Sunday, September 30, 2007

The Rule of 72. How long will it take to double my money?

There is a simple mathematical rule that I learned as a kid that I have never forgotten -- the "Rule of 72." This is a very simple way to find out how long it will take to double your money at a certain interest rate.

Here's how the rule works.

Suppose you want to know how long it will take to double your money at 10% interest. 72 divided by 10 is 7.2. Therefore it will take 7.2 years to double your money if you are earning 10%. If you are earning 7%, it will take a little over 10 years. (Because 72 divided by 7 is roughly 10.) There is no need to figure the exact calculation, because this rule is only for estimating -- but what a powerful little rule it is.

Thanks to several readers who pointed out that this rule is mathematically derived from the "natural log" of 2 which is 0.69. So, it could technically be the "Rule of 69," although 70 or 72 seems easier for quick estimating.

1 comment:

James said...

Just to be clear, the "rule of 72" is meant to tell you how long it will take to double your money after taxes .
In certain examples, such as IRA accounts, the "rule of 69" would be the rule to follow. Here is the math behind the "rule of 69":

Nearly all investments are compounded continuously. The equation for continuous compounding is:

Balance = Investment*e^(rate*time)

If we are only concerned with the time required to double the investment, and not the actual amount invested or returned, then we can divide both sides by the Investment, and replace the quotient on the left with the target value 2, yielding:

2 = e^(rate*time)

taking the natural log of both sides and rounding off gives us:

.69 = rate*time

To solve for time we simply divide both sides by the rate; and because most financial rates are two digit decimals (7% = .07 for example), we can ignore the decimals (they just divide out) which gives us:

69/rate = time

Now we consider two things:

1. The natural log of 2 is actually slightly greater than .69 (remember I rounded down). This means dividing 69 by the rate will actually give us the time it takes to almost double.

2. More importantly, taxes have not been considered, and who really cares how long it takes to double before Big Brother gets his cut?

If we assume taxation to be about 5% for capital gains, then we simply replace 2 with 2.05 in the second step, and proceed from there. I will leave this exercise to the reader, who will see that the "rule of 72" does not short change us due to rounding.