Lecture 12/13
Bond Pricing and
the Term Structure of Interest Rates
Alexander K. Koch∗
Department of Economics, Royal Holloway, University of London
January 14 and 21, 2008
In addition to learning the material covered in the reading and the lecture, students should In
addition to learning the material covered in the reading and the lecture, students should
• understand how spot and forward rates are derived;
• be familiar with theories of the term structure;
• be able to price default-free bonds.
Required reading: Bodie, Kane, and Marcus (2008) (Chapter 15)
Supplementary reading: Elton and Gruber (1995) (Chapter 20)
∗
E-mail: Alexander.Koch@rhul.ac.uk.
1
Positive yield curve
6
5.5
09/12/2004
09/11/2004
08/10/2004
5
4.5
Yield (%)
4
3.5
3
2.5
2
1.5
1
1M
3M
6M
1Y
2Y
3Y
5Y
7Y
10Y
20Y
Maturity
Figure 1: Positive yield curve (source: U.S. Treasury.)
1
The term structure of interest rates
In lecture 11 we discussed basic principles of bond pricing, assuming for simplicity that the
same constant interest rate can be used to discount future cash flows. In reality, however,
we observe that yields vary with the maturity of bonds. That is, interest rates have a term
structure. This is manifested in the shape of the yield curve, a plot of bond yields against
maturities. Figure 1 reproduces such a plot of yields of US treasury securities for 9 December
2004 and dates in the two preceding months.1 Two features are apparent. First, the yield
curve varies over time. Second, yields are lower for short-term maturities than for longer
maturities. This shape is the ’normal’ shape that the yield curve has and is called a positive
yield curve. Figures 2-4 depict other possible shapes observed in the market. We will discuss
later how these different shapes can arise. Finally, Figure 5 depicts the UK yield curve in
January for the years 2005-2008.
1.1
The term structure under certainty
As in the last lecture, we focus here on interest rates on default-free bonds. Moreover, we
start our analysis of the term structure of interest rates by supposing that investors know for
certain the path of future interest rates.
Example 1
1
2
Consider the sequence of 1-year (1Y) interest rates given in table 1.
Source: U.S. Treasury.
Data is available at http://www.treasury.gov/offices/domestic-finance/
debt-management/interest-rate/yield.html. UK yield curve date is available from the Bank of England
at http://213.225.136.206/statistics/yieldcurve/index.htm
2
This example is taken from the readings: Bodie, Kane, and Marcus (2008), Chapter 15.
2
Negative (inverted) yield curve
6.5
US 27/11/2000
6
Yield (%)
5.5
5
4.5
4
UK 04/01/2007
3.5
3
1M
3M
6M
1Y
2Y
3Y
Maturity
5Y
7Y
10Y
20Y
Figure 2: Negative (inverted) yield curve (source: U.S. Treasury/Bank of England.)
Flat yield curve (1 March 1990)
9
8.5
Yield (%)
8
7.5
7
6.5
6
1M
3M
6M
1Y
2Y
3Y
5Y
7Y
10Y
20Y
10Y
20Y
Maturity
Figure 3: Flat yield curve (source: U.S. Treasury.)
Hump shaped yield curve (29 March 2000)
8
7.5
Yield (%)
7
6.5
6
5.5
5
1M
3M
6M
1Y
2Y
3Y
5Y
7Y
Maturity
Figure 4: Hump-shaped yield curve (source: U.S. Treasury.)
3
Yield curves 2005-2008
6.00
5.50
Yield (%)
5.00
4.50
4.00
04
04
04
04
3.50
3.00
Jan
Jan
Jan
Jan
05
06
07
08
2.50
1M
3M
6M
1Y
2Y
3Y
5Y
7Y
10Y
20Y
Maturity
Figure 5: UK yield curve in January 2005-2008 (source: Bank of England.)
year
1Y interest rate rt
0
0.08
1
0.10
2
0.11
3
0.11
Table 1: Sequence of 1-year interest rates
What are the fair values of 1Y, 2Y, 3Y, and 4Y zero-coupon bonds with face value £ 1,000?
As in lecture 11, we simply compute the present value, now applying the appropriate discount
factors for the respective periods. Denote the fair value of an n-year zero-coupon bond by
ZBn :
ZB1 =
ZB2 =
ZB3 =
ZB4 =
£ 1, 000
= £ 925.93
1 + r1
£ 1, 000
= £ 841.75
(1 + r1 ) (1 + r2 )
£ 1, 000
= £ 758.33
(1 + r1 ) (1 + r2 ) (1 + r3 )
£ 1, 000
= £ 683.18.
(1 + r1 ) (1 + r2 ) (1 + r3 ) (1 + r4 )
These prices imply a term structure of interest rates that we can extract by looking at the
yields to maturities of the zero-coupon bonds:
Yield to maturity
ZB1
ZB2
ZB3
ZB4
8.000%
8.995%
9.660%
9.993%
4
yield curve
12
spot rate
1Y interest rate
11.5
11
Yield (%)
10.5
10
9.5
9
8.5
8
7.5
7
1
2
3
4
Maturity
Figure 6: Yield curve from example 1.
The yield to maturity on an n-year zero-coupon bond is called n-year spot rate, yn . Plotting
the spot rates (i.e. the YTM of zero-coupon bonds) from the example we obtain the pure (or
zero-coupon) yield curve plotted in figure 6.
What is the relation between the 1Y interest rates and the spot rate? In the case of example
1 we see immediately in figure 6 that the yield curve (i.e., the spot rate curve) is not equal to
the sequence of 1Y interest rates, plotted as a dotted line.
Let us explore this relation further. The n-year spot rate is the YTM of an n-year zero-coupon
bond. Since there is no uncertainty, the return on a £ 1 investment in a 2Y zero-coupon bond
has to equal that on a £ 1 investment in a 1Y bond that is then rolled over after one year at
the future 1Y rate:
1
1
=
(1.08) (1.10)
(1 + y2 )2
⇒
1 + y2 = [1.08 · 1.10]1/2 .
In general, in our special case of certain future rates, the return on an n-year investment has
to be equal to that obtained on a 1-year investment that is rolled over at the future 1Y rates
for n-years:
1 + yn = [(1 + r1 ) · (1 + r2 ) · . . . · (1 + rn )]1/n .
(1)
That is, the n-year spot rate is the geometric average of the 1Y interest rates for the next n
years.
1.2
The term structure under uncertainty
So far we assumed that future 1Y interest rates were known with certainty. However, in reality
this is not the case, of course. Nevertheless, we can compute a zero-coupon yield curve from
5
observed bond prices and extract from it interest rates that we can lock in today for a future
period.3 Such a rate for an n-period investment starting at date t and ending at date t+n is
called a forward rate and we will denote it by fn,t .
Example 2 The following prices for zero-coupon bonds with face value £ 1,000 are currently
observed:
£
ZB1
ZB2
ZB3
ZB4
925.9259
841.6800
761.6539
688.0035
Compute all possible forward rates.
To find the 1Y forward rate f1,1 , i.e., the interest rate that we can lock in today for a 1Y loan
starting in one year, we construct a £ 1,000 synthetic forward contract, using zero-coupon
bonds. For these we know the yields to maturity since we know the 1Y and 2Y spot rates.
cash flow
t=0
t=1
long one 1Y zero bond
−925.93 = −ZB1 = − 1,000
1+y1
+1, 000
short x 2Y zero bond
+925.93 = x · ZB2
sum
0
The above table implies that x = 1 + f1,1 and x =
−1, 000 · x
+1,000
ZB1
ZB2
t=2
−1, 000 · x
≈ 1.1001. That is, the forward rate
f1,1 is 10.01 percent.
Another way to see this is to exploit the no-arbitrage condition stating that an investment in
the 2Y zero bond must have the same return as an investment in a 1Y zero bond rolled over
in the second year at an interest rate locked in today at f1,1 :
=ZB1
=ZB2
z }| {
1
1, 000
·
1 + y1 1 + f1,1
⇔
z
=
f1,1 =
}| {
1, 000
(1 + y2 )2
ZB1
(1 + y2 )2
−1=
− 1.
ZB2
1 + y1
To compute the 2Y forward rate f2,1 , we can employ the same strategy of constructing a
synthetic portfolio. Alternatively, we can exploit the no-arbitrage condition stating that an
investment in a 3Y zero bond must have the same return as an investment in a 1Y zero bond
3
See Bodie, Kane, and Marcus (2008), Section 15.6 for details on estimating the pure yield curve from bond
data.
6
t
1
2
3
4
ZBt
£ 925.9259
£ 841.6800
£ 761.6539
£ 688.0035
yt
8.00 %
9.00 %
9.50 %
9.80 %
n
fn,t
1
10.01 %
2
10.26 %
3
10.41 %
10.51 %
10.70 %
Table 2: Forward rates for example 2
rolled over after the first year for a two-year investment at the forward rate f2,1 :
=ZB3
=ZB1
z }| {
1
1, 000
·
1 + y1 (1 + f2,1 )2
⇔
f2,1
z
}| {
1, 000
=
(1 + y3 )3
1
1
ZB1 2
(1 + y3 )3 2
− 1.
=
−1=
ZB3
1 + y1
To compute the 2Y forward rate f2,2 , we can again exploit the no-arbitrage condition stating
that an investment in the 4Y zero bond must have the same return as an investment in a 2Y
zero bond rolled over after the second year for a two-year investment at the forward rate f2,2 :
=ZB4
=ZB2
z
}| {
1
1, 000
·
2
(1 + y2 ) (1 + f2,2 )2
⇔
f2,2
z
}| {
1, 000
=
(1 + y4 )4
1
1
ZB2 2
(1 + y4 )4 2
=
−1=
− 1.
ZB4
(1 + y2 )2
Repeating this exercise, it is easy to convince yourself that the general formula for computing
the n-period forward rate for a starting date t is given by:
fn,t =
ZBt
ZBt+n
1
n
−1=
(1 + yt+n )t+n
(1 + yt )t
n1
− 1.
Find the other 1Y and 2Y forward rates and compare with table 2.
Forward rates versus expected future spot rates
Future interest rates are random variables. Hence, in general the spot rate that will realize in
the future will not equal today’s forward rate. For example, the realized 2Y-spot rate in two
years, y2 will usually not be equal to today’s forward rate f2,2 . The simplest theory on the
term structure of interest rates is the expectations hypothesis. It starts from the premise
that investors are indifferent between a long-term investment and a short-term investment that
7
is rolled over if the two strategies give the same expected rate of return over the investment
horizon. Thus, the expectations hypothesis asserts that forward rates are equal to the market’s
’consensus’ expectation of future spot rates, i.e. that fn,t = E[yn for date t].
Can forward rates really reflect something like a ’consensus’ expectation of market participants
on future interest rates? The answer to this question is “not necessarily”. To see why, consider
the following example:
Example 3 The current 1Y spot rate is r1 = 0.08 and the 1Y forward rate for in one year is
f1,1 = 0.08 and for in two years is f1,2 = 0.10. Suppose that the market consensus expectation
for the 1Y spot rate in one year’s time is 8 percent and for the 1Y spot rate in two year’s time
is 10 percent.
Short-term investors
Suppose an investor wants to earn money on her funds but knows that she will need cash in
two years. Buying a 2Y bond matches the investment horizon of this short-term investor and
guarantees a rate of return of 8 percent, since ZB2 =
FV
(1+r1 ) (1+f1,1 ) .
In contrast, buying the 3Y
bond exposes her to price risk after two years. Again, the bond is priced as if the forward rate
was the interest rate prevailing in the future. However, the price of the bond after 2 years will
move depending on the interest rate that will actually realize for the third year. The bond will
face value
. Only if the then realized
then be a 1Y zero bond priced at
1+realized 1Y spot rate in year 2
interest rate corresponds exactly to f1,2 , which in this example is also today’s expectation of the
future 1Y spot rate, will she have the same return as with the short-term investment. Only if
the realized 1Y spot rate turns out to exceed today’s expectations does the investor gain relative
to the safe strategy. Therefore, a risk-averse short-term investor will only be willing to hold
long-term bonds if the forward rate exceeds the expected rate of return by a risk premium: i.e.,
if f1,2 > E[r1 for date 2].
Long-term investors
In contrast, an investor wishing to have cash available only after three years faces a reinvestment rate risk when investing in 2Y bonds and rolling over at the uncertain third-period
rate. Buying a 3Y zero-coupon bond can lock in an interest rate of 8 percent for the first two
years and 10 percent for the third year, since ZB3 =
FV
(1+r1 ) (1+f1,1 )(1+f1,2 ) .
In contrast, buying
the 2Y zero-coupon bond and then rolling over the proceeds leaves the investor with the same
interest rate for the first two years but an uncertain interest rate for the third year. If the
1Y rate turns out to be less than the forward rate, the A risk-averse long-term investor who
shares the market’s belief that the future 1Y spot rate will be equal to today’s forward rate for
that period will not be willing to buy short-term bonds. To gain relative to the safe long-term
8
strategy the expected 1Y rate must exceed the one that can be locked in today, implying a negative risk premium on the forward rate. Long-term investors hold short-term bonds only if
f1,2 < E[r1 for date 2].
The preceding example demonstrated why forward rates need not reflect the market’s consensus estimate of future interest rates. Depending on the investment horizon of market
participants, the forward rates can either be above or below the ’market’s estimate’.
Let us return now briefly to the different possible shapes of the yield curve, illustrated by
the examples in figures 1-4. An inverted (or negative) yield curve might indicate that market
participants expect interest rates to decline. However, most of the time one can observe the
’normal’ shape of an upward sloping (or positive) yield curve. A competing theory to the
expectations hypothesis for explaining the yield curves rationalizes this shape by a preference
of investors for short investment horizons (the preceding example on short-term investors
illustrates the logic). According to the liquidity preference theory long-term bonds need to
offer a higher yield than short-term ones to convince investors to buy them and face the price
risk of these bonds. That is, forward rates exceed the market expectation of future spot rates
by a positive risk premium. In the example we have seen that long-term investors actually
might require a negative premium on forward rates relative to the market expectation. A third
theory explaining the term structure of interest rates allows for such different types of investors.
The market segmentation hypothesis asserts that investors have a preferred habitat. For
example, some institutional investors such as banks prefer short-term securities while others
such as insurance companies prefer the long end of the market. Yields in each market segment
are then determined by the supply and demand in that market. Investors might be convinced
to leave their preferred habitat if they receive a positive or negative premium relative to
the expectation about future rates, depending on the maturity and the investor’s investment
horizon, as illustrated in example 3.
Nominal vs real yields
It is worth pointing out that changes in expected nominal interest rates can either be due to
changes in expected inflation or changes in expected real interest rates. As Figure 7 illustrates
for UK data from Hanuary 2002, the nominal yields and the real yields are quite different.
For example, a three year investment of £ 100 would return in January 2005 £ 116.5, but once
accounting for expected inflation the investment would be anticipated to return in real terms
only £ 107.9.
To how differences in inflation expectations can give rise to different shapes of the nominla
9
Nominal/real yields and implied inflation (UK -January 2002)
6.00
nominal
real
implied inflation
5.50
5.00
4.50
Yield (%)
4.00
3.50
3.00
2.50
2.00
1.50
1.00
3Y
5Y
7Y
10Y
20Y
Maturity
Figure 7: Real vs Nominal yield curve and implied inflation (source: Bank of England.)
yield curve, suppose that investors are demanding a constant real interest rate for a 1-year
investment at any future date. Then, the nominal rates will depend on the expected inflation
rates. Specifiaclly, the relation between the nominal interest rate and the real interest rate
over a given period is as follows:4
1+ nominal interest rate
.
1+inflation rate
1 + real interest rate =
The real interest rate ’discounts’ the nominal rate using the inflation rate. For example,
suppose that the 1Y real interest rate is expected to remain at 5 percent for all future dates.
Moreover, the market expects the annual inflation rate to rise from 1 percent in the first year
to 1.5 percent in the second and third year and then to 2 percent in the fourth year. Table
3 gives examples for nominal, real and inflation rates over the different investment horizons.
As we can see, even with constant real interest rates a positive yield curve can result from
expected increases in inflation.
Summary of important results
Let us briefly pause to review our results.
1. Given prices, ZBt , for zero-coupon bonds with maturities t and face value FV we can
compute the t-period spot rate, yt by computing their respective yields to maturity:
yt =
4
FV
ZBt
1
t
− 1.
Often the relation given is approximated using the Fisher equation: real interest rate≈ nominal interest
rate-inflation rate.
10
investment
annual rates (in %) over investment period
period
nominal rate
real rate
inflation rate
1
6.05
5
1.00
2
6.31
5
1.25
3
6.40
5
1.33
4
6.57
5
1.50
Table 3: Nominal, real and inflation rates over different investment periods.
2. The zero-coupon (or pure) yield curve is a graphical representation of these spot rates.
3. Using the spot rates or zero bond prices, we can compute the n-period forward rate for
a starting date t as:
fn,t =
2
ZBt
ZBt+n
1
n
−1=
(1 + yt+n )t+n
(1 + yt )t
n1
− 1.
Bond pricing
Given the spot rates and forward rates derived from the pure yield curve, we can price bonds
easily. The technique for determining the fair value of a bond is to treat each part of the cash
flow as a separate zero-coupon bond.
Example 4
5
The current spot rates are as follows: 1Y – 8 percent, 2Y – 8.995 percent, and
3Y – 9.66 percent. What is the fair price of a 3Y bond with face value £ 1,000 and annual
coupon payments at coupon rate 8 percent?
We ’strip’ the bond into its year 1, year 2, and year 3 cash flows, which each are discounted
with the appropriate spot rate:
£ 80
£ 80
£ 1, 080
+
+
2
1 + y1 (1 + y2 )
(1 + y3 )3
= £ 960.41.
fair value =
3
Credit risk
So far we ignored default risk. As pointed out in lecture 11, bonds are rated by rating agencies
in categories of default risk. These ratings provide benchmarks for valuing bonds based on
the prevailing government bond yields and spreads for the respective rating. Figure 8 plots
5
This example is taken from the readings: Bodie, Kane, and Marcus (2008), Chapter 15.
11
6
December 13, 2004
A Corporate Bonds
5.5
AA Corporate Bonds
AAA Corporate Bonds
5
spread
Yield (%)
4.5
4
3.5
US Treasury Bonds
3
2.5
2
2
5
10
20
30
Maturity
Figure 8: US yield curves for different credit ratings 2004 (source: Yahoo!Finance)
6.5
January 10, 2008
A Corporate Bonds
6
5.5
AA Corporate Bonds
AAA Corporate Bonds
Yield (%)
5
4.5
4
AAA spread
3.5
3
2.5
US Treasury Bonds
2
2
5
10
20
Maturity
Figure 9: US yield curves for different credit ratings 2008 (source: Yahoo!Finance)
the yield curves for US treasury bonds and corporate bonds with ratings AAA, AA, and A
for December 2004.6 A comparison with the situation in January 2008 in Figure 9 shows
that the risk premium inherent in corporate bond yields has risen significantly. This is a
consequence of the recent turmoil in financial markets (subprime crisis which spread to the
UK with the sinking of Northern Rock). Many financial institutions had invested in the
structured products exposed to the problems in subprime mortage markets. However, the
investment strategies used and the assets involved tend to be very complex, so that there
was and still is much uncertainty in the marekt as to the exposure of individual financial
institutions. This uncertainty has made banks reluctant to lend to each other: first, because
no one could know for certain whether the bank one is lending money to will not find losses
6
Data source: Yahoo!Finance http://bonds.yahoo.com/rates.html.
12
Crisis in the UK Interbank Market
7
Northern Rock crisis
1M LIBOR
Base rate
6.5
6
interest rate (%pa)
5.5
5
4.5
4
3.5
Central banks inject liquidity
3
2.5
Jan-08
Nov-07
Jul-07
Sep-07
Mar-07
May-07
Jan-07
Nov-06
Sep-06
Jul-06
Mar-06
May-06
Jan-06
Nov-05
Sep-05
Jul-05
May-05
Jan-05
Mar-05
Nov-04
Jul-04
Sep-04
May-04
Jan-04
Mar-04
2
Figure 10: Crisis in the UK interbank money market (source: Data Stream)
related to the subprime crisis; second, because a bank itself may find it needs the extra cash
and may then have problems finding funds in the interbank market. This has lead to a crisis
in interbank money markets, as illustrated in the large jump in spread between the Bank
of England base rate7 and the 1 month interbank offered rate (Libor) shown in Figure 10.
The following newspaper quote from the Telegraph illustrates this: “The three-month London
interbank offered rate (Libor) dropped 10.375 basis points to 5.89pc - just 39 basis points above
the current base rate of 5.5pc. The spread has not been so narrow since before the crisis erupted
on August 9. At its height in September, Libor was 6.9pc - 115 points above the then base rate
of 5.75pc. Short-term borrowing costs have fallen sharply since central banks in the US, UK,
Switzerland, Canada and the eurozone last month pledged to pump up to $ 600bn (£ 300m)
of emergency funding into the markets. The measures were taken to relieve pressure in the
system and see the banks through the tricky year-end period, when many were unwilling to
lend to protect their capital positions.” (source: Telegraph online January 4th 2008, http:
//www.telegraph.co.uk/money/main.jhtml?xml=/money/2008/01/03/cnlibor103.xml).
References
Bodie, Zvi, Alex Kane, and Alan J. Marcus, 2008, Investments (Irwin McGraw-Hill: Chicago).
Elton, Edwin J., and Martin J. Gruber, 1995, Modern Portfolio Theory and Investment Analysis (Wiley: New York).
7
The rate paid by the Bank of England on commercial bank reserves held with it.
13