## What is the Rule of 72?

The Rule of 72 is a mathematical principle that estimates the time it will take for an investment to double in value. The mention of math might make your jaw clench, but the Rule of 72 is actually a very basic formula that anyone can use.

Simply take the number 72 and divide it by the interest earned on your investments each year to get the number of years it will take for your investments to grow 100%. Or to drop by 50%.

However, you can only apply this rule to compounding growth or decay. In other words, you can only use it for investments that earn compound interest, not simple interest. With simple interest, you only earn interest on the principal amount you invest. Compound interest is “interest earned on interest”: It accrues on accumulated interest, in addition to the principal.

Because interest is essentially being added into your principal, and used as the base for fresh interest calculations, compounding makes your investment grow exponentially. Meaning: As interest accrues and the quantity of money increases, the rate of growth becomes faster.

It doesn't have to be investment interest; anything that augments your principal creates “the magic of compounding.” For example, if you reinvest the dividends you earn on your investments, your earnings are being compounded. Therefore, the Rule of 72 applies.

On the other hand, if you choose to withdraw your dividends rather than reinvest them, your earnings might not compound, and the Rule of 72 wouldn't work.

## How to calculate the Rule of 72

To calculate the Rule of 72, all you have to do is divide the number 72 by the rate of return. You can use the formula below to calculate the doubling time in days, months, or years, depending on how the interest rate is expressed. For example, if you input the annualized interest rate, you'll get the number of years it will take for your investments to double.

You'll notice the formula uses the “approximately equals” symbol (≈) rather than the regular “equals” symbol (=). That's because this formula offers an estimate rather than an exact amount, and it's most accurate when used on investments that earn a typical rate of 6% to 10%.

While usually used to estimate the doubling time on a growing investment, the Rule of 72 can also be used to estimate halving time on something that's depreciating.

For example, you can use the Rule of 72 to estimate how many years it will take for a currency's buying power to be cut in half due to inflation, or how many years it will take for the total value of a universal life insurance policy to decline by 50%. The formula works exactly the same either way — simply plug in the inflation rate instead of the rate of return, and you'll get an estimate for how many years it will take for the initial amount to lose half its value.

## 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.

## Who Came Up With the Rule of 72?

The Rule of 72 dates back to 1494 when Luca Pacioli referenced the rule in his comprehensive mathematics book called Summa de Arithmetica. Pacioli makes no derivation or explanation of why the rule may work, so some suspect the rule pre-dates Pacioli's novel.2

## 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:

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.

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