A:

The "Rule of 72" is a simplified way to determine how long an investment will take to double, given a fixed annual rate of interest. By dividing 72 by the annual rate of return, investors can get a rough estimate of how many years it will take for the initial investment to duplicate itself.

For example, the rule of 72 states that \$1 invested at 10% would take 7.2 years ((72/10) = 7.2) to turn into \$2. In reality, a 10% investment will take 7.3 years to double ((1.10^7.3 = 2).

When dealing with low rates of return, the Rule of 72 is fairly accurate. This chart compares the numbers given by the rule of 72 and the actual number of years it takes an investment to double.

 Rate of Return Rule of 72 Actual # of Years Difference (#) of Years 2% 36.0 35 1.0 3% 24.0 23.45 0.6 5% 14.4 14.21 0.2 7% 10.3 10.24 0.0 9% 8.0 8.04 0.0 12% 6.0 6.12 0.1 25% 2.9 3.11 0.2 50% 1.4 1.71 0.3 72% 1.0 1.28 0.3 100% 0.7 1 0.3

Notice that, although it gives a quick rough estimate, the rule of 72 gets less precise as rates of return become higher.

## 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 of a power; for example, the opposite of 10³ is log base 10 of 1000.

The rule of 72 uses the natural log, sometimes called the inverse of e.

where ln (base e) = 1

and reverse is e1 = 2.718281828.

The natural 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 (TVM) 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 state the future value as 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) = number of time periods x ln(1 + interest rate). 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.

## How to Adjust the Rule of 72 for Higher Accuracy?

The rule of 72 can be made more accurate by adjusting it back to more closely resembling the compound interest formula – effectively transforming the rule of 72 into the rule of 69.3.

With the aid of a calculator, there isn't really any reason to substitute 72 in for 69.3. In fact, many investors prefer to use the rule of 69 rather than the rule of 72. For maximum accuracy – especially for continuous compounding interest rate instruments – use the rule of 69.3.

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.

## How to Calculate the Rule of 72 Using Matlab?

The calculation of the rule of 72 in Matlab involves running a simple command of "years = 72/return", where the variable "return" is the rate of return on investment and "years" is the result for the rule of 72. The rule of 72 can be also used to determine how long it takes for money to halve in value for a given rate of inflation. For example, if the rate of inflation is 4%, a command "years = 72/inflation", where variable "inflation" is defined as "inflation = 4", produces 18 years.

(To learn more, see Understanding the Time Value of Money.)

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