Understanding the Rule of 72

The Rule of 72 can help estimate the impact of exponential growth, such as in compound savings accounts (where interest is added to the principal at regular intervals). It can also estimate the impact of exponential decay (such as how your money depreciates due to inflation). This calculation is a simplified version of the original logarithmic formula, allowing you to roughly estimate the time needed to double or halve your funds without the need for a scientific calculator or logarithm table. Importantly, when calculating growth, the rule ignores expenses or taxes that may affect your returns.

The formula is: Years to double = 72 / Rate

Where Rate = Investment Return Rate

History of the Rule of 72

In 1494, Italian mathematician Luca Pacioli first mentioned the significance of the number 72 in his work "Summa de arithmetica, geometria, proportioni et proportionalita." Pacioli believed that one could infer how many years it would take for money to double using the number 72.

The Rule of 72 was formulated nearly a century later, based on the standard compound interest formula: A = P (1 + r/n) ^ nt. "A" represents the interest earned plus your principal; "P" is the principal or initial investment; "r" is the rate in decimal form; "n" is the number of compounding periods; "t" represents time (in years).

To double your money, you can use A = 2 and P = 1, resulting in 2 = 1 (1 + r/n) ^ nt.

If we assume compounding annually, we can replace n with 1; yielding 2 = 1 (1 + r/1)^1*t; simplifying this equation to 2 = (1 + r)^t.

Now, taking natural logarithms on both sides to further simplify the equation: ln 2 = ln (1 + r)^nt.

Next, use the power rule to reduce the exponent, ln 2 = t * ln (1 + r).

The natural logarithm of 2 is approximately 0.693. For small values of r, ln (1 + r) ≈ r. In other words, 0.693 ≈ t * r.

Multiplying both sides by 100 to use integer rates, you get 69.3 ≈ t * r (where r is in percentage).

Finally, to isolate t as the number of years required for doubling investment, we can divide by 100r to get 69.3 / r ≈ t (where r is in percentage).

Since 69.3 is a cumbersome number, statisticians and investors unanimously agreed to use the next closest integer with many divisible factors, which is 72. Thus, dividing 72 by the rate gives approximately how long (in years) it takes for an investment to double.

What Does the Rule of 72 Reveal?

To estimate how long money takes to double, it relates to the compound interest formula. Since most people can't compute this formula without a calculator, the Rule of 72 serves as a shortcut for roughly estimating the time for an investment to double.

A significant distinction of this rule is that it doesn't use simple interest (your initial investment multiplied by the rate multiplied by time), but compound interest (interest earned on the initial investment plus interest previously earned). In other words, the Rule of 72 assumes that every time interest is paid on your investment, you reinvest that money.

Compound interest can help your investments grow faster, and the Rule of 72 tells you approximately how long it takes.

How to Use the Rule of 72?

While deriving the Rule of 72 requires more mathematical knowledge, its application involves only division. You can estimate the doubling time of almost any investment by dividing 72 by the annual growth rate. You should use integer rates rather than percentages or decimals.

For example, suppose you have a $1 investment with an annual fixed rate of 6%; 72 divided by 6 equals 12. Therefore, your $1 will take 12 years to grow to $2.

The Rule of 72 can also tell you about the decay of money. For instance, if the inflation rate is 8%, dividing 72 by 8 means your money will lose half its value in about 9 years; alternatively, if the inflation rate drops to 6%, your money will halve in value in 12 years.

Differences Between the Rule of 69, 70, and 72

The Rule of 72 is best suited for annual rates, while the Rule of 70 is more suitable for semi-annual compounding.

For example, suppose your investment has a rate of 4%, compounded semi-annually or twice a year.

Using the Rule of 72, you get 72 / 4 = 18 years. With the Rule of 70, you get 70 / 4 = 17.5 years.

Ultimately, if you perform original logarithmic calculations, it actually takes about 17.501 years for your money to double. Thus, the Rule of 70 is closer.

The Rule of 69 is for continuous compounding (extreme compounding, where you reinvest interest as frequently as possible). For example, comparing investments with a daily compounding rate of 3%.

According to the Rule of 72, your money doubles in 24 years (72 / 3 = 24); using the Rule of 70, your money doubles in about 23.3 years (70 / 3 = 23.3); however, the Rule of 69 indicates your money doubles in 23 years (69 / 3 = 23).

Finally, the compound interest formula shows your money actually doubles in about 23.1 years. So, the Rule of 69 is closest to the original logarithmic calculation.

When to Use the Rule of 72?

The Rule of 72 can help you quickly compare the futures of different compounded investments, aiding in visualizing your funds.

For example, an investment with a 3% annual rate will take about 24 years to double; whereas, an investment with a 4% annual return will take about 18 years. A 1% difference could mean a 6-year disparity.

These investments may carry different levels of risk; however, the Rule of 72 can assist in planning whether these investments align with your retirement timetable and goals.

Is the Rule of 72 Effective?

The Rule of 72 is a rough estimate of how long it takes for your funds to double using compound interest formula.

If you apply the Rule of 72, you'll find that comparing 7 to the original formula, the 2 rule is most suitable for interest rates between 6% and 10%.

For lower rates, the Rule of 72 tends to slightly overestimate the time needed for your money to double; for higher rates, it tends to slightly underestimate the time needed for your money to double.