To determine the future value of the retirement account when the individual reaches age 60, we can use the future value of an annuity formula since the individual is making regular contributions to the account.
The formula for the future value of an ordinary annuity 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
Given:
- Age of the individual = 52 years
- Age when reaching age 60 = 60 years
- Total contribution years = 60 - 52 = 8 years
- Payments per year = 4 (because contributions are made quarterly)
- Total number of payments (\(n\)) = 8 years × 4 = 32
- Payment amount (\(P\)) = $1300
- Annual interest rate = 8.5% = 0.085
- Quarterly interest rate (\(r\)) = \(\frac{0.085}{4} = 0.02125\)
Calculate the future value:
Substituting these values into the formula:
\[ FV = 1300 \times \frac{(1 + 0.02125)^{32} - 1}{0.02125} \]
Calculating \(1 + r\):
\[ 1 + 0.02125 = 1.02125 \]
Calculating \((1 + r)^n\):
\[ (1.02125)^{32} \approx 1.85956 \]
Now calculate \((1 + r)^n - 1\):
\[ 1.85956 - 1 = 0.85956 \]
Putting it all together:
\[ FV = 1300 \times \frac{0.85956}{0.02125} \]
Calculating \(\frac{0.85956}{0.02125} \approx 40.5\):
\[ FV \approx 1300 \times 40.5 \approx 52650 \]
Thus, the value of the retirement account when the individual reaches age 60 will be approximately $52,650.
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