Chapter 5b Recommended End-of-Chapter Problems and Solutions
9. Calculate the duration of a $1,000 4-year bond with an 8% coupon (annual payments) that
is currently selling at par. Assume the length of each discount period is 1 year.
The following grid is useful for the duration calculation:
________________________________________________________________
1
2
3
4
PV at Market
Period
Cash Flows
Rate of 8%
1x3
1
$80
$ 74.07
$ 74.07
2
$80
$ 68.59
$ 137.18
3
$80
$ 63.81
$ 190.53
4
$1,080
$ 793.83
$3,175.32
Price =
$1,000.00
$3,577.10
Duration =
Summation of time weighted PVs $3,577.10
=
= 3.577 years
Price
$1,000
Early cash flows (high reinvestment risk) will be weighted at a low value, thus lowering duration. If
the bond is held 3.577 years, the investor will earn the yield to maturity, 8%. If held to maturity, price risk is
eliminated, but realized yield will be higher/lower than 8% depending on reinvestment rates.
12. Calculate the duration for a $1000, 4-year bond with a 4.5% annual coupon, currently
selling at par. Use the duration to estimate the percentage change in the bond’s price for a decrease in
the market interest rate to 3.5%. Assume the length of each discount period is 1 year.
1
Period
1
2
3
4
2
Cash Flows
$45
$45
$45
$1045
3
PV at 4.5%
4
1x3
$43.06
$41.21
$39.43
$876.30
$1000.00
$43.06
$82.42
$118.29
$3505.20
$3748.97
5
PV at 3.5%
$43.47
$42.01
$40.59
$910.66
$1036.73
Duration is the sum of the discounted, time-weighted cash flows divided by the price of the bond: $3748.97/
$1000 or 3.75 years. Per equation 5.8, the estimated price change is (-3.75)(-0.01/1.045)(100) =3.588% which
implies $1,035.88. However, the actual price change to $1,036.73 or 3.673%. The 0.085% or 85 cents
difference is due to convexity.
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a.
Suppose a 6% $10,000 bond (whose last semi-annual coupon occurred 42 days ago) is
quoted on the web at 96.4578. How much would 4000 of them cost assuming 181 days in the current
coupon period?
42(300) 

Cost  4000 9645.78 
181 

 38,861,573.04
What would be the error if in our calculations we were off by one day in our count since the
last coupon payment?
4000*1*300/181 = $6,629.83
b.
Assume a 4.5% bond (next semi-annual coupon payment in 52 days) whose Full Price
is $412.34772. What is the bond’s Clean Price assuming 183 days in the current coupon period?
131(22.50)
183
 396.2411 (or in the media 39.62411)
Clean Price  412.34772 
c.
What is the duration of a 6% bond (semi-annual coupon payments) that matures 27
months for now whose required yield is 7%? Just succinctly set up the math and load with parameter
values. Assume that the length of each discount period is 0.5 year.
Use
D
 time-weighted PV of each cash flow
PB
Then
.25(30)
.75(30) 1.25(30) 1.75(30) 2.25(1030)




.5
(1.035)
(1.035)1.5 (1.035) 2.5 (1.035)3.5 (1.035) 4.5
D
PB
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d.
This exercise is about the versatility of the ordinary annuity formula. What is the
monthly payment on a 4%, 15-year, $160,000 mortgage? Solve the ordinary annuity formula for A,
then load.
1  (1  r )  n 
PV  A 

r


A
A
r PV
1  (1  r )  n
.04 12  (160000)
1  (1  .04 12)  (15*12)
ans. $1,183.50
e.
Consider a $100,000 bond portfolio, $20,000 of which is in D = 4 bonds,
$50,000 of which is in D = 6 bonds, and $30,000 of which is in D = 7 bonds. What is the
portfolio’s duration? To raise the portfolio’s duration to 6.4, how much of D = 4 bonds
should be sold to buy D = 7 bonds?
Portfolio D = .2(4) + .5(6) + .3(7) = 5.9 years
6.4 = (.2 –x)(4) + .5(6) + (.3 + x)(7)
= .8 -4x + 3.0 + 2.1 +7x
0.5 = 3x
x = .5/3
= .16667
or
$16,667
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