Questions: Calculate the present value of the following annuities, assuming each annuity payment is made at the end of each compounding period. (FV of 1, PV of 1. FVA of 1, and PVA of 1) (Use tables, Excel, or a financial calculator. Round your answers to 2 decimal places.) Annuity Payment Annual Rate Interest Compounded Period Invested Present Value of Annuity --------------- 4,700 6.0% Semiannually 3 years 9,700 8.0% Quarterly 2 years 3,700 10.0% Annually 5 years

Calculate the present value of the following annuities, assuming each annuity payment is made at the end of each compounding period. (FV of 1, PV of 1. FVA of 1, and PVA of 1) (Use tables, Excel, or a financial calculator. Round your answers to 2 decimal places.) Annuity Payment Annual Rate Interest Compounded Period Invested Present Value of Annuity --------------- 4,700 6.0% Semiannually 3 years 9,700 8.0% Quarterly 2 years 3,700 10.0% Annually 5 years
Transcript text: Calculate the present value of the following annuities, assuming each annuity payment is made at the end of each compounding period. (FV of $1, PV of $1. FVA of $1, and PVA of $1) (Use tables, Excel, or a financial calculator. Round your answers to 2 decimal places.) \begin{tabular}{|c|c|r|c|c|c|} \hline & \begin{tabular}{c} Annuity \\ Payment \end{tabular} & \begin{tabular}{c} Annual \\ Rate \end{tabular} & \begin{tabular}{c} Interest \\ Compounded \end{tabular} & \begin{tabular}{c} Period \\ Invested \end{tabular} & \begin{tabular}{c} Present Value of \\ Annuity \end{tabular} \\ \hline 1. & $\$ 4,700$ & $6.0 \%$ & Semiannually & 3 years & \\ \hline 2. & 9,700 & $8.0 \%$ & Quarterly & 2 years & \\ \hline 3 & 3,700 & $10.0 \%$ & Annually & 5 years & \\ \hline \end{tabular}
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Solution

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Solution Steps

To calculate the present value of the annuities, we can use the formula for present value of an annuity:

\[ PV = Pmt \times \left(1 - (1 + r)^{-n}\right) / r \]

where:

  • \( PV \) is the present value of the annuity
  • \( Pmt \) is the annuity payment
  • \( r \) is the periodic interest rate
  • \( n \) is the total number of compounding periods

We will convert the annual rate to the periodic rate based on the compounding frequency before plugging the values into the formula.

Step 1: Calculate the present value of Annuity 1

Given:

  • Annuity payment (\(Pmt_1\)) = $4700
  • Periodic interest rate (\(r_1\)) = 0.03
  • Total number of compounding periods (\(n_1\)) = 3 * 2 = 6

Using the formula for present value of an annuity: \[ PV_1 = Pmt_1 \times \frac{1 - (1 + r_1)^{-n_1}}{r_1} \]

Substitute the values: \[ PV_1 = 4700 \times \frac{1 - (1 + 0.03)^{-6}}{0.03} \]

Calculate \(PV_1\) to four significant digits: \[ PV_1 \approx \boxed{25460.8} \]

Step 2: Calculate the present value of Annuity 2

Given:

  • Annuity payment (\(Pmt_2\)) = $9700
  • Periodic interest rate (\(r_2\)) = 0.02
  • Total number of compounding periods (\(n_2\)) = 2 * 4 = 8

Using the formula for present value of an annuity: \[ PV_2 = Pmt_2 \times \frac{1 - (1 + r_2)^{-n_2}}{r_2} \]

Substitute the values: \[ PV_2 = 9700 \times \frac{1 - (1 + 0.02)^{-8}}{0.02} \]

Calculate \(PV_2\) to four significant digits: \[ PV_2 \approx \boxed{71057.2} \]

Final Answer

\[ PV_1 \approx \boxed{25460.8} \] \[ PV_2 \approx \boxed{71057.2} \]

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