Kevin deposits a fixed monthly amount into an annuity to accumulate a future value of 75,000 dollars in 16 years at an APR of 3.8% compounded monthly. How much will he ultimately deposit, and how much is interest?

Step-by-step solution

1. 1

   Identify the periodic rate and the number of periods. Monthly compounding splits the APR into twelve equal periods:

   $i = \frac{0.038}{12} = 0.0031666667, \qquad n = 16 \times 12 = 192$

2. 2

   Write the future-value formula and rearrange for PMT. From $FV = PMT \cdot \dfrac{(1+i)^n - 1}{i}$,

   $PMT = FV \cdot \frac{i}{(1+i)^n - 1}$

3. 3

   Evaluate the accumulation factor to full precision.

   $(1.0031666667)^{192} = 1.83499060$

   A truncated value such as $1.835571$ would inflate the denominator and give PMT \approx \284.23$— about$20$cents low, which compounds into a$$40$ error in the deposit total.

4. 4

   Compute the monthly payment.

   $PMT = 75000 \times \frac{0.0031666667}{0.83499060} = \$284.43$

5. 5

   Total the deposits and subtract.

   $284.43 \times 192 = \$54{,}610.56, \qquad 75000 - 54610.56 = \$20{,}389.44$

6. 6

   **Compare with the 17-year, $85{,}000 case.** There,$$297.11$a month over$204$months yields$$24{,}389.56$of interest — a larger share, because the extra year of compounding does more work than the extra principal. Interest is about$27.2%$of the balance here versus$28.7%$ there.