Rule of 72

Have you heard of ‘Rule of 72’? You might come across this rule in article or book on investment. What is ‘Rule of 72’?

Definition from Wikipedia

In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69.3 are methods 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.

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 half 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 unit of currency 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 ½ in 72/3 = 24 years, and then to just ¼ the value in 48 years, compare 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, 5, 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 72 provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.

Low rates / daily compounding
For continuous compounding, 69.3 gives accurate results for any rate (this is because In(2) is about 69.3%; see derivation below). 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 0.2 off). This because, as above, the rule of 72 is only approximation that is accurate for interest rates from 6% to 10%. Outside that range the error will vary from 2.4% to -14%. 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.)

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