Rule of 72 Totally Explained

Everything about Rule Of 70 totally explained

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 compounding

The 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 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 (for example 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's 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. ť

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   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 ofInterest(given) $i$	Periods (given) $n$	Periods to Double $d=72/i$	Number of Doubling Periods, $m=n/d$	Final Divisor $f=2^m-1$	Savings Required per Period, $S=$720,000/d/f$	Actual Amount Saved $S*n$	Actual Interest Accumulated	Total
9%	8	8	1	1	$90,000.00	$720,000.00	$361,893.28	$1,081,893.28
9%	16	8	2	3	$30,000.00	$480,000.00	$599,211.14	$1,079,211.14
9%	24	8	3	7	$12,857.14	$308,571.41	$767,582.95	$1,076,154.38
9%	32	8	4	15	$6000.00	$192,000.00	$880,801.89	$1,072,801.89
9%	40	8	5	31	$2903.23	$116,129.00	$953,105.41	$1,069,234.45

Rate of Interest (given) $i$	Periods (given) $n$	Periods to Double $d=72/i$	Number of Doubling Periods, $m=n/d$	Final Divisor $f=2^m-1$	Savings Required per Period, $S=$720,000/d/f$	Actual Amount Saved $S*n$	Actual Interest Accumulated	Total
12%	6	6	1	1	$120,000.00	$720,000.00	$370,681.41	$1,090,681.41
12%	12	6	2	3	$40,000.00	$480,000.00	$601,164.37	$1,081,164.37
12%	18	6	3	7	$17,142.86	$308,571.00	$761,823.10	$1,070,394.53
12%	24	6	4	15	$8,000.00	$192,000.00	$866,670.96	$1,058,670.96
12%	30	6	5	31	$3,870.97	$116,129.00	$930,164.93	$1,046,293.96
12%	36	6	6	63	$1,904.76	$68,571.00	$964,949.89	$1,033,521.31

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).

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 doesn't derive or explain the rule, and it's thus assumed that the rule predates Pacioli by some time.
Roughly translated:

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