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# Rule of 72

## What Is the Rule of 72?

The Rule of 72 is a quick, useful formula that is popularly used to estimate the number of years required to double the invested money at a given annual rate of return.

While calculators and spreadsheet programs like Microsoft's Excel have inbuilt functions to accurately calculate the precise time required to double the invested money, the Rule of 72 comes in handy for mental calculations to quickly gauge an approximate value. Alternatively, it can compute the annual rate of compounded return from an investment given how many years it will take to double the investment.

### Key Takeaways

• The Rule of 72 is a simplified formula that calculates how long it'll take for an investment to double in value, based on its rate of return.
• The Rule of 72 applies to compounded interest rates and is reasonably accurate for interest rates that fall in the range of 6% and 10%.
• The Rule of 72 can be applied to anything that increases exponentially, such as GDP or inflation; it can also indicate the long-term effect of annual fees on an investment's growth.

## The Formula for the Rule of 72

﻿ \begin{aligned} &\text{Years to Double} = \frac{ 72 }{ \text{Interest Rate} } \\ &\textbf{where:}\\ &\text{Interest Rate} = \text{Rate of return on an investment} \\ \end{aligned}﻿

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## How to Use the Rule of 72

The Rule of 72 could apply to anything that grows at a compounded rate, such as population, macroeconomic numbers, charges, or loans. If the gross domestic product (GDP) grows at 4% annually, the economy will be expected to double in 72 / 4 = 18 years.

With regards to the fee that eats into investment gains, the Rule of 72 can be used to demonstrate the long-term effects of these costs. A mutual fund that charges 3% in annual expense fees will reduce the investment principal to half in around 24 years. A borrower who pays 12% interest on their credit card (or any other form of loan that is charging compound interest) will double the amount they owe in six years.

The rule can also be used to find the amount of time it takes for money's value to halve due to inflation. If inflation is 6%, then a given purchasing power of the money will be worth half in around 12 years (72 / 6 = 12). If inflation decreases from 6% to 4%, an investment will be expected to lose half its value in 18 years, instead of 12 years.

Additionally, the Rule of 72 can be applied across all kinds of durations provided the rate of return is compounded annually. If the interest per quarter is 4% (but interest is only compounded annually), then it will take (72 / 4) = 18 quarters or 4.5 years to double the principal. If the population of a nation increases at the rate of 1% per month, it will double in 72 months, or six years.

## Rule of 72 FAQs

### Who Came Up With the Rule of 72?

People love money, and they love to see it grow even more. Getting a rough estimate of how much time it will take to double your money also helps the average Joe or Jane to compare different investment options. However, mathematical calculations that project an investment's appreciation can be complex for common individuals to do without the help of log tables or a calculator, especially those involving compound interest.

The Rule of 72 offers a useful shortcut. It's a simplified version of a logarithmic calculation that involves complex functions like taking the natural log of numbers. The rule applies to the exponential growth of an investment based on a compounded rate of return.

### How Do You Calculate the Rule of 72?

Here's how the Rule of 72 works. You take the number 72 and divide it by the investment's projected annual return. The result is the number of years, approximately, it'll take for your money to double.

For example, if an investment scheme promises an 8% annual compounded rate of return, it will take approximately nine years (72 / 8 = 9) to double the invested money. Note that a compound annual return of 8% is plugged into this equation as 8, and not 0.08, giving a result of nine years (and not 900).

If it takes nine years to double a $1,000 investment, then the investment will grow to$2,000 in year 9, $4,000 in year 18,$8,000 in year 27, and so on.

### How Accurate Is the Rule of 72?

The Rule of 72 formula provides a reasonably accurate, but approximate, timeline—reflecting the fact that it's a simplification of a more complex logarithmic equation. To get the exact doubling time, you'd need to do the entire calculation.

The precise formula for calculating the exact doubling time for an investment earning a compounded interest rate of r% per period is:

﻿ \begin{aligned} &T = \frac{ \ln( 2 ) }{ \ln \left ( 1 + \frac{ r } { 100 } \right ) } \simeq \frac{ 72 }{ r } \\ &\textbf{where:}\\ &T = \text{Time to double} \\ &\ln = \text{Natural log function} \\ &r = \text{Compounded interest rate per period} \\ &\simeq = \text{Approximately equal to} \\ \end{aligned}﻿

To find out exactly how long it would take to double an investment that returns 8% annually, you would use the following equation:

• T = ln(2) / ln (1 + (8 / 100)) = 9.006 years

As you can see, this result is very close to the approximate value obtained by (72 / 8) = 9 years.

### What Is the Difference Between the Rule of 72 and the Rule of 73?

The Rule of 72 primarily works with interest rates or rates of return that fall in the range of 6% and 10%. When dealing with rates outside this range, the rule can be adjusted by adding or subtracting 1 from 72 for every 3 points the interest rate diverges from the 8% threshold. For example, the rate of 11% annual compounding interest is 3 percentage points higher than 8%.

Hence, adding 1 (for the 3 points higher than 8%) to 72 leads to using the Rule of 73 for higher precision. For a 14% rate of return, it would be the rule of 74 (adding 2 for 6 percentage points higher), and for a 5% rate of return, it will mean reducing 1 (for 3 percentage points lower) to lead to the Rule of 71.

For example, say you have a very attractive investment offering a 22% rate of return. The basic rule of 72 says the initial investment will double in 3.27 years. However, since (22 – 8) is 14, and (14 ÷ 3) is 4.67 ≈ 5, the adjusted rule should use 72 + 5 = 77 for the numerator. This gives a value of 3.5 years, indicating that you'll have to wait an additional quarter to double your money compared to the result of 3.27 years obtained from the basic Rule of 72. The period given by the logarithmic equation is 3.49, so the result obtained from the adjusted rule is more accurate.

For daily or continuous compounding, using 69.3 in the numerator gives a more accurate result. Some people adjust this to 69 or 70 for the sake of easy calculations.

The Rule of 72 applies to cases of compound interest, and not to the cases of simple interest.

The interest rate charged on an investment or a loan broadly falls into two categories—simple or compounded.

• Simple interest is determined by multiplying the daily interest rate by the principal amount and by the number of days that elapse between payments. It is used for calculating interest on investments where the accumulated interest is not added back to the principal.
• For compound interest, the interest is calculated on the initial principal and also on the accumulated interest of previous periods of a deposit. Compound interest can be thought of as “interest on interest,” and it will make the invested money grow to a higher amount at a faster rate compared to that from the simple interest, which is calculated only on the principal amount.