James deposits a fixed monthly amount into an annuity to accumulate a future value of 85,000 dollars in 17 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

   Set up the ordinary-annuity future value formula. Deposits are made at the end of each period, so

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

   with $i = \tfrac{0.038}{12} = 0.0031666667$ (the monthly rate) and $n = 17 \times 12 = 204$ (the number of deposits).

2. 2

   Compute the accumulation factor carefully. This is the step that decides the accuracy of everything else:

   $(1.0031666667)^{204} = 1.90594762$

   Carrying only four decimals here (e.g. $1.9059$) shifts the final deposit total by several dollars, which is why the problem asks for at least six decimal places in intermediate work.

3. 3

   Solve for the monthly deposit.

   $PMT = FV \cdot \frac{i}{(1+i)^n - 1} = 85000 \times \frac{0.0031666667}{0.90594762} = \$297.11$

4. 4

   Total the deposits. James makes $204$ payments of \297.11$:

   $297.11 \times 204 = \$60{,}610.44$

5. 5

   Find the interest by subtraction. Everything in the account that James did not deposit is interest:

   $85000 - 60610.44 = \$24{,}389.56$

   So about $28.7\%$ of the final balance comes from compounding.

6. 6

   Check the answer forwards. Growing \297.11$per month for$204$months at$0.0031666667$:$297.11 \times \tfrac{0.90594762}{0.0031666667} = 297.11 \times 286.08872 = $84{,}999.82$— the exact$PMT$is$297.110631$, so rounding down by$0.00063$per payment leaves a shortfall of about$0.18$.