Lecture 7: Term Structure Models
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Bond basics
A zero-coupon n period bond is a claim to a sure payoff of 1 at
(n)
time t + n. The price is denoted Pt and it satisfies the
recursion:
(n)
(n−1)
Pt
= Et Mt+1 Pt+1
,
(0)
Pt
= 1.
We define the yield of a bond with maturity n at time t through
the equation
1
(n)
,
Pt = (n) n
1 + Yt
i.e. it is the per period (e.g. per year) return that you get if you
buy a bond today and hold it until it matures.
The holding period return is the return if you buy a bond of
maturity n at time t and sell it back at time t + 1 :
(n−1)
(n)
Rt+1
=
Pt+1
P
(n)
.
Yield curve with iid stochastic discount factor
Suppose that {Mt+1 } is iid. Then the yield curve is constant and
(n)
flat, i.e. yt = yt = y for all t, n.
Proof: Let p= E(M ). Start
with n = 2 and show
(n)
(n−1)
= Et Mt+1 pn−1 = p.pn−1 = pn .
Pt = Et Mt+1 Pt+1
(n)
If Pt
(n)
= pn , then yt
= y with 1 + y =
1
P
.
Implication: if the short-rate is constant, all yields are constant,
and hence bonds of all maturities have risk-free returns (Whereas
in general, only the one-period bond has a return that is
risk-free.)
Nominal discount factors
Arbitrage:
Pt
Dtnom
= Et Mt+1
,
qt
qt+1
where qt =price index (CPI). Hence, inflation is πt+1 = qt+1 /qt .
Define a nominal SDF
nom
Mt+1
=
Mt+1
,
πt+1
The bond pricing recursion holds for nominal bonds using this
nominal SDF:
(n)
nom (n−1)
Pt = Et Mt+1
Pt+1
.
Comovement
All yields (long and short) are highly correlated - they tend to
move up and down together a lot; more precisely, one can do
principal components to find the factors which move yields.
The first, by far most important factor is the “level”: all yields
move up and down together; second, there is a “slope” effect i.e.
long term yields and short term yields move in opposite
direction; last, there is a “curvature” effect i.e. the concavity of
the yield curve changes somewhat.
0
5
10
15
Monthly Nominal Treasury Yields (1985:2013)
1985m1
1990m1
1995m1
2000m1
date
2005m1
treas03m
treas02y
treas05y
treas10y
2010m1
2015m1
0
2
4
6
8
Daily Nominal Treasury Yields (1996:2013)
01jan199601jan199801jan200001jan200201jan200401jan200601jan200801jan201001jan201201jan2014
date
treas03m
treas02y
treas05y
treas10y
treas15y
treas30y
0
2
4
6
8
Daily Forward Rates (1996:2013)
01jan199601jan199801jan200001jan200201jan200401jan200601jan200801jan201001jan201201jan2014
date
ftreas02y1
ftreas09y1
ftreas04y1
Term Spread
On average the yield curve is somewhat upward sloping; i.e. the
yield on long-term bonds is larger than on short-term bonds.
The slope of the yield curve, i.e. the term spread measured as the
difference between a long rate (≥ 5 years) and a short rate (≤ 1
year) is correlated with the business cycle: an inverted yield
curve predicts a recession, and at the trough of the recession, the
yield curve is steeply upward sloping.
-1
0
1
2
3
4
Monthly Term Spreads (1985:2013)
1985m1
1990m1
1995m1
TS_10y_2y
2000m1
date
2005m1
TS_10y_3m
2010m1
2015m1
Volatility
Long-term bond prices are fairly volatile; the std dev of the 10y
return is about 8% per year, i.e. almost half that of stocks. In
terms of yields, the std dev of yields as a function of maturity is
hump-shaped.
n (years)
1
2
3
4
5
E(hpr)
5.83
6.15
6.40
6.40
6.36
T able1
s.e.
.42
.54
.69
.85
.98
σ(hpr)
2.83
3.65
4.66
5.71
6.58
Expectation hypothesis (EH)
The EH states that the expected log (holding period) returns on
all bonds is the same:
(n)
(1)
(1)
Et hprt+1 = Et hprt+1 = yt , alln ≥ 0.
The N-period (log) yield is the average of expected future
one-period (log) yields:
(N )
yt
=
1
(1)
(1)
(1)
Et yt + yt+1 + ... + yt+N −1
N
.
The forward rate equals the expected future spot rate (in logs):
(1)
ftN →N +1 = Et yt+N .
Weaker version of EH
A weaker form of the EH is that these relations hold “up to a
constant”, i.e.
(n)
(1)
Et hprt+1 = yt+1 + constant,
1
(N )
(1)
(1)
(1)
yt
=
Et yt + yt+1 + ... + yt+N −1 + constant,
N
(1)
ftN →N +1 = Et yt+N + constant,
where the constant may depend on maturity n but not on time t.
Key point: the EH is almost assuming risk-neutrality - expected
returns should be the same on all assets. (It is not quite that,
because it is in logs instead of levels.)
Implication and test of EH:
Under the EH, if the long-term yield is high today relative to the
short yield, it must be that the short yield will rise in the future,
so that if you invest in short rate only every period you will end
up getting the same return at the end.
In the data, the expectation hypothesis does not work very well:
the expected return on bonds is forecastable,
(n)
(1)
Et hprt+1 − yt+1 = α + βXt
i.e. there are times when investing in long-term bonds brings
excess returns.
Intuition
One way to summarize the results is to go back to the example
explaining the EH – in the data, on average, when short < long,
the short yield does not increase enough in the future, so there is
a positive excess return to borrowing short term and buying
long-term bonds.
More precisely, the variable Xt that researchers use to predict
returns on long-term bonds is usually based on current yields or
forward rates. Fama and Bliss use the difference between the
forward rate at time t + n and the current short rate, to forecast
the maturity n bond excess return. Cochrane and Piazzesi find
that a particular combination of forward rates forecast all
maturities of excess bonds returns. (Refs: Fama and Bliss (1988),
Campbell and Shiller (1991), Cochrane and Piazzesi (2005)).
Testing the EH
The expectation hypothesis is often tested through to the
following equation:
n
yt − yt1
n−1
n
+ εt+1 .
yt+1 − yt = α + βn
n−1
The expectation hypothesis implies that βn = 1. In the data,
βn < 1, often negative, and decreasing with the horizon n.
Table: Expectation Hypothesis Tests
n=2
n=3
n=4
n=5
Slope Coefficients - 1961-1979
−1.03
−1.52
−1.55
−1.43
[0.65]
[0.71]
[0.83]
[0.96]
Slope Coefficients - 1988-2006
0.61
−0.13
−0.19
−0.21
[0.89]
[0.90]
[0.97]
[1.02]
Summing up
Just like for stocks, we need a model which explains:
1
the mean return on long-term bonds, relative to short-term bonds,
2
the volatility of long-term bond returns
3
the variation over time in the expected returns.
A simple affine model:
Assume conditional log-normality of Mt+1 and Ptn then for
n>1
1
pnt = Et (mt+1 + pn−1,t+1 ) + vart (mt+1 + pn−1,t+1 )
2
and
1
p1t = Et mt+1 + vart (mt+1 )
2
Suppose
mt+1 = −xt + et+1
where
xt+1 = (1 − φ) µ + φxt + vt+1
Also assume
et+1 = βvt+1 + ηt+1
where vt+1 and ηt+1 are iid uncorrelated.
Simplified version:
For simplicity set ηt+1 = 0 so that
mt+1 = −xt + βvt+1
(it only affects the level of the term structure). In this case mt+1
is an ARMA(1,1) (sum of AR1 and white noise).
Note if β = 1/φ we have
mt+1 = −
xt+1 (1 − φ)
+
µ
φ
φ
which may be interpreted as renormalized consumption growth,
hence a form of CARRA.
One period yield:
For the one period yield
y1t = −p1t
so
σ2
2
Since short-rate inherits AR1 dynamics we can think of xt as the
process for the short-rate process itself.
y1t = xt − β 2
N period yield
Guess a solution of the form
−pnt = An + Bn xt
Since the yield on an n-period bond is
ynt = −
pnt
n
we are guessing that the yield on any maturity is an affine
function of xt .
Implications
With this guess we have
pnt = Et [mt+1 + pn−1,t+1 ]
= Et [−xt + βvt+1 − An−1 − Bn−1 xt+1 ]
and
vart [mt+1 + pn−1,t+1 ] = [β + Bn−1 ]2 σ 2
So
−An − Bnt xt = −xt − An−1 − Bn−1 (1 − φ) µ − Bn−1 φxt
+
[β + Bn−1 ]2 σ 2
2
Solution:
For n = 0, 1
A0 = B 0 = 0
σ2
A1 = −β 2 , B1 = 0
2
Equating coefficients we have:
1 − φn
1−φ
= (1 − φ) µBn−1 − (β + Bn−1 )2 σ 2 /2
Bn = 1 + φBn−1 =
An − An−1
Interpretation:
The coefficient Bn measures the sensitivity of the bond price to
an increase in the short-rate.
Bond prices fall when short rates rise. The higher the maturity,
the greater the response.
Expected excess holding period return:
The expected holding period return is
Et rn,t+1 = Et pn−1,t+1 − pn,t
h
i
2 2
= − (1 − φ) µBn−1 + (β + Bn−1 ) σ /2 + xt
so all expected holding period returns increase one for one with
the short-rate.
The excess holding period return is defined as
vart (rn,t+1 )
2
vart (xt+1 )
Et rn,t+1 − y1t +
2
This implies
Et rn,t+1 − y1t +
= −covt (rn,t+1 , mt+1 )
= Bn−1 covt (xt+1 , mt+1 )
2 σ2
Bn−1
= −Bn βσ 2
2
We can interpret Bn σ as the amount of risk for an n period
holding period return and βσ as the price of risk.
Et rn,t+1 − y1t +
Interpretation:
Assume that
mt+1 = log δ − γ∆ct+1
then
γ (∆ct+1 − Et ∆ct+1 ) = βvt+1
If β > 0 a shock to vt+1 represents positive news about future
consumption growth which then follows a persistent AR1
process. (i.e. β governs the covariance between consumption
innovations and expected future consumption growth).
So if β > 0 a positive shock signals that future consumption
growth and hence interest rates will be high. A rise in expected
future interest rates implies a reduction in current bond prices.
So bond returns covary negatively with current consumption
growth – they are therefore a good hedge and carry a negative
risk premium.
Forward rates:
The forward rate satisfies
fnt = pnt − pn+1,t
= −p1t + (Et pn,t+1 − pn+1,t + p1t ) − (Et pn,t+1 − pnt )
= y1t + Et rn+t,t+1 − y1t − (Et pn,t+1 − pnt )
where
pn,t+1 − pnt = −Bn Et ∆xt+1
This implies
fnt
1 − φn 2 σ 2
=µ− β+
+ φn (xt − u)
1−φ
2
With this specification we can get rising, humped-shaped or
inverted yield curves depending on xt .
If xt = µ the yield curve is initially rising relatively steeply
owing to the squared term.
Conditional Heteroskedasticity
Cox-Ingersoll-Ross in discrete time:
√
−mt+1 = xt + xt βvt+1
xt+1 = (1 − φ) µ + φxt +
√
xt vt+1
We now have
2
2σ
p1t = Et mt+1 = −xt 1 − β
2
2
so A1 = 0, B1 = 1 − β 2 σ2 and the recursion
Bn = 1 + φBn−1 − (β + Bn−1 )2
σ2
2
An − An−1 = (1 − φ) µBn−1
In the limit Bn solves a quadratic but is roughly equal to
before so it is positive and increasing in n.
1
1−φ
as
Expected excess log bond return:
We now have
vart (xt+1 )
= Bn−1 covt (xt+1 , mt+1 )
2
σ2
2
Et rn,t+1 − y1t + Bn−1
xt
= −Bn−1 xt βσ 2
2
√
We can again interpret Bn−1 xt σ as the amount of risk and
√
β xt σ as the price of risk.
2
Et rn,t+1 − y1t + Bn−1
Thus, in this model, the price of risk rises with the (square root)
of the level of interest rates.
-1
0
1
2
3
4
Ten-year term premium vs ten minus 2 year term spread
01jan199601jan199801jan200001jan200201jan200401jan200601jan200801jan201001jan201201jan2014
date
term_premium
term_spread
8
0
-1
2
0
4
treas10y
term_premium
1
6
2
Ten-year term premium vs ten-year treasury yield
01jan1996
01jan1998
01jan2000
01jan2002
01jan2004
01jan2006
01jan2008
01jan2010
01jan2012
01jan2014
date
term_premium
treas10y