Question

It takes ten years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly

Answer 1

To verify whether a monthly deposit of $100 at an interest rate of 5%, compounded monthly, will amount to $15,528.23 over 10 years, we can use the future value of an annuity formula:

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

Where:
- \( FV \) = future value of the annuity
- \( P \) = amount of each deposit (payment)
- \( r \) = interest rate per period
- \( n \) = total number of payments

For this scenario:
- \( P = 100 \) (monthly deposit)
- Annual interest rate = 5%, so monthly interest rate \( r = \frac{0.05}{12} = \frac{0.05}{12} \approx 0.0041667 \)
- Total payments over 10 years = \( 10 \times 12 = 120 \)

Substituting the values:

\[
FV = 100 \times \frac{(1 + 0.0041667)^{120} - 1}{0.0041667}
\]

Calculating \( (1 + 0.0041667)^{120} \):

\[
(1 + 0.0041667)^{120} \approx 1.647009
\]

Now, substituting this value back into the formula:

\[
FV = 100 \times \frac{1.647009 - 1}{0.0041667}
\]

Calculating:

\[
\frac{1.647009 - 1}{0.0041667} = \frac{0.647009}{0.0041667} \approx 155.28
\]

\[
FV \approx 100 \times 155.28 \approx 15528
\]

Thus, the future value \( FV \) is approximately $15,528, which confirms your statement. Therefore, a $100 monthly deposit at an interest rate of 5% compounded monthly will indeed amount to $15,528.23 after 10 years.