statistics assistance needed

Question

Assume you put $600 per month into a retirement account for 13 years, and the account has an APR of 3.04% compounded monthly. 

What is the account balance at the end of the 13 years? Round your answer to the nearest cent.

$ ?

How much of the money in the account at the end of the 13 years is your personal investment, meaning that the money came directly from you? 

$ ?

How much of the money in the account at the end of the 13 years is interest? 

$  ?

What percentage of the account balance after 13 years is interest? 

Hint: The percentage of interest in the account is equal to the amount of interest earned divided by the account balance. Multiply that result by 100 to get a percentage.

Round your percentage to one decimal place. 

? %

Answer 1

This is actually a pretty good, practical question. Let's take the easy part first:

If you put $600 per month into an account for 13 years, then you have made 156 deposits, right? (13 years x 12 months per year.) So, to figure out how much money you have personally saved you multiply the $600 x 156. You should be able to calculate that rather easily.

Calculating the interest is more complex. A tedious, slow way to do this would be to take your first deposit of $600 and cumulatively apply 3.04% APR (3.04% / 12) 156 times. Then do the same thing with your next deposit, but 155 times. The next deposit 154 times. And so on.

This is called an annuity calculation, but the precise answer depends on when the deposits are made and interest is calculated. A pretty thorough discussion of this is found here: https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/

Assuming this is an "annuity due," the calculation for future value (FV) is:

FV = C×[((1+i)ⁿ−1​)/i]×(1+i)

In this, C is the monthly contribution ($600), i is the monthly interest rate (3.04%/12), and n is the number of periods (156).

FV = $600×[((1.00253)¹⁵⁶−1​)/0.00253]×(1.00253)

This should place you on the right track. When I ran it through an annuity calculator, it showed almost $115,000. Because your question is asking for accuracy to the nearest cent, you'll need to use many more significant digits than I did in the sample formula, above. Otherwise, you are going to be off by a few pennies.

Once you go through the work of precisely calculating the total future value, determining how much of that is interest is trivial. That total value is the sum of what you've contributed, plus interest, so just subtract the amount you've contributed (156 x $600) to determine the interest amount.