## Monday, June 20, 2011

### The Rule of 72: Compound Interest in Action

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.

2p = p × (1.06)^12
2p ≈ 2.012p

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.

2pp × (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. :))