E-M rule

The Eckart–McHale second-order rule (the E-M rule) provides a multiplicative correction for the rule of 69.3 that is very accurate for rates from 0% to 20%, whereas the rule is normally only accurate at the lowest end of interest rates, from 0% to about 5%.

To compute the E-M approximation, multiply the rule of 69.3 result by 200/(200−r) as follows:

${\displaystyle t\approx {\frac {69.3}{r}}\times {\frac {200}{200-r}}}$.

For example, if the interest rate is 18%, the rule of 69.3 gives t = 3.85 years, which the E-M rule multiplies by ${\displaystyle {\frac {200}{182}}}$ (i.e. 200/ (200−18)) to give a doubling time of 4.23 years. As the actual doubling time at this rate is 4.19 years, the E-M rule thus gives a closer approximation than the rule of 72.

To obtain a similar correction for the rule of 70 or 72, one of the numerators can be set and the other adjusted to keep their product approximately the same. The E-M rule could thus be written also as

${\displaystyle t\approx {\frac {70}{r}}\times {\frac {198}{200-r}}}$ or ${\displaystyle t\approx {\frac {72}{r}}\times {\frac {192}{200-r}}}$

In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate.

Padé approximant

The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula:

${\displaystyle t\approx {\frac {69.3}{r}}\times {\frac {4r+600}{r+600}}}$

which simplifies to:

${\displaystyle t\approx {\frac {1386r+207900}{5r^{2}+3000r}}}$

Periodic compounding

For periodic compounding, future value is given by:

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

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:

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

which is the following condition:

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

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

${\displaystyle {\begin{array}{ccc}\ln((1+r)^{t})&=&\ln 2\\t\ln(1+r)&=&\ln 2\\t&=&{\frac {\ln 2}{\ln(1+r)}}\end{array}}}$

A simple rearrangement shows:

${\displaystyle {\frac {\ln {2}}{\ln {(1+r)}}}={\bigg (}{\frac {\ln 2}{r}}{\bigg )}{\bigg (}{\frac {r}{\ln(1+r)}}{\bigg )}}$

If r is small, then ln(1 + r) approximately equals r (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 ${\displaystyle f(r)={\frac {r}{\ln(1+r)}}}$. The function ${\displaystyle f(r)}$ is shown to be accurate in the approximation of ${\displaystyle t}$ for a small, positive interest rate when ${\displaystyle r=.08}$ (see derivation below). ${\displaystyle f(.08)\approx 1.03949}$, and we therefore approximate time ${\displaystyle t}$ as:

${\displaystyle t={\bigg (}{\frac {\ln 2}{r}}{\bigg )}f(.08)\approx {\frac {.72}{r}}}$

Written as a percentage:

${\displaystyle {\frac {.72}{r}}={\frac {72}{100r}}}$

This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). ${\displaystyle 100r}$ is ${\displaystyle r}$ written as a percentage.

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

${\displaystyle {\begin{array}{ccc}{\frac {0.693}{r-r^{2}/2}}&=&{\frac {69.3}{R-R^{2}/200}}\\&&\\&=&{\frac {69.3}{R}}{\frac {1}{1-R/200}}\\&&\\&\approx &{\frac {69.3(1+R/200)}{R}}\\&&\\&=&{\frac {69.3}{R}}+{\frac {69.3}{200}}\\&&\\&=&{\frac {69.3}{R}}+0.3465\end{array}}}$

Replacing the "R" in R/200 on the third line with 7.79 gives 72 on the numerator. This shows that the rule of 72 is most accurate for periodically compounded interests around 8%. Similarly, replacing the "R" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%.

Alternatively, the E-M rule is obtained if the second-order Taylor approximation is used directly.

Continuous compounding

For continuous compounding, the derivation is simpler and yields a more accurate rule:

${\displaystyle {\begin{array}{ccc}(e^{r})^{p}&=&2\\e^{rp}&=&2\\\ln e^{rp}&=&\ln 2\\rp&=&\ln 2\\p&=&{\frac {\ln 2}{r}}\\&&\\p&\approx &{\frac {0.693147}{r}}\end{array}}}$