Betting Against Beta

Transcription

1 Being Agains Bea Andrea Frazzini and Lasse H. Pedersen * This draf: Sepember 13, 2010 Absrac. We presen a model in which some invesors are prohibied from using leverage and oher invesors leverage is limied by margin requiremens. The former invesors bid up high-bea asses while he laer agens rade o profi from his, bu mus delever when hey hi heir margin consrains. We es he model s predicions wihin U.S. equiies, across 20 global equiy markes, for Treasury bonds, corporae bonds, and fuures. Consisen wih he model, we find in each asse class ha a beingagains-bea (BAB) facor which is long a leveraged porfolio of low-bea asses and shor a porfolio of high-bea asses produces significan risk-adjused reurns. When funding consrains ighen, beas are compressed owards one, and he reurn of he BAB facor is low. * Andrea Frazzini is a AQR Capial Managemen, Two Greenwich Plaza, Greenwich, CT 06830, andrea.frazzini@aqr.com. Lasse H. Pedersen is a New York Universiy, NBER, and CEPR, 44 Wes Fourh Sree, NY ; lpederse@sern.nyu.edu; web: hp:// We hank Cliff Asness, Nicolae Garleanu. Ma Richardson, Rober Whielaw, Michael Mendelson, Michael Kaz, Aaron Brown and Gene Fama for helpful commens and discussions as well as seminar paricipans a Columbia Universiy, New York Universiy, Yale Universiy, Emory Universiy, and he 2010 Annual Managemen Conference a Universiy of Chicago Booh School of Business. 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 lever or de-lever i o sui heir risk preferences. However, many invesors such as individuals, pension funds, and muual funds are consrained in he leverage hey can ake, and herefore over-weigh 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 as 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 he same expeced reurn a a lower volailiy raher han iling o a large 90% allocaion o socks. The demand for exchange-raded funds (ETFs) wih leverage buil in presens furher evidence ha many invesors canno use leverage direcly. This behavior of iling owards high-bea asses suggess ha risky high-bea asses require lower risk-adjused reurns han low-bea asses, which require leverage. Consisenly, 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 addiional quesions arise: how can an unconsrained arbirageur exploi his effec i.e., how do you be agains bea and wha is he magniude of his anomaly relaive o he size, value, and momenum effecs? Is being agains bea rewarded in oher asse classes? How does he reurn premium vary over ime and in he cross secion? Which invesors be agains bea? We address hese quesions by considering a dynamic model of leverage consrains and by presening consisen empirical evidence from 20 global sock markes, Treasury bond markes, credi markes, and fuures markes. Our model feaures several ypes of agens. Some agens canno use leverage and, herefore, over-weigh 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 implies a flaer securiy marke line (as in Black (1972)), where he slope Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 2

3 depends on he ighness (i.e., Lagrange muliplier) of he funding consrains on average across agens. 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 long a porfolio of low-bea asses, leveraged o a bea of 1, and shor a porfolio of high-bea asses, de-leveraged o a bea of 1. For insance, he BAB facor for U.S. socks achieves a zero bea by being long $1.5 of low-bea socks, shor $0.7 of high-bea socks, wih offseing posiions in he risk-free asse o make i zero-cos. 1 Our model predics ha BAB facors have 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. When he leveraged agens hi heir margin consrain, hey mus de-lever, and, herefore, he model predics ha he BAB facor has negaive reurns during imes of ighening funding liquidiy consrains. Furher, he model predics ha he beas of securiies in he cross secion are compressed owards 1 when funding liquidiy risk rises. Our model hus exends Black (1972) s 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. Consisen wih he model s predicion, we find significan reurns o being agains bea wihin each of he major asse classes globally. We show ha beingagains-bea facors produce negaive reurns when credi consrains are more likely o be bindings and we also documen he model-implied bea compression during imes of illiquidiy. To perform hese empirical ess, 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 provides broad evidence ha he flaness of he securiy marke line is no isolaed o he U.S. sock marke, 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 bu 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. We firs consider he BAB facor 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 0.75 beween 1926 and To pu his facor reurn in perspecive, noe ha his is abou wice he Sharpe raio of he value effec over he same period and 40% higher han he Sharpe raio of momenum. I has a 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 global equiies: combining socks in each of he non-us 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 aversion is he underlying driver and is a general phenomenon as in our model, hen he effec should also exis in oher markes. We examine BAB facors in oher major asse classes. For U.S. Treasuries, he BAB facor is long a leveraged porfolio of low-bea ha is, shor mauriy bonds, and shor a de-leveraged porfolio of long-daed bonds. This porfolio produces highly significan risk-adjused reurns wih a Sharpe raio of This profiabiliy of shoring long-erm bonds may seem in conras o he mos wellknown erm premium in fixed income markes. There is no paradox, however. The erm premium means ha invesors are compensaed on average for holding longerm bonds raher han T-bills due o he need for mauriy ransformaion. The erm premium exis a all horizons, hough: Invesors are compensaed for holding 1- year bonds over T-bills as well as hey are compensaed for holding 10-year bonds. Our finding is ha he compensaion per uni of risk is in fac larger for he 1-year Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 4

5 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 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 a 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 high-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 riskadjused reurns. We es he model s predicion ha he cross-secional dispersion of beas is lower during imes of high funding liquidiy risk, which we proxy by he TED spread empirically. Consisen wih he bea-compression predicion, we find ha he dispersion of beas is significanly lower when he TED spread is high, and his resul holds across a number of specificaions. Furher, we also find evidence consisen wih he model s predicion ha he BAB facor realizes a posiive marke bea when liquidiy risk is high. Lasly, we es he model s ime-series predicions ha he BAB facor should realize a high reurn when lagged illiquidiy is high, when conemporaneous liquidiy improves, and when here is a large spread beween he ex ane bea of he long side of he porfolio and he shor side of he porfolio. Consisen wih he model, we find ha high conemporaneous TED spreads predics BAB reurns negaively, and he ex ane bea spread predics BAB reurns posiively. 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. I could be consisen wih he model if a high TED spread means ha invesors funding consrains are ighening, perhaps as heir banks diminish credi availabiliy over ime. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 5

6 Our resuls shed new ligh on he relaion 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)) so he consrainedborrowing CAPM has a beer fi (Gibbons (1982), Kandel (1984), Shanken (1985)). Socks wih high idiosyncraic volailiy have realized low reurns (Ang, Hodrick, Xing, Zhang (2006, 2009)), 2 bu we find ha he bea effec holds even 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 highmargin asses have higher required reurns, especially during imes of funding illiquidiy (Garleanu and Pedersen (2009), Ashcraf, Garleanu, and Pedersen (2010)). Garleanu and Pedersen (2009) find empirically ha deviaions of he Law of One Price arises when high-margin asses become cheaper han low-margin asses, and Ashcraf, Garleanu, and Pedersen (2010) find he prices increase when cenral bank lending faciliies lower margins. Furher, 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 boh leverage and margin consrains, and by esing is implicaions across broad cross-secion of securiies across all he major asse classes. 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-V es he heory s cross-secional and ime series predicions across asse classes, and Secion VI concludes. Appendix A conains all proofs and Appendix B provides a number of addiional empirical resuls and robusness ess. 2 This effec disappears when conrolling for he maximum daily reurn over he pas monh (Bali, Cakici, and Whielaw (2010)) and oher measures of idiosyncraic volailiy (Fu (2009)). Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 6

7 I. Theory i=1,...,i We consider an overlapping-generaions (OLG) economy in which agens are born each period and live for wo periods. Agens rade securiies s=1,...,s, where securiy s has x * i shares ousanding. Each ime period, young agens choose a porfolio of shares x=(x 1,...,x S ), invesing he res of heir wealh W i a he risk-free reurn r f, o maximize heir uiliy: i f max x '( E P 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, and γ i is agen i s risk aversion. Agen i is subjec o he following porfolio consrain: m x P W (2) i s s i s This consrain says ha some muliple m i 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 may no only be precluded from using leverage, bu also need o 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. Oher agens ye may be able o use leverage, bu face margin consrains. For insance, if an agen faces a margin requiremen of 50%, hen his m i is 0.50 since his means ha he can inves a mos in asses worh wice his wealh. A smaller margin requiremen m i naurally means ha he agen can ake larger posiions. We noe ha our formulaion assumes for simpliciy ha all securiies have he same margin requiremen. This may be rue when comparing securiies wihin he same Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 7

8 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 where ψ i is he Lagrange muliplier of he porfolio consrain. This gives he opimal posiion i 1 1 f i x E P 1 1 r i P (5) The equilibrium condiion now follows from summing over hese posiions: 1 1 f x* E P 1 1 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.) This gives he equilibrium price: Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 8

9 P 1 E P x 1 * f r (7) Translaing his ino he reurn of any securiy r i 1 P i 1 / P i 1, he reurn on he M marke 1 r, and using he usual expression for bea, s cov s 1, M 1 / var M r r r 1 ge he following resuls. (All proofs are in Appendix A.), we Proposiion 1. (i) The equilibrium required reurn for any securiy s is: s f s E r 1 r (8) where he risk premium is M 1 E r 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 ). Alpha decreases s in he securiy s marke bea,. (iii) For a diversified efficien porfolio, he Sharpe raio is highes for an efficien porfolio wih bea less han 1 and decreases in lower beas. s for higher beas and increases for i As in Black s CAPM wih resriced borrowing (in which m 1for 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, here he inercep is increased by he weighed average of he Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 9

10 agens Lagrange mulipliers. You may wonder why zero-bea asses require reurns in excess of he risk free rae? The reason is ha ying up your capial in such asses prevens you from making profiable rades ha you would like o pursue bu canno if you are consrained. Furher, if unconsrained agens buy a lo of hese securiies, hen, from heir perspecive, his risk is no longer idiosyncraic since addiional exposure o such asses would increase he risk of heir porfolio. Hence, in equilibrium even zero-bea risky asses mus offer higher reurns han he risk-free rae. (Asses ha have zero covariance o Markowiz s (1952) angency porfolio held by an unconsrained agens do earn he risk free rae, on he oher hand, bu he angency porfolio is no he marke porfolio in his equilibrium.) The porfolio consrains furher imply a lower slope.of he securiy marke line, ha is, a lower compensaion for a marginal increase in sysemaic risk. This is because consrained agens need his access o high un-leveraged reurns and herefore are willing o accep less high reurns for high-bea asses. We nex consider he properies of a facor ha goes long low-bea asses and shor high-bea asses. For his, le w L be he relaive porfolio weighs a porfolio of L low-bea asses wih reurn r 1 wl ' r 1 and consider similarly a porfolio of high- H bea asses wih reurn r 1. The beas of hese porfolios are denoed L and H, where L. We hen consruc a being-agains-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, 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. Furher, he BAB facor provides he excess reurn on a zero-cos porfolio like 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. 3 The model has several predicions regarding he BAB facor: 3 A naural BAB facor is he zero-covariance porfolio of Black (1972) and Black, Jensen, and Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 10

11 Proposiion 2. (i) The expeced excess reurn of he zero-cos BAB facor is posiive: E (10) H L BAB r 1 0 L H H and increasing in he bea spread L H L and he funding ighness. (ii) A igher porfolio consrain, ha is, an increase in m for some of k, leads o a conemporaneous loss for he BAB facor k r BAB k m 0 (11) and an increase in is fuure required reurn: E r m BAB 1 k 0 (12) The firs par of he proposiion says ha a marke-neural porfolio ha is long leveraged low-bea securiies and shor higher-bea securiies should earn a posiive expeced reurn on average. The size of he expeced reurn depends on he spread in beas and he how binding porfolio consrains are in he marke, as capured by he average of he Lagrange mulipliers,. The second par of he proposiion considers he effec of a shock o he porfolio consrains (or margin requiremens), m k, which can be inerpreed as a 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 Scholes (1972) do. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 11

12 worsening of funding liquidiy, a liquidiy crisis in he exreme. Such a funding liquidiy shock resuls in losses for he BAB facor as is required reurn increases. This happens as agens may need o de-lever heir bes agains bea or srech even furher o buy he high-bea asses. This shows ha he BAB facor is exposed o funding liquidiy risk i loses when porfolio consrains become more binding. Furher, he marke reurn ends o be low during such liquidiy crises. Indeed, a higher m k increases he required reurn of he marke and reduces he conemporaneous marke reurn. Hence, while he BAB facor is marke neural on average, liquidiy shocks can lead o correlaion beween BAB and he marke. Anoher way of saying his is ha low-bea securiies fare poorly during imes of increased illiquidiy relaive o heir beas, while high-bea securiies fare less poorly han heir beas would sugges ( bea compression ): 4 Proposiion 3. s P The percenage price sensiiviy wih respec o funding shocks / s is he same P for all securiies s. A higher independen variance of funding shocks compresses beas of all securiies owards 1, and he bea of he BAB facor increases if his is unanicipaed. In addiion o he asse-pricing predicions ha we have derived, funding consrains naurally also affec agens porfolio choices. In paricular, he more consrained invesors il owards riskier securiies in equilibrium, whereas less consrained agens il owards safer securiies wih higher reward per uni of risk. To see his, we wrie nex period s securiy values as M M P E P b P E P e (13) 4 Garleanu and Pedersen (2009) finds 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 highmargin 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 so, in his case, higher funding liquidiy risk means ha beas are compressed owards one. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 12

13 where b is a vecor of marke exposures and e is a vecor of noise ha is uncorrelaed wih he marke. Wih his, we have he following naural resul for he agens posiions: Proposiion 4. Unconsrained agens hold risk free securiies and 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 are idenical expec ha s has a larger marke exposure han k, b s k b, hen any consrained agen j wih greaer han average Lagrange j muliplier, j agen wih., holds more shares of s han k, while he reverse is rue for any We nex urn o he empirical evidence for Proposiions 1-3. We leave a formal es of Proposiion 4 for fuure research, alhough we discuss some suggesive evidence in he conclusion. II. Daa and Mehodology The daa in his sudy are colleced from several sources. The sample of U.S. and global socks includes 50,826 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 December Beas are compued wih respec o he CRSP value weighed marke index. The global 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 1984 and December We assign individual issues o heir corresponding markes based on he locaion of he primary exchange. Beas are compued wih Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 13

14 respec o he corresponding MSCI local marke index 5. All reurns are in USD and excess reurns are above he US Treasury bill rae. We consider alphas wih respec o he marke and US facor reurns based on size (SMB), book-o-marke (HML), momenum (UMD), and liquidiy risk. 6 We also consider a variey of oher asses and Table II conains he lis insrumens and he corresponding daa availabiliy ranges. We obain U.S. Treasury bond daa from he CRSP US Treasury Daabase. Our analysis focuses on 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 December Reurns are an equal-weighed average of he unadjused holding period reurn for each bond in he porfolios. 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. 7 Our analysis focused on monhly reurns (in excess of he 1- monh Treasury bill) on 4 aggregae US credi indices wih mauriy ranging from one o en years and nine invesmen grade and high yield corporae bond porfolios wih credi risk ranging from AAA o Ca-D and Disressed 8. The daa cover he period beween January 1973 and December 2009 alhough he daa 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 is colleced from a variey of sources and conains daily reurns on fuures, forwards or swaps 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 nex mos liquid conrac. The 5 Our resuls are robus o he choice of benchmark (local vs. global). We repor hese ess in he Appendix. 6 SMB, HML, UMD are from Ken French s websie and he liquidiy risk facor is from WRDS. 7 The daa can be downloaded a hps://live.barcap.com 8 The disress index was provided o us by Credi Suisse. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 14

15 rolling dae s convenion differs across conracs and depends on he relaive liquidiy of differen mauriies. The daa cover he period beween 1963 and 2009, alhough he daa availabiliy varies depending on he asse class. For more deails on he compuaion of reurns and daa sources see Moskowiz, Ooi, and Pedersen (2010), Appendix A. For equiy indexes, counry bonds and currencies, beas are compued wih respec o a GDP-weighed porfolio, and, for commodiies, 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 Garleanu and Pedersen (2009) and ohers). The TED spread is defined as he difference beween he hree-monh EuroDollar LIBOR rae on he hree-monh U.S. Treasuries rae. Our TED daa run from December 1984 o December Esimaing Beas We esimae pre-ranking beas from rolling regressions of excess reurns on excess marke reurns. Whenever possible we use daily daa raher han monhly since he accuracy of covariance esimaion improves wih he sample frequency (see Meron (1980)). If daily daa is available we use 1-year rolling windows and require a leas 200 observaions. If we only have access o monhly daa we use rolling 3- year windows and require a leas 12 observaions 9. Following Dimson (1979) and Fama and French (1992) we esimae beas as he sum of he slopes in a regression of he asse s excess reurn of he curren and prior marke excess reurns: K f ˆ ˆ M f k k k k 0 r r r r ˆ ˆ TS K k0 ˆ k (14) The addiional lagged erms capure he effecs of non-synchronous rading. We 9 Daily reurns are no available for our sample of US Treasury bonds, US corporae bonds and US credi indices. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 15

16 include lags up o K = 5 rading days. When he sample frequency is monhly, we include a single lag. Finally, in order o reduce he influence of ouliers, we follow Vasicek (1973) and Elon, Gruber, Brown, and Goezmann (2003) and shrink he XS bea esimaed using he ime-series ( ) owards he cross-secional mean ( ) TS i ˆ ˆTS (1 ) ˆ XS w w (15) i i i i For simpliciy, raher han having asse-specific and ime-varying shrinkage facors as in Vasicek (1973), we se w = 0.5 and XS bu our resuls are very similar eiher way. 10 =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 securiy beas. 11 The amoun of shrinkage does affec he choice of he hedge raio in consrucing zero-bea porfolios since i deermines he relaive size of he long and he shor side necessary o keep he hedge porfolios bea-neural a formaion. To accoun for he fac ha hedge raios can be noisy, our inference is focused on realized abnormal reurns so ha any mismach beween ex ane and realized beas is picked up by he realized loadings in he facor regression. Our resuls are robus o alernaive bea esimaion procedures as we repor in he Appendix. Consrucing Being-Agains-Bea Facors We consruc simple porfolios ha are long low bea securiies and shor high bea securiies, hereafer BAB facors. To consruc each BAB facor, all securiies in an asse class (or wihin a counry for global equiies) are ranked in ascending order on he basis of heir esimaed bea. The ranked socks are assigned The Vasicek (1973) Bayesian shrinkage facor is given by 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 here is large dispersion of beas in he cross secion. Pooling across all socks, in our US equiy daa, he shrinkage facor w has a mean (median) of 0.51 (0.49). 11 Using alernaive rolling window, lag lengh or differen shrinkage facors does no aler our main resuls. We repor robusness checks in he Appendix. i, TS Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 16

17 o one of wo porfolios: low bea and high bea. Securiies are weighed by he ranked beas and he porfolios are rebalanced every calendar monh. Boh porfolios are rescaled o have a bea of one a porfolio formaion. The BAB is he zero-cos zero-bea porfolio (9) ha is long he low-bea porfolio and shors he high-bea porfolio. For example, on average he U.S. sock BAB facor is long $1.5 worh of low-bea socks (financed by shoring $1.5 of risk free securiies) and shor $0.7 worh of high-bea socks (wih $0.7 earning he risk-free rae). III. Being Agains Bea in Each Asse Class Cross secion of sock reurns We now es how he required premium varies in he cross-secion of beasored securiies (Proposiion 1) and he hypohesis ha long/shor BAB facors have posiive average reurns (Proposiion 2). 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 fla securiy marke line. Hence, consisen wih Proposiion 1 and wih Black (1972), alphas decline almos monoonically from low-bea o high-bea porfolios. Indeed, alphas decline boh when esimaed relaive o a 1-, 3-, 4-, and 5-facor model. Also, Sharpe raios decline monoonically from low-bea o high-bea porfolios. As we discuss in deail below, declining alphas and Sharpe raios across bea sored porfolios is a general phenomenon across asse classes. As a overview of hese resuls, he Sharpe raios of all he bea-sored porfolios considered in his paper are ploed in Figure B1 in he Appendix. The righmos column of Table III repors reurns of he being-agains-bea (BAB) facor of Equaion (9), ha is, a porfolio ha is long a levered baske of low-bea socks and shor a de-levered baske of high-bea socks such as o keep he porfolio bea-neural. 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 0.69% per monh (-saisic = 6.55). Addiionally Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 17

18 adjusing reurns for Carhar s (1997) momenum-facor, he BAB porfolio earns abnormal reurns of 0.55% per monh (-saisic = 5.12). 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 0.46% per monh (-saisic = 2.93) 12. 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. Figure B2 in he Appendix plos he annual abnormal reurns of he BAB sock porfolio. We nex consider bea-sored porfolios for global socks. 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 global porfolios where all global socks are pooled ogeher (Table IV), and we consider resuls separaely for each counry (Table V). The global porfolio is counry neural ha is socks are assignee o low (high) bea baske wihin each counry. 13 The resuls for our pooled sample of global equiies in Table IV mimic he U.S. resuls: Alphas and Sharpe raios of he bea-sored porfolios decline (alhough no perfecly monoonically) wih beas, and he BAB facor earns risk-adjused reurns beween 0.42% and 0.71% per monh depending on he choice of risk adjusmen wih -saisics ranging from 2.22 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 16 ou of 19, displaying a srikingly consisen paern across equiy markes. The BAB reurns are saisically significanly posiive in 9 counries. Of course, he small number of socks in our sample in many of he counries (wih some counries having only a few dozen securiies raded) makes i difficul o rejec he null hypohesis of zero reurn in each individual facor. Figure B3 in he Appendix plos he annual abnormal reurns of he BAB global porfolio. 12 Noe ha Pasor and Sambaugh (2003) liquidiy facor is available on WRDS only beween 1968 and 2008 hus cuing abou 50% of our observaions. 13 We keep he global porfolio counry neural since we repor resuls for equiy indices BAB separaely in able IX. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 18

19 Tables B1 and B2 in he Appendix repor facors loadings. On average, he U.S. BAB facor invess $1.52 long ($1.58 for Global BAB) and $0.71 shor ($0.84 for Global BAB). The larger long invesmen is mean o make he BAB facor marke neural since he long socks have smaller beas. The U.S. BAB facor realizes a small posiive marke loading, indicaing ha our ex-ane bea are measured wih noise. The oher facor loadings indicaes ha, relaive o high-bea socks, low-bea socks are likely o be smaller, have higher book-o-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 Appendix repors furher ess and addiional robusness checks. We spli he sample by size and ime periods. We conrol for idiosyncraic volailiy (boh level and changes) and repor resuls for alernaive definiion of beas. All he resuls ell a consisen sory: equiy bea-neural porfolios ha be agains beas earn significan risk-adjused reurns. Treasury Bonds Table VI repors resuls for US 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: 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 fla securiy marke line is no limied o he cross secion of sock reurns. Indeed, consisen wih Proposiion 1, alphas decline monoonically wih bea. Likewise, Sharpe raios decline monoonically from 0.73 for low-bea (shor mauriy) bonds o 0.27 for high-bea (long mauriy) bonds. Furher, he bond BAB porfolio delivers abnormal reurns of Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 19

20 0.16% per monh (-saisic = 6.37) wih a large annual Sharpe raio of Figure B4 in he Appendix plos he annual ime series of reurns. 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 on 1.65% per year. One way o achieve his reurn arge is o inves $1 in a porfolio of 10-year bonds as seen in Table VI. If insead he agen invess in 1-year Treasuries hen he would need o inves $4.76 if all mauriies had he same Sharpe raio. This is because 10-year Treasures are 4.76 imes more volaile han 1-year Treasuries. Hence, he agen would need o borrow an addiional $3.76 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. This is he case empirically. Empirically, he reurn arge can be achieved wih by invesing $2.7 in 1-year bonds. While a consrained invesor may sill prefer an unleveraged 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 2.7 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. In Table VII, he es asses are monhly excess reurns of corporae bond indexes wih mauriy ranging from 1 o 10 years. Table VII panel A shows ha he credi BAB porfolio delivers abnormal reurns of 0.13% per monh (-saisic = 4.91) wih a large annual Sharpe raio of Furher, alphas and Sharpe raios decline monoonically, wih Sharpe raios ranging from 0.79 o 0.64 from low bea (shor mauriy) o high bea (long mauriy bonds). Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 20

21 Panel B of Table VII repors resuls for porfolio of US credi indices where we ry 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 excess reurn on Barclay s US governmen bond index. We consruc es asses by going long he corporae bond index and hedging his posiion by shoring he appropriae amoun of he governmen bond index: CDS f f ( ) ˆ USGOV f r r r r 1( r r ), where ˆ 1 is he slope coefficien esimaed in an expanding regression using daa up o monh -1. One inerpreaion of his reurns series is ha i approximaely mimics he reurns on a Credi Defaul Swap (CDS). We compue marke reurns by aking equally weighed average of hese hedged reurns, and compue beas and BAB porfolios as before. Abnormal reurns are compued wih respec o a wo facor model: 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 (un-hedged) BAB facor for US credi indices in he righmos column of Table VII panel B. The addiion of he un-hedged 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 B of Table VII ell he same sory as Panel A: he CDS BAB porfolio delivers significan reurns of 0.08% per monh (-saisics = 3.65) and Sharpe raios decline monoonically from low bea o high bea asses. Figure B5 in he Appendix plos he annual ime series of reurns. Las, in Table VIII 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 (0.56% per monh wih a -saisics = 4.02), and declining alphas and Sharpe raios across bea sored porfolios. Figure B6 in he Appendix plos he annual ime series of reurns. Equiy indexes, counry bond indexes, foreign exchange and commodiies Table IX repors resuls for equiy indexes, counry bond indexes, foreign exchange and commodiies. The BAB porfolio delivers posiive reurn in each of he Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 21

22 four asse classes, wih annualized Sharpe raio ranging from 0.22 o The magniude of reurns is large, bu he BAB porfolios in hese asses are much more volaile and, as a resul, we are only able o rejec he null hypohesis of zero average reurn for global equiy indexes. We can, however, rejec he null hypohesis of zero reurns for combinaion porfolios han include all or some combinaion of he four asse classes, aking advanage of diversificaion. We consruc a simple equally weighed BAB porfolio. To accoun for differen volailiy across he four asse classes, in monh we rescale each reurn series o 10% annualized volailiy using rolling 3-year esimae up o moh -1 and hen equally weigh he reurn series and heir respecive marke benchmark. This corresponds o a simple implemenable porfolio ha arges 10% BAB volailiy in each asse classes. We repor resuls for an All fuures combo including all four asse classes and a Counry Selecion combo including only Equiy indices, Counry Bonds and Foreign Exchange. The BAB All Fuures and Counry Selecion deliver abnormal reurn of 0.52% and 0.71% per monh (-saisics = 4.50 and 4.42). Figure B7 in he Appendix plos he annual ime series of reurns. To summarize, he resuls in Table III IX srongly suppor he predicions ha alphas decline wih bea and BAB facors earn posiive excess reurns in each asse class. Figure A1 illusrae he remarkably consisen paern of declining Sharpe raios in each asse class. Clearly, he fla securiy marke line, documened by Black, Jensen, Scholes (1972) for U.S. socks, is a pervasive phenomenon ha we find across markes and asse classes. Puing all he BAB facors ogeher produces a large and significan abnormal reurn of 0.77% per monh (-saisics of 8.8) as seen in Table IX panel B. This evidence is consisen wih of a model in which some invesors are prohibied from using leverage and oher invesors leverage is limied by margin requiremens, generaing posiive average reurn of facors ha are long a leveraged porfolio of low-bea asses and shor a porfolio of high-bea asses. To furher examine his explanaion of wha appears o be a pervasive phenomenon, we nex urn o ess he cross-secional ime-series predicions of he model. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 22

23 IV. Bea Compression In his secion, we ess Proposiion 3 ha beas are compressed owards 1 during imes wih shocks o funding consrains. This model predicion generaes wo esable hypoheses. The firs is a direc predicion on he cross-secional of beas: he cross-secional dispersion in beas should be lower when individual credi consrains are more likely o be binding. The second is a predicion on he condiional marke beas of BAB porfolios: alhough bea neural a porfolio formaion (and on average), a BAB facor should end o realize posiive marke exposure when individual credi consrains are more likely o be binding. We presen resuls for boh predicions in Table X. We use he TED spread as a proxy of funding liquidiy condiions. Our ess rely on he assumpion ha high levels of TED spread (or, similarly, high levels of TED spread volailiy) correspond o imes when invesors are more likely o face shocks o heir funding condiions. Since we expec ha funding shocks affec he overall marke reurn, we confirm ha he monhly correlaion beween he TED spread (eiher level or 1-monh changes) and he CRSP value weighed index is negaive, around -25%. We es he model s predicions abou he dispersion in beas using our samples of US and Global equiies which have he larges cross secions of securiies. The sample runs from December 1984 (he firs available dae for he TED spread) o Table X, Panel A shows he cross-secional dispersion in beas in differen ime periods sored by likelihood of binding credi consrains for U.S. socks. Panel B shows he same for global socks. Each calendar monh we compue cross-secional sandard deviaion, mean absolue deviaion and iner-quinile range in beas for all socks in he universe. We assign he TED spread ino hree groups (low, medium, and high) based on full sample breakpoins (op and boom 1/3) and regress he imes series of he cross-secional dispersion measure on he full se of dummies (wihou inercep). Table X shows ha, consisen wih Proposiion 3, he crosssecional dispersion in beas is lower when credi consrains are more likely o be Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 23

24 biding. The average cross-secional sandard deviaion of US equiy beas in periods of low spreads is 0.47 while he dispersion shrinks o 0.35 in igh credi environmen and he difference is highly saisical significan (-saisics = ). The ess based on he oher dispersion measures and he global daa all ell a consisen sory: he cross-secional dispersion in bea shrink a imes where credi is more likely o be raioned. Panel C and D repors condiional marke beas of he BAB porfolios based on he credi environmen for, respecively, U.S. and global socks. We run facor regression and allow loadings on he marke porfolio o vary as funcion of he realized TED spread. The dependen variable is he monhly reurn of he BAB porfolio. The explanaory variables are he monhly reurns of he marke porfolio, Fama and French (1993) mimicking porfolios and Carhar (1997) momenum facor. Marke beas are allowed o vary across TED spread regimes (low, neural and high) using he full se of TED dummies. We are ineresed in esing he hypohesis ha ˆ MKT high ˆ MKT where ˆ MKT ( ˆ MKT ) is he condiional marke bea in imes when credi low high low consrains are more (less) likely o be binding. Panel B repors loading on he marke facor corresponding o differen ime periods sored by he credi environmen. We include he full se of explanaory variables in he regression bu only repor he marke loading. The resuls are consisen wih Proposiion 3: alhough he BAB facor is boh ex ane and ex pos marke neural on average, he condiional marke loading on he BAB facor is funcion of he credi environmen. Indeed, recall from Table III ha he realized average marke loading is an insignifican 0.03, while Table X shows ha when credi is more likely o be raioned, he BAB-facor bea rises o The righmos column shows ha variaion in realized beween igh and relaxed credi environmen is large (0.51), and we are safely able o rejec he null ha ˆ MKT ˆ MKT (-saisics 3.64). Conrolling for 3 or 4 facors does no aler he resuls, alhough loadings on he oher facors absorb some he difference. The resuls for our sample of global equiies are similar as shown and panel D. To summarize, he resuls in Table X suppor he predicion of our model ha here is bea compression in imes of funding liquidiy risk. This can be high low Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 24

25 undersood in wo ways. Firs, more discoun-rae volailiy ha affecs all securiies he same way compresses bea. A deeper explanaion is ha, as funding condiions ge worse, all prices end o go down, bu high-bea asses do no drop as much as heir ex-ane bea suggess because he securiies marke line flaens a such imes, providing suppor for high-bea asses. Conversely, he flaening of he securiy marke line makes low-bea asses drop more han heir ex-ane beas sugges. V. Time Series Tess In his secion, we es Proposiion 2 s predicions for he ime-series of he BAB reurns. When funding consrains become more binding (e.g., because margin requiremens rise), he required BAB premium increases and he realized BAB reurns becomes negaive. We ake his predicion o he daa using he TED spread as a proxy of funding condiions as in Secion IV. Figure 2 shows he realized reurn on he U.S. BAB facor and he (negaed) TED spread. We plo 3-years rolling average of boh variables. The figure shows ha he BAB reurns end o be lower in periods of high TED spread, consisen wih Proposiion 2. We nex es he hypohesis in a regression framework for each of he BAB facors across asse classes, as repored in Table XI. The firs column simply regresses he U.S. BAB facor on he conemporaneous level of he TED spread. Consisen wih Proposiion 2, we find a negaive and significan relaion, confirming he relaion ha is visually clear in Figure 2. Column (2) has a similar resul when conrolling for a number of conrol variables. The conrol variables are he marke reurns, he 1-monh lagged BAB reurn, he ex ane Bea Spread, and he Shor Volailiy Reurns. The Bea Spread is equal o ( ) / and measures he bea difference beween he long and S L S L shor side of he BAB porfolios. The Shor Volailiy Reurns is he reurn on a porfolio ha is shor closes-o-he-money, nex-o-expire sraddles on he S&P500 index, and measures shor o aggregae volailiy. Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 25

26 In columns (3) and (4), we decompose he TED spread ino is level and change: The Change in TED Spread is equal o TED in monh minus he median spread over he pas 3 years while Lagged TED Spread is he median spread over he pas 3 years. We see ha boh he lagged level and conemporaneous change in he TED spread are negaively relaed o he BAB reurns. If he TED spread measures ha agens funding consrain (given by in he model) are igh, hen he model predics a negaive coefficien for he change in TED and a posiive coefficien for he lagged level. Hence, he coefficien for he lagged level is no consisen wih he model under his inerpreaion of he TED spread. If, insead, a high TED spread indicaes ha agens funding consrains are worsening, hen he resuls could be consisen wih he model. Under his inerpreaion, a high TED spread could indicae ha banks are credi consrained and ha banks over ime ighen oher invesors credi consrains, hus leading o a deerioraion of BAB reurns over ime, if his is no fully priced in. Columns (5)-(8) of Table XI repors panel regressions for global sock BAB facors, and columns (9)-(12) for all he BAB facors. These regressions include fixed effec and sandard errors are clusered by dae. We consisenly find a negaive relaionship beween BAB reurns and he TED spread. In addiion o he TED spread, he ex ane Bea Spread, ( ) /, is of S L S L ineres since Proposiion 2 predics ha he ex ane bea spread should predic BAB reurns posiively. Consisen wih he model, Table XI shows ha he esimaed coefficien for he Bea Spread is posiive in all six regressions where i is included, and saisically significan in hree regressions ha conrol for he lagged TED spread. To ensure ha hese panel-regression esimaes are no driven by a few asse classes, we also run a separae regression for each BAB facor on he TED spread. Figure 3 plos he -saisics of he slope esimae on he TED spread. Alhough we are no always able o rejec he null of no effec for each individual facor, he slopes esimaes display a consisen paern: we find negaive coefficiens in 16 ou of he 19 asse classes, wih Credi and Treasuries being he excepions. Obviously he excepions could be jus noise, bu posiive reurns o BAB porfolios during Being Agains Bea - Andrea Frazzini and Lasse H. Pedersen Page 26