To determine how much you will have when you retire at age 65 after making annual deposits into a mutual fund that compounds annually, we can use the future value of an ordinary annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = annual deposit (payment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of years (number of deposits)
Given values:
- \( P = 1440 \)
- \( r = 8.8% = 0.088 \)
- \( n = 65 - 22 = 43 \) (years of deposits)
A.) Calculate the future value:
Plugging in the values into the formula:
\[ FV = 1440 \times \frac{(1 + 0.088)^{43} - 1}{0.088} \]
Calculating \( (1 + 0.088)^{43} \):
\[ (1.088)^{43} \approx 31.55722 \]
Now we can substitute this back into the formula:
\[ FV = 1440 \times \frac{31.55722 - 1}{0.088} \] \[ FV = 1440 \times \frac{30.55722}{0.088} \] \[ FV = 1440 \times 346.554 \] \[ FV \approx 498,000.00 \]
So when rounded to the nearest whole number: \[ FV \approx 498,000 \]
B.) Calculate the interest earned:
To find the interest earned, we first need to calculate the total amount deposited and then subtract it from the future value.
Total amount deposited: \[ Total\ Deposits = P \times n = 1440 \times 43 = 61,920 \]
Interest earned: \[ Interest = FV - Total\ Deposits \] \[ Interest = 498,000 - 61,920 \] \[ Interest = 436,080 \]
Final Results:
A.) You will have approximately $498,000 when you retire at age 65 (rounded to the nearest whole number).
B.) The interest earned will be $436,080.
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