The doubling time formula is used in finance to calculate the amount of time that it takes for a certain amount of money to double in 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 doubling time?

There are several methods of doubling time calculation.

1. Doubling time can be calculated by dividing the natural logarithm of 2 by the natural logarithm of the exponent of growth (1+ rate per period).
• $Doubling\;time = \frac{ln2}{ln(1+r)}$
2. Doubling time can also be calculated with help of the rule of 72, the rule of 70 or the rule of 63.9. These methods might seem much easier 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.
• $Doubling\;time = \frac{Rule\;number}{Rate\;per\;period}$

In all those methods, 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.

### Examples of doubling time calculation

Example 1. Let’s assume you want to invest your money and want to find out how long it would take it to double. The annual interest rate is 12%. Your earnings are compounded monthly.

#### Step 1. Monthly rate

Since the earnings are compounded monthly and the interest rate is annual, we should first calculate the monthly interest rate.

$Monthly\;interest\;rate = \frac{12}{12} = 1\%$

#### Step 2. Doubling time

$Doubling\;time = \frac{ln2}{ln 1.01} = 69.66\;months$

That means it would take you 69.66 months to double your money or about 5.8 years, which is 5 years and 7 months. (0.8 years is 6.7 months).

We can also calculate the doubling time using the rule of 72, the rule of 70 and the rule of 69.3.

$Doubling\;time_{72} = \frac{72}{1} = 72\;months$$Doubling\;time_{70} = \frac{70}{1} = 70\;months$$Doubling\;time_{69.3} = \frac{69.3}{1} = 69.3\;months$

The rule of 69.3 is the most accurate method because the result is the closest to the original doubling time calculation. In this case, the rate was 1% so it was easy to use the rule of 69.3 method.

Example 2. Let’s assume you want to invest your money and want to find out how long it would take it to double. The annual interest rate is 5%. Your earnings are compounded annually.

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In this example, it would take you 14.2 years to double your money.

[latex]Doubling\;time_{72} = \frac{72}{5} = 14.4\;years$

$Doubling\;time_{70} = \frac{70}{5} = 14\;years$$Doubling\;time_{69.3} = \frac{69.3}{1} = 13.86\;years$

As I mentioned before, 72 is better for annual compounding, so it’s the most accurate method in this case.