# How does compounding of annual interest work?

I have tried reading up on it, but I just find it confusing.

Apparently, if you started investing \$150/month at a 15% annual return and kept reinvesting the profits, here is what you would have accumulated:

``````In 30 years - \$1,051,000.00
In 40 years - \$4,700,000.00
``````

If you decided to invest \$250/month at a 15% annual return and kept reinvesting the profits, in 40 years, you would have accumulated `\$7,800,000.00`.

If you want to have your child's college education paid for, invest \$100/month at a 15% rate of return starting the day your child's born.

``````By age 19, it would be worth \$119,000.00
By age 50, it would be worth \$9,600,000.00
By age 60, it would be worth \$39,200,000.00
By age 70, it would be worth \$158,000,000.00
``````

How are all these being worked out?

• Are you looking for the formula to calculate compounding interest, or an explanation of why it adds up so quickly? – JohnFx Aug 2 '11 at 14:09
• Both ideally, but when I was typing up this question, I was more looking for a formula, which produces the same results as above. – oshirowanen Aug 2 '11 at 14:13

The general concept is that your money will grow at an accelerating rate because you start getting interest paid on your returns in addition to the original investment.

As a simple example, assume you invest \$100 and get 10% interest per year paid annually.

-At the end of the first year you have your \$100 + \$10 interest for a total of \$110.
-So you start the second year with \$110 and so 10% would be \$11 for a total of \$121.
-The third year you start with \$121 so 10% would be \$12.10 for a total of \$133.10

See how the amount it goes up each year increases? If we were talking a higher initial amount or a larger number of years that can really add up. That is essence is compound interest.

Most of the complicated looking formulas you see out there for compound interest are just shortcuts so you don't have to iteratively go through the above exercise a bunch of times to find out how much you would have after some number of years.

This formula tells you how much you would have(A) after a certain number of years(t) at a given interest rate(r) assuming they pay interest n times per year, for example you would use 12 for n if it paid interest monthly instead of yearly. P represents the amount you started out with. If you keep investing monthly (as shown in your example) instead of just depositing it and letting it sit, you have to use a more complicated formula. Finance people refer to this as calculating the future value of an annuity.

That formula looks like this:

A = PMT [((1 + r)N - 1) / r] x (1+r)

A : Is the amount you would have at the end of the time period.

N : The number of compounding periods (months if you get interest calculated monthly)

PMT : The total amount you are putting in each period (N)

r: Just like before, the interest rate you are getting paid. Be sure to adjust this to a monthly number if N represents months (divide APR by 12)*

*Most interest rates are quoted as APR, which is the annualized interest rate not counting compounding. Don't confuse this with APY, which has compounding built into it and is not appropriate for use in this formula.

r (monthly interest rate) = 15% APR / 12 = .0125
n = 30 years * 12 months/year = 360 months

A = \$150 x [((1 + .0125)360 - 1) / .0125] x (1+.0125)
A = \$1,051,473.09 (rounded)

• Great answer, but perhaps you could elaborate a bit more on realistic interest rates. It's highly unlikely that you will get 10-15% annual returns for 30-40 years at a time. Compounding interest is incredibly powerful but I think a little context can make projections more realistic :) – BlackJack Aug 2 '11 at 14:55
• @JohnFX, I have tried to convert that formula `A = PMT [((1 + r)N - 1) / r]` to a spreadsheet here, but when I enter in the variables shown in the original question, I am not getting the same results: spreadsheets.google.com/spreadsheet/…. I've probably misunderstood something, or converted the formula incorrectly?comments may only be edited for 5 minutes(click on this box to dismiss) – oshirowanen Aug 2 '11 at 15:03
• Sorry, you may have done that while I was still doing edits. I had to go figure out how to make the N a superscript. Be sure tha tyou are raising (1+r) to the power of N and not multiplying by it. – JohnFx Aug 2 '11 at 15:08
• @BlackJack - You are absolutely right. Those would be insane returns. I was just trying to use the OPs numbers in the second example and round numbers for easy math in the first. – JohnFx Aug 2 '11 at 15:10
• @user2662: Sorry for the confusion. I've made some edits to assume that you are paying the \$150 at the START of each month instead of the END of each month, which makes a big difference. That was probably why your numbers were not adding up. – JohnFx Aug 2 '11 at 15:21

Here's how I have worked it out. Different answer to the one expected. Pretty sure it's right though.

• When I enter 1.17149 as the monthly rate, I get \$834,980 as the future value after 30 years. Close. Still curious where we diverge. – JTP - Apologise to Monica Nov 9 '13 at 23:13
• The actual precision version used in the calculation is 1.171491691985338 %. That might fix it. (It's rounded for display.) – Chris Degnen Nov 10 '13 at 0:26
• You are exactly right. The number is correct. My mistake was missing the initial \$150 deposit. When I changed the mode of calculating to account for this, I'm within a few cents of your number. – JTP - Apologise to Monica Nov 10 '13 at 1:13
• Of course, \$9931.77 different. I just missed spotting that. – Chris Degnen Nov 10 '13 at 1:24