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9 7 0 1
Estimating the Cost
of U.S. Indexed Bonds
by Silverio Foresi, Alessandro Penati,
and George Pennacchi
FEDERAL RESERVE BANK
OF CLEVELAND
Estimating the Cost of U.S. Indexed Bonds
Silverio Foresi¤
Alessandro Penatiy
George Pennacchiz
Initial Draft: August 4, 1996
This Revision: January 7, 1997
Abstract: This paper presents an equilibrium bond pricing model driven by two
stochastic factors: the real interest rate and the expected rate of in‡ation. The
model’s parameters are estimated using a maximum likelihood technique based
on a Kalman …lter. Data on nominal U.S. Treasury securities and Survey of
Professional Forecasters predictions of the GDP de‡ator are employed to identify
the separate e¤ects of real and nominal variables. The market prices of real
interest rate risk and in‡ation risk are estimated, which allows us to construct
yield curves for nominal and indexed U.S. Treasury securities. The relative costs
of nominal and indexed bonds can then be assessed.
We are grateful for the comments of seminar participants at the Federal Reserve
Bank of Cleveland.
¤
Department of Finance, New York University, (212) 998-0358, sforesi@stern.nyu.edu.
Department of Economics, Bocconi University and University of Sassari, 31-2-8394044,
apenati@mbox.vol.it.
z
Department of Finance, University of Illinois, (217) 244-0952, gpennacc@uiuc.edu.
y
Estimating the Cost of U.S. Indexed Bonds
1
1
Introduction
For many years, economists have proposed that the United States government issue bonds having payments indexed to a price level.1 Until recently,
most government o¢cials had not shared economists’ attraction for indexed
bonds. However, the current political pressure to reduce Federal budget
de…cits may have led some government policymakers, including o¢cials at
the U.S. Treasury Department, to re-think their view. A likely reason for
this change of heart is the current belief that indexed bonds could reduce the
Federal government’s debt servicing costs.
On May 16, 1996, the U.S. Treasury Department announced plans to
begin selling bonds with payments indexed to in‡ation. In its proposal, the
Treasury states that the primary reason for o¤ering these bonds is to reduce
the expected interest costs paid by the Federal government and to provide
an alternative debt instrument that many investors could …nd attractive:
Because the Treasury, rather than the investor, would bear the
in‡ation risk on an in‡ation-protection security, the Department
expects that the prices at which it would sell this new type of
security would capture some or all of the in‡ation risk premium
charged by investors on conventional Treasury securities. In other
words, investors should be willing to pay extra for a security on
which the issuer, rather than the investor, bears the risk of higher
than expected in‡ation. Consequently, the expected interest costs
to the Treasury of in‡ation protection securities should be lower
than those on conventional Treasury securities.2
While it is generally presumed that nominal bonds bear an in‡ation risk
premium, there has been very little research on modeling and estimating this
premium, especially for U.S. dollar securities. Using data on the prices of
existing nominal and indexed government debt, Foresi, Penati, and Pennacchi (1996) provide estimates of in‡ation risk premia, but only for the United
Kingdom and Sweden where indexed bonds have already been issued. For
the U.S., the only work of which we are aware is the recent paper by Campbell and Shiller (1996). They estimate bond risk premia in the context of
1
Early in this century, Irving Fisher (1911) urged that ”purchasing power bonds” be
issued by the Federal government. Since then, a growing number of American economists,
especially those in academia, have supported this idea.
2
Federal Register 61(98), May 20, 1996, p.25165.
Estimating the Cost of U.S. Indexed Bonds
2
a multi-factor Capital Asset Pricing Model, where risk factors are proxied
using data on stock and bond market returns as well as changes in aggregate
consumption.
If ”saving” an in‡ation risk premium is the U.S. Treasury’s primary motive for issuing indexed bonds, it would be worthwhile to understand the
nature and size of this premium.3 Better knowledge of this premium could
in‡uence the proportion of total government debt that the Treasury might
decide to index. Quantifying this in‡ation premium, as well as the risk premium from changes in real interest rates, is the primary goal of the present
paper. In doing so, we will also provide estimates of the market prices of
indexed bonds of various maturities.
It would appear di¢cult to estimate an in‡ation risk premium prior to
the actual issuance of index bonds. The di¢culty lies in disentangling the
e¤ects of real returns and in‡ation from only nominal bond prices. The
approach taken in this paper is to use survey data on forecasts of in‡ation,
in addition to nominal bond price data, to identify these separate e¤ects.
By proxying market expectations with survey forecasts, along with use of
a powerful empirical technique, we are able to estimate a large number of
parameters that are necessary for modeling nominal and indexed bond yield
curves.
The analysis of this paper ignores some theoretical and practical considerations that could a¤ect the value of indexed bonds. First, the market
prices of real interest rate and in‡ation risk that we estimate from historical
data are assumed to remain unchanged following the introduction of indexed
bonds. This ignores possible general equilibrium e¤ects on the value of nominal and indexed securities if and when indexed securities become widely
available. If indexed bonds provide a new hedging vehicle which improves
…nancial market completeness, market risk premia may change. Predicting
the size and direction of this change would require a more detailed model of
3
An arguement against treating the savings of an in‡ation risk premium as a bene…t of
indexed bonds is that the reduction in the Treasury’s costs simply re‡ects a re-distribution
of cash‡ows between taxpayers and bondholders. One could even argue that on a riskadjusted basis, the value of this redistribution is zero. Some counter-arguments exist,
however. First, by issuing indexed bonds, the in‡ation risk absorbed by the government
might be better allocated across all taxpayers, which could lead to more e¢cient risksharing between individual bondholders and taxpayers. Second, if issuing indexed bonds
reduces the average cost of servicing debt, it could allow a reduction in other, moredistortionary revenue sources, such as income taxes.
Estimating the Cost of U.S. Indexed Bonds
3
the economy than that presented in this paper. Second, our model ignores
the e¤ect of taxes on the relative values of nominal and indexed bonds. The
value of indexed bonds will be in‡uenced by whether the government chooses
to tax their capital gains.4 If capital gains from index adjustments are taxed,
then the real after-tax payments of indexed bonds will still be a¤ected by
in‡ation. Third, we neglect that, in practice, indexed bonds are not fully
indexed, but have payments tied to a previous period’s price index. This
means that there is an indexation lag, that is, a period prior to the bond’s
payment date when indexed bonds lack in‡ation protection. The original
U.S. Treasury proposal is for this period to be three months. Future versions
of this paper will explicitly account for valuation e¤ects of this indexation
lag.
The plan of the paper is as follows. In Section 2 we present a two factor
model for valuing nominal and indexed bonds. The two factors represent
the levels of the real interest rate and expected in‡ation. Section 3 describes
our data and empirical method. The model is cast in state space form and
its parameters are estimated by maximum likelihood using a Kalman …lter
to recursively compute the likelihood function. The estimation results are
presented in Section 4. There we also characterize nominal and indexed bond
yield curves and provide a measure of the in‡ation risk premium. Section 5
concludes.
2
Valuing Nominal and Indexed Bonds
As in Constantinides (1992) and Turnbull and Milne (1991), we start from a
nominal pricing kernel, M, a stochastic discount factor which regulates prices
of contingent claims. The existence of such a pricing kernel is equivalent to
the absence of pure arbitrage opportunities.5 The time t; nominal value,
V (t; ¿), of a contingent claim which entitles the owner to receive a possibly
uncertain monetary cash ‡ow at time t + ¿ , H(t + ¿ ), satis…es the pricing
relation:
·
¸
M(t + ¿ )
V (t; ¿) = Et
H(t + ¿ )
(1)
M(t)
These quantities are nominal, that is, expressed in currency units. We
de…ne the general price level as p(t). The relationship between nominal and
4
5
Currently, the U.S. Treasury proposes to tax capital gains.
Du¢e (1996) provides a detailed derivation of this result.
Estimating the Cost of U.S. Indexed Bonds
de‡ated (real) prices is summarized by the following:
·
¸
M(t + ¿ ) p(t + ¿) H(t + ¿ )
v(t; ¿) = Et
M(t)
p(t) p(t + ¿ )
·
¸
m(t + ¿ )
= Et
h(t + ¿)
m(t)
4
(2)
where the contingent claim’s real price, v(t; ¿ ) ´ V (t; ¿ )=p(t), and the real
cash ‡ow, h(t + ¿ ) ´ H(t + ¿)=p(t + ¿ ), satisfy a pricing equation analogous
to that of the nominal quantities but using a real pricing kernel, m(t) ´
M(t)p(t).
The general price level, p(t), depends on an economy’s transaction technology, as well as the monetary and …scal policy of its government. In models
with endogenous in‡ation, the covariance of in‡ation with consumption, and
hence the in‡ation risk premium, can be positive or negative depending on
investors’ hedging attitudes towards in‡ation, which in turn are model speci…c. We choose not to derive the demand for and supply of money from
…rst principles, with an eye to leaving the sign and size of the covariance of
in‡ation with other sources of risk to be free to take on any feasible values.
2.1
A Two-Factor Gaussian Model
We write the law of motion for the price level p(t) as
dp
= ¼dt + ¾p dzp
p
(3)
Our notation will denote a shock to variable x by dzx , a standard Wiener
process, and denote the instantaneous correlation of the shocks to variables
x and y by ½xy . Our term structure model is a¤ected by real and nominal
shocks. The real variable, which re‡ects variations in average productivity,
is the real interest rate on a bond which pays an instantaneously riskless real
return. This riskless real rate of return is denoted by r(t). The nominal
variable is ¼(t), the expected instantaneous in‡ation rate. The joint process
for r and ¼ is
dr = (a1 + b11 r + b12 ¼)dt + ¾ r dzr
d¼ = (a2 + b21 r + b22 ¼)dt + ¾ ¼ dz¼
(4)
Estimating the Cost of U.S. Indexed Bonds
5
· or, ¸more compactly, denoting the vector of state variables as s(t) ´
r(t)
,
¼(t)
ds = (a + Bs)dt + ¾dZ
(5)
·
¸
·
¸
· 2
¸
P
a1
b11 b12
¾r
½r¼ ¾r ¾ ¼
|
where a ´
,B´
, and ¾¾ ´
=
.
a2
b21 b22
½r¼ ¾ r ¾¼ ¾ 2¼
a, B, and § are assumed to be constants. The above process for the state
variables is a special case of the model in Langetieg (1980). Note that r and
¼ can depend on each other, both instantaneously (½r¼ 6= 0) and in terms of
their expected changes (b12 6= 0,b21 6= 0). However, they do not depend on
the price level, p, which is a requirement for the absence of money illusion.
The above assumptions imply that s(t) has a bivariate normal distribution
with mean and covariance matrix
Et [s(t + ¿)] = °(¿) + ª(¿ )s(t)
Z ¿
X
|
Covt [s(t + ¿) s(t + ¿)] ´ Q(¿ ) =
ª(!)
ª(!)| d!
(6)
0
where the
) ´ eB¿ , and the 2x1 vector °(¿ ) ´ ¡(¿)a, where
R ¿ 2x2 matrix ª(¿
¡1
¡(¿ ) ´ 0 ª(!)d! = B (ª(¿ ) ¡ I). In addition, if the (real parts of the)
eigenvalues of matrix B are negative, then s(t) has a stationary
· ss ¸ distribution
r
with an unconditional expected (steady state) value of
= ¡B¡1 a:
¼ ss
The nominal pricing kernel M is given by
¡
dM
= idt + Ár dzr + Á¼ dz¼
M
(7)
where Ár and Á¼ are constants which represent the market prices of risk from
changes in real interest rates and in‡ation, respectively. It is straightforward
to obtain this particular pricing kernel as the nominal marginal utility of a
representative investor in a Cox, Ingersoll, and Ross (1985b)-type production
economy (where the representative investor has a time-separable logarithmic
utility function and the production technology has constant returns to scale)
by suitably choosing the stochastic process governing the mean and volatility
of production and in‡ation.6
6
For example, see Constantinides (1991), Pennacchi (1991), and Sun (1991).
Estimating the Cost of U.S. Indexed Bonds
6
It is well known that the expected rate of change of the nominal pricing
kernel is equal to minus the instantaneous maturity nominal interest rate,
i(t), a property that equation (7) satis…es by construction.7 Analogously,
the expected rate of change of the real pricing kernel is equal to minus the
instantaneous real interest rate, Et[dm=m] = ¡rdt. Since
·
¸
dM
i(t) = ¡Et
(8)
M
·
¸
d(m=p)
= ¡Et
m=p
"
µ ¶2 #
dm dp dm dp
dp
¡
¡
+
= ¡Et
m
p
m p
p
i(t) = r(t) + ¼(t) + ¾ mp ¡ ¾ 2p
the nominal interest rate, i, and the real interest rate, r, are related through
an extended Fisher relation.
We can now compute the prices of pure discount (zero-coupon) bonds.
Let N (t; ¿ ) be the date t nominal price of a nominal bond that pays $1 with
certainty at date t + ¿ . Its value is given by equation (1) with H(t + ¿ ) =
1. Because of assumed normality of the state variables, the expectation of
pricing kernel can be computed as follows:
N(t; ¿) = Et [exp fln M(t + ¿) ¡ ln M(t)g]
(9)
·
½Z t+¿
¾¸
= Et exp
d ln M(t + !)
t
·
½ R t+¿
¾¸
¡ t (i(!) + 21 [Á2r + Á2¼ + 2½r¼ ¾r ¾ ¼ ])d!
R t+¿
= Et exp
¡ t (Ár dzr + Á¼ dz¼ )
= Et [exp fAn (t; ¿ )g]
¾
½
1
= exp Et [An (t; ¿)] + V art [An (t; ¿ )]
2
= Jn (¿) exp f¡wn ¡(¿ )s(t)g
where the 1x2 vector wn ´ [ 1 1 ], and Jn (¿) is an expression, independent
of s(t), that given in the Appendix.
In a similar manner, we can value a zero-coupon indexed bond. Let
R(t; ¿ ) be the date t real price of an indexed bond that pays $p(t+ ¿ ) at date
7
See Theorem 1 in Cox, Ingersoll, and Ross (1985a).
Estimating the Cost of U.S. Indexed Bonds
7
t + ¿ .8 Its value is given by equation (2) with H(t + ¿ ) = p(t + ¿ ), that is,
h(t + ¿) = 1.
R(t; ¿ ) = Et [exp fln m(t + ¿ ) ¡ ln m(t)g]
(10)
·
½Z t+¿
¾¸
= Et exp
d ln m(t + !)
t
¾¸
·
½ R t+¿
¡ t (r(!) + ¾ mp + ½r¼ ¾r ¾ ¼ + 12 [Á2r + Á2¼ ¡ ¾ 2p ])d!
R t+¿
= Et exp
¡ t (Ár dzr + Á¼ dz¼ ¡ ¾p dzp )
= Et [exp fAr (t; ¿ )g]
½
¾
1
= exp Et [Ar (t; ¿ )] + V art [Ar (t; ¿)]
2
= Jr (¿ ) exp f¡wr ¡(¿ )s(t)g
where the 1x2 vector wr ´ [ 1 0 ], and Jr (¿ ) is an expression, similar to
Jn (¿) and also independent of s(t), that is given in the Appendix.
We conclude this section by calculating another variable that will used in
our empirical work. The rate of in‡ation between date t and date ¿ that is
expected at date t equals
Et [p(t + ¿)=p(t)] = Et [exp fln p(t + ¿ ) ¡ ln p(t)g]
(11)
·
½Z t+¿
¾¸
= Et exp
d ln p(t + !)
t
·
½ Z t+¿
¾¸
Z t+¿
1 2
¾ p dzp
= Et exp ¡
(¼(!) ¡ ¾ p )d! +
2
t
t
= Et [exp fAp (t; ¿)g]
½
¾
1
= exp Et [Ap (t; ¿ )] + V art [Ap (t; ¿ )]
2
= Jp (¿ ) exp f¡wp ¡(¿ )s(t)g
where the 1x2 vector wp ´ ¡[ 0 1 ], and Jp (¿ ) is an expression, independent
of s(t), that is also given in the Appendix.
8
This assumes no indexation lag. Due to delays in the collection of price information,
price indices are reported with a lag. Thus, for practical reasons, indexed bond payments
can be linked only to a price index for a prior date. The di¤erence between the payment
date and the date of the price index equals the indexation lag. For U.K. government indexlinked gilts, the lag is eight months. The current U.S. Treasury proposal is for a three
month lag. Future versions of the paper will value indexed bonds with an adjustment for
this lag.
Estimating the Cost of U.S. Indexed Bonds
3
8
The Empirical Technique
3.1
Data
Our bond price data consists of zero-coupon bond prices derived from nominal U.S. Treasury bill, note, and bond quotes using the smoothed Fama-Bliss
method. See Bliss (1996) for a description of this …tting method and an
analysis of its performance compared to other …tting methods. Bond prices
were observed on the last trading day of each month over the period January
1970 to November 1995. At each observation date, we selected zero-coupon
bond prices having the following eight maturities: 41 , 12 , 1, 2, 3, 5, 7, and 10
years.9 These bond prices would then correspond to the theoretical nominal
zero-coupon bond prices given in equation (9) above.
In addition to this bond data, we used forecasts of the GDP de‡ator obtained from the Survey of Professional Forecasters. The American Statistical
Association in conjunction with the National Bureau of Economic Research
began conducting this quarterly survey in November of 1968, and it was taken
over by the Federal Reserve Bank of Philadelphia in 1990.10 The survey asks
professional forecasters for their prediction of the GDP de‡ator one, four,
seven, ten, and 13 months into the future. The surveys are taken during
the second month of each quarter and released at the end of that month or
early in the third month of the quarter. As emphasized by Keane and Runkle (1990), this survey is limited to professional forecasters who are likely to
have a strong incentive to utilize information in an e¢cient manner. If so,
this data may match the expectations of in‡ation embedded in the market
prices of bonds better than other survey data on in‡ation expectations.
We used the median forecast of the survey respondents for each of these
…ve di¤erent horizons. In principle, if we divide this forecast by the GDP
de‡ator corresponding to the time that the forecast is made, this measure
would match our theoretical measure of Et [p(t + ¿ )=p(t)] given by equation
(11). However, because of reporting lags and subsequent revisions in GDP
de‡ator data, it is not clear what measure of p(t) the survey participants had
in mind at the time of their forecasts. This problem can be overcome if we
divide the four, seven, ten, and 13 month forecasts by the one month forecast.
9
Because markets for longer maturity bonds tend to be thin, we limited the longest
bond maturity to 10 years so as to minimize zero-coupon bond …tting errors.
10
See Croushore (1993, 1996) for a detailed description of this survey data and an
analysis of its forecasting performance.
Estimating the Cost of U.S. Indexed Bonds
9
1
This would correspond to the theoretical expression Et[p(t + ¿ )=p(t + 12
)] for
4
7 10
13
¿ = 12 , 12 , 12 , and 12 years. Using equation (11), this theoretical expression
equals
·
¸
·
¸
¾
½
1
Jp(¿ )
1
Et p(t + ¿ )=p(t + ) =
) s(t) :(12)
1 exp ¡wp ¡(¿ ) ¡ ¡(
12
12
Jp ( 12
)
3.2
Estimation Method
Our model implies that nominal bond prices, given by equation (9), and
in‡ation forecasts, given by equation (12), are both exponential functions of
a linear combination of the state variables. Thus, the natural logs of bond
prices and in‡ation forecasts will be linear in the state variables. Denote
the date t continuously-compounded yield on a nominal zero-coupon bond
having ¿ periods until maturity as yn (t; ¿) ´ ¡ ¿1 ln[N(t; ¿ )]. Similarly, denote
the date t forecast of the continuously-compounded, annualized in‡ation rate
1
1
between date t + 12
and date t+¿ as yp (t; ¿ ) = lnfEt [p(t+¿ )=p(t+ 12
)]g=(¿ ¡
1
).
Because
our
observed
zero-coupon
bond
prices
are
…tted
from
bid and
12
ask quotes of coupon bonds, we assume that the observed yn (t; ¿ ) equals its
theoretical value with measurement error:
yn (t; ¿) = jn (¿ ) + ®n (¿ )s(t) + "n (t)
(13)
where the scalar jn ´ ¡ ¿1 ln[Jn (¿ )] and the 1x2 vector ®n ´ ¿1 wn ¡(¿ ). Likewise, because our observed Survey of Professional Forecasters median forecast of in‡ation may deviate from the ”true” market expectation of in‡ation,
we assume that the observed yp (t; ¿ ) measures its theoretical value with error:
yp(t; ¿ ) = jp (¿ ) + ®p (¿ )s(t) + "p (t)
(14)
1
1
where the scalar jp ´ ln[Jp (¿ )=Jp ( 12
)]=(¿ ¡ 12
) and the 1x2 vector ®p ´
1
1
¡wp[¡(¿ ) ¡ ¡( 12
)]=(¿ ¡ 12
). Since we observe eight di¤erent maturity bond
prices and four di¤erent horizon in‡ation forecasts at each date, we can
”stack” eight versions of equation (13) and four versions of equation (14) to
obtain a 12x1 vector y(t) ´ [yn (t; ¿ 1 ) ¢ ¢ ¢ yn (t; ¿ 8 ) yp (t; ¿ 1 ) ¢ ¢ ¢ yp (t; ¿ 4 )]| which
equals
y(t) = j + ®s(t) + "(t)
(15)
where j is a 12x1 vector with elements equal to the jn (¿ )’s and jp(¿ )’s, ® is
a 12x2 vector with rows equal to the ®n (¿ )’s and ®p (¿ )’s, and "(t) is a 12x1
vector with elements equal to the "n (t)’s and "p (t)’s.
Estimating the Cost of U.S. Indexed Bonds
10
Equations (5) and (15) comprise a state space system. The unobserved
state variables, r(t) and ¼(t), follow the ”state transition” equation (5), which
is equivalent to a discrete-time bivariate AR(1) process. The ”measurement” equation (15) equates observables y(t) to a function of the state variables
plus measurement noise, "(t). For estimation purposes, we assume that this
measurement noise has a serially uncorrelated, mean-zero, multivariate normal distribution with a date t covariance matrix that is diagonal with the
…rst eight diagonal elements (corresponding to bond yields) equal to ¾2"n and
the last four diagonal elements (corresponding to expected in‡ation rates)
equal to ¾ 2"p .
Given these distributional assumptions, maximum likelihood estimates of
the model parameters can be obtained using a Kalman …lter to recursively
compute the likelihood function. This is a relatively powerful estimation procedure in that both cross-sectional (equation 15) and time-series (equation
5) model restrictions are imposed on the data. For details of this procedure,
including how bond prices observed at a monthly frequency are optimally
combined with in‡ation forecasts observed at a quarterly frequency, see Pennacchi (1991 p.66 and Appendix B).11
4
Results
4.1
Term Structure Parameter Estimates
The two-factor term structure model involves a large number of parameters (…fteen). For a few of these parameters, we have strong priors as to
what values they should take because they can be directly estimated by
other means. In particular, the variance of the price level, ¾2p , and its covariance with the real pricing kernel, ¾ mp , can be reasonably directly estimated. For a production economy with a representative, logarithmic utility investor, such as Cox, Ingersoll, and Ross (1985b), ¾ mp equals minus
the covariance between the growth rate of the price level and the growth
rate of real output (capital). Thus, using data on the GDP de‡ator and
real GDP over the period 1959.Q1-1996.Q1, we estimated ¾ p = :02107 and
11
It should be noted that this paper imposes all theoretical cross-equation restrictions
on the parameters of the measurement equations. In particular, the elements of the vector
j are not assumed to be an arbitrary constants, as in Pennacchi (1991), but are restricted
to take values implied by the model. The Appendix details the theorectical values for the
elements of j.
Estimating the Cost of U.S. Indexed Bonds
11
¾mp = ½mp ¾m ¾ p = :2866(:01426)(:02107) and …xed the parameters to these
values prior to estimation. Recall from equation (8) that the nominal interest rate is given by i(t) = r(t) + ¼(t) + ¾mp ¡ ¾ 2p. Thus, our estimate of
¾mp ¡ ¾ 2p = ¡:000358 reduces the nominal interest rate by 3.5 basis points
over the real rate plus the drift of the price level process.
The only other parameter that we restricted was the implied steady state
instantaneous real interest rate, rss . As discussed in Merton (1980), precise
estimation of expected real returns on …nancial assets generally requires a
very long time series. Thus, we felt r ss could be constrained to a reasonable
value, and we tried both 2% and 2:5%.
The parameter estimates for the two cases, rss = 2% and rss = 2:5% are
given in the …rst two columns of Table 1. With the possible exception of the
estimates for ¼ ss , there is little di¤erence in the results for these two cases.
The estimates of the elements of matrix B are each statistically di¤erent
from zero and, together, imply a stationary state variable process12 . The
relatively large size of the o¤-diagonal elements (b12 ¼ ¡:4, b21 ¼ :8) indicate
substantial serial dependence between the real rate and expected in‡ation.
Regarding the elements of the covariance matrix, §, the estimated standard
deviations, ¾r and ¾¼ , suggest that changes in the expected instantaneous
rate of in‡ation are more volatile than changes in the instantaneous real
interest rate. Further, these changes are highly positively correlated (½r¼ ¼
:82).
The market prices of risk from changes in real interest rates and in‡ation,
Ár and Á¼ , are approximately the same magnitude, ¼ ¡:23. Their negative
signs are what would be expected, as this implies that bond returns have positive real interest rate and in‡ation risk premia. However, these estimates
are not statistically signi…cant at the 5% level (though the real premium is
borderline). The implied steady state in‡ation rates, ¼ss = 2:75%, 3:0%,
may seem a bit low, but are still within a reasonable range. Lastly, the standard deviations of the measurement errors, ¾ bonds and ¾f orecast , indicate that
a typical error in the theoretical model’s …t of the observed annualized bond
yields is 16 basis points, while a typical error in the model …tting the annualized in‡ation forecasts is around 134 basis points. Thus, the model seems to
be …tting the bond yields quite well, but the survey in‡ation forecasts rather
poorly.
A measure of the mean-reversion of the state variables can be calculated
12
That is, the real parts of the eignevalues of B are negative.
Estimating the Cost of U.S. Indexed Bonds
12
from the point estimates for B and §. Table 1 gives the half-lives for r(t)
and ¼(t), that is, the expected time it takes for these state variables to return
one half way back to their steady state levels following a deviation. Mean
reversion for expected in‡ation (half-life ¼ 1.1 years) is relatively stronger
than that for the real interest rate (half-life ¼ 6.5 years). Figures 1 and
2 illustrate the state variables mean-reverting tendencies by plotting their
impulse response functions for a one-standard deviation shock from their
steady state variables.13 Figure 1 is for the case in which a shock of ¾i to
state variable i is coincident with a shock of ½ij ¾ i ¾j to state variable j, that
is, it considers the likely correlation between shocks of the state variables. In
contrast, Figure 2 considers independent shocks, so that an initial shock of ¾ i
to state variable i leads to no immediate change in state variable j. It is clear,
especially from Figure 2, that the state variables are highly dependent upon
each other. Note especially that an independent upward shock to in‡ation
leads to a signi…cant expected fall in the real interest rate. Hence, while the
two state variables are instantaneously positively correlated, implying high
volatility in short term rates, the expected future opposite movement in the
state variables will lead to longer-term yields (being risk-adjusted averages
of expected short rates) reacting in a much less volatile manner.
4.2
Indexed Bonds and the In‡ation Yield Premium
Based on the parameter estimates for column two of Table 1, Figure 3 plots
the term structure of interest rates for both nominal bonds and indexed
bonds assuming that the initial state variables are equal to their steady state
values. The nominal and indexed bond yields curves both are upward sloping
and appear reasonable. An intuitive measure of the premium attributable to
in‡ation risk is the spread between equivalent maturity nominal and indexed
bond yields, less the expected in‡ation rate over the life of the bonds. Since
both state variables start from their steady states levels, their expected values
for any future horizon equal their current values, so that this premium can
be calculated as
yn (t; ¿) ¡ yr (t; ¿) ¡ (¼ ss + ¾ mp ¡ ¾2p )
(16)
By construction, this in‡ation yield premium equals zero for the instantaneous maturity nominal and real yields. From Figure 3, we see that this
13
All …gures in the paper re‡ect parameter estimates for the case of rss = 2:5%.
Estimating the Cost of U.S. Indexed Bonds
13
yield premium is hump-shaped, reaching a maximum of 53.76 basis points at
a maturity of 12.6 years. Thus, based on these model estimates, the size of
the in‡ation risk premium, translated in terms of a yield premium, could be
approximately one-half of one percent for longer-maturity bonds. This is one
potential measure of the government savings attributable to issuing indexed
bonds relative to nominal bonds.
These results should be treated with caution in that there are indications
that the parameter estimates do not produce realistic yield curves for some
values of the state variables su¢ciently far from their steady states. For
example, in Figure 4 we plot nominal and indexed bond yield curves for
r(t) = rss , but ¼(t) equal to 100, 200, and 300 basis points above its steady
state level. Because b12 ¼ ¡:43 is signi…cantly negative, higher expected
in‡ation leads to an expected decline in future real rates, which is indicated
by the lowering of the indexed bond’s yield curve. In addition, because
b21 ¼ :81 is signi…cantly positive, this lowering of the real rate reduces the
expected future values of ¼. The net e¤ect is that an independent upward
shock in the current in‡ation rate increases short term nominal interest rates,
but may actually lower longer term nominal rates, producing rather extreme
inverted nominal yield curves. A related anomaly is the highly positive value
for the instantaneous correlation coe¢cient, ½r¼ ¼ :82. The vast majority of
previous studies of the relationship between real interest rates and expected
in‡ation have found signi…cant negative correlation between these variables.14
In one sense, our estimation results are consistent with the state variables
moving in opposite directions, but the timing is di¤erent. As Figure 2 shows,
an independent upward shock to in‡ation is expected to reduce future short
term interest rates, so that long-term yields will not adjust one-for-one with
an increase in ¼. This suggests that our parameter estimation procedure
may be having di¢culty identifying whether this opposite movement of the
state variables is occurring instantaneously (which would occur if ½r¼ ; < 0),
or intertemporally (which would occur if one or more of the o¤-diagonal
elements of B were < 0).
14
See Pennacchi (1991) and the references therein. Theoretically, one would expect
negative correlation, at least when the nominal interest rate is low. Note that the real
interest rate and expected in‡ation could, individually, become negative, but the nominal
interest rate (equal to their sum plus ¾mp ¡ ¾ 2p ) cannot, else currency would dominate
bonds. If the nominal rate is positive, but near zero, a fall in one state variable (e.g., the
real interest rate) would need to be o¤set by a rise in the other (e.g., expected in‡ation).
See Black (1995) for more discussion.
Estimating the Cost of U.S. Indexed Bonds
4.3
14
Results for a Restricted Model
In an attempt to remedy the yield curve peculiarities of the previous section,
we re-estimated the model under the restriction b12 = b21 = 0, that is, that B
is a diagonal matrix. This restriction removes the intertemporal, but not the
instantaneous, dependence between the state variables. The results for the
case of rss = 2:0% and rss = 2:5% are given in columns three and four of Table
1. The estimates of b11 and b22 are both negative and statistically signi…cant,
though b11 is quite small and implies very weak mean-reversion for the real
interest rate. Both ¾ r and ¾ ¼ are slightly larger than in the unrestricted
case and, most interestingly, ½r¼ switches from being signi…cantly positive to
being signi…cantly negative.15 As mentioned earlier, this signi…cant negative
correlation between real rates and expected in‡ation is consistent with the
bulk of previous empirical work. The restriction on B also has sizable e¤ects
on the estimates of the risk premia. Ár declines, but Á¼ rises and both are
now statistically signi…cant. There is not much di¤erence in the estimated
steady state in‡ation rate, ¼ ss : it falls slightly to 2.9%. Importantly, the
restricted likelihood function does not decline very much: a likelihood ratio
test cannot reject the diagonality of B at reasonable con…dence levels.
Figure 5 illustrates the implications of these restricted estimates for the
nominal and indexed bond yield curves and the in‡ation yield premium. The
indexed yield curve is ‡atter than the previous case, while the nominal yield
curve is reasonably upward sloping. This results in an in‡ation yield premium
that is signi…cantly larger than that of the unrestricted case. Here, the in‡ation yield premium for a 10 year zero-coupon bond is 219 basis points versus
53 basis points for the unrestricted case. Also, while not displayed here, the
restricted case’s indexed and nominal yield curves do not exhibit extreme
inverted shapes when the state variables deviate from their steady states.
In this sense, these restricted yield curves have more realistic characteristics
than the unrestricted ones.
Having estimated the parameters for what appears to be a more realistic
model, we can now compute the best estimates of the levels of the (unobserved) state variables over the sample period, a process called smoothing.16
15
While the results are not presented here, we also estimated the model under the single
restriction that either b12 or b21 equals zero. However, under these assumptions, the
correlation coe¢cient remained positive and statistically signi…cant: ½r¼ ¼ :16 when only
b12 = 0; and ½r¼ ¼ :36 when only b21 = 0.
16
For a description of smoothing, see Harvey (1981).
Estimating the Cost of U.S. Indexed Bonds
15
The results of this are illustrated in Figure 6, which plots the estimated times
series for the real interest rate, expected in‡ation, and, for the sake of comparison, the Fama-Bliss three-month bond yield. Notably, there are periods
during the 1970’s and 1990’s when the instantaneous real interest rate became negative. The negative correlation of real rates and expected in‡ation
is also apparent from this …gure. Figure 7 plots the same estimated time
series of expected in‡ation, but compares it to the actual level of in‡ation
given by the actual quarterly change in the GDP de‡ator. There was an extended period of time in the late 1970’s when actual in‡ation appeared to be
systematically higher than investors’ expectations. Similarly, actual in‡ation
was frequently lower than what was expected during the late 1980’s and early
1990’s. However, we caution against concluding that these episodes indicate
irrational expectations. Investors may require a signi…cant accumulation of
evidence before a change in monetary policy is viewed to be credible.
In summary, this restricted model produces sensible term structures of
real and nominal interest rates and implies dynamics for real interest rates
and in‡ation that are consistent with prior research. Given that its restrictions cannot be statistically rejected, it should be preferred to its unrestricted
counterpart.
5
Conclusion
This paper provides a framework for valuing bonds having payments indexed
to a price level. Using survey forecasts of in‡ation as a proxy for market
expectations, we were able to identify the separate in‡uences of real returns
and in‡ation embedded in nominal bond prices. In addition, the paper’s
maximum likelihood technique, which imposes both cross-sectional and timeseries restrictions on the sample of observations, allowed us to calculate a
large number of model parameters relatively accurately. Based on these
parameter estimates, we could construct nominal and indexed bond yield
curves and calculate investors’ required premia for risk from changes in both
real interest rates and in‡ation.
The term structures implied by the unrestricted parameter estimates appeared to display excessive curvature when in‡ation was signi…cantly above
its steady state level. By restricting the state variable process in a way
that reduced intertemporal dependence, we obtained nominal and indexed
bond yield curves that appeared more realistic. Importantly, this state vari-
Estimating the Cost of U.S. Indexed Bonds
16
able process restriction could not be statistically rejected by the data, which
leads us to conclude that it represents an improvement over the unrestricted
version. Based on our results for this restricted model, we found that the
size of an in‡ation yield premium to be approximately 220 basis points for a
10 year zero-coupon bond. Subject to the modeling and estimation caveats
previously discussed, this suggests that the potential government budgetary
”savings” from issuing indexed bonds are substantial.
Estimating the Cost of U.S. Indexed Bonds
17
Appendix
This Appendix gives the values for Jn (¿ ), Jr (¿), and Jp(¿ ) presented
in section 2.1 of the text. Their values can be derived from the results of
Langetieg (1980). For i = n, r, or p, Ji (¿ ) is given by the following formula:
ln[Ji ] = ¡¿ (ui ¡ wi B¡1a) ¡ wi ¡(¿ )B ¡1 a
(17)
1
+di wi B¡1 (¡(¿ ) ¡ I ¢ ¿ )© + wi ¡(¿ )-¡(¿ )| wi
2
1
¡ wi (B¡1¡(¿ )- + -¡(¿ )| (B¡1 )| )wi|
2
1
+ wi B¡1 §(B¡1)| wi| ¿
2
·
¸
¾r Ár
where © ´
, un = ¾mp ¡ ¾2p , ur = 0, up = ¡ 21 ¾ 2p , dn = dr = 1,
¾¼ Á¼
and dp = 0. The 2x2 matrix - is de…ned as follows. Let ¸i , i = 1,2, be
the eigenvalues of B, and let C be a 2x2 matrix whose columns are the
corresponding eigenvectors. De…ne §¤ ´ C ¡1§(C ¡1 )| , which is a 2x2 matrix
with elements ¾ ¤ij and then de…ne §¤¤ as a 2x2 matrix with elements equal
to ¾ ¤ij =(¸i + ¸j ). Then - ´ C§¤¤C | . Note that once - is calculated, the
covariance matrix of the state variable process, Q(¿ ), given in equation (6)
can be calculated: Q(¿ ) = ª(¿)-ª(¿)| ¡ -.
Estimating the Cost of U.S. Indexed Bonds
18
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@FedFRASER