Tuesday, July 10, 2018

The Rule of 72 (English)

The Rule of 72 is a great mental math shortcut to estimate the effect of any growth rate, from quick financial calculations to population estimates. Here’s the formula:

Years to double = 72 / Interest Rate

This formula is useful for financial estimates and understanding the nature of compound interest.

a * (1+R)^N = 2a, ln(1+R) ~ R, where

a - initial amount of capital;
R - interest rate, for example 10%, means R=0.1;
Interest Rate - in percents, 10%, for example,
N - number of years, 12 years, for example.

In addition to being nicely divisible, 72 has an important advantage which most people don't realize: It's on the right side of 100 ln(2).

The exact "rule" for N% interest is N log(2) / log (1 + N/100), which has taylor series 100 log(2) + log(2)/2 N + O(N^2) ≈ 69.315 + 0.34657 N - 0.0005776 N^2 + ...

For N approaching 0, the exact "rule" becomes the "rule of 100 log 2"; but for larger N increases slightly; the "rule of 72" is exactly correct for ~7.84687% interest, and for 15% interest it only gets as far as a "rule of 74.4".

That said, the power series gives us a way to get a significantly more accurate result: Divide the annual percentage interest rate into 832 months, then add 4 months. For any interest rate between 1% and 40%, this result will be accurate to within 3 days.

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