# Rule of 72 Redeux

I frequently have to work on “time value of money” problems. Good approximate solutions to these problems can often be obtained by using the “Rule of 72.” The Rule of 72 will tell you approximately how many interest payment intervals are required for an investment to double in value by evaluating the following formula. $N = \frac{{72}}{{r\left[ \% \right]}}$

Where N is the number of payment intervals and r is the interest rate in percentage. I just accepted this formula for years until I had a mathematician friend tell me that the Rule of 72 should really be the “Rule of 69.” He did not go into detail and I decided to think a bit more about the equation.

To determine the doubling interval of an investment, the basic equation that must be solved is the following. ${A_0} \cdot {\left( {1 + r} \right)^N} = 2 \cdot {A_0}$

The exact solution is straightforward and can be derived as follows. ${A_0} \cdot {\left( {1 + r} \right)^N} = 2 \cdot {A_0}$ $\log \left( {{{\left( {1 + r} \right)}^N}} \right) = \log \left( 2 \right)$ $N \cdot \log \left( {1 + r} \right) = \log \left( 2 \right)$ $N = \frac{{\log \left( 2 \right)}}{{\log \left( {1 + r} \right)}} \doteq \frac{{69}}{r}$

I now understand where my friend's comment came from. So I decided to graph the error in the Rule of 72 and Rule of 69 from the exact solution to see which is closer. The result surprised me.

Observe that the Rule of 72 has less error than the Rule of 69 for interest rates greater than 3.4%. These are common interest rates and I began to wonder if the Rule of 72 was written the way it is for a reason. Like always, I began to play with the equation. I soon saw that the Rule of 72 could be obtained by continuing the previous derivation with a slightly changed logarithm approximation. $N = \frac{{\log \left( 2 \right)}}{{\log \left( {1 + r} \right)}} \doteq \frac{{69}}{{r - \frac{{{r^2}}}{2}}} = \frac{{69}}{{r \cdot \left( {1 - \frac{r}{2}} \right)}} \doteq \frac{{69 \cdot \left( {1 + \frac{r}{2}} \right)}}{r}$

We can make the error in this equation near zero for interest rates around 8% by substituting 8% in for r in numerator. The numerator function changes more slowly than the denominator function and an approximation is worthwhile there. $N \doteq \frac{{69 \cdot \left( {1 + \frac{{8\% }}{2}} \right)}}{r} = \frac{{72}}{r}$

This gives us the Rule of 72 and explains why the error goes to zero for interest rates around 8%.

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