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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.)

Ask by Guzman Rogers. in the United States
Jan 24,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The prices of the zero-coupon bond at different maturities are as follows: - **20 years:** \$306.56 - **35 years:** \$126.30 - **70 years:** \$15.95 - **100 years:** \$2.71

Solution

To calculate the price of a zero-coupon bond using a semiannual convention, we can use the formula: \[ P = \frac{F}{(1 + r/n)^{nt}} \] Where: - \( P \) = price of the bond - \( F \) = par value of the bond - \( r \) = annual yield (as a decimal) - \( n \) = number of compounding periods per year - \( t \) = number of years to maturity Given: - \( F = 1000 \) - \( r = 0.06 \) - \( n = 2 \) (since it's semiannual) - \( t \) will vary based on the maturity. ### a. Maturity of 20 years 1. Substitute the values into the formula: - \( t = 20 \) \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 20}} \] 2. Calculate \( P \): Let's perform the calculation. First, calculate \( r/n \): \[ \frac{0.06}{2} = 0.03 \] Now calculate \( nt \): \[ 2 \times 20 = 40 \] Now substitute these values into the formula: \[ P = \frac{1000}{(1 + 0.03)^{40}} \] Now, we will calculate \( (1 + 0.03)^{40} \) and then find \( P \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\left(1+0.03\right)^{40}\) - step1: Add the numbers: \(1.03^{40}\) - step2: Convert the expressions: \(\left(\frac{103}{100}\right)^{40}\) - step3: Use the properties of exponents: \(\frac{103^{40}}{100^{40}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{40} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{40}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{40}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{40}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{40}}{100^{40}}}\) - step4: Rewrite the expression: \(\frac{10^{83}}{103^{40}}\) The price of the zero-coupon bond with a maturity of 20 years is approximately: \[ P \approx 306.56 \] Thus, the price of the bond is: \[ \$ 306.56 \] ### Now, let's calculate the price for the other maturities: ### b. Maturity of 35 years 1. Substitute \( t = 35 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 35}} \] 2. Calculate \( nt \): \[ 2 \times 35 = 70 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{70}} \] ### c. Maturity of 70 years 1. Substitute \( t = 70 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 70}} \] 2. Calculate \( nt \): \[ 2 \times 70 = 140 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{140}} \] ### d. Maturity of 100 years 1. Substitute \( t = 100 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 100}} \] 2. Calculate \( nt \): \[ 2 \times 100 = 200 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{200}} \] Now, let's perform the calculations for maturities of 35, 70, and 100 years. Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{200}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{200}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{200}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{200}}{100^{200}}}\) - step4: Rewrite the expression: \(\frac{10^{403}}{103^{200}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{140} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{140}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{140}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{140}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{140}}{100^{140}}}\) - step4: Rewrite the expression: \(\frac{10^{283}}{103^{140}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{70} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{70}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{70}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{70}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{70}}{100^{70}}}\) - step4: Rewrite the expression: \(\frac{10^{143}}{103^{70}}\) Here are the calculated prices for the zero-coupon bond at different maturities: ### b. Maturity of 35 years The price of the bond is approximately: \[ P \approx 126.30 \] Thus, the price of the bond is: \[ \$ 126.30 \] ### c. Maturity of 70 years The price of the bond is approximately: \[ P \approx 15.95 \] Thus, the price of the bond is: \[ \$ 15.95 \] ### d. Maturity of 100 years The price of the bond is approximately: \[ P \approx 2.71 \] Thus, the price of the bond is: \[ \$ 2.71 \] ### Summary of Prices: - Maturity of 20 years: \$306.56 - Maturity of 35 years: \$126.30 - Maturity of 70 years: \$15.95 - Maturity of 100 years: \$2.71

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The Deep Dive

To find the price of a zero-coupon bond, you can use the formula: \[ P = \frac{F}{(1 + r/n)^{nt}} \] Where: - \( P \) = price of the bond - \( F \) = face value of the bond (\$1,000) - \( r \) = annual yield (0.06) - \( n \) = number of compounding periods per year (2 for semiannual) - \( t \) = number of years to maturity For a maturity of 20 years, the calculations would be: \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 20}} = \frac{1000}{(1 + 0.03)^{40}} = \frac{1000}{(1.03)^{40}} \] Calculating this gives: \[ P = \frac{1000}{3.262} \approx 306.59 \] So the price of the bond if the maturity is 20 years is approximately **\$306.59**. Now, let’s compute the prices for the other maturities. For 35 years: \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 35}} = \frac{1000}{(1 + 0.03)^{70}} = \frac{1000}{(1.03)^{70}} \] Calculating gives: \[ P \approx \frac{1000}{7.612} \approx 131.00 \text{ (rounded to nearest cent)} \] For 70 years: \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 70}} = \frac{1000}{(1 + 0.03)^{140}} \] Calculating gives: \[ P \approx \frac{1000}{32.450} \approx 30.85 \] For 100 years: \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 100}} = \frac{1000}{(1 + 0.03)^{200}} \] Calculating gives: \[ P \approx \frac{1000}{108.141} \approx 9.24 \] So the prices are as follows: - Maturity 20 years: **\$306.59** - Maturity 35 years: **\$131.00** - Maturity 70 years: **\$30.85** - Maturity 100 years: **\$9.24** For the fun part! Adding a bit of flair: Ever wondered how zero-coupon bonds got their funky name? They’re like that friend who only shows up for parties but doesn’t bring anything upfront; they might not pay you dividends while they’re hanging around, but at maturity, they'll surprise you with that hefty payout! Want a real-world application? Think about saving for a long-term goal, like a dream vacation or a child's education. Zero-coupon bonds can be a fantastic tool since they’re designed to mature at a given date, ensuring you'll have a set amount waiting for you when you're ready to take that vacation or pay for college!

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