Questions: Find the future value of each annuity due. Then determine how much of this value is from contributions and how much is from interest. Payments of 410 made at the beginning of each quarter for 13 years at 3.8% compounded quarterly The future value of the annuity due is . (Do not round until the final answer. Then round to the nearest cent as needed.)

Find the future value of each annuity due. Then determine how much of this value is from contributions and how much is from interest.

Payments of 410 made at the beginning of each quarter for 13 years at 3.8% compounded quarterly

The future value of the annuity due is .
(Do not round until the final answer. Then round to the nearest cent as needed.)

Transcript text: Find the future value of each annuity due. Then determine how much of this value is from contributions and how much is from interest. Payments of $\$ 410$ made at the beginning of each quarter for 13 years at $3.8 \%$ compounded quarterly The future value of the annuity due is $\$$ $\square$ . (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution

Solution Steps

Step 1: Calculate the Future Value of the Annuity Due (FVAD)

The formula to calculate the future value of an annuity due is given by: \[ FVAD = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \] Substituting the given values: \[ FVAD = 410 \times \left( \frac{(1 + 0.038)^52 - 1}{0.038} \right) \times (1 + 0.038) = 66686.63 \]

Step 2: Calculate the Total Contributions

The total contributions can be calculated by multiplying the payment amount by the number of periods. \[ \text{Total Contributions} = P \times n = 410 \times 52 = 21320 \]

Step 3: Calculate the Amount from Interest

The amount from interest can be found by subtracting the total contributions from the future value of the annuity due. \[ \text{Amount from Interest} = FVAD - \text{Total Contributions} = 66686.63 - 21320 = 45366.63 \]