Rule of 72

The 72 rule is a rule of thumb from the calculation of interest. The rule specifies approximately how many years an interest-bearing investment doubled in nominal value. For this purpose, dividing 72 by the percentage of the annual rate of the invested amount, hence the name of the rule. Variants of the 72 rule are the 70 rule and the 69's rule.

Formula

The time ( in years) in which an investment with interest rate doubled (in percent), according to the 72 rule:

One can use the same formula to evaluate which rate is required to double the capital predetermined time:

Examples

In what time will a sum which is invested at an interest rate of each year, doubled?

What interest rate is required to double a capital in period years?

Of course, the 72 rule can be applied not only to the calculation of interest, but to any type of exponential growth. For example, the generation time, ie the time until a population doubled, with an annual population growth of about years.

Derivation

After the compound interest formula, the end value of a fixed-income investment with initial capital is at an interest rate of ( in percent) after a term of years at an annual interest rate

If, then, applies the logarithm on both sides of the equation and solving for up to the number of years as a result to the double

After magnitude for small to converge (see Taylor series ) and is calculated as the approximate formula

You now approaches or by, one speaks of the 69 rule or 70 rule. As a rule of thumb but the approximation has been proven by, among other things because the number has many small divisors. For the '69 - rule can be found in the literature, a modification of the form

Obtained by Taylor expansion of the logarithm function up to second order.

Accuracy

The following table compares the estimates according to the 72er, the 70, the 69 - rule and the modified '69 - rule with the actual values ​​for typical interest rates.