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.

 

[CellCoree]

I understand your frustration with trying to figure out compounding interest using Excel or complex equations. However, there are simpler mathematical formulas that can accurately calculate the growth of investments with regular contributions.

One such formula is the Future Value of an Annuity formula, which takes into account both the initial investment and regular contributions over a period of time. This formula is:

FV = P * [(1 + r)^n - 1] / r

Where:
FV = Future Value
P = Initial Investment
r = Interest Rate
n = Number of Periods

In your example, the initial investment is $100 and the regular contribution is also $100 per year, so the formula would be:

FV = $100 * [(1 + 0.08)^18 - 1] / 0.08

This gives a future value of $4,144.63, which matches your Excel calculation.

To understand why your initial equation gave a slightly different value, it's important to note that compounding interest is calculated based on the previous balance, not the initial investment. So, the correct formula would be:

FV = P * (1 + r)^n + C * [(1 + r)^n - 1] / r

Where:
FV = Future Value
P = Initial Investment
r = Interest Rate
n = Number of Periods
C = Regular Contribution

In your case, C = $100 and P = 0, so the second part of the equation becomes:

$100 * [(1 + 0.08)^18 - 1] / 0.08 = $144.63

Adding this to the initial investment of $100 gives a total of $4,144.63, which again matches the correct formula.

I hope this helps to simplify and understand the calculations for compounding interest. It's important to remember that regular contributions play a significant role in the growth of investments and should not be overlooked in calculations.