Doubling Time Formula

What is Doubling Time?

Doubling time is referred 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,

where

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

In the case of continuous compounding formula, 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 69.

However, the above formula is also modified as the rule of 72 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 70 in vogue, which is used just for the ease of calculation.

Doubling Time Calculation (Step by Step)

• Step 1: Firstly, determine the rate of annual return for the given investment. The annual rate of interest is denoted by ‘r.’
• Step 2: 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)
• Step 3: 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
• Step 4: 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%

#1 – Daily Compounding

Since daily compounding, therefore n = 365

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

• = ln 2 / [365 * ln (1 + 10%/365)
• = 6.9324 years

#2 – Monthly Compounding

Since monthly compounding, therefore n = 12

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

• = ln 2 / [12 * ln (1 + 10%/12)
• = 6.9603 years

#3 – Quarterly Compounding

Since quarterly compounding, therefore n = 4

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

• = ln 2 / [4 * ln (1 + 10%/4)
• = 7.0178 years

#4 – Half  Yearly Compounding

Since half yearly compounding, therefore n = 2

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

•  = ln 2 / [2 * ln (1 + 10%/2)
• = 7.1033 years

#5 – Annual Compounding

Since annual compounding, therefore n = 1,

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

•  = ln 2 / [1 * ln (1 + 10%/1)
• = 7.2725 years

#6 – Continuous Compounding

Since continuous compounding,

Doubling time = ln 2 / r

•  = ln 2 / 10%
• = 6.9315 years

Therefore, the calculation for various compounding periods will be –

The above example shows that the doubling time depends not only on the rate of annual return of the investment but also on no. of compounding periods per year and it increases with the increase in the frequency of compounding per year.

Relevance and Use

It is important that an investment analyst understands the concept of doubling time because it helps them to roughly estimate how many years it will take for the investment to double in value. Investors, on the other hand, use this metric to evaluate various investments or the growth rate for a retirement portfolio. In fact, it finds application in the estimation of how long a country would take to double its real gross domestic product (GDP).

Recommended Articles

This article has been a guide to Doubling Time and its meaning. Here we learn the formula to calculate doubling time for different compounding periods along with some practical examples along with downloadable excel templates. You may learn more about excel modeling from the following articles –

• Formula of Monthly Compound Interest
• Daily Compound Interest
• Effective Annual Rate Formula
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