Figuring out compounding interest

[drymetal]

I talk to people a lot about the power in investing their money. I've always relied on Excel to figure out things though and I'm getting sick of it. So I figured there was a way to do it simpler with math than making gigantic lists that detailed every month and year a person invests money.

So, let's say I have 10,000 and will expect an 8% yearly return on it. I figured out a formula or whatever that will give me the correct answer quickly:

10,000 * 1.08^n

Or say it was for 20 years: 10,000 * 1.08^20

This is great. But it doesn't do a whole lot because people generally contribute money regularly to their investments. Which gets me to my question...


I wanted to keep it simple. Let's say a person has $100. They invest it and can expect to earn 8% every year. Additionally, they add an additional $100 every year. The answer I got in Excel was $4,044.63 after 18 years.

After countless months beating my head against a wall and talking to my cat, I came up with this:

100(1+.08)18+100[((1+.08)18-1)/.08]

However, that equals $4,144.63. And to be honest, I don't remember how the heck I came up with that crazy looking equation. :(

But, it is giving me the wrong answer! By $100! I must be doing something right. lol

Can anyone help me simplify and understand this? Thanks!

 

[drag12]

Your formula is correct, the difference is that you are assuming that the person invests $10,000 plus an additional $100 on day one. The formula that excel uses is starting the yearly $100 investments at the end of the first year.

 

[drymetal]

I don't understand. Is there just a regular formula with x's and y's and all those happy letters that does this? You know, where I can just plug the numbers in. The formula above I forgot how I came up with it.

The answer isn't as important to me as understanding it. Not that I don't want an answer - I do. But I need to understand it. Understanding it is paramount to me. I hope by learning the why - I can figure out equations on my own easier in the future.

 

[rcgldr]

If the yearly investment and the interest rate are fixed, you could use power series to solve this:

let a = 1.08

you want to calculate the sum a^18 + a^17 + ... + a^1

multiply by a = a^19 + a^18 + ... a^2

subtract the original equation:

 a^19 + a^18 + ... + a^2
            a^18 + ... + a^2 + a^1
--------------------------------
 a^19                            - a^1

So the result is (a - 1)(a^18 + a^17 + ... + a^1) = (a^19 - a^1)

To get the original number divide by (a-1)

(a^18 + a^17 + ... + a^1) = (a^19 - a^1)/(a-1)

For your case you have 100 x (1.08^19 - 1.08) / (1.08 - 1) ~= 4044.6263239

Although this is nice for doing algebra, it's probably better to use a spread sheet, to handle variations in monthly deposits, changes in interest rates, and also allowing for interest that is compounded monthly (or daily) instead of yearly.