To calculate the future value of the retirement account, we will use the formula for the future value of an ordinary annuity since the individual is making periodic contributions at the end of each quarter.
The formula for the future value of an ordinary annuity is:
\[ FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = amount of each payment (contribution)
- \( r \) = interest rate per period
- \( n \) = total number of payments
Given:
- The individual makes quarterly payments of \( P = 1300 \).
- The annual interest rate is \( 8.5% \), so the quarterly interest rate is:
\[ r = \frac{8.5%}{4} = \frac{0.085}{4} = 0.02125 \]
- The individual is contributing from age 52 to age 65. This is a period of 13 years, and since they are contributing quarterly, the total number of contributions (payments) is:
\[ n = 13 \text{ years} \times 4 \text{ quarters/year} = 52 \text{ quarters} \]
Plugging the values into the formula:
\[ FV = 1300 \left(\frac{(1 + 0.02125)^{52} - 1}{0.02125}\right) \]
First, calculate \( (1 + 0.02125)^{52} \):
\[ (1 + 0.02125)^{52} \approx 1.129959 \]
Now plug this value back into the formula:
\[ FV = 1300 \left(\frac{1.129959 - 1}{0.02125}\right) \]
\[ FV = 1300 \left(\frac{0.129959}{0.02125}\right) \]
\[ FV = 1300 \times 6.115086 \]
\[ FV \approx 7949.71 \]
So, the value of the account when the individual reaches age 65 will be approximately \($7949.71\).
Hence, rounding to the nearest cent, the final amount is:
\($7,949.71\)
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