Learn how to use the compound interest formula and how it’s related to the rule of 72.
In the last several articles we’ve talked about the compound interest formula and the rule of 72. Now that we know what these things are, it’s time to talk a bit more about how, why, and when you can use them. While we’re at it, we’ll also finally solve the mystery of why the rule of 72 works in the first place. And we’ll even see that sometimes it doesn’t actually work at all!
SPONSOR: http://stitcher.com/math. Use the code MATH to be entered to win $1000.
Recap: the Compound Interest Formula and the Rule of 72?
As we’ve found, you can use the compound interest formula to calculate how long it will take money to grow. Which means that you can use it to figure out how long it might take for your money to double. But hang on a minute! You might be thinking: “Isn’t that exactly what the rule of 72 is for? Doesn’t the rule of 72 just say that you can find the doubling time of an investment by dividing the number 72 by the interest rate?” Why yes, yes it does. And the rule of 72 certainly works wonderfully…so long as you’re interested in the doubling time.
But what if you’re interested in knowing how long it will take to grow your money by 20%? Or 50%? Or how about the time it takes to triple your money? Well, for that you need something more than the rule of 72. And that something is the compound interest formula that we discovered in the last article.
How to Use the Compound Interest Formula
Let’s say you have $1000 invested in an account earning 5% annual interest and you want to figure out how long it’s going to take to triple your money. In other words, how long will it take for your account to be worth $3000? Well, the compound interest formula says that:
FV = PV x (1 + rate)^years
In this case we can plug our present value PV = $1000, the future value that we’re interested in FV = $3000, and the interest rate = 0.05 (the decimal form of 5%) into this equation to figure out how many years it’s going to take. After plugging in and simplifying, we’re left with the equation:
1.05^years = 3
You can solve this equation using algebra and something called logarithms, or you can solve it using trial and error by trying different numbers of years until you get close enough to the answer. Today, we’ll stick to the latter method and save logarithms for a future date. In this case, 22 years gives 1.05^22 = 2.93 (which being a bit less than 3 is too small) and 23 years gives 1.05^23 = 3.07 (which being a bit more than 3 is too big). So the real answer must be somewhere in between. As it turns out, and as you can check by plugging numbers in, it takes about 22.5 years for an account earning 5% interest to triple.
How is the Compound Interest Formula Related to the Rule of 72?
Instead of using the compound interest formula to figure out how long it takes your money to triple, we could have used it to figure out how long it takes your money to double—exactly what we do using the rule of 72. Does that mean that the two things are related? Yes! Let’s return to the compound interest formula to see why. The formula says that
FV = PV x (1 + rate)^years
Now, if we’re talking about the doubling time as we are with the rule of 72, then FV / PV = 2…since that’s what it means to double! So for the rule of 72, the compound interest formula looks like:
(1 + rate)^years = 2
Now, we’re not going to go into all the details here, but it turns out that you can use logarithms and an approximation having to do with logarithms to turn this into the equation that you know as the rule of 72:
years = 72 / rate
If you’re interested in seeing more of the details, check out the bonus section at the very end of the article. But remember, the details aren’t all that important right now. The important thing is to understand that the rule of 72 isn’t some bit of magic—it’s an approximation to its bigger and more powerful sibling: the compound interest formula.