Rule of 72: Explanation and Generalization

Haven’t you ever wanted to calculate mentally the number of years required to double your invested money at a given fixed interest rate? Obviously you can make this calculation with spreadsheet programs, but to quickly find an approximate value, the Rule of 72 comes handy in mental calculations. Rule of 72 is a way of calculating fast the number of years is takes for an investment or your money to double.

\[\text{Years to Double} = \frac{72}{\text{Interest Rate}}\]

For example, you heard of an interesting investment opportunity with a fixed interest rate of 9%. Given this fact, using the Rule of 72, you can calculate with just a division, that 72/9 = 8 years are required to double your invested amount.

Different rates may require a different numerator than 72

An important notice is that Rule of 72 is an approximation of a more complex logarithmic equation. This approximation can be applied effectively to a specific range of rates (6-10%). However, there is a way to make it more flexible by adding or subtracting 1 from 72 every 3 points the interest rate diverges from the 8% threshold. Expressed in an equation: \(T = \frac{72 + (r-8)/3}{r}\).

For continuous compounding, the rule of 69 would be more suitable for any rate, since the actual calculation depends on \(ln(2)\) which is approximate 69.3%. (see Derivation). Daily compounding is close enough to continuous compounding, thus 69, 69.3, 70 are better than 72 for daily compounding.

Examples:

• Interest rate 24%

  • No adjustment: \(T = ln(2)/ln(1+24\%) = 3.22 \; \text{years}\; \neq 3 = \frac{72}{24}\)

  • Adjustment, to be more accurate: \((24-8)/3 = 16/3 = 5.3 = 5\) We add 5 to 72, resulting in: \(\frac{77}{24} = 3.21\)

• Interest rate 4%

  • No adjustment: \(T = ln(2)/ln(1+4\%) = 17.673 \; \text{years}\; \neq 18 = \frac{72}{4}\)

  • Adjustment, to be more accurate: \((4-8)/3 = -4/3 = -1.3 = -1\) We subtract 1 from 72, resulting in: \(\frac{71}{4} = 17.75\)

History

Luca Pacioli (1445–1514), an italian Mathematician and collaborator of Leonardo DaVinci had a reference to the rule in the Summa de arithmetica (Venice, 1494. Fol. 181, n. 44). Even though he presents the rule as an estimation of the doubling time of an investment, he does not derive or explain the rule. Thus its assumed that the rule predates Pacioli by some time. ³

Derivation

Considering the fact that Rule of 72 is a simplification of a more complex logarithmic equation, we can’t rely on this rule when precise calculations are needed. However, it can be used for fast mental calculations, as a pretty close approximation. The actual calculation that one can use to be precise is the following: \(Τ_d = \frac{ln(2)}{ln(1+\frac{1}{100})} \approx \frac{72}{r}\)

where \(T_d\): Time to double, \(ln\): Natural log function, \(r\) : Compounded interest rate per period.

Let’s check the power of the rule of 72: \(T = ln(2)/ln(1+9\%) = 8.043 \;\text{years,}\) which is pretty close to 72/9 = 8 that rule of 72 gives.

Periodic Compounding

\[FV = PV \cdot (1+r)^T\]

where FV: future value, PV: present value, T: number of time periods, r: interest rate per time period.

If the FV is double the PV, the following condition is met: \((1+r)^T = 2\) Let’s solve this equation for T:

\[ln((1+r)^T) = ln2 \Rightarrow T\ ln(1+r) = ln2 \Rightarrow T = \frac{ln2}{ln(1+r)}\]

A simple rearrangement shows:

\[\frac{ln2}{ln(1+r)} = (\frac{ln2}{r})(\frac{r}{ln(1+r)})\]

For r close to 0, ln(1+r) approximately equals r. Therefore, the second factor grows slowly for small r. For a small, positive interest rate r=8%, \(\frac{r}{ln(1+r)} = 1.0395\).

We thus approximate T as: \(T = 1.0395 \cdot \frac{ln2}{r}\,=\,\frac{0.72}{r}\,=\,\frac{72}{R}\) Where R is the interest rate expressed as a percentage R%

Using the second-order Taylor approximation, we can obtain a more accurate formula called E-M Rule (Eckart-McHale) .

Continuous compounding

\[(e^r)^p = 2 \Rightarrow \\ \Rightarrow e^{rp} = 2 \Rightarrow ln\ e^{rp} = ln\ 2 \Rightarrow rp = ln\ 2 \Rightarrow p=\frac{ln2}{r} \Rightarrow \\ \Rightarrow p \approx \frac{0.693147}{r}\]

E-M Rule (Eckart-McHale) second order rule

This rule provides a multiplicative correction for the rule of 69.3 that is very accurate to rates 0-20% (whereas the rule is normally only accurate at the lowest end of interest rates, 0-5%). \(t = \frac{69.3}{r} \cdot \frac{200}{200-r}\)

What about the years needed to triple (or more) your investment ?

Here comes the Rule of 114 and the rule of 144, applied the same way as the rule of 72. Dividing 114 by the interest rate value, will result in the number of years required to triple your investment. (Divide 144 by interest rate to find time until quadruple). For example, with an interest rate of 6%, you would need 114/6=19 years to triple and 144/6=24 years to quadruple your initial investment.

If you want to calculate how long it will take to make your investment X times more, you could use the precise calculation to result in another rule of thumb.

Starting from \((1+r)^T = X\) (See Derivation), we have \(ln((1+r)^T) = ln(X) \Rightarrow T\ ln(1+r) = ln(X) \Rightarrow T = \frac{ln(X)}{ln(1+r)} \Rightarrow \\ \Rightarrow T\,=\,\frac{ln(X)}{ln(1+r)} = \frac{ln(X)}{r}\cdot \frac{r}{ln(1+r)}\,=\,...\,=\,1.0395\cdot \frac{ln(X)}{r} \,=\,103.95\cdot \frac{ln(X)}{R} \\\)

\[T\,=\,103.95\cdot \frac{ln(X)}{R}\]

• x2 (double) : \(T = 103.95\cdot \frac{ln(2)}{R} \,=\, \frac{72}{R}\)

• x3 (triple): \(T = 103.95\cdot \frac{ln(3)}{R} \,=\, \frac{114}{R}\)

• x4 (quadruple) : \(T = 103.95\cdot \frac{ln(4)}{R} \,=\, \frac{144}{R}\)

  etc,

Key takeaways

• It can be used as a way to calculate fast the number of years it would take for an investment to double your money, only for compound interest.
• With an easy calculation you can generate another rule of thumb for X times your money.
• It applies to compounded interest rates and is reasonably accurate for interest rates that fall in the range of 6% and 10%.
• For interest rates out of 6-10% range, you can adjust the rule to be more accurate depending on the given interest rate. However, 72 is more convenient, since it has many small divisors (1, 2, 3, 4, 6, 8, 9, and 12).
• It can also be applied to anything that increases exponentially (such as GDP or inflation).

References:

[1] https://www.investopedia.com/terms/r/ruleof72.asp
[2] https://www.investopedia.com/terms/r/rule-of-70.asp
[3] https://en.wikipedia.org/wiki/Rule_of_72
[4] https://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/