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maturity date approaches.

Low-coupon (discount) bond

$850 Maturity date

0 10 20 30
Time (years)
Valuing Bonds 267

Plots of bond prices as a
function of the interest rate.
Long-term bond prices are
$2,500 30-year bond
more sensitive to the interest
3-year bond
rate than prices of short-term

Bond price



0 2% 4% 6% 8% 10%
Interest rate

We have seen that bond prices fluctuate as interest rates change. In other words, bonds
exhibit interest rate risk. Bond investors cross their fingers that market interest rates
will fall, so that the price of their bond will rise. If they are unlucky and the market in-
The risk in bond prices due
terest rate rises, the value of their investment falls.
to fluctuations in interest
But all bonds are not equally affected by changing interest rates. Compare the two
curves in Figure 3.6. The red line shows how the value of the 3-year, 6 percent coupon
bond varies with the level of the interest rate. The blue line shows how the price of a
30-year, 6 percent bond varies with the level of interest rates. You can see that the 30-
year bond is more sensitive to interest rate fluctuations than the 3-year bond. This
should not surprise you. If you buy a 3-year bond when the interest rate is 5.6 percent
and rates then rise, you will be stuck with a bad deal”you have just loaned your money
at a lower interest rate than if you had waited. However, think how much worse it would
be if the loan had been for 30 years rather than 3 years. The longer the loan, the more
income you have lost by accepting what turns out to be a low coupon rate. This shows
up in a bigger decline in the price of the longer-term bond. Of course, there is a flip side
to this effect, which you can also see from Figure 3.6. When interest rates fall, the
longer-term bond responds with a greater increase in price.

Suppose that the interest rate rises overnight from 5.6 percent to 10 percent. Calculate
Self-Test 6
the present values of the 6 percent, 3-year bond and of the 6 percent, 30-year bond both
before and after this change in interest rates. Confirm that your answers correspond
with Figure 3.6. Use your financial calculator.

Look back for a moment to Figure 3.2. The U.S. Treasury bonds are arranged in order
of their maturity. Notice that the longer the maturity, the higher the yield. This is usu-
ally the case, though sometimes long-term bonds offer lower yields.

The yield curve. A plot of Treasury Yield Curve
yield to maturity as a Yields as of 4:30 p.m. Eastern time
function of time to maturity 7
for Treasury bonds on July
23, 1999.

Yield to maturity (%)

July 23, 1999
Dec. 31, 1997
4 Dec. 31, 1996

3 6 1 2 5 10 30
mos. yr. maturities

In addition to showing the yields on individual bonds, The Wall Street Journal also
shows a daily plot of the relationship between bond yields and maturity. This is known
as the yield curve. You can see from the yield curve in Figure 3.7 that bonds with 3
Graph of
months to maturity offered a yield of about 4.75 percent; those with 30 years of matu-
the relationship between time
rity offered a yield of just over 6 percent.
to maturity and yield to
Why didn™t everyone buy long-maturity bonds and earn an extra 1.25 percentage
points? Who were those investors who put their money into short-term Treasuries at
only 4.75 percent?
Even when the yield curve is upward-sloping, investors might rationally stay away
from long-term bonds for two reasons. First, the prices of long-term bonds fluctuate
much more than prices of short-term bonds. Figure 3.6 illustrates that long-term bond
prices are more sensitive to shifting interest rates. A sharp increase in interest rates
could easily knock 20 or 30 percent off long-term bond prices. If investors don™t like
price fluctuations, they will invest their funds in short-term bonds unless they receive a
higher yield to maturity on long-term bonds.
Second, short-term investors can profit if interest rates rise. Suppose you hold a 1-
year bond. A year from now when the bond matures you can reinvest the proceeds and
enjoy whatever rates the bond market offers then. Rates may be high enough to offset
the first year™s relatively low yield on the 1-year bond. Thus you often see an upward-
sloping yield curve when future interest rates are expected to rise.

Earlier we drew a distinction between nominal and real rates of interest. The cash flows
on the 6 percent Treasury bonds are fixed in nominal terms. Investors are sure to receive
an interest payment of $60 each year, but they do not know what that money will buy
them. The real interest rate on the Treasury bonds depends on the rate of inflation. For
Valuing Bonds 269

example, if the nominal rate of interest is 5.6 percent and the inflation rate is 3 percent,
then the real interest rate is calculated as follows:
1 + nominal interest rate 1.056
(1 + real interest rate) = = = 1.0252
1 + inflation rate 1.03
Real interest rate = .0252, or 2.52%
Since the inflation rate is uncertain, so is the real rate of interest on the Treasury bonds.
You can nail down a real rate of interest by buying an indexed bond, whose payments
are linked to inflation. Indexed bonds have been available in some countries for many
years, but they were almost unknown in the United States until 1997 when the U.S.
Treasury began to issue inflation-indexed bonds known as Treasury Inflation-Protected
Securities, or TIPS. The cash flows on TIPS are fixed, but the nominal cash flows (in-
terest and principal) are increased as the consumer price index increases. For example,
suppose the U.S. Treasury issues 3 percent, 2-year TIPS. The real cash flows on the 2-
year TIPS are therefore
Year 1 Year 2
Real cash flows $30 $1,030

The nominal cash flows on TIPS depend on the inflation rate. For example, suppose in-
flation turns out to be 5 percent in Year 1 and a further 4 percent in Year 2. Then the
nominal cash flows would be
Year 1 Year 2
$30 — 1.05 = $31.50 $1,030 — 1.05 — 1.04 = $1,124.76
Nominal cash flows

These cash payments are just sufficient to provide the holder with a 3 percent real rate
of interest.
As we write this in mid-1999, three-year TIPS offer a yield of 3.9 percent. This yield
is a real interest rate. It measures the amount of extra goods your investment will allow
you to buy. The 3.9 percent real yield on TIPS is 1.7 percent less than the 5.6 percent
yield on nominal Treasury bonds.5 If the annual inflation rate proves to be higher than
1.7 percent, you will earn a higher return by holding TIPS; if the inflation rate is lower
than 1.7 percent, the reverse will be true. The nearby box discusses the case for invest-
ments in TIPS.
Real interest rates depend on the supply of savings and the demand for new invest-
ment. As this supply“demand balance changes, real interest rates change. But they do
so gradually. We can see this by looking at the United Kingdom, where the government
has issued indexed bonds since 1982. The red line in Figure 3.8 shows that the (real) in-
terest rate on these bonds has fluctuated within a relatively narrow range.
Suppose that investors revise upward their forecast of inflation by 1 percent. How
will this affect interest rates? If investors are concerned about the purchasing power of
their money, the changed forecast should not affect the real rate of interest. The nomi-
nal interest rate must therefore rise by 1 percent to compensate investors for the higher
inflation prospects.
The blue line in Figure 3.8 shows the nominal rate of interest in the United Kingdom
since 1982. You can see that the nominal rate is much more variable than the real rate.
When inflation concern was near its peak in the early 1980s, the nominal interest rate

Real and nominal yields to
maturity on government
bonds in the United 14 Nominal Yield
Kingdom. Real Yield

Yield to maturity (%)





1/29/82 1/29/85 1/29/88 1/29/91 1/29/94 1/29/97

was almost 10 percent above the real rate. As we write this in mid-1999, inflation fears
have eased and the nominal interest rate in the United Kingdom is only 21„2 percent above
the real rate.

Our focus so far has been on U.S. Treasury bonds. But the federal government is not the
only issuer of bonds. State and local governments borrow by selling bonds.6 So do cor-
porations. Many foreign governments and corporations also borrow in the United
States. At the same time U.S. corporations may borrow dollars or other currencies by
issuing their bonds in other countries. For example, they may issue dollar bonds in Lon-
don which are then sold to investors throughout the world.
There is an important distinction between bonds issued by corporations and those is-
sued by the U.S. Treasury. National governments don™t go bankrupt”they just print
more money.7 So investors do not worry that the U.S. Treasury will default on its bonds.
However, there is some chance that corporations may get into financial difficulties and
may default on their bonds. Thus the payments promised to corporate bondholders rep-
resent a best-case scenario: the firm will never pay more than the promised cash flows,
but in hard times it may pay less.
The risk that a bond issuer may default on its obligations is called default risk (or
credit risk). It should be no surprise to find that to compensate for this default risk
companies need to promise a higher rate of interest than the U.S. Treasury when bor-
rowing money. The difference between the promised yield on a corporate bond and the
RISK The risk that a bond
issuer may default on its

6 These municipal bonds enjoy a special tax advantage; investors are exempt from federal income tax on the
coupon payments on state and local government bonds. As a result, investors are prepared to accept lower
yields on this debt.
7 But they can™t print money of other countries. Therefore, when a foreign government borrows dollars, in-

vestors worry that in some future crisis the government may not be able to come up with enough dollars to
repay the debt. This worry shows up in the yield that investors demand on such debt. For example, during the
Asian financial crisis in 1998, yields on the dollar bonds issued by the Indonesian government rose to 18 per-
centage points above the yields on comparable U.S. Treasury issues.

A New Leader in the Bond Derby?
economy falls into deflation, you™ll get the face value of
With Wall Street pundits fixated on deflation, the idea of
the bonds back at maturity.


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