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= The Rule of 72 =

= An analytical approach =

= Robert L. Weiss, P.E. =

Copyright © 2017, Robert L. Weiss
The 'Rule of 72' is a simplified way to determine how long an investment will take to double, given a fixed annual rate of interest. By dividing 72 by the annual rate of return, investors can get an estimate of how many years it will take for the initial investment to duplicate itself.

The purpose of this report is to present an analytical derivation of the basic principles and assumptions used in the formation of The Rule of 72. The motivation for this report came about when the author became interested in the foundation or analytical derivative to support the validity of The Rule of 72. In searching the literature, there he found an abundance of material about The Rule of 72. However, no clear justification for its legitimacy was found and this condition germinated the motivation to prepare this report.

The Rule of 72 has existed for a long time. An early reference to the principle involved is found in the book Summa de arithmetica, geometria, proportioni et proportionalita. The English translation of this title is Summary of arithmetic, geometry, proportions and proportionality. The author Luca Pacioli, in the year 1494, presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule.

Now getting back to the purpose of this report, in this derivation, the original principal is designated as S and the rate of interest is designated as r. At the end of the first period of time for which the rate of interest applies, the accumulated value has become:

$$S_1 = S + Sr$$$$S_1 = S(1 + r)$$

At the end of the second period of time, the accumulated value has become:

$$S_2 = S(1 + r)(1 + r)$$

At the end of the third period of time, the accumulated value has become:

$$S_3 = S(1 + r)(1 + r)(1 + r)$$

At the end of the nth period of time, the accumulated value has become:

$$S_n = S(1 + r)^n$$

In order for some future value to be twice that of the original value, the following condition must exist.

$$(1 + r)^n = 2$$

In the solution of this equation, logarithms come to our rescue and are utilized. Specifically, natural logarithms sometimes called napierian logarithms named after John Napier. This equation is solved by taking the natural logarithm of each side of the equation as follows.

$$ln(1 + r)^n = ln2$$$$n*ln(1 + r) = ln2$$

Thus far in this analysis, no assumptions or approximations have been made. This equation is exact. As we continue, however, some approximations will have to be made. The natural logarithm of 2 is a known constant, thus this equation may be written:

$$n*ln(1 + r) = 0.693147$$$$n = \frac{0.693147}{ln(1 + r)}$$

This equation is accurate to 6 decimal places. The natural logarithm of (1+r) is an important special function in mathematics. This function can be expanded in a convergent Taylor series. The first four terms of this series are as follows:

$$ln(1 + r) = r - \frac{r^2}{2} + \frac{r^3}{3} - \frac{r^4}{4}$$

For the purpose of this analysis, the first two terms of this expansion are used. Thus the above equation becomes:

$$n = \frac{0.693147}{r(1 - \frac{r}{2})}$$

In the initial statement of the rule of 72, it is clearly stated that this is a simplified rule meaning that there are no second order terms to deal with. With this in mind, some average value of r needs to be inserted into the r in the bracket of this equation. Some average value of 6% or 8% might be a reasonable value to substitute in at this point. However, an average value is already built into the rule of 72 so that will be used.

$$r = 2\biggl(1 - \frac{ln2}{0.72}\biggr) = 0.074591$$

When this value is substituted for r in the brackets of the above equation, the result becomes:

$$n = \frac{0.72000}{r}$$

At the outset of this analysis, r was defined as the rate of return applicable for the duration of the time n. The value of r is now defined as the annual interest rate in percent R and the value n becomes years N. The above equation is now written:

$$N = \frac{72}{R}$$

Which has become known as The Rule of 72. Please be reminded that The Rule of 72 is an approximation only. It should be obvious from this analysis that it is always an approximation but most accurate in the vicinity of 7.5%.

Copyright © 2017, Robert L. Weiss