Rule of 72

Choice of rule

The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%); the approximations are less accurate at higher interest rates.

For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72. For higher annual rates, 78 is more accurate.

Note: The most accurate value on each row is in bold.

History

An early reference to the rule is in the Summa de arithmetica (Venice, 1494. Fol. 181, n. 44) of Luca Pacioli (1445–1514). He presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule, and it is thus assumed that the rule predates Pacioli by some time.

A voler sapere ogni quantità a tanto per 100 l'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale. (emphasis added).

Roughly translated:

In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.

Derivation

Periodic compounding

For periodic compounding, future value is given by:

${\displaystyle FV=PV\cdot (1+r/100{)}^t}$

where ${\displaystyle PV}$ is the present value, ${\displaystyle t}$ is the number of time periods, and ${\displaystyle r}$ stands for the interest rate per time period.

The future value is double the present value when:

${\displaystyle FV=2\cdot PV}$

which is the following condition:

${\displaystyle (1+r/100{)}^t=2}$

This equation is easily solved for ${\displaystyle t}$:

${\displaystyle \begin{array}{cc}\ln ((1+r/100{)}^t)& =\ln 2\\ t\cdot \ln (1+r/100)& =\ln 2\\ t& =\frac{\ln 2}{\ln (1+r/100)}.\end{array}}$

A simple rearrangement shows

${\displaystyle \frac{\ln 2}{\ln (1+r/100)}=\frac{\ln 2}{r/100}\cdot \frac{r/100}{\ln (1+r/100)}.}$

If ${\displaystyle r/100}$ is small, then ${\displaystyle \ln (1+r/100)}$ approximately equals ${\displaystyle r/100}$ (this is the first term in the Taylor series). That is, the latter factor grows slowly when ${\displaystyle r}$ is close to zero.

Call this latter factor ${\displaystyle (r/100)/\ln (1+r/100)=f(r)}$. The function ${\displaystyle f(r)}$ is shown to be accurate in the approximation of ${\displaystyle t}$ for a small, positive interest rate when ${\displaystyle r=8}$ (see derivation below). ${\displaystyle f(8)\approx 1.039}$, and we therefore approximate time ${\displaystyle t}$ as:

${\displaystyle t(r)=\frac{\ln 2}{r/100}\cdot f(8)\approx \frac{0.72}{r/100}=\frac{72}{r}.}$

This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below).

In order to derive a more precise adjustment, it is noted that ${\displaystyle \ln (1+r/100)}$ is more closely approximated by ${\displaystyle r/100-\frac{1}{2}(r/100{)}^2}$ (using the second term in the Taylor series). ${\displaystyle 0.693/\left(r/100-\frac{1}{2}(r/100{)}^2\right)}$ can then be further simplified by Taylor approximations:

${\displaystyle \begin{array}{cc}t(r)=& \frac{0.693}{r/100-\frac{1}{2}(r/100{)}^2}=\frac{69.3}{r-r^2/200}=\frac{69.3}{r}\frac{1}{1-r/200}\approx \frac{69.3}{r}(1+r/200)=\frac{69.3}{r}+\frac{69.3}{200}\\ t(r)=& \frac{69.3}{r}+\mathrm{0.3465.}\end{array}}$

Replacing the ${\displaystyle r}$ in ${\displaystyle r/200}$ with 7.79 gives 72 in the numerator. This shows that the rule of 72 is most accurate for periodically compounded interests around 8 %. Similarly, replacing the ${\displaystyle r}$ in ${\displaystyle r/200}$ with 2.02 gives 70 in the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2 %.

As a sophisticated but elegant mathematical method to achieve a more accurate fit, the function ${\displaystyle t(r)=\ln (2)/\ln (1+r/100)}$ is developed in a Laurent series at the point ${\displaystyle r=0}$. With the first two terms one obtains:

${\displaystyle \begin{array}{cc}t(r)\approx & \frac{100\cdot \ln 2}{r}+\frac{\ln 2}{2}\\ t(r)\approx & \frac{69.3147}{r}+0.346574,\text{ or rounded:}\\ t(r)\approx & \frac{69}{r}+\mathrm{0.35.}\end{array}}$

Continuous compounding

In the case of theoretical continuous compounding, the derivation is simpler and yields to a more accurate rule:

${\displaystyle \begin{array}{cc}\exp \left(\frac{r}{100}\cdot t\right)& =\frac{FV}{PV}=2\\ \frac{r}{100}\cdot t& =\ln 2\\ t& =\frac{100\cdot \ln 2}{r}\approx \frac{69.3147}{r}\end{array}}$