 # The Rule of 72 Doesn’t Work!

The Rule of 72 is often used by finance people to show how long it will take an investment to double. It’s a quick and easy way to see how a few percentage points will affect the growth of your investment over time. When you are considering two investments a half of a percentage point might not seem like a big difference. But when you see how it affects the doubling time you might change your mind. A savings account that earns 2% will take 34.69 years to double, while an account earning 2.5% will only take 27.75 years. That’s a 7 year difference.

With the savings rates being driven down by the economy it makes a huge difference to us savers. A few years ago I was able to get a CD earning 5%. Now I’m lucky to see 3%. While 2% might not seem like a big deal, it will take me a full 10 years longer to double my money.

## What is the Rule of 72?

The Rule of 72 is a quick and easy way to see how long it will take your money to double at a given interest rate. You take 72 and divide it by the interest rate. So for example, if you are earning 2% it will take 36 years for your money to double (72/2). The amount of money that is doing the doubling doesn’t matter. It will take \$1 to turn to \$2 the same amount of time it will take \$100 to turn to \$200 if both sets are earning the same interest rate.

## What’s Good About the Rule of 72?

What’s great about the Rule of 72 is that you don’t need a scientific calculator to determine a rough estimate of when your money will double. To calculate the actual formula for doubling time you need a calculator that can do logarithms. Since most people don’t carry around such a calculator, it’s nice to have a simple, easy to use equation.

## But the Rule of 72 Doesn’t Work!

First off we have the issue of compounding frequency. How often the interest is added to the balance will affect how quickly the original amount will double. When considering interest rates between 2% and 20% the Rule of 72 only provides an accurate estimate when interest is compounding annually or semi-annually and at very limited interest rates. I’ve determined a result within .03 years (about 11 days) of the actual doubling time to be considered accurate.
The rule of 72 is accurate in the following situations:

• Compounded annually with interest rates between about 7% and 9%.
• Compounded semi-annually with rates between about 13% and 20%.

Let’s take our example above of earning 2%, which sadly is about what you are probably earning on your savings account right now. The Rule of 72 says it will take your balance 36 years to double. Your savings account probably compounds interest monthly, so at 2% it would actually take 34.69 years. Which is about a year and 3 months quicker than the Rule of 72 says it would take.

## So what works better? The Rule of 70!

The Rule of 70 provides more accurate doubling times for more interest rates between 2% and 20% and for more compounding frequencies than the Rule of 72. Simply divide 70 by the interest rate rather than 72.
The Rule of 70 is accurate in the following situations:

• Compounding annually with rates of 2% and 2.25%.
• Compounding semi-annually with rates between 2.5% and 5.5%.
• Compounding quarterly with rates between 5.75% and 11.75%.
• Compounding monthly with rates between 11.25% and 20%. As you can see, the Rule of 70 works much more often than the Rule of 72. And even when the Rule of 70 is providing it’s least accurate results it is still often times more accurate than the Rule of 72. For example, lets look back at our savings account earning 2%. The Rule of 70 says that our money should double in 35 years. We’ve already determined that the actual doubling time is 34.69. So the rule of 70 is only about 3 months off, while the Rule of 72 was a full year and 3 months off. Clearly, the Rule of 70 is a more useful tool for estimating doubling time than the Rule of 72 ever was. For a more detailed analysis, download the Rule of 72 vs. 70 (quarter percentages)

So what’s your opinion? Which is better? The Rule of 72 or 70?

1. Frank says

I’ll stick with 72.

Usually the interest rate used on deposits (and loans) is in the form of APR, which means that annual is the correct compounding period to use. And with annual, as your spreadsheet shows, 72 is more accurate above 5% and not that shabby below. Further, 72 has more factors and so is easier to use to estimate compounding in your head, which is the whole point.

2. BayouJosh says

I graduated with a math minor, so I know enough to appreciate it, but not nearly enough to use it. I always enjoy seeing the logic behind the formulas though. Thanks.

3. Jared Grubb says

Im an engineer and a math nerd, so I enjoy it 🙂
In case you want to see the beauty:

Y = ln(2) / (n * ln(1 + r/n))

where Y = number of years, n = how often interest compounds in a year (e.g, n=12 for monthly), and r is rate (as a decimal, 5% means r = 0.05). So then, if we want a “rule” to use as an approximate guide, we take that and multiply by the rate (so that when we divide by the rate later, we get the number of years).

I personally always use “Rule of 70” for my own use, as it tends to be pretty good for typical APR’s and compounding schemes.

4. BayouJosh says

Great post Ashley. Very interesting.

Jared, please tell me you used a calculator to do all that and that your head hurt when you were done. 🙂 If not, you might want to try laying off the algebra before bedtime! Never really thought of it this way though, so thanks!

~bayou

5. Jared Grubb says

If interest were to accrue continuously (meaning every smidgen of a second, rather than daily, weekly, monthly, etc), then it’s actually the “Rule of 69.31” (the natural logarithm of 2).

If you move from “continuously” to daily, or weekly, interest accrues slower, so you have to increase the 69.31 upwards. And the adjustment depends on both the rate and the accrual period. For example:
5% daily => Rule of 69.32
20% daily => Rule of 69.33
5% monthly => Rule of 69.46
20% monthly => Rule of 69.89
5% annually => Rule of 71.03
20% annually => Rule of 76.04 