Doubling Time Formula

What is Doubling Time?

Doubling time refers to the time period required to double the value or size of investment, population, inflation etc and is calculated by dividing the log of 2 by the product of number of compounding per year and the natural log of one plus the rate of periodic return.

Doubling Time Formula

Mathematically, the doubling time formula is represented as,

Doubling Time = ln 2 / [n * ln (1 + r/n)]

You are free to use this image on your website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Doubling Time Formula (wallstreetmojo.com)

where

r = rate of annual return
n = no. of compounding period per year

In the case of continuous compounding formulaContinuous Compounding FormulaThe continuous compounding formula depicts the interest received when constant compounding is done for an infinite number of periods. The four variables used for its computation are the principal amount, time, interest rate and the number of the compounding period.read more, the calculation of doubling time in terms of years is derived by dividing the natural log of 2 by the rate of annual return (since (1 + r/n) ~ e^r/n).

Doubling time = ln 2 / [n * ln e^r/n]

= ln 2 / [n * r / n]
= ln 2 / r

where r = rate of return

The above formula can be further expanded as,

Doubling time = 0.69 / r = 69 / r% which is known as rule of 69Rule Of 69The Rule of 69 is a common rule for estimating the time it will take to double an investment with a continuous compounding interest rate. It does not provide an exact time, but it does provide a near approximation without relying on a mathematical formula.read more.

However, the above formula is also modified as the rule of 72The Rule Of 72Rule of 72 is an estimated approach of calculating the time required to double the invested amount at a fixed interest rate. This is determined as a ratio of 72 to the annual interest rate. read more because practically continuous compounding is not used, and hence 72 gives a more realistic value of the time period for less frequent compounding intervals. On the other hand, there is also the rule of 70Rule Of 70The “Rule of 70” refers to the total time it takes to double a quantity or value. It simply means how long it will take to double the money, investments, or profit assuming all other factors remain constant.read more in vogue, which is used just for the ease of calculation.

Doubling Time Calculation (Step by Step)

Follow the below steps:

Firstly, determine the rate of annual return for the given investment. The annual rate of interest is denoted by ‘r.’
Next, try to figure out the frequency of compounding per year, which can be 1, 2, 4, etc., corresponding to annual compounding, half-yearly, and quarterly, respectively. The number of compounding periods per year is denoted by ‘n.’ (The step is not required for continuous compounding)
Next, the rate of periodic return is calculated by dividing the rate of annual return by the number of compounding periods per year. Rate of periodic return = r / n
Finally, in case of discrete compounding, the formula in terms of years is calculated by dividing the natural log of 2 by the product of no. of compounding period per year and the natural log of one plus the rate of periodic return as Doubling time = ln 2 / [n * ln (1 + r/n)]
On the other hand, in the case of continuous compounding, the formula in terms of years is derived by dividing the natural log of 2 by the rate of annual return as,
Doubling time = ln 2 / r

Example

Let us take an example where the rate of annual return is 10%. Calculate the doubling time for the following compounding period:

Daily
Monthly
Quarterly
Half Yearly
Annual
Continuous

Given, Rate of annual return, r = 10%