Tài liệu Inflation-Indexed Bonds and the Expectations Hypothesis pdf

Copyright © 2011 by Carolin E. Pflueger and Luis M. Viceira
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Inflation-Indexed Bonds and
the Expectations Hypothesis

Carolin E. Pflueger
Luis M. Viceira




Working Paper

11-095

Inflation-Indexed Bonds and the Expectations
Hypothesis
Caro lin E. Pflueger and Luis M. V iceira
1
1
Pflueger: Harvard Business School, Boston MA 02163. Email cpfl Viceira:
Harvard Business School, Bosto n MA 02163 and NB ER. Email We are grateful to
seminar participants at the HBS-Harvard Economics Finance Lunch, John Campbell, Graig Fantuzzi,
Josh Gottlieb, Robin Greenwood and Jeremy Stein for helpful comments and suggestions. We are
also grateful t o Martin Duffell and Anna Christie from the UK Debt Mana gement Office for their
help providing us with UK bond da t a. This material is based upon work supported by the Harvard

Business School Researc h Funding.
Abstract
This paper empirically analyze s the Ex pectations Hypothesis (EH) in inflat ion -
indexed(orreal)bondsandinnominalbondsintheUSandintheUK.Westrongly
reject the EH in inflation-indexed bonds, and also confirm and update the existing
evidence rejecting the EH in nominal bonds. This rejection implies that the risk
premium o n both real and nom inal bonds varies predictably ove r t ime. We also find
strong evidence that the spread bet we en the nominal and the real bond risk prem ium,
or the break even in flation risk p re mium , also varies over time. We argu e that the time
variation in real bond risk prem ia mostly likely reflects both a chan ging real interest
rate risk pre m iu m and a ch an g ing liquidity risk premium, and that the variab ility in
the nominal bond risk premia reflects a changing inflation risk prem iu m . We estimate
significant tim e series variability in the magnitude and sign of bond risk premia.
Key Words: TIP S, Breakeven In flation, Retu rn Predictability, Bon d R is k Pre-
mia
1Introduction
This article conducts an emp irical exploration of the magn itu de and time variation
of risk premia in inflation-indexed and nominal governmen t bonds, using data on US
Treasury bonds and UK gilts. Understanding bond risk premia is fundamen tal in
thinking about the term structure of interest rates. It is also of first order importance
for bond issuers, since public debt constitutes one of the main sources of govern-
ment financing, an d for users, w hether central banks or investors. Cen tral ba nks use
go v ernm ent bonds as a k ey instrument in the execution of monetary policy, while
investors u se them as t he anc hor of t heir fixe d income allocations.
The most common form of go v ernm en t bonds are nominal bonds that pa y fixed
coupons a nd principal. Ho wever, in recen t times governments around the world,
including the U.S. Treasury, have started issuing significant amounts of inflation-
indexed bonds (C am pbell, Shiller, and Viceira 2009) . Inflation-indexed bonds, which
in the U.S. are known as Treasury Infla tion Protected Securities (TIPS), are bonds
whose coupons and p rincipal adjust automatically with the ev olution of a consumer

price index
2
. They aim to pay in v estors a fixed inflation -ad ju sted co upon and prin c i-
pal. For this reason they are also kn own as r eal bonds, and their y ields are typically
considered the best proxy for the term structure of r eal interest ra tes in t he economy.
Although government bo nds i n large stable economics are generally free from
defa ult risk, they ex pose inve stors to other risks. Investors holding either inflation-
indexed or n ominal government bonds are ex posed to the risk of chang ing real interest
rates. Fo r any inve stor th e riskless asset is an inflation-indexed bond whose cash
flow s match his consump tion plan (Cam pbell a nd Viceira 2001, Wachter 2003). If
future real in terest rates are uncertain, investors will view bonds not match ing the
timing and length of the ir c on s umption plans as risk y, leading to a risk prem ium
for holding suc h bonds. This real in terest rate risk prem ium will be a function of
in vestors’ risk tolerance, and it can be time-varying if investors’ toleran ce for risk
c han ges ove r the bu siness cycle (Cam pbell and Coch rane 1999, Wa chter 2006). A
time-varying correlation of real interest rates w ith investor well-being c an also make
the real in terest rate risk prem ium vary over t ime (Ca mpbell, Sunderam, and Viceira
2010).
2
In the US, TIPS paymen ts are linked to the Cons umer Price I ndex for All Urban C onsumers
(CPI-U). The relevant index i n the UK is the Retail Price Index (R PI).
1
In add ition to r eal interest ra te risk, nominal gove rnment bonds e xpose i nvestors
to in flation risk. Whe n future inflation is uncertain, the coupons and principal of
nominal bonds can suffer from the eroding effect s of inflationary surprises. If infla-
tion is negatively correlated with economic conditions, as in times of s ta gflation, the
real payoffs of nominal bonds will tend t o decline when e conomic c onditions worsen.
Risk averse inv estors will therefore demand a positive inflation risk premium for hold-
ing nomina l bonds. Bu t if in flation is positively correlated w ith econ omic cond ition s,
nominal bonds will have hedging value to risk-a verse investors (Piazzesi and Schneider

2007, Campbell, Sunderam , and V iceira 2010). By con trast, inflation-indexed bonds
are n ot exposed to i n flation risk, since their coupons and principal adjust automati-
cally with in flatio n.
3
The s tartin g point of our e mpirical invest igation of bond risk premia is the expec-
tations hy pothesis of interest r ates (EH for short). The EH postulates that t he risk
premium o n long-term bonds, or the expected excess return on long-term bonds over
short-term bonds, should be constant ov er time. If the E H holds for inflation - in de x ed
bonds, this implies that the real interest rate risk premium is constan t. In that case
the yield on long- ter m inflation-indexed bonds is equa l to the a verage expected short-
term real interest rate over the life of t he bond plus a constant. Inve stors cannot earn
predictable returns b y shifting between long-maturity and short-maturity r eal bonds.
The implications of the EH for nominal bonds are stronger. If it holds, both
the in flation r isk p remium and the real interest rate risk premium ar e co nstant
4
.In
that case the yield on long-term nominal bonds i s eq u al to the ave rage e x pected fu-
ture shor t-term nominal interest rate up to a constant. A rejection of the nominal
expectations hypothesis can be the result of a time-va rying inflation risk premium ,
a time-varying real interest rate risk pre mium, or both. W i t hout inde pende nt o b-
servationofrealbondpricesitishardtodistinguishbetweenthosesourcesoftime
variation in nominal bo nd risk premia.
In o ur analys is we adopt a flexib le empirical appro ach that does not rely on a
tigh tly param eterized model
5
. The EH has been t ested and rejected on U.S. nominal
3
Tax regulations in some countries, including the US, mak e the after-tax income and capital gains
from inflat ion-indexed bonds not fully inflation indexed. This effect can be exacer bated at t imes of
high accelerating inflation. See Section 2.

4
Unless we are in the unlikely case wher e time-variation in the inflation risk premium and the
real interest rate risk premium exactly cancel each other out.
5
See Adrian and Wu (2009), Buraschi and Jiltsov (2004), Campbell, Sunderam, and Viceira
(2010), C hristensen, Lopez, and Rudebusc h (2010) and Evans (2003) for formal models o f the term
2
Treasury bonds n um erous times, but previous tests for inflation-indexed bonds only
had significan tly shorter samples at their disposal and w ere not able to reject the
expectations hy pothesis (Barr and Campbell 1997). Campbell and S hiller (1991)
present regression r esults for different combinations of maturities and holding periods
and resounding ly reject the expectations hy pothesis for U.S. nominal bonds. Fama
and Bliss (1987), Coch rane and Piazzesi (2005) and o th ers have also presented robust
empirical eviden ce th at nominal Treasu r y bond risk premia vary ove r time. Howe ver,
Cam pbell (1999 ) presents evidence th at the ex pectations hy pothesis is harder to reject
on nominal g overnmen t bonds in a cross-section of other developed econ om ies.
The structure of this article is as follow s. Section 2 describes the m echa nics of
inflation-indexed bonds. Section 3 formalizes the expectations h ypothesis of the term
structure of interest rat es and expected inflation. Section 4 tests the expectations
hypothesis in real and nominal bonds. Section 5 prov ides evidence on real and nomin al
bond return pred ictability from the tent- shaped linear combination of nom ina l interes t
rates p roposed by Cochrane and Piazzesi (20 05). Section 6 shows es tim ates of bond
risk premia and their variation o ver time. Finally, sectio n 7 offers some concluding
remarks and su ggestions for future research.
2Inflation-Indexed Bonds
Inflation-indexed bonds ha ve been available in the UK since 1983 and in the US since
1997. US inflation-inde xe d bonds are called Treasur y Inflation Protected Securities
(TIPS). They are designed to appro ximate real bonds with pa youts that are constant
despite inflatio n sur p rise s . T he y are quot ed in term s of a real in terest rate and are
issued mostly at long maturities greater than 10 years. The principal on t hese bonds

gro w s with a p re-specified price index, whic h i n the U.S. is the Consumer Price Index
(CPI-U) a nd in the UK is the Retail Price Index ( RPI). T he coupons are eq ual to the
inflation-adjusted principal on the bo nd times a fixed c ou pon rate. Th us the coupons
on these bonds als o fluctuate w ith inflat ion.
Of course, the price index migh t not grow over time. For instance the CPI de-
creased b y a lmost 4 % betwe en September 2008 and December 2008. In the US, the
final paym e nt of principal on a TIPS is p rotected against deflat ion and it can nev er
structure of interest rates that analyze and estimate inflation and real interest rate risk premia using
data on both real and nominal bonds.
3
be sm aller t han the st ated nom inal value at i ssuance. However, its co upons are no t:
the inflation-adjusted value of the principal for coupon computation purposes can fall
below t he initial value at issu an ce. In con tras t, neith er t h e pr in cipa l no r the coupons
on inflation-linked gilts i n the UK are protected from deflation.
Further details complicate the pricing of these bonds. Since inflation figures in the
US an d in th e U K are publish e d w ith a lag, the p rin cipal value of inflation-in de x ed
bonds ad ju sts with a 3 month lag. UK inflation-linke d gilts tha t we re issued prior to
the financial year 2005-06 follo w an 8 month lagged indexin g procedure while more
recent iss ues follo w a 3 month lagged methodology. The tax treatment of th ese bonds
also differs. In the UK principal adjustments of inflation-linke d g ilts are not tax ed .
This gives inflation-linked gilts a tax a d vantage over nominal gilts, a la rge r share
of whose cash flo w s come in the form of taxab le nominal coupon pay m ents. In the
US, on the other hand, inflation-adjustments of pr incipal are considered ordinary
incom e for tax purposes. A s a result the tax obligation s associated with holding a
TIPS increase when inflationishighsothatonanafter-taxbasisU.S.TIPSarenot
fully index e d to inflation. More details on TIPS can be f ou n d in Viceira (2001), Roll
(2004) an d Gurkay n ak, S ack, and Wright (2010). Campbell and Sh iller (1996) offer a
discussion of the t axation of inflation-inde xe d bonds. Campbell, Sh iller, and Viceira
(2009) pro v ide an ov erview of the history of inflation-indexed bonds in the US and
the UK.

3 The Expectations Hypo thesis
The expectations h ypothesis of t he term structure of i nterest r ates says that the ex-
cess return on an -periodbondovera1-period bond should be constant ov er time.
There should n ot be any particularly good time to hold short-term or long-term bonds.
Equivalently, the e xpectations hypothesis says th at if short yield s are ant icipa ted to
rise in the future then this should already be reflected in current long yields. The
expectations hypothesis is usually stated for nom inal bonds. We formulate expecta-
tions h y potheses for nom ina l bonds, real bonds and for inflation expectations. We
show that these d ifferen t interpretations of the expectations hypothesis a re closely re-
lated to real in tere s t rate r isk , inflation risk an d liquidity premia and de rive empiric al
predictions that we will test subsequen tly.
4
3.1 Bond Notation and D efinitions
We start by establishing some n o tation and definitions that will be used t hroughout
the article. We d en ote b y 
$

thelogpriceofazero-coupon-period nominal bond,
and by 
$

the bond’s log (or continu ou sly compounded) yield. For zero-coupon
bonds , log price and yield are relate d accordin g to

$

= −
µ
1


¶

$

 (1)
The yield spread is the difference between a long-term yield and a short-term yield,

$

= 
$

− 
$
1

The log return on a zer o-coupon -periodbondifitisheldforoneperiodand
sold before matu rity is given by the ch an ge in its price , i.e.

$
+1
= 
$
−1+1
− 
$

= 
$


− ( − 1) 
$
−1+1
 (2)
where t he second equ alit y follows imm ediately from (1).
We use the superscript  to denote the corresponding quantities for both US
and UK inflation-indexed bonds. Inflation-indexed bonds are comm only quoted in
terms o f real yields. That is 


is the real log price of an indexed bond, 


is
the real yield and 

+
is the real one-period log r etur n. The nominal one-period log
return on an inflation-indexed bond is then giv en by 

+1
+ 
1
 where 
1
denotes
the 1-period log inflation rate f ro m period  to period  +1.
3.2 N ominal Expectation s Hypothesis
The nominal E H states that the e xpected log excess return on long-term nom inal
bonds over short-term nominal bonds, or bond risk p rem ium, i s constant over time:

E

£

$
+1
− 
$
1
¤
= 
$

 (3)
where the constant bond risk premium 
$

can depend on maturit y . The adv a n-
tage of formulating the expectations h ypothesis in log s is that the log expectations
5
hypothesis for one h o ld in g period is consiste nt with the log expectations hypothesis
for any other holding period.
6
The EH can be represented in a number of differen t w ays that obtain by simple
algebraic manipulation of (2) and (3).
7
A popular equivalen t representation of the
nominal EH relates the yield on a n-period zero-coupon nominal bond at time  to
expected fu ture short-term nominal interest rates:


$

= 
$

+
1

E

−1
X
=0

$
1+
 (4)
Equation (4) says that the curren t n-period yield should be equal to the expected
aver age short yields ove r its maturit y up to a time- invariant consta nt 
$

. The constan t

$

is sim ply the average of bond risk pr e mia for maturities up to  periods, i.e., 
$

=
P


=2

$

. A second equiva lent representation of the nominal EH relates changes in
long-term yield s to the yield spread
E

£

$
−1+1
− 
$

¤
=
µ

$
−1
−

 − 1

$

¶
+

1
 − 1

$

 (5)
Although these alternativ e equiva lent represen tation s of the EH p rovide useful in-
tuition to unders tan d the mean in g and imp lication s of the EH, we choose to work
with the return-based definition (3) in our empirical exploration of the EH. This
approach allow s for transparent interpretatio n of empirical results in terms of re-
turn predictab ility, and it is flexib le en ou gh to easily ac comm odate a com p lementary
analysis of liqu id ity prem ia and supp ly pressures in t he bond market.
The E H says t hat 
$
+1
− 
$
1
cannot be predicted. Ho wever, early tests of the EH
based o n (5) found robust evidence that the nom inal te rm spread–or an equivalen t
com bination of forw ard rates–predicts nominal excess returns positiv ely (Campbell
and Shiller 1991, Fa m a and Bliss 1987). That is, whe never the term spread is high
6
Another version of the expectations hypothesis, the so-called pure expectations hypothesis
(PEH), considers a formulation of (3) in terms of simple returnsasopposedtolog returns, and
sets expected excess simple returns to zero (Campbell, Lo, and M acKinlay 1997). The in tuition of
the PEH is that if investors are risk neutral then they should adjust positions un til the expected
one-period returns for short n ominal bonds and long nominal bonds are equalized. The log EH (3)
is less constraining in that it allows for a non-zero bond risk premium.
7

For a detailed discussion of equivalent formula tions of the expectations hypothesis see Campbell,
Lo, and MacKinlay (1997, Chapter 10) or Cochrane (2005, Chapter 19).
6
the risk premium on long nominal bonds is higher.
8
Building on th is prior work, we
test in our data whether the term spread contains a time-varying risk premium b y
running the regression test

$
+1
− 
$
1
= 
$
+ 
$

$

+ 
$
+1
 (6)
where 
$
=0under the n ull that the EH hold s . O f course, failing to reject 
$
=0

in (6) does not imp ly that we fail to r eject the EH, as other state variab les might
forecast bond excess returns. Thus w e also include in (6) other variables that hav e
been show n to forec ast bond exce ss returns in our empir i cal analy sis.
3.3 Real Expectations H ypothesis
The EH has traditionally been fo rmulated and tested i n terms o f nominal bonds but
it appears at least as plausible to formulate the hypothesis in terms of real bonds.
The nomina l EH supposes th at the bond r isk premium on nominal bonds, consisting
of both infla tion risk and real in terest rate risk, is constant o ver time. The EH for
inflation-indexe d bonds is strictly weaker in that it only supposes that real in teres t
rate risk is constant.
Expressed in terms of re t urns th e EH for in flation-indexed zero-coupon bonds says
that


£


+1
− 

1
¤
= 


 (7)
Analogously to the nom inal EH we t est the real EH by testin g w hether th e real
term spread predicts e x cess re turns on real bonds:



+1
− 

1
= 

+ 




+ 

+1
(8)
where 


= 


−

1
is the TIPS bond spread . T h e real EH hy pothesis implies
that the coefficien t of real excess log returns of inflation-indexed bonds on the real
term sp read should be zero. If 

6=0then we can infer t hat th e real yield reflects
time-varying real risk prem ia and 



is t ime-varying. The TIPS bond sprea d is a
natural candid ate regre ssor d ue to its simila rity to the nominal bond spre ad . Since
TIPS are not exposed to i nflation su rp r ises the TIPS yield spread s h ou ld not reflec t
8
This is equivalent to finding a negative slope in a regression of cha nges in the yield o n long-term
bonds on 
$

( − 1).
7
inflation risk, although it might reflect other risk premia such as real interest rate
risk and liquidity prem ia. Hence, if any of these prem ia are important in dr ivin g the
rejection of the nomin al expectations hypothesis they would be likely t o be reflected
in term s of a no nzero coefficient 


3.4 Break ev en Inflation a nd the Inflation Expectation Hy -
pothesis
We now l ook at the difference between nominal and indexed yields, known by bond
market participants as “breakeven inflation:”


= 
$

− 



(9)
Most simply -period break even inflation is the inflation rate ove r t h e n e xt  periods
that wou ld make a nominal bond and an inde xed bond of maturity  earn the exact
same buy-and-hold return. The nominal bond outperforms the inflation-indexed bond
if realize d infla tion o ve r the life of the bonds turns out to be smaller t h an breake ven
inflation, and underperforms it if realized inflation turns ou t to be larger.
Bond market participan ts often use bre akeven i nflation as a measure of expected
inflation. Ho wever, the iden tification of breakeven inflationwithexpectedinflation
is not entirely correct. In order to understand this point, it is useful to think about
the c omponen ts of bond y ields, both nominal and inflation-index ed . Econom ic logic,
formally corroborated b y models of the term structure of in terest rates,
9
suggests that
we can deco mpose the yield—or equivalently, the p rice – on an inflation-inde xed bond
in t o a componen t that reflects cu rrent expectations o f future real interest rates, and a
componentthatreflects a real int e r es t rate risk prem iu m . Similarly, we can think of
the yield on a nom inal bond as composed of the yield on a real bond plus additional
components reflecting expectations of fut ure inflation and an inflation risk pre mium .
Th us the spread bet ween both yields, or breakeven inflation , reflect s both expected
inflation and the inflation risk premium embedded in the nomin al bond yield.
If institu tion al, beha vioral, or liq u idity factors affect the nominal bond market and
the inflation-indexed bond mark et differen tly, breakeven inflation will also re flect the
differ ential impact of these factors on yie lds (Pflueger an d Viceira 2010). For exam ple,
we think of the market for inflation-indexed bonds to be less liquid than the market for
9
See references in footnote 5.
8
nom in al bonds. I f inves tors ap ply a liquid ity d isc ou nt to th e price of inflation - in dexed
bonds, o r a liquidity premium to the pri ce of nominal bonds, break even inflation w ill
be lowe r than it would be otherw ise, sinc e prices and yie lds mo ve inversely.

10
When
changes in the liquidity differential a r e correlated with aggregate econom ic con ditions,
breakeven inflation will also re flect an additional liquidity risk premium.
Of course, expected inflation, the inflation risk prem ium , the liqu id ity differen-
tial, a nd the liquidity risk pre mium need not be constan t over time, c ausing rea lized
breakeven inflation to move over time. Mo re importantly, time va ria tion in the in-
flation risk prem iu m or in the liquidity prem iu m can also c au se the expected ex cess
return on breakeven inflation, or th e d ifference between the excess return on nominal
bonds and the excess return on inflation-indexed bonds o f identical maturit y, to vary
o ver time. Mechanically, th e excess r eturn on breakeven inflation is g iven b y


+1
≡
¡

$
+1
− 
$
1
¢
−
¡


+1
− 


1
¢
= 

− ( − 1) 
−1+1
− 
1
 (10)
Under the assump tion of constant inflation and liquid ity risk prem ia , the left-hand
side of equa tion (10) equals a con stant plus the expression 


− ( − 1) 

−1+1
−

1
,where


denote s n-period expected inflation at time . This express ion is zero
when inflation expectations are rational. To see this, note that realized in flation
verifies


− ( − 1) 
−1+1
− 

1
=0 (11)
since both 
+1
and ( − 1) 
−1+1
+ 
1
denote cumulative inflation from time
 to time  + . Th erefo re under rational e x pectations equation (11) must also hold
ex-ant e.
We call the join t hypothesis of rational inflatio n expectations and a constan t
inflation risk premium t he in flation expectations hypothesis. By analogy with our
tests o f t he nominal a nd real EH, we run the r egression


+1
= 

+ 




+ 

+1
 (12)
10
Campbell, Shiller, and Vice ira (2009) document an episode o f “flight to liquidit y” during t he

recent financial crisis. In the Fall of 2008 the price of nominal Treasury bonds increased rapidly,
while the price of TIPS declined, causing a dramatic narrowing of breakeven inflation, whic h at
some point became even negative. They pr ovide evidence that this change in prices did not reflect
a sudden change i n t he outlook for in flation towards massive deflation, but r ather an increase in
the liquidity differential between both markets, as investors around the world flew into nominal
Treasuries.
9
where 


= 

− 
1
is the b reakeven inflation spread, and test whether 

=0.If
the inflation r isk p rem ium or th e liquidity risk pre mium are tim e varyin g, and th ey
arecorrelatedwiththebreakeveninflation spread, we would expect to find a nonzero
regression slope coefficie nt 

in (12). In particular, the break even spread 


shou ld
reflec t the inflation r isk p remium contained in the nominal y ield spread 
$

.
Since the break even inflation spread, the nom ina l term spread, and the real term

spread are mech an ically related by 
$

= 


+ 


, it a lso makes sense to t est
whether both the real term spread and the break ev e n inflationspreadjointlyforecast
the return on breakev en inflation . It is im portant to note that neither (12) nor the
expanded v ersion of the equation that includes 


are redundant with re spect to
the standard EH re gre ssions (8) an d (6), ex cep t of c ou rse in t h e trivial case where the
spreads d o not forecast bond excess return s and thus all slope coefficients are z ero.
4 Testin g th e E x pecta tion s H y pothesis in Re al a n d
Nom inal Gov ernm ent Bond s
4.1 Data
We conduct tests of the real and nominal EH u sing governm en t bond data from both
the U S and UK . For th e US we use an ex pan d ed version of th e Gurkayna k, S ack , an d
Wr ight (2007) and Gur ka yn ak , Sack, and Wright ( 2 010, GS W henceforth ) data set.
GSW hav e constructed a zero-coupon yield curve starting in January 1961 for nominal
bonds and starting in Janu ar y 1999 for TIPS by fitting a smoothed yield curve. We
expand their d a ta back to 1951 using the McCulloc h, Houston, and Kw on (1993)
data for US nominal zero coupon yields from Janu a ry 1951 through December 1 9 60.
The GSW data set c ontains constant maturity yie lds for maturities of 2 to 20 years.
Our empirical tests will focus on the 10-year nom inal and real yields, because this

maturity bracket has th e l on gest and most co ntinuous history of T IP S outstanding.
We measure U.S. i nflation with the all-urban seasonally adjusted CPI, and the short-
term nomin al interest rate with the 3 mo nth T -bill rate from the Fama-Bliss riskless
in t erest rate file from CRSP. TIPS pa y outs are linked to the all-urban non seasonally
adjusted CPI and our results become sligh tly stronger when using the non seasonally
adjusted CPI i n stead.
10
For the UK w e use zero-coupon yield curves from t he B ank o f England. Anderson
and Sleath (2001) desc ribe the spline- b ase d technique s use d to estimate the yield
curves. Nomin al yields are available starting in 1970 for 0.5 to 20 years to maturity.
Real yields are available startin g i n 1985 for 2.5 to 25 years to maturit y. We foc u s on
the 20-year nominal and real yields because they are available from 1985, while other
maturities are available only since 1991.We me asure inflation by th e non seaso n ally -
adjusted Retail Price Inde x, whic h serves as the measu re of inflation for inflation
indexed bond payou ts.
In all r egressions we approximate 
$
−1+1
and 

−1+1
with 
$
+1
and 

+1
.
Because neither the US nor the UK go vern m ents issue inflation-ind ex ed bills, we need
to resort to an empirical procedure to build a hypothetical short-term real in terest

rate. We describe this procedure in Section 4.2. Finally, a lthough our yi eld data sets
are available at a monthly frequency, we sample our data at a quarterly frequency
in order to reduce the influence of high-frequency noise in observed inflation and
short-term nominal interest rate volatility in o ur tests.
4.2 Construction of the Short-Te rm R e al Interest Rate
Wh ile thr e e -m onth nom in al T-b ills are issu ed in the US an d in the UK, there exists
no equivalen t sh ort-term instrument with fixed real p ayoffs. Apart from technical
difficulties , the deman d would probably not exist simp ly because inflation risk in
both countries has been historically negligible over such a short horizon. Ho wev er, we
need a proxy for a short-term real ra te for o ur tes ts of the expectations hypothesis.
We follow Cam pbell and S hiller’s (1996) analy sis of hypothetical TIPS to c on stru ct
an ex-an te measure of the short-term real interest rate.
We start by ass umin g zero inflation risk and liquidity premium over 1 quarter.
Therefore, the ex-an te short-term real interest rate is given by


1
= 
$
1
− 

1

Next w e assume that inflation expectations o ve r the next quarter are rational and
proxy for the ex-ante real short rate as th e fittedvaluefromtheregressionofthis
quarter’s realized real rat e 
$
1
−

1+1
onto last q uarter’s r ealized real rate 
$
1−1
−
1
,
the nominal short ra te 
$
1
, a n d annual inflationuptotime
Table 1 show s the mon thly predictiv e regressions for the US and the UK. It reports
11
the point estim ates of the slope coefficients as well as Newey-We st heteroskedasticity
and autocorrelation consistent (h.a.c.) stand ard errors wi th four lags in pare nthesis.
The table sho w s that the m ain determinant of the ex-an te real rate is the nominal rate,
with a positive coefficient of about 0.5 in both the US and the UK. The regressions can
explain 44% of th e real interest rate variatio n in the US and 18% of the real interest
rate variation in the UK , respectiv ely, and the regressors are jointly significant in
both r egressions.
Figure 1 shows the predicted and r ealized US real short rate togeth er with the
nominal short rate. It sho w s that the predicted real short rate very muc h just follows
the nominal short r ate a nd smooth s o ut fluctuation s in the e x- post real ra te caused
by short-term vola tility in realized inflatio n. The estimate is conservative in th e sense
that it bare ly relies on lag ged realized inflation and it does not attempt to predict
high-frequency fluctuation s in inflation.
Table 2 presents summ a ry statistics for inflation, short-term nominal and real in-
terest rates, n ominal and real yield spreads, brea keven inflation, and bond returns for
the US (P anel A) and the U K (Panel B). Because the sample period and bond matu-
rit y in each table are different, it is hard to do com parisons across panels. N onetheless,

the average excess return on nominal bonds is similar across both c ount ries, w hile the
a verage excess return on inflation-indexed bonds have been larger in the US. Bond
return vo latilities and co rrelations are generally comparab le across both countries,
contr olling f or maturity d ifferentials. T h e averag e exc e ss r etu r n s and volatilities r e -
ported in Table 2 imply sample Sharpe ratios on US real a nd nominal bonds of 0 .392
and 0.542, respectiv e ly. Th ese are larg er than the corresponding S harpe r atios for
UK real and nominal bon ds, at 0.236 an d 0.179 respectiv e ly.
4.3 The Nominal Expectations Hypothesis in the U S
Tab les 3 report tests of the nominal E H in the U S u sing our p referred retur n-based
regression test (6). Th u s we test whether nominal log exces s returns are predicta ble
from the nominal term spread. The objective of this table is to analyze changes in
the predictab ility of n om inal bond return s since C ampbell and Shille r (199 1) reported
tests of the nom in al EH. Campbell a nd Shiller (199 1) foun d th at t h ey were able to
clearly reject the expectations h y pothesis for almost all of their maturit y combina-
tions for the sam p le period 1952-1987. Table 3 reru n s those s am e regressions f or our
full sample period (1951.12-2009.12) with the 1 0-year constant maturity zero -coupon
12
bond. Fo r comp aris on we also r eport th e m for the C ampbell-Shiller sample period and
the sample period from 1987 until present.
11
The table reports the poin t estimates
of the slope coefficien ts and Newey-Wes t standard errors with 3 lags.
Ta ble 3 show s that the full time period 1952-2009 yields an even stronger rejection
of the expectations hy pothesis than the earlier sa mp le period 1952-1987. At the same
time, using the second part of the sample only it is harder to re ject the expectations
hypothesis at conv entional si gnificance lev els. Stock and Wa tson (2002) documen t a
break in the mid-1980s in a number of macroeconomic time series. If the predictability
of bond r eturns is link ed to macroeconomic processes, it is conceivable that bond
return predictability also experienced a break at th e same time.
Following this intuition, the last colum n of Table 3 examine s more closely wh eth e r

there was a stru ctu ral change in bond r etur n predictability in 1987. We add the term
spread intera c t ed with a du m my for the second sub period, 
$

× 
1987−2008
to the
regression in order to allo w for differen t slope coefficients before 1987 and a fter 1987.
The int eraction term does no t ent er sign ificantly the regression, ind icating that w e
cannot reject the hypothesis of a stable relationship across sub samples. M or eover,
the full sam p le period and t he Campbell-Shiller sa m p le period yield ver y similar
regression coefficien ts and the coefficien t is more accurately measured using the full
sample period.
Thus the addition of the 1987-200 9 period to the early sample period con trib utes
to reinforce the empirical evidence to wards rejection of the nominal EH. It also em -
phasizes the difficulty of d e tec tin g this type of bond re tu rn pr ed ic tab ility in smaller
sample sizes, even if the s a mple c omprises mo re than 20 years of data . This qualifi-
cation is pa r ticu larly important for our subs equ e nt a n alys is of the real E H, because
ev en our longest series of inflation-in dex ed bonds only spans a 24 y ear period from
1985 to 20 09.
4.4 Expectations Hypothesis Real and Nominal
Tab le 4 present our main regression tests for the nominal, real and inflation expec-
tations hypothesis in the US and in the UK. Columns 1 through 4 in each p anel in
11
Campbell and Shiller (1991) used the M cCulloch, H uston, and Kwon (1993) data of zero coupon
yields for their e ntire period so that our resu lts differ slightly from theirs due to our different da ta
source.
13
the table report coefficien ts from the return-based regressions (6), (8), (12) and its
expanded version, respectively. The data consists of mon thly data of overlapping

quarterly returns. Newey-West standard errors are based on 3 lags to adjust for
o verlapping returns.
P an el A reports t he regression tests for the U S data from 1999 to 2009. A ccording
to the nominal EH the coefficientincolumn1shouldbezero. Wecannotreject
the nominal E H over this particular time period at conventional significance levels.
However, th is rejection is som ew hat marginal–the significance level is 15%–a nd the
results in Tab le 3 indicate that this m ay well be related to the sh ort sample size rather
than a c h ange in th e pr edictive relationship. Column 2 of the panel tests whether
excess returns on inflation-indexed bonds are p redictable. The coefficient on the real
spread is large compared to the coefficient on the nominal spread reported in column
1, and statistically significant at the 1% level. T his strong rejection of the real EH is
particular ly strikin g in light of the w ea k reje ction of the nominal EH in column 1. It
suggests that f actors specific to the TIPS market such a s liquidity might be driving
expected e x cess re turn on TIPS, besides real interest rate risk.
12
Column3ofPanelAteststheinflationEHintheUS.Wefind that the breakeven
spread predicts th e d ifference in nominal and inflation- in de xed bond excess retu r n s.
Assuming that bond market pa rticipants’ infla tion expectations are rational and that
liquidity prem ia are con sta nt , this result is consistent with a time-varying inflation
risk premiu m. Column 4 also contr ols f or the real term s prea d in the regression and
show s that adding this va riable does not affect the predictive po wer or the coefficien t
estimate of the breakeven spread. T h ese results sug gest that w h en th e br eakeven
spread increases, inflation risk also increases and in ve s tors dem a nd a high er inflation
risk p rem ium from n o minal bonds.
In terestingly, the real term spread appears to predict breakeven returns nega-
tiv ely in the US. Thus a widening of the real term spread f o recasts a decrease in the
spread bet ween nominal bond risk premia and inflation- in de xed bond r is k premia.
One m ight expect that if the real term spre ad p roxies for the real inter e st rate risk
prem iu m, its coefficient should be zero; that is, increases in the real in terest rate
risk premium should affect appro ximately in the sam e proportion the prices of both

inflation-indexed bonds and nominal bonds. Th e rejection of the nu ll hypothesis th at
12
Results ommitted here to save space show that, interestingly, the nominal term spread does
not forecast TIPS exc ess returns, w hile the real term s pread does n ot forecast nominal bond excess
returns.
14
it is zero suggests t ha t the effect of the real term spread on breakeven inflation returns
might be related to liquid ity factors (Pflueger and Viceira 2010).
P anel B in Ta ble 4 reports the corresponding results for the U K bond mark et.
The patte rn of resu lts is remarkably cons istent with the results sh own in Panel A for
the US bond marke t, with the excep tion th at the real term sp read d oes not appear
to pre dict breakeven returns.
Overall, the results in Table 4 prov ide stron g support for the predictabilit y of
nominal and real bond returns and for the predictability of their differ ence in US
and UK bond data. These results pro vide suppor t for the hypothesis that the risk
premiumonnominalbondsisdrivenbybothatime-varyinginflation r is k p re m ium
and a time-varying real interest rate risk premium. An increa se in breakeven inflation
forecas ts positively a n in crease in nominal bond risk pr emia rela tive to inflat ion-
indexed bond risk prem ia. T h e US results also show t he striking result t h at the real
term spread f orecasts negativ ely the spread between the nominal bond risk prem ium
and the inflation-indexed bond risk premium. Pflueger and Viceira (2010) presen t
empiric al evid e n ce that this is the res u lt of a time-varying liquidity risk pre mium in
inflation-indexed bonds.
5 Coc hrane a nd Piazzesi Bond R eturn Pred ictabil-
ity an d t he Infla t io n Risk Pr emium
In recent w ork Cochran e and Piazzesi (2005, CP) show that a tent -shaped linear
combination of nom inal forward rates is a good predictor of excess nom inal bond
returns for a wide range of bond maturities. T heir findings imply that no m inal bond
risk prem ia are h igh when intermediate-term no minal in terest r ates a re high relative
to both shorter-term and l on ger-term rates. In the context of a non -linear m odel of

the term structure of interest rates, Campbell, Sunderam, and Viceira (2010) interpret
their findings as refle c ting a time-varying transitory inflation risk premium.
We explore this in terpretation by examining w hether the CP ten t-shaped com-
bination of nominal forward rates forecasts inflation-indexed bond excess returns in
addition to nom i nal bond excess retur ns. If the Cochrane Piaz ze si factor reflects
inflation r isk premia i t s hould n ot predict returns on inflation-indexed bonds.
15
We constru c t t h e CP factor from o ne- to five-year Fama-Bliss zero cou pon nominal
bond yields, a vaila ble from C RSP. Let 
$

deno te the log 1 y ear nominal forward
rate at tim e  for loans between  − 1 and  y ears in the future. We obtain the
CP factor using the optimal w eights found in Cochrane , P iazzesi (2005) as 

=
−214
$
1
+081
$
2
+300
$
3
+080
$
4
−208
$

5
 Unfortunately we do not hav e enough
ric hn ess in the term structu re of TIPS rates to construct a CP variable based on TIPS
yields. We a ls o limit our analysis to t h e US.
Pane l A in Table 5 reproduces the CP predictability results for o u r data set, u sin g
the 1952-2009 sam p le period and a one-quarter forecasting horizon. C olu m n 1 in the
panel sho ws t hat CP is significant and f orecasts nominal bond excess retur ns with a
respectable 
2
of 4% a t a quarterly horizon. Ho wev er, column 2 in t he pa nel sho ws
that the variable loses its statistical significanceoncewecontrolfortheyieldspread.
P anel B in Table 5 show s that CP does not forecast nominal bond excess returns,
TIPS excess returns, and breakeven infla tion over our 199 9-2009 sample period. Panel
B also sho ws that the inc lusion of CP in the nominal and real EH regressions does
not c hange o ur basic results. The factor en ters significantly and with a positive sign
only in the l ast column. Comparing this to column 4 of Table 4 shows t h at CP a lso
increas es the 
2
from 20% to 27% When C P is high , nominal bond exces s re turns
areexpectedtobelargerthaninflation-indexed bond excess returns. This result is
consisten t with Cam pbell, Sund eram, and Viceira (2 010)’s interpret ation of C P as a
proxy for a t ime-varying inflation risk premium .
6 Histo r ica l F itt e d Risk P r emia
We no w look at the fitted risk premia in order to better u nderstand the economic
signific an ce of the bond return predictability examined in this article. Table 6 sho ws
the mean a nd standard de viation of the fitted excess log returns from the EH regres-
sions sho wn in Ta ble 4.
13
P anel A reports results for the US, while Panel B reports
results f or the U K .

P anel A in Table 6 shows that TIPS hav e had a high ave rage risk premium over our
sample period. This premium is also larger than the a vera ge risk prem ium o n nom inal
13
Since the regressions include a constant, the mean of the fitted values coincide with the mean
excess log return reported in Table 2.
16
bonds, which results in a negative average breakeven inflation risk p re miu m . Pfluege r
and Viceira ( 2010 ) show e v idenc e tha t th is negative p rem ium is mostly d riven by a
positive liquidity risk premium in TIPS, no t by a negative inflation risk prem ium in
nominal Treasury bonds. In fact, Panel B sho ws that the average break ev en inflation
risk premium in the UK is positiv e, consisten t with the notion that the inflatio n risk
premium is positive on a ve rage. Column s 2 and 3 in each panel show that bond
risk premia exhibit significant variability over time, although this va riab ility is small
relative to the ove rall variability of realized bond excess retur ns.
Figure 2 illustrates the time series of the fitted bond risk premia and their difference–
the breakeven inflation risk premium–in the US (P anel A) and in the UK (P anel B).
Panel A in the figure shows that the nominal and TIPS risk premia ha ve gener-
ally mo ved together, and that they exhibit significant varia bility o ver time . Bond
risk premia declined during the period following th e stock m arket cra sh of t he ear ly
2000’s, increased during 2002 and 2003, and declined and became even negativ e in
the subsequ ent period until t hey i n creased again at t he on set o f t he re cent financial
crisis.
However, the breakeven inflation risk prem iu m also sh ows significant time series
variation, implying that the magnitu de of the change s in nominal and real bond risk
prem ia w as not identical. T he re were times at which they even moved in opposite
directions. The break even inflation risk premium w as marked ly negative at the be-
ginning of the sample, when TIPS were first issued and in v estors might not hav e
been familiar with them, and during the recen t financial crisis, when the TIPS risk
premiu m increa sed dramatic ally.
Panel B in Figure 2 shows the time series of the fitted UK risk premia. T he

nominal, rea l and breakeven risk pr emia ha ve moved together ove r the period 1985 to
2009 and, consistent with the results shown in Ta ble 6, t he nominal bond risk prem ium
has been abov e the real bond risk premium for m ost of the sa mple. In c on trast to the
US bond market, both the nominal bond risk prem ium and the real bond risk prem ium
shot up during the financial crisis. The nominal bond risk premium increased more
than the real bond r isk premiu m, c ausing the breakeve n inflation risk premium to
increase during the crisis. As i n the case of the US bond market, the risk prem iu m on
UK bonds, both real and nomina l, is not necessarily po sitive at all times. There are
periods of negative bond risk premia, mo st notably the turn of the 1990’s for both
real and nominal bonds and the period 200 4-2008 for real bonds.
17
7Conclusion
This article explores t he EH of the term structure of interest rates in the US and in
the UK government rea l an d nomin a l bond m arkets. It docum ents pr e d ictability o f
excess returns in inflation-indexedbondsandinnominalbondsinbothmarkets,thus
rejecting the E H , both real and nominal. While retu rn pred ict ab ility in US Treasury
nominal bonds has been well-documented in the past, to our know ledge this is the
first article to provide direct empirical eviden c e for predict ability of returns in real
bonds. We also find robust evidence that breakev en inflation returns, or the spread
bet ween n o minal bond excess r eturns and inflation-indexed bond e xcess returns, are
predictable.
The rejection of the r eal E H i m plies that inve stors in the i nflation-indexed bond
market face a tim e-varying risk premium that re flects a time- var ying real interes t rate
risk premium and possibly also a time-varying liquidity premium. The r ejection of t he
nominal EH and pa rticularly the rejection t hat e xpected breakev en inflatio n returns
are constant suggests that inflation risk, and the pre mium that i n vestors demand for
bearing it, also varies o ver time. R eal and nomin al bond risk prem ia appear to be
positively related to the real and nominal term spread, respectiv ely. When the real
ter m spread increases, expect ed returns on inflation- in de xed bond return s increa se
and, interestingly, real bonds are also expected t o outperform no m inal bonds. When

the n om inal term spread increases, expected excess r eturns on nom inal bonds increase.
The evidence against the real and nom in al EH suggests that increases in the yields
of long-term bonds, whether real or nom in a l, do not necessarily imply that expected
future sho r t-te rm inte re st rate s have risen . T h e increa s e in yields, or the decline in
bond prices, could be the result of an increase in the risk of long-term bonds and
the risk premium that inv estors demand for holding them. Thus investors should
be cautious in in terpreting increasing yields in long-term bonds as a signal of f uture
higher interest rates.
In recen t wo rk, Ca mpbell, Sunde ram, and Viceira (2010) show tha t bond risk
prem ia can m ove ove r tim e and take either sign depending on whether investors see
bonds as sa fe ass ets or risky assets. Our estimates show significa nt variat io n ove r
time of rea l and nominal bond risk premia, w ith periods of positive bond risk premia
and periods of negativ e bond risk pr emia , s u gg esting a chang ing in ve stor perception
of the risk of real a nd no m inal bonds.
18
In particular, the risk premium on TIPS has been large on average since they
were first issued in 1997 , more so th a n the average risk p rem ium on nomina l Treasury
bonds, but there have been periods where it has been negative, notably the period
between 2004 and the onset of the financial crisis. Th e historical large positiv e average
of the TIPS risk premium appears to be driven by two particular periods: the years
follow ing the creation of TIPS in the late 1990’s and most recently du rin g the recent
fin ancial crisis. Campbell, S hiller, and Viceira (2009) and Pflu eger and Viceira (2010)
show evidence that these episodes are linked to periods in which the TIPS mark et w as
particu larly illiquid, and inve stors might have dema nd e d a large liquidity premium
for h olding them.
Our estimates also suggest that in vestors demand a risk premium on nominal
bonds that also varies o ve r time. Cons istent with the evidence in Ca mp bell, Sunderam
and Viceira (2010), this premium might reflect the c h anging perception of inflation
risk by investors.
Our results suggest sev eral directions for future research. First, they suggest a

more detailed analysis of t he economic sources of ris k in real an d nominal bonds, along
the lines of Campbell, Sund eram, and Viceira (2010). Second , one could also explore
if the return predictability in the inflation-ind exe d bond market is the result of price
pressure and supply imbalances caused by limited arbitrage and preferred-habitat
investors in the bond market, along the lines of the preferred-habitat h ypothesis of
Modigliani and Sutch (1966), formalized in Vayanos and V ila (200 9) and G reenwood
and Vayan os (2008, 2009).
Arguab ly the inflation -ind exed bond marke t is a natura l candidate to look for seg-
mentation effects in the bond marke t. Just as investors m ight differ in their preference
for bond matu r itie s, they might also differ in their preference for holding inflation-
indexed or nominal bonds. For example, som e investors such as traditional defined-
benefit pensio n fund s in the US with a matu re liability structur e h ave l ia bilitie s wh ich
are mostly nominal, while other investors suc h as less mature d efined-benefitpension
funds or individuals investing for retiremen t face liabilities whic h a re mostly indexed.
Pflueger and Viceira (2010) f u rth er explore this hy pothesis.
Finally, it would be intere sting to explore the imp lication s of our findings for
portfolio man agem ent and pension inves tin g a n d how these im p lic ation s va ry by in -
vestm e nt horizon and the i nvestor’s s ha re of real and nomin al liabilities.
19
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