Question

At the age of 22, you decide to deposit $120 at the end of each month into a mutual fund that compounds monthly at a rate of 8.8%. How much will you have when you retire at age 65? (round to the nearest whole number).

Answer 1

To find out how much you will have when you retire at age 65 after making monthly deposits into a mutual fund, we can use the future value of an annuity formula:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

where:
- \( FV \) is the future value of the annuity (the total amount accumulated when you retire),
- \( P \) is the regular deposit amount (monthly deposit),
- \( r \) is the monthly interest rate,
- \( n \) is the total number of deposits (total number of months).

Let's break down the given information:

- \( P = 120 \) (the amount deposited each month),
- The annual interest rate is 8.8%, therefore the monthly interest rate \( r = \frac{8.8\%}{12} = \frac{0.088}{12} = 0.0073333 \) (approximately),
- You will deposit this amount from age 22 to age 65, which is \( 65 - 22 = 43 \) years. In months, this is \( n = 43 \times 12 = 516 \) months.

Now we can plug the numbers into the formula:

\[
FV = 120 \times \frac{(1 + 0.0073333)^{516} - 1}{0.0073333}
\]

Calculating \( (1 + 0.0073333)^{516} \):

\[
(1 + 0.0073333)^{516} \approx 17.4315
\]

Now, substituting this back into the formula:

\[
FV \approx 120 \times \frac{17.4315 - 1}{0.0073333}
\]

Calculating the fraction:

\[
\frac{17.4315 - 1}{0.0073333} \approx \frac{16.4315}{0.0073333} \approx 2231.56
\]

Now calculating \( FV \):

\[
FV \approx 120 \times 2231.56 \approx 267787.2
\]

Rounding to the nearest whole number gives:

\[
\text{Total amount at retirement} \approx 267787
\]

Thus, when you retire at age 65, you will have approximately **$267,787**.