The Rule of 72

An easy way to caculate how fast your money will grow.

The rule of 72 is the fastest way of finding how fast any sum will double at any particular interest rate. Simply divide 72 by the rate of interest, and convert that into years. For example, the average large-cap stock has grown 10 percent annually. How long would a well diversified portfolio take to double? Divide 72 by 10 and get 7.2 years. Every 7.2 years, the average large-cap stock doubles! What about inflation? we assume 3% inflation, so 72/3=24, which means every 24 years, the value of one dollar is equal to fifty cents. If we assume 4% inflation, the value of one dollar is halved in 18 years. That's a big difference, and it's important. Our portfolio that just doubled in value in 7.2 years is now taking 10.3 years to truly double (72/(10%-3%)=10.3 years). This is a very useful rule when considering long term investments, so just remember: 72/(rate of appreciation/depreciation)=years to double/half.

Why the Rule of 72 Works

For those of you who are interested, the following shows why the rule of 72 works.

Let's start with continuous interest, which is a bit easier to work with:

Start with the Pert formula (where P=Principal, e=mathematical constant, r=interest rate, and t=time)2P=Pert
Simplify and take the natural log of both sidesln(2)=rt
Solve for tt=ln(2)/r
Since ln(2)=.69t=.69/r

So the with continuous interest, we would use the rule of 69.

Using yearly interest is a bit harder:

Start with the yearly interest formula, where t=number of years2P=P(1+r)t
Simplify and take the natural log of both sidesln(2)=t(ln(1+r))
Solve for tt=ln(2)/ln(1+r)
We know that ln(2)=.69; ln(1+r) is just below 1, given an r around .05-.15, sot~.72/r

This shows that the rule of 72 is an approximation, but it comes pretty close when your interest rate is between 5% and 10%.

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