W E B E X T E N S I O N 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 BOND RISK 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 lump-sum 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%, 10-year 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. 1For the sake of simplicity, we assume that the firm can buy a fraction of a bond. 1 2. The firm buys $4,224 of 9%, 40-year 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 30-year 9% bonds when rd ¼ 7% ¼ $5; 252 þ $5; 272 ¼ $10; 524 In this situation, the firm has excess capital at the end of the 10-year period. 3. The firm buys $4,224 of 9%, 10-year 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. 4. The firm buys $4,224 of 9%, 40-year bonds; rates rise to 12% immediately after the purchase a ...

1. W E B E X T E N S I O N 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 BOND RISK
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 lump-sum 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

2. 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%, 10-year 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

3. must contribute
additional funds.
1For the sake of simplicity, we assume that the firm can buy a
fraction of a bond.
1
2. The firm buys $4,224 of 9%, 40-year 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
30-year
9% bonds
when rd ¼ 7%
¼ $5; 252 þ $5; 272
¼ $10; 524
In this situation, the firm has excess capital at the end of the 10-
year period.
3. The firm buys $4,224 of 9%, 10-year bonds; rates rise to 12%
immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼

4. 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.
4. The firm buys $4,224 of 9%, 40-year bonds; rates rise to 12%
immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼ $6; 671 þ
Value of
30-year
9% bonds
when rd ¼ 12%
¼ $6; 671 þ $3; 203
¼ $9; 874
This time, a shortfall occurs.
Here are some generalizations drawn from the examples.
1. If interest rates fall and the portfolio is invested in relatively

5. short-term bonds,
then the reinvestment rate penalty exceeds the capital gains and
so a net short-
fall occurs. However, if the portfolio had been invested in
relatively long-term
bonds, then a drop in rates would produce capital gains that
would more than
offset the shortfall caused by low reinvestment rates.
2. If interest rates rise and the portfolio is invested in relatively
short-term bonds,
then gains from high reinvestment rates will more than offset
capital losses,
and the final portfolio value will exceed the required amount.
However, if the
portfolio had been invested in long-term bonds, then capital
losses would more
than offset reinvestment gains, and a net shortfall would result.
If a company has many cash obligations expected in the future,
then the
complexity of estimating the effects of interest rate changes is
obviously exac-
erbated. Still, methods have been devised to help deal with the
risks associated
with changing interest rates. Several methods are discussed in
the following
sections.
2 Web Extension 5C: A Closer Look at Bond Risk: Duration
5.2 IMMUNIZATION
Bond portfolios can be immunized against interest rate and
reinvestment rate risk,

6. much as people can be immunized against the flu. In brief,
immunization involves
selecting bonds with coupons and maturities such that the
benefits or losses from
changes in reinvestment rates are exactly offset by losses or
gains in the prices of
the bonds. In other words, if a bond’s reinvestment rate risk
exactly matches its inter-
est rate price risk, then the bond is immunized against the
adverse effects of changes
in interest rates.
To see what is involved, refer back to our example of a firm
that buys $4,224 of
9% Treasury bonds to meet a $10,000 obligation 10 years hence.
In the example, we
see that if the firm buys bonds with a 10-year maturity and
interest rates remain con-
stant, then the obligation can be met exactly. However, if
interest rates fall from 9%
to 7%, then a shortfall will occur because the coupons received
will be reinvested at a
rate of 7% rather than the 9% reinvestment rate required to
reach the $10,000 tar-
get. But suppose the firm had bought 40-year rather than 10-
year bonds. A decline in
interest rates would still have the same effect on the
compounded coupon payments,
but now the firm would hold 9% coupon, 30-year bonds in a 7%
market 10 years
hence, so the bonds would have a value greater than par. In this
case, the bonds
would have risen by more than enough to offset the shortfall in
compounded interest.
Can we buy bonds with a maturity somewhere between 10 and
40 years such that the

7. net effect of changes in reinvested cash flows and changes in
bond values at year 10
will always be positive? The answer is “yes,” as we explain
next.
5.3 DURATION
The key to immunizing a portfolio is to buy bonds that have a
duration equal to the
years until the funds will be needed. Duration cannot be defined
in simple terms like
maturity, but it can be thought of as the weighted average
maturity of all the cash
flows (coupon payments plus maturity value) provided by a
bond, and it is exception-
ally useful to help manage the risk inherent in a bond portfolio.
The duration for-
mula and an example of the calculation are provided below, but
first we present
some additional points about duration.
1. Duration is similar to the concept of discounted payback in
capital budgeting in
the sense that, the longer the duration, the longer funds are tied
up in the bond.
2. To immunize a bond portfolio, buy bonds that have a duration
equal to the
number of years until the funds will be needed. In our example,
the firm should
buy bonds with a duration of 10 years.
3. Duration is a measure of bond volatility. The percentage
change in the value of a
bond (or bond portfolio) will be approximately equal to the
bond’s duration mul-
tiplied by the percentage-point change in interest rates.2

8. Therefore, a 2-
percentage-point increase in interest rates will lower the value
of a bond with a
10-year duration by about 20%, but the value of a 5-year
duration bond will fall
by only 10%. As this example shows, if Bond A has twice the
duration of Bond B
then Bond A also has twice the volatility of Bond B. A
corporate treasurer (or any
other investor) who is concerned about declines in the market
value of his or her
2Actually, duration measures the negative of the percentage
change in bond price. Duration is a measure
of the bond price’s elasticity with respect to the interest rate,
but this elasticity measure is valid only for
small changes in the interest rate.
Web Extension 5C: A Closer Look at Bond Risk: Duration 3
portfolio should buy bonds with low durations. (This is
important even if the in-
vestor buys a bond mutual fund.)
4. The duration of a zero coupon bond is equal to its maturity,
but the duration of
any coupon bond is less than its maturity. (Remember, the
duration is a weighted
average maturity of the cash flows, the only cash flow from a
zero occurs at ma-
turity, and coupon bonds have cash flows prior to maturity.)
And with all else
held constant, the higher the coupon rate the shorter the
duration, because a

9. high-coupon bond provides significant early cash flows even if
it has a long
maturity.
Duration is calculated using this formula:
Duration ¼
∑
N
t¼1
tðCFtÞ
ð1 þ rdÞt
∑
N
t¼1
CFt
ð1 þ rdÞt
¼
∑
N
t¼1
tðCFtÞ
ð1 þ rdÞt
VB
(5C-1)
Here rd is the required return on the bond, N is the bond’s years
to maturity, t is
the year each cash flow occurs, and CFt is the cash flow in year

10. t (CFt = INT for t <
N and CFt = INT + M for t = N, where INT is the interest
payment and M is the
principal payment). Notice that the denominator of Equation
5C-1 is simply the
value of the bond, VB.
To simplify calculations in Excel, we define the present value
of cash flow t (PV of
CFt ) as
PV of CFt ¼
CFt
ð1 þ rdÞt
We can rewrite Equation 5C-1 as
Duration ¼
∑
N
t¼1
tðPVofCFtÞ
VB
(5C-2)
To illustrate the duration calculation, consider a 20-year, 9%
annual coupon
bond bought at its par value of $1,000. It provides cash flows of
$90 per year
for 19 years plus $1,090 in the 20th year. To calculate duration,
we used an Excel
model as shown in Figure 5C-1; see the worksheet Web 5C in

11. the file Ch05 Tool
Kit.xls for details.
Column 1 of Figure 5C-1 gives the year each cash flow occurs,
Column 2 gives the
cash flows, Column 3 shows the PV of each cash flow, and
Column 4 shows the product
of t and the PV of each cash flow. The sum the PVs in Column 3
is the value of
the bond, VB. The duration is equal to the value of the bond
divided by the sum of
Column 4.
This 20-year bond’s 9.95 duration is close to that of our
illustrative firm’s 10-year
liability. So, if the firm bought a portfolio of these 20-year
bonds and then reinvested
4 Web Extension 5C: A Closer Look at Bond Risk: Duration
the coupons as they came in, then the accumulated interest
payments, plus the
value of the bond after 10 years, would be close to or exceed
$10,000 regardless of
whether interest rates rose, fell, or remained constant at 9%. See
Ch05 Tool Kit.xls
for an example. There is also an Excel function for duration; the
Tool Kit also de-
monstrates its use.
Unfortunately, other complications arise. Our simple example
looked at a single
interest rate change that occurred immediately after funding. In
reality, interest rates

12. change every day, which causes bonds’ durations to change;
this, in turn, requires
that bond portfolios be rebalanced periodically to remain
immunized.
FIGURE 5C-1 Duration
7
A B C D E
Inputs
Years to maturity = 20
9.00%
$90.0
$1,080
9.00%
t
(1)
1
2
3
4
5
6
7
8
9
10
11
12
13

13. 14
15
16
17
18
19
20
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$1,090 $194.49
VB =
Duration = Sum of t(PV of CFt)/VB =
$1,000,00
Sum of

14. t(PV of CFt) =
$17.50
$19.08
$20.80
$22.67
$24.71
$26.93
$29.36
$32.00
$34.88
$38.02
$41.44
$45.17
$49.23
$53.66
$58.49
$63.76
$69.50
$75.75
$82.57 82.57
151.50
208.49
255.03
292.47
321.98
344.63
361.34
372.95
380.17
383.66
383.98
381.63
377.05

15. 370.63
362.69
353.54
343.43
332.58
3,889.79
$9,990.11
CFt
(2)
PV of CFt
(3)
t(PV of CFt)
(4)
Coupon rate =
Annual payment =
Par value = FV =
Going rate, r =
8
9
10
11
12
13
14
15

16. 16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39 9.95
resource
See the worksheet Web
5C in Ch05 Tool Kit.xls on
the textbook’s Web site.
Web Extension 5C: A Closer Look at Bond Risk: Duration 5