A monthly annuity must accumulate a future value of $85,000 in 17 years at 3.8% APR compounded monthly. How much is deposited in total, and how much is interest?

Step-by-step solution

1. 1

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

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

   where $i$ is the periodic rate and $n$ the number of deposits.

2. 2

   Convert the APR to a monthly rate and count the periods. Compounding is monthly, so

   $i = \frac{0.038}{12} \approx 0.0031666667, \qquad n = 17 \times 12 = 204$

3. 3

   Solve the formula for $PMT$. Rearranging,

   $PMT = FV \cdot \frac{i}{(1+i)^n - 1} = 85000 \cdot \frac{0.0031666667}{(1.0031666667)^{204} - 1}$

4. 4

   Evaluate the growth factor. $(1.0031666667)^{204} \approx 1.905948$, so the denominator is $0.905948$ and

   $PMT \approx 85000 \cdot \frac{0.0031666667}{0.905948} \approx 297.11$

   The monthly deposit is about \297.11$.

5. 5

   Total the deposits. Over $204$ months,

   $297.110626 \times 204 = 60{,}610.57$

   Note the unrounded $PMT$ is used here. If the deposit is actually rounded to the cent, \297.11 \times 204 = $60{,}610.44$and the balance lands a few cents short of$$85{,}000$.

6. 6

   Subtract to get the interest. The account reaches \85{,}000$, so the interest is the part that was not deposited:

   $85000 - 60610.57 = 24{,}389.43$