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A THREE-FACTOR ECONOMETRIC MODEL
OF THE U.S. TERM STRUCTURE
by
Frank F. Gong and Eli M. Remolona
Federal Reserve Bank of New York
Research Paper No. 9619
July 1996
This paper is being circulated for purposes of discussion and comment only.
The contents should be regarded as preliminary and not for citation or quotation without
permission of the author. The views expressed are those of the author and do not necessarily
reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045
This draft, July 1996
Not for quotation
A Three-Factor Econometric Model
of the U.S. Term Structure
FRANK F. GONG and ELIM. REMOLONA 1
Capital Markets Function
Federal Reserve Bank of New York
New York, NY 10045
Tel: (212}720-6943(Gong)
(212) 720-6328 (Remolona}
E-mail: frank.gong@frbny.sprint.com
eli. remolona@frbny.sprint.com
Abstract
We estimate and test a model of the U.S. term structure that fits
both the time series of interest rates and the cross-sectional shapes of the
yield and-volatility curves. In the model, three unobserved factors drive
a stochastic discount process that prices assets so as to rule .out.arbitrage .
opportunities. The resulting bond yields are conveniently affine in the
factors. We use monthly zero-coupon yield data from January 1986 to
March 1996 and estimate the model by applying a Kalman filter that
takes into account the model's no-arbitrage restrictions and using only
three maturities at a time. The parameter estimates describe a first
factor that reverts slowly to a fixed mean and a second factor that
reverts relatively quickly to a time-varying mean serving as the third
factor. The estimates are robust to the choice of maturities, suggesting
that these factors give us an adequate model.
JEL Classification Codes: E43, Gl2, Gl3.
Keywords: Term structure, pricing kernel, affine yields, mean reversion,
time-varying mean, Kalman filter.
1 We
thank Richard Cantor, John Clark, Arturo Estrella, Jean Helwege, Jim Mahoney,
and Tony Rodrigues for helpful comments.
A Three-Factor Econometric Model
of the U.S. Term Structure
FRANK F. GONG and ELIM. REMOLONA
Federal Reserve Bank of New York
I. Introduction
A challenge of equilibrium models of the term structure of interest rates is
to reconcile the time-series dynamics of interest rates with the cross-sectional
shapes of the yield and volatility curves. Backus and Zin (1992) and Campbell,
Lo, and MacKinlay (1994, hereafter CLM) have pointed out that such models
have been estimated to be consistent with either the time series or the cross .
section of bond yields but not both. 2 When the models are. based on timeseries data, the constructed term structures fail to match the shapes of actual
curves. 3 When the models are based on cross-section data, parameter values
. must vary over time. ,with. shifts in the term structure. 4 In this paper, we ...
specify and estimate an equilibrium model of the U.S. term structure that is
consistent with both time-series and cross-section data. The model we propose
2
Campbcll. Lo. and Mackinlay state, "But in simple term structure models, there also
appear to be systematic differences between the parameter values needed to fit cross-section
term structure data and the parameter values implied by the time-series behavior of interest
rates."
3
Chan, Karolyi, Longstaff, and Sanders (1992), Ait-Sahalia (1995), Eom (1995), and
Stanton (1996) provide alternative estimates of the time-series parameters.
4
To market participants, it is more critical that their model be consistent with the cross
section than with the time series, particularly for pricing contingent claims. However, as
Black and Karasinski (1991) point out, relying solely on cross section data means having a
different model from one moment to the next. The most popular cross-section models are
Ho and Lee (1986), Black, Derman, and Toy (1990), Hull and White (1990), and Heath,
Jarrow, and Morton (1992).
1
is one that requires three factors to price bonds consistently across the term
structure and produce the actual shapes of yield and volatility curves.
The number of factors required for an adequate term structure model is
an important issue. The point is to build a consistent model with as few
factors as possible. Litterman and Scheinkman (1991) show that three factors
can explain nearly all the variation in bond returns .. They interpret their
factors as representing the level of interest rates, the slope of the yield curve,
and the curvature of the yield curve. However, they do not ensure that their
factor loadings are consistent with no arbitrage. Gong and Remolona (1996)
are careful to impose no arbitrage when they fit three alternative two-factor
models to U.S. quarterly yield data. However, they fail to find a model that is
adequate for explaining the whole term structure, and they conclude that at
least three factors would be required for that purpose. Chen (1996) proposes
but does not estimate a three-factor model in which the future short-term
interest rate is determined by its current value, its time0 varying.mean, and its
stochastic volatility.
We follow Backus and Zin (1992) and CLM (1994) by specifying a term
structure model in terms of a stochastic discount process. We use this process,
known as a pricing kernel, to consistently price bonds of different maturities so
that we avoid arbitrage opportunities. To maintain tractability, we write the
model to satisfy Duffie and Kan's (1993) conditions for affine yields. In this
model, three unobserved factors drive the pricing kernel: one factor reverts
over time to a fixed mean while a second factor reverts to a time-varying mean
that serves as the third factor. Such a model will produce reduced forms that
nest most factor models in the literature. 5 To capture the hump in the yield
5
Examples of these models are the one-factor models ofVasicek (1977) and Cox, Ingersoll,
2
curve, we follow Gong and Remolona (1996) by pricing a risk associated with
the volatility of the time-varying mean.
To estimate and test the model, we use monthly U.S. Treasury zero-coupon
yield data from January 1986 to March 1996. The model lends itself to estimation by a Kalman filter, and .we apply the technique in a way that takes
account of the model's arbitrage conditions. The yields ,as functions of the factors serve as the measurement equations and the factors' stochastic processes
as the transition equations. The arbitrage conditions impose restrictions between the measurement and transition equations. To test the adequacy of
three factors·, we estimate the model using only three yields at a time - a
short-term rate, a medium-term rate, and a long-term rate. The estimates
turn out to be robust to the choice of maturities, suggesting that three factors
are adequate.
The fit between the estimated model and actual yield and volatility curves
strikes -us as ,impressive. In ,particular, the·. unconditional yield and volatility, . ., .
curves we construct from our parameter estimates capture the hump in the
average yield curve and the flatness of the average volatility curve for the
sample period. The shapes of these curves are most sensitive to the rates
of mean reversion in the factors, the price of risk, and the volatility of the
time-varying mean. Our estimates describe a first factor that reverts rather
slowly to a fixed mean and a second factor that reverts relatively quickly to
a time-varying mean. The first factor's slow mean reversion decides the yield
curve's slope near the long end and the volatility curve's slope across most of its
length. The second factor's fast mean reversion determines the yield curve's
and Ross (1985) and the two-factor models of Brennan and Schwartz (1979), Schaefer and
Schwartz (1984), Longstaff and Schwartz (1992), and Campbell, Lo, and Mackinlay (1994).
3
slope near the short end; The price of risk and the third factor's volatility
impart curvature to the yield curve. We estimat e a relatively high price of risk
(in absolute value) and a relatively low volatility, and these estimates produce
just enough curvature to capture the hump in the actual yield curve.
In what follows, we .begin with a..brief discussion of pricing-kerneLaffine-.
yield models of the term structure. In Section III, we then illustra te with
monthly data the inadequacy of two-factor affine yield models, as Gong and
Remolona have demons trated with quarterly data. We specify our three-factor
model in Section IV, estimat e it in Section V, and evaluate the robustness of
the estimates in Section VI. We discuss the role of the various parame ters in
shaping the yield and volatility curves in Section VIL We propose further work
in section VIII.
II. Theor y: Affine Yield Pricin g Kerne l Mode ls
A. Background Literature
Theoretical work with equilibrium models, notably by Vasicek (1977) and
Cox, Ingersoll, and Ross (1985, hereafter CIR), show how the term structur e
at a moment in time would reflect regularities in interest rate movements over
time. In the simplest such models, the short-te rm interest rate is the single
factor driving movements in the term structur e. Vasicek assumes that the
short-rate's volatility is constant, while CIR assume that it is proport ional to
the square root of the short rate itself. The absence of arbitrage requires that
the ratio of expected excess return to return volatility be the same for different
bonds. This arbitrag e condition, the assumption of lognormal bond prices, and
4
either Vasicek's or CIR's short-rate volatility produce an affine-yield solution
in which all bond yields (or log bond prices) are linear functions of the shortterm rate. Such linearity simplifies the pricing of fixed-income securities and
contingent claims. In the two-factor models of Brennan and Schwartz (1979),
Schaefer and Schwartz (1984), Longstaff and Schwartz (1992), and CLM (1994)
and in Chen's (1996)- three-factor model, similar assumptions generate bond
yields that are also linear in the factors. Duffie and Kan (1993) establish the
conditions that produce such affine yields in general.
Rather than model the short-term interest rate directly, Backus and Zin
(1992) and CLM (1994) focus on the stochastic discount process or the pricing
kernel used to price assets in general. Arbitrage opportunities are avoided
by applying the same pricing kernel to different assets. In this approach,
the factors are unobservable state variables that serve to forecast discount
rates. In principle, the factors can be related to observable macroeconomic
fundamentals, as Gong and Remolona (1996) try to do. Pricing kernel models
can also be ,specified -so that bond yields _are affine in the factors and, with a
linear transformation, affine in the short rate as well. We describe below such
a pricing-kernel affine-yield model with K factors.
B. The Pricing Kernel
The pricing kernel approach relies on a no-arbitrage condition common to
intertemporal asset pricing models. 6 In the case of zero-coupon bonds, the
price of an n-period bond is
(1)
6Singleton
(1990) provides a critical survey of these models, particularly their empiri-
cal performance. Duffie (1992) relates arbitrage conditions to concepts of optimality and
equilibrium.
5
where M,+1 is the stochastic discount factor. The condition expresses the price
of the bond as the expected discounted value of the bond's next-period price .
. It rules out arbitrage opportunities by applying the same discount factor to•
all bonds. We will model Pnt by modelling the stochastic process for M,+1,
a process called the pricing kernel. 7 Indeed we can solve ( 1) forward to get
Pnt
= E,[Mt+l•·•Mt+nl, which specifies bond prices to be simply functions
of
the future discount factors. By convention we normalize Poi= 1 to ensure the
equality of a bond's price at maturity to its par value.
C. K-Factor Affine Yields
For our affine yield models, we assume that M,+1 is conditionally lognormal,
bond prices are jointly lognormal with M,+ 1 , and bond yields are linear in the
factors that forecast M,+1The assumption of joint lognormality allows us to take logs of (1) and write
it as
(2)
where lower-case letters denote logarithms of upper-case letters. Furthermore,
ifwe have K factors, xu, x 2,, ••• , XKt, that forecast mt+ 1 , an affine yield model
can be written as
(3)
Since the n-period bond yield is Ynt
= -pn,/n, yields will also be linear in the
factors. The coeficients An, B 1n, B 2n, ... , BKn will depend on the stochasti c
7The
term "pricing kernel" is due to Sargent (1977). In consumption-based equilibrium
models, Mt+ 1 would represent the marginal rate of substitutio n between present and nextperiod consumption (Lucas 1978, for example).
6
processes of
X1t,
x 2fr .. ,XKt• Since the number of. factors is usually smaller
than the number of maturities on the curve, the factor structure would imply restrictions across coefficients for bond prices of different maturities. In
practice, specifying An, Bin, B2n, ... , BKn involves solving (2) based on the
stochastic processes of x 11 , x 21 , .... , XKt and verifying that (3) holds.
III. Failure of Two-Factor Affine Models
A. Previous results
In the effort to reconcile time-series data with cross-section data on interest
rates, models with fewer than three factors have not fared well. Backus and Zin
(1992) and CLM (1994) argue that the basic problem with one-factor models
is that the yield curve's steep slope near the short end requires swift mean
reversion by the factor while the curve's flat. slope near the long end .requires ..
slow mean reversion. The. flat slope of the volatility curve also requires slow
mean reversion. Gong and Remolona (1996) demonstrate that the problem is
not solved with two factors either. They find that the data favor models in
which one of the factors is a time-varying mean, which serves to produce the
characteristic hump in the U.S. yield curve around the one-year to two-year
maturities. The other factor must then revert rapidly to this mean to create
a steep yield curve near the short end but revert slowly .to create both a flat
yield curve near the long end and a flat volatility curve.
In this section, we replicate Gong and Remolona's results with a somewhat
different data set. We use monthly data on U.S. zero-coupon Treasury yields
from January 1986 to March 1996. Gong and Remolona used quarterly data
7
on those yields from 1984 Ql to 1995 Q4. As they did, we try to fit a twofactor additive model and a two-factor time-varying mean model using two
yield maturities at a time and using a Kalman filter that takes into account
the models' no-arbitrage restrictions. If two factors are adequate, then we
should be able to estimate the same model with any two maturities. We
discuss the estimation procedure in more detail in Section V, where we turn.
to the estimation of a three-factor model.
B. A two-factor additive model
In the additive model, the conditional expectation of the negative of the
log stochastic discount factor depends on two factors that enter additively:
(4)
where wt+ 1 represents the unexpected change in the log stochastic discount
factor and will be related to risk. The shock wt+ 1 has mean zero and a variance
i-hat: will be specified, to -depend on .the stochastic processes- of-the two Jactors -··• (
X1t
and x 21 . Each of these factors follows a univariate AR(l) process with
heteroscedastic shocks described by a square-root process
X1,t+1
-
X2,1+1 -
+ ,P1X1,t + xUu1,1+1
</i2)0 + </i2x2,1 + xUu2,1+1
(1 - </i1)µ
(1 -
(5)
where 1 - q, 1 and I - ¢,2 are the rates of mean reversion, µ and 0 are the longrun means to which the factors revert, and
u1,t+1
and
u2,t+1
are shocks with
mean zero, volatilities o-f and o-~ and covariance o-12 • We specify the shock to
m 1+i to be proportional to the shock to x 1 ,1+i, which in turn depends on the
level of X21:
(6)
8
where
>. represents the market price of risk. When.>. is.negative, bond returns
are inversely correlated with the stochastic discount factor and risk premia are
positive. The model can be solved to produce affine yields
(7)
where An,B1n, and B2n depend on ¢1,¢2,µ,0,>.,o-f,o-?, and 0-12-
C. A time-varying mean model
In the time-varying mean two-factor model, one factor directly affects expectations -of the stochastic discount factor for the immediate period, while
the second factor affects expectations of the ultimate destination of the discount factor. Specifically, the model specifies the conditional expectation of
the negative of the log stochastic discount factor to depend directly on one
factor, but this factor reverts over time to a second factor:
+ Wt+l
-m,+1
=
Xt
Xt+l
=
(1 - ¢1)µ,
µt+l
=
(1 - ¢2)0 + ¢2µ1
Wt+l
-
), 0.5
µ,
+ ¢1x, + µ~· 5 u1,1+1
+ µ~· 5u2,t+l,
(8)
U2,t+l
where 1 - </,1 and 1 - </, 2 are rates of mean reversion, but we have a single
parameter for the mean, 0. The factor shocks u1,t+ 1 and u2,1+1 have mean
zero, volatilities o-f and o-~ and covariance o-12 . The conditional expectation of
the stochastic discount factor depends only on the first factor, while its shock
depends on the second factor. The solution to the model produces affine yields
(9)
where An, B 1n, and B2n depend on ¢1, ¢2, 0, >., o-f, o-t and 0"129
D. The poor fit of two-factor models
With only two factors, the constructed yield and volatility curves tend
to match the actual curves only at the maturities used in estimation. More
over, the estimates are not robust to the choice of maturities. Figs. 1 to 6
compare the curves implied by the estimated models to the actual averaage .
curves for the sample period, In Figs. 1 and 2, the yield and volatility curves
implied by the models are based on estimates using only three-month and
two-year yields. The models produce yields that fall below the actual yields
for maturities longer than two years, with the additive model even producing
negative ·yields for maturities longer than eight years. At the long end, the
volatilities implied by the additive model are too high and those implied by
the time-varying mean model too low. In Fig. 3, the models are estimated
on two-year and ten-year yields, and the implied yields are too low near the
short end and too high in the maturities between two and ten years. In Fig. 5,
the models are estimated on three-month and ten-year yields,.and the implied·
yields are too 'high in the intermediate maturities. The evidence suggests.that ·
two factors will not give us an adequate model.
IV. A Three-factor Model
We now propose a three-factor model to fit both the time-series dynamics
of interest rates and the cross-sectional shapes of the term structure. We
follow Backus and Zin (1992), CLM (1994), and Gong and Remolona (1996)
by specifying the model in terms of a pricing kernel. To maintain tractability,
we write the model to satisfy Duffie and Kan's (1993) conditions for affine
yields. In this model, three unobserved factors drive the pricing kernel: one
10
factor reverts over time to a fixed mean while a second factor reverts to .a '"
time-varying mean that serves as the third factor. The model is a combination
of CLM's two-factor additive model and Gong and Remolona's time-varying .
mean model. From the outset, we specify the model in discrete time to avoid
possible problems in estimating a continuousstime model with discrete-time
data.
A. Model specification
Three unobservable factors drive the pricing kernel. Two of the factors
directly affect expectations of the stochastic discount factor for the next period,
while the third factor affects the ultimate destination of the stochastic discount
factor. Specifically, the conditional expectation of the negative of the log
stochastic discount factor depends on the sum of two factors:
(10)
where wt+ 1 rep~esent1, the unexpected change in .the log stochastic discount.
factor and will be related to risk. The shock w,+ 1 has mean zero and a variance
that will be specified to depend on the time-varing mean of the second factor.
Each of these factors follows a univariate AR(l) process with heteroscedastic
shocks described by a square-root process:
X1,t+l -
(1 - ¢1)0 + ¢1X1,, + µ~·
X2,<+1 -
(1 - ¢2)µ,
µ,+1
-
(1 -
5
ul,t+l
+ ¢2x2,, + µ~· 5u2,t+1
¢3)µ + ¢3µ, + µ~· u3,t+1,
(11)
5
where 1- ¢1, 1-¢2, and 1-¢3 are the rates of mean reversion, 0 andµ are the
long-run means to which the factors revert, and u1,i+1, u2,t+1, and
shocks with mean zero, volatilities o-1 , o-2 and
11
o-3
and covariances
U3,t+1
are
0-12, 0-13, 0"23-
,
It is importa nt to allow correlation between factor shocks, if we hope to relate
the factors to fundamentals, which may not be orthogonal.
As in Gong and Remolona (1996), we specify the shock to m +1 to be
1
proport ional to the shock to µ,+1, which in turn depends on the level ofµ,:
(12)
where A represents the market price of risk. When A is negative, bond returns
are inversely correlated with the stochastic discount factor and risk premia are
positive.
We now verify that yields are affine in the factors so that we can write
(13)
The normalization Pot
= 0 gives us coefficients of Ao = B10 = B20 = B30 = 0.
We can then derive the one-period yield or short rate as
(14)
which is also linear in the factors, with the coefficients A1 = 0, Bu
and B 31
= -½>-
2
= B21 = 1,
of
We can also verify that the yield of an n-perio d bond is linear in the factors
with the coefficients restricted by (see Append ix A)
An -
An-1 + (1 - ¢1)BB1,n-l
-
1 + ¢1B1,n-l
B2,n -
1 + ¢2B2,n-l
B1,n
B3,n
-
¢3B3,n-l
+ (1 -
+ (1 -
¢3)µB3,n-l
(15)
¢2)B2,n-l
12
2
1[(.\ + B3,n-1) a~
+ Btn-10"i + B{n-10"~
+ 2(.\ + B3,n-1)B1,n-!O"J3 + 2(.\ + B3,n-1)B2,n-!0"23+ 2B1,n-1B2,n-10"u],
The coefficients Bin, B2n, and B 3n are factor loadings for X1t, x 21 , and µ 1.
The coefficient An represents the pull of the factors to their means µ and 0.
These recursive equations impose cross-sectional restrictions to be satisfied by
twelve parameters: ,Pi, ¢2, <p3, 0, µ, ai, a 2 , a 3 , o-12 , o-13 , o-23 , and .\.
To price fixed-income options, we need a consistent volatility curve. In the
case of our model, such a curve is derived from the conditional variance of the
n-period yield:
We would have a downward-sloping volatility curve given that ¢ 1 , ¢ 2 , and ¢3
are less than unity. Mean reversion by the factors serves to dampen yield
volatilities as maturity is lengthened.
A linear transformation of this model will give us a reduced form which
which expresses the yield for a given maturity as a linear function of some other
yield, the conditional variance of some yield, and the conditional expectation
of some yield. This follows from the fact that yields, conditional variances,
and conditional expectations are different linear functions of the factors. Such
a reduced form will nest the one-factor models of Vasicek (1977) and CIR
(1985) and the two-factor models of Brennan and Schwartz (1979), Brennan
and Schaefer (1984), and Longstaff and Schwartz (1994). One such reduced
form will also look like Chen's (1996) three-factor model.
The model also allows us to measure term premia. We can derive term
13
premia in the form of the expected excess bond return:
EtPn-1,t+I - Pnt
Y1t
= -.>.(B1,n-10"13 + B2,n-10"23 + Ba,n-10"Dµt
21 [B21,n-1 1712 + sz2,n-10"22 + sz3,n-10"32
(17)
where
II -
(B1,n-10"13
+ B2,n-10"23 + B3,n-10";)
(18)
r - 21 (B21,n-1 1712 + sz2,n-1 1722 + Bz3,n-10"32
+
2B1,n-1B2,n-10"12
+ 2B1,n-1B3,n-10"13 + 2B2,n-1Ba,n-10"23)µt
(19)
The first term II represents a risk premium that depends on the covariance
between the stochastic discount factor and bond returns, while the second term
r
represents Jensen's inequality arising from the use of logarithms. Positive
term prernia require that .>. be so negative that
r
->-> -II
Note also that if o-3
(20)
= 0, we will have homoscedastic shocks, term premia will
be constant, and the pure expectations hypothesis will hold.
V. Estimating the Model
A. Econometric Approach
Our econometric approach allows us to estimate the parameters of a pricing
kernel without directly observing it or the three factors that are supposed to
drive it. Moreover, to confront the issue of model adequacy, we estimate the
14
model using yields of only three maturities at a time; If a three-factor model is.
to explain the movements of the whole term structure; then we should be able
to estimate it with only three maturities, and the model should be robust to
the choice of maturities. Our approach is a special application of the Kalman
filter. In applying this technique, we make use of observed yields, which would
reflect the dynamics of the factors, and we use .only three yields at a time
-c.
a short-term yield, a medium-term yield, and a long-term yield. The yields
as affine functions of the factors serve as the measurement equations of the
Kalman filter and the factors' stochastic processes as the transition equations.
The model's arbitrage conditions, however, imply strong restrictions between
the measurement and transition equations, and we take careful account of
these restrictions.
Recent efforts to reconcile time series data with cross section data on interest rate have tended to rely on the GM M approach. In practice this approach
places cross-section restrictions on only some of the unconditional moments.
Backus ·am:! ·Zin .. {19943. place,·cross-section restrictions. on yields up to the
10-year maturity, but the restrictions are placed on only the first moments.
Gibbons and Ramaswamy (1993) have cross section restrictions on both the
first and the second moments when they fit and test the CIR model. Longstaff
and Schwartz (1992) estimate a model with the short rate and its volatility
as the two factors. In testing their overidentifying restrictions, however, they
fit a reduced form that takes account of only four out of the six parameters
implied by their structural model.
For our purposes, the Kalman filter is the appropriate estimation procedure. The technique is effective in exploiting conditional moments, which
constitute essential information when one is trying to estimate the dynamics of
15
three unobserved factors on the basis of observed yields for only three maturities. The technique is especially suitable for estimating term structure models,
because it allows the imposition of the arbitrage restrictions. Jegadeesh and
Pennacchi (1996) use the Kalman filter in estimating a two-factor term structure model using data on Eurodollar futures, although they use more than
two yields at a time. Gong and Remolona (1996) use the technique in their
exploration of the U.S. term structure with two-factor models, and they use
only two yjelds at a time.
The work builds on Gong and Remolona, and in spirit, it is close to Backus
'and Zin {1994)-in that we use the observed yields to determine the dynamics
of the underlying stochastic discount factor. Our work differs from Backus and
Zin in an important respect: they estimate a reduced form in the sense that
they study various ARMA processes for the stochastic discounting factor .. ·We
estimate a structural model by specifying the underlying factors. that drive the.
movements of the stochastic discounting factor. An ARM A(3, 2) yield process
is generated by our three AR( 1) factors. 8
B. Data and Summary Statistics
We obtained end-of-month U.S; zero-coupon Treasury yield data for maturities of one-year and longer from J.P. Morgan and Company and for maturities
of three months and six months from the Federal Reserve Bank of New York.
The sample period is 1986:1 to 1996:3.9 In the case of the Federal Reserve
data, each zero curve is generated by fitting a cubic spline to prices and maturities of about 160 outstanding coupon-bearing U.S. Treasury securities. The
8
Engel (1984) derives the sums, products, and time aggregations of ARMA processes.
9The
data are available on request.
16
securities are limited to off0 the0 run Treasuries to eliminate the most liquid ·securities and reducethe possible effect of liquidity premia. Fisher, Nychka, and.
Zervos (1995) explain the procedure in detail.
Summary statistics for the yields with maturities of 3 months, 6 months, 1
year, 2, 5, and 10 years for the sample period 86:1-96:3 are reported in Table 1.
The average term structure is upward sloping, with mean yields ranging from
5.54% to 7.80%. Its slope is steep near the short end and flat near the long
end. This term structure is some what hump-shaped, with the hump located
near the two year maturity. Overall, the volatility curve slopes downward very
gradually. 'The yields across the curve are all very persistent, with first-order
monthly auto-correlations of 0.94-0.99.
To evaluate robustness, we use four different combinations of yield maturities to estimate the parameters. The four different combinations .are: 3-month,
2-year, and 10-year yields; 3-month, 1-year, and 10-year yields; 6-month, 2year,·and l0°year yields; and 3-month, 2°year, and 5-year yields. If the threefactor model is adequate for explaining the. whole term-structure, we should
get similar parameter estimates and implied yield and volatility curves from
the different combinations of maturities.
C. Kalman Filtering and Maximum Likelihood Estimation
We now show how to fit the three-factor model to U;S. zero-coupon rates
data, using three-yields at a time.
We write the model in the linear state-space form, with the measurement
17
equation
a1
Yl,t
Ym,t
-
am
+
an
Yn,t
b2,1
bs,1
X1,t
b1,m b2,m
b3,m
X2,t
b1,n
b3,n
µt
b1,1
b2,n
V1,t
+
Vz,t
(21)
V3,t
where Yl,t, Ym,t, and Yn,t are zero-coupon yields at time t with maturities I, m,
and n and
Vt
is a measurement error assumed to be i.Ld. as
Vt~
N(
0
e~
0
0
0
0
e22
0
0
0
0 e23
0
0
),
(22)
The transition equation is
(1 - </>i)0
X!,t+l
Xz,t+l
µt+l
-
0
<h
+
0 </,2 1 - </,2
0
(1 - q,3)µ
0
X1,t
Xz,t
q,3
µt
U1,t+1
+ µ~·5
Uz,t+l
u3,t+l
with shocks to the state variable Xt+l distributed as
0
a-2
1
(23)
0
a-3
3
In standard linear state-space models, no restrictions link the measurement
equation and the transition equation. This time, however, the measurement
equation comes from the transition equation and the no-arbitrage conditions,
and the restrictions are given by equation (15).
After putting the restrictions into the measurement equations, the preceding models can be estimated by maximum likelihood using the Kalman filter.
18
The· algorithm is discussed in Appendix -C. For more detailed discussions of .
the Kalman filtering procedure, see, for example, Hamilton (1994).
VI. Estimates of the Model and Robustness
Our estimates allow us to test the adequacy of our model and to differentiate among the three factors that are supposed to be driving the model.
We informally test the model's adequacy by comparing alternative parameter
estimates based on different combinations of yield maturities and by examining how well the unconditional yield and volatility curves produced by the
estimates fit the actual average curves for the sample period. We differentiate among the factors primarily by comparing their estimated rates of mean
reversion. Table 3 reports our parameter estimates based on the four alter- .·.
native combinations of yield maturities. It is immediately apparent that the
different combinations of maturities provide very similar parameter estimates.
These estimates characterize a•first ·factor that -reverts slowly to fixed mean, a
second factor that reverts rapidly to the third factor, which itself reverts more
rapidly to its own mean than does the first factor and more slowly than does
the second factor.
Since
W<'
estimate the model with the observed yields for only. three matu-
rities at a time. we may evaluate the model by comparing estimates based on
other maturities. If a three-factor model held, then the estimates should be
robust to the choice of maturities. Table 3 compares estimates based on the
following maturity combinations: (1) the three-month, two-year, and ten-year
yields; (2) the three-month, one-year, and ten-year yields; (3) the six-month,
two-year, and ten-year yields; and (4) the three-month, two-year, and five19
year yields. The differences among ·the alternative parameter estimates are
statistically insignificant. The similarity among the alternative estimates .. is
,· particularly impressive for the parameters that are tightly estimated, .such as
the persistence parameters
</>1 ,
</>2, and </>3. The parameter estimates are never
more then two standard errors apart even based on the smallest standard error
of 0.006 for the estimate of </>1 using the three-month, two-year, and ten-year
yields.
Table 4 reports estimates based on two subsamples, one for January 1986 to
December 1980 and one for January 1991 to March 1996. Again the parameter
estimates areremarkably similar across subsamples.
We may also evaluate the model by seeing how well it reproduces the
rest of the terni' structure. Our Kalman-filter procedure would tend to cause.,
the term structure to match at the maturities used to estimate the model.
However, the procedure will not assure a fit with the rest of the term structure
unless the yields for other maturities reflect the dynamics of the same three
factors. Fig. 7 compares the unconditional yield curves we construct from our
alternative model estimates with the actual average yield curve for the sample
period. Similarly, Fig. 8 compares the unconditional volatility curves from the
alternative estimates with the actual average volatility curve. On the whole,
the fit between the estimated model and actual yield and volatility curves is
remarkable, especially compared to the results of the-two factor models. Inparticular, the implied yield and volatility curves capture the hump in the
average yield curve and the flatness of the average volatility curve for the
sample period.
How do the factors differ? Recall that the rate of mean reversion is given
by 1 - </,1 for the first factor, 1 - </>2 for the second factor, and 1 - q>3 for the
20
third factor: As Table 3 reports, the first factor reverts to its fixed mean at the
rather slow rate of one percent a month. This rate of mean reversion implies
a near unit root process. 10 The-second .factor reverts to its time-varying mean . •
at the rate of about 12 percent a month or a rate 12 times faster than that
of the first factor. The third factor, which is the time-varying mean, reverts
to its own mean at the rate of five percent, which is five times faster than
the first factor but less than half as fast as the second factor. Such mean
reversion rates are critical determinants of the shape of the term structure.
The volatility estimates suggest that the first factor has the smallest shocks
while the second factor has the largest ones. Shocks to the first factor are
not significantly correlated with shocks to the second or third factor. The
second and third factors, however, have shocks that are significantly positively
correlated.
VII. How the Factors Shape the Term Structure
How do the three factors succeed in reproducing the actual shapes of the
term structure? The average U.S. yield curve can be characterized as having
a steep slope near the short end, a flat slope near the long end, and something
like a hump around the one-year to two-year maturities. The average U.S.
volatility curve can be characterized as downward sloping but with a rather
flat slope. Our estimates of the three-factor model suggest that the first factor
accounts for yield curve's flat slope near the long end and the volatility curve's
flat slope for most of its length. The second factor accounts for the yield curve's
10 We
can rule out a unit root process, however, because such a process implies negative
yields at long maturities. If ¢ 1
= 1, we will have
(17)-(19) must eventually turn negative.
21
Bin
= n,
and the term premium in
steep slope near the short end. The third factor combined with the price of
risk impart just enough curvature to account for the yield curve's hump.
The slopes of the yield curve near either end are the easiest characteristics
of the term structure to explain. The first factor determines the slope near
the long end, because it is the factor with the slowest mean reversion. It
would have 'the most influence on the long end because. a shock to the factor.
would persist the longest and would be the shock most likely to be reflected in
long-term yields. The second factor determines the slope near the short end,
because it is the factor with the most rapid mean reversion, and it would have
its greatest effect on the shorter term yields. These explanations are consistent
with the factor loadings shown in Fig. 9. The picture portrays the first factor
as having an effect on yields that decays very slowly as maturity is lengthened
and the second factor as having an effect that dies down so swiftly that its
effect at the ten-year maturity is only a fifth of that of the first factor.
The hump in the yield curve is apparently a feature associated with the
time-varying mean and the effect it has on the risk premium. Such an association is evident in time-varying mean models with two factors as well as
those with three factors. In these models, the time-varying mean factor induces hetorescedasticity in volatility, and this source of risk is priced. In the
present three-factor model, the loading for this factor, as shown in Fig. 9, is
initially negative then becomes positive around the 16-month maturity, which
is roughly the location of yield curve's hump. To produce the right curvature,
however, requires the right mean reversion rate as well as the right volatility
and price of risk. Fig. 10, for example, shows the implied yield curve produced with the factor having a mean reversion rate of one percent instead of
five percent. The curve overshoots the average one-year yield and undershoots
22
the yields between the two-year and ten-year maturities.
In general, a downward sloping volatility curve requires mean reversion and
a flat curve slow mean reversion. To produce the right shape for this curve,
however, requires slow mean reversion for the first factor only. Fig. 11, for
example, shows that a slow rate of mean reversion for the time-varying mean
· induces a volatility curve that over.shoots volatilities for maturities between
two years and ten years.
VIII. Conclusion
We believe we have an adequate econometric model of the U.S. term structure. It is a model of a pricing kernel that serves to consistently price bonds
of different maturities so that arbitrage opportunities do not arise. Three fac-..
tors drive this pricing kernel: one factor reverts over time to a fixed mean,
a second factor reverts to a time-varying mean, and the time-varying mean ·
itself is a mean-reverting factor that induces time-varying term premia. With
alternative estimates. of the model using a special Kalman filter and only three
maturities at a time, we find that different combinations of maturities produce
the same three factors, particularly as characterized by their rates of mean reversion:, The estimates also reproduce the actual average yield and volatility
curves for the sample period, suggesting that the yields not used in estimation
also reflect the time-series dynamics of the same three factors.
The estimates describe three factors with very different rates of mean reversion. The first factor reverts to its fixed mean at the rate of about one
percent a month, the second factor reverts to its time-varying mean about 12
times faster; and the third factor reverts to its own mean five times faster than
the first factor but less than half as fast as the second factor. Something seems
23
key about these parameter values, because small deviations from our range of
estimates produce very bad yield and volatility curves. Each factor has a role
in the shapes of the term structure. The first factor explains the yield curve'.s.
flat slope near the long end and the volatility curve's flat slope for most of its
length. The second factor explains the yield curve's steep slope near the short
end. The time-varying mean produces the yield curve's hump.
There is more work to be done. Having decided that three factors are
adequate, we would like to estimate the model using all the maturities available
at once. Such an estimate will be more efficient than the ones reported in
· this paper and -will be appropriate for testing the significance of the arbitrage
restrictions. We would also like ·to use such a model to forecast changes in
short-term rates and long-term rates over different time horizons to see whether
controlling for time-varying term premia by means of the model would support
the expectations hypothesis.
24
Appendix A
A 1. Mode/ I: Recursive Restrictions
We start with the general pricing equation:
Pnt
1
= E,(mt+l + Pn-1,t+t) + 2Var,(m1+1 + Pn-1,1+1)
(24)
The short rate is derived by setting Po,t = 0:
Ytt
= -plt
-
-E,(m,+1) -
-
Xt,t
+ X2,t -
1
2Var,(m 1+1)
1 2 2
2). <Y3µt,
showing the short rate to be linear in the factors.
Now we guess that the price of .an n-period bond is affine:
(25)
We verify that there exist An, B 1,n, B 2,n, and B 3,n that satisfy the general
pricing equation:
E,(mt+l
+ Pn-1,t+i)
-
-An-1 - (1 - r/>1)8B1,n-1 - (1 - r/>a)µBa,n-1
+ r/>1B1,n-1)x1,t - (1 + r/>2B2,n-1)X2,t
(26)
[(1 - r/>2)B2,n-l + r/>aBa,n-1]µ,
- [(>. + Ba,n-1)2<Y5 + Bf.n-1<Yi + B~,n-i<Y~
+ 2(>. + Ba,n-1)B1,n-1<Y13 + 2B1,n-1B2,n-1<Y12
(1
Var,(mt+l
+ Pn-1,t+1)
(27)
Now substitute (26) (27) into (24) and match coefficients of equations (24)
and (25), we have
25
B1,n -
1 + ef>1B1,n-l
B2,n -
1 + ef>2B2,n-l
B3,n
=
q>3B3,n-l
+ (1 -
ef>2)B2,n-l
1[(>. + B3,n-1J20-5 + Btn-10"i + B{n-10"~
+
2(,\ + B3,n-1)B1,n-10"13 + 2(,\ + B3,n-1)B2,n-10"23 + 2B1,n-1B2,n-10"12],
26
Appendix B
BJ. The Term Premia
Term premia can be derived from the expected excess bond return over the
short rate:
Et
Pn-1,t+l - Pn,t - Ytt
+
An
=
(A,. -An-1 - B1,n-1(l -
+
+
+ B1,nX1,t + B2,nX2,t + B3,n/l,t
1 2 2
Xt,t - X2,t + A 173/l,t
2
1-
(B1,n -
<h)O -
B3,n-1/l,(l - <f>3)µ)
+ (B2,n -
</>1B1,n-tlX1t
1-
ef>2B2,n-1)X2t
+ i>- 2 u; - <f>3B3,n-1 - (1 - </>2)B2,n-1)/l,t
->.(Bt,n-11713 + B2,n-11723 + B3,n-117;)µt
(B3n
21 [B21,n-1 1712 + B22,n-1 1722 + B23,n-1 1732
+ 2B1,n-1B2,n-11712 + 2B1,n-1B3,n-tl713 + 2B2,n-1B3,n-11723]/l,t
B2.
(28)
Expected Change in the Short Rate
The conditional expectation of the short rate n periods in the future is
EtYl,t+n
Et
Xt,t+l -
(}
X2,t+t -
fl,
/l,t+t - µ
-
-
Etxl,t+n
+ E1X2,t+n - 21 A2 1732 E t/l,t+n
</>1
0
0
¢2
0
0
0
1-
Xtt
'
</>2
<f>3
27
-0
X2,t - fl,
/l,t - µ
n
X1,t+n - 0
E,
X2,t+n -
-
µ
µt+n - µ
-
¢1
0
0
¢2 1 - ¢2
X2,t - µ
0
0
µ, -µ
¢,f
0
0
¢,n+l _ ¢,n+l)
<P2 ..!.::b..(
¢2-¢3
2
3
X2,t - µ
0
0
µ, - µ
0
X1 •t
ef,3
-
0
0
X1,t - 0
<Ps
We can then write
where
Ctn
=
[1- <Pi-
:2~~3 (¢2+1 - 1)- 2
¢,;+
~A al(l - <Ps)]µ
1- ¢,~
f31n
f32n
=
1-
1n
=
;
¢2
~ ~
2
3
(¢,2+i
-
¢,3+1 )
+ ~A 2 a~(l - ¢,3)- (1- ¢2)
28
Appendix C: The Kalman Filter Algorithm
For the state-space models in section II, the measurement and transition
equations can be written in the following matrix form:
Measurement Equation:
Yt
where v,
= A + BX, + v,
~ N(O, R).
Transition Equation:
(29)
where
Ut+ilt
~ N(O, Q,).
The Kalman filter algorithm of this state-space model is the following:
1. Initialize the state-vector S,:
The recursion begins with a guess S1 1o, usually given by
(30)
The associated MS E is
Pi10 -
E[(S1 - S11o)(S1 - 8110)']
Var(S1 ).
The initial state S1 is assumed to be N(S110, P110),
2. Forecast y,:
29
Let I, denote the information set at time t. Then
Ytlt-1 -
A+ BE[S,II,_1]
A+ BStlt-1•
(31)
The forcasting MS E is
(32)
3. Update the inference about S, given It:
Note,that, since:S1 -and Yt are related by specification, knowing Yt can help
to update S,1t-I by the following:
Write
s, == stit-1
Yt -
+ (st - st1,-1)
A+ BS,1,-1
+ B(St -
Stlt-1) + Vt
(33)
We have the following joint distribution:
(34)
Thus,
S,1, Ptlt _
E[S,ly,, lt-1]
Stlt-1 + Ptlt-1B'(BPtlt-1B' +
Rt 1(y, -
BS,1,-1 - A)
(35)
E[(S, - Stjt)(St - Stltl']
(36)
30
4. Forecast S,+ 1 given J,:
s,+11, P,+w
E[st+1I1,1
= Fs, 1,
=
E[(S<+1 - s,+1 I,)(S,+1 - s,+1 I,)'l
-
F P,1,F' + Q,
(37)
(38)
5. Maximum Likelihood Estimation of Parameters
The likelihood function can be built up recursively
T
log L(YT)
= I: log f(y,if,_1),
(39)
t=I
where
f(y,II,-1)
=
(271't 112 IH'P,1,-1H + Rl- 1' 2
* exp{-~(y, - A - BS,1,-1)'(B'P,1,-1B + Rt 1(y, - A for
t
=
1,2, ... ,T
BS,1,-il}
(40)
Parameter estimates can then be based on the numerical maximization of
the likelihood function.
31
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34
Table 1
Zero-Coupon Yields: Summary Statistics
Monthly Observations ( 86: 1-96:3 )
Maturity
Mean
Std Dev
1st-Auto. Corr
3 Months
5.54
1.63
0.99
6 Months
5.65
1.59
0.99
1 Year
6.26
1.69
0.98
2 Year
6.63
1.53
0.97
5 Year
7.29
1.23
0.96
10 Year
7.80
1.01
0.94
Table2
Correlations Between Zero-Coupon Yields
6-month
1-year
2-year
5-year
Maturity
3-month
3-month
1.000
6-month
0.995
1.000
I-year
0.960
0.980
1.000
2-year
0.938
0.962
0.995
1.000
5-year
0.856
0.888
0.946
0.973
1.000
JO-year
0.739
0.774
0.849
0.894
0.969
10-year
1.000
Table3
Parameter Estimates for the Three-factor Model ·using Different Maturities
3mon-2yr-10vr
3mon-lvr-10vr
6mon-2vr-10vr
3mon-2vr-5vr
e
0.291
(2.482)
0.700
(2.442)
1.668
(5.165)
0.347
(7.10)
µ
10.875
(8.920)
10.521
(9.048)
10.148
(16.070)
6.900
(16.600)
Q) I
0.987
(0.006)
0.989
(0.007)
0.990
(0.010)
0.992
(0.008)
<!>2
0.849
(0.055)
0.893
(0.103)
0.876
(0.157)
0.890
(0.193)
Q)3
0.958
(0.052)
0.950
(0.057)
0.939
(0.140)
0.942
(0.348)
'-
-12.143
(1.496)
-12.078
(4.186)
-15.384
(2.149)
-13.122
(1.512)
01
0.008
(0.044)
0.010
(0.008)
0.011
(0.003)
0.005
(0.012)
Oz
0.116
(0.076)
0.093
(0.065)
0.132
(0.175)
0.133
(0.256)
03
0.093
(0.014)
0.095
(0.026)
0.083
(0.043)
0.100
(0.087)
P12
0.045
(2.227)
0.145
(3.857)
0.022
(0.334)
-0.020
(18.040)
P13
-0.061
(1.165)
-0.008
(1.520)
-0.036
(1.294)
-0.120
(21.140)
P2J
0.492
(0.144)
0.727
(0.105)
0.583
(0.569)
0.715
(0.194)
el
0.0000
(0.0001)
0.060
(0.030)
0.004
(0.005)
0.003
(0.003)
e2
0.377
(0.109)
0.440
(0.145)
0.443
(I.I 79)
0.371
(1.189)
e3
0.509
(0.089)
0.506
(0.090)
0.522
(0.514)
0.412
(1.408)
llf
40
44
13
61
'
Table4
Parameters Estimates of the Three-factor Model for Different Sample Periods Using
3-Month, 2-Year, and 10-Year Maturities
86:1-96:3
86:1-90:12
91:1-96:3
0.291
(2.482)
0.426
(34.31)
0.314
(12.16)
10.875
(8.920)
8.045
(29.85)
15.041
(85.74)
<!>1
0.987
(0.006)
0.999
(0.004)
0.994
(0.013)
<!>2
0.849
(0.055)
0.895
(0.258)
0.883
(0.487)
<f,3
0.958
(0.052)
0.950
(0.065)
0.973
(0.212)
i..
-12.143
(l.496)
-13.970
(7.325)
-10.417
(1.879)
01
0.008
(0.044)
0.009
(0.003)
0.011
(0.016)
02
0.116
(0.076)
0.140
(0.207)
0.138
(0.143)
03
0.093
(0.014)
0.087
(0.009)
0.124
(0.047)
P12
0.045
(2.227)
0.018
(1.192)
0.026
(6.758)
P13
-0.061
(1.165)
0.197
(2.276)
-0.024
(2.895)
p23
0.492
(0.144)
0.540
(0.143)
0.763
(0.226)
el
0.000
(0.0001)
0.000
(0.0001)
0.000
(0.0002)
Ci
0.377
(0.109)
0.418
(0.722)
0.644
(2.514)
~
0.509
(0.089)
0.492
(0.325)
0.691
(2.515)
Mean Log-Likelihood
0.325
0.550
0.111
e
µ
8
♦
♦
6
4
2
"'
iii
'C
0
>
-2
..
..
. . . . . • . - . . . . - - • . - - - • . . ·"t.. - - - - - •• - - • . - - - . • . - - • .
---r
• · • · • · r[nLAd
--r[nLTm
~
-4
. . . . . . . . . - . . • . - - - - • - . - - . - • - • - •• - - . • • !' .. .,. - - - • • • .
-6
. . . . . . . . . . . . - - - .... - .. - ..... - -
-8
·······-·····················-····· .............
, ...
..... - .. - . ._.. .., - ...
••
-10
Maturity in Months
FIG. 1: Actual and Implied Yield Curves by the Two-factor Additive and
Time-varying Mean Models Using 3-month and 2-year Maturities
3..-------------------------,
2.5
.
-. .
C
.2
2
ii
C
.
"E
'C
C
1.5
..
...... - ....... -.... ·. :: .-.. ·.. :: ..-.. -. : : -·-·=-·-:; .: .. ·.. :
♦
♦
1
.l!l
en
•..• :..,. .. - - . . . . . . . . . . . . - . - . - - - .... - - - . - . - .... - ;
0.5
~
.•
.
.-.. -.:; . :.·. :. ••· .
.
♦
•
• · • • • ·std[nLAd
--std[n] Tm
...... ····· ..........:-:-:-:
...~
.. ---=-··~···:. . :.··.:..:..:..···~···
. :. ·
0 .::::::::::::
(\J
(I)
st
st
std
Maturity in Months
FIG. 2: Actual and Implied Volatility Curves by the Two-factor Additive and
Time-varying Mean Models Using 3-month and 2-year Maturities
20
15
10
"'
,::,
~
5
1--+--r
0
• • • • • · r[nLAd
1--r[nLTm
-5
-10
-15
...... - . - ............ - ...... - .... - . - - - . - -
·/
.
·····················-················-···-
•
-20
Maturity in Months
Fig.3: Implied and Actual Yield Curves by the Two-factor Additive and Time
Varying Mean Models Using Maturities of 2- and 10-year Yields
12
10
·············································
8
····························-················
C
0
;:
=
•
GI
..,
6
.,
4
C
,::,
,::,
C
····-·······························-······-
std
• • • • • ·std[nLAd
1---std[n] Tm
1/J
2
0 "" . ""
.,.. .,.. .,..
Maturity in Month
Fig. 4: Actual and Implied Volatility Curves by the Two-factor Additive and
Time-varying Mean Models Using 2- and 10-year Maturities
12
10
8
Ill
'ti
ai
►
6
4
2
------ -...... . . -.... . . . . ......
..
.
···;.'··············-····-·······
•
•
•
•
- - - ~- - - .•- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .•
--:-r
• • · • • · r{nLAd
--r{n] Tm
-·····························-······-----
··-················-······················
0
Maturity in Months
FIG.5: Actual and Implied Yield Curves by the Two-factor Additive and
Time-varying Mean Models Using 3-month and 10-year Maturities
C
-..
.2
>
C
.
.-
'E
'ti
C
en
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
········
·········-·······················-
•
• • • • • · std[n]_Ad
--std(n) Tm
···········-··········-··················-·
,-...-.:t...-CDt.n
T""
C\I
C\I
t')
Maturity in Months
FIG.
std
s: Actual and Implied Volatility Curves by the Two-factor Additive and
Time-varying Mean Models Using 3-month and 10-year Maturities
9..------------------------------------,
:::::::::::; ::~~~:~~-:~;;~:~;;~;-~:·:.-: ~-:~:·.~.:~~~~~~~-:~~,~~~:~~:
6
0
,:J
➔
.........
5
ai
>
... - . / ~ " . - - - ... - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-
...................
....... - .....
4
-
.....
- -
.....
-
................
- - -
........
-
- - - -
........... - .......... .
. . . . . - - - - - . . . . . . . . - .. - . . . . . . . . . . .
3
............. - ............ - ...................
2
··············-···························•·····························-
1
~
................................... .
- - - -
.......
- - -
. ..................
-
.
-
.......... .
..
O I I 11111111111111111111111111111111111111111 111111111111111 I I 111111111 11 1111111 11 1111111111111 11 111 11 111111 11 11111111111 11
v m N
m o Nv Nm ~
N m o v m N m o v m N m o v m N m o v m N m o
.,- .,- N
~
~
~
~
~
~
w w w ~ ~ m m m m m .,o .,o .,o .,.,N
T"'
,T""'
Maturity In Months
FIG. 7: Actual and Implied Yield Curves by the Three-factor Model Using Four Combinations
of Three Maturities Each: 3mon, 2yr & 10yr; 3mon, 1yr & 10yr; Smon, 2yr & 10yr; and 3mon,
2yr & 5yr
~Actualr
······r3210
--r3110
--r6210
-··-r325
--
1.8 - , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~
1.6 - 1 - ~ . . . . . . . . . . . . . . . . -,...._. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.21········::::::::····•·•;~
.
~
1.4
.J
§
CD
C
1!
.!J
1
~
....... .
c•• -·-·~•·~•
....
·......
··
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -.~ ..
0.8
C
s 0.6
Ill
.......... - . . . . . . . . . . . . . . .
...............
-
.....
-
-
. . . . .
......
-
. . .......
-
.................
.......... .
....
.. :-::.-.·.·.~·.•.--.:~~-:::::.
..........
- - -
j
- _
-
.............
-
..
. . . . . . . . . . . - - . . . . . . . ... .
0.4
-I · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2
-I································.··· .................................. .
0
I I I 111111111111111111IIII111111111111111111111111111111 I I I I I I I I I II 111111111 II 11111111111111111111111111111111111111111111
~
m ,N ID O ~ m N
,- N N N ~
ID
~
O
V
~
V
m
N
V ~
ID
~
O
W
~
W
m N
~
W
ID
~
O
00
N ID O ~ m N ID 0
m m
00 m m T
O "O" ,O
r
r
N
-,-,-,-,-
~
Maturity In Months
FIG. 8: Actual and Implied Volatility Curves by the Three-factor Model Using Four
Combinations of Three Maturities each: 3mon, 2yr & 1Oyr; 3mon, lyr & 1Oyr; 6mon, 2yr &
1Oyr; and 3mon, 2yr & Syr
1--Actual std
--std3210
--std3110
· · · · · · std6210
- · · - std325
1
~
.
. . . .. . . . . .
:::\
\
::::::···-·-:···-:-,·-". .. :.... ::.·······:: ...::: .: .....:..... ... ::....... :.
:;
0.4 -1 • - - - - ••• "-· • • • . . • . • • . • • . • • • • • • . . • • • . • _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ________ _
· 0.2
,, • '
0
-
,, •
~
~
- •• - •. - •• - - •• _..., •. ·--. ··- =-·. - •• .:. : • - .. - - • - •• - .. - •. -
I1111111111 IIJJ fl 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111++1-t+t+-i
6
1r 16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
01
96 101 106 111 116 1i1
-0.2 ~ · · · · _/. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . .
I
'
-0.4 -I · ·/·• · · · . . . . . . . . .
-0.6
.
.................
-I,',i· · ..... . ..... - ..
- ..
. ...... .
. . . . . .
..................... ..
. - ......... .
.. ...
. .....
. .......
-
-0.8 .,___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___J
Maturity In Months
FIG. 9. Factor Loadings of Yields from the Estimated Three-Factor Model Using 3-month, 2year and 10-year Maturities
-- -- -- -~
-- ----·---····
--··-9- .-- -----·
---- --·· ···· ···· ··
-----8
7
-
"' 6
5
.!! 4
► 3
2
.. .
- - - - - - ~-
- - -- - -
--
'ti
··
···· ···· ·-·· ---- --·- ··-· ···- ··-- ---· -······
······· ···· ···· ···· ···- ···· ·--- --- --··
··-· ····· -··- ---- ---- ---- ---· ···- ····- ··-·
1
0
---
sr
0> o"'
co -a., co
Maturity in Month s
C\I
,-
a,
,-
Other Parameters as In the ThraeFIG. 10. Simulated Yield Curves Using Phi3=0.99 and All
1(1.year Matur ities
and
factor Modal Estimated at 3-month,'2•year,
3
C
0
2.5
2
0 1.5
"E
1
"Cl
C
0.5
UI
0
,::
-;.,
.
-.
•
- - -- - -
---
:::~::: :~::: ~::::: :::~
C\I
st"
a,
st"
(0
u,
<')
(0
0
...
- - -- - --- -- -
--Ac tual std
-std [n]
...
...
Matur ity in Month s
0.99and Other Parameters in
FIG.11. Simulated Volati lity Curve Using phi1=0.97, phi2-and 10-yea r Matur ities
,
2-year
the Three -facto r Model Estimated at 3-month,
@FedFRASER