Rule of 72

There is a curious and helpful trick that allows us to mentally estimate annual compound interest amounts, where we are interested in doubling our money.

The Rule of 72 works as follows.

If we want to know how long it will take for our money to double, just divide `72` by the interest rate.

So for example, if the interest rate is `10%`,

72 ÷ 10 = 7.2 years

So it will take just over `7` years to double our money.

If the interest rate is `8%`, to double our money it will take

72 ÷ 8 = 9 years

Here is a graph showing $1 doubling at different interest rates.

Finding Interest

We can use the Rule of 72 the other way around too. Say we have a 15 year time span and we want to double our money in that time. What interest rate do we need so that the money will double?

Answer: 72 ÷ 15 = 4.8%

How Does Rule of 72 Work?

From the last section (Interesting Interest), the amount of money we have after investing P dollars for t years at r% interest (as a decimal) is given by:

A = P(1 + r)^t

We want to know how long it takes to double our $A to $2A.

2A = A(1 + r)^t

Cancelling gives:

2 = (1 + r)^t

Using logarithms to solve this equation, we have (recall `ln` means `log_e`):

ln 2 = t ln(1 + r)

`t=(ln\ 2)/(ln(1+r))`

We can find the value of the right hand side for different values of r. When we multiply these values by r, an interesting thing occurs − the values are very near `72`.

Example

If `r = 3% = 0.03`, then:

`t=(ln\ 2)/(ln(1+r))=(ln\ 2)/(ln (1.03))=23.45`

This means it would take more than 23 years to double our money at an interest rate of 3%.

Now multiplying 23.45 by 3, we find

`23.45 xx 3 = 70.35`.

We see that this (value of years) `xx` (interest rate) is quite close to 72.

Range of interest rates

Let’s now do the same for a range of typical interest rates from r = 2 through to r = 14.

We get:

Rate	Years	Rate `xx` Years
2%	35.00	70.01
3%	23.45	70.35
4%	17.67	70.69
5%	14.21	71.03
6%	11.90	71.37
7%	10.24	71.71
8%	9.01	72.05
9%	8.04	72.39
10%	7.27	72.72
11%	6.64	73.06
12%	6.12	73.40
13%	5.67	73.73
14%	5.29	74.06

(Rounded to 2 decimal places)

We observe that the values in the last column are near `72`. So we can approximate t (the time it takes to double our money for a given interest rate, r) as:

`t~~72/r`

Equivalently, we can approximate r (the interest rate needed to double our money for a given interest time, t) as:

`r~~72/t`

Here is a graph of the curve `r=72/t` with the values given in the above table (the pink dots). We can see the curve very closely approximates the situation for a wide range of interest rates.

Conclusion

The Rule of 72 gives us an easy "back of the envelope" calculation for the time it will take to double our money. It is good for a range of typical interest rates, from `5%` to about `12%`. Even for high interest rates like `20%`, the value is `76.04`.

If you are interested, go back to Interesting Interest.