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.
1
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
2
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
3
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