Betting against Beta (and Gamma) Using Government Bonds

Transcription

1 Federal Reserve Bank of New York Saff Repors Being agains Bea (and Gamma) Using Governmen Bonds J. Benson Durham Saff Repor No. 708 January 2015 his paper presens preliminary findings and is being disribued o economiss and oher ineresed readers solely o simulae discussion and elici commens. he views expressed in his paper are hose of he auhor and do no necessarily reflec he posiion of he Federal Reserve Bank of New York or he Federal Reserve Sysem. Any errors or omissions are he responsibiliy of he auhor.

2 Being agains Bea (and Gamma) Using Governmen Bonds J. Benson Durham Federal Reserve Bank of New York Saff Repors, no. 708 January 2015 JEL classificaion: G10, G12, G15 Absrac Purporedly consisen wih risk pariy (RP) asse allocaion, recen sudies documen compelling low risk rading sraegies ha exploi a persisenly negaive relaion beween Sharpe raios (SRs) and mauriy along he U.S. reasury (US) erm srucure. his paper exends his evidence on being agains bea wih governmen bonds (BAB gov ) in four respecs. Firs, ou-of-sample ess sugges ha excess reurns may have waned somewha recenly and ha he paern seems mos pronounced for USs given daa on en oher previously unexamined governmen bond markes. Second, BAB gov appears robus when hedged ex-ane agains covariance wih equiies and hereby does no resemble selling volailiy, bu hese resuls noneheless belie a possible ension raher han consisency beween leverage consrains and lowrisk invesing: namely, ha invesors bid longer-daed US prices higher (lower) under BAB (RP). hird, he fac ha Gaussian affine erm srucure models of USs also imply an invered SR schedule suggess ha invesors canno, in fac, realize abnormal reurns if hey are fully hedged o he underlying model facors, and BAB gov excess reurns are indeed no robus o exane consrains on exposure o he yield curve s principal componens. Fourh, some evidence suggess ha previous BAB gov resuls reflec coskew preferences, alernaive BAB gov sraegies hedged o coskew risks ex-ane forgo subsanial reurns, and here is no indicaion ha invesors can earn excess reurns being agains gamma. However, he sign of invesors coskew preferences in governmen bond markes remains ambiguous. Key words: asse pricing, erm srucure, risk pariy, being agains bea Durham: Federal Reserve Bank of New York ( jbenson.durham@ny.frb.org). he auhor hanks seminar paricipans a he Federal Reserve Bank of New York for helpful commens. he views expressed in his paper are hose of he auhor and do no necessarily reflec he posiion of he Federal Reserve Bank of New York or he Federal Reserve Sysem.

3 Inroducion Mouning evidence suggess ha low-risk invesing delivers superior resuls, noably boh across asse classes, in he conex of risk pariy (RP), as well as wihin hem, so-called being agains bea (BAB). Such findings conradic he inuiive noion ha higher expeced as well as realized reurns compensae invesors for aking risk, a leas as measured by he second momen of asse reurns. he effeciveness of hese invesmen sraegies reflecs a persisenly inverse relaion beween Sharpe raios (SRs) and bea, defined as he covariance of asse reurns wih marke porfolio reurns, and correspondingly, an insufficienly upwardsloped securiy marke line (SML). Besides a comprehensive sudy of several asse classes ha documens hese paerns (Frazzini and Pedersen, 2014), subsequen research repors ha excess reurns from BAB using global equiies are robus no only o size and momenum bu also o indusry classificaions (e.g., Asness e al., 2014). Even hough sudies have long noed he modes slope of he SML (e.g., Black, 1972; Black e al., 1972), which is surprising hrough he lens of he sandard wo-momen CAPM, he empirical lieraure on low-risk invesing seems far less exensive compared o oher anomalies such as value, size, and momenum. Moreover, arguably furher analyses of any financial marke anomaly end o graviae disproporionaely o risky asses, shares in paricular. 1 However, BAB resuls for he U.S. reasury (US) marke (BAB gov ) are among he mos compelling, as only hree of 28 oher BAB sraegies across asse classes have higher SRs, 2 and he SR for BAB gov, 0.81, is afer all jus as large if no greaer in fac han ha repored for U.S. equiies, 0.78 (Frazzini and Pedersen, 2014). 3 Given hese compelling findings, analyses of low-risk invesing migh evolve wih equivalen aenion o fixed income insrumens, 4 bu o dae his lieraure seems silen on jus why low-risk invesing works for governmen bonds, he defaul-risk-free asse class, as disinc from he mechanisms behind equiy marke behavior. Whaever he asse class, BAB should work as long as SRs decrease in bea, and he salien feaures of he hisorical US erm srucure daa do seem amenable o he sraegy. 5 Jus why migh compensaion per uni of risk decline wih mauriy and hereby make BAB gov sraegies profiable for invesors who are willing and able o lever? he moivaion for addiional analyses of BAB gov ranscends he obvious relevance for invesors, and low-risk excess reurn paerns are noeworhy for 1 Compare he number of sudies on, say, momenum in equiy as opposed o bond markes. 2 Frazzini and Pedersen (2014) do no examine non-us governmen bond markes. 3 Noe, however, ha he sample for equiies (USs) begins in 1926 (1952). Also, he corresponding excess reurn for BAB gov is 17 bps per monh (Frazzini and Pedersen, 2014, able 6, pg. 14), compared wih 70 bps for BAB using U.S. equiies (able 3, pg. 13), again given a longer sample for socks. 4 Early indicaions seem consisen wih prior paerns. For example, Baker e al. (2011), Asness e al. (2014), Chow e al. (2014), and Walkshäusl (2014) exclusively address he low-risk anomaly or BAB in he conex of equiies. 5 Duffee (2010), Fama and French (1993), and Campbell and Viceira (2001) each documen high (low) SRs a he shor (long) end of he curve. 1

4 moneary policymakers in a leas wo respecs. Firs, a key mechanism underlying BAB excess reurns are leverage consrains, which have obvious implicaions for broader financial condiions. his noion advances Black s (1972) relaxed CAPM wih resriced borrowing and suggess ha a sufficien number of individual as well as insiuional invesors, who noably seek high reurns, are limied in heir abiliy o borrow. hese resricions hereby bid up prices on high-bea asses, concomianly lower required reurns, and flaen he SML. BAB gov paerns in paricular highligh no he general level per se bu he schedule of (relaive) erm premiums across yield curves, an arguably negleced subjec ha can be relevan o marke monioring as well as cenral bank open marke operaions. For example, he schedule of forward erm premiums, or perhaps raher SRs, seems highly relevan o cenral banks ha endeavor o remove duraion risk mos efficienly from financial markes wih ourigh purchases of governmen bonds. 6 Second, he underlying empirical regulariy again, elevaed SRs a he fron end of he erm srucure may represen a puzzle if no an affron o cenral bankers who painsakingly hone heir communicaion sraegies. Invesors migh no only chronically require greaer compensaion per uni of volailiy around business cycle flucuaions bu also amid corresponding uncerainy abou he response of moneary policy, quie possibly o varying degrees over ime and across space. herefore, he exen o which SRs are greaer a shorer mauriies migh be assessed in a similar vein as, for example, evidence ha disanhorizon forward raes are excessively sensiive o macroeconomic news announcemens (Gürkaynak e al., 2005), a finding ha arguably reflecs unanchored expecaions around cenral bankers objecives. Evidence on BAB gov or simple analyses of SRs deailed furher in subsequen secions below may be relevan in his conex. A reasonable prior migh be ha more invered SR schedules correspond o less ighly moored expecaions around moneary policy goals or, perhaps more accuraely, nearer-erm objecives. Excess BABgov reurns, hen, conceivably represen compensaion for policy-relaed risks. In addiion o his shared relevance for invesors and policymakers alike, four lingering quesions moivae an exclusive, closer look a BAB gov. he firs issue is he simple bu pressing imperaive o es BAB gov ou of sample, which includes some furher analyses of he U.S. daa, including addiional focus on ime-variaion of reurns, as well as an exension o 10 oher previously unexamined governmen bond markes, including Germany, France, Neherlands, Belgium, Ialy, Spain, Japan, UK, Canada, and Swizerland. Beyond assessing he breadh of he sraegy, he cross-secional evidence may help uncover condiional informaion relevan o more efficien BAB gov sraegies. he second issue sems from he general noion ha he mechanisms ha drive BAB may fundamenally differ across asse classes, wih possible implicaions for he broader, aggregae issue of RP. Again, Frazzini and Pedersen (2014), drawing on Black (1972), argue ha leverage aversion primarily 6 See, say, Bernanke (2010). 2

5 generaes he resuls. his mechanism seems persuasive wihin risky asse classes, bu leverage aversion wih respec o governmen bond invesors may appear a odds wih RP iself, given he unique role of he (global) risk-free asse class in porfolios. 7 o consider any ension beween BAB and RP, as well as o evaluae flighs-o-qualiy (FQs) more broadly as an alernaive or complemenary explanaion for BAB gov, he following furher examines he covariance of BAB gov no vis-à-vis he wihin-asse-class bea bu wih respec o benchmarks ha include he yardsick risky asse class. Whereas addiional evidence on BAB gov covariance addresses possible mis-specificaion broadly reminiscen of he Roll (1977) criique, a hird issue regards poenial under-specificaion. Gaussian affine erm srucure models (GASMs), a more common asse-pricing framework for governmen bonds han he CAPM afer all, nes BAB gov bu hardly imply ha reurns are anomalous. Closed-form soluions o GASMs are flexible enough o capure he invered schedule beween SRs and mauriy (e.g., Duffee, 2010), wihou he (however persuasive) amendmens o he CAPM based on leverage aversion in Frazzini and Pedersen (2014). he problem for BAB gov as a proper anomaly is ha GASM-implied hedge raios by consrucion oulaw arbirage and herefore excess reurns, apar from porfolios formed from model-implied fiing errors (e.g., Duare e al., 2006). Levered porfolios ha are long low-beas based solely on he invered SR schedule, and comprehensively hedged wih respec o he underlying model facors, simply canno produce a free lunch. Perhaps no GASM calibraion comprises a saisfacory es given esimaion issues such a parameer sabiliy and sample selecion, bu he affine model framework readily raises he suspicion ha any anomalous aspecs of BAB gov reurns derive from an under-specified pricing kernel. Simple regressions of ex-pos BAB gov reurns on he principle componens (PCs) of he erm srucure as well as reurns on porfolios ha hedge broader exposure o PCs ex-ane are informaive on his score. he fourh issue addresses he fac ha he risk in low-risk invesing refers o he second and no he hird momen of asse reurns. Beyond leverage aversion heory and oher mechanisms, he following examines wheher BAB gov reurns owe in par o invesors coskew preferences. Persuasive behavioral argumens ha migh accoun for low-risk paerns for equiies, including over-confidence and represenaiveness, seem much less relevan for bonds. Noneheless, BAB gov porfolios may have demonsrably favorable coskew characerisics ha command a (negaive) premium, which in urn may help accoun for he low-second-momen risk phenomenon. Correspondingly, consrained BAB gov sraegies ha neuralize coskew exposures may generae lower reurns. And, jus as leverage-consrained invesors may reach for yield and hereby bid up prices o earn second-momen-based premiums, o boos gains hey may 7 See also Adrian e al. (2014), he firs sudy o conduc cross-secional asse pricing ess ha direcly include in he pricing kernel measures of leverage, namely from financial inermediary balance shee daa from he Federal Reserve s Flow of Funds daa. 3

6 similarly pressure required reurns on asses wih greaer coskew risks lower. his begs he quesion of wheher invesors can earn excess reurns being agains gamma wih governmen bonds (BAG gov ). Daa and Empirical Design hese empirical analyses sar wih a brief evaluaion of previous findings using Cener for Research in Securiy Prices (CRSP) daa on USs. Minor weaks o Frazzini and Pedersen (2014) include use of daily in addiion o monhly daa (as hey do for socks), marke-capializaion-based insead of equal weighs across mauriies, and alernaive windows for esimaion of he ex-ane (rolling) beas of 12 and 36 (12 and 59) monhs for he daily (monhly) frequency. he resuls also consider alernaives for shrinking he beas, following Vasicek (1973), 8 alhough he close relaion beween bea and mauriy (and duraion), as well as less suspeced measuremen error of mauriy, may sugges less need o make hese adjusmens for governmen bonds. Also, he relevan beas are based on an aggregaion of individual CUSIPs from he CRSP daa ino eigh mauriy caegories, which very closely bu no precisely follows Frazzini and Pedersen (2014, able 5, pg. 14). 9 Subsequen secions below examine sraegies based on linear programming and non-linear opimizaions wih key consrains, bu he formaion of he BAB gov porfolios for his secion follows he convenion inroduced in Frazzini and Pedersen (2014), as in 1 1 r r r r r (1) BABgov L f H f 1 L 1 H 1 where 1 1 r r w L H L H and L H ˆ L H w, wih r ( ˆ )he column vecor of reurns (beas) for each mauriy, and r f is he risk-free rae (i.e., he one-monh bill rae), as defined in Frazzini and Pedersen (2014) and in urn Asness e al. (2014). 10 All in all, daily (monhly) informaion is available for a sufficien cross 8 he shrinkage facor from Vasicek (1973) follows ˆ ˆ S 1 ˆ i wii wi wih w 1 where 2 2 is, XS 4 XS 1 i i, S i, S XS is he ime-series, S, (cross-secional, XS) variance of he esimaed beas. Frazzini and Pedersen (2014) cie his formula bu fix wi 0.6, based on heir resuls on U.S. equiy daa, and se ˆ XS 1 for all asses, including USs. 9 As noed in able 2, hese include 1 monh o 9 monhs, 9 monhs o 2 years, 2 o 3 years, 3 o 4 years, 4 o 5 years, 5 o 6.5 years, 6.5 years o 12 years, and 12+ years. Varying US issuance paerns presen some challenges for covering some mauriy buckes over exended periods. For example, he reasury issued no 30-year bonds from November 1978 hrough February Also,

7 secion of mauriies from June 1961 (Ocober 1957) hrough December Assuming a 12-monh lag o calculae L H based on daily (monhly) daa, he firs observaion he esimae of BAB gov following (1) for June 1962 is based on reurns from June 1961 hrough May he final observaion, December 2013, refers o esimaes of L H derived from reurns observed from December 2012 hrough November his produces a oal of 620 monhs. As noed previously, wih he obvious moivaion o explore he breadh of he sraegy, he analyses cover 10 addiional governmen bond markes and use zero-coupon yield curves from Bloomberg. 12 hese (monhly) daa are only comprehensively available from around he mid-1990s, and herefore he inferences addiionally comprise ou-of-sample emporal ess o a considerable degree. 13 Also, he inernaional panel evaluaes a broader scope of mauriies, 14 noably beyond he 10-year-plus caegory in Frazzini and Pedersen (2014) o include he 20- and 30-year secors in each case for a finer parsing of higher-bea mauriies. 15 Resuls: Sensiiviy Analyses hese modes revisions o he empirical design produce resuls ha are largely consisen wih Frazzini and Pedersen (2014) wih respec o USs. able 1 repors he resuls based on daily and monhly reurns using CRSP daa, wih and wihou shrinkage of he beas, and alernaive lag lenghs for esimaion of he beas, namely one and hree years for daily daa and one and five years for monhly daa. Wihou excepion across hese eigh specificaions, he SRs for BAB gov exceed hose on he US index as well as he sock marke, and he excess reurns in some cases are greaer han hose on shares. Alhough he resuls using 1-year lags based on eiher daily or annual reurns are less pronounced, he magniude of he findings, w k z z L H ' where k 21 1 n z z z rank i, 1 n denoes a vecors of ones of lengh n, and x +(-) indicae he posiive (negaive) elemens of he vecor x. 11 Noe ha he resuls below refer only o monhly daa from June 1961 o ensure consisency wih he esimaes based on daily daa. However, earlier daa do no meaningfully change he inferences. 12 Longer ime series of fied yield curves exis for a few cases, bu Bloomberg perhaps provides he mos comprehensive coverage for he purpose of consisen comparisons. 13 he Bloomberg (Index) mnemonics for, say, 10-year yields, are C91010Y (Germany), C91510Y (France), C92010Y (Neherlands), C90010Y (Belgium), C90510Y (Ialy), C90210Y (Spain), C10510Y (Japan), C11010Y (UK), C10110Y (Canada), C25610Y (Swizerland), and C08210Y (U.S.). 14 o esimae realized reurns on, say, he 10-year zero, furher spline-based esimaes of 9-year, 11-monh zero-coupon yields are required, and he calculaions oherwise follow Adrian e al. (2013). In addiion, marke capializaion weighs are unavailable, and herefore he esimaes of marke reurns assume equal weigh across mauriies, following Frazzini and Pedersen (2014). 15 Noe also ha Frazzini and Pedersen (2014) do no repor daa on he longes mauriy bonds from Augus 1962 o December Also, lenghening he cross-secion wih finer delineaions of duraion migh be very useful for assepricing ess. 5

8 based on longer lags are similar o Frazzini and Pedersen (2014). For example, as noed in able 1 as well as Figure 1, daily daa based on shrunk beas esimaed wih a 3-year lag, which implies 596 monhs of reurns from May 1964 hrough December 2013, produce an average (calendar-ime, daily) excess reurn (SR) of abou 4.0 basis poins (bps) (1.01), compared o 2.7 bps (0.47) for he US marke and 4.0 bps (0.19) for equiies. he long (shor) side of he sraegy on average has a bea of abou 0.43 (1.85) or concomianly a required leverage raio of 2.35 (0.54), and he ex-pos bea (alpha) is very close o zero (posiive), as expeced and consisen wih Frazzini and Pedersen (2014). hese resuls, in urn, reflec a clear underlying negaive relaion beween SRs (alpha) and mauriy, which declines from abou ( ) o ( ) from he 1M-9M o he 10Y+ mauriy caegory, as noed in he op panel of able 2. Beyond his corroboraion of previous resuls, ime variaion in BAB gov reurns meris furher consideraion. Figure 1 also shows he rolling (geomeric) 5-year mean reurn from he sraegy, he black line, which is noably lower oward he end of he sample and approaches zero, and indeed he SR from 1992 decreases o 0.74 from Alhough no necessarily indicaive of a vanishing or disappearing anomaly per se, a reasonable prior is ha he resuls migh no be as compelling for he cross-secion of 10 oher markes, given ha daa are only available from he 1990s. Indeed, as noed in he las column of able 3, he corresponding Bloomberg daa for he US (considering he 36-monh esimaion lag lengh for bea), which again are monhly and also include he 20- and 30-year secors, produce a lower SR (0.353) over he January 1995 o November 2014 sample. hen again, he reurns o BAB gov based on hese limied Bloomberg daa noneheless sugges a comparaively profiable sraegy, as he SRs during his paricular sample for he US and equiy markes, as well as for a balanced porfolio, 16 are clearly lower a 0.161, 0.129, and 0.196, respecively. In addiion, excess BAB gov (calendar-ime, monhly) excess reurns are 61.6 bps, compared o 29.7, 57.4, and 43.6 bps, respecively. However, he same canno be said as assuredly for a number of oher markes, and herefore he resuls o BAB gov may be somewha sensiive o no only emporal bu also spaial ou-of-sample ess. able 3 also repors ha BAB gov has a lower (greaer) SR han he local governmen bond (sock) marke in each of he 10 cases. Also, BAB gov excess reurns exceed hose on he corresponding equally-weighed governmen bond in all cases excep Canada, bu he differences are arguably minimal compared o he U.S., wih perhaps he lone excepion of daa on Japan, which produces an excess reurn on BAB gov of 50.2 bps, compared o 20.1 bps on he consruced governmen bond index over he sample. Also, BAB gov excess reurns are greaer han hose on shares for six of he 10 non-u.s. cases Germany, France, Neherlands, Belgium, Ialy, and 16 he balanced porfolio is an equally-weighed average of reurns on he local sock marke, measured by he MSCI local currency, gross oal reurn index (e.g., GDDU), and he governmen bond index, in urn an equally-weighed average of he 10 mauriy poins, which include 3 and 6 monhs as well as 1, 2, 3, 5, 7, 10, 20, and 30 years. he Bloomberg counry (Index) codes follow GR (Germany), FR (France), NE (Neherlands), BE (Belgium), I (Ialy), SP (Spain), JN (Japan), UK (Unied Kingdom), CA (Canada), SZ (Swizerland), and US (Unied Saes). 6

9 Spain bu on balance, hese resuls seem o conras meaningfully wih hose of he U.S., even allowing for he differing emporal coverage of he CRSP and Bloomberg daa. Furhermore, his general inference is insensiive o assumpions regarding lag lenghs, namely for 12- and 60-monhs, in addiion o he 36-monh lengh for bea esimaions summarized in able 3. o consider some explanaions for he discrepancy, again BAB gov profiabiliy ress on a downwardsloping schedule of SRs along he erm srucure, and as Figure 2 indicaes, his paern seems common bu perhaps no universal given his cross secion of governmen bond markes. he schedule for he U.S., assigned o he back of he surface, is he mos clearly downward sloping, bu Germany, say, posiioned a he fron, displays a noiceably less pronounced bu noneheless consisen paern. In addiion, Figure 3 corresponds o Figure 2 bu focuses on comparisons wih Germany, Japan, he U.K, and Swizerland and shows a similarly inverse bu weaker relaion compared o he U.S. along he curve, which in urn reflecs less compelling BAB gov excess reurns. 17 his cross-secional evidence may be consrucive in idenifying hresholds ha signal abnormal BAB gov reurns, bu he resuls also may moivae a closer examinaion of he ransmission mechanisms relevan for BAB gov, or perhaps USs in paricular as a disinc global asse class. More on Covariance: From FQs o ension beween BAB and RP Indeed, he dearh of compelling evidence across oher governmen bond markes migh reflec unique demand for longer-daed $U.S. denominaed asses ha ranscends any leverage mechanism. Insofar as longer- as disinc from shorer-daed USs benefi from (global) safe-haven flows, or more precisely provided invesors anicipae his safe-haven demand, BAB gov reurns may owe less o leverage consrains per se. In fac, some sudies find ha GASM-based erm premiums are negaively correlaed wih he VIX (Li and Wei, 2012; Durham, 2008), or ha joinly-esimaed erm and equiy risk premiums are negaively correlaed (Durham, 2013a), findings ha a firs blush sugges ha BAB gov reurns should be lower precisely during bad imes. 18 However, his is only rue if decreases in erm premiums, noably scaled by risk, are greaer a he back end of he erm srucure. he resuls may hinge on invesors percepions and expecaions of he severiy of shocks during more benign FQs, invesors shed credi bu no duraion exposure, whereas during more severe shocks hey jeison boh credi and duraion risk for he relaive safey of shor-daed governmen paper. he former (laer) would sugges ha BAB does (does no) indeed 17 Higher SRs a he very fron end of he US as opposed o oher sovereign curves may owe more o he numeraor raher han he denominaor. A he very fron end of he erm srucure, perhaps surprisingly, greaer reurns are a subsanial par of he sory, however furher ou he yield curve i appears ha greaer U.S. volailiy plays he larger role. 18 Ilmanen (2011) argues ha BAB reurns resemble he profile of volailiy selling, which pays off in good raher han bad imes. 7

10 resemble volailiy selling. 19 In any case, praciioners migh worry ha by underweighing he long end, BAB gov foregoes meaningful insurance agains risky asses, and herefore excess reurns reflec compensaion for such exposure. 20 More broadly, he single-facor asse-pricing model behind previous BAB gov resuls is possibly misspecified, perhaps broadly reminiscen of he Roll (1977) criique, no because he relevan marke is unobservable bu raher given ha he narrow wihin-asse-class beas behind BAB gov exclude informaion abou covariance wih he risky asse class. wo ses of rudimenary esimaes are informaive. he firs is simply o examine BAB gov reurn loadings wih respec o broader porfolios,, including, firs, an markecapializaion-weighed index of USs and socks from he CRSP daase and, second, a pure equiy porfolio, as in BAB gov r r (2) able 2 also liss hese resuls, namely he ex-pos beas wih respec o boh measures of for BAB gov as well as he eigh mauriy buckes, alongside he corresponding bea for he US marke index wih respec o. In shor, BAB gov appears o hedge agains he balanced and equiy porfolios jus as well as, if no beer han, any secor of he erm srucure or he marke as a whole. Boh relevan beas are negaive ( and ), whereas he remaining esimaes along he erm srucure (from he 1-9 monh o he 12-year pluse caegories) are sricly posiive, albei similarly close o zero. Of course, a covariance marix migh also suffice, bu hese simple loadings seem inconsisen wih he view ha broader FQs produce BAB gov excess reurns. he second consrucs an alernaive BAB gov rading sraegy ha similarly levers (de-levers) he long- (shor-) side o have a bea of one as in Frazzini and Pedersen (2014), bu under he linear programming consrain ha he long side has no greaer bea wih respec o, alernaively, he balanced or he pure equiy porfolio han he shor side of he rade. More formally, he objecive funcion and consrain follow min w wl ˆ w w L s.. ˆ ˆ L S (3) 19 Peso problems may lurk in he background, as i could be ha even during a lenghy sample mild FQs obained, increasing he demand for longer- relaive o shorer-daed reasuries. Wha maers for BAB and leverage aversion is wheher invesors perceive longer-daed reasuries as a beer hedge agains risky asses, which in urn depresses (expeced) SRs furher ou he erm srucure. herefore, insead of, or in addiion o, leverage aversion, invesors wih balanced mandaes may be paying an insurance premium. 20 Frazzini and Pedersen (2014) repor wihin-asse-class ex-pos BAB beas and do no address wheher he long side has covariance properies ha command posiive premium. 8

11 where is he vecor of (US, wihin-asse-class) beas along he erm srucure, he weighs for he shor side follow max ws, ws, and reurns. 21 he reurns follow is he vecor of beas wih respec o he porfolios ha include sock 1 1 r w r r w r r BAB gov f f L S ˆ w ˆ L ws (4) he resuling porfolio herefore is no only neural o he US marke bea bu also o he shor side s covariance wih. 22 Pu anoher way, (3) represens BAB, bu wihou decreased relaive proecion agains sock marke declines o achieve low-risk governmen bond invesing, wihou sock marke covariance bes. Consisen wih he simple beas in able 2, he resuls in able 4 clearly sugges ha BAB gov does no increase exposure o he yardsick risky asse class, and he resuls conradic he noion ha BAB gov owes o FQ-relaed preferences for longer-daed USs. For example, he SRs for he consrained sraegy wih respec o he balanced (sock) porfolio is 1.22 (1.09), which compares quie favorably wih he resuls of he unconsrained linear programming SRs of abou Also, he drag on average (daily) excess reurns is modes, wih reurns of 10.1 bps and 8.8 bps, compared o he unconsrained porfolio excess reurns of 10.6 bps. 23 ables 6 and 7, which include he excess reurns and SRs, respecively, for he consrained sraegies across he 10 addiional governmen bond markes ell similar sories, wih only a few excepions. As such, hese resuls conradic he general noion ha BAB consiues volailiy selling. Insead BAB gov appears o pay off jus as well as a generic porfolio of USs in bad imes. Arguably, hese BAB gov resuls consiue furher crucial ou-of-sample confirmaion of RP, which ress on only one draw from hisory (hough admiedly a fairly long one), following Asness e al. (2012, pg. 48). However, even if he sraegy does no appear o reflec selling volailiy, BAB gov may sill represen a 21 An equaliy consrain e.g., ˆ w w L S ˆ is oo resricive. he objecive is o insure ex-ane ha he BAB gov porfolio does no afford less of a hedge han he shor side. 22 he linear programming consrain in effec forces he long side o include any secors of he reasury curve ha hedge covariance a leas as well as he shor side. 23 In addiion, he shor end of he US curve on average, bu no always over he full sample, has a lower covariance wih shares. ha is, he consrain binds a imes, evidenced by he fac ha he average bea of he shor side is abou (0.672), compared o he average bea for he unconsrained porfolio. Also no he fac ha reurns are near zero for he mos recen five years or so in he sample, when he long-end may have proved a decidedly beer hedge wih respec o shares near he zero lower bound. (In general, such a consrain is perhaps preferable o he ordinal weighing scheme for he sandard BAB gov porfolio, insofar as required leverage is much lower. herefore, invesors migh consider linear-programming wih sock marke covariance consrains insead of he somewha arbirary weighing scheme in Frazzini and Pedersen (2014), which may exclusively serve o lower leverage requiremens by adding relaively higher-bea asses o he long side.) 9

12 special if no problemaic case for RP. Consider invesors who paricipae exclusively in he US marke, and make all assumpions consisen wih BAB. hese paricipans may well overweigh he back end of he erm srucure, increasingly so wih leverage consrains. Such preferences push prices (yields) on, say, 10-year USs higher (lower), all else equal. Now consider anoher se of invesors wih balanced mandaes who paricipae in boh he US and he U.S. sock marke. he same leverage consrains bind, bu noably now across raher han wihin asse classes, as in he broader conex of RP. hese invesors may well overweigh no longer-daed reasuries bu riskier shares o boos reurns. Preferences push sock prices higher and expeced equiy reurns lower, bu hey also may pressure bond prices (expeced reurns) lower (higher) amid reaching for yield. herefore, invesors wih balanced mandaes (and especially hose also wih longerduraion liabiliies) may offse hose wih narrower consrains, and heerogeneous preferences buffe longerdaed yields. Wihou furher clarificaion, BAB says leverage-consrained invesors buy US duraion o boos reurns, whereas RP implies hey sell i o do so, 24 and he ne effec on longer-daed yields appears ambiguous. Unlike BAB evidence on risky asse classes, BAB gov is perhaps no so much any confirmaion as a conradicion for RP, absen a complicaed unold segmenaion sory ha reconciles heerogeneous invesor preferences. 25 GASMs and Under-Specificaion Besides mis-specificaion and ension beween BAB and RP, also consider poenial underspecificaion. ha is, alhough he CAPM and is exensions are common benchmarks for shares, he same canno be said for governmen bonds. Insead assess governmen bond SRs in he conex of GASMs, which afer all produce closed-form expressions for SRs (Duffee, 2010). 26 Again, BAB works if SRs decrease in bea (or duraion), and he relevan analyical quaniy is simply he parial derivaive of SRs wih respec o mauriy. 27 Whereas he CAPM mus be amended o incorporae borrowing consrains and capure he relaion, sandard GASMs do no. Wih he excepion of some simplisic models, such as Meron (1973), 28 calibraed GASMs are flexible enough o produce downward-sloping SR schedules wih respec o mauriy, bu noably under condiions ha preclude arbirage. As such, he model parameers, paricularly he generic marke price(s) of risk, migh very well sugges more general mechanisms besides leverage aversion. 24 Indeed, Asness e al. (2012, pg. 49) noe ha under BAB safer shor-mauriy U.S. reasuries offer higher riskadjused reurns han do riskier long-mauriy ones bu (w)ih respec o RP, bonds are he low-bea asse and socks he high-bea asse, and he benefi ha we documened is anoher empirical success of he heory. 25 Baker e al. (2011, pg. 7) noe ha balanced fund mandaes, wihou leverage consrain as well as perhaps bea raher han a marke-weigh allocaion o shares, conceivably buy low-bea socks (and presumably longer-daed reasuries), bu hey also sugges ha such mandaes, a abou wo percen of he oal, indeed hardly comprise he represenaive invesor capable of realigning asse prices. 26 A GASM migh have a CAPM represenaion, bu noneheless he issue is under-specificaion of he pricing kernel. 27 As Frazzini and Pedersen (2014, pg. 3) noe, erm premiums exiss a all horizons beyond he insananeous shor rae. 28 See Appendix 1. 10

13 o repea, BAB gov resuls are based on a univariae asse-pricing framework (e.g., Frazzini and Pedersen, 2014), bu he lieraure on erm srucure models ubiquiously considers muliple facors, including no jus he level bu also he slope and curvaure of he erm srucure as well as oher laen as well as spanned and un-spanned macroeconomic facors. Moreover, and unforunaely for he prospecs of a free lunch, he observaion ha esimaed GASMs produce he salien feaures of he daa, including SRs ha decline wih mauriy (Duffee, 2010), hardly confirms bu raher precludes excess reurns from BAB gov. Reconsider a BAB gov sraegy ha is long he fron end, P L, o exploi elevaed GASM-based SRs, wih corresponding shor posiions, P S, of required quaniy,, o offse he exposure of he long side o each of he underlying model facors, x. 29 Insead of levering he long and shors sides up and down o a bea equal s s o one, as in (1), he hedge raios, 1 and 2, for a 2-facor model, which can be rivially exended o n- facors, follow S S 1 L P1 P1 P s 1 x1 x2 x1 s S S L 2 P2 P 2 P n1 x1 x2 x 2 nn n1 (5) Bu, hese porfolio weighs, very much akin o hose in, say, Langeieg (1980), produce a cerain reurn ha canno exceed he risk-free rae wihou arbirage (and in urn lead o he bond pricing equaion). In oher words, he relevan parial derivaives on he righ hand side of (4) are derived under he no-arbirage resricion, and herefore BAB gov can only work by he coincidence ha GASM-implied yields are persisenly lower (higher) han observed yields a he fron (back) end of he erm srucure, or if model fiing errors are somehow a funcion of mauriy. he very fac ha GASMs readily reproduce he required SR schedule suggess ha BAB gov porfolios ulimaely canno produce excess reurns, if comprehensively hedged. Of course, o asser ha any anomaly mus compensaive invesors for some unknown bu rue risk is unconsrucive, bu he issue is ha GASM-implied SRs may quesion more han confirm BAB gov as an anomaly per se. Any empirical illusraion of hese resricions, besides comprising a somewha pedanic exercise, is bese by he fac ha arguably no calibraion of a GASM is saisfacory, given esimaion issues such a 29 ha is, he porfolio, which includes as many bonds as underlying risks, is P saisfy L S P P 0. x x 11 L P S, wih he weighs chosen o

14 parameer sabiliy, no o menion sample selecion in forming rading sraegies. 30 However, wo simple ess wih much less srucure migh address some of he relaed issues, paricularly ha he asse-pricing model on which BAB is based is no as much mis-specified, as in he case of covariance wih risky asses, as under-specified wih respec o he mulivariae naure of bond yields. he firs is o examine BAB gov reurn loadings on he (firs hree) principal componens of he yield curve, 31 denoed by he vecor following able 2 also includes hese resuls, and noably BAB PC PC PC, 32 gov r PC (6) PC (0.0004) is safely significan wihin he 95 percen band and is jus as large in magniude as he corresponding esimae of ex-pos wih respec o overall US marke reurns ( ). 33 Moreover, he loadings on each PC are wihin he range of esimaes of he corresponding ex-pos beas for he eigh secions of he yield curve, and herefore BAB gov appears o be robus, wih no increased (conemporaneous) exposure o he underlying facors of he erm srucure. However, a second es is more punishing and comprises an alernaive BAB gov rading sraegy ha similarly levers (de-levers) he long- (shor-) side based on he bea wih respec o he US marke, bu under he consrain ha he long and shor sides have equivalen exposure o US marke PCs, following min w wl s.. ˆPC w, w, L, ˆ ˆ PC L i (7) where ˆ PC is he n m marix of ex-ane bea exposures across he erm srucure o he firs m PCs esimaed from s o 1, w, is he marke weigh of he i h of n mauriy caegories a ime, and reurns, i hen, generally follow (4). 34 he lower panels of able 4 include he resuls for hree consrains, including 30 In oher words, one could examine wheher he gap beween GASM-based fied and observed yields i.e., violaions of he no arbirage resricion decline wih duraion, bu uncerainy around parameer esimaion and sample selecion would complicae inferences from such an invesigaion. 31 Duffee (2010, pg. 3) suggess ha GASMs aribue he downward-sloping SRs o level and slope risk. ha is, invesors are compensaed for he risk ha he erm srucure jumps up; all bonds face his risk. Invesors are also compensaed for he risk ha ha slope of he erm srucure falls. Long mauriy bonds hedge his risk, while shormauriy bonds are exposed o his risk. If so, exposure o slope risk in paricular migh accoun for BAB gov reurns given he underweigh o he back end of he yield curve. 32 he PCs are derived from 1-, 2-, 3-, 4-, 5-, 10-, 20-, and 30-year fied zero-coupon yields over he full sample based on Gürkaynak e al. (2007). 33 able 2 assigns 0 o alpha esimaes wihin he 95 percen confidence inerval. 34 ha is, he PC beas for mauriy caegory i follow 12

15 slope (S); slope and curvaure (S-C); and level, slope, and curvaure (L-S-C). he SR for he sraegy wih he slope bea consrain (0.387) is favorable wih respec o eiher he sock or he US marke, bu he figure is much lower han ha for he unconsrained BAB sraegy (1.27), wih subsanially lower reurns, 1.4 bps compared o 10.6 bps (4.0 bps for he sandard sraegy). he igher consrains ha include curvaure as well as level exposure produce minimal excess reurns, 0.4 bps and 0.2 bps, respecively, and unremarkable SRs. In addiion, he cross-secional evidence similarly suggess ha PC-based consrains are puniive, as he SRs lised in able 6 under he S, S-C, and L-S-C consrains are negligible. ellingly also, he weighed beas of he long sides given ex-ane hedges for slope and curve no o menion for he level, slope, and curve increasingly approach uniy, which may simply sugges ha invesors canno simulaneously BAB and hedge hese PCs o meaningful effec. hus in shor, alhough ex-pos BAB gov reurns do no seem o load on conemporaneous PCs, ex-ane hedges agains exposure o he underlying PCs appear very cosly, which in urn raises quesions abou under-specificaion. Ye whaever hese empirical resuls, again GASM-implieddownward-sloping SR schedules do no necessarily imply excess BAB gov reurns. Skew Preferences he previous discussion of mis- and under-specificaion is confined o risk wih respec o he second momen of reurns. Anoher consideraion regarding BAB in general as well as BAB gov in paricular is invesors skew preferences, 35 which migh bu need no necessarily sem from managerial delegaion. 36 Indeed, he lieraure on hird-momen preferences wih reference o fla SMLs and high risk-adjused reurns on low-second-momen-risk asses is hardly new. As Kraus and Lizenberger (1976) repored in heir early sudy of he hree-momen CAPM, higher bea porfolios exhibi greaer marke coskew, which in urn requires a premium ha migh accoun for he lower-han-expeced second-momen-based slope of he PC PC i, i i, PC i, r where refers o he alernaive lagged windows. An ex-ane slope bea, for example, is he coefficien on he regression of, say, 2- o 3-year mauriy reurns on he second PC of he yield curve, esimaed over he, say, 12-monh rolling window. 35 Anoher possible explanaion for abnormal reurns on low-risk asses refers o delegaed managemen incenives, namely benchmarking (Baker e al., 2011), arguably represenaive of a broad limi o arbirage. Alhough absolue risk, or raher covariance wih marginal uiliy, should maer for invesors, ypical indusry pracice is o assess managerial skill wih respec some benchmark. In urn, alhough low-risk sraegies produce quie favorable SRs, informaion raios (IRs) may pale in comparison, and herefore even smar managed money does no offse he price impac of (irraional) demand for high-bea asses. 36 More recenly, aleb (2004) makes he opposie case for negaive skew preferences, bu only parially in he conex of delegaed asse managers who prefer blowups over bleeding. o he exen ha he represenaive invesor conforms o uiliy funcions broadly consisen wih prospec heory, he agen-based mechanisms ha Baker e al. (2011) describe are neiher necessary nor sufficien condiions for BAB. 13

16 SML. 37 Expeced excess reurns, bu also coskew, wih 0, as in e i E r, are a funcion of no only covariance wih marke porfolio reurns e ˆ 3 ˆ i i i sign Mm E r (8) where follows cov ri rm m ˆ i 3 E r m m 2 (9) 3 and M m is he hird cenral momen of marke reurns. 38 Kraus and Lizenberger (1976) posi ha high bea is a source for gamma, 39 and as such BAB migh seem less anomalous in he conex of a higher-momen CAPM. 40 hus, low-risk invesing wih respec o he second momen of reurns migh comprise high-risk invesing regarding he hird, and he lieraure does no address wheher BAB gov migh embed coskew risks. 41 Even so, much of he inuiion for skew preferences for risky asse classes seems less germane o governmen bonds, given he usual focus on he poenial for excepional upside raher han downside proecion. For example, Mion and Viorkink (2007) find ha volaile socks are more posiively skewed, and herefore command higher prices, owing o heir loery-like poenial cash flows. o a large degree, hese mechanisms sem from represenaiveness, which hardly seems relevan o longer-daed reasuries ha may benefi from FQs. Consider also he lieraure ha invokes overconfidence in he conex of skew. Dieher e al. (2002) argue consisen wih Miller (1977) ha, given impedimens o shor selling, 37 ransacion coss are anoher consideraion under his general rubric, and indeed bid-ask spreads can be paricularly puniive a he very shor end of he US bill curve (Duffee, 2010), again where he long side of BAB reasury porfolios are noably overweigh. Bu a he same ime, as Duffee (2010) and Figure 4 illusrae, he inverse relaion beween SRs and mauriy (or bea) is more monoonic han disconinuous and does no owe exclusively o he fron end. In addiion, he resuls in Frazzini and Pedersen (2014) do no res on exclusively overweighing he shores mauriies, as insead he low and he high bea porfolios include every secor along he erm srucure, albei wih greaer weighs on he exremes. 38 See Appendix 2 for rudimenary measuremen deails. 39 Kraus and Lizenberger (1976, pg. 1098) regress gamma on bea and repor a coefficien greaer han uniy (i.e., 4.5), and hey infer ha invesors have an aversion o increases in bea as a direc measure of sysemaic sandard deviaion and have a preference for increases in bea as a surrogae for proporionally greaer increases in sysemaic skewness (given posiive marke skewness). 40 An analogous explanaion for BAB success wih respec o shares refers o he observaion ha high-bea, volaile socks also end o be small and illiquid, which presumably in he conex of Baker e al. (2011) suggess ha insiuional invesors canno exploi he anomaly. An alernaive reasoning, as he following oulines, is ha skew preferences for loery-like payoffs accoun for lower expeced reurns. 41 Pu anoher way, explici consideraion of he coskew of BAB reurns should shed ligh on he exen o which leverage consrains drive he anomaly, as Ilmanen for example pis leverage consrains and skew preferences as alernaive explanaions. 14

17 opimiss influence over socks prices increases wih uncerainy, evidenced by disagreemen among invesors. herefore, volaile high-bea socks are overbid o he exen opimiss disproporionaely se prices, again under he assumpion ha oherwise offseing invesors have a general relucance or inabiliy o shor socks, noably no inconsisen wih leverage aversion. 42 he same mechanism may work for longerdaed reasuries, bu he logic conradics he documened posiive empirical correlaion (and analyical resul from GASM closed-form soluions) beween uncerainy and erm premiums (i.e., required reurns over shor raes). Convenion suggess ha greaer volailiy lowers, raher han increases, expeced reurns in governmen bond markes. Noneheless, coskew preferences may help accoun for BAB gov, bu hrough disinc channels. 43 I may seem de rigueur o es wheher any fixed-income arbirage sraegy resembles picking up nickels in fron of seamrollers (Duare e al., 2006), bu more imporanly, fixed income securiies may have fundamenal characerisics ha generae asymmeric reurn disribuions, as of course bond prices (nominal ineres raes) face an upper (lower) bound a par (zero). In addiion, pull o par is arguably more pronounced nearer he shor end of he erm srucure, which may be especially relevan for BAB gov. In fac, Fujiwara e al. (2013) repor condiional negaive skew using daily governmen bond excess reurns, paricularly a shorer mauriies, 44 which may imply greaer second-momen-based SRs a shorer horizons consisen wih BAB gov. Also, Durham (2008) argues ha expeced reurns, as measured by ex-ane GASMbased erm premiums in he U.S., posiively correlae wih Eurodollar-opion-implied shor-erm-ineres-rae skew bu does no address he schedule of SRs. 45 Ye o be sure, coskew raher han uncondiional skew should affec asse reurns. he quesion is wheher he coskew of longer-daed bonds makes hem pay off disproporionaely well, again per uni of duraion risk, compared o shorer-daed securiies in bad imes amid asymmeric shocks o boh he risk-free and he risky asse classes. he coskew of reurns boh wihin he reasury marke, as in he BAB evidence, or across asse classes, more relevan o RP may complemen 42 See Hong and Sraer (2012) also in his general conex. 43 he argumen is no ha he back end of he reasury curve has (poenial) loery-like reurn characerisics. Bu perhaps in conras o Ilmanen (2011, pg. 389), downside proecion from reasuries migh noneheless manifes hrough coskew preferences, in addiion o volailiy, correlaion, and jump premiums. 44 Even so, hey also find ha he Bowley coefficien, defined in Appendix 2 as a measure of skew nearer he cener of he disribuion and noably furher from he ails, is ofen posiive, which suggess ha negaive sandard skew in he sample owes o ouliers (i.e., he small probabiliy of a large and negaive excess reurn). 45 hree-momen CAPM ess for USs are also informaive wih respec o he ubiquious use of GASM, in which by definiion skew canno be priced. 15

18 or supplan leverage aversion heory behind BAB gov, 46 as invesors may no be so much reaching for yield amid leverage consrains as paying for heir preferred coskew exposure. 47 wo benchmarks for coskew are relevan, which reflecs he previous discussion of covariance. he firs is wihin asse classes, o follow Frazzini and Pedersen (2014), and he second is across hem, which addresses he noion ha skew preferences may operae hrough broader (balanced) asse allocaion channels. From a praciioner s perspecive, he quesion is wheher BAB gov unwiingly enails coskew risks, wih respec o eiher he US or he sock marke, and oward ha end, able 2 also repors BAB coskew wih differen enors along he governmen bond curve. Considering he US benchmark firs, again o address direcly he original BAB findings, ends o increase wih mauriy, no unlike, and ranges from an ex-pos (ex-ane) esimae of 0.50 (0.87) for he 1-o-9-monh caegory o 1.86 (1.40) for he longes duraion caegory. Noably, he ex-pos esimae is 0.44 for he BAB gov porfolio, and he key issue is conribuion o uncondiional US marke porfolio skew, which is posiive according o boh sandard (0.536) and Bowley skew (0.093) measures. 48 herefore, under he convenional assumpion ha invesors prefer posiive porfolio skew, commands a negaive premium, 49 and as such skew preferences may help accoun for previous BAB gov resuls (Frazzini and Pedersen, 2014), because longer mauriies bes conribue o he overall posiive il of US marke reurns over his sample. However, beyond he iniial BAB gov resuls, he second broader se of benchmarks imply no clear inference, as he corresponding esimaes of wih respec o he broader balanced porfolio of bonds and socks produce ambiguous relaions. BAB gov reurns have negaive coskew wih he balanced porfolio (-.0572), which has negaive uncondiional sandard (-0.973) as well as Bowley skew (-0.069), and hereby implies a posiive premium. Also, he for BAB gov is safely wihin he range across he erm srucure (from for he 3Y-4Y secor o for he 1M-9M caegory). Arguably, he conribuion of BAB o broader balanced porfolio skew as opposed o US marke skew is he more relevan consideraion for praciioners, paricularly wih balanced mandaes, bu he former resuls are noneheless relevan o he original BAB gov evidence. In any case, besides ex-pos BAB gov reurns, 46 Noe ha on average if longer-run reasuries favorably conribue o porfolio coskew, hey mus have posiive coskew when he marke is skewed posiively and negaive coskew when he marke is skewed negaively. Boh scenarios command a premium i.e., lower yields (higher prices) for duraion as reasury duraion reinforces benevolen posiive skew in he firs case bu offses undesirable negaive skew in he second. 47 he resuls and moivaion from Barone-Adesi e al. (2004) migh be analogous. hey repor coskew characerisics ha command a posiive (negaive) risk premium and hereby accoun for abnormally high (low) small (large) cap sock reurns, in perhaps he same way coskew migh help accoun for BAB gov excess reurns. 48 his finding is also consisen wih Chiang (2008), who uses monhly daa from 1976 hrough 2005 on a broader se of U.S. fixed income secors, including USs, corporae bonds, and MBS. 49 he negaive condiional skew on BAB reurns (-0.09) is broadly consisen wih picking up nickels in fron of seamrollers, whereas all poins on he erm srucure are posiively skewed. Bu, he Bowley coefficien is posiive, and besides, coskew maers for asse pricing. 16

19 anoher imperaive is o evaluae he cos of hedging coskew risk ex-ane, similar o he covariance and PC exposures discussed previously. able 7 repors he resuls from consrained opimizaions from low-risk governmen bond invesing wihou coskew bes. hese porfolios are long low-bea bonds as before, bu subjec o he consrain ha he long and shor sides have a weighed-average of uniy o mach he marke porfolio. he consrains follow boh linear programming as well as non-linear opimizaion. he long side for he former consrain, given ha is addiive like (Kraus and Lizenberger, 1976, pg. 1089), follows 50 min w wl s.. n 1 L, i, i, i 1 L, ˆ w ˆ n w ˆ 1 (10) where ˆ is he vecor of esimaes across he erm srucure, and w i, is he marke weigh of he i h of n duraion buckes a ime. he long side for he laer opimizaion, wih he same objecive funcion and a non-linear equaliy consrain, follows cov min w wl s.. wl, rrm, m, E r m, m, L, 3 ˆ (11) where r is he n 1 vecor of reurns across he erm srucure, and rm, is he corresponding marke reurn wih mean m, Correspondingly, he objecive funcion and he linear consrain for he shor side follow max w ˆ S, ws s.. n 1 ˆ ˆ S, i, i, i 1 w n w 51 Similarly, he objecive funcion and he linear consrain for he shor side follow max w ˆ S, ws s.. ws, rrm,, m Erm, m, 2 3 cov

20 he resuls sugges a meaningful drag on excess reurns, wih an offseing decline in volailiy ha sill resuls in favorable SRs. As lised in able 7, BAB gov excess reurns wih non-linear (linear) gamma consrains wih respec o he US marke is abou 1.2 (1.7) bps per monh, compared o 10.6 (4.0) bps per monh for he unconsrained (sandard) sraegy. he SR is abou 0.40 (0.58), compared o 1.27 (1.01) for he unconsrained porfolio (sandard mehod). he resuls are similar given he balanced porfolio benchmark, and he coss of he ex-ane coskew hedge are evidenced by an excess reurn of 1.3 (1.6) bps per monh under he nonlinear (linear) consrains, wih a corresponding SR of abou 0.40 (0.41). Reurning o ables 6 and 7, he resuls are broadly similar for he cross-secion of governmen bond markes, alhough perhaps he reducion in excess reurns and SRs are somewha more pronounced compared o he corresponding unconsrained opimizaions. Being agains gamma (BAG) and he marke price of coskew risk hese resuls on he hird-momen risks of BAB gov porfolios are somewha ambiguous. On he one hand, some resuls sugges ha BAB gov excess reurns (Frazzini and Pedersen, 2014) do compensae invesors for coskew exposure, bu on he oher, able 7 also suggess some residual excess reurns o he sraegy. Ye anoher naural and relaed empirical quesion is wheher, similar o bea, invesors can be agains gamma (BAG). Jus as BAB is long low risk, so is BAG, bu wih respec o he hird as opposed o he as second momen. Insead of shoring high-bea asses, he rading sraegy is o shor asses wih posiive (negaive) coskew, bu only when uncondiional marke skew is negaive (posiive). he corresponding long side overweighs asses wih negaive (posiive) coskew when marke skew is uncondiionally negaive (posiive). o grasp inuiion furher, recall ha under leverage consrains invesors reach for yield and bid up prices of high-bea asses, more formally given ha 0. By similar logic given he same leverage aversion, if skew preferences are also priced wih 0, hen o earn furher premiums and boos reurns under consrains, invesors migh similarly overprice asses ha have posiive (negaive) coskew wih he marke when he marke is negaively (posiively) skewed, all else equal. he long side of anoher rading sraegy follows 3 min w ˆ L, if Mm, 0 wl 3 max w ˆ L, if Mm, 0 w L s.. L, w ˆ 10 (12) 18