Zero-coupon bond. Addison Company will issue a zero-coupon bond this coming month. The bond's projected yield is \( 6 \% \). If the par value is \( \$ 1,000 \), what is the bond's price using a semiannual convention if a. the maturity is 20 years? b. the maturity is 35 years? c. the maturity is 70 years? d. the maturity is 100 years? a. What is the price of the bond using a semiannual convention if the maturity is 20 years? \( \$ \square \) (Round to the nearest cent.)

To find the price of a zero-coupon bond using a semiannual convention, you can use the present value formula. The price \( P \) of a zero-coupon bond can be calculated using the formula: \[ P = \frac{F}{(1 + r/n)^{nt}} \] where: - \( F \) is the par value of the bond (in this case, \$1,000), - \( r \) is the annual yield (6% or 0.06), - \( n \) is the number of compounding periods per year (2 for semiannual), - \( t \) is the number of years to maturity. ### a. Maturity is 20 years: Given: - \( F = 1,000 \) - \( r = 0.06 \) - \( n = 2 \) - \( t = 20 \) \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 20}} = \frac{1000}{(1 + 0.03)^{40}} = \frac{1000}{(1.03)^{40}} \] Calculating \( (1.03)^{40} \): \[ (1.03)^{40} \approx 3.262 \] Then: \[ P \approx \frac{1000}{3.262} \approx 306.12 \] So, the price of the bond when maturity is 20 years is approximately **\$306.12.** ### b. Maturity is 35 years: Using \( t = 35 \): \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 35}} = \frac{1000}{(1.03)^{70}} \] Calculating \( (1.03)^{70} \): \[ (1.03)^{70} \approx 7.612 \] Then: \[ P \approx \frac{1000}{7.612} \approx 131.67 \] So, the price of the bond when maturity is 35 years is approximately **\$131.67.** ### c. Maturity is 70 years: Using \( t = 70 \): \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 70}} = \frac{1000}{(1.03)^{140}} \] Calculating \( (1.03)^{140} \): \[ (1.03)^{140} \approx 57.774 \] Then: \[ P \approx \frac{1000}{57.774} \approx 17.29 \] So, the price of the bond when maturity is 70 years is approximately **\$17.29.** ### d. Maturity is 100 years: Using \( t = 100 \): \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 100}} = \frac{1000}{(1.03)^{200}} \] Calculating \( (1.03)^{200} \): \[ (1.03)^{200} \approx 340.667 \] Then: \[ P \approx \frac{1000}{340.667} \approx 2.94 \] So, the price of the bond when maturity is 100 years is approximately **\$2.94.**