# The Rule of 72: A Mathematical Thing of Beauty

As an admitted math geek, I never tire of learning new tricks that simplify otherwise complex calculations. Far and away my favorite trick is known as "the rule of 72," which states that if you are given an annualized average rate of return of X%, the estimated number of years required for the initial investment to double is 72 divided by X. For example, if you expect to earn an 8% return, your investment will double in approximately nine years. For a 12% return, the doubling time is about six years.

The discovery of the rule of 72 is credited to the Italian mathematician, Luca Pacioli (1445-1517), who pioneered the use of the double-entry accounting system, a game-changing concept in finance. A Franciscan friar, Pacioli mentored Leonardo da Vinci in mathematics, and much of Pacioli's influence can be seen in da Vinci's works, particularly in his use of the golden ratio, which Pacioli named the "divine proportion." Below is a portrait of Pacioli of uncertain attribution, possibly painted by da Vinci.

While recently combing through the returns of the IFA Index Portfolios, we came across two instances of the rule of 72. IFA Index Portfolio 20 had an annualized return of exactly 5% from January 1, 1999 to March 31, 2013 (a 14.25 year period), during which an initial investment almost exactly doubled in value. From the rule of 72, the expected doubling time would have been 14.4 years.  The table and chart below from the IFA Returns Calculator display the results.

IFA Index Portfolio 92 had an annualized return of 10.04% from January 1, 2003 to February 28, 2010 (a 7.16-year period). Again, the observed result is very close to what we would have expected from the rule of 72, as seen in the table band chart below from the IFA Returns Calculator.

While the rule of 72 provides a good approximation for returns at or below 15%, its accuracy begins to break down for returns above 15%. More details, including a mathematical derivation of the rule of 72 based on Taylor series expansions can be found in this article published by the Association for Financial Counseling and Planning Education.