discoun facor 1/ (1 r ) rises (falls) due o new informaion abou and. Then: i (i) The condiional reurn beas 1 of all securiies are compressed oward 1 (more dispersed). (ii) The condiional bea of he BAB porfolio becomes posiive (negaive), even hough i is marke neural relaive o he informaion se used for porfolio formaion. In addiion o he asse-pricing predicions ha we have derived, funding consrains naurally affec agens porfolio choices. In paricular, more-consrained 6 Garleanu and Pedersen (2009) find a complemenary resul, sudying securiies wih idenical fundamenal risk bu differen margin requiremens. They find heoreically and empirically ha such asses have similar beas when liquidiy is good, bu when funding liquidiy risk rises he high-margin securiies have larger beas, as heir high margins make hem more funding sensiive. Here, we sudy securiies wih differen fundamenal risk, bu he same margin requiremens. In his case, higher funding liquidiy risk means ha beas are compressed oward one. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 13

14 invesors il oward riskier securiies in equilibrium whereas less-consrained agens il oward safer securiies wih higher reward per uni of risk. To sae his resul, we wrie nex period s securiy payoffs as M M M M P E P b P E P e (13) where b is a vecor of marke exposures, and e is a vecor of noise ha is uncorrelaed wih he marke. We have he following naural resul for he agens posiions: Proposiion 5 (consrained invesors hold high beas). Unconsrained agens hold a porfolio of risky securiies ha has a bea less han 1; consrained agens hold porfolios of securiies wih higher beas. If securiies s and k s k are idenical excep ha s has a larger marke exposure han k, b b, hen any consrained agen j wih greaer-han-average Lagrange muliplier, j, holds j more shares of s han k; he reverse is rue for any agen wih. We nex provide empirical evidence for Proposiions 1-5. Beyond maching he daa qualiaively, Appendix C illusraes how well a calibraed model can quaniaively mach he magniude of he esimaed BAB reurns. II. Daa and Mehodology The daa in his sudy are colleced from several sources. The sample of U.S. and inernaional equiies includes 55,600 socks covering 20 counries, and he summary saisics for socks are repored in Table I. Sock reurn daa are from he union of he CRSP ape and he Xpressfeed Global daabase. Our U.S. equiy daa include all available common socks on CRSP beween January 1926 and March 2012, and beas are compued wih respec o he CRSP value-weighed marke index. Excess reurns are above he U.S. Treasury bill rae. We consider alphas wih respec o he marke facor and facor reurns based on size (SMB), book-o-marke Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 14

15 (HML), momenum (UMD), and (when available) liquidiy risk. 7 The inernaional equiy daa include all available common socks on he Xpressfeed Global daily securiy file for 19 markes belonging o he MSCI developed universe beween January 1989 and March We assign each sock o is corresponding marke based on he locaion of he primary exchange. Beas are compued wih respec o he corresponding MSCI local marke index. 8 All reurns are in USD, and excess reurns are above he U.S. Treasury bill rae. We compue alphas wih respec o he inernaional marke and facor reurns based on size (SMB), book-o-marke (HML) and momenum (UMD) from Asness and Frazzini (2011) 9 and (when available) liquidiy risk. We also consider a variey of oher asses: Table II conains he lis of insrumens and he corresponding ranges of available daa. We obain U.S. Treasury bond daa from he CRSP U.S. Treasury Daabase, using monhly reurns (in excess of he 1-monh Treasury bill) on he Fama Bond porfolios for mauriies ranging from 1 o 10 years beween January 1952 and March Each porfolio reurn is an equal-weighed average of he unadjused holding period reurn for each bond in he porfolio. Only non-callable, non-flower noes and bonds are included in he porfolios. Beas are compued wih respec o an equally weighed porfolio of all bonds in he daabase. We collec aggregae corporae bond index reurns from Barclays Capial s Bond.Hub daabase. 10 Our analysis focuses on he monhly reurns (in excess of he 1-monh Treasury bill) of four aggregae U.S. credi indices wih mauriy ranging from 1 o 10 years and nine invesmen-grade and high-yield corporae bond porfolios wih credi risk ranging from AAA o Ca-D and Disressed. 11 The daa cover he period beween January 1973 and March 2012, alhough he daa 7 SMB, HML, and UMD are from Ken French s daa library, and he liquidiy risk facor is from WRDS. 8 Our resuls are robus o he choice of benchmark (local vs. global). We repor hese ess in he Appendix. 9 These facors mimic heir U.S counerpars and follow Fama and French (1992, 1993, 1996). See Asness and Frazzini (2011) for a deailed descripion of heir consrucion. The daa can be downloaded a hp:// 10 The daa can be downloaded a hps://live.barcap.com. 11 The disress index was provided o us by Credi Suisse. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 15

16 availabiliy varies depending on he individual bond series. Beas are compued wih respec o an equally weighed porfolio of all bonds in he daabase. We also sudy fuures and forwards on counry equiy indexes, counry bond indexes, foreign exchange, and commodiies. Reurn daa are drawn from he inernal pricing daa mainained by AQR Capial Managemen LLC. The daa are colleced from a variey of sources and conains daily reurn on fuures, forwards, or swap conracs in excess of he relevan financing rae. The ype of conrac for each asse depends on availabiliy or he relaive liquidiy of differen insrumens. Prior o expiraion, posiions are rolled over ino he nex mos-liquid conrac. The rolling dae s convenion differs across conracs and depends on he relaive liquidiy of differen mauriies. The daa cover he period beween January 1963 and March 2012, wih varying daa availabiliy depending on he asse class. For more deails on he compuaion of reurns and daa sources, see Moskowiz, Ooi, and Pedersen (2012), Appendix A. For equiy indexes, counry bonds, and currencies, he beas are compued wih respec o a GDP-weighed porfolio, and for commodiies, he beas are compued wih respec o a diversified porfolio ha gives equal risk weigh across commodiies. Finally, we use he TED spread as a proxy for ime periods where credi consrain are more likely o be binding (as in Garleanu and Pedersen (2011) and ohers). The TED spread is defined as he difference beween he hree-monh EuroDollar LIBOR rae and he hree-monh U.S. Treasuries rae. Our TED daa run from December 1984 o March Esimaing Ex-ane Beas We esimae pre-ranking beas from rolling regressions of excess reurns on marke excess reurns. Whenever possible, we use daily daa raher han monhly as he accuracy of covariance esimaion improves wih he sample frequency (Meron (1980)). 12 Our esimaed bea for securiy is given by 12 Daily reurns are no available for our sample of U.S. Treasury bonds, U.S. corporae bonds, and U.S. credi indices. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 16

17 (14) where and are he esimaed volailiies for he sock and he marke and is heir correlaion. We esimae volailiies and correlaions separaely for wo reasons. Firs, we use a 1-year rolling sandard deviaion for volailiies and a 5-year horizon for he correlaion o accoun for he fac ha ha correlaions appear o move more slowly han volailiies. 13 and overlapping 3-day log reurns, Second, we use 1-day log reurns o esimae volailiies r 3d i, 2 k0 i ln(1 ), for correlaion o conrol for non-synchronous rading (which obviously only affecs correlaions). We require a leas 6 monhs (120 rading days) of non-missing daa o esimae volailiies and a leas 3 years (750 rading days) of non-missing reurn daa for correlaions. If we only have access o monhly daa, we use rolling 1 and 5-year windows and require a leas 12 and 36 observaions. Finally, o reduce he influence of ouliers, we follow Vasicek (1973) and Elon, Gruber, Brown, and Goezmann (2003) and shrink he ime-series esimae of XS bea ( ) oward he cross-secional mean ( ): TS i r k ˆ ˆTS w (1 w) ˆ i i i i XS (15) For simpliciy, raher han having asse-specific and ime-varying shrinkage facors as in Vasicek (1973), we se w = 0.6 and XS bu our resuls are very similar eiher way. 14 =1 for all periods and across all asses, We noe ha our choice of he shrinkage facor does no affec how securiies are sored ino porfolios since he common shrinkage does no change he ranks of he securiy beas. However, he amoun of shrinkage affecs he consrucion of he 13 See, for example, De Sanis and Gerard (1997). 14 The Vasicek (1973) Bayesian shrinkage facor is given by 2 w 1 / ( ) where is he i i, TS i, TS XS variance of he esimaed bea for securiy i, and 2 XS is he cross-secional variance of beas. This esimaor places more weigh on he hisorical imes series esimae when he esimae has a lower variance or when here is large dispersion of beas in he cross secion. Pooling across all socks in our U.S. equiy daa, he shrinkage facor w has a mean of i, TS Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 17

18 BAB porfolios since he esimaed beas are used o scale he long and shor sides of porfolio as seen in Equaion (9). To accoun for he fac ha noise in he ex-ane beas affecs he consrucion of he BAB facors, our inference is focused on realized abnormal reurns so ha any mismach beween ex-ane and (ex-pos) realized beas is picked up by he realized loadings in he facor regression. Of course, when we regress our porfolios on sandard risk facors, he realized facor loadings are no shrunk as above since only he ex-ane beas are subjec o selecion bias. Our resuls are robus o alernaive bea esimaion procedures as we repor in he Appendix. We compue beas wih respec o a marke porfolio, which is eiher specific o an asse class or he overall world marke porfolio of all asses. While our resuls hold boh ways, we focus on beas wih respec o asse-class-specific marke porfolios since hese beas are less noisy for several reasons. Firs, his approach allows us o use daily daa over a long ime period for mos asse classes, as opposed o using he mos diversified marke porfolio for which we only have monhly daa over a limied ime period. Second, his approach is applicable even if markes are segmened. As a robusness es, Table B8 in he Appendix repors resuls when we compue beas wih respec o a proxy for a world marke porfolio comprised of many asse classes. We use he world marke porfolio from Asness, Frazzini, and Pedersen (2011). 15 The resuls are consisen wih our main ess as he BAB facors earn large and significan abnormal reurns in each of asse classes in our sample. Consrucing Being-Agains-Bea Facors We consruc simple porfolios ha are long low-bea securiies and ha shor-sell high-bea securiies, hereafer BAB facors. To consruc each BAB facor, all securiies in an asse class are ranked in ascending order on he basis of heir esimaed bea. The ranked securiies are assigned o one of wo porfolios: lowbea and high-bea. The low (high) bea porfolio is comprised of all socks wih a 15 See Asness, Frazzini, and Pedersen (2011) for a deailed descripion of his marke porfolio. The marke series is monhly and ranges from 1973 o Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 18

19 bea below (above) is asse-class median (or counry median for inernaional equiies). In each porfolio, securiies are weighed by he ranked beas (i.e., lowerbea securiies have larger weighs in he low-bea porfolio and higher-bea securiies have larger weighs in he high-bea porfolio). The porfolios are rebalanced every calendar monh. More formally, le z be he n 1 vecor of bea ranks z rank ( ) a i i porfolio formaion, and le z z n be he average rank, where n is he number ' 1 n / of securiies and 1 n is an n 1 vecor of ones. The porfolio weighs of he low-bea and high-bea porfolios are given by ( ) ( ) (16) where k is a normalizing consan and x and x indicae he posiive and negaive elemens of a vecor x. Noe ha by consrucion we have ' 1nwH 1 and 1 ' w 1. To consruc he BAB facor, boh porfolios are rescaled o n L have a bea of one a porfolio formaion. The BAB is he self-financing zero-bea porfolio (8) ha is long he low-bea porfolio and ha shor-sells he high-bea porfolio. 1 1 r r r r r (17) H BAB L f H f 1 L 1 1 where r r w, r r w, w, and L ' 1 1 L H ' 1 1 H L ' L w. H ' H For example, on average, he U.S. sock BAB facor is long $1.4 of low-bea socks (financed by shor-selling $1.4 of risk-free securiies) and shor-sells $0.7 of high-bea socks (wih $0.7 earning he risk-free rae). Daa Used o Tes he Theory s Porfolio Predicions We collec muual fund holdings from he union of he CRSP Muual Fund Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 19

20 Daabase and Thompson Financial CDA/Specrum holdings daabase, which includes all regisered domesic muual funds filing wih he SEC. The holdings daa run from March 1980 o March We focus our analysis on open-end, acively managed, domesic equiy muual funds. Our sample selecion procedure follows ha of Kacperzczyk, Sialm, and Zheng (2008), and we refer o heir Appendix for deails abou he screens ha were used and summary saisics of he daa. Our individual invesors holdings daa was colleced from a naionwide discoun brokerage house and conains rades made by abou 78,000 households in he period from January of 1991 o November of This daase has been used exensively in he exising lieraure on individual invesors. For a deailed descripion of he brokerage daa se, see Barber and Odean (2000). Our sample of buyous is drawn from he M&A and corporae evens daabase mainained by AQR/CNH Parners. 16 The daa conain various daa iems, including iniial, subsequen announcemen daes, and (if applicable) compleion or erminaion dae for all akeover deals where he arge is a U.S. publicly raded firm and where he acquirer is a privae company. For some (bu no all) deals, he acquirer descripor also conains informaion on wheher he deal is a Leveraged or Managemen Buyou (LBO, MBO). The daa run from January 1963 o March Finally, we download holdings daa for Berkshire Hahaway from Thomson Financial Insiuional (13f) Holding Daabase. The daa run from March 1980 o March III. Being Agains Bea in Each Asse Class We now es how he required reurn varies in he cross-secion of bea-sored securiies (Proposiion 1) and he hypohesis ha long/shor BAB facors have posiive average reurns (Proposiion 2). As an overview of hese resuls, he alphas of all he bea-sored porfolios considered in his paper are ploed in Figure 1. We see ha declining alphas across bea-sored porfolios are general phenomena across asse classes. (Figure B1 in he Appendix plos he Sharpe raios of bea-sored porfolios, which also shows a consisenly declining paern.) 16 We would like o hank Mark Michell for providing us wih his daa. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 20

21 Figure 2 plos he annualized Sharpe raios of he BAB porfolios in he various asse classes. We see all he BAB porfolios deliver posiive reurns, excep for a small insignificanly negaive reurn in Ausrian socks. The BAB porfolios based on large numbers of securiies (U.S. socks, Inernaional socks, Treasuries, credis) deliver high risk-adjused reurns relaive o he sandard risk facors considered in he lieraure. We discuss hese resuls in deail below. Socks Table III repors our ess for U.S. socks. We consider 10 bea-sored porfolios and repor heir average reurns, alphas, marke beas, volailiies, and Sharpe raios. The average reurns of he differen bea porfolios are similar, which is he well-known relaively fla securiy marke line. Hence, consisen wih Proposiion 1 and wih Black (1972), he alphas decline almos monoonically from he low-bea o high-bea porfolios. Indeed, he alphas decline when esimaed relaive o a 1-, 3-, 4-, and 5-facor model. Moreover, Sharpe raios decline monoonically from low-bea o high-bea porfolios. The righmos column of Table III repors reurns of he being-agains-bea (BAB) facor, i.e., a porfolio ha is long leveraged low-bea socks and ha shorsells de-leveraged high-bea socks, hus mainaining a bea-neural porfolio. Consisen wih Proposiion 2, he BAB facor delivers a high average reurn and a high alpha. Specifically, he BAB facor has Fama and French (1993) abnormal reurns of 0.73% per monh (-saisic = 7.39). Furher adjusing reurns for Carhar s (1997) momenum-facor, he BAB porfolio earns abnormal reurns of 0.55% per monh (-saisic = 5.59). Las, we adjus reurns using a 5-facor model by adding he raded liquidiy facor by Pasor and Sambaugh (2003), yielding an abnormal BAB reurn of 0.55% per monh (-saisic = 4.09, which is lower in par because he liquidiy facor is only available during half of our sample). We noe ha while he alpha of he long-shor porfolio is consisen across regressions, he choice of risk adjusmen influences he relaive alpha conribuion of he long and shor sides of he porfolio. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 21

22 Our resuls for U.S. equiies show how he securiy marke line has coninued o be oo fla for anoher four decades afer Black, Jensen, and Scholes (1972). Furher, our resuls exend inernaionally. We consider bea-sored porfolios for inernaional equiies and laer urn o alogeher differen asse classes. We use all 19 MSCI developed counries excep he U.S. (o keep he resuls separae from he U.S. resuls above), and we do his in wo ways: We consider inernaional porfolios where all inernaional socks are pooled ogeher (Table IV), and we consider resuls separaely for each counry (Table V). The inernaional porfolio is counry neural, i.e., he low (high) bea porfolio is comprised of all socks wih a bea below (above) is counry median. 17 The resuls for our pooled sample of inernaional equiies in Table IV mimic he U.S. resuls: he alpha and Sharpe raios of he bea-sored porfolios decline (alhough no perfecly monoonically) wih he beas, and he BAB facor earns risk-adjused reurns beween 0.28% and 0.64% per monh depending on he choice of risk adjusmen, wih -saisics ranging from 2.09 o Table V shows he performance of he BAB facor wihin each individual counry. The BAB delivers posiive Sharpe raios in 18 of he 19 MSCI developed counries and posiive 4-facor alphas in 13 ou of 19, displaying a srikingly consisen paern across equiy markes. The BAB reurns are saisically significanly posiive in 6 counries, while none of he negaive alphas is significan. Of course, he small number of socks in our sample in many of he counries makes i difficul o rejec he null hypohesis of zero reurn in each individual counry. Table B1 in he Appendix repors facor loadings. On average, he U.S. BAB facor goes long $1.40 ($1.40 for Inernaional BAB) and shor-sells $0.70 ($0.89 for Inernaional BAB). The larger long invesmen is mean o make he BAB facor marke-neural because he socks ha are held long have lower beas. The BAB facor s realized marke loading is no exacly zero, reflecing he fac ha our exane beas are measured wih noise. The oher facor loadings indicae ha, relaive o high-bea socks, low-bea socks are likely o be larger, have higher book-o- 17 We keep he inernaional porfolio counry neural because we repor he resul of being agains bea across equiy indices BAB separaely in Table VIII. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 22

23 marke raios, and have higher reurn over he prior 12 monhs, alhough none of he loadings can explain he large and significan abnormal reurns. The BAB porfolio s posiive HML loading is naural since our heory predics ha low-bea socks are cheap and high-bea socks are expensive. The Appendix repors furher ess and addiional robusness checks. In Table B2, we repor resuls using differen window lenghs o esimae beas and differen benchmarks (local, global). We spli he sample by size (Table B3) and ime periods (Table B4), we conrol for idiosyncraic volailiy (Table B5) and repor resuls for alernaive definiion of he risk-free rae (B6). Finally, in Table B7 and Figure B2 we repor an ou-of-sample es. We collec pricing daa from DaaSream and for each counry in Table I we compue a BAB porfolio over sample period no covered by he Xpressfeed Global daabase. 18 All of he resuls are consisen: equiy porfolios ha be agains beas earn significan risk-adjused reurns. Treasury Bonds Table VI repors resuls for U.S. Treasury bonds. As before, we repor average excess reurns of bond porfolios formed by soring on bea in he previous monh. In he cross secion of Treasury bonds, ranking on beas wih respec o an aggregae Treasury bond index is empirically equivalen o ranking on duraion or mauriy. Therefore, in Table VI, one can hink of he erm bea, duraion, or mauriy in an inerchangeable fashion. The righmos column repors reurns of he BAB facor. Abnormal reurns are compued wih respec o a one-facor model where alpha is he inercep in a regression of monhly excess reurn on an equally weighed Treasury bond excess marke reurn. The resuls show ha he phenomenon of a flaer securiy marke line han prediced by he sandard CAPM is no limied o he cross secion of sock reurns. Indeed, consisen wih Proposiion 1, he alphas decline monoonically wih bea. Likewise, Sharpe raios decline monoonically from 0.73 for low-bea (shor mauriy) bonds o 0.31 for high-bea (long mauriy) bonds. Furhermore, he bond 18 DaaSream inernaional pricing daa sar in 1969 while Xpressfeed Global coverage sars in Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 23

24 BAB porfolio delivers abnormal reurns of 0.17% per monh (-saisic = 6.26) wih a large annual Sharpe raio of Since he idea ha funding consrains have a significan effec on he erm srucure of ineres may be surprising, le us illusrae he economic mechanism ha may be a work. Suppose an agen, e.g., a pension fund, has $1 o allocae o Treasuries wih a arge excess reurn of 2.9% per year. One way o achieve his reurn arge is o inves $1 in a porfolio of Treasuries wih mauriy above 10 years as seen in Table VI, P7. If he agen invess in 1-year Treasuries (P1) insead, hen he would need o inves $11 if all mauriies had he same Sharpe raio. This higher leverage is needed because he long-erm Treasures are 11 imes more volaile han he shor-erm Treasuries. Hence, he agen would need o borrow an addiional $10 o lever his invesmen in 1-year bonds. If he agen has leverage limis (or prefers lower leverage), hen he would sricly prefer he 10-year Treasuries in his case. According o our heory, he 1-year Treasuries herefore mus offer higher reurns and higher Sharpe raios, flaening he securiy marke line for bonds. Empirically, shor-erm Treasuries do in fac offer higher risk-adjused reurns so he reurn arge can be achieved by invesing abou $5 in 1-year bonds. While a consrained invesor may sill prefer an un-leveraged invesmen in 10-year bonds, unconsrained invesors now prefer he leveraged low-bea bonds, and he marke can clear. While he severiy of leverage consrains varies across marke paricipans, i appears plausible ha a 5-o-1 leverage (on his par of he porfolio) makes a difference for some large invesors such as pension funds. Credi We nex es our model using several credi porfolios and repor resuls in Table VII. In Panel A, columns (1) o (5), he es asses are monhly excess reurns of corporae bond indexes by mauriy. We see ha he credi BAB porfolio delivers abnormal reurns of 0.11% per monh (-saisic = 5.14) wih a large annual Sharpe raio of Furhermore, alphas and Sharpe raios decline monoonically. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 24

25 In columns (6) o (10), we aemp o isolae he credi componen by hedging away he ineres rae risk. Given he resuls on Treasuries in Table VI, we are ineresed in esing a pure credi version of he BAB porfolio. Each calendar monh, we run 1-year rolling regressions of excess bond reurns on he excess reurn on Barclay s U.S. governmen bond index. We consruc es asses by going long he corporae bond index and hedging his posiion by shor-selling he appropriae CDS f f amoun of he governmen bond index: ( ) ˆ USGOV f r r r r 1( r r ), where is he slope coefficien esimaed in an expanding regression using daa from he ˆ 1 beginning of he sample and up o monh -1. One inerpreaion of his reurns series is ha i approximaes he reurns on a Credi Defaul Swap (CDS). We compue marke reurns by aking he equally weighed average of hese hedged reurns, and we compue beas and BAB porfolios as before. Abnormal reurns are compued wih respec o a wo-facor model where alpha is he inercep in a regression of monhly excess reurn on he equally weighed average pseudo-cds excess reurn and he monhly reurn on he Treasury BAB facor. The addiion of he Treasury BAB facor on he righ-hand side is an exra check o es a pure credi version of he BAB porfolio. The resuls in Panel A of Table VII columns (6) o (10) ell he same sory as columns (1) o (5): he BAB porfolio delivers significan abnormal reurns of 0.17% per monh (-saisics = 4.44) and Sharpe raios decline monoonically from lowbea o high-bea asses. Las, in Panel B of Table VII, we repor resuls where he es asses are credi indexes sored by raing, ranging from AAA o Ca-D and Disressed. Consisen wih all our previous resuls, we find large abnormal reurns of he BAB porfolios (0.57% per monh wih a -saisics = 3.72) and declining alphas and Sharpe raios across bea-sored porfolios. Equiy Indexes, Counry Bond Indexes, Currencies, and Commodiies Table VII repors resuls for equiy indexes, counry bond indexes, foreign exchange and commodiies. The BAB porfolio delivers posiive reurns in each of he four asse classes, wih an annualized Sharpe raio ranging from 0.11 o We Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 25

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