Question

Bond X is a premium bond making semiannual payments. The bond has a coupon rate of 7.5 percent, a YTM of 6 percent, and 13 years to maturity. Bond Y is a discount bond making semiannual payments. This bond has a coupon rate of 6 percent, a YTM of 7.5 percent, and also 13 years to maturity. Assume the interest rates remain unchanged and a $1,000 par value.
Required:
a. What are the prices of these bonds today?
b. What do you expect the prices of these bonds to be in one year?
c. What do you expect the prices of these bonds to be in three years?
d. What do you expect the prices of these bonds to be in eight years?

Answer 1

Answer:

a. What are the prices of these bonds today?

price of bond X:

0.03 = {37.5 + [(1,000 - MV)/26]} /  [(1,000 + MV)/2]

0.03 x [(1,000 + MV)/2] = 37.5 + [(1,000 - MV)/26]

0.03 x (500 + 0.5MV) = 37.5 + 38.46 - 0.03846MV

15 + 0.015MV = 75.96 - 0.03846MV

0.05346MV = 60.96

MV = 60.96 / 0.05346 = $1,140.29

price of bond Y:

0.0375 = {30 + [(1,000 - MV)/26]} /  [(1,000 + MV)/2]

0.0375 x [(1,000 + MV)/2] = 30 + [(1,000 - MV)/26]

0.0375 x (500 + 0.5MV) = 30 + 38.46 - 0.03846MV

18.75 + 0.01875MV = 68.46 - 0.03846MV

0.05721MV = 49.71

MV = 49.71 / 0.05721 = $868.90

b. What do you expect the prices of these bonds to be in one year?

price of bond X:

0.03 = {37.5 + [(1,000 - MV)/24]} /  [(1,000 + MV)/2]

0.03 x [(1,000 + MV)/2] = 37.5 + [(1,000 - MV)/24]

0.03 x (500 + 0.5MV) = 37.5 + 41.67 - 0.04167MV

15 + 0.015MV = 79.17 - 0.04167MV

0.05667MV = 64.17/0.05667 = $1,132.29

price of bond Y:

0.0375 = {30 + [(1,000 - MV)/24]} /  [(1,000 + MV)/2]

0.0375 x [(1,000 + MV)/2] = 30 + [(1,000 - MV)/24]

0.0375 x (500 + 0.5MV) = 30 + 41.67 - 0.04167MV

18.75 + 0.01875MV = 71.67 - 0.04167MV

0.06042MV = 52.92

MV = 52.92 / 0.06042 = $875.87

c. What do you expect the prices of these bonds to be in three years?

price of bond X:

0.03 = {37.5 + [(1,000 - MV)/20]} /  [(1,000 + MV)/2]

0.03 x [(1,000 + MV)/2] = 37.5 + [(1,000 - MV)/20]

0.03 x (500 + 0.5MV) = 37.5 + 50 - 0.05MV

15 + 0.015MV = 87.5 - 0.05MV

0.065MV = 72.5

MV = 72.5 / 0.065 = $1,115.38

price of bond Y:

0.0375 = {30 + [(1,000 - MV)/20]} /  [(1,000 + MV)/2]

0.0375 x [(1,000 + MV)/2] = 30 + [(1,000 - MV)/20]

0.0375 x (500 + 0.5MV) = 30 + 50 - 0.05MV

18.75 + 0.01875MV = 80 - 0.05MV

0.06875MV = 61.25

MV = 61.251 / 0.06875 = $890.91

d. What do you expect the prices of these bonds to be in eight years?

price of bond X:

0.03 = {37.5 + [(1,000 - MV)/10]} /  [(1,000 + MV)/2]

0.03 x [(1,000 + MV)/2] = 37.5 + [(1,000 - MV)/10]

0.03 x (500 + 0.5MV) = 37.5 + 100 - 0.1MV

15 + 0.015MV = 137.5 - 0.1MV

0.115MV = 122.5

MV = 122.5 / 0.115 = $1,065.22

price of bond Y:

0.0375 = {30 + [(1,000 - MV)/10]} /  [(1,000 + MV)/2]

0.0375 x [(1,000 + MV)/2] = 30 + [(1,000 - MV)/10]

0.0375 x (500 + 0.5MV) = 30 + 100 - 0.1MV

18.75 + 0.01875MV = 130 - 0.1MV

0.11875V = 111.25

MV = 111.25 / 0.11875 = $936.84