Depositing $700 each quarter for 20 years at 7.2% APR compounded quarterly: how much is paid in, and how much is interest?

Step-by-step solution

1. 1

   Count the deposits. Quarterly compounding means $n = 4$ deposits per year, so over $t = 20$ years

   $nt = 4 \times 20 = 80 \text{ deposits}$

2. 2

   Total the money paid in. This part needs no interest formula at all — it is just the deposit times the count:

   $700 \times 80 = 56{,}000$

3. 3

   Apply the future value of an annuity formula.

   $FV = PMT \cdot \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$

   The quarterly rate is $\frac{0.072}{4} = 0.018$.

4. 4

   Evaluate. $(1.018)^{80} \approx 4.166988$, so

   $FV = 700 \cdot \frac{4.166988 - 1}{0.018} = 700 \cdot 175.943778 \approx 123{,}160.64$

5. 5

   Subtract to isolate the interest.

   $123160.64 - 56000 = 67{,}160.64$

   More than half the final balance is interest — that is the effect of 20 years of compounding.