Financial Accounting Home Page

Non Accounting Home Page

Learn the Rule of 72

Why the Rule of 72 works

Introduction

There is a lot to think about when we start to learn about and use compound interest. One of the questions that bothers a lot of people is the answer to the question, "How long will it take me to double my money?" Of course, it is possible to answer that question fairly readily once we have understood the basics of compound interest formulae. However, what people really want to know is, "Is there an easy way to work out how long it will take me to double my money?"

Good news! There is an easy way to estimate the doubling point and here it is! Are you ready to find out how simple it is?

To estimate how long it would take to double you money on an investment just divide 72 by the percentage rate you are earning on your investment; and that's it.

For example, imagine you have a savings account with £500 deposited in it. The rate of interest is 4% per year. So the doubling point, the length of time it will take you to double your £500 to become £1,000 is:

72 divided by 4 = 18 years

If the rate of interest were 6%, we would estimate the doubling point to be 72/6=12 years.

And so on.

Please note, to save time and space, we are only discussing the rule of 72 in the context of an investment; but the method and mathematics are exactly the same when we deal with a loan.

Accuracy

Does the rule of 72 give us the exact doubling point, though? Well, that's a direct question and the direct answer is No!

Not much use then, really, is it? Well, we have said all along that it gives us an estimate; and if we take a look at the table that follows, we'll see how accurate/inaccurate it is:

	Actual doubling points	Rule of 72 estimate	% error
1.0%	69.6607	72.0000	3.36%
2.0%	35.0028	36.0000	2.85%
4.0%	17.6730	18.0000	1.85%
6.0%	11.8957	12.0000	0.88%
8.0%	9.0065	9.0000	-0.07%
10.0%	7.2725	7.2000	-1.00%
20.0%	3.8018	3.6000	-5.31%
30.0%	2.6419	2.4000	-9.16%
40.0%	2.0600	1.8000	-12.62%
50.0%	1.7095	1.4400	-15.77%

So, not that bad is it? In fact, if we concentrate on the rates of interest from 1% to around 30%, the estimation errors are not that wild; and since we know that the rule of 72 only gives estimates, we have nothing to be afraid of, have we.

As a matter of interest, if we were to investigate how wild the rule of 72 might be, if the rate of interest were, say, 400%, the estimation error would be just over 58%; but then we don't normally pay or earn 400% interest do we?

Some Maths

If you are of a nervous disposition, you might want to look away now! Let's take a look at some of the mathematics behind the doubling point.

We know that the rule of 72 is a decent estimator of the doubling point. We also see from the table above that there are actual values for the doubling point: how did we calculate the actual doubling points?

Compound interest formula: P(1 + r/100)^n

That is when we invest an amount of money, P, at a rate of interest, r, for a number of years, n, we can calculate the current balance on our investment by using that formula. It's a standard formula and we can find it in many accounting, finance, banking and business studies books; and the symbol "^" means raised to the power of: for example 2^2 means 2 squared, 2^4 means 2 to the power of 4 ...

We want to find out when our investment will be double what we started with. We started with P, so we want to know when we will have 2P (I know, I know, there's a joke in there: 2P or not 2P, that is the question ... OK, OK, done that now!)

Assuming that our rate of interest is 10%, our formula, is P(1 + r/100)^n = 2P

We cancel the P's to get: (1 + r/100)^n = 2

Simplify to get 1.1^n = 2

Now we're stumped because we can't possible continue: we are trying to find 'n' but to find 'n', we need to know 'n'!

Tarraaaa! We can use natural logarithms at this point. Mathematicians will tell us that ln(a^b) = b*ln(a)

So, n*ln(1.1) = ln(2)

Giving us n*(0.09531) = 0.693147

And finally n = 0.693147/0.09531 = 7.2725527

Which means that at 10%, your money doubles in about 7.272 years. So the rule of 72 gave us a good estimate.

For you to do: use the rule of 72 and the mathematical method to evaluate the doubling points for 5%, 14% and 21%

Check your answers here!

	Actual doubling points	Rule of 72 estimate	% error
5%	14.2067	14.4000	1.36%
14%	5.2901	5.1429	-2.78%
21%	3.6363	3.4286	-5.71%

Conclusion

Overall, the rule of 72 is a neat little trick that can be useful in situations where we might need to do some pretty quick calculations but when we might not be glued to our spreadsheet or our calculator. As as rule of thumb method it's a good one since within a relatively wide range of interest rates it fairly accurate.

Duncan Williamson
24 February 2002 revised August 2003