Editing Rule of 72 (section)

==Derivation==

===Periodic compounding ===
For [[Compound interest#Periodic compounding|periodic compounding]], [[future value]] is given by:
:<math>FV = PV \cdot (1+r/100)^t</math>
where <math>PV</math> is the [[present value]], <math>t</math> is the number of time periods, and <math>r</math> stands for the [[interest rate]] per time period.

The future value is double the present value when:
:<math>FV = 2 \cdot PV</math>

which is the following condition:
:<math>(1+r/100)^t = 2\,</math>

This equation is easily solved for <math>t</math>:
:<math>
  \begin{align}
    \ln((1+r/100)^t) & = \ln 2                  \\
    t \cdot \ln(1+r/100)   & = \ln 2                  \\
    t           & = \frac{\ln 2}{\ln(1+r/100)}
  \end{align}
</math>

A simple rearrangement shows
:<math>\frac{\ln{2}}{\ln{(1+r/100)}}=\frac{\ln2}{r/100} \cdot \frac{r/100}{\ln(1+r/100)}</math>.

If <math>r/100</math> is small, then <math>\ln(1 + r/100)</math> [[Natural logarithm#Derivative.2C Taylor series|approximately equals]] <math>r/100</math> (this is the first term in the [[Taylor series#Calculation of Taylor series|Taylor series]]). That is, the latter factor grows slowly when <math>r</math> is close to zero.

Call this latter factor <math>(r/100)/\ln(1+r/100) = f(r)</math>. The function <math>f(r)</math> is shown to be accurate in the approximation of <math>t</math> for a small, positive interest rate when <math>r=8</math> (see derivation below). <math>f(8)\approx1.03949</math>, and we therefore approximate time <math>t</math> as:

:<math>t(r)=\frac{\ln2}{r/100} \cdot f(8) \approx \frac{0.72}{r/100} = \frac{72}{r}</math>.

This approximation increases in accuracy as the [[Compound Interest|compounding of interest]] becomes continuous (see derivation below).

In order to derive a more precise adjustment, it is noted that <math>\ln(1+r/100)</math> is more closely approximated by <math>r/100-\tfrac{1}{2}(r/100)^2</math> (using the second term in the [[Taylor series]]). <math>0.693/\left(r/100-\tfrac{1}{2}(r/100)^2\right)</math> can then be further simplified by Taylor approximations:<ref>{{cite web |url=https://www.wolframalpha.com/input?i=taylor+series+1%2F%281-r%2F200%29 |title=Taylor series of 1 / (1 - r/200) |website=WolframAlpha |access-date=January 3, 2025}}</ref>

:<math>t(r) = \frac{0.693}{r/100 - \tfrac{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}</math>
:<math>t(r) = \frac{69.3}{r} + 0.3465</math>.

Replacing the <math>r</math> in <math>r/200</math> 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 <math>r</math> in <math>r/200</math> 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 <math>t(r) = \ln(2)/\ln(1+r/100)</math> is [[Series expansion|developed]] in a [[Laurent series]] at the point <math>r = 0</math>.<ref>{{cite web |url=https://www.wolframalpha.com/input?i=laurent+series+ln%282%29%2Fln%281%2Br%2F100%29 |title=Laurent series of ln(2) / ln(1 + r/100) |website=WolframAlpha |access-date=January 2, 2025}}</ref> With the first two terms one obtains:

: <math>t(r) \approx \frac{100 \cdot \ln 2}{r} + \frac{\ln 2}{2}</math>
: <math>t(r) \approx \frac{69.3147}{r} + 0.346574</math> &nbsp;or rounded
: <math>t(r) \approx \frac{69}{r} + 0.35</math>.

=== Continuous compounding ===
In the case of theoretical [[Compound interest#Continuous compounding|continuous compounding]], the derivation is simpler and yields to a more accurate rule:
:<math>
  \begin{align}
       \exp\left(\tfrac{r}{100} \cdot t\right) & = \frac{FV}{PV} = 2 \\
       \tfrac{r}{100} \cdot t & = \ln 2 \\
       t & = \frac{100 \cdot \ln 2}{r} \approx \frac{69.3147}{r}
  \end{align}
</math>