A:

The rule of 72 is best used to estimate compounding periods that are factors of two (2, 4, 12, 200 and so on). This is because the rule of 72 â€“ and its more accurate cousins, the rule of 70 and rule of 69.3 â€“ is meant to calculate how long it takes any exponentially growing variable to double in number. The actual equation is very simple: Length of time until value doubles = 72 / (percentage rate of growth).

For instance, consider an investment valued at \$10,000 with a compounding interest rate of 8%. Using the rule of 72, you can estimate the amount of time until the investment doubles like so: Time = 72 / 8 = 9 years. The investment should be worth approximately \$20,000 in eight years.

The rule of 72 is most commonly seen in finance as a time value of money calculation, although it has some practical use in biology and physics for various naturally compounding populations. It can also be inverted to find halving times for exponential decay.

## The Rule of 72 and Natural Logs

To understand how the rule of 72 allows you to estimate compounding periods, you have to understand natural logarithms. In mathematics, the logarithm is the opposite concept as a power; for example, the opposite of 10Â³ is log base 3 of 10.

The rule of 72 uses the natural log, sometimes called the inverse of e. This logarithm can be generally understood as the amount of time needed to reach a certain level of growth with continuous compounding.

A time value of money formula is normally written as: FV = PV x (1 + interest rate)^number of time periods.

To see how long it will take an investment to double, you can substitute the future value for 2 and the present value as 1: 2 = 1 x (1 + interest rate)^number of time periods. Simplify, and you get 2 = (1 + interest rate)^number of time periods.

To remove the exponent on the right-hand side of the equation, take the natural log of each side: ln(2) = ln(1 + interest rate) x number of time periods. This can be simplified again because the natural log of (1 + interest rate) equals the interest rate as the rate gets continuously closer to zero.

In other words, you are left with: ln(2) = interest rate x number of time periods. The natural log of 2 is equal to 0.693 and, after dividing both sides by the interest rate, you get: 0.693 / interest rate = number of time periods.

If you multiply the numerator and denominator on the left hand side by 100, you can express each as a percentage. This makes: 69.3 / interest rate percent = number of time periods.

## Rules of 69.3, 70 and 72

For maximum accuracy, you should use the rule of 69.3 to estimate how long it will take an investment to double with compound interest. Unfortunately, it isn't easy to do mental math with 69.3 and 70 comparatively few factors.

The number 72 has many convenient factors, including 2, 3, 4, 6 and 9. This makes it easier to use the rule of 72 for a close approximation of compounding periods.

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