Question

1. Bond X is a 5 percent coupon bond. Bond Y is a 10 percent coupon bond. Both bonds have 8 years to maturity, make semiannual payments, and have a yield to maturity of 10 percent. If the interest rate suddenly falls by 1 percent, what is the percentage price change of these bonds? What about if the interest rate rises by 1 percent? What does this problem tell you about interest rate risk of lower-coupon bonds? (20 points)

Answer 1

Bond X:

Face Value = $1,000

Annual Coupon Rate = 5.00%
Semiannual Coupon Rate = 2.50%
Semiannual Coupon = 2.50% * $1,000
Semiannual Coupon = $25

Time to Maturity = 8
Semiannual Period to Maturity = 16

If interest rate is 10%:

Annual Interest Rate = 10%
Semiannual Interest Rate = 5%

Price of Bond = $25 * PVIFA(5%, 16) + $1,000 * PVIF(5%, 16)
Price of Bond = $25 * (1 - (1/1.05)^16) / 0.05 + $1,000 / 1.05^16
Price of Bond = $729.06

If interest rate increases by 1%:

Annual Interest Rate = 11%
Semiannual Interest Rate = 5.50%

Price of Bond = $25 * PVIFA(5.5%, 16) + $1,000 * PVIF(5.5%, 16)
Price of Bond = $25 * (1 - (1/1.055)^16) / 0.055 + $1,000 / 1.055^16
Price of Bond = $686.14

Percentage Change in Price = ($686.14 - $729.06) / $729.06
Percentage Change in Price = -5.89%

If interest rate decreases by 1%:

Annual Interest Rate = 9%
Semiannual Interest Rate = 4.50%

Price of Bond = $25 * PVIFA(4.5%, 16) + $1,000 * PVIF(4.5%, 16)
Price of Bond = $25 * (1 - (1/1.045)^16) / 0.045 + $1,000 / 1.045^16
Price of Bond = $775.32

Percentage Change in Price = ($775.32 - $729.06) / $729.06
Percentage Change in Price = 6.35%

Bond Y:

Face Value = $1,000

Annual Coupon Rate = 10.00%
Semiannual Coupon Rate = 5.00%
Semiannual Coupon = 5.00% * $1,000
Semiannual Coupon = $50

Time to Maturity = 8
Semiannual Period to Maturity = 16

If interest rate is 10%:

Annual Interest Rate = 10%
Semiannual Interest Rate = 5%

Price of Bond = $50 * PVIFA(5%, 16) + $1,000 * PVIF(5%, 16)
Price of Bond = $50 * (1 - (1/1.05)^16) / 0.05 + $1,000 / 1.05^16
Price of Bond = $1,000.00

If interest rate increases by 1%:

Annual Interest Rate = 11%
Semiannual Interest Rate = 5.50%

Price of Bond = $50 * PVIFA(5.5%, 16) + $1,000 * PVIF(5.5%, 16)
Price of Bond = $50 * (1 - (1/1.055)^16) / 0.055 + $1,000 / 1.055^16
Price of Bond = $947.69

Percentage Change in Price = ($947.69 - $1,000.00) / $1,000.00
Percentage Change in Price = -5.23%

If interest rate decreases by 1%:

Annual Interest Rate = 9%
Semiannual Interest Rate = 4.50%

Price of Bond = $50 * PVIFA(4.5%, 16) + $1,000 * PVIF(4.5%, 16)
Price of Bond = $50 * (1 - (1/1.045)^16) / 0.045 + $1,000 / 1.045^16
Price of Bond = $1,056.17

Percentage Change in Price = ($1,056.17 - $1,000.00) / $1,000.00
Percentage Change in Price = 5.62%

Price of bond will increase if the interest rate decreases and Price of bond will decrease if the interest rate increases.