To calculate the future value of the retirement account, we will use the future value of an annuity formula since the individual is making regular contributions at the end of each quarter.
The formula for the future value of an annuity compounded at regular intervals is:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \(FV\) = future value of the annuity
- \(P\) = payment amount per period
- \(r\) = interest rate per period
- \(n\) = total number of payments
Step 1: Identify the components
- Payment per quarter (\(P\)): $1300
- Annual interest rate: 8.5% = 0.085
- Interest rate per quarter (\(r\)): \( \frac{0.085}{4} = 0.02125 \)
- Time period: 60 - 52 = 8 years
- Total number of payments (\(n\)): \( 8 \times 4 = 32 \) (since payments are made quarterly)
Step 2: Substitute the values into the formula
Now we substitute the values into the formula:
\[ FV = 1300 \times \frac{(1 + 0.02125)^{32} - 1}{0.02125} \]
Step 3: Calculate \( (1 + r)^n \)
First, calculate \( (1 + 0.02125)^{32} \):
\[ (1 + 0.02125)^{32} = 1.02125^{32} \approx 1.836738 \]
Step 4: Calculate the future value
Now, substitute back into the formula:
\[ FV = 1300 \times \frac{1.836738 - 1}{0.02125} \]
Calculate \( 1.836738 - 1 \):
\[ 1.836738 - 1 = 0.836738 \]
Now, divide this by the interest rate per period:
\[ \frac{0.836738}{0.02125} \approx 39.375 \]
Finally, calculate the future value:
\[ FV = 1300 \times 39.375 \approx 51287.50 \]
Final Result
Thus, when the individual reaches age 60, the value of the account will be approximately $51,287.50 (rounded to the nearest cent).
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