Rule of 72

• definition by Wikipedia

In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69.3 all refer to a method for estimating an investment's doubling time, or halving time. These rules apply to exponential growth and decay respectively, and are therefore used for compound interest as opposed to simple interest calculations.

History

An early reference to the rule is in the Summa de Arithmetica (Venice, 1494. Fol. 181, n. 44) of Fra Luca Pacioli (1445–1514). He presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule, and it is thus assumed that the rule predates Pacioli by some time.

"In wanting to know for any percentage, in how many years the capital will be doubled, you bring to mind the rule of 72, which you always divide by the interest, and the result is in how many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; obtaining 12, and in 12 years the capital will be doubled.

Using the Rule to Estimate Compounding Periods

To estimate the number of periods required to double an original investment, divide the most convenient "rule-quantity" by the expected growth rate, expressed as a percentage.

For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the "rule of 72" gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives 8.0432 years.

Similarly, to determine the time it takes for the value of money to halve at a given rate, divide the rule quantity by that rate.

To determine the time for money's buying power to halve, financiers simply divide the "rule-quantity" by the inflation rate. Thus at 3.5% inflation using the rule of 70, it should take approximately 70/3.5 = 20 years for the value of a dollar to halve.

To estimate the impact of additional fees on financial policies (eg. Mutual fund fees and expenses, Loading and Expense Charges on Variable universal life insurance investment portfolios), divide 72 by the fee. For example, if the Universal Life policy charges a 3% fee over and above the cost of the underlying investment fund, then the total account value will be cut to 1/2 in 72 / 3 = 24 years, and then to just 1/4 the value in 48 years, compared to holding the exact same investment outside the policy.

Choice of rule

The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. However, depending on the rate and compounding period in question, other values will provide a more appropriate choice.

"Typical" rates / annual compounding

The rule of 71 provides a good approximation for annual compounding, and for compounding at "typical rates" (from 6% to 10%).

Low rates / daily compounding

For continuous compounding, 69.3 gives accurate results for any rate (this is because ln(2) is about 69.3%). Since daily compounding is close enough to continuous compounding, for most purposes 69.3 - or 70 - is used in preference to 72 here. For lower rates than those above, 69.3 would also be more accurate than 72.

Adjustments for higher rates

For higher rates, a bigger numerator would be better (e.g. for 20%, using 76 to get 3.8 years would be only about 0.002 off, where using 72 to get 3.6 would be about 2.002 off). This is because, as above, the rule of 72 is only an approximation that is accurate for interest rates from 6% to 10%. Outside that range the error will vary from 2.4% to −14.0%. For every three percentage points away from 8% the value 72 could be adjusted by 1.

A similar accuracy adjustment for the rule of 69.3 - used for high rates with daily compounding - is as follows:

E-M Rule

The Eckart-McHale Second Order Rule, "the E-M Rule", gives a multiplicative correction to the Rule of 69.3 or 70 (but not 72). The E-M Rule's main advantage is that it provides the best results over the widest range of interest rates. Using the E-M correction to the rule of 69.3, for example, makes the Rule of 69.3 very accurate for rates from 0%-20% even though the Rule of 69.3 is normally only accurate at the lowest end of interest rates, from 0% to about 5%.

To compute the E-M approximation, simply multiply the Rule of 69.3 (or 70) result by 200/(200-r) as follows:

For example, if the interest rate is 18% the Rule of 69.3 says t = 3.85 years. The E-M Rule multiplies this by 200/(200-18), giving a doubling time of 4.23 years, where the actual doubling time at this rate is 4.19 years. (The E-M Rule thus gives a closer approximation than the Rule of 72.)

Similarly, the 3rd order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula:

Illustrative Comparison

This table compares the three rules, using periodic compounding, and illustrates the error of the estimation over a range of typical values.