# 5. Rule of 72 Definition¶

## 5.1. 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 excel sheets 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.

关键要点

The Rule of 72 is a simplified way to estimate the doubling of an investment’s value, based on a logarithmic formula. The Rule of 72 can be applied to investments, inflation or anything that grows, such as GDP or population. The formula is useful for understanding the effect of compound interest.

The Formula for the Rule of 72 Is

Years to Double = 72 / Interest Rate where: Interest Rate = Rate of return on an investment

## 5.2. How to Calculate the Rule of 72¶

If an investment scheme promises an 8% annual compounded rate of return, it will take approximately (72 / 8) = 9 years 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).

The formula has emerged as a simplified version of the original 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.

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

T = ln(2)/ln(1+100r) ≃ 72 / r where: T=Time to doubleln=Natural log function r=Compounded interest rate per period ≃=Approximately equal to

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, which is very close to the approximate value obtained by (72 / 8) = 9 years

Since people cannot do logarithmic functions instantly without the help of log tables or scientific calculators, they can rely on the simpler version that uses the factor of 72 and gets almost the same result. If it takes 9 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.

## 5.3. What Does the Rule of 72 Tell You?¶

People love money, and they love it more to see the money getting double. Getting a rough estimate of how much time it will take to double the money also helps the average Joe to compare investments. However, mathematical calculations can be complex for common individuals to compute how much time is required for their money to double from a particular investment that promises a certain rate of return. The Rule of 72 offers a useful shortcut since the equations related to compound interest are too complicated for most people to do without a calculator. Simple Versus Compound 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.

In the case of 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.

Simply put, since the interest portion gets accumulated in case of compound interest, it raises the principal value with each passing month and leads to higher exponential returns overall. By not withdrawing the interest every month, the investor is increasing the principal value which helps him earn more interest.

It contrasts with simple interest where the investor withdraws the interest every month and keeps the principal amount consistent leading to comparatively lower returns. The Rule of 72 applies to cases of compound interest, and not to the cases of simple interest.

## 5.4. Examples of How to Use the Rule of 72¶

The unit does not necessarily have to be invested or loaned money. 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 his credit card (or any other form of loans which is charging compound interest) will double the amount he owes 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 (72 ÷ 6) = 12 years. 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. If the interest per quarter is 4%, then it will take (72 / 4) = 18 quarters or 4.5 years to double the principal. If the population of a nation increases as the rate of 1% per month, it will double in 72 months, or six years.

## 5.5. Variations in Applying the Rule of 72¶

The Rule of 72 is reasonably accurate for interest rates 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 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 14% rate of return, it would be the rule of 74 (adding 2 for 6 percentage points higher), and for 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 scheme 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.

Amid all the variations suggested for better estimations, one can rely on the basic Rule of 72 to make the quick mental calculation for roughly assessing when their money or loan amount would double.