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# Zero Coupon Bonds: Calculating Price, Interest, and Value

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1) You purchased a zero-coupon bond that has a face value of \$1,000, five years to maturity and a yield to maturity of 7.3%. It is one year later and similar bonds are offering a yield to maturity of 8.1%. You will sell the bond now. You have a tax rate of 40% on regular income and 15% on capital gains. Calculate the following for this bond.

a) The purchase price of the bond
b) The current price of the bond
c) The imputed interest income
d) The capital gain (or loss) on the bond
e) The before-tax rate of return on this investment
f) The after-tax rate of return on this investment

2) You have purchased a bond for \$973.02. The bond has a coupon rate of 6.4%, pays interest annually, has a face value of \$1,000, 4 years to maturity, and a yield to maturity of 7.2%. The bond's duration is 3.6481 years. You expect that interest rates will fall by .3% later today.

* Use the modified duration to find the approximate percentage change in the bond's price. Find the new price of the bond from this calculation.
* Use your calculator to do the regular present value calculations to find the bond's new price at its new yield to maturity.
* What is the amount of the difference between the two answers?

Why are your answers different? Explain the reason in words and illustrate it graphically.

#### Solution Preview

See attached file for graph.

1) You purchased a zero-coupon bond that has a face value of \$1,000, five years to maturity and a yield to maturity of 7.3%. It is one year later and similar bonds are offering a yield to maturity of 8.1%. You will sell the bond now. You have a tax rate of 40% on regular income and 15% on capital gains. Calculate the following for this bond.

a) The purchase price of the bond
The purchase price is the PV of the face value of the bond discounted at yield to maturity.
Face Value \$1,000
Maturity 5 years
YTM 7.30%

Value of the Bond \$703.07

b) The current price of the bond
Face Value \$1,000
Remaining Time to maturity 4 years
YTM 8.10%

Value of the Bond \$732.31

c) The imputed interest income
Price at original YTM 754.3990141
Imputed Interest Income ...

#### Solution Summary

The solution uses the provided information for bonds to calculate the purchase price of given bonds, current price of bonds, inputed interest income, capital gain or loss, before-tax rate of return, and after-tax rate of return on investment.

\$2.19

## A) What is the annual yield-to-maturity of each bond if the yield is compounded annually? b) Consider the prices of zero coupon bonds above, what is the fair price of a coupon bond that pays a coupon of \$10 at year 1, \$10 at year 2, \$10 at year 3 and repays the principal of \$100 at year 3? c) What is the annual yield-to-maturity of the coupon bond in b)?

These are the prices for zero coupon bonds per \$1,000 of face value:
Maturity Price
1 yr \$951.5
2 yr \$895.6
3 yr \$825.4

Show all work and formulae
a) What is the annual yield-to-maturity of each bond if the yield is compounded annually? Show work and formulas

b) Consider the prices of zero coupon bonds above, what is the fair price of a coupon bond that pays a coupon of \$10 at year 1, \$10 at year 2, \$10 at year 3 and repays the principal of \$100 at year 3?

c) What is the annual yield-to-maturity of the coupon bond in b)?

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