Question

Q. Suppose you buy a 30-year, 7.5% (annual payment) coupon bond when its yield to maturity is 7.67% and you plan to hold it for 20 years. Your forecast is that the bond’s yield to maturity will be 8% when it is sold and that the reinvestment rate on the coupons will be 6% for the first 10 years and 7% for the next 10 years.

a. What is the initial price of the bond when you buy it?

b. What will be the sales price of the bond after 20 years?

c. What will be the realized compound return?

Answer 1

a) Suppose the face value of the bond is P. Then the annual coupon is 0.075*P and the redemption value is P.

By the bond pricing formula,

Purchase price = 0.075*P/1.0767 +0.075*P/1.0767² + ..... + 0.075*P/1.0767³⁰ + P/ 1.0767³⁰

= 0.069657* P + 0.064695 *P + ..... + 0.00817*P + 0.108933*P

=0.9803* P

So, the bond will be priced at apx 98.03%of the face value.

b) After 20 years when the bond is sold, it will have 10 years to maturity and a YTM of 8%

So ,we have

Selling price after 20 years = 0.075*P/1.08 +0.075*P/1.08² + ..... + 0.075*P/1.08¹⁰ + P/ 1.08¹⁰

= 0.069444* P + 0.0643*P + .... +0.03474*P + 0.4632*P

= 0.9664 * P

Selling price of bond will be 96.64% of the face value.

c) The first coupon payment of 0.075*P (received at the end of 1st year) shall be re-invested at a rate of 6% for 9 years and then again at 7% for the next 10 years

So, at the end of 20 years,1st coupon amount will accumulate to = 0.075*P * 1.06⁹ * 1.07¹⁰

Similarly, the 2nd  coupon amount will accumulate to = 0.075*P * 1.06⁸ * 1.07¹⁰

^...

the 9th coupon amount will accumulate to = 0.075*P * 1.06¹ * 1.07¹⁰

the 10th coupon amount will accumulate to = 0.075*P * 1.07¹⁰

the 11th coupon amount will accumulate to = 0.075*P * 1.07⁹

^...

the 19th coupon amount will accumulate to = 0.075*P * 1.07¹

the 20th coupon amount will accumulate to = 0.075*P (just received at the end of 20 years)

Final selling price = 0.9664 * P (just received at the end of 20 years)

So,total value accumulated at the end of 20 years

= 0.075*P * 1.06⁹ * 1.07¹⁰ + ......+0.075*P * 1.06¹ * 1.07¹⁰+ 0.075*P * 1.07¹⁰+ ......+ 0.075*P + 0.9664 *P

= 3.94728 * P

If r is the realised compound return , then

Invested amount * (1+r)²⁰ = accumulated amount

=> 0.99803* P * (1+r)²⁰ = 3.94728 *P

=> (1+r)²⁰ = 4.0268

=> 1+r = 4.0268^(1/20) = 1.072131

=> r =   7.21% which is the realised compound return