Question

An investor contributes $1,000.00 per quarter in an account that pays 6.72% APR with quarterly compounding. The first contribution will be made next quarter. What is the future value of this account in 17.00 years?

Answer 1

To calculate the future value of an investment with regular contributions and compound interest, 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
- \(P\) = payment amount per period (contribution per quarter)
- \(r\) = interest rate per period
- \(n\) = total number of payments

Given:
- Quarterly contribution \(P = 1000\)
- Annual nominal interest rate \(APR = 6.72\%\)
- Interest rate per quarter \(r = \frac{6.72}{100} \div 4 = 0.0168\)
- Total time in years = 17
- Total number of quarters \(n = 17 \times 4 = 68\)

Let's substitute these values into the future value formula:

1. **Calculate \(r\)**:

\[
r = \frac{6.72}{100} \div 4 = 0.0168
\]

2. **Calculate \(n\)**:

\[
n = 17 \times 4 = 68
\]

3. **Substitute values into the formula**:

\[
FV = 1000 \times \frac{(1 + 0.0168)^{68} - 1}{0.0168}
\]

4. **Calculate \((1 + r)^n\)**:

\[
(1 + 0.0168)^{68} \approx (1.0168)^{68} \approx 3.294084
\]

5. **Subtract 1**:

\[
3.294084 - 1 = 2.294084
\]

6. **Calculate \(\frac{(1 + r)^n - 1}{r}\)**:

\[
\frac{2.294084}{0.0168} \approx 136.906
\]

7. **Calculate the future value \(FV\)**:

\[
FV = 1000 \times 136.906 \approx 136906
\]

The future value of the account after 17 years of quarterly contributions of $1,000 at a 6.72% APR with quarterly compounding is approximately **$136,906.00**.