Rule of 72

In finance, the rule of 72, the rule of 70 and the rule of 69 are methods for estimating an investment's doubling time. The rule number is divided by the interest percentage per period to obtain the approximate number of periods (usually years) required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available.^[1]

These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference, 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible. Felix's Corollary provides a method of estimating the future value of an annuity using the same principles.^{[citation needed]}

There are a number of variations to the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of r per period is:

T={\frac {\ln(2)}{\ln(1+r)}}

.

where T is the number of periods required. The formula above can be used for more than calculating the doubling time. If you want to know the tripling time, for example, simply replace the constant 2 in the numerator with 3. As another example, if you want to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.

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 unit of currency to halve.

To estimate the impact of additional fees on financial policies (e.g., 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 exactly the 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. It 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.

For continuous compounding, 69 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, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72.

Rate	Actual Years	Rule of 72	Rule of 70	Rule of 69.3	72 adjusted	E-M rule
0.25%	277.605	288.000	280.000	277.200	277.667	277.547
0.5%	138.976	144.000	140.000	138.600	139.000	138.947
1%	69.661	72.000	70.000	69.300	69.667	69.648
2%	35.003	36.000	35.000	34.650	35.000	35.000
3%	23.450	24.000	23.333	23.100	23.444	23.452
4%	17.673	18.000	17.500	17.325	17.667	17.679
5%	14.207	14.400	14.000	13.860	14.200	14.215
6%	11.896	12.000	11.667	11.550	11.889	11.907
7%	10.245	10.286	10.000	9.900	10.238	10.259
8%	9.006	9.000	8.750	8.663	9.000	9.023
9%	8.043	8.000	7.778	7.700	8.037	8.062
10%	7.273	7.200	7.000	6.930	7.267	7.295
11%	6.642	6.545	6.364	6.300	6.636	6.667
12%	6.116	6.000	5.833	5.775	6.111	6.144
15%	4.959	4.800	4.667	4.620	4.956	4.995
18%	4.188	4.000	3.889	3.850	4.185	4.231
20%	3.802	3.600	3.500	3.465	3.800	3.850
25%	3.106	2.880	2.800	2.772	3.107	3.168
30%	2.642	2.400	2.333	2.310	2.644	2.718
40%	2.060	1.800	1.750	1.733	2.067	2.166
50%	1.710	1.440	1.400	1.386	1.720	1.848
60%	1.475	1.200	1.167	1.155	1.489	1.650
70%	1.306	1.029	1.000	0.990	1.324	1.523

History

An early reference to the rule is in the Summa de Arithmetica (Venice, 1494. Fol. 181, n. 44) of 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 sarà 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 sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale. (emphasis added).

Roughly translated:

In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.

Adjustments for higher accuracy

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.

t\approx {\frac {72+(r-8)/3}{r}}

or for the same result, but simpler:

t\approx {\frac {70+(r-2)/3}{r}}

t\approx {\frac {69.3}{r}}+0.33

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%–20%. 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 result by 200/(200-r) as follows:

t\approx {\frac {69.3}{r}}\times {\frac {200}{200-r}}

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.)

Note that if the numerator here is simply 69.3 times 200. As long as the product stays constant, the factors can be modified arbitrarily. The E-M rule could thus be written also as

t\approx {\frac {70}{r}}\times {\frac {198}{200-r}}

or

t\approx {\frac {72}{r}}\times {\frac {192}{200-r}}

in order to keep the product mostly unchanged. In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rule of 70 (respectively 72) is most precise.

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:

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

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 by 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 the $12,000 of payments by 1.5. This is an application of Felix's corollary because 12 (= #periods) times 6 (= %interest) is 72. Likewise, 8 periodic thousand dollar payments at 9% will result in 1.5 times the $8000, or $12,000.

The equation is:

T\approx P\times N\times 1.5

when

R={\frac {72}{N}}

where:

$T$ is the total ending balance.
$P$ is the periodic payment amount.
$N$ is the number of payments.
$R$ is the interest rate multiplied by $100$ ; that is, the interest rate can be expressed as $R\%$ per period.

Accuracy

Felix'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.

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 because it is divisible by 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 or Rate pct x $1MM)	Amount Saved	Actual Interest Accumulated	Total
2%	36	$20,000	$720,000	$340,685.09	$1,060,685.09
4%	18	$40,000	$720,000	$346,849.18	$1,066,849.18
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

Other Rules

Extending the Rule of 72 out further, other approximations can be determined for tripling and quadrupling. To estimate the time it would take to triple your money, one can use 114 instead of 72. And for quadrupling, use 144.

Derivation

Periodic compounding

For periodic compounding, future value is given by:

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

where $PV$ is the present value, $t$ is the number of time periods, and $r$ stands for the interest rate per time period.

The future value is double the present value when the following condition is met:

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

This equation is easily solved for $t$ :

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

If $r$ is small, then $ln(1+r)\,$ approximately equals $r$ (this is the first term in the Taylor series). Together with the approximation $ln(2)\approx 0.693147$ , this gives:

t\approx {\frac {0.693147}{r}}

which improves in accuracy as the compounding of interest becomes continuous (see derivation below).

However, humans tend to prefer performing mental calculations with percentages, so the formula is often restated as follows:

{\begin{array}{ccc}t&\approx &{\frac {0.693147}{r}}\\&&\\&=&{\frac {0.693147}{R\%}}\\&&\\&=&{\frac {0.693147}{R{\frac {1}{100}}}}\\&&\\&=&{\frac {0.693147\cdot 100}{R}}\\&&\\&=&{\frac {69.3147}{R}}\\&&\\&\approx &{\frac {70}{R}}\end{array}}

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

{\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}}+0.34\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 precise for periodically composed interests around 8%.

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:

{\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}}

References

^ Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. pp. 153–154. ISBN 0471506362.

External links

The Scales Of 70 – extends the rule of 72 beyond fixed-rate growth to variable rate compound growth including positive and negative rates.
A note on the rule of 72 or how long it takes to double your money, The Investment Analysts Society of South Africa
Rule of 72 Calculator, moneychimp.com
Rule of 72 Learning Tool, directinvesting.com

[1] Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. pp. 153–154. ISBN 0471506362.

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