Rule of 72
The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling.
[2] These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations.
They can also be used for decay to obtain a halving time.
The choice of number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible.
There are a number of variations to the rules that improve accuracy.
The formula above can be used for more than calculating the doubling time.
If one wants to know the tripling time, for example, replace the constant 2 in the numerator with 3.
As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.
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.
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.
The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12.
It 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.
For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below.
Note: The most accurate value on each row is in bold.
An early reference to the rule is in the Summa de arithmetica (Venice, 1494.
A voler sapere ogni quantità a tanto per 100 l'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato.
Esempio: Quando l'interesse è a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale.
(emphasis added).Roughly translated: In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled.
Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.For periodic compounding, future value is given by: where
stands for the interest rate per time period.
The future value is double the present value when: which is the following condition: This equation is easily solved for
as: This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below).
In order to derive a more precise adjustment, it is noted that
can then be further simplified by Taylor approximations:[4] Replacing the
This shows that the rule of 72 is most accurate for periodically compounded interests around 8 %.
on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2 %.
As a sophisticated but elegant mathematical method to achieve a more accurate fit, the function
is developed in a Laurent series at the point
[5] With the first two terms one obtains: In the case of theoretical continuous compounding, the derivation is simpler and yields to a more accurate rule:
