Question

A 30-year maturity bond has a 6.7% coupon rate, paid annually. It sells today for $881.17. A 20-year maturity bond has a 6.2% coupon rate, also paid annually. It sells today for $893.1. A bond market analyst forecasts that in five years, 25-year maturity bonds will sell at yields to maturity of 7.7% and 15-year maturity bonds will sell at yields of 7.2%. Because the yield curve is upward sloping, the analyst believes that coupons will be invested in short-term securities at a rate of 7.2%. What rate of return does each bond offer over the 5-year period?

Answer 1

Answer:

Rate of return

30 year bond =  42%

20 year bond = 45%

Explanation:

First of all find current yield on 30 year maturity bond

We will use PV of annuity formula to calculate current YTM

Coupon Payment = 6.7% x 1000 = $67

$881.17 =( $67( 1- ( 1 + r )^-30 ) / r ) + ( 1000 / ( 1 + r )^30 )

r = 0.0773 = 7.73%

Current YTM is 7.73%

Now calculate the current yield for 20 years maturity bond

Coupon Payment = 6.2% x 1000 = $62

893.1 = ( ( $62 x ( 1 - ( 1 + r )^-20 ) / r ) + ( 1000 / ( 1 + r )^20 )

r = 0.0723 = 7.23%

As given

5 years from now the YTM on 30 Year bond will be 7.70% and on 20 Year bond will be 7.20%.

Now calculate

Price of the 30 year bond Bond after 5 year at YTM of 7.7%

Price of the Bond = ( $67 x ( 1 - ( 1 + 0.077 )^-(30-5) ) / 0.077 )+( 1000 / ( 1 + 0.077 )^(30-5) ) = $890.46

Price of the 20 year bond Bond after 5 year at YTM of 7.2%

Price of the Bond = ((6.7%*1000)*(1-(1+0.072)^-15)/0.072)+(1000/(1+0.072)^15)

( $62 x ( 1 - ( 1 + 0.072 )^-(20-5) ) / 0.072 )+( 1000 / ( 1 + 0.072 )^(20-5) ) = $910.06

Increase in price of 30 year bond = $890.46 - $881.17 = $9.29

Increase in price of 30 year bond = $910.06 - $893.1 = $16.96

Future value of Coupon payment for 5 years

30 year bond = 67 x ( 1.072^5 -1 ) / 0.072 = $386.84

20 year bond = 62 x ( 1.072^5 -1 ) / 0.072 = $357.97

Total return = FV of Coupon payment + Price increase

30 year bond = $386.84 + $9.29 = $396.13

20 year bond = $357.97 + $16.96 = $374.93

Rate of return =  

30 year bond = $396.13 / $881.17 = 0.45 = 45%

20 year bond = $374.93 / $893.1 = 0.42 = 42%