Question

​(Bond valuation​) You are examining three bonds with a par value of $1,000 ​(you receive $1,000 at​ maturity) and are concerned with what would happen to their market value if interest rates​ (or the market discount​ rate) changed. The three bonds are:

Bond A—a bond with 4 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually.

Bond B —a bond with 12 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually.

Bond C —a bond with 15 years left to maturity that has an annual coupon interest rate of 8 ​percent, but the interest is paid semiannually.

What would be the value of these bonds if the market discount rate were:

a. 8 percent per year compounded​ semiannually?

b. 6 percent per year compounded​ semiannually?

c. 17 percent per year compounded​ semiannually?

d. What observations can you make about these​ results?

Answer 1

a. Value of bonds if discount rate is 8% per year compounded semiannually:

Bond A = Number of years = 4 x 2 = 8

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 8% / 2 = 4%

= $ 40 / 1.04¹ + $ 40 / 1.04² + $ 40 / 1.04³ + $ 40 / 1.04⁴ + $ 40 / 1.04⁵ + $ 40 / 1.04⁶ + $ 40 / 1.04⁷ + $ 40 / 1.04⁸ + $ 1,000 / 1.04⁸

= $ 1,000

Bond B = Number of years = 12 x 2 = 24

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 8% / 2 = 4%

= $ 40 / 1.04¹ + $ 40 / 1.04² + $ 40 / 1.04³ + $ 40 / 1.04⁴ + $ 40 / 1.04⁵ + $ 40 / 1.04⁶ + $ 40 / 1.04⁷ + $ 40 / 1.04⁸ + $ 40 / 1.04⁹ + $ 40 / 1.04¹⁰+ $ 40 / 1.04¹¹+ $ 40 / 1.04¹² + $ 40 / 1.04¹³ + $ 40 / 1.04¹⁴ + $ 40 / 1.04¹⁵ + $ 40 / 1.04¹⁶ + $40 / 1.04¹⁷ + $ 40 / 1.04¹⁸ + $ 40 / 1.04¹⁹ + $ 40 / 1.04²⁰ + $ 40 / 1.04²¹ + $ 40 / 1.04²² + $ 40 / 1.04²³ + $ 40 / 1.04²⁴ + $ 1,000 / 1.04²⁴

= $ 1,000

Bond C = Number of years = 15 x 2 = 30

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 8% / 2 = 4%

= $ 40 / 1.04¹ + $ 40 / 1.04² + $ 40 / 1.04³ + $ 40 / 1.04⁴ + $ 40 / 1.04⁵ + $ 40 / 1.04⁶ + $ 40 / 1.04⁷ + $ 40 / 1.04⁸ + $ 40 / 1.04⁹ + $ 40 / 1.04¹⁰+ $ 40 / 1.04¹¹+ $ 40 / 1.04¹² + $ 40 / 1.04¹³ + $ 40 / 1.04¹⁴ + $ 40 / 1.04¹⁵ + $ 40 / 1.04¹⁶ + $40 / 1.04¹⁷ + $ 40 / 1.04¹⁸ + $ 40 / 1.04¹⁹ + $ 40 / 1.04²⁰ + $ 40 / 1.04²¹ + $ 40 / 1.04²² + $ 40 / 1.04²³ + $ 40 / 1.04²⁴ + $ 40 / 1.04²⁵ + $ 40 / 1.04²⁶ + $ 40 / 1.04²⁷ + $ 40 / 1.04²⁸ + $ 40 / 1.04²⁹ + $ 40 / 1.04³⁰ + $ 1,000 / 1.04³⁰

= $ 1,000

b. Value of bonds if discount rate is 6% per year compounded semiannually:

Bond A = Number of years = 4 x 2 = 8

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 6% / 2 = 3%

= $ 40 / 1.03¹ + $ 40 / 1.03² + $ 40 / 1.03³ + $ 40 / 1.03⁴ + $ 40 / 1.03⁵ + $ 40 / 1.03⁶ + $ 40 / 1.03⁷ + $ 40 / 1.03⁸ + $ 1,000 / 1.03⁸

= $ 1,070.20 Approximately

Bond B = Number of years = 12 x 2 = 24

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 6% / 2 = 3%

= $ 40 / 1.03¹ + $ 40 / 1.03² + $ 40 / 1.03³ + $ 40 / 1.03⁴ + $ 40 / 1.03⁵ + $ 40 / 1.03⁶ + $ 40 / 1.03⁷ + $ 40 / 1.03⁸ + $ 40 / 1.03⁹ + $ 40 / 1.03¹⁰+ $ 40 / 1.03¹¹+ $ 40 / 1.03¹² + $ 40 / 1.03¹³ + $ 40 / 1.03¹⁴ + $ 40 / 1.03¹⁵ + $ 40 / 1.03¹⁶ + $40 / 1.03¹⁷ + $ 40 / 1.03¹⁸ + $ 40 / 1.03¹⁹ + $ 40 / 1.03²⁰ + $ 40 / 1.03²¹ + $ 40 / 1.03²² + $ 40 / 1.03²³ + $ 40 / 1.03²⁴ + $ 1,000 / 1.03²⁴

= $ 1,169.36 Approximately

Bond C = Number of years = 15 x 2 = 30

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 6% / 2 = 3%

= $ 40 / 1.03¹ + $ 40 / 1.03² + $ 40 / 1.03³ + $ 40 / 1.03⁴ + $ 40 / 1.03⁵ + $ 40 / 1.03⁶ + $ 40 / 1.03⁷ + $ 40 / 1.03⁸ + $ 40 / 1.03⁹ + $ 40 / 1.03¹⁰+ $ 40 / 1.03¹¹+ $ 40 / 1.03¹² + $ 40 / 1.03¹³ + $ 40 / 1.03¹⁴ + $ 40 / 1.03¹⁵ + $ 40 / 1.03¹⁶ + $40 / 1.03¹⁷ + $ 40 / 1.03¹⁸ + $ 40 / 1.03¹⁹ + $ 40 / 1.03²⁰ + $ 40 / 1.03²¹ + $ 40 / 1.03²² + $ 40 / 1.03²³ + $ 40 / 1.03²⁴ + $ 40 / 1.03²⁵ + $ 40 / 1.03²⁶ + $ 40 / 1.03²⁷ + $ 40 / 1.03²⁸ + $ 40 / 1.03²⁹ + $ 40 / 1.03³⁰ + $ 1,000 / 1.03³⁰

= $ 1,196 Approximately

c. Value of bonds if discount rate is 17% per year compounded semiannually:

Bond A = Number of years = 4 x 2 = 8

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 17% / 2 = 8.5%

= $ 40 / 1.085¹ + $ 40 / 1.085² + $ 40 / 1.085³ + $ 40 / 1.085⁴ + $ 40 / 1.085⁵ + $ 40 / 1.085⁶ + $ 40 / 1.085⁷ + $ 40 / 1.085⁸ + $ 1,000 / 1.085⁸

= $ 746.24 Approximately

Bond B = Number of years = 12 x 2 = 24

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 17% / 2 = 8.5%

= $ 40 / 1.085¹ + $ 40 / 1.085² + $ 40 / 1.085³ + $ 40 / 1.085⁴ + $ 40 / 1.085⁵ + $ 40 / 1.085⁶ + $ 40 / 1.085⁷ + $ 40 / 1.085⁸ + $ 40 / 1.085⁹ + $ 40 / 1.085¹⁰+ $ 40 / 1.085¹¹+ $ 40 / 1.085¹² + $ 40 / 1.085¹³ + $ 40 / 1.085¹⁴ + $ 40 / 1.085¹⁵ + $ 40 / 1.085¹⁶ + $40 / 1.085¹⁷ + $ 40 / 1.085¹⁸ + $ 40 / 1.085¹⁹ + $ 40 / 1.085²⁰ + $ 40 / 1.085²¹ + $ 40 / 1.085²² + $ 40 / 1.085²³ + $ 40 / 1.085²⁴ + $ 1,000 / 1.085²⁴

= $ 545.32 Approximately

Bond C = Number of years = 15 x 2 = 30

Coupon payment = 8% / 2 = 4% x 1000 = $ 40

Discount Rate = 17% / 2 = 8.5%

= $ 40 / 1.085¹ + $ 40 / 1.085² + $ 40 / 1.085³ + $ 40 / 1.085⁴ + $ 40 / 1.085⁵ + $ 40 / 1.085⁶ + $ 40 / 1.085⁷ + $ 40 / 1.085⁸ + $ 40 / 1.085⁹ + $ 40 / 1.085¹⁰+ $ 40 / 1.085¹¹+ $ 40 / 1.085¹² + $ 40 / 1.085¹³ + $ 40 / 1.085¹⁴ + $ 40 / 1.085¹⁵ + $ 40 / 1.085¹⁶ + $40 / 1.085¹⁷ + $ 40 / 1.085¹⁸ + $ 40 / 1.085¹⁹ + $ 40 / 1.085²⁰ + $ 40 / 1.085²¹ + $ 40 / 1.085²² + $ 40 / 1.085²³ + $ 40 / 1.085²⁴ + $ 40 / 1.085²⁵ + $ 40 / 1.085²⁶ + $ 40 / 1.085²⁷ + $ 40 / 1.085²⁸ + $ 40 / 1.085²⁹ + $ 40 / 1.085³⁰ + $ 1,000 / 1.085³⁰

= $ 516.39 Approximately

d. We can see that whenever the coupon rate is equal to the discount rate the bonds value will always be equal to its par value as what we have seen in part a.

Whenever the discount rate decreases the value of the bond increases as what we have seen in part b

Whenever the discount rate increases the value of the bond decreases as what we have seen in part c

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