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.

Contents

1 Workings
2 Choice of rule

2.1 \"Typical\" rates / annual compounding
2.2 Low rates / daily compounding
2.3 Adjustments for higher rates

3 Derivation

3.1 Periodic compounding
3.2 Continuous compounding

4 See also
5 External links

Workings

To estimate the number of periods required to double an original investment, divide the "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 70 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.

Choice of rule

The value 72 is a convenient choice of numerator, since it has many small divisors: 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 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 ] (approx)

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

[ t = \frac ] (approx)

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 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. ]

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. ]

Continuous compounding

For continuous compounding the derivation is simpler:

[\ 2=(e^r)^p]

implies

[\ pr=\ln(2)]

[p= \frac = \frac.]

Using 100r to get percentages and taking 70 as a close enough approximation to 69.3147:

[p= \frac.]

External links

[Exponentialist article "The Scales Of 70"], which 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 in reverse], personal finance article.
[Rule of 72] explained in plain English.

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