Answer :
Sure, let's break down the solution to find the present value of the given ordinary annuity step by step. Here's the scenario:
- The payment amount is [tex]$9347. - Payments are made semiannually. - The duration is 8 years. - The annual interest rate is 10%, compounded semiannually. We'll use the present value formula for an ordinary annuity, which is: \[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Where: - \(PV\) is the present value we need to find. - \(PMT\) is the payment amount. - \(r\) is the interest rate per period. - \(n\) is the total number of periods. Step-by-Step Solution: 1. Determine the interest rate per period: Since the interest is compounded semiannually: \[ r = \frac{10\%}{2} = 5\% = 0.05 \] 2. Calculate the total number of periods: Given that payments are made semiannually for 8 years: \[ n = 2 \times 8 = 16 \] 3. Plug the values into the present value formula: \[ PV = 9347 \times \left( \frac{1 - (1 + 0.05)^{-16}}{0.05} \right) \] 4. Solve the expression inside the parenthesis: \[ (1 + 0.05)^{-16} = (1.05)^{-16} \] Now calculate \(1 - (1.05)^{-16}\): \[ 1 - (1.05)^{-16} \approx 1 - 0.4581 \approx 0.5419 \] 5. Divide by the interest rate per period: \[ \frac{0.5419}{0.05} \approx 10.838 \] 6. Multiply by the payment amount: \[ PV = 9347 \times 10.838 \approx 101300.63 \] So, the present value of the annuity is approximately \$[/tex]101,300.63.
- The payment amount is [tex]$9347. - Payments are made semiannually. - The duration is 8 years. - The annual interest rate is 10%, compounded semiannually. We'll use the present value formula for an ordinary annuity, which is: \[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Where: - \(PV\) is the present value we need to find. - \(PMT\) is the payment amount. - \(r\) is the interest rate per period. - \(n\) is the total number of periods. Step-by-Step Solution: 1. Determine the interest rate per period: Since the interest is compounded semiannually: \[ r = \frac{10\%}{2} = 5\% = 0.05 \] 2. Calculate the total number of periods: Given that payments are made semiannually for 8 years: \[ n = 2 \times 8 = 16 \] 3. Plug the values into the present value formula: \[ PV = 9347 \times \left( \frac{1 - (1 + 0.05)^{-16}}{0.05} \right) \] 4. Solve the expression inside the parenthesis: \[ (1 + 0.05)^{-16} = (1.05)^{-16} \] Now calculate \(1 - (1.05)^{-16}\): \[ 1 - (1.05)^{-16} \approx 1 - 0.4581 \approx 0.5419 \] 5. Divide by the interest rate per period: \[ \frac{0.5419}{0.05} \approx 10.838 \] 6. Multiply by the payment amount: \[ PV = 9347 \times 10.838 \approx 101300.63 \] So, the present value of the annuity is approximately \$[/tex]101,300.63.