Editing Rule of 72 (section)

{{distinguish|72-year rule}}
{{short description|Methods of estimating the doubling time of an investment}}

In [[finance]], the '''rule of 72''', the '''rule of 70'''<ref name=Meadows/> and the '''rule of 69.3''' are methods for estimating an [[investment]]'s doubling time. 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.  Although [[scientific calculator]]s and [[spreadsheet]] programs have functions to find the accurate doubling time, the rules are useful for [[mental calculation]]s and when only a basic [[calculator]] is available.<ref>{{Cite book|title=All the Math You'll Ever Need|author=Slavin, Steve|publisher=[[John Wiley & Sons]]|year=1989|isbn=0-471-50636-2|pages=[https://archive.org/details/allmathyoullever00slav/page/153 153–154]|url-access=registration|url=https://archive.org/details/allmathyoullever00slav/page/153}}</ref>

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 [[exponential decay|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. For periodic compounding, the ''exact'' doubling time for an [[interest rate]] of ''r'' [[percent]] per period is

:<math>t = \frac{\ln(2)}{\ln(1+r/100)}\approx \frac{72}{r}</math>,

where ''t'' is the number of periods required. 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.