Betting Against Beta

Transcription

1 Beng Agans Bea Andrea Frazzn and Lasse Heje Pedersen * Ths draf: February 4, 2013 Absrac. We presen a model wh leverage and margn consrans ha vary across nvesors and me. We fnd evdence conssen wh each of he model s fve cenral predcons: (1) Snce consraned nvesors bd up hgh-bea asses, hgh bea s assocaed wh low alpha, as we fnd emprcally for U.S. eques, 20 nernaonal equy markes, Treasury bonds, corporae bonds, and fuures; (2) A beng-agans-bea (BAB) facor, whch s long leveraged lowbea asses and shor hgh-bea asses, produces sgnfcan posve rsk-adjused reurns; (3) When fundng consrans ghen, he reurn of he BAB facor s low; (4) Increased fundng lqudy rsk compresses beas oward one; (5) More consraned nvesors hold rsker asses. * Andrea Frazzn s a AQR Capal Managemen, Two Greenwch Plaza, Greenwch, CT 06830, e-mal: andrea.frazzn@aqr.com; web: hp:// Lasse H. Pedersen s a New York Unversy, Copenhagen Busness School (FRIC Cener), AQR Capal Managemen, CEPR, and NBER, 44 Wes Fourh Sree, NY ; e-mal: lpederse@sern.nyu.edu; web: hp:// We hank Clff Asness, Aaron Brown, John Campbell, Ken Danel, Gene Fama, Ncolae Garleanu, John Heaon (dscussan), Mchael Kaz, Owen Lamon, Mchael Mendelson, Mark Mchell, Ma Rchardson, Tuomo Vuoleenaho and Rober Whelaw for helpful commens and dscussons as well as semnar parcpans a Columba Unversy, New York Unversy, Yale Unversy, Emory Unversy, Unversy of Chcago Booh, Kellogg School of Managemen, Harvard Unversy, NBER Behavoral Economcs 2010, he 2010 Annual Managemen Conference a Unversy of Chcago Booh School of Busness, he 2011 Bank of Amerca/Merrll Lynch Quan Conference and he 2011 Nomura Global Quanave Invesmen Sraeges Conference. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 1

2 A basc premse of he capal asse prcng model (CAPM) s ha all agens nves n he porfolo wh he hghes expeced excess reurn per un of rsk (Sharpe rao), and lever or de-lever hs porfolo o su her rsk preferences. However, many nvesors such as ndvduals, penson funds, and muual funds are consraned n he leverage ha hey can ake, and hey herefore overwegh rsky secures nsead of usng leverage. For nsance, many muual fund famles offer balanced funds where he normal fund may nves 40% n long-erm bonds and 60% n socks, whereas he aggressve fund nvess 10% n bonds and 90% n socks. If he normal fund s effcen, hen an nvesor could leverage and acheve a beer rade-off beween rsk and expeced reurn han he aggressve porfolo wh a large l owards socks. The demand for exchange-raded funds (ETFs) wh embedded leverage provdes furher evdence ha many nvesors canno use leverage drecly. Ths behavor of lng oward hgh-bea asses suggess ha rsky hgh-bea asses requre lower rsk-adjused reurns han low-bea asses, whch requre leverage. Indeed, he secury marke lne for U.S. socks s oo fla relave o he CAPM (Black, Jensen, and Scholes (1972)) and s beer explaned by he CAPM wh resrced borrowng han he sandard CAPM (Black (1972, 1993), Brennan (1971), see Mehrlng (2005) for an excellen hsorcal perspecve). Several quesons arse: How can an unconsraned arbrageur explo hs effec,.e., how do you be agans bea? Wha s he magnude of hs anomaly relave o he sze, value, and momenum effecs? Is beng agans bea rewarded n oher counres and asse classes? How does he reurn premum vary over me and n he cross secon? Who bes agans bea? We address hese quesons by consderng a dynamc model of leverage consrans and by presenng conssen emprcal evdence from 20 nernaonal sock markes, Treasury bond markes, cred markes, and fuures markes. Our model feaures several ypes of agens. Some agens canno use leverage and herefore overwegh hgh-bea asses, causng hose asses o offer lower reurns. Oher agens can use leverage bu face margn consrans. They underwegh (or shor-sell) hgh-bea asses and buy low-bea asses ha hey lever up. The model Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 2

3 mples a flaer secury marke lne (as n Black (1972)), where he slope depends on he ghness (.e., Lagrange mulpler) of he fundng consrans on average across agens (Proposon 1). One way o llusrae he asse-prcng effec of he fundng frcon s o consder he reurns on marke-neural beng agans bea (BAB) facors. A BAB facor s a porfolo ha holds low-bea asses, leveraged o a bea of 1, and ha shors hgh-bea asses, de-leveraged o a bea of 1. For nsance, he BAB facor for U.S. socks acheves a zero bea by holdng $1.4 of low-bea socks and shor-sellng $0.7 of hgh-bea socks, wh offseng posons n he rsk-free asse o make self-fnancng. 1 Our model predcs ha BAB facors have a posve average reurn and ha he reurn s ncreasng n he ex-ane ghness of consrans and n he spread n beas beween hgh- and low-bea secures (Proposon 2). When he leveraged agens h her margn consran, hey mus de-lever. Therefore, he model predcs ha, durng mes of ghenng fundng lqudy consrans, he BAB facor realzes negave reurns as s expeced fuure reurn rses (Proposon 3). Furhermore, he model predcs ha he beas of secures n he cross secon are compressed oward 1 when fundng lqudy rsk s hgh (Proposon 4). Fnally, he model mples ha more consraned nvesors overwegh hgh-bea asses n her porfolos whle less consraned nvesors overwegh low-bea asses and possbly apply leverage (Proposon 5). Our model hus exends Black s (1972) cenral nsgh by consderng a broader se of consrans and dervng he dynamc me-seres and cross-seconal properes arsng from he equlbrum neracon beween agens wh dfferen consrans. We fnd conssen evdence for each of he model s cenral predcons. To es Proposon 1, we frs consder porfolos sored by bea whn each asse class. We fnd ha alphas and Sharpe raos are almos monooncally declnng n bea n each asse class. Ths fndng provdes broad evdence ha he relave flaness of he 1 Whle we consder a varey of BAB facors whn a number of markes, one noable example s he zero-covarance porfolo nroduced by Black (1972) and suded for U.S. socks by Black, Jensen, and Scholes (1972), Kandel (1984), Shanken (1985), Polk, Thompson, and Vuoleenaho (2006), and ohers. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 3

4 secury marke lne s no solaed o he U.S. sock marke bu ha s a pervasve global phenomenon. Hence, hs paern of requred reurns s lkely drven by a common economc cause, and our fundng consran model provdes one such unfed explanaon. To es Proposon 2, we consruc BAB facors whn he U.S. sock marke, and whn each of he 19 oher developed MSCI sock markes. The U.S. BAB facor realzes a Sharpe rao of 0.78 beween 1926 and March To pu hs BAB facor reurn n perspecve, noe ha s Sharpe rao s abou wce ha of he value effec and 40% hgher han ha of momenum over he same me perod. The BAB facor has hghly sgnfcan rsk-adjused reurns, accounng for s realzed exposure o marke, value, sze, momenum, and lqudy facors (.e., sgnfcan 1, 3, 4, and 5-facor alphas), and realzes a sgnfcan posve reurn n each of he four 20-year sub-perods beween 1926 and We fnd smlar resuls n our sample of nernaonal eques; Indeed, combnng socks n each of he non-us counres produces a BAB facor wh reurns abou as srong as he U.S. BAB facor. We show ha BAB reurns are conssen across counres, me, whn decles sored by sze, whn decles sored by dosyncrac rsk, and robus o a number of specfcaons. These conssen resuls sugges ha concdence or daamnng are unlkely explanaons. However, f leverage averson s he underlyng drver and s a general phenomenon, as n our model, hen he effec should also exs n oher markes. Hence, we examne BAB facors n oher major asse classes. For U.S. Treasures, he BAB facor s a porfolo ha holds leveraged low-bea,.e., shormaury, bonds and ha shor-sells de-leveraged hgh-bea long-erm bonds. Ths porfolo produces hghly sgnfcan rsk-adjused reurns wh a Sharpe rao of Ths profably of shor-sellng long-erm bonds may seem o conradc he well-known erm premum n fxed ncome markes. There s no paradox, however. The erm premum means ha nvesors are compensaed on average for holdng long-erm bonds raher han T-blls because of he need for maury ransformaon. The erm premum exss a all horzons, however: Jus lke nvesors are Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 4

5 compensaed for holdng 10-year bonds over T-blls, hey are also compensaed for holdng 1-year bonds. Our fndng s ha he compensaon per un of rsk s n fac larger for he 1-year bond han for he 10-year bond. Hence, a porfolo ha has a leveraged long poson n 1-year (and oher shor-erm) bonds and a shor poson n long-erm bonds produces posve reurns. Ths resul s conssen wh our model n whch some nvesors are leverage consraned n her bond exposure and, herefore, requre lower rsk-adjused reurns for long-erm bonds ha gve more bang for he buck. Indeed, shor-erm bonds requre remendous leverage o acheve smlar rsk or reurn as long-erm bonds. These resuls complemen hose of Fama (1986) and Duffee (2010), who also consders Sharpe raos across maures mpled by sandard erm srucure models. We fnd smlar evdence n cred markes: A leveraged porfolo of hgh-raed corporae bonds ouperforms a de-leveraged porfolo of low-raed bonds. Smlarly, usng a BAB facor based on corporae bond ndces by maury produces hgh rskadjused reurns. We es he me-seres predcons of Proposon 3 usng he TED spread as a measure of fundng condons. Conssen wh he model, a hgher TED spread s assocaed wh low conemporaneous BAB reurns. The lagged TED spread predcs reurns negavely, whch s nconssen wh he model f a hgh TED spread means a hgh ghness of nvesors fundng consrans. Ths resul could be explaned f hgher TED spreads mean ha nvesors fundng consrans would be ghenng as her banks dmnsh cred avalably over me, hough hs s speculaon. To es he predcon of Proposon 4, we use he volaly of he TED spread as an emprcal proxy for fundng lqudy rsk. Conssen wh he model s bea-compresson predcon, we fnd ha he dsperson of beas s sgnfcanly lower when fundng lqudy rsk s hgh. Lasly, we fnd evdence conssen wh he model s porfolo predcon ha more consraned nvesors hold hgher-bea secures han less consraned nvesors (Proposon 5). On he one hand, we sudy he equy porfolos of muual funds and ndvdual nvesors, whch s lkely o be consraned. Conssen wh he model, we fnd ha hese nvesors hold porfolos wh average beas above 1. On Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 5

6 he oher sde of he marke, we fnd ha leveraged buyou (LBO) funds, acqure frms wh average beas below 1 and apply leverage. Smlarly, lookng a he holdngs of Berkshre Hahaway, we see ha Warren Buffe bes agans bea by buyng low-bea socks and applyng leverage. Our resuls shed new lgh on he relaonshp beween rsk and expeced reurns. Ths cenral ssue n fnancal economcs has naurally receved much aenon. The sandard CAPM bea canno explan he cross-secon of uncondonal sock reurns (Fama and French (1992)) or condonal sock reurns (Lewellen and Nagel (2006)). Socks wh hgh bea have been found o delver low rsk-adjused reurns (Black, Jensen, and Scholes (1972), Baker, Bradley, and Wurgler (2010)); hus, he consraned-borrowng CAPM has a beer f (Gbbons (1982), Kandel (1984), Shanken (1985)). Socks wh hgh dosyncrac volaly have realzed low reurns (Falkensen (1994), Ang, Hodrck, Xng, Zhang (2006, 2009)), 2 bu we fnd ha he bea effec holds even when conrollng for dosyncrac rsk. Theorecally, asse prcng models wh benchmarked managers (Brennan (1993)) or consrans mply more general CAPM-lke relaons (Hndy (1995), Cuoco (1997)), n parcular he margn-capm mples ha hgh-margn asses have hgher requred reurns, especally durng mes of fundng llqudy (Garleanu and Pedersen (2009), Ashcraf, Garleanu, and Pedersen (2010)). Garleanu and Pedersen (2009) show emprcally ha devaons of he Law of One Prce arses when hghmargn asses become cheaper han low-margn asses, and Ashcraf, Garleanu, and Pedersen (2010) fnd ha prces ncrease when cenral bank lendng facles lower margns. Furhermore, fundng lqudy rsk s lnked o marke lqudy rsk (Gromb and Vayanos (2002), Brunnermeer and Pedersen (2010)), whch also affecs requred reurns (Acharya and Pedersen (2005)). We complemen he leraure by dervng new cross-seconal and me-seres predcons n a smple dynamc model ha capures leverage and margn consrans and by esng s mplcaons across a broad cross-secon of secures across all he major asse classes. Fnally, n a follow up research, Asness, Frazzn and Pedersen (2011) repor evdence of a low- 2 Ths effec dsappears when conrollng for he maxmum daly reurn over he pas monh (Bal, Cakc, and Whelaw (2010)) and when usng oher measures of dosyncrac volaly (Fu (2009)). Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 6

7 bea effec across asse classes conssen wh our heory. The res of he paper s organzed as follows. Secon I lays ou he heory, Secon II descrbes our daa and emprcal mehodology, Secons III-VI es Proposons 1-5, and Secon VII concludes. Appendx A conans all proofs, Appendx B provdes a number of addonal emprcal resuls and robusness ess, and Appendx C provdes a calbraon of he model. I. Theory We consder an overlappng-generaons (OLG) economy n whch agens =1,...,I are born each me perod wh wealh rade secures s=1,...,s, where secury s pays dvdends W and lve for wo perods. Agens s and has x * s shares ousandng. 3 Each me perod, young agens choose a porfolo of shares x=(x 1,...,x S ), nvesng he res of her wealh a he rsk-free reurn r f, o maxmze her uly: f max x'( E P 1 1 (1 r ) P ) x' x (1) 2 where P s he vecor of prces a me, Ω s he varance-covarance marx of P 1 1 consran:, and γ s agen s rsk averson. Agen s subjec o he followng porfolo (2) m x P W s s s Ths consran requres ha some mulple m of he oal dollars nvesed he sum of he number of shares x s mes her prces P s mus be less han he agen s wealh. 3 The dvdends and shares ousandng are aken as exogenous. We noe ha our modfed CAPM has mplcaons for a corporaon s opmal capal srucure, whch suggess an neresng avenue of fuure research beyond he scope hs paper. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 7

8 The nvesmen consran depends on he agen. For nsance, some agens smply canno use leverage, whch s capured by m =1 (as Black (1972) assumes). Oher agens no only may be precluded from usng leverage bu also mus have some of her wealh n cash, whch s capured by m greaer han 1. For nsance, m = 1/(1-0.20)=1.25 represens an agen who mus hold 20% of her wealh n cash. For nsance, a muual fund may need some ready cash o be able o mee daly redempons, an nsurance company needs o pay clams, and ndvdual nvesors may need cash for unforeseen expenses. Oher agens ye may be able o use leverage bu may face margn consrans. For nsance, f an agen faces a margn requremen of 50%, hen hs m s Wh hs margn requremen, he agen can nves n asses worh wce hs wealh a mos. A smaller margn requremen m naurally means ha he agen can ake greaer posons. We noe ha our formulaon assumes for smplcy ha all secures have he same margn requremen, whch may be rue when comparng secures whn he same asse class (e.g., socks) as we do emprcally. Garleanu and Pedersen (2009) and Ashcraf, Garleanu, and Pedersen (2010) consder asses wh dfferen margn requremens and show heorecally and emprcally ha hgher margn requremens are assocaed wh hgher requred reurns (Margn CAPM). We are neresed n he properes of he compeve equlbrum n whch he oal demand equals he supply: x x* (3) To derve equlbrum, consder he frs order condon for agen : 0 E P (1 r ) P x P (4) f 1 1 where ψ s he Lagrange mulpler of he porfolo consran. Solvng for x gves he opmal poson: Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 8

9 1 1 f x E P r P (5) The equlbrum condon now follows from summng over hese posons: 1 1 f x* E P r P (6) where he aggregae rsk averson γ s defned by 1/ γ = Σ 1/ γ, and s he weghed average Lagrange mulpler. (The coeffcens sum o 1 by defnon of he aggregae rsk averson.) The equlbrum prce can hen be compued: P E P x 1 1 * f 1r (7) Translang hs no he reurn of any secury M he marke 1 r 1 P 1 1 / P 1, he reurn on r, and usng he usual expresson for bea, s cov s 1, M 1 / var M r r r 1, we oban he followng resuls. (All proofs are n Appendx A, whch also llusraes he porfolo choce wh leverage consrans n a mean-sd dagram.) Proposon 1 (hgh bea s low alpha). () The equlbrum requred reurn for any secury s s: s f s E r 1 r (8) where he rsk premum s E r M 1 r and s he average Lagrange f mulpler, measurng he ghness of fundng consrans. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 9

10 s s () A secury s alpha wh respec o he marke s (1 ). The alpha s decreases n he bea,. () For an effcen porfolo, he Sharpe rao s hghes for an effcen porfolo wh a bea less han 1 and decreases n beas. s for hgher beas and ncreases for lower As n Black s CAPM wh resrced borrowng (n whch m 1 for all agens), he requred reurn s a consan plus bea mes a rsk premum. Our expresson shows explcly how rsk prema are affeced by he ghness of agens porfolo consrans, as measured by he average Lagrange mulpler. Indeed, gher porfolo consrans (.e., a larger ) flaen he secury marke lne by ncreasng he nercep and decreasng he slope. Whereas he sandard CAPM mples ha he nercep of he secury marke lne s r f, he nercep here s ncreased by he weghed average of he agens Lagrange mulplers. One may wonder why zero-bea asses requre reurns n excess of he rsk-free rae? The reason s ha yng up capal n such asses prevens a consraned nvesor from makng oher profable rades. Furhermore, f unconsraned agens buy a consderable amoun of hese secures, hen, from her perspecve, hs rsk s no longer dosyncrac snce addonal exposure o such asses would ncrease he rsk of her porfolo. Hence, n equlbrum, even zero-bea rsky asses mus offer hgher reurns han he rsk-free rae. Asses ha have zero covarance o Tobn s (1958) angency porfolo held by an unconsraned agen do earn he rsk-free rae, bu he angency porfolo s no he marke porfolo n our equlbrum. Indeed, he marke porfolo s he weghed average of all nvesors porfolos, ha s, an average of he angency porfolo held by unconsraned nvesors and rsker porfolos held by consraned nvesors. Hence, he marke porfolo has hgher rsk and expeced reurn han he angency porfolo, bu a lower Sharpe rao. The porfolo consrans furher mply a lower slope of he secury marke lne, ha s, a lower compensaon for a margnal ncrease n sysemac rsk. The Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 10

11 slope s lower because consraned agens need hs access o hgh un-leveraged reurns and are herefore wllng o accep less hgh reurns for hgh-bea asses. We nex consder he properes of a facor ha goes long low-bea asses and shor-sells hgh-bea asses. To consruc such a facor, le w be he relave L L porfolo weghs for a porfolo of low-bea asses wh reurn r 1 wl ' r and 1 consder smlarly a porfolo of hgh-bea asses wh reurn r H. The beas of hese 1 porfolos are denoed and L L H, where. We hen consruc a bengagans-bea (BAB) facor as: H 1 1 r r r r r (9) H BAB L f H f 1 L 1 1 Ths porfolo s marke neural, ha s, has a bea of zero: he long sde has been leveraged o a bea of 1, and he shor sde has been de-leveraged o a bea of 1. Furhermore, he BAB facor provdes he excess reurn on a self-fnancng porfolo, such as HML and SMB, snce s a dfference beween excess reurns. The dfference s ha BAB s no dollar neural n erms of only he rsky secures snce hs would no produce a bea of zero. 4 The model has several predcons regardng he BAB facor: Proposon 2 (posve expeced reurn of BAB). The expeced excess reurn of he self-fnancng BAB facor s posve E (10) H L BAB r 1 0 L H 4 A naural BAB facor s he zero-covarance porfolo of Black (1972) and Black, Jensen, and Scholes (1972). We consder a broader class of BAB porfolos snce we emprcally consder a varey of BAB porfolos whn varous asse classes ha are subses of all secures (e.g., socks n a parcular sze group). Therefore, our consrucon acheves marke neuraly by leveragng (and deleveragng) he long and shor sdes raher han addng he marke self as Black, Jensen, and Scholes (1972) do. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 11

12 H L and ncreasng n he ex ane bea spread and fundng ghness L H. Ths proposon shows ha a marke-neural BAB porfolo ha s long leveraged low-bea secures and shor hgher-bea secures earns a posve expeced reurn on average. The sze of he expeced reurn depends on he spread n he beas and how bndng he porfolo consrans are n he marke, as capured by he average of he Lagrange mulplers. The nex proposon consders he effec of a shock o he porfolo consrans (or margn requremens), m k, whch can be nerpreed as a worsenng of fundng lqudy, a cred crss n he exreme. Such a fundng lqudy shock resuls n losses for he BAB facor as s requred reurn ncreases. Ths happens because agens may need o de-leverage her bes agans bea or srech even furher o buy he hgh-bea asses. Thus, he BAB facor s exposed o fundng lqudy rsk, as loses when porfolo consrans become more bndng. Proposon 3 (fundng shocks and BAB reurns). k A gher porfolo consran, ha s, an ncrease n m for some of k, leads o a conemporaneous loss for he BAB facor r BAB k m 0 (11) and an ncrease n s fuure requred reurn: E r m BAB 1 k 0 (12) Fundng shocks have furher mplcaons for he cross secon of asse reurns and he BAB porfolo. Specfcally, a fundng shock makes all secury prces drop Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 12

13 ogeher (ha s, s he same for all secures s). Therefore, an ncreased fundng rsk compresses beas owards one. 5 If he BAB porfolo consrucon s based on an nformaon se ha does no accoun for hs ncreased fundng rsk, hen he BAB porfolo s condonal marke bea s affeced. Proposon 4 (bea compresson). () Suppose ha dvdends are..d. over me and wealh and margn requremens are ndependen over me. Then, a hgher varance of fundng lqudy a me generaed by a hgher varance of margn requremens m or wealh W compresses reurn beas of all secures oward 1. 1 () The BAB porfolo s marke neural relave o he nformaon se used for porfolo formaon, bu, f he varance of fundng shocks ncreases (decreases) mmedaely afer he formaon of he BAB porfolo, s condonal marke bea wll be posve (negave). In addon o he asse-prcng predcons ha we have derved, fundng consrans naurally affec agens porfolo choces. In parcular, he more consraned nvesors l oward rsker secures n equlbrum whereas less consraned agens l oward safer secures wh hgher reward per un of rsk. To sae hs resul, we wre nex perod s secury payoffs as M M M M P E P b P E P e (13) where b s a vecor of marke exposures, and e s a vecor of nose ha s uncorrelaed wh he marke. We have he followng naural resul for he agens posons: 5 Garleanu and Pedersen (2009) fnd a complemenary resul, sudyng secures wh dencal fundamenal rsk, bu dfferen margn requremens. They fnd heorecally and emprcally ha such asses have smlar beas when lqudy s good, bu when fundng lqudy rsk rses, he hghmargn secures have larger beas as her hgh margns make hem more fundng sensve. Here, we sudy secures wh dfferen fundamenal rsk, bu he same margn requremens. In hs case, hgher fundng lqudy rsk means ha beas are compressed oward one. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 13

14 Proposon 5 (consraned nvesors hold hgh beas). Unconsraned agens hold a porfolo of rsky secures ha has a bea less han 1; consraned agens hold porfolos of secures wh hgher beas. If secures s and k s k are dencal excep ha s has a larger marke exposure han k, b b, hen any consraned agen j wh greaer-han-average Lagrange mulpler, j, holds j more shares of s han k; he reverse s rue for any agen wh. We nex provde emprcal evdence for Proposons 1-5. Beyond machng he daa qualavely, Appendx C llusraes how well a calbraed model can mach he quanes ha we esmae emprcally. II. Daa and Mehodology The daa n hs sudy are colleced from several sources. The sample of U.S. and nernaonal eques ncludes 55,600 socks coverng 20 counres, and he summary sascs for socks are repored n Table I. Sock reurn daa are from he unon of he CRSP ape and he Xpressfeed Global daabase. Our U.S. equy daa nclude all avalable common socks on CRSP beween January 1926 and March 2012 and beas are compued wh respec o he CRSP value-weghed marke ndex. Excess reurns are above he US Treasury bll rae. We consder alphas wh respec o he marke facor and facor reurns based on sze (SMB), book-o-marke (HML), momenum (UMD), and (when avalable) lqudy rsk. 6 The nernaonal equy daa nclude all avalable common socks on he Xpressfeed Global daly secury fle for 19 markes belongng o he MSCI developed unverse beween January 1989 and March We assgn each sock o s correspondng marke based on he locaon of he prmary exchange. Beas are compued wh respec o he correspondng MSCI local marke ndex. 7 6 SMB, HML, and UMD are from Ken French s daa lbrary, and he lqudy rsk facor s from WRDS. 7 Our resuls are robus o he choce of benchmark (local vs. global). We repor hese ess n he Appendx. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 14

15 All reurns are n USD, and excess reurns are above he U.S. Treasury bll rae. We compue alphas wh respec o he nernaonal marke and facor reurns based on sze (SMB), book-o-marke (HML) and momenum (UMD) from Asness and Frazzn (2011) 8 and (when avalable) lqudy rsk. We also consder a varey of oher asses, and Table II conans he ls nsrumens and he correspondng daa avalably ranges. We oban U.S. Treasury bond daa from he CRSP U.S. Treasury Daabase, usng monhly reurns (n excess of he 1-monh Treasury bll) on he Fama Bond porfolos for maures rangng from 1 o 10 years beween January 1952 and March Each porfolo reurn s an equal-weghed average of he unadjused holdng perod reurn for each bond n he porfolo. Only non-callable, non-flower noes and bonds are ncluded n he porfolos. Beas are compued wh respec o an equally weghed porfolo of all bonds n he daabase. We collec aggregae corporae bond ndex reurns from Barclays Capal s Bond.Hub daabase. 9 Our analyss focuses on he monhly reurns (n excess of he 1-monh Treasury bll) of four aggregae U.S. cred ndces wh maury rangng from one o en years and nne nvesmen grade and hgh yeld corporae bond porfolos wh cred rsk rangng from AAA o Ca-D and Dsressed. 10 The daa cover he perod beween January 1973 and March 2012 alhough he daa avalably vares dependng on he ndvdual bond seres. Beas are compued wh respec o an equally weghed porfolo of all bonds n he daabase. We also sudy fuures and forwards on counry equy ndexes, counry bond ndexes, foregn exchange, and commodes. Reurn daa are drawn from he nernal prcng daa mananed by AQR Capal Managemen LLC. The daa are colleced from a varey of sources and conans daly reurn on fuures, forwards, or swap conracs n excess of he relevan fnancng rae. The ype of conrac for each asse depends on avalably or he relave lqudy of dfferen nsrumens. Pror 8 These facors mmc her U.S counerpars and follow Fama and French (1992, 1993, 1996). See Asness and Frazzn (2011) for a dealed descrpon of her consrucon. The daa can be downloaded a hp:// 9 The daa can be downloaded a hps://lve.barcap.com. 10 The dsress ndex was provded o us by Cred Susse. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 15

16 o expraon, posons are rolled over no he nex mos lqud conrac. The rollng dae s convenon dffers across conracs and depends on he relave lqudy of dfferen maures. The daa cover he perod beween January 1963 and March 2012, wh varyng daa avalably dependng on he asse class. For more deals on he compuaon of reurns and daa sources, see Moskowz, Oo, and Pedersen (2012), Appendx A. For equy ndexes, counry bonds, and currences, he beas are compued wh respec o a GDP-weghed porfolo, and for commodes, he beas are compued wh respec o a dversfed porfolo ha gves equal rsk wegh across commodes. Fnally, we use he TED spread as a proxy for me perods where cred consran are more lkely o be bndng (as n Garleanu and Pedersen (2011) and ohers). The TED spread s defned as he dfference beween he hree-monh EuroDollar LIBOR rae and he hree-monh U.S. Treasures rae. Our TED daa run from December 1984 o March Esmang Ex-ane Beas We esmae pre-rankng beas from rollng regressons of excess reurns on marke excess reurns. Whenever possble, we use daly daa raher han monhly as he accuracy of covarance esmaon mproves wh he sample frequency (Meron (1980)). 11 Our esmaed bea for secury s gven by (14) where and are he esmaed volales for he sock and he marke and s her correlaon. We esmae volales and correlaons separaely for wo reasons. Frs, we use a 1-year rollng sandard devaon for volales and a 5-year horzon for he correlaon o accoun for he fac ha ha correlaons appear o move consderably more slowly han volales. 12 Second, we use 1-day log reurns o 11 Daly reurns are no avalable for our sample of U.S. Treasury bonds, U.S. corporae bonds, and U.S. cred ndces. 12 See, for example, De Sans and Gerard (1997). Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 16

17 esmae volales and overlappng 3-day log reurns, r 3d, 2 k0 ln(1 ), for correlaon o conrol for non-synchronous radng (whch obvously only affecs correlaons). We requre a leas 6 monhs (120 radng days) of non-mssng daa o esmae volales and a leas 3 years (750 radng days) of non-mssng reurn daa for correlaons. If we only have access o monhly daa, we use rollng 1 and 5- year wndows and requre a leas 12 and 36 observaons. Fnally, o reduce he nfluence of oulers, we follow Vascek (1973) and Elon, Gruber, Brown, and Goezmann (2003) and shrnk he me-seres esmae of XS bea ( ) owards he cross-seconal mean ( ): TS r k ˆ ˆTS w (1 w) ˆ XS (15) For smplcy, raher han havng asse-specfc and me-varyng shrnkage facors as n Vascek (1973), we se w = 0.6 and XS bu our resuls are very smlar eher way. 13 =1 for all perods and across all asses, We noe ha our choce of he shrnkage facor does no affec how secures are sored no porfolos snce he common shrnkage does no change he ranks of secury beas. However, he amoun of shrnkage affecs he consrucon of he BAB porfolos snce he esmaed beas are used o he sze he long and he shor sdes o make he porfolo marke neural a formaon. To accoun for he fac ha nose n he ex-ane beas affecs he consrucon of he BAB facors, our nference s focused on realzed abnormal reurns so ha any msmach beween ex-ane and (ex pos) realzed beas s pcked up by he realzed loadngs n he facor regresson. Of course, when we regress our porfolos on sandard rsk facors, he realzed facor loadngs are no shrunk as above snce only he ex-ane beas are subjec o selecon bas. Our resuls are robus o alernave The Vascek (1973) Bayesan shrnkage facor s gven by w 1 / ( ) where s he, TS, TS XS varance of he esmaed bea for secury, and 2 XS s he cross-seconal varance of beas. Ths esmaor places more wegh on he hsorcal mes seres esmae when he esmae has a lower varance or when here s large dsperson of beas n he cross secon. Poolng across all socks n our U.S. equy daa, he shrnkage facor w has a mean of 0.61., TS Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 17

18 bea esmaon procedures as we repor n he Appendx. We compue beas wh respec o a marke porfolo, whch s eher specfc o an asse class or he overall world marke porfolo of all asses. Whle our resuls hold boh ways, we focus on beas wh respec o asse-class-specfc marke porfolos snce hese beas are less nosy for several reasons. Ths approach allows us o use daly daa over a long me perod for mos asse classes, as opposed o usng he mos dversfed marke porfolo for whch we only have monhly daa over a lmed me perod. Moreover, hs approach s ndependen of assumpons abou wha he overall marke porfolo s, and s applcable even f markes are segmened. As a robusness es, Table B8 n he Appendx repors resuls when we compue beas wh respec o a proxy for a world marke porfolo comprsed of many asse classes. We use he world marke porfolo from Asness, Frazzn, and Pedersen (2011). 14 The resuls are conssen wh our man ess as he BAB facors earn large and sgnfcan abnormal reurns n each of asse classes n our sample. Consrucng Beng-Agans-Bea Facors We consruc smple porfolos ha are long low-bea secures and ha shor-sell hgh-bea secures, hereafer BAB facors. To consruc each BAB facor, all secures n an asse class are ranked n ascendng order on he bass of her esmaed bea. The ranked secures are assgned o one of wo porfolos: lowbea and hgh-bea. The low (hgh) bea porfolo s comprsed of all socks wh a bea below (above) s asse class medan (or counry medan for nernaonal eques). In each porfolo, secures are weghed by he ranked beas (.e., lowerbea secures have larger weghs n he low-bea porfolo and hgher-bea secures have larger weghs n he hgh-bea porfolo). The porfolos are rebalanced every calendar monh. More formally, le z be he n 1 vecor of bea ranks z rank ( ) a porfolo formaon, and le z z n be he average rank, where n s he number ' 1 n / 14 See Asness, Frazzn, and Pedersen (2011) for a dealed descrpon of hs marke porfolo. The marke seres s monhly and ranges from 1973 o Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 18

19 of secures and 1 n s an n 1 vecor of ones. The porfolo weghs of he low-bea and hgh-bea porfolos are gven by w k ( z z ) h w k ( z z ) L (16) where k s a normalzng consan k z z and x and x ndcae he posve ' 1 n / 2 and negave elemens of a vecor x. Noe ha by consrucon we have 1 ' w 1 and 1 ' w 1. To consruc he BAB facor, boh porfolos are rescaled o have a n L bea of one a porfolo formaon. The BAB s he self-fnancng zero-bea porfolo (8) ha s long he low-bea porfolo and ha shor-sells he hgh-bea porfolo. 1 1 r r r r r (17) H BAB L f H f 1 L 1 1 n H 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 s long $0.7 of low-bea socks (fnanced by shor-sellng $0.7 of rsk-free secures) and shor-sells $1.4 of hgh-bea socks (wh $1.4 earnng he rsk-free rae). Daa Used o Tes he Theory s Porfolo Predcons We collec muual fund holdngs from he unon of he CRSP Muual Fund Daabase and Thompson Fnancal CDA/Specrum holdngs daabase, whch ncludes all regsered domesc muual funds flng wh he SEC. The holdngs daa run from March 1980 o March We focus our analyss on open-end acvely managed domesc equy muual funds. Our sample selecon procedure follows ha of Kacperzczyk, Salm, and Zheng (2008), and we refer o her Appendx for deals abou he screens ha were used and summary sascs of he daa. Our ndvdual nvesors holdngs daa was colleced from a naonwde dscoun brokerage house and conans rade made by abou 78,000 households n Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 19

20 he perod from January of 1991 o November of Ths daase has been used exensvely n he exsng leraure on ndvdual nvesors. For a dealed descrpon of he brokerage daa se, see Barber and Odean (2000). Our sample of buyous s drawn from he M&A and corporae evens daabase mananed by AQR/CNH Parners. 15 The daa conan varous daa ems ncludng nal, subsequen announcemen daes, and (f applcable) compleon or ermnaon dae for all akeover deals where he arge s a U.S. publcly raded frm and where he acqurer s a prvae company. For some (bu no all) deals, he acqurer descrpor also conans nformaon on wheher he deal s a Leveraged or Managemen Buyou (LBO, MBO). The daa run from January 1963 o March Fnally, we download holdngs daa for Berkshre Hahaway from Thomson Fnancal Insuonal (13f) Holdng Daabase. The daa run from March 1980 o March III. Beng Agans Bea n Each Asse Class We now es how he requred reurn vares n he cross-secon of bea-sored secures (Proposon 1) and he hypohess ha long/shor BAB facors have posve average reurns (Proposon 2). As an overvew of hese resuls, he alphas of all he bea-sored porfolos consdered n hs paper are ploed n Fgure 1, and he Sharpe raos are ploed n Fgure B1 n he Appendx. We see ha declnng alphas and Sharpe raos across bea-sored porfolos are general phenomena across asse classes as we dscuss n deal below. Socks Table III repors our ess for U.S. socks. We consder 10 bea-sored porfolos and repor her average reurns, alphas, marke beas, volales, and Sharpe raos. The average reurns of he dfferen bea porfolos are smlar, whch s he well-known relavely fla secury marke lne. Hence, conssen wh Proposon 1 and wh Black (1972), he alphas declne almos monooncally from he low-bea o hgh-bea porfolos. Indeed, he alphas declne when esmaed 15 We would lke o hank Mark Mchell for provdng us wh hs daa. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 20

21 relave o a 1-, 3-, 4-, and 5-facor model. Moreover, Sharpe raos declne monooncally from low-bea o hgh-bea porfolos. The rghmos column of Table III repors reurns of he beng-agans-bea (BAB) facor, ha s, a porfolo ha s long leveraged low-bea socks and ha shorsells de-leveraged hgh-bea socks, hus mananng a bea-neural porfolo. Conssen wh Proposon 2, he BAB facor delvers a hgh average reurn and a hgh alpha. Specfcally, he BAB facor has Fama and French (1993) abnormal reurns of 0.73% per monh (-sasc = 7.39). Furher adjusng reurns for Carhar s (1997) momenum-facor, he BAB porfolo earns abnormal reurns of 0.55% per monh (-sasc = 5.59). Las, we adjus reurns usng a 5-facor model by addng he raded lqudy facor by Pasor and Sambaugh (2003), yeldng an abnormal BAB reurn of 0.55% per monh (-sasc = 4.09, whch s lower n par because he lqudy facor s only avalable durng half of our sample). We noe ha whle he alpha of he long-shor porfolo s conssen across regressons, he choce of rsk adjusmen nfluences he relave alpha conrbuon of he long and shor sdes of he porfolo. Fgure 2 plos he annualzed Sharpe rao of he sock BAB porfolo and he BAB porfolos n he oher asse classes. Our resuls for U.S. eques show how he secury marke lne has connued o be oo fla for anoher four decades afer Black, Jensen, and Scholes (1972). Furher, our resuls exend nernaonally. We consder bea-sored porfolos for nernaonal eques and laer urn o alogeher dfferen asse classes. We use all 19 MSCI developed counres excep he U.S. (o keep he resuls separae from he U.S. resuls above), and we do hs n wo ways: We consder nernaonal porfolos where all nernaonal socks are pooled ogeher (Table IV), and we consder resuls separaely for each counry (Table V). The nernaonal porfolo s counry neural, ha s, he low (hgh) bea porfolo s comprsed of all socks wh a bea below (above) s counry medan. 16 The resuls for our pooled sample of nernaonal eques n Table IV mmc he U.S. resuls: he alpha and Sharpe raos of he bea-sored porfolos declne 16 We keep he nernaonal porfolo counry neural because we repor he resul of beng agans bea across equy ndces BAB separaely n Table VIII. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 21

22 (alhough no perfecly monooncally) wh he beas, and he BAB facor earns rsk-adjused reurns beween 0.28% and 0.64% per monh dependng on he choce of rsk adjusmen, wh -sascs rangng from 2.09 o Table V shows he performance of he BAB facor whn each ndvdual counry. The BAB delvers posve Sharpe raos n 18 of he 19 MSCI developed counres and posve 4-facor alphas n 13 ou of 19, dsplayng a srkngly conssen paern across equy markes. The BAB reurns are sascally sgnfcanly posve n 6 counres, whle none of he negave alphas s sgnfcan. Of course, he small number of socks n our sample n many of he counres makes dffcul o rejec he null hypohess of zero reurn n each ndvdual counry. Tables B1 n he Appendx repors facor loadngs. On average, he U.S. BAB facor goes long $1.40 ($1.40 for Inernaonal BAB) and shor-sells $0.70 ($0.89 for Inernaonal BAB). The larger long nvesmen s mean o make he BAB facor marke-neural because he socks ha are held long have lower beas. The BAB facor s realzed marke loadng s no exacly zero, reflecng he fac ha our ex ane beas are measured wh nose. The oher facor loadngs ndcae ha, relave o hgh-bea socks, low-bea socks are lkely o be larger, have hgher book-omarke raos, and have hgher reurn over he pror 12 monhs, alhough none of he loadngs can explan he large and sgnfcan abnormal reurns. The BAB porfolo s posve HML loadng s naural snce our heory predcs ha low-bea socks are cheap and hgh-bea socks are expensve. The Appendx repors furher ess and addonal robusness checks. In Table B2 we repor resuls usng dfferen wndow lenghs o esmae beas and dfferen benchmarks (local, global). We spl he sample by sze (Table B3) and me perods (Table B4), we conrol for dosyncrac volaly (Table B5) and repor resuls for alernave defnon of he rsk-free rae (B6). Fnally, n Table B7 and Fgure B2 we repor an ou of sample es. We collec prcng daa from DaaSream and for each counry n Table I we compue a BAB porfolo over sample perod no covered by he Xpressfeed Global daabase DaaSream nernaonal prcng daa sar n 1969 whle Xpressfeed Global coverage sars n Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 22

23 All of he resuls are conssen: equy porfolos ha be agans beas earn sgnfcan rsk-adjused reurns. Treasury Bonds Table VI repors resuls for U.S. Treasury bonds. As before, we repor average excess reurns of bond porfolos formed by sorng on bea n he prevous monh. In he cross secon of Treasury bonds, rankng on beas wh respec o an aggregae Treasury bond ndex s emprcally equvalen o rankng on duraon or maury. Therefore, n Table VI, one can hnk of he erm bea, duraon, or maury n an nerchangeable fashon. The rghmos column repors reurns of he BAB facor. Abnormal reurns are compued wh respec o a one-facor model where alpha s he nercep n a regresson of monhly excess reurn on an equally weghed Treasury bond excess marke reurn. The resuls show ha he phenomenon of a flaer secury marke lne han predced by he sandard CAPM s no lmed o he cross secon of sock reurns. Indeed, conssen wh Proposon 1, he alphas declne monooncally wh bea. Lkewse, Sharpe raos declne monooncally from 0.73 for low-bea (shor maury) bonds o 0.31 for hgh-bea (long maury) bonds. Furhermore, he bond BAB porfolo delvers abnormal reurns of 0.17% per monh (-sasc = 6.26) wh a large annual Sharpe rao of Snce he dea ha fundng consrans have a sgnfcan effec on he erm srucure of neres may be surprsng, le us llusrae he economc mechansm ha may be a work. Suppose an agen, e.g., a penson fund, has $1 o allocae o Treasures wh a arge excess reurn of 2.9% per year. One way o acheve hs reurn arge s o nves $1 n a porfolo of Treasures wh maury above 10 years as seen n Table VI, P7. If he agen nvess n 1-year Treasures (P1) nsead, hen he would need o nves $11 f all maures had he same Sharpe rao. Ths hgher leverage s needed because he long-erm Treasures are 11 mes more volale han he shor-erm Treasures. Hence, he agen would need o borrow an addonal $10 o lever hs nvesmen n 1-year bonds. If he agen has leverage lms (or prefers lower leverage), hen he would srcly prefer he 10-year Treasures n hs case. Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 23

24 Accordng o our heory, he 1-year Treasures herefore mus offer hgher reurns and hgher Sharpe raos, flaenng he secury marke lne for bonds. Emprcally, shor-erm Treasures do n fac offer hgher rsk-adjused reurns so he reurn arge can be acheved by nvesng abou $5 n 1-year bonds. Whle a consraned nvesor may sll prefer an un-leveraged nvesmen n 10-year bonds, unconsraned nvesors now prefer he leveraged low-bea bonds, and he marke can clear. Whle he severy of leverage consrans vares across marke parcpans, appears plausble ha a 5-o-1 leverage (on hs par of he porfolo) makes a dfference for some large nvesors such as penson funds. Cred We nex es our model usng several cred porfolos and repor resuls n Table VII. In Panel A, columns (1) o (5), he es asses are monhly excess reurns of corporae bond ndexes by maury. We see ha he cred BAB porfolo delvers abnormal reurns of 0.11% per monh (-sasc = 5.14) wh a large annual Sharpe rao of Furhermore, alphas and Sharpe raos declne monooncally. In columns (6) o (10), we aemp o solae he cred componen by hedgng away he neres rae rsk. Gven he resuls on Treasures n Table VI, we are neresed n esng a pure cred verson of he BAB porfolo. Each calendar monh, we run 1-year rollng regressons of excess bond reurns on excess reurn on Barclay s U.S. governmen bond ndex. We consruc es asses by gong long he corporae bond ndex and hedgng hs poson by shor-sellng he approprae CDS f f amoun of he governmen bond ndex: ( ) ˆ USGOV f r r r r 1( r r ), where s he slope coeffcen esmaed n an expandng regresson usng daa from he ˆ 1 begnnng of he sample and up o monh -1. One nerpreaon of hs reurns seres s ha approxmaes he reurns on a Cred Defaul Swap (CDS). We compue marke reurns by akng he equally weghed average of hese hedged reurns, and we compue beas and BAB porfolos as before. Abnormal reurns are compued wh respec o a wo-facor model where alpha s he nercep n a Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 24

25 regresson of monhly excess reurn on he equally weghed average pseudo-cds excess reurn and he monhly reurn on he Treasury BAB facor. The addon of he Treasury BAB facor on he rgh-hand sde s an exra check o es a pure cred verson of he BAB porfolo. The resuls n Panel A of Table VII columns (6) o (10) ell he same sory as columns (1) o (5): he BAB porfolo delvers sgnfcan abnormal reurns of 0.17% per monh (-sascs = 4.44) and Sharpe raos declne monooncally from lowbea o hgh-bea asses. Las, n Panel B of Table VII, we repor resuls where he es asses are cred ndexes sored by rang, rangng from AAA o Ca-D and Dsressed. Conssen wh all our prevous resuls, we fnd large abnormal reurns of he BAB porfolos (0.57% per monh wh a -sascs = 3.72) and declnng alphas and Sharpe raos across bea-sored porfolos. Equy Indexes, Counry Bond Indexes, Currences, and Commodes Table VII repors resuls for equy ndexes, counry bond ndexes, foregn exchange and commodes. The BAB porfolo delvers posve reurns n each of he four asse classes, wh an annualzed Sharpe rao rangng from 0.11 o We are only able o rejec he null hypohess of zero average reurn for equy ndexes, bu we can rejec he null hypohess of zero reurns for combnaon porfolos ha nclude all or some combnaon of he four asse classes, akng advanage of dversfcaon. We consruc a smple equally weghed BAB porfolo. To accoun for dfferen volaly across he four asse classes, n monh, we rescale each reurn seres o 10% annualzed volaly usng rollng 3-year esmaes up o monh -1, and hen we equally wegh he reurn seres and her respecve marke benchmark. Ths porfolo consrucon generaes a smple mplemenable porfolo ha arges 10% BAB volaly n each of he asse classes. We repor resuls for an All fuures combo ncludng all four asse classes and a Counry Selecon combo ncludng only Equy ndces, Counry Bonds and Foregn Exchange. The BAB All Fuures and Counry Selecon delver abnormal reurn of 0.25% and 0.26% per monh (-sascs = 2.53 and 2.42). Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 25

26 Beng Agans All of he Beas To summarze, he resuls n Table III VIII srongly suppor he predcons ha alphas declne wh bea and BAB facors earn posve excess reurns n each asse class. Fgure 1 llusraes he remarkably conssen paern of declnng alphas n each asse class, and Fgure 2 shows he conssen reurn o he BAB facors. Clearly, he relavely fla secury marke lne, documened by Black, Jensen, Scholes (1972) for U.S. socks, s a pervasve phenomenon ha we fnd across markes and asse classes. Averagng all of he BAB facors produces a dversfed BAB facor wh a large and sgnfcan abnormal reurn of 0.54% per monh (sascs of 6.98) as seen n Table VIII Panel B. IV. Tme Seres Tess In hs secon, we es Proposon 3 s predcons for he me-seres of BAB reurns: When fundng consrans become more bndng (e.g., because margn requremens rse), he requred BAB premum ncreases, and he realzed BAB reurns become negave. We ake hs predcon o he daa usng he TED spread as a proxy of fundng condons. The sample runs from December 1984 (he frs avalable dae for he TED spread) o March Table IX repors regresson-based ess of our hypoheses for he BAB facors across asse classes. The frs column smply regresses he U.S. BAB facor on he conemporaneous level of he TED spread measured a he end of he perod. 18 Conssen wh Proposon 3, we fnd a negave and sgnfcan relaonshp. We noe, however, ha he model s predcon as a paral dervave assumes ha he curren fundng condons change whle everyhng else reman unchanged, bu emprcally oher hngs do change. Hence, our es reles on an assumpon ha such varaon of oher varables does no lead o a bas. 18 We noe ha we are vewng he TED spread smply as a measure of cred condons, no as a reurn. Hence, he TED spread a he end of he reurn perod s a measure of he cred condons a ha me (even f he TED spread s a dfference n neres raes ha would be earned over he followng me perod). Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 26

27 To parally address hs ssue, column (2) provdes a smlar resul when conrollng for a number of oher varables. The conrol varables are he marke reurn (o accoun for possble nose n he ex ane beas uses for makng he BAB porfolo marke neural), he 1-monh lagged BAB reurn (o accoun for possble momenum n BAB), he ex-ane Bea Spread, he Shor Volaly Reurns, and he Lagged Inflaon. The Bea Spread s equal o ( ) / and measures he ex S L S L ane bea dfference beween he long and shor sde of he BAB porfolos, whch should posvely predc he BAB reurn as seen n Proposon 2. Conssen wh he model, Table IX shows ha he esmaed coeffcen for he Bea Spread s posve n all specfcaons, bu no sascally sgnfcan. The Shor Volaly Reurns s he reurn on a porfolo ha shor-sells closes-o-he-money, nex-oexpre sraddles on he S&P500 ndex, capurng poenal sensvy o volaly rsk. Lagged Inflaon s equal o he 1-year U.S. CPI nflaon rae, lagged 1 monh, whch s ncluded o accoun for poenal effecs of money lluson as suded by Cohen, Polk, and Vuoleenaho (2005), alhough we do no fnd evdence of hs effec. In columns (3) and (4), we decompose he TED spread no s 1-monh lagged level and 1-monh change. We see ha boh he lagged level and conemporaneous change n he TED spread are negavely relaed o he BAB reurns. If he TED spread measures he ghness of fundng consrans (gven by n he model), hen he model predcs a negave coeffcen for he conemporaneous change n TED (eqn. (11)) and a posve coeffcen for he lagged level (eqn. (12)). Hence, he coeffcen for he lagged level s no conssen wh he model under hs nerpreaon of he TED spread. If, nsead, a hgh TED spread ndcaes ha agens fundng consrans are worsenng, hen he resuls would be easer o undersand. Under hs nerpreaon, a hgh TED spread could ndcae ha banks are cred-consraned and ha banks ghen oher nvesors cred consrans over me, leadng o a deeroraon of BAB reurns over me (f nvesors don foresee hs). Columns (5)-(8) of Table IX repor panel regressons for nernaonal sock BAB facors and columns (9)-(12) for all he BAB facors. These regressons nclude Beng Agans Bea - Andrea Frazzn and Lasse H. Pedersen Page 27