A bond with a face value of \( \$ 18,000 \) and a \( 3.1 \% \) interest rate (compounded semiannually) will mature in 14 years. What is a fair price to pay for the bond today? A fair price to buy the bond at would be \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

To find the fair price to pay for the bond today, we can use the present value formula for bonds. The price of a bond is the present value of its future cash flows, which consist of the interest payments and the face value at maturity. ### Known Conditions: - Face Value (\( F \)): \( 18000 \) - Interest Rate (\( r \)): \( 3.1\% \) or \( 0.031 \) - Compounding Frequency: Semiannually (2 times a year) - Number of Years (\( t \)): \( 14 \) ### Steps to Calculate the Fair Price: 1. **Calculate the number of periods (\( n \))**: \[ n = t \times \text{compounding frequency} = 14 \times 2 = 28 \] 2. **Calculate the interest payment (\( PMT \))**: \[ PMT = \frac{F \times r}{\text{compounding frequency}} = \frac{18000 \times 0.031}{2} \] 3. **Calculate the present value of the interest payments**: The present value of an annuity formula is: \[ PV_{PMT} = PMT \times \left(1 - (1 + i)^{-n}\right) / i \] where \( i = \frac{r}{\text{compounding frequency}} = \frac{0.031}{2} \). 4. **Calculate the present value of the face value**: The present value of a lump sum formula is: \[ PV_{F} = \frac{F}{(1 + i)^{n}} \] 5. **Add the present values to find the fair price**: \[ Price = PV_{PMT} + PV_{F} \] Now, let's perform the calculations step by step. ### Step 1: Calculate \( n \) \[ n = 14 \times 2 = 28 \] ### Step 2: Calculate \( PMT \) \[ PMT = \frac{18000 \times 0.031}{2} \] ### Step 3: Calculate \( i \) \[ i = \frac{0.031}{2} \] ### Step 4: Calculate \( PV_{PMT} \) and \( PV_{F} \) Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{18000\times 0.031}{2}\) - step1: Multiply the numbers: \(\frac{558}{2}\) - step2: Reduce the fraction: \(279\) Calculate or simplify the expression \( 0.031 / 2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.031}{2}\) - step1: Convert the expressions: \(\frac{\frac{31}{1000}}{2}\) - step2: Multiply by the reciprocal: \(\frac{31}{1000}\times \frac{1}{2}\) - step3: Multiply the fractions: \(\frac{31}{1000\times 2}\) - step4: Multiply: \(\frac{31}{2000}\) Calculate or simplify the expression \( 18000 / (1 + 0.0155)^{28} \). Calculate the value by following steps: - step0: Calculate: \(\frac{18000}{\left(1+0.0155\right)^{28}}\) - step1: Add the numbers: \(\frac{18000}{1.0155^{28}}\) - step2: Convert the expressions: \(\frac{18000}{\left(\frac{2031}{2000}\right)^{28}}\) - step3: Simplify the expression: \(\frac{18000}{\frac{2031^{28}}{2000^{28}}}\) - step4: Rewrite the expression: \(\frac{18000\times 2000^{28}}{2031^{28}}\) Calculate or simplify the expression \( 279 * (1 - (1 + 0.0155)^{-28}) / 0.0155 \). Calculate the value by following steps: - step0: Calculate: \(\frac{279\left(1-\left(1+0.0155\right)^{-28}\right)}{0.0155}\) - step1: Add the numbers: \(\frac{279\left(1-1.0155^{-28}\right)}{0.0155}\) - step2: Convert the expressions: \(\frac{279\left(1-\left(\frac{2031}{2000}\right)^{-28}\right)}{0.0155}\) - step3: Subtract the numbers: \(\frac{279\times \frac{2031^{28}-2000^{28}}{2031^{28}}}{0.0155}\) - step4: Multiply the numbers: \(\frac{\frac{31\times 2031^{28}-31\times 2000^{28}}{3^{26}\times 677^{28}}}{0.0155}\) - step5: Convert the expressions: \(\frac{\frac{31\times 2031^{28}-31\times 2000^{28}}{3^{26}\times 677^{28}}}{\frac{31}{2000}}\) - step6: Multiply by the reciprocal: \(\frac{31\times 2031^{28}-31\times 2000^{28}}{3^{26}\times 677^{28}}\times \frac{2000}{31}\) - step7: Rewrite the expression: \(\frac{31\left(2031^{28}-2000^{28}\right)}{3^{26}\times 677^{28}}\times \frac{2000}{31}\) - step8: Reduce the numbers: \(\frac{2031^{28}-2000^{28}}{3^{26}\times 677^{28}}\times 2000\) - step9: Multiply the fractions: \(\frac{\left(2031^{28}-2000^{28}\right)\times 2000}{3^{26}\times 677^{28}}\) - step10: Multiply: \(\frac{2031^{28}\times 2000-2000^{29}}{3^{26}\times 677^{28}}\) Calculate or simplify the expression \( 6298.688836 + 11701.311164 \). Calculate the value by following steps: - step0: Calculate: \(6298.688836+11701.311164\) - step1: Add the numbers: \(18000\) Now that we have completed the calculations, let's summarize the results: 1. **Interest Payment (\( PMT \))**: \[ PMT = 279 \] 2. **Interest Rate per Period (\( i \))**: \[ i = 0.0155 \] 3. **Present Value of Interest Payments (\( PV_{PMT} \))**: \[ PV_{PMT} \approx 6298.69 \] 4. **Present Value of Face Value (\( PV_{F} \))**: \[ PV_{F} \approx 11701.31 \] 5. **Total Price of the Bond**: \[ Price = PV_{PMT} + PV_{F} \approx 6298.69 + 11701.31 = 18000 \] Thus, the fair price to buy the bond today would be \( \$ 18,000.00 \).