Learn how mathematical series and the compound interest formula can be used in the real world to calculate how fast money grows in recurring investments.
How Does Money Grow in Recurring Investments?
Now that we all remember what mathematical series are, we’re ready to see how they apply to the real world. In this particular case, the series we’re talking about is a finite series, and it’s related to the idea of a recurring investment that we talked about in the last article. As you’ll recall, a recurring investment is one in which money is periodically put into an interest-earning account.
To see how this works, let’s review the basic way to calculate the value of such an investment some number of years after you start saving your money. In particular, let’s say that you invest $2000 at the start of every year into an account earning 5% interest. Using the compound interest formula, we find that the value of the account at the end of the first 3 years is:
Value at End of Year 1 = $2000 x 1.05^1 = $2100
Value at End of Year 2 = $2000 x 1.05^1 + $2000 x 1.05^2 = $4305
Value at End of Year 3 = $2000 x 1.05^1 + $2000 x 1.05^2 + $2000 x 1.05^3 = $6620.25
Are you noticing a pattern? If you look closely, you’ll see that this sequence of expressions for the value of the investment at the end of each year form a mathematical series where each term is the amount of money invested each year—$2000—multiplied by the interest rate raised to a higher and higher integer power.
How Mathematical Series Solve Real World Problems
As it turns out, we can use this fact to create a formula analogous to the compound interest formula that allows us to find the value of the account any number of years in the future. Why is that such a big deal? Because it lets us skip all the work of calculating the value at the end of each year, and instead jump right to the final value in the future. We won’t go through every last detail right now (because it gets a little complicated), but if you’re curious to see those details they’re available as a bonus section at the end of this article.
[[AdMiddle]For now, all you need to know is that since the expressions for the value of a recurring investment form a mathematical series, it’s possible to use some cool (but slightly more advanced) math to figure out what that series is going to add up to. The end result is a formula for the value of a recurring investment at any point in the future. The formula says:
(future value) = (previous value) x [ (1+rate) / rate ] x [ (1+rate)^years - 1 ]
Yes, I know that’s a mouthful, but try it out and you’ll see that it really does work for 3, 5, 20, 100, or whatever number of years you want! It’s a very handy formula to have in your bag of mathematical tricks.