Here’s a handy rule of thumb for calculating compound interest:
If you want to know how many years it will take your money to double, if it grows at a yearly rate of r, just divide 72 by r.
For example, how long would it take for your money to double if it grew at a yearly rate of 5%? years. And, sure enough, !
So let’s say you’re trying to figure out how much money you’ll have saved for retirement if you save now $100,000 and it grows at an annual rate of 5% for the next 30 years? It will double every 14.4 years, so in 28.8 years it will double twice, so it will be a little more than $400,000. How’d we do? . Pretty good for something you can do in your head!
Probably obvious, but this trick can also convert between a “doubling time” and an interest rate. If I tell you that your money will double in 10 years, you know the interest rate is about . And, sure enough, . Very close!
How does it work?
What’s the equation we’re trying to solve?
- r is the interest rate
- y is the doubling time (the number of years it takes to double)
- and 2 is because we want our money to double
We need to turn an exponent into multiplication/division, which usually means taking the log of both sides. Let’s try it:
So far, everything we’ve done is exact. Now it’s time to make a few approximations:
The first one is trivial, you can just check it with a calculator. Why, though, is the second one true?
You can think about it this way. . And the derivative of at 0 is also 1. So, if you zoom in really close around 0, it looks like a straight line with a y-intercept of 1 and a slope of 1. Which means, for small values of r, . And if we take the natural log of both sides, we get .
So, using what we have so far, we can say:
This works just fine, especially for really small values of r. For example, how long would it take for your money to double at a 1% interest rate? 69.3 years right? . Close!
So where does 72 come from?
Well, gets to be a worse approximation as gets large. In particular, is an overestimate for .
So, when we divide by in , we’re dividing by something that’s too large. For “normal” interest rate values - say, 8% - is about 4% bigger than . So, to adjust for that fact, we can just make numerator 4% bigger as well. What’s ? 0.72!