**PERSONAL FINANCE 101**

If you can earn returns on your investment at 6% per year, how much will it take for your money to double? The "Rule of 72" is a simple tool that we can use to answer this question. According to the rule, an investment earning

*r*% per year would

*approximately*take 72 ÷

*r*years to double; in our example, the investment would double in approximately 72 ÷ 6 = 12 years.

*Rule of 72*

Number of years for an investment to double = 72 ÷

*r*The Rule of 72 is a direct application of compound interest, which we have discussed in the previous post. Recall that if you invest an amount

*p*at compound interest

*r*per year, you will get the following amount after

*t*years:

Total amount after

*t*years =*p*× (1 +*r*)^*t*Plugging in the result of the example above into this equation, we'll see how closely the rule approximates the

*true*answer.

2

*p*=*p*×*(1.06)^12*2

*p ≈*2.012*p*

So you see, for practical purposes, the Rule of 72 actually produces good-enough results. It wouldn't take much more effort if you insist on getting an exact answer to the question, though: all it takes are a few algebraic manipulations to the compound interest formula.

2

*p*=*p*× (1 +*r*)^*t*2 = (1 +

*r*)^*t**ln*2 =

*t*×

*ln*(1 +

*r*)

*t*= [

*ln*2] ÷ [

*ln*(1 +

*r*)]

*t*= 0.6931 ÷ [

*ln*(1 +

*r*)]

Since

*ln*(1 +*r*) is approximately equal to*r*when*r*is small, we get*t*= 0.6931 ÷

*r*

If we redefine

*r*% as the interest rate or percentage return of the investment in question (as we did when we introduced the Rule of 72 earlier), we can restate the equation as*t*= 69.31 ÷

*r*

Some people prefer using the "Rule of 69" or "Rule of 70" instead since these lead to more accurate estimates, as can be seen from the above equation. However, a lot more people prefer using 72 since it makes clean, mental division by many more numbers--1, 2, 3, 4, 6, 8, 9, and 12--possible; whatever accuracy is lost is more than made up for in convenience.

The simplicity of the rule leads to many practical applications; playing around with the concept will let you answer related questions like "How much returns should I earn if I want to double my money in three years?" You can even try to figure out similar rules for

*tripling*or*quadrupling*your investment. Do I hear "challenge accepted?" (googling the answers defeats the purpose, of course. :))