Question

The following is a list of prices for zero-coupon bonds with different maturities but same par value of $1,000: the 1-year zero bond sells at $925.15, the 2-year zero bond sells at $862.57, the 3-year zero bond sells at $788.66, and the 4-year zero bond sells at $711.00. You have purchased a 4-year maturity bond with a 9% coupon rate paid annually. The bond has a par value of $1,000. What would be the price of the bond one year from now if the implied forward rates stay the same? (Please, do not round intermediate calculations. Round the final answer only to 2 decimal places.

Answer 1

In case of zero coupon bond, the spot rate is the yield to maturity.

Price of zero coupoon bond = Par value / (1 + YTM)^n

Where,

n = number of years

from above we can calculate the YTM of the zero coupon bond which is spot rates

1 year Zero coupon bond

$925.15 = $1,000 / (1 + YTM) ; YTM = 8.09058%

2 year Zero coupon bond

$862.57 = $1,000 / (1 + YTM)^2 ; YTM = 7.672011%

3 year Zero coupon bond

$788.66 = $1,000 / (1 + YTM)^3 ; YTM = 8.235583%

4 year Zero coupon bond

$711 = $1,000 / (1 + YTM)^4 ; YTM = 8.901184%

This are the spot rate of the respective years

Now we will calculate the forward price from the post rate

f(1,1) which represet forward rate after one year for one year which is calculated as follows:

1 + f(1,1) = (1.07672011)^2 / (1.08090580)

f(1,1) = 7.255063%

so as

[ 1 + f(1,2) ]^2 = (1.08235583)^3 / (1.08090580)

f(1,2) = 8.308158%

[ 1 + f(1,3) ]^3 = (1.08901184)^4 / (1.08090580)

f(1,3) = 9.172734%

now we have the forward rate we can calculate the price of the bond after 1 year using forward rates as discount rates.

Year	Cash flows	Discounting factor using forward rate	Present value
2	$       90.00	0.932356918	1/(1.07255063)^1	$              83.91
3	$       90.00	0.852467166	1/(1.08308158)^2	$              76.72
4	$       90.00	0.768524011	1/(1.09172734)^3	$              69.17
4	$ 1,000.00	0.768524011	1/(1.09172734)^3	$            768.52
Value of bond after one year	$            998.33

therefor value of the bond after one year will be $998.33.