This preview shows pages 1–3. Sign up to view the full content.
In our discussion of interestrate risk, we saw that when interest rates change, a bond
with a longer term to maturity has a larger change in its price and hence more interest
rate risk than a bond with a shorter term to maturity. Although this is a useful general
fact, in order to measure interestrate risk, the manager of a financial institution needs
more precise information on the actual capital gain or loss that occurs when the inter
est rate changes by a certain amount. To do this, the manager needs to make use of the
concept of duration, the average lifetime of a debt security’s stream of payments.
The fact that two bonds have the same term to maturity does not mean that they
have the same interestrate risk. A longterm discount bond with ten years to maturity,
a socalled
zerocoupon bond
, makes all of its payments at the end of the ten years,
whereas a 10% coupon bond with ten years to maturity makes substantial cash pay
ments before the maturity date. Since the coupon bond makes payments earlier than
the zerocoupon bond, we might intuitively guess that the coupon bond’s
effective matu
rity
, the term to maturity that accurately measures interestrate risk, is shorter than it is
for the zerocoupon discount bond.
Indeed, this is exactly what we find in example 1.
APPLICATION
Rate of Capital Gain
Calculate the rate of capital gain or loss on a tenyear zerocoupon bond for which the
interest rate has increased from 10% to 20%. The bond has a face value of $1,000.
Solution
The rate of capital gain or loss is
2
49.7%.
g
5
where
P
t
1
1
5
price of the bond one year from now
5
5
$193.81
P
t
5
price of the bond today
5
5
$385.54
$1,000
(1
1
0.10)
10
$1,000
(1
1
0.20)
9
P
t
1
1
2
P
t
P
t
1
APPENDIX TO CHAPTER
4
Measuring InterestRate
Risk: Duration
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThus:
g
5
g
52
0.497
49.7%
But as we have already calculated in Table 2 in Chapter 4, the capital gain on the
10% tenyear coupon bond is
2
40.3%. We see that interestrate risk for the tenyear
coupon bond is less than for the tenyear zerocoupon bond, so the effective maturity
on the coupon bond (which measures interestrate risk) is, as expected, shorter than the
effective maturity on the zerocoupon bond.
Calculating Duration
To calculate the duration or effective maturity on any debt security, Frederick Macaulay,
a researcher at the National Bureau of Economic Research, invented the concept of
duration more than half a century ago. Because a zerocoupon bond makes no cash pay
ments before the bond matures, it makes sense to define its effective maturity as equal
to its actual term to maturity. Macaulay then realized that he could measure the effec
tive maturity of a coupon bond by recognizing that a coupon bond is equivalent to a
set of zerocoupon discount bonds. A tenyear 10% coupon bond with a face value of
$1,000 has cash payments identical to the following set of zerocoupon bonds: a $100
oneyear zerocoupon bond (which pays the equivalent of the $100 coupon payment
made by the $1,000 tenyear 10% coupon bond at the end of one year), a $100 two
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '08
 DISTAD

Click to edit the document details