Question

2. You own a 20-year, $1000 face value bond paying 8% coupon annually. What should be the market price of the bond so that its Yield to Maturity is exactly 10%?

You also own a 30-year, $1000 face value bond paying 9% coupon annually. If your required rate of return is 9%, what should be the value of the bond?

Answer 1

The market price of the bond is computed as shown below:

= $ 80 / 1.10¹ + $ 80 / 1.10² + $ 80 / 1.10³ + $ 80 / 1.10⁴ + $ 80 / 1.10⁵ + $ 80 / 1.10⁶ + $ 80 / 1.10⁷ + $ 80 / 1.10⁸ + $ 80 / 1.10⁹ + $ 80 / 1.10¹⁰ + $ 80 / 1.10¹¹ + $ 80 / 1.10¹² + $ 80 / 1.10¹³ + $ 80 / 1.10¹⁴ + $ 80 / 1.10¹⁵ + $ 80 / 1.10¹⁶ + $ 80 / 1.10¹⁷ + $ 80 / 1.10¹⁸ + $ 80 / 1.10¹⁹ + $ 80 / 1.10²⁰ + $ 1,000 / 1.10²⁰

= $ 829.73 Approximately

The value of the bond is computed as follows:

= $ 90 / 1.09¹ + $ 90 / 1.09² + $ 90 / 1.09³ + $ 90 / 1.09⁴ + $ 90 / 1.09⁵ + $ 90 / 1.09⁶ + $ 90 / 1.09⁷ + $ 90 / 1.09⁸ + $ 90 / 1.09⁹ + $ 90 / 1.09¹⁰ + $ 90 / 1.09¹¹ + $ 90 / 1.09¹² + $ 90 / 1.09¹³ + $ 90 / 1.09¹⁴ + $ 90 / 1.09¹⁵ + $ 90 / 1.09¹⁶ + $ 90 / 1.09¹⁷ + $ 90 / 1.09¹⁸ + $ 90 / 1.09¹⁹ + $ 90 / 1.09²⁰ + $ 90 / 1.09²¹ + $ 90 / 1.09²² + $ 90 / 1.09²³ + $ 90 / 1.09²⁴ + $ 90 / 1.09²⁵ + $ 90 / 1.09²⁶ + $ 90 / 1.09²⁷ + $ 90 / 1.09²⁸ + $ 90 / 1.09²⁹ + $ 90 / 1.09³⁰ + $ 1,000 / 1.09³⁰

= $ 1,000

This also proves the fact that whenever the coupon rate and the required rate of return is equal, the value of the bond is equal to the face value of the bond.

Feel free to ask in case of any query relating to this question

Answer 2

1/ 80(1-(1.1)^-20)/0.1 + 1000/1.1^20 = 829.728 

2/ coupon rate = YTM = 9% 

so PV = FV = 1000 

Answer 3

SOLUTION :

Yield = k = 10% = 0.1

=> (1 + k) = 1.1

n = 20 years

Coupon amount = 1000 *0.08 = 80 ($0 per year.

So,

Market price of the bond

= PV of future cash flows 

= Coupon amount  ( (1+k)^n - 1)/(k(1+k)^n)  + Face Value / (1+k)^n

= 80(1.1^20 - 1)/(0.1*1.1^20) + 1000/1.1^20

= 829.73 ($) (ANSWER).

For the second bond, coupon rate = yield rate = 9%

Hence, value of the bond 

= Face value 

= 1000 ($) (ANSWER).

[ NOTE : It can be checked as follows :

Value of the bond

= Coupon amount  ( (1+k)^n - 1)/(k(1+k)^n)  + Face Value / (1+k)^n

= 1000 *0.09(1.09^30 - 1)/(0.09*1.09^30) + 1000/1.09^30

= 1000 ($) (ANSWER) ]