Bond Risk Premia. November 6, Abstract

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1 Bond Risk Premia John H. Cochrane and Monika Piazzesi November 6, 2001 Absrac This paper sudies risk premia in he erm srucure. We sar wih regressions of annual holding period reurns on forward raes. We find ha a single facor, which is a en-shaped funcion of forward raes, can predic one-year bond excess reurns wih an R 2 up o 45%. Though he reurn forecasing facor has a clear business cycle correlaion, i does no forecas oupu, and macro variables do no forecas bond reurns. The reurn forecasing facor does forecas sock reurns, abou as much as i would a 7 year duraion bond. Is forecas power is reained in he presence of he dividend price raio and he yield spread. We relae hese ime-varying expeced reurns o covariances wih various shocks, which is he same as finding he marke prices of risk ha jusify a yield VAR as an affine erm srucure model. The ime-varying expeced reurn can be enirely accouned for by a ime-varying risk premium for level-shocks in yields, and almos enirely accouned for by a ime-varying risk premium for moneary policy shocks. How could such an imporan facor have been missed? The reurn forecasing facor does very lile for undersanding yields. Convenional wo or hree facor models provide an excellen approximaion for yields, bu a poor approximaion for expeced reurns. Also, bond yields do no follow a monhly AR(1), wih a paern ha suggess measuremen error. Hence, if you follow he usual approach in erm srucure analysis, saring wih a monhly k-facor model chosen o minimize pricing errors, and hen finding implied annual reurn forecass, you compleely miss he forecasabiliy of annual reurns. Universiy of Chicago and NBER. Graduae School of Business, Universiy of Chicago, 1101 E. 58h S. Chicago IL john.cochrane@gsb.uchicago.edu. I graefully acknowledge research suppor from he Graduae School of Business and from an NSF gran adminisered by he NBER. UCLA and NBER. Anderson School, 110 Weswood Plaza, Los Angeles CA 90095, piazzesi@ucla.edu We hank Lars Hansen and seminar paricipans a he Sanford Insiue for Theoreical Economics 2001 for helpful commens. 1

2 1 Inroducion This paper sudies risk premia in he erm srucure of ineres raes. We sar by exending Fama and Bliss (1987) classic regressions. Fama and Bliss found ha he spread beween forward raes and one-year raes can predic excess bond reurns. Campbell and Shiller (1991) also find ha he slope of he erm srucure forecass bond reurns. We find ha one paricular linear combinaion of forward raes predics excess bond reurns even beer. I raises he R 2 in excess reurn forecasing regressions from abou 17% o abou 45%. Furhermore, he same linear combinaion of forward raes predics bond reurns a all mauriies, where Fama and Bliss relae each bond s reurn o a separae forward-spo spread. This finding paves he way for a simple represenaion of he erm srucure of ineres raes, in which we can use a small number of linear combinaions of yields as sae variables, raher han requiring each yield or forward rae as a separae sae variable, in order o forecas ha mauriy s reurn. Our reurn-forecasing facor is a en-shaped linear combinaion of forward raes. In a horse race, our reurn-forecasing facor compleely drives ou he separae forward-spo spreads used by Fama and Bliss. Expeced reurns should be relaed o covariances muliplied by risk premia. We find ha covariance wih a level shock o yields, muliplied by a ime-varying risk premium proporional o he reurn-forecasing facor, describes he ime-variaion in expeced bond reurns. We also sudy shocks o expeced bond reurns, inflaion shocks and moneary policy shocks, in an aemp o find a fundamenal shock o explain he ime-variaion in reurns. We find ha shocks o expeced reurns and inflaion do no help, bu a ime-varying premium for exposure o moneary policy shocks can explain he bulk of he ime-varying expeced bond reurn. These resuls correspond wih inuiion. The uncondiional mean excess reurn rises wih mauriy: long bonds reurn a bi more, on average, han shor bonds. The much larger ime-varying componen of expeced reurns also varies sysemaically wih mauriy. Long bonds expeced excess reurns load more heavily on our reurn-forecasing facor han do shor bonds expeced excess reurns. If we wish o explain hese facs wih a facor risk premium (he same for all bonds) muliplied by a covariance of each bond wih a shock, hen we mus find a shock ha affecs all bond reurns in he same direcion, and affecs long bond reurns more han shor bond reurns. Tha is exacly he feaure of a level shock; if he yield curve shifs up, bonds of successively longer mauriy (duraion) are successively more affeced. The moneary policy shock works precisely because i produces a similar level effec in bond reurns. A moneary policy shock moves all yields up or down ogeher; i hus produces a larger change in long-erm bond reurns han shor-erm bond reurns. This larger covariance wih long-erm bond reurns can, when muliplied by a facor risk premium, produce a larger expeced reurn for long erm bond reurns. (Why moneary policy shocks move long erm bonds so much is an ineresing puzzle, bu one we do no address here.) The ineresing par of his resul is he srong ime-variaion in bond expeced excess 2

3 reurns, and hence in he facor risk premium. Rises in yields mechanically produce declines in reurns, so he sign of he covariance beween reurns and he shock canno change. Hence, given ha expeced excess reurns are someimes posiive and someimes negaive, he facor risk premium mus change sign. Bonds whose prices will decline when here is a moneary policy shock someimes earn a posiive expeced reurn, and hus have lower prices han prediced by expecaions logic; hose same bonds, subjec o he same price decline when here is a moneary policy shock, will a oher imes earn a negaive expeced reurn, and hus have higher prices han prediced by expecaions logic. I all depends on he sae of he economy, as refleced in our en-shaped linear combinaion of forward raes. Our ime-varying risk premium specificaion resuls in an affine erm srucure model based on a VAR represenaion for bond yields. Thus, we have consruced an affine model ha can compleely capure he predicabiliy of bond reurns, and i exacly reproduces he prices of bonds (or heir linear combinaions) used as sae variables. Wheher or no bond reurn predicabiliy is consisen wih affine models has been a conenious poin in he lieraure. (See Fisher 1998, Duare 1999, Duffie 2001 and Dai and Singleon 2001.) We show ha, almos rivially, one can consruc an affine model o mirror complex paerns of bond reurn predicabiliy including hose we find in he daa. Why have exensive invesigaions of he erm srucure of ineres raes missed his reurn forecasing facor? Mos erm srucure analyses are performed by firs fiing an approximae k facor model in high frequency daa, and hen (if a all) looking a implied one-year forecasabiliy. Two ineresing feaures of he daa imply ha his procedure will miss reurn forecasabiliy. Firs, a monhly auoregression raised o he 12h power compleely misses he forecasabiliy of reurns a he one year horizon. The monhly bond daa do no follow an AR(1). Monhly yields are closely approximaed by an ARMA(1,1),which sugges an underlying AR(1) plus i.i.d. measuremen error. Though he deviaion from an AR(1) is small, he 12h power magnifies small misspecificaions. Second, he reurn forecasing facor is almos compleely unimporan for describing prices or yields. Level and slope facors unrelaed o he reurn forecasing facor can fi yields wih a high degree of accuracy. (Analyses ha include mauriies less han a year ofen find a hird curvaure facor, in order o reconcile mauriies longer han a year wih hose less han a year. This facor does no have he same shape as our reurnforecasing facor, whose weighs are symmeric around he 3 year forward rae.) For his reason, he reurn-forecasing facor is no recovered by radiional facor analysis or maximum likelihood esimaion of erm srucure models. You have o look a excess reurn forecass o see i. Reduced facor represenaions are sill ineresing of course. We find ha a hree facor represenaion, using a level and a slope facor, deriving ulimaely from he VAR shock covariance marix, and a reurn facor deriving from he expeced reurn regressions, does an excellen job of represening all his informaion in he erm srucure. 3

4 Our invesigaion is a lile unusual in ha we examine condiionally homoskedasic discree ime models, raher han coninuous ime models wih heeroskedasic shocks as is common in he erm srucure lieraure. Our specificaion is closer o he single index or laen variable models used by Hansen and Hodrick (1983) and Gibbons and Ferson (1985) o capure ime-varying expeced reurns. This fac has an imporan implicaion: hough many affine models use condiionally heeroskedasic shocks o produce curved paerns in he erm srucure and ime-varying expeced reurns, one does no have o use heeroskedasic shocks o obain hese resuls. For he expeced reurn - bea quesions we address in his paper, condiional homoskedasiciy is no likely o have a major effec on he resuls. The covariances of bond reurns wih yield curve shocks are really driven by he arihmeic of duraion, and do no change sign. Thus, we will have o undersand expeced reurns ha change sign wih a risk premium ha changes sign, raher han a covariance ha changes sign muliplied by a consan risk premium. Thus, while ime-varying covariances wih yield shocks (really, ime-varying variances of he yield shocks) can be par of he sory, hey mus be a secondary par, and ime-varying risk premia mus be he mos imporan par of he sory. Of course, since cov(r, η)[λ 0 (1 + λ 1 x )] = cov [r, η(1 + λ 1 x )] λ 0,wherer = reurn, η =shock,x = sae variable, and λ 0, λ 1 = parameers, much of wha one can express wih consan covariances and a ime varying risk premium can be expressed as a changing covariance and a consan risk premium, wih differen shocks. Wheher modeling condiional heeroskedasiciy is really imporan, in he end, will have o be judged by consrucing such a model and seeing wheher i gives an imporanly differen characerizaion of expeced reurns. 4

5 2 Fama-Bliss and beyond 2.1 Noaion We use he following noaion. Denoe he log price of a n year discoun bond a ime by p (n) p (n) =logpriceofn year discoun bond a ime. We use parenheses o disinguish mauriy from exponeniaion in he superscrip. The log yield is y (n) = 1 n p(n). We wrie he log forward rae a ime for loans beween ime + n 1and + n as f (n 1 n) = p (n 1) p (n) and we wrie he log holding period reurn from buying an n year bond a ime and selling i as an n 1yearbondaime +1as We summarize he excess reurn by hpr (n) +1 = p (n 1) +1 p (n). hprx (n) +1 hpr (n) +1 y (1) We use he same leers wihou n index o denoe vecors across mauriy, e.g. h i 0 y = y (1) y (2) y (3) y (4) y (5) h i 0 hprx +1 = hprx (2) +1 hprx (3) +1 hprx (4) +1 hprx (5) +1 h i 0 f = y (1) f (1 2) f (2 3) f (3 4) f (4 5) 2.2 Fama-Bliss regressions Fama and Bliss (1987) run a regression of one-year excess reurns on long-erm bonds agains he forward-spo spread for he same mauriy. The expecaions hypohesis predics a coefficien of zero nohing should forecas bond excess reurns. The firs panel of Table 1 updaes Fama and Bliss regressions o include more recen daa. We see in he one-year reurn regression ha he forward-spo spread moves essenially onefor-one wih expeced excess reurns on long erm bonds he expecaions hypohesis is exacly wrong a he one year horizon. 5

6 1 year reurns Change in y (1) mauriy b σ(b) R 2 b σ(b) R Table 1. Fama-Bliss regressions The 1 year reurns regression is ³ hprx (n) +1 = a + b f (n 1 n) y (1) + ε +1. The Change in y (1) regression is ³ y (1) +n 1 y (1) = a + b f (n 1 n) y (1) + ε +n 1. Sandard errors use he Hansen-Hodrick GMM correcion for overlap. Sample 1964:1-1999:12. Fama and Bliss also run a regression of muli-year changes in he one-year rae agains forward-spo spreads. The expecaions hypohesispredicsacoefficien of 1.0 he forwardraeshouldbeequaloheexpecedfuuresporae(plusajensen sinequaliy erm). Corresponding o he failure in he lef hand panel, he righ hand panel of Table 1 shows ha he 1-year forward rae (from year one o year wo, hence he n =2row) has essenially no power o forecas changes in he one-year rae one year from now. However, moving down he rows in he righ hand column, longer and longer forward raes correspond more and more o changes in spo raes, so ha a 4-year forward rae is wihin one sandard error of moving one-for-one wih he expeced change in he spo raes. This success for he expecaions hypohesis means ha he 5-year forward-spo spread does no forecas he four year reurn on 5-year bonds, hough i does forecas he one-yearreurnonsuchbonds 1. Fama and Bliss regressions are driven by robus sylized facs in he daa. When forward raes are higher han he one-year rae, all raes ofen rise subsequenly, as prediced by he expecaions hypohesis. However, his rise may ake 3 years or more o happen; here can be several years in which he forward raes are above he one-year rae before he ineres rae rise akes place. During hese years, holders of long-erm bonds make money. The period since 1987 is a grea ou-of-sample success for Fama and Bliss. The regressions have held up well since publicaion, unlike many oher anomalies. In paricular, forward-spo spreads were high in , bu ineres raes declined, and so long-erm bond holders made money. They los money when ineres raes rose in 1994, bu Fama-Bliss rading rule sill made money on average in he pos-publicaion sample. 1 Here and below, we use Fama and Bliss sar dae of 1964:01, and we do no use he more recenly available 1952:6-1963:12 daa. A visual inspecion of he earlier daa suggess a lo more measuremen error, which is naural given he hinner selecion of bonds and less ineres rae movemen. Also, he resuls are quie differen for his period for example, he Fama-Bliss coefficiens are all -1 raher han +1 so a a minimum one needs o hink of a ime-varying model o include he period. 6

7 2.3 The reurn-forecasing facor emerges While Fama and Bliss specificaion is he mos sensible for exploring he expecaions hypohesis and is failures, we are more ineresed in characerizing expeced excess bond reurns. For his purpose, here is no reason why only he 4-year forward rae spread should be imporan for forecasing he expeced reurns on 4-year bonds. Oher spreads may maer. Table 2 follows up on his hough by regressing he one-year reurn on long-erm bonds on all of he forward raes separaely. Coefficiens R 2 n a y (1) f (1 2) f (2 3) f (3 4) f (4 5) f f f,f +f 1/12 1/12 level Sandard errors Table 2. Regression of one year holding period reurns on forward raes, 1964: :12. The regression equaion is hprx (n) +1 = a + β 1 y (1) + β 2 f (1 2) β 5 f (4 5) + ε (n) +1 Sandard errors correc for overlapping daa. In he R 2 column, f repors R 2 from his regression. f, f 1 repors R 2 from a regression wih an 12 addiional monhly lag of all righ hand variables. (f +f 1/12 )/2 repors R 2 from a regression using a one-monh moving average of righ hand variables. level repors he R 2 from a regression using he level, no log, excess reurn on he lef hand side, e hpr(n) (1) +1 y. These regressions pick far more han he mached forward-spo spread as he bes regressor for holding period reurns. For example, he firs line of Table 2 suggess ha he f (2 3) f (4 5) spread is jus as imporan as Fama and Bliss variable, he f (1 2) y (1) spread, for forecasing he one-year reurns of wo-year bonds. These regressions more han double he R 2 from below 0.18 in Table 1 o across all mauriies. The 5 year rae R 2 is paricularly dramaic, jumping from 0.07 in Table 1 o 0.38 in Table 2.The op panel of Figure 1 graphs he regression coefficiens as a funcion of he mauriy on he righ hand side each row of Table 2 is a solid line of he graph. (For now, ignore he boom panel and he dashed line in he op panel.) The plo makes he paern clear he same funcion of forward raes forecass holding period reurns a all mauriies. Longer 7

8 Coefficiens of excess reurns and y(1) on forward raes y Coefficiens implied by resriced model Figure 1: Coefficiens in a regression of holding period excess reurns on he one-year yield and 4 forward raes. The op panel presens unresriced esimaes from Table 2. The boom panel presens resriced esimaes from a single-facor model, from Tables 4 and 5. The legend (2, 3, 4, 5) refers o he mauriy of he bond whose excess reurn is forecas. The x axis gives he mauriy of he forward rae on he righ hand side. The dashed line in he op panel gives he negaive of he regression coefficiens of he one yearyieldonhesamerighhandvariables. mauriies jus have greaer loadings on his same funcion. The paern of coefficiens suggess a en-shaped facor. We can, of course, run excess reurns on bond yields raher han forward raes. The fied values of he regression are exacly he same, since forward raes are linear funcions of yields. The paern of regression coefficiens is less prey. One migh worry abou logs vs. levels; ha acual excess reurns are no forecasable, bu ha he coefficiens in Table 2 only reflec 1/2σ 2 erms and condiional heeroskedasiciy. 2 We repeaed he regressions using acual excess reurns, e hpr(n) (1) +1 y on he lef hand side. The coefficiens are nearly idenical. The las column of Table 2 repors he R 2 from hese regressions, and hey are in fac slighly higher han he R 2 for he regression in logs. 2 We hank Ron Gallan for raising his imporan quesion. 8

9 2.3.1 Shor rae forecas Fama and Bliss also run regressions of changes in shor raes on forward-spo spreads. Such regressions are imporan, since he wo ingrediens of any erm srucure model are shor rae forecass plus risk premia. Table 3 presens regressions ha forecas he one-year rae using all he available forward raes. Again, hese resuls conras srongly wih he updaed Fama-Bliss regressions in Table 1. The R 2 in Table 1 was 0.001%, using he 2 year forward-spo spread as a righ hand variable. (The remaining rows in he righ half of Table 1 look a horizons longer han a year as well as using longer mauriy forward raes as regressors.) Using all of he forward raes in Table 3, he R 2 jumps o a subsanial 26%. Whereas i appeared ha he one-year change in he one-year rae was compleely unpredicable, i now appears ha all he forward raes aken ogeher have subsanial power o predic one-year changes in one-year raes. The coefficien of one-year rae changes on he lagged one-year rae is sill close o zero. There is a near-uni roo in ineres raes. Wheher one runs he regression in levels or changes makes no difference, of course, excep for he inerpreaion and value of R 2,andbyadifference of 1.0 on he coefficien on y (1). lhv y (1) f (1 2) f (2 3) f (3 4) f (4 5) R 2 y (1) +1 y (1) y (1) Sandard errors (same for boh regressions): Table 3. Regression of one year yields on forward raes, 1964: :12. The regression equaion is lhv +1 = a + β 1 y (1) + β 2 f (1 2) β 5 f (4 5) + ε +1 where lhv is eiher he level or he change in he one-year rae y (1) +1 as indicaed. Sandard errors correc for overlapping daa. The one-year yield regression conains no informaion ha is no already conained in he holding period reurn regressions. The holding period reurn of wo year bonds, which are sold as one year bonds nex year, conains a forecas of nex year s one-year rae. Mechanically, hprx (2) +1 = p (1) +1 p (2) y (1) = y (1) +1 p (2) + p (1) = y (1) +1 + f (1 2). (1) Thus, he regression of he one-year yield on our variables should give exacly he negaive of he coefficiens of he wo year holding period reurn on he same variables, wih a 1.0 difference in he coefficien on f (1 2). We include in Figure 1 he negaive of he 9

10 one-year yield forecasing coefficiens from he second row of Table 3, and you can see his paern exacly. More deeply, he ideniy (1) implies ha he forward-spo spread equals he change in yield plus he holding period excess reurn, and hence, using any se of forecasing variables, ³ ³ E y (1) +1 y (1) = f (1 2) y (1) E hprx (2) +1. (2) (Fama and Bliss use his ideniy as well.) In Fama and Bliss regressions, he forwardspo spread corresponds almos one o one o changes in expeced reurns boh componens on he righ hand side vary, bu hey vary in equal amouns, so he one-year rae is unpredicable. Now we have variables ha forecas he holding period reurns beyond he forward-spo spread. (2) implies ha hose variables mus also forecas changes in he spo rae. In his way, he forecasabiliy of he spo rae documened in Table 3 does no mean ha he expecaions hypohesis is working, i means ha he spo rae mus be predicable precisely because he expecaions hypohesis is no working Addiional lags We invesigaed wheher addiional lags of forward raes help o forecas bond reurns. One addiional monhly lag does ener wih boh saisical and economic significance. InTable2,wereporheR 2 of his regression, in he column labeled f,f 1/12. The R 2 rise by abou 0.05 o Raher han add hem o a able, Figure 2 plos he coefficiens from hese regressions. You can see ha he shape of he coefficiens is roughly he same a he firs and second lag. The daa seem o wan a one-monh moving average of forward raes o predic bond ³ reurns. We ran ³ a regression wih his resriced specificaion, i.e. hprx (n) +1 on y (1) + y (1) 1/12 /2, f (1 2) + f (1 2) 1/12 /2, ec. Figure 2 includes a plo of he coefficiens, and Table 2 includes he R 2 in he column f + f 1/12 /2. TheR 2 is lowered by no more han 0.01 by his addiional resricion, and i is no rejeced saisically, so his seems he bes way o include he lagged informaion. This finding suggess measuremen error in he forward raes, so ha he rue forward rae is beer recovered by he moving average. Addiional monhly lags or a one year lag add lile o he regression. Despie he small increase in forecas power available from an addiional lag, we focus our aenion on specificaions ha use only he curren variables f, as his drasically simplifies he analysis. Then we reurn o a reamen of he exra lags, while reconciling hese annual horizon regressions wih a monhly VAR represenaion for bond yields. 10

11 Curren One monh lag Average of curren and one monh lag Figure 2: Coefficiens in a regression of bond excess reurns on he one year yield and 1 o 4 year forward raes, including an exra one-monh lag. The op panel plos he coefficiens of hprx (n) +12 on forward raes a ime, while he middle panel plos he coefficiens on forward raes a ime 1/12. The boom panel presens he coefficiens of hprx (n) +1 on a one monh moving average of forward raes a and a 1/12. Sample A single facor for bond expeced reurns The paern of coefficiens in Figure 1 cries for us o describe expeced excess reurns of bonds on all mauriies in erms of a single facor, as follows. ³ hprx (n) +1 = a n + b n γ 0 + γ 1 y (1) + γ 2 f (1 2) γ 5 f (4 5) + ε (n) +1 (3) b n and γ n are no separaely idenified by his specificaion, since you can double all he bs and halve all he γs. We normalize he coefficiens by imposing ha he average value of b n is one, and he average value of a n are zero 1 4 5X b n =1; n=2 5X a n =0 n=2 Wih his normalizaion, we can fi (3) in wo sages. Firs, we esimae γ by running 11

12 he regression 1 4 5X n=2 hprx (n) +1 = γ 0 + γ 1 y (1) + γ 2 f (1 2) γ 5 f (4 5) + ε +1 (4) = γ 0 + γ 0 f + ε +1. The second equaliy inroduces he noaion γ, f for corresponding 4 1 vecors. Then, we can esimae he a n,b n by running he four regressions hprx (n) +1 = a n + b n (γ 0 + γ 0 f )+ε (n) +1, n =2, 3, 4, 5. We use GMM sandard errors o correc for he fac ha γ 0 f is a generaed regressor, along wih serial correlaion due o overlap. We consider he addiional resricion a n =0 ha he inerceps as well as he slope coefficiens follow he single-facor model. This procedure is consisen. While one can esimae he parameers wih somewha greaer asympoic efficiency (essenially, using he esimaed covariance marix o find a weighed sum in (4)) we prefer he clariy of he wo-sage OLS procedure. This is a resriced model. We describe he (4 mauriies (5 righ hand variables + 5 inerceps) = 25 unresriced regression coefficiens wih (4 as + 4 bs + 6 γs - 2 normalizaions) = 12 parameers, or, if a n = 0 wih 9 parameers. The essence of he resricion is ha a single linear combinaion of forward raes γ 0 + γ 0 f is he sae variable for ime-varying expeced reurns of all mauriies. Tables 4 and 5 presens he esimaed values of γ, a and b. γ 0 γ 1 γ 2 γ 3 γ 4 γ 5 R 2 Esimae Sd. error Table 4. Esimaed common facor in bond expeced reurns. The regression is 1 4 5X n=2 hprx (n) +1 = γ 0 + γ 1 y (1) + γ 2 f (1 2) γ 5 f (4 5) + ε +1 γ 0 has unis of annual percen reurn The γ 1 γ 5 esimaes in Table 4 are jus abou wha one would expec from inspecion of Figure 1. The loadings b n of expeced reurns on he common facor presened in Table 5 increase smoohly wih mauriy. The R 2 intable5arehesameasintable3owo significan digis. This fac indicaes ha he cross-equaion resricions implied by he model (3) ha bonds of each mauriy are forecas by he same porfolio of forward raes do no damage o he forecas abiliy. 12

13 Mauriy n a n s.e. a n σ2 n b n s.e. s.e. OLS R Table5. Esimaeofeachexcessreurn s loading on he reurn-forecasing facor. The regression is hprx (n) +1 = a n + b n (γ 0 + γ 0 f )+ε (n) +1 where hprx denoes bond reurn less one year rae, γ are he esimaes from Table 4, and f denoes he vecor of all forward raes. a n has unis of percen annual reurn. The s.e. columns are GMM sandard errors. They correc for he fac ha γ is esimaed, by considering his esimae ogeher wih he regression of Table 4 as a single GMM esimaion. The s.e. OLS column gives convenional sandard errors including he Hansen-Hodrick correcion for overlap. The sandard errors ha correc for he fac ha γ is a generaed regressor are much smaller han he s.e. OLS convenional (equaion-by-equaion) sandard errors ha rea γ as a fixed number. The second se of regressions, each holding period reurn on he common facor, canno impose he resricion P n b n = 4. Tha resricion is imposed in sample by he firs regression. Imposing ha resricion in sample removes (places on γ) he larges, common, source of sample variaion in b n. Therefore, he correc sandard errors for esimaes ha impose he resricion P n b n = 4 in each sample are smaller han P he sandard errors ha would occur if γ were known, in which case he resricion n b n = 4 would no hold in each sample. The boom panel of Figure 1 plos he coefficiens of expeced reurns on each of he forward raes implied by he resriced model, i.e. for each n,ipresens b n γ 1 b n γ 2 b n γ 3 b n γ 4. Comparing his plo wih he unresriced esimaes of he op panel, you can see ha he one facor model almos exacly capures he unresriced parameer esimaes. Table 5 suggess ha we eliminae he consans in he individual regressions as well, i.e. ha he inercep in each bond reurn regression is well modeled as he slope coefficien imes he inercep in he average reurn regression b n γ 0. Thisleavesaruly one-facor model, hprx (n) +1 = b n (γ 0 + γ 0 f )+ε +1. (n) The coefficiens a n intable5areiny. Theyareanorderofmagniudebelowhesandard errors, so ha hey are individually significan. Figure 3 plos he inerceps from he unresriced regressions and heir sandard error bars along wih he inerceps b n γ 0 from he resriced regression and you can see he excellen fi. Following his hunch, we repeaed he wo sep esimaion of Table 4 and Table 5 wih a n =0. Theγ esimaes and sandard errors are of course exacly he same, since 13

14 Inercep (annual %) Mauriy Figure 3: Resriced and unresriced inerceps. The unresriced inerceps are from he regressions hprx (n) +1 = a n + β 0 nf + ε (n) +1. The resriced inerceps are b n γ 0 from he regressions hprx (n) +1 = b n (γ 0 + γ 0 f )+ε +1, (n) where γ 0 and γ are esimaed from P n=2 hprx(n) +1 = γ 0 + γ 0 f + ε +1. Error bars are +/- 2 sandard errors from he unresriced regression. hey are esimaed in he firs sep. The b n coefficiens, sandard errors, and R 2 are he same o he decimals indicaed in Table 4 and 5. However, overidenifying resricions ess presened below rejec his specificaion, so we keep he inerceps a n. If his really is he single facor for expeced excess reurns, i should drive ou oher forecasing variables, and he Fama-Bliss slope variables in paricular. Table 6 presens a muliple regression. In he presence of he Fama-Bliss forward-spo spread, he coefficiens and significance of he regression on he reurn-forecasing facor from Table 5 are unchanged. The R 2 is also unaffeced, meaning ha he addiion of he Fama- Bliss forward-spo spread does no help o forecas bond reurns. On he oher hand, in he presence of he reurn-forecasing facor, he Fama-Bliss slope is desroyed as a forecasing variable. The coefficiens decline from 1 or even more o almos exacly zero, and are insignifican. Clearly, he reurn-forecasing facor subsumes all he predicabiliy of bond reurns capured by he Fama-Bliss forward-spo spread. 14

15 mauriy n b n σ(b n ) c n σ(c n ) R Table 6. Muliple regression of holding period reurns on he reurnforecasing facor and Fama-Bliss slope. The regression is ³ hprx (n) +1 = a n + b n (γ 0 + γ 0 f )+c n f (n 1 n) y (1) + ε (n) +1. Sandard errors correced by GMM for overlap. Figure 4 plos he forecas of he holding period excess reurns on hree year bonds implied by he Fama-Bliss regression of Table 1 (op), he forecas from he regression on he reurn-forecasing facor from Table 3 (middle, i.e. a 3 + b 3 (γ 0 + γ 0 f )) and he acual holding period reurns (boom). For many episodes, you can see ha he reurnforecasing facor and he forward-spo spread agree. This paern is paricularly visible in he hree swings from 1975 o The reurn-forecasing facor is correlaed wih he forward-spo spread. However, you can also see he much beer fi of he regression using he reurn-forecasing facor in he middle. In paricular, he fi is much beer hrough he urbulen early 1980s and he mid 1990 s. The improved R 2 is no driven by spurious forecasing of one or wo unusual daa poins. Sambaugh (1988) ran similar regressions of 2-6 monh bond excess reurns on 1-6 monh forward raes. Sambaugh s coefficiens are quie similar o he paern in Figure 1. (See Sambaugh s Figure 2, p. 53.) In he basic regression, Sambaugh found ha he mached-mauriy forward-spo spread rae he Fama-Bliss variable remained he single sronges predicor for excess reurns in his muliple regression. However, Sambaugh righly suspeced measuremen error if a bill has a bad price, hen he spurious spread gives rise o a spurious reurn in he nex period. Sambaugh hen used a slighly differen bill as predicor and prediced variable. This specificaion resuled in esimaes ha look a lo like Figure 1. Sambaugh soundly rejeced a one or wo facor represenaion of his forecas Tess Thissecionisverypreliminary hisishemehod,buwedon rushenumbers We need a es of he one-facor model and a es of he consan resricions. The underlying momens are he regression forecas errors muliplied by forward raes (righ hand variables), µ ε E +1 =0 (5) ε +1 f 15

16 20 15 Fama-Bliss γ f Ex-pos reurns Figure 4: Fied and acual holding excess reurns of hree year bonds. Top: Fied value using Fama-Bliss regression, 3 year forward-spo spread. Middle: Fied value using he resriced regression on all forward raes. Boom: ex-pos excess reurns. The forecass in he op wo lines are graphed a he dae of he reurn; he forecas made a 1 is graphed a year o line up wih he ex-pos reurn a year. The op and boom graphs are shifed up and down 15% for clariy. where ε +1 denoes he 4 1 vecor of holding period reurn regression residuals, and f denoes he 5 1 vecor of he one-year yield and four available forward raes. The unconsrained regression of Table 2 ses all of hese momens o zero in each sample. The single facor model wih consans (a n 6= 0) ses only cerain linear combinaions of hese momens o zero γ 0 : E [1 0 4ε +1 ]=0 (6) γ : E [(1 0 4ε +1 ) f ]=0 a : E [ε +1 ]=0 (7) b : E [ε +1 (γ 0 f )] = 0 (8) where 1 4 denoes a 4 1 vecor of ones.(we have indicaed which parameer is idenified by each momen before he colon.) We used he momens (6) o compue he second se of sandard errors in Table 5. 16

17 The single facor model wih no addiional consans (a = 0)ses γ 0 : E [1 0 4ε +1 ]=0 (9) γ : E [(1 0 4ε +1 ) f ]=0 (10) b : E [ε +1 (γ 0 + γ 0 f )] = 0 (11) For boh resriced models, we can compue he χ 2 es ha he remaining momens in (5) are zero, which we denoe he J T es in Table 7. Denoing he sample momens by g T he es is gt 0 cov(g T ) + g T χ 2 rank(cov(g T )) where + denoes a pseudo-inverse. (Deails in he appendix.) We can also use he variance covariance marix of esimaed parameers from less resriced models o es he parameer resricions of more resriced models, which we label a Wald es in Table 7. Table 7 collecs our es resuls. The single facor model wih free inerceps seems a grea success, wih a χ 2 value of 5 and 16 degrees of freedom. However, he Wald es of is parameer resricions is decisively rejeced wih an enormous p value. The single facor model wih resriced inerceps fails is overidenifying resricions es wih a 77 χ 2 value and 16 degrees of freedom. The Wald es of he same parameer resricions, based on he unconsrained parameer variance-covariance marix, also dramaically rejecs wih a 478 χ 2 value. Puzzlingly, he addiional resricion of his model, ha he inerceps a are zero, is decisively no rejeced, wih a χ 2 value of only 0.014, exacly as Figure 3 suggess. Bu if his, is only exra resricion, is no rejeced, why does he model wih a = 0 fare so much worse han he model wih a 6= 0? The nex row suggess he answer: when you consrain he inercep, i affecs he slope coefficiens; hese have much smaller sandard errors han he inercep, so he slope resricions are violaed when we consrain he inercep. Wald ess based on he parameer variance covariance marix from he single facor model wih a 6= 0 raher han he compleely unresriced regressions pain a differen picure. Here, he a = 0 resricion is no rejeced, (finally!) a a sensible p-value of 54%. This ime he slope resricions are also no rejeced, as we migh have expeced given he good vidual indicaion of he fi. However, he join es ha he inerceps are zero and he small changes in he slope coefficiens ha resul when he inercep is resriced o zero now rejecs. 17

18 Tes ype Hypohesis χ 2 dof %p 1. Single facor model wih a 6= 0 J T Wald from unresriced β = bγ 0 1, Single facor model wih a =0 J T Wald from unresriced, a =0andβ = bγ Wald from unresriced, α = bγ 0 (i.e. a =0) Wald from unresriced, β = bγ 0 1, Wald from a 6= 0model, a = Wald from a 6= 0model, b unr = b resr Wald from a 6= 0model, a =0andb unr = b resr Table7. Modeless. Thesinglefacormodelishprx +1 = a + b(γ 0 + γ 0 f )+ε +1 In he firs panel i is esimaed imposing a = 0 in he second panel i is esimaed allowing a 6= 0(whichalsoaffecs he b esimaes, leading o he difference beween b unr and b resr ). The unresriced model is hprx +1 = α+βf +ε +1. J T ess are ess ha he momens no se o zero in esimaion are in fac zero afer accouning for sampling errors. Wald ess use he variance covariance marix of parameer esimaes in less resriced models o es he resricions of more resriced models. In summary, hese ess do no pain a clear picure. Wald and overidenifying resricions ess do no agree, and Wald ess from differen models do no agree. We suspec ha he asympoic saisics based on a momen marix and 12 monhly lags in he specral densiy marix are simply no reliable in our sample Addiional Lags Following up on he unconsrained regressions wih addiional monhly lags in Figure 2, we run bond reurns on addiional lags of he sae variable γ 0 f. Table 8 presens he resuls. Regression 1 repeas he regression of holding period excess reurns on (γ 0 f )from Table 5 for comparison. In he second regression, we add an addiional lag γ 0 f 1/12. The R 2 now jumps up o , nearly equal o he values from he unconsrained wo-lag regression in Table 2. Once again, he single facor seems o capure all of he informaion in all 5 forward raes. The coefficiens in he second regression are abou half of he coefficiens in he firs regression, and he new coefficiens have he same paern across mauriies. The daa again sugges γ 0 f + f 1/12 /2asasae variable, and he hird regression checks his specificaion. The addiional consrain on he coefficiens makes no difference whaever o he R 2,andhecoefficiens hemselves are very close o he value in he firs regression. 18

19 The fourh regression invesigaes an addiional lag. The paern of coefficiens seems similar, and he coefficiens seem o be dying off. Though he addiional coefficiens are saisically significan (no shown), adding a second monhly lag raises he R 2 by no more han Adding a one-year lag (no repored) does absoluely nohing for he R 2 of he regression. We conclude ha he moving average of he firs wo lags is a good robus specificaion, hough one may wan o consider addiional lags wih an auoregressive paern. We argue below ha his paern suggess an ARMA(1,1) model for monhly yields induced by i.i.d. measuremen error. (1) (2) (3) (4) γ 0 f R 2 γ 0 f γ 0 f 1/12 R 2 γ 0 (f +f 1/12 ) 2 R 2 γ 0 f γ 0 f 1/12 γ 0 f 1/12 R 2 hprx (2) hprx (3) hprx (4) hprx (5) Table8. Esimaeofeachexcessreurn s loading on he reurn-forecasing facor. The lef hand variable is shown in each row heading and he righ hand variables are shown in he column headings. γ are he esimaes from Table 4. OLS on overlapping monhly daa Subsamples Table 9 repors P a breakdown by subsamples of a regression of average holding period reurns n=2 hprx(n) +1 on yields and forwards. The firs se of columns run he average reurn on he yields and forwards separaely. The second se of columns runs he average reurn on γ 0 f where γ are esimaed from he full sample. This regression moderaes he endency o find spurious forecasabiliy wih 5 righ hand variables in shor ime periods. The firs row reminds us of he full sample resul he prey en-shaped coefficiens and he 0.40 R 2. Of course, if you run a regression on is own fied value you ge a coefficien of 1.0. The second se of rows break down he regression ino he period before, during, and afer he momenous period 1979:8-1982:10, when he Fed changed operaing procedures, ineres raes were very volaile, and inflaion became much less volaile. The broad paern of coefficiens is he same before and afer. The R 2 is a lile lower in he inflaionary period. This suggess ha real holding period excess reurns are beer forecas by yield curve movemens in an environmen such as afer he grea moneary experimen, in which real ineres rae movemens are more imporan han inflaion in driving he erm srucure. The 0.77 R 2 looks dramaic in he experimen, bu his period really only has hree daa poins and 5 righ hand variables. When we consrain 19

20 he paern of he coefficiens in he second se of columns, he R 2 ishesameashe earlier period. The hird se of rows break down he regression by decades. Again, we see he paern of he coefficiens is quie sable. The R 2 is wors in he 70s, a decade dominaed by inflaion. I is a dramaic 0.70 in he 90s, and even 0.51 when we consrain he coefficiens γ o heir full sample values. Again, his suggess ha he forecass have greaes power when reurn shocks are real raher han nominal. y (1) f (1 2) f (2 3) f (3 4) f (4 5) R 2 γ 0 f R : : : : : : : : : : : : : : : : Table 9. Subsample analysis of average reurn forecasing regressions. The firs se of columns presen regression 1 5X hprx (n) +1 = γ γ 0 f + ε +1 n=2 P The second se of columns repor a regression n=2 hprx(n) +1 = a+b (γ 0 f )+ ε +1 using he γ parameer from he full sample regression, as presened in he op row. Overlapping annual forecass using monhly daa. 2.5 Macroeconomics and bond reurn forecass Figure 4 already shows ha he reurn-forecasing facor is highly correlaed wih he slope of he erm srucure, which is well known o be associaed wih recessions (Fama and French 1989) and o forecas oupu growh (Harvey 1989, Sock and Wason 1989, Esrella and Hardouvelis 1991, Hamilon and Kim 1999). We discover a surprising difference beween he reurn forecasing facor and he erm srucure slope. The reurn forecasing facor, like he slope, is highly correlaed wih business cycle measures. However, he forecasing relaions are los. Business cycle measures have no power alone, and even less in compeiion wih he reurn forecasing facor, o forecas bond reurns. Worse, he reurn forecasing facor loses he slope s abiliy o forecas oupu. Apparenly, he componen of he slope of he erm srucure ha forecass excess reurns has nohing o do wih he componen ha forecass oupu. 20

21 2.5.1 Correlaion beween he reurn forecas and business cycles Figure 5 presens he reurn forecasing facor ogeher wih he unemploymen rae and he NBER peaks and roughs. The reurn-forecasing facor is closely associaed wih business cycles, high in bad imes and low in good imes. The graph shows he very nice correlaion beween he reurn forecasing facor and recessions. As Fama and French (1989) documen for he yield curve slope, he ime-varying expeced reurn is clearly relaed o business cycles. 4 Reurn forecas Unemploymen Figure 5: Reurn forecasing facor γ 0 f and unemploymen rae. Boh series are ransformed o [x E(x)]/σ(x) so ha hey fi on he same graph. The eeh a he boom represen NBER business cycles. Ineresingly, he correlaion is also eviden a lower frequencies han usual business cycles. The reurn forecasing facor increases hroughou he 70s and decreases hroughou he 80s, mirroring he unemploymen rae as i does many measures of a decade long drop in produciviy during ha period. The bond reurn forecasing facor is a level variable raher han a growh rae variable. I is high when he level of unemploymen is high, or he level of income is low, raher han being high during recessions defined as periods of poor GDP growh. The reurn forecasing facor is correlaed wih many oher recession indicaors as well, including indusrial producion growh, Leau and Ludvigson s (1999) consumpion/wealh raio, he invesmen/gdp raio, and so on. I is much less correlaed wih inflaion. We presen he graph for unemploymen as i has he highes correlaion among he cyclical indicaors we examined. 21

22 2.5.2 Macroeconomic forecass of bond reurns Given he high correlaion beween he reurn facor and he unemploymen rae, a naural quesion is wheher we can use unemploymen or oher macro variables o forecas excess reurns on bonds. The answer is no, or a leas no among he variables we have ried so far. This is an unforunae resul for economic inerpreaion. I would be much nicer if we could undersand he reurn forecasing facor as a simple mirror of macroeconomic condiions. I appears insead ha he bond marke uses addiional informaion o forecas bond reurns. On he oher hand, i is a forunae resul for our empirical analysis: i means we can sick o he model E (hprx +1 )=a+bγ 0 f wih grea accuracy, even in VAR sysems ha include macroeconomic variables. P Table 10 conrass regressions of he average one year bond excess reurn n=2 hprx(n) +1 on he reurn forecasing facor γ 0 f, on he unemploymen rae U and oher macroeconomic variables. The firs par of he able reminds us of he 0.40 and 0.45 R 2 when we forecas bond excess reurns from γ 0 f. Despie is beauiful correlaion wih he reurn forecasing facor, unemploymen forecass bond excess reurns wih an R 2 of only In a muliple regression i does no affechesizeandsignificance of he γ 0 f coefficien, and only raises he R 2 o The Sock-Wason (1989) leading index is designed o forecas oupu growh a a 6 monh horizon. Alas, i forecass bond excess reurns wih an even lower R 2 of 0.01 and has no effec in a muliple regression. Leau and Ludvigson s (2001) consumpionwealh raio, which forecass income growh and sock reurns, does no beer. Finally, cpi inflaion is jus as useless as he variables. A large variey of macroeconomic variables do no beer. γ 0 f γ 0 f 1 R 2 γ 0 f U R (8.0) (1.9) (6.4) (5.3) (7.7) (-1.5) γ 0 f XLI R 2 γ 0 f cay R 2 γ 0 f cpi R (-0.6) (2.3) (7.4) (-1.4) (7.3) (1.1) (8.5) (-0.9) P Table 10. Forecass of average bond reurns n=2 hprx(n) +1. U =he unemploymen rae. XLI = Sock-Wason leading indicaor. cay = he Leau-Ludvigson consumpion-wealh raio using end of period wealh. cpi is inflaion, he one-year P growh in he cpi index. We esimae γ 0 f by running he regression n=2 hprx(n) +1 = a + γ 0 f + ε +1 in a firs sage. Overlapping 22

23 annual forecass, 1964: :12 Sandard errors correced for overlap and heeroskedasiciy by GMM Term srucure forecass of oupu growh The slope of he erm srucure slope forecass oupu growh as well as bond reurns. How does he reurn forecasing facor γ 0 f forecas oupu growh? Table 11 presens regressions. The lef hand panel forecass indusrial producion, while he righ hand panel forecass growh in Sock and Wason s coinciden index. The able verifies ha he erm srucure slope y (5) y (1) forecass boh oupu growh measures, wih saisical significance and R 2 of The Sock-Wason leading index, which includes erm srucure variables as well as a variey of oher macroeconomic variables, does even beer, wih sunning saisics and R 2 of Surprisingly, hough, he reurn forecasing facor is a miserable failure a forecasing oupu growh. The coefficiens are iny and insignifican, he R 2 almos vanish. The reurn facor is correlaed wih he yield spread, and he reurn facor forecass bond reurns much beer, bu i noneheless loses any abiliy o forecas oupu growh. Apparenly, he componen of he yield spread ha forecass oupu growh is uncorrelaed wih he componen ha forecass bond excess reurns. indusrial producion coinciden index γ 0 f y (5) y (1) LI R 2 γ 0 f y (5) y (1) LI R (0.19) (0.29) (-3.0) (-2.5) (9.3) (10.5) (1.7) (7.2) (-3.0) (2.2) (-3.0) (7.1) Table 11. Regression forecass of one-year indusrial producion growh and one-year growh in he Sock-Wason coinciden index on he bond reurn forecasing facor γ 0 f, he erm spread y (5) y (1), and he Sock-Wason leading index. overlapping annual forecass, 1964: :12 Sandard errors correced by GMM Forecasing sock reurns The slope of he erm srucure forecass sock reurns, as emphasized by Fama and French (1989). Table11.1 evaluaes how well our reurn forecasing facor forecass sock reurns. 23

24 The firs 4 regressions remind us of reurn forecasabiliy from he dividend price raio and erm spread. Regressions 1 and 2 sudy he dividend price raio. Unil he 1990s, he dividend price raio was a srong reurn forecaser, wih a 14% R 2. The long boom of he 1990s has cu down his forecasabiliy dramaically, especially in our raher shor sample (for hese purposes) saring only in Of course, one good crash will resore he d/p forecasabiliy. The erm spread in he hird regression forecass he VWsockreurnwiha4.6coefficien one percenage poin erm spread corresponds o.4.6 percenage poin increase in sock reurn. The R 2 is only 6.2% however. The fourh regression shows ha he erm spread and dividend price raio forecas differen componens of reurns, since he coefficiens are unchanged in muliple regressions and he R 2 increases, hough o a sill low 8.8%. Regression # d/p y (5) y (1) γ 0 f R 2 (%) (0.81) 2: (2.43) (1.8) (0.97) (2.0) (2.7) (0.69) (1.82) 7 (1.00) (0.37) (2.79) 8 y (1),f (1 2),f (2 3),f (3 4),f (4 5) 13.7 Table Sock reurn forecass. The lef hand variable is he one-year reurn on he value-weighed NYSE sock reurn, less he one year bond yield. The righ hand variables are as indicaed in he column headings. Overlapping monhly observaions of annual reurns, The dividend price raio is based on he reurn wih and wihou dividends for he preceding year. T saisics in parenheses. Sandard errors are correced for overlap. The fifh regression inroduces he reurn forecasing facor. I is significan, which neiher d/p (in his sample) nor he erm spread are, and a 8.9%, is R 2 is slighly higher han ha of he erm spread and d/p combined. The coefficien is The reurn forecasing facor is he average expeced reurn across 2-5 year bonds. The 5 year bond in Table 5 had a coefficien of 1.43 on he reurn forecasing facor. Thus, he sock reurn coefficienisjusabouwhawouldexpecofa6or7yearduraionbond, which is perfecly sensible. 24

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