Dr.
Maddah ENMG 624 Financial Eng’g I 11/18/19
Chapter 4 The Term Structure of Interest Rate
The yield curve
Long bonds tend to offer higher yields than short bonds of
the same quality.
The yield curve display yield as a function of time to
maturity.
The yield is constructed based on yields of available bonds of
a given quality class.
A rising yield curve is normally shaped. This occurs most
often.
If long bonds happen to have lower yields than short bonds
then the result is an inverted yield curve.
(Source: http://en.wikipedia.org/wiki/Yield_curve)
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(Source: The Lebanon Brief, October 20, 2007. Blom Bank)
When studying a particular bond it is useful to place it as a
point in the plot of the yield curve.
The term structure
Term structure theory is based on the observation that interest
rate depends on the length of time the money is held.
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The spot rates
Spot rates are the basic interest rates defining the term
structure.
The spot rate st is the interest rate charged for money held
from present till year t.
For example, a 1-year deposit will grow by a factor of (1+s1).
A 2-year deposit will grow by a factor of (1+s2)2 .
In general, a t-year investment grows by a factor of (1+st)t .
Compounding rules applies to spot rates. For example, under
a compounding of m times per year, a t-year deposit will
grow by a factor of (1+st/m)mt .
Under continuous compounding a t-year deposit will grow by
a factor of e s t .
t
Discount factors and present values can then be determined
in the usual way.
For example, with yearly compounding, the present value of
a cash flow stream x = (x0, x1, …., xn) is
n
PV d k xk , where dk = 1/(1+sk)k .
k 0
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Spot rates curve
Spot rates can be determined from the yields of zero-coupon.
bonds
Spot rates
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10
st 6
0
0 2 4 6 8 10 12 14 16 18 20
t
If not enough, zero-coupon bonds are available (especially
long ones), the spot rate curve can be determined from the
prices of coupon-bearing bonds.
For example, suppose you have a 1-year zero-coupon bond
and a 2-year bond paying a coupon C2 every year. The yield
of the first coupon (P1/F1) gives the spot rate s1.
Then, the spot rate s2 can be determined from the equation
C2 C F2
P2 2
1 s1 (1 s2 ) 2 .
Spot rates can also be found by “subtraction” of two bonds
with different coupons to construct a zero-coupon bond.
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Examples
Forward rates
Forward rates are interest rates for money to be borrowed
between two future dates, under terms agreed upon today.
E.g., suppose there are two ways of investing $1 for 2 years
(i) Deposit it in a 2-year bank account where it will grow to
(1+s2)2 at the end of the two years.
(ii) Deposit it in a 1-year bank account where it will grow to
(1+s1) at the end of the first year, and then deposit the
proceeds for one more year at a rate f1,2 (yielding
(1+s1)(1+f1,2) at the end of the two years).
In this case, f1, 2 is the forward rate between years 1 and 2.
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Invoking the comparison principle implies that
(1 s2 )2
(1 s2 ) (1 s1 )(1 f1,2 ) f1,2
2
1.
1 s1
The use of the comparison principle can be justified through
an arbitrage argument.
Arbitrage is earning money without investing anything.
If (1+s2)2 < (1+s1)(1+f1,2), then one can borrow $1 for two
years and invest it according to (ii) and make an arbitrage
profit of (1+s1)(1+f1,2) − (1+s2)2 after two years.
If (1+s2)2 > (1+s1)(1+f1,2), then one can borrow $1 for one
year and invest it according to (i). Then, at the end of the first
year, borrow another (1+s1) dollars to pay the first loan. This
will yield an arbitrage profit of (1+s2)2 − (1+s1)(1+f1,2) .
Such an arbitrage scheme cannot exist in the market because
many people will jump on it leading to closing the gap.
This arbitrage argument assumes that there are no transaction
costs and that borrowing and lending rates are identical. This
is a reasonable approximation.
In general, the forward rate ft ,t is the annual interest rate
1 2
charged for borrowing money between times t1 and t2, t1 < t2 .
Forward rates deduced from spot rates are termed implied
forward rates to distinguish them from market forward rates.
The implied forward rate between year i and year j satisfies
(1+sj)j = (1+si)i(1+fi,j)j−i , which implies that
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1/( j i )
(1 s j ) j
fi , j i
1.
(1 si )
For m period-per-year compounding, the implied forward
rate (per year) between periods i and j satisfies
(1+sj/m)j = (1+si/m)i(1+fi,j/m)j−i, which implies that
1/( j i )
(1 s j / m) j
fi , j m i
m.
(1 si / m)
Under continuous compounding, the implied forward rate
(per year) between times t1 and t2 satisfies
f t1 ,t 2 (t 2 t1 )
e
st 2 t 2 st1t1
e e , which implies that
st2 t2 st1 t1
ft1 ,t2 .
t2 t1
Note that (at any compounding) the spot rate at time t can be
seen as the forward rate between time 0 and t, f0, t = st .
Term structure explanations
The spot rate curve is almost never flat but usually slopes
upward.
Why is this curve not just flat at a common interest rate?
There three standard explanations for this: Expectation
theory, liquidity preference, and market segmentation.
We adopt the expectation theory explanation.
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Expectation theory
This theory explains the shape of the spot rate curve based on
expectations of what rates will be in the future.
E.g., the theory argues that most people in the market believe
that the 1-year rate next year will be higher than the current
1-year rate.
The expectation hypothesis expresses this expectation in
terms of forward rates.
E.g., according to this hypothesis, the forward rate, f1, 2 , is
exactly equal to market expectation of what the 1-year rate
will be next year, s1′. That is, s1′ = f1, 2 .
More generally, the hypothesis is sn-1′ = f1, n .
The main weakness of expectation theory is that it implies
that spot rates always increase, which is not always true.
Expectation dynamics
The expectation hypothesis leads to useful tools.
Spot rate forecasts: Under the expectation hypothesis, the
k-year spot rate i years from now is
sk(i ) fi ,k i .
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Specifically, if the current spot rates are s0 = f0,1, s2 = f0,2, …,
sn = f0,n, then forecasts for spot rates for years 1 to n−1 are
s1(i ) s2(i ) …. sn(i)2 sn(i)1 sn(i )
Year, i
0 f0,1 f0,2 … f0,n-2 f0,n-1 f0,n
1 f1,2 f1,3 … f1,n-1 f1,n
2 f2,3 f2,4 … f2,n-2
n−2 fn-2,n-1 fn-2,n
n−1 fn-1,n
The discount factor between years i and j is
j i
1
di , j .
1 fi , j
Note that di,k = di,j dj,k .
Short rates
Short rates are the forward rates spanning a single time
period. The short rate at year k is
rk f k ,k 1 .
Short rates are as fundamental as spot rates because a
complete set of short rates fully defines a term structure.
The spot rate can be obtained from short rates as follows.
(1 s k ) k (1 r0 )(1 r1 ) (1 rk 1 )
s k [(1 r0 )(1 r1 ) (1 rk 1 )]1/ k 1.
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Similarly, the forward rates can be obtained from the short
rates as follows.
(1 f i , j ) j i (1 ri )(1 ri 1 ) (1 r j 1 )
f i , j [(1 ri )(1 ri 1 ) (1 r j 1 )]1/ j i 1.
A useful feature of short rates (under the expectation
hypothesis) is that they do not change from year to year.
(spot rates do change.)
If the short rates now are r0, r1, …, rn, then the short rates
next year are r1, …, rn .
Examples
To see how short rates work, we will go together over re-generating
Table 4.2 in the text in Excel. The example is available here.
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Duration under the term structure
Under the term structure, duration is defined as sensitivity to
linear shifts in the spot rate curve.
Specifically, the duration of a bond is the sensitivity of the
bond value relative to when the spot rates shift from s1, s2,
…, sn, to s1 + , s2 + , …, sn + .
Shifts of the spot rate curve
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10
st 6
0
0 2 4 6 8 10 12 14 16 18 20
t
Under continuous compounding, the Fisher-Weil duration of
a cash flow stream with cash flows xti at time ti, i = 1, …, n is
n
1
t x e
sti ti
DFW i ti ,
PV i 0
where PV xti e
sti ti
.
i 0
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Let P() be the value (price) of the stream when the spot rate
curve shifts by Then,
n
P( ) xti e
( sti ) ti
.
i 0
(Observe that with no shift this value is P(0) = PV.)
Upon differentiation,
dP ( ) n
x t i (s t i )e ti i .
( s )t
d i 0
Then,
1 dP ( )
D FW .
P (0) d 0
Under discrete compounding, with m period-per-year
compounding,
n
xk
P ( ) .
k 0 [1 (s k ) / m ]
k
Then,
dP ( ) n
(k / m )x k
d
k 0 [1 (s k ) / m ]
k 1
0 0
n
(k / m )x k
k 1
.
k 0 [1 s k / m ]
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Then, the quasi-modified duration is then defined as
n
1 dP ( ) (k / m )x k (1 s k / m ) ( k 1)
DQ k 0
.
P (0) d 0 n
x
k 0
k (1 s k / m ) k
Immunization Idea and Example
Immunization can now be done similar to Chapter 3 but by
matching DFW (or DQ) instead of DM.
This can be done for structuring a portfolio of bonds with
different yields.
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