WEB EXTENSION
5C
A Closer Look at Bond Risk:
Duration
T
his extension explains how to manage the risk of a bond portfolio using the con
cept of duration.
5.1 B
OND
R
ISK
In our discussion of bond valuation in Chapter 5, we discussed interest rate and rein
vestment rate risk.
Interest rate (price) risk
is the risk that the price of a debt secu
rity will fall as a result of increases in interest rates, and
reinvestment rate risk
is the
risk of earning a less than expected return when debt principal or interest payments
are reinvested at rates that are lower than the original yield to maturity.
To illustrate how to reduce interest rate and reinvestment rate risks, we will consider
a firm that is obligated to pay a worker a lumpsum retirement benefit of $10,000 at the
end of 10 years. Assume that the yield curve is horizontal, the current interest rate on all
Treasury securities is 9%, and the type of security used to fund the retirement benefit is
Treasury bonds. The present value of $10,000, discounted back 10 years at 9%, is
$10,000(0.4224) = $4,224. Therefore, the firm could invest $4,224 in Treasury bonds
and expect to be able to meet its obligation 10 years hence.
1
Suppose, however, that interest rates change from the current 9% rate immedi
ately after the firm has bought the Treasury bonds. How will this affect the situation?
The answer is,
“
It all depends.
”
If rates fall, then the value of the bonds in the port
folio will rise, but this benefit will be offset to a greater or lesser degree by a decline
in the rate at which the coupon payment of 0.09($4,224) = $380.16 can be reinvested.
The reverse would hold if interest rates rose above 9%. Here are some examples (for
simplicity, we assume annual coupons).
1.
The firm buys $4,224 of 9%, 10year maturity bonds; rates fall to 7% immediately
after the purchase and remain at that level:
Portfolio value at
the end of 10 years
¼
Future value of
10 interest payments
of $380
:
16 each
compounded at 7%
þ
Maturity
value
¼
$5
;
252
þ
$4
;
224
¼
$9
;
476
Therefore, the firm cannot meet its $10,000 obligation, and it must contribute
additional funds.
1
For the sake of simplicity, we assume that the firm can buy a fraction of a bond.
1
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The firm buys $4,224 of 9%, 40year bonds; rates fall to 7% immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼
$5
;
252
þ
Value of
30year
9% bonds
when r
d
¼
7%
¼
$5
;
252
þ
$5
;
272
¼
$10
;
524
In this situation, the firm has excess capital at the end of the 10year period.
3.
The firm buys $4,224 of 9%, 10year bonds; rates rise to 12% immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼
Future value of
10 interest payments
of $380
:
16 each
compounded at 12%
þ
Maturity
value
¼
$6
;
671
þ
$4
;
224
¼
$10
;
895
This situation also produces a funding surplus.
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 Spring '10
 MR.Wroshr
 Interest Rates, duration

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