In finance, the rule of 72, the rule of 71, the rule of 70 and the
rule of 69.3 are methods 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.
The Eckart-McHale Rule ("the E-M Rule") provides a multiplicative correction to these approximate results, while Felix's
Corollary provides a method of estimating the future value of an annuity using the same principles.
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 72 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%; see derivation below). 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 0.2 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.
(approx)
A similar accuracy adjustment for the rule of 69.3 - used for high rates with daily compounding - is as follows:
(approx)
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:
(approx)
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:
(approx)
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 |
Derivation
Periodic compounding
For periodic compounding, future value is
given by
where PV is the present value, t is the number of time periods, and
r stands for the interest rate per time period.
Now, suppose that the money has doubled, then FV = 2PV.
Substituting this in the above formula and cancelling the factor PV on both side yields
This equation is easily solved for t:
If r is small, then ln(1+r) approximately equals r (this is the first term in the Taylor series). Together with the approximation ln(2) ≈ 0.693147, this gives
The relation approaches equality as the compounding of interest becomes continuous
(see derivation below).
In order to derive the E-M rule, we use the fact that ln(1+r) is more closely approximated by r - r^2/2
(using the second term in the Taylor series).
Continuous compounding
For continuous compounding the derivation is simpler:
implies
or
Using 100r to get percentages and taking 70 as a close enough approximation to 69.3147:
Felix's Corollary to the Rule of 72
Felix's Corollary provides a method of approximating the
future value of an annuity (a series of
regular payments), using the same principles as the Rule of 72. The corollary states that future value of an annuity whose
percentage interest rate and number of payments multiply to be 72 can be approximated by multiplying the sum of the payments
times 1.5.
As an example, 12 periodic payments of $1000 growing at 6% per period will be worth approximately $18,000 after the last
period. This can be calculated by multiplying 1.5 times the $12,000 of payments. This is an application of Felix's collorary
because 12 times 6 is 72. Likewise, 8 periodic thousand dollar payments at 9% will result in 1.5 times the $8000, or $12,000.
Accuracy
Felix's Corollary has similar accuracy issues as the Rule of 72; it is reasonably accurate in the 6% to 12% range (especially
in the 8% to 9% range), and progressively loses accuracy at smaller or larger values. In addition, an adjustment needs to be
considered in the cases where non-integer payments are required (such as at 7% or 10% or 12.5% interest). In such cases, a
fractional last payment must be made as you would expect. As an example, at 10% interest, 7.2 periodic payments must be made. In
normal cases, whole payments are made at the beginning of a period. It's not entirely obvious as to when the .2 payment must be
made. But for purposes of approximation, the corollary works quite well.
Applications of Felix's corollary
Millionaire's estimation
The millionaire's estimation is a simple savings calculator, posing the question "How much
must I save per year to have saved $1,080,000?" Of course, the annual interest rate is a factor. In the original challenge, the
number $1,080,000 was chosen due to its multiplicative relation to the number 72.
Using Felix's corollary, one can estimate that by saving two-thirds of the total, in periodic deposits, the interest will take
care of the rest (since 1.5 times two-thirds will equal the desired goal). So the goal becomes to set aside $720,000 in equal
periodic deposits, such that it grows to approximate the target amount of $1,080,000.
Rate of
Interest
(given) |
Periods,
(calculated
72/Rate) |
Savings Required
per Period,
(calculated
$720,000/Periods) |
Amount
Saved |
Actual Interest
Accumulated |
Total |
| 6% |
12 |
$60,000 |
$720,000 |
$352,928.26 |
$1,072,928.26 |
| 8% |
9 |
$80,000 |
$720,000 |
$358,925.00 |
$1,078,925.00 |
| 9% |
8 |
$90,000 |
$720,000 |
$361,893.28 |
$1,081,893.28 |
| 12% |
6 |
$120,000 |
$720,000 |
$370,681.41 |
$1,090,681.41 |
Combining the rule of 72 and Felix's corollary
Advanced calculations can also be performed, combining the Rule of 72 and its corollary.
For instance, using an annual 9% rate (which is often cited as an average stock market rate of return), the answer to the
Millionaire's Estimate problem is that you must save $90,000 per year for 8 years to accumulate the desired target. But if the
time horizon is 16 years at the same interest rate, then one must combine the Rule of 72 and the Corollary to arrive at the
estimated target annual savings rate.
It is solved (without a calculator) as follows: Target savings is $1,080,000, through fixed payments over 16 years, with a 9%
annual interest rate. The amount accumulated in the first 8 years will double during the second eight years with no additional
contributions (using the Rule of 72). And the amount of contributions accumulated during the second 8 years will need to
accumulate to some value so that when you multiply it by 3 (that is, add in the first 8 years' contributions, doubled), it
reaches $720,000. So $240,000 (or $720,000 divided by 3) needs to be deposited evenly over each 8 year period, or $30,000 per
year ($240,000 divided by 8).
In summary, 8 annual contributions of $30,000 starting in year 1 will grow to $360,000 after year 8 (using the Corollary,
$240,000 times 1.5), and will double to $720,000 after year 16 (using the Rule of 72). The 8 annual contributions in years 9
through 16 will likewise grow to $360,000 (using the Corollary). The sum of $720,000 and $360,000 provide the target savings of
$1,080,000 at the end of year 16. The yearly required savings can be quickly calculated as $720,000 divided by 8, divided by
3.
Likewise, other estimations can be performed, combining the Rule of 72 and its Corollary. For 24 years at 9%, the yearly
amount can be quickly estimated as $720,000, divided by 8, divided by 7. For 32 years at 9%, use $720,000 divided by 8, divided
by 15. For each 8-year period involved in the calculation (when the interest rate is 9%), the final divisor is doubled and incremented (that is, the divisor is {1, 3, 7, 15, 31, ...} when the savings period is
{8, 16, 24, 32, 40, ...} years).
Typically, one is solving for Savings Required Per Period, given a Rate of Interest, a Number of Periods, and a targeted
accumulated savings of $1,080,000. This is shown in the tables below:
Rate of
Interest
(given)
i |
Periods
(given)
n |
Periods
to Double
d = 72 / i |
Number of
Doubling
Periods,
m = n / d |
Final
Divisor
f = 2m - 1 |
Savings Required
per Period,
$720,000 / d / f |
Amount
Saved |
Actual Interest
Accumulated |
Total |
| 9% |
8 |
8 |
1 |
1 |
$90,000 |
$720,000 |
$361,893.28 |
$1,081,893.28 |
| 9% |
16 |
8 |
2 |
3 |
$30,000 |
$720,000 |
$359,211.14 |
$1,079,211.14 |
| 9% |
24 |
8 |
3 |
7 |
$12,857.14 |
$720,000 |
$356,154.14 |
$1,076,154.14 |
| 9% |
32 |
8 |
4 |
15 |
$6000 |
$720,000 |
$352,801.89 |
$1,072,801.89 |
| 9% |
40 |
8 |
5 |
31 |
$2903.23 |
$720,000 |
$349,235.99 |
$1,069,235.99 |
Rate of
Interest
(given)
i |
Periods
(given)
n |
Periods
to Double
d = 72 / i |
Number of
Doubling
Periods,
m = n / d |
Final
Divisor
f = 2m - 1 |
Savings Required
per Period,
$720,000 / d / f |
Amount
Saved |
Actual Interest
Accumulated |
Total |
| 12% |
6 |
6 |
1 |
1 |
$120,000 |
$720,000 |
$370,681.41 |
$1,090,681.41 |
| 12% |
12 |
6 |
2 |
3 |
$40,000 |
$720,000 |
$361,164.37 |
$1,081,164.37 |
| 12% |
18 |
6 |
3 |
7 |
$17,142.86 |
$720,000 |
$350,394.53 |
$1,070,394.53 |
| 12% |
24 |
6 |
4 |
15 |
$8,000 |
$720,000 |
$338,670.96 |
$1,058,670.96 |
| 12% |
30 |
6 |
5 |
31 |
$3,870.97 |
$720,000 |
$326,293.96 |
$1,046,293.96 |
| 12% |
36 |
6 |
6 |
63 |
$1,904.76 |
$720,000 |
$313,521.31 |
$1,033,521.31 |
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.
| “ |
A voler sapere ogni quantita a tanto per 100 l'anno, in quanti anni sara 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 sara raddoppiato. Esempio: Quando l'interesse e a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12,
e in 12 anni sara raddoppiato il capitale. (emphasis added). |
” |
Roughly translated:
| “ |
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. |
” |
See also
External links
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