The rule of 72, and may I introduce the rule of 108?

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Goal of this article

Here we will answer the question, where does the rule of 72 come from? And how can we derive similar rules for other tripling or other multiplicative factors?

Origin of 72 for doubling time of an investment

The rule of 72 is an estimate for the doubling time of an investment at a given rate of return.

$$ \left[ \text{Doubling time} \right] = \frac{72}{\left[ \text{Rate of return as a percent}\right]} $$

As an example, for a 6% return, the rule of 72 says an investment will double in 12 years.

$$ \left[ \text{Doubling time}\right] = \frac{72}{6\text{% per year}} = 12 \text{ years} $$

An investment at a fixed rate of return grows exponentially according to the simple differential equation from the article on what it takes to save $1 million.

$$ \begin{align} \frac{dV}{dt} &= rV\\ \int^{V_f}_{V_0} \frac{dV}{rV} &= \int^t_0 t\\ V_f = V_0 e^{rt} \end{align} $$

To find the doubling time, set \(V_f = 2V_0\).

$$ t_\text{double} = \frac{\log(2)}{r} $$

Notice that \(\log(\)2\() =\) 0.693147 and if we want a formula that does not require the conversion from percent to dimensionless fraction, we have to multiply by 100.

$$ t_\text{double} = \frac{69.3}{r_\text{as percent}} $$

The only trouble is that no reasonable rate of return divides cleanly into 69, except for 1 and 3. However, 72 can be cleanly divided by 1, 2, 3, 4, 6, 8, 9, and 12 making it a much more convenient number for a back-of-the-envelope estimation. That is the story of how 69 became 72.

This begets the question - "if the rule of 72 is actually the rule of 69.3, what multiplicative factor does 72 correspond to?" The answer is "pretty much 2."

$$\begin{align} \left. rt \right| _\text{72rule} = \log\left(\frac{V_f}{V_0}\right) &= 0.72\\ \frac{V_f}{V_0} &= e^{0.72}\\ \frac{V_f}{V_0} &= 2.0544 \approx 2 \\ \end{align}$$

The purpose of the rule of 72 is for back of the envelope calculations and for that, a <3% error is acceptable.

Rules for the doubling, tripling, and quadrupling of investments

Using this logic we can extend the rule. How long does it take our money to triple?

$$ t_\text{triple} = \frac{\log(3)}{r} = \frac{1.0986}{r} \approx \frac{108}{r_\text{as percent}} $$

Approximate 109.86 as 108 since 109 and 110 are only evenly divisible by 1 and 5 of the first 12 integers respectively. However, 108 is evenly divisible by 1, 2, 3, 4, 6, 9, and 12; the result being 108, 54, 36, 27, 18, 12, and 9 respectively.

This gives us the following table. The quadrupling time is just double the doubling time.

Doubling time	Tripling time	Quadrupling time
\(\displaystyle \frac{72}{r_\text{as percent}}\)	\(\displaystyle\frac{108}{r_\text{as percent}}\)	\(\displaystyle\frac{144}{r_\text{as percent}} \)