Question

A 20-year maturity bond with par value \(1,000 makes semiannual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is:

a. \)950

b. \(1,000

c. \)1,050

Short answer

a. When the bond price is $950, the bond equivalent is 8.52% and the effective annual yield to maturity is 8.70%

b. When the bond price is $1000, the bond equivalent is 8% and the effective annual yield to maturity is 8.16%

c. When the bond price is $1050, the bond equivalent 3.76% and the effective annual yield to maturity is 7.66%

Step-by-step solution

Calculation of bond equivalent and effective annual yield to maturity of the bond if the bond price is $950

Available inputs: n = 40, FV = 1000, PV = –950, PMT = 40.

The yield to maturity on a semi-annual basis =

Yield To Maturity = (Face Value/Current Bond Price)^(1/Years To Maturity)−1

= (1000 / -950) )^(1/40)−1

= 4.26%.

This implies a bond equivalent yield to maturity of: 4.26% x 2 = 8.52%

Effective annual yield to maturity i = [1 + (r/n)]ⁿ – 1

(1.0426)² – 1 = 0.0870 = 8.70%

Calculation of bond equivalent and effective annual yield to maturity of the bond if the bond price is $1000

b. Since the bond is selling at par, the yield to maturity on a semi-annual basis is the same as the semi-annual coupon, 4%.

The bond equivalent yield to maturity is 8%.

Effective annual yield to maturity i = [1 + (r/n)]ⁿ – 1

= (1.04)² – 1 = 0.0816 = 8.16%

Calculation of bond equivalent and effective annual yield to maturity of the bond if the bond price is $1050

Available inputs: n = 40, FV = 1000, PV = –1050, PMT = 40.

The yield to maturity on a semi-annual basis =

Yield To Maturity = (Face Value/Current Bond Price)^(1/Years To Maturity)−1

= (1000 / -1050) )^(1/40)−1

= 7.52%.

This implies a bond equivalent yield to maturity semi-annually: 7.52% / 2 = 3.76%

Effective annual yield to maturity i = [1 + (r/n)]ⁿ – 1

(1.0376)² – 1 = 0.0766 = 7.66%