Bond Prices, Interest Rate Changes, and Size of Bond Coupon
Question: Consider four bonds each with semiannual
payments, a 10year maturity and a face value of $1,000 redeemed at maturity. The bonds differ in semiannual coupon –
$100, $50, $25, and $0.
What are bond values at issue date if the yield to maturity
is 3 percent?
What are bond values at issue date if the yield to maturity
is 6 percent?
How did the change in interest rates impact the change in
bond prices across the four bonds?
What are the durations of the four bonds?
How do bond durations appear to vary with changes in bond
prices stemming from an interest rate change in this example?
Calculations:
Here are the bond prices at a 3 percent interest rate.
Bond Prices at 3.0 percent interest rate


Maturity SemiAnnual Payments

20

20

20

20

Coupon Payment

100

50

25

0

Par Value

1000

1000

1000

1000

YTM

0.03

0.03

0.03

0.03

PV

$2,459.33

$1,600.90

$1,171.69

$742.47

Here are bond prices at 6 percent interest rate.
Bond Prices at 6 percent interest rate


Maturity SemiAnnual Payments

20

20

20

20

Coupon Payment

100

50

25

0

Par Value

1000

1000

1000

1000

YTM

0.06

0.06

0.06

0.06

PV

$2,041.42

$1,297.55

$925.61

$553.68

Here are bond prices changes for when the interest rate goes
from 3 percent to 6 percent.
Change in bond prices when interest rates go from 3 to 6 percent


PV @ 3 percent

$2,459.33

$1,600.90

$1,171.69

$742.47

PV @ 6 percent

$2,041.42

$1,297.55

$925.61

$553.68

Diff

($417.91)

($303.35)

($246.07)

($188.79)

Percent Diff.

17.0%

18.9%

21.0%

25.4%

Here are bond duration estimates (weighted average of times
cash flow is received) for the four bonds.
Bond Duration When Interest Rates are at 3 percent


Settlement Date

1Jan18

1Jan18

1Jan18

1Jan18

Maturity Date

1Jan28

1Jan28

1Jan28

1Jan28

Coupon rate

10.00%

5.00%

2.50%

0.00%

Yield

3.00%

3.00%

3.00%

3.00%

Frequency

2

2

2

2

Duration

7.32

8.17

8.88

10.00

Note that the higher duration bonds realized a large percent
decline in bond price when interest rates rose from 3 percent to 6 percent.
Notes:
The bond price is the negative of the PV of the cash flows. The PV function itself provides a negative
number reflecting that the investor is paying for the bond and at the time of
purchase money is leaving the investor’s hands.
A bond duration is the weighted average of times when the
cash flows of the bonds are received. The
duration of a zerocoupon bond is equal to its maturity as there is only one
cash flow date, the maturity of the loan.
Bonds with higher coupon rates have lower durations because
cash flows are received earlier. An
increase in the coupon rate leads to lower duration.
Bond durations are a measure of the sensitivity of the price
of bonds to interest rate changes. When
interest rate changes are very small the duration can provide a close estimate
to the impact of a change in interest rates on bond prices. This approximation
is less exact for large interest rate changes.
Authors Note: Please
consider reading my finance blog. Start
with this essay on problems with the U.S. retirement system.
Also, my book on student debt is available at Kindle and at
Amazon. Please go here and purchase it.
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