The Rule of 72 formula is one of the simple and quick methods that are used to calculate an investment’s doubling time. Doubling time is the amount of time that it takes for a certain amount of money to double in its value.

Doubling time is applied not only to money but also to other resources and investments, inflation, consumption of goods and services, population growth and many other things that grow exponentially over time.

Exponential growth means the bigger something gets, the faster it grows or, in other words, the amount that is being added is proportional to the amount that was already present, i.e. the additional rate is constant.

In finance, doubling time is used to find out how much time it takes for an investment or a sum of money to double its value at a constant growth rate.  The larger the rate of growth, the faster the doubling time.

### How to calculate the doubling time using the rule of 72?

Doubling can be quickly calculated using the rule of 72, the rule of 70 or the rule of 63.9. These methods might seem much easier than the original doubling time formula since you only have to divide the rule number by the rate per period.

However, standard doubling time formula is more accurate, and you may consider using the rule of 72/rule of 70/rule of 69.3 when you can’t use a scientific calculator or computer programs. The original doubling time formula is:

$Doubling\;time = \frac{ln2}{ln(1+r)}$

it’s important to notice that r is the rate per period. That means if in some cases compounding happens monthly or quarterly and the given rate is annual, you should calculate your monthly or quarterly rate. Doubling time will be expressed in months/quarters.

### How to choose the rule number

• 72 is more convenient.
• 72 number is a convenient choice because you can divide 72 by many small numbers: 1,2,3,4,6,8,9,12. So, this number is helpful for mental calculations and small rates.
• 72 is less accurate for higher rates.
• 72 is good for calculating annual compounding (Compounding is a process of reinvesting interest or capital gains to generate more earnings. An investment then generates profit not only from its initial principal but also from interest or capital gains from the previous periods.).
• 69.3 or 70 are more accurate.
• 69.3 is the most accurate number to use because ln(2) is approximately 69.3, but because this number is not convenient for mental calculations you can round it up to 70.
• 69.3 or 70 is good for calculating continuous compounding and daily compounding.

Here are some doubling time numbers calculated using the original doubling time formula, the rule of 72, the rule of 70 and the rule of 69.3:

 Rate % Original formula Rule of 72 Rule of 70 Rule of 69.3 0.5 138.98 144 140 138.6 1 69.66 72 70 69.3 2 35 36 35 35 3 23.45 24 23.33 23.1 4 17.67 18 17.5 17.33 5 14.21 14.4 14 13.86 6 11.9 12 11.67 11.55 7 10.25 10.29 10 9.9 8 9 9 8.75 8.66 9 8.04 8 7.78 7.7 10 7.27 7.2 7 6.93 11 6.64 6.55 6.36 6.3 12 6.12 6 5.83 5.78

### Examples of doubling time calculation using the rule of 72

Example 1. How long will it take to double your money if the interest rate is 8% and it is compounded annually?

$Doubling\;time_{72} = \frac{72}{8} = 9\;years$$Doubling\;time_{70} = \frac{70}{8} = 8.75\; years$$Doubling\;time_{69.3} = \frac{69.3}{8} = 8.66\; years$$Doubling\;time = \frac{ln2}{ln1.08} = 9\; years$

It will take 9 years to double. In this example, the rule of 72 is the most accurate and easiest method of calculation.

Example 2. How long will it take to double your money if the interest rate is 14% and it is compounded monthly?

First, we need to derive the monthly rate because compounding happens monthly.

$Monthly\;rate = \frac{14}{12} = 1.17\%$$Doubling\;time_{72} = \frac{72}{1.17} = 61.54\; months$$Doubling\;time_{70} = \frac{70}{1.17} = 59.83\; months$$Doubling\;time_{69.3} = \frac{69.3}{1.17} = 59.23\; months$$Doubling\;time = \frac{ln2}{ln1.0117} = 59.59\; months$

According to the original (and the most accurate) formula, it will take 59.59 months or almost 5 years. However, for a quick estimation of this number, we can use the rule of 70, which happens to be the most accurate, comparing to the rule of 72 and 69.3.