Rule of 72
In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69.3 all refer
to a method for estimating an investment's doubling time, or halving time. These rules apply to
exponential growth and decay respectively, and are therefore used for compound interest as opposed
to simple interest calculations.
History
An early reference to the rule is in the Summa de Arithmetica (Venice, 1494. Fol. 181, n. 44)
of Fra 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.
"In wanting to know for any percentage, in how many years the capital will be doubled, you
bring to mind the rule of 72, which you always divide by the interest, and the result is in how
many years it will be doubled. Example: When the interest is 6 percent per year, I say that one
divides 72 by 6; obtaining 12, and in 12 years the capital will be doubled.
Using the Rule to Estimate Compounding Periods
To estimate the number of periods required to double an original investment, divide the most
convenient "rule-quantity" by the expected growth rate, expressed as a percentage.
For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum,
the "rule of 72" gives 72/9 = 8 years required for the investment to be worth $200; an exact
calculation gives 8.0432 years.
Similarly, to determine the time it takes for the value of money to halve at a given rate,
divide the rule quantity by that rate.
To determine the time for money's buying power to halve, financiers simply divide the
"rule-quantity" by the inflation rate. Thus at 3.5% inflation using the rule of 70, it should
take approximately 70/3.5 = 20 years for the value of a dollar to halve.
To estimate the impact of additional fees on financial policies (eg. Mutual fund fees and
expenses, Loading and Expense Charges on Variable universal life insurance investment portfolios),
divide 72 by the fee. For example, if the Universal Life policy charges a 3% fee over and above
the cost of the underlying investment fund, then the total account value will be cut
to 1/2 in 72 / 3 = 24 years, and then to just 1/4 the value in 48 years, compared to holding
the exact same investment outside the policy.
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. However, depending on the rate and compounding period in question,
other values will provide a more appropriate choice.
"Typical" rates / annual compounding
The rule of 71 provides a good approximation for annual compounding, and for compounding at
"typical rates" (from 6% to 10%).
Low rates / daily compounding
For continuous compounding, 69.3 gives accurate results for any rate (this is because ln(2)
is about 69.3%). Since daily compounding is close enough to continuous compounding, for most
purposes 69.3 - or 70 - is used in preference to 72 here. For lower rates than those above,
69.3 would also be more accurate than 72.
Adjustments for higher rates
For higher rates, a bigger numerator would be better (e.g. for 20%, using 76 to get 3.8
years would be only about 0.002 off, where using 72 to get 3.6 would be about 2.002 off). This
is because, as above, the rule of 72 is only an approximation that is accurate for interest
rates from 6% to 10%. Outside that range the error will vary from 2.4% to −14.0%. For every
three percentage points away from 8% the value 72 could be adjusted by 1.
A similar accuracy adjustment for the rule of 69.3 - used for high rates with daily
compounding - is as follows:
E-M Rule
The Eckart-McHale Second Order Rule, "the E-M Rule", gives a multiplicative correction to
the Rule of 69.3 or 70 (but not 72). The E-M Rule's main advantage is that it provides the best
results over the widest range of interest rates. Using the E-M correction to the rule of 69.3,
for example, makes the Rule of 69.3 very accurate for rates from 0%-20% even though the Rule of
69.3 is normally only accurate at the lowest end of interest rates, from 0% to about 5%.
To compute the E-M approximation, simply multiply the Rule of 69.3 (or 70) result by
200/(200-r) as follows:
For example, if the interest rate is 18% the Rule of 69.3 says t = 3.85 years. The E-M Rule
multiplies this by 200/(200-18), giving a doubling time of 4.23 years, where 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.)
Similarly, the 3rd order Padé approximant gives a more accurate answer over an even larger
range of r, but it has a slightly more complicated formula:
Illustrative Comparison
This table compares the three rules, using periodic compounding, and illustrates the error
of the estimation over a range of typical values.
Rate of
Interest |
Actual
Years |
Rule of 72
Estimate |
Rule of 70
Estimate |
Rule of 69.3
Estimate |
E-M Rule
Estimate |
| 0.25% |
277.605 |
288.000 |
280.000 |
277.200 |
277.547 |
| 0.5% |
138.976 |
144.000 |
140.000 |
138.600 |
138.947 |
| 1% |
69.661 |
72.000 |
70.000 |
69.300 |
69.648 |
| 2% |
35.003 |
36.000 |
35.000 |
34.650 |
35.000 |
| 3% |
23.450 |
24.000 |
23.333 |
23.100 |
23.452 |
| 4% |
17.673 |
18.000 |
17.500 |
17.325 |
17.679 |
| 5% |
14.207 |
14.400 |
14.000 |
13.860 |
14.215 |
| 6% |
11.896 |
12.000 |
11.667 |
11.550 |
11.907 |
| 7% |
10.245 |
10.286 |
10.000 |
9.900 |
10.259 |
| 8% |
9.006 |
9.000 |
8.750 |
8.663 |
9.023 |
| 9% |
8.043 |
8.000 |
7.778 |
7.700 |
8.062 |
| 10% |
7.273 |
7.200 |
7.000 |
6.930 |
7.295 |
| 11% |
6.642 |
6.545 |
6.364 |
6.300 |
6.667 |
| 12% |
6.116 |
6.000 |
5.833 |
5.775 |
6.144 |
| 15% |
4.959 |
4.800 |
4.667 |
4.620 |
4.995 |
| 18% |
4.188 |
4.000 |
3.889 |
3.850 |
4.231 |
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