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Rule of 72
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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 half 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 unit of currency 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 compoundingThe rule of 72 provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.
Low rates / daily compoundingFor 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 ratesFor 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 ruleThe 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 comparisonThis table compares the three rules, using periodic compounding, and illustrates the error of the estimation over a range of typical values.
DerivationPeriodic 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 72Felix'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.
AccuracyFelix's Corollary has accuracy issues similar to 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 corollaryMillionaire's estimationThe 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.
Combining the rule of 72 and Felix's corollaryAdvanced 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 when the savings period is 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:
The final example in the table above demonstrates that one who saves just over $1900 per year for 36 years at 12% will accumulate over a million dollars - a plausible plan for an aggressive investor to accumulate wealth from age 19 to 55. Likewise, at 9%, saving just over $2900 per year will accumulate to over one million dollars from age 20 to 60 (or any 40 year span).
HistoryAn 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.
Roughly translated:
See also
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