Rule of 72

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In finance, the rule of 72, 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. The Eckart-McHale Rule ("the E-M Rule") provides a multiplicative correction to these approximate results.

1 Workings
2 Choice of rule
3 Derivation
- 3.1 Periodic compounding
- 3.2 Continuous compounding
4 See also
5 External links

[edit] Workings

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 at 9% per annum, the "rule of 72" gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation, using time value of money, 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.

For instance, 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, it should take approximately 70/3.5 = 20 years for the value of a dollar to halve.

[edit] 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.

[edit] "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%).

[edit] 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.

[edit] 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.

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

A similar accuracy adjustment for the rule of 69.3 - used for high rates with daily compounding - is as follows:

$t = \frac{69.3147 + r/3}{r}$ (approx)

[edit] 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:

$t = \frac{69.3}{r} \times \frac{200}{200-r}$ (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.)

[edit] 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

[edit] Derivation

[edit] 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 discount 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

$2 = (1+r)^t.\,$

This equation is easily solved for t:

$t = \frac{\ln 2}{\ln(1+r)}.$

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

$t = \frac{0.693147}{r}.$