Derivation of the rule of 72 (or 70 or 69)

The rule of 72 states that to find the number of years that it will take your investment to double, you divide 72 by the interest rate.

The formula for compound interest is:

A = P(1+r)^t

where A is the size of your account
P is the amount of the principle invested
r is the interest rate
t is the number of time periods (we'll assume years and that r is given as an APR)

the account has doubled when A = 2P, so...

2P = P(1+r)^t

dividing both sides by P...

2 = (1+r)^t

taking the natural logarithm of both sides (the "ln" function on a calculator)

ln(2) = ln((1+r)^t)

there's a property of logarithms where log (a^b) = b*log(a), so we can reduce the above to:

ln(2) = t*ln(1+r)

Two things to note at this point:

1. ln(2) = 0.69314718055994530941723212145818

2. ln(1+r) is roughly equal to r. For r<10%, ln(1+r) is within 5% of r (that is using such an approximation introduces a 5% error). For r<20%, ln(1+r) is within 10% of r

so approximating...

0.693 = t*r

but r is given as a percentage, (15% = .15), so to do the math using whole numbers

69.3 = rt

so you can calculate how long it takes to double your money at a given interest rate using 69.3

72 is a nice number because it has many perfect divisors (2,3,4,6,8,9,12,18,24,36)

using 72 instead of 69.3 introduces about a 4% error, but that error goes in the same direction as the error introduced by using the approximation ln(1+r) = r, so they cancel each other out somewhat.

You can use the rule of 69 (=3*23), 70 (=5*14, =7*10), or 72 to do this. As we used to say when I worked in defense, "It's good enough for government work."

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