Up Rules Of Population Rule Of 70 and Rule Of 72 Compared The Scales Of e External Links: The Rule Of 70 - Professor E. Kwan Choi, Department of Economics Iowa State University The Rule of 72 - Better Explained

The Rule Of 70 and The Rule Of 72 Compared

Introduction

The heuristic tool the "Rule of 70" is derived from the Natural Logarithm of 2, which is  0.693147. When dealing in percentages, this becomes  69.3147. You will find an occasional reference to the more accurate Rule of 69.3 (or the Rule of 69), but both the Rule of 70 and the Rule of 72 are far more widely used.

The Rule Of 70 is normally explained only in terms of positive growth rates. It can be stated thus: if you divide the growth rate (expressed as a percentage) for a given period into 70 then you will get the crude population doubling period for that population (expressed in the same time units as used for the rate).

In finance, the Rule Of 72 is used in a similar fashion, except that you divide the rate into 72 rather than 70. Rather than obtaining a population doubling period, you would typically calculate an investment doubling period. In finance, the Rule Of 72 is probably used in preference to the Rule Of 70 as 72 has more whole number divisors (72, 36, 24, 18, 12, 9, 8 and 1) than 70 (70, 35, 14, 10, 7 and 1). Hence, the Rule of 72 is easier to explain to potential customers than the Rule of 70.

The Comparison

Table 1 (an extension of Professor Choi's own comparison in his article The Rule Of 70) compares the accuracy of the Rule Of 70 to the Rule Of 72.

Assuming the growth rate to be positive, the Rule Of 70 is more accurate up to 4%, you can use either at 5% (though the Rule Of 72 is slightly more accurate), and the Rule Of 72 is more accurate from 6% to 10%. Overall, accuracy declines as the growth rate increases.

Assuming the growth rate to be negative, the Rule Of 70 is always more accurate than the Rule Of 72.

 Growth Rate Doubling Period Halving Period f(X) + f(1/X) Rule Of 70 Rule Of 72 Accurate Value f(X) Rule Of 70 Rule Of 72 Accurate Value f(1/X) 1 70 72 69.66071689 70 72 -68.96756394 0.693153 2 35 36 35.00278878 35 36 -34.30961849 0.693170 3 23.33333333 24 23.44977225 23.33333333 24 -22.75657306 0.693199 4 17.5 18 17.67298769 17.5 18 -16.97974802 0.693240 5 14 14.4 14.20669908 14 14.4 -13.51340733 0.693292 6 11.66666667 12 11.89566105 11.66666667 12 -11.20230558 0.693355 7 10 10.28571429 10.24476835 10 10.28571429 -9.551337509 0.693431 8 8.75 9 9.006468342 8.75 9 -8.312950414 0.693518 9 7.777777778 8 8.043231727 7.777777778 8 -7.349614958 0.693617 10 7 7.2 7.272540897 7 7.2 -6.578813479 0.693727

Table 1 - The Rule Of 70 and The Rule Of 72 compared to accurate doubling and halving periods. Red doubling / halving periods are closer to the actual accurate values Overall, the Rule Of 70 is more accurate than the Rule Of 72.

Calculating Accurate Doubling and Halving Periods

Assume you ignore the Rule Of 70 and the Rule Of 72, and simply want to know the accurate doubling and halving periods for a given rate - how is this done? How does negative growth and positive growth compare?

Taking a growth rate of r, the accurate doubling period is calculated as:

f(X) = Ln 2 / Ln (1 + (r / 100))

For a growing population, exponential factor f(X) will always be positive.

Taking a growth rate of r, the accurate halving period is calculated as:

f(1/X) = Ln 2 / Ln (1 - (r / 100))

For a shrinking population, exponential factor f(1/X) will always be negative.

Natural Logarithms - A New Approximation

Note the slight but consistent difference between the accurate doubling and halving periods for any given growth rate.

Curiously enough this difference is very close to the Natural Logarithm of 2 (Ln 2 = 0.693147), which is where we get the Rule Of 70 in the first place.  This suspicion was confirmed by examining similar Rules Of Population for trebling / thirding (Rule Of 110) and quadrupling / quartering (Rule Of 140) which seems to point to a scientific law of nature.

However it should be noted that, the greater the rate, the further away the sum of the accurate doubling and halving periods is to the Natural Logarithm of 2. So instead what we have is a very accurate approximation:

The Natural Logarithm of exponential factor F is defined as the sum of the accurate positive period required to grow by factor F and the accurate negative period required to shrink by factor 1/F for any given rate

So, if the factor is 2, then (for any given rate) the Natural Logarithm of 2 is defined as the sum of the accurate positive doubling period and the accurate negative halving period.

That is:

Ln(2) = [Ln 2 / Ln (1 + (r / 100))] + [Ln 2 / Ln (1 - (r / 100))]

Or

Ln 2 = f(X) + f(1/X)

This can be written:

f(X) = ABS( f(1/X)) + Ln 2

which translates into English as:

For any given (constant) rate, the Doubling Period is equal to the (absolute value of the) Halving Period plus the Natural Logarithm of 2.

Rule Of 70 - Positive and Negative Growth Compared

So, the actual doubling periods and halving periods for any given rate are not quite as predicted by either the Rule Of 70 or the Rule Of 72.

Suppose instead that we apply a constant growth rate (r) for the period (P) called for by the Rule Of 70 or Rule Of 72, what happens? For a given growth rate, how does the imbalance between positive and negative growth look after the growth period required by the Rule Of 70 or the Rule Of 72?

Applying the Natural Logarithm of the given growth rate (r) for the period called for by the Rule Of 70, we get:

 r P = 70 / r LN(1+r/100) LN(1-r/100) Approx. Doubling Factor (X) Approx. Halving Factor (Y) 1 70 0.00995 -0.01005 2.006763 0.494839 2 35 0.019803 -0.0202 1.99989 0.493075 3 23.33333 0.029559 -0.03046 1.993128 0.491293 4 17.5 0.039221 -0.04082 1.986477 0.489493 5 14 0.04879 -0.05129 1.979932 0.487675 6 11.66667 0.058269 -0.06188 1.973491 0.485838 7 10 0.067659 -0.07257 1.967151 0.483982 8 8.75 0.076961 -0.08338 1.960911 0.482107 9 7.777778 0.086178 -0.09431 1.954767 0.480212 10 7 0.09531 -0.10536 1.948717 0.478297

Table 2 - Approximate Doubling Factor (X) and Approximate Halving Factor (Y) for a given rate, assuming periods called for by the Rule Of 70. The factors in red are more accurate than those calculated for the Rule Of 72. At 5%, there is only 0.001117 difference between the Rule Of 70 Approx. Doubling Factor and the Rule Of 72 Approx Doubling factor

Table 2 shows what would happen if the given growth rates were applied for the periods called for by the Rule Of 70.

Taking a growth rate of r, the Approx. Doubling Factor ( value X) is calculated as:

X = e(LN(1+r/100)*P)

I refer to X here as the Approximate Doubling Factor, as the Rule Of 70 is a rule of thumb for calculating doubling periods. However, the X factor is the accurate multiplying factor that would be applied to a population (or amount of money) if the growth rate r was applied for the period P.

Taking a growth rate of r, the Approx. Halving Factor  ( value Y) is calculated as:

Y = e(LN(1-r/100)*P)

I refer to Y here as the Approximate Halving Factor, as the Rule Of 70 is a rule of thumb for calculating halving periods (though it used more often for calculating doubling periods). However, the Y factor is the accurate multiplying factor that would be applied to a population (or amount of money) if the growth rate r was applied for the period P. Being less than 1, it is effectively a dividing factor.

These growth factors are generically referred to as exponential factors throughout the Exponentialist web site.

Also, note that though the Rule Of 70 is always more accurate than the Rule Of 72 for negative growth (or shrinkage), it is always less accurate for negative growth than it is for positive growth.

Rule Of 72 - Positive and Negative Growth Compared

Applying the Natural Logarithm of the given growth rate (r) for the period called for by the Rule Of 72, we get:

 r P = 72 / r LN(1+r/100) LN(1-r/100) Approx. Doubling Factor (X) Approx. Halving Factor (Y) 1 72 0.00995 -0.01005 2.047099 0.484991 2 36 0.019803 -0.0202 2.039887 0.483213 3 24 0.029559 -0.03046 2.032794 0.481417 4 18 0.039221 -0.04082 2.025817 0.479603 5 14.4 0.04879 -0.05129 2.018952 0.477771 6 12 0.058269 -0.06188 2.012196 0.47592 7 10.28571 0.067659 -0.07257 2.005548 0.474051 8 9 0.076961 -0.08338 1.999005 0.472161 9 8 0.086178 -0.09431 1.992563 0.470253 10 7.2 0.09531 -0.10536 1.98622 0.468324

Table 3 - Approximate Doubling Factor (X) and Approximate Halving Factor (Y) for a given rate, assuming periods called for by the Rule Of 72. The factors in red are more accurate than those calculated for the Rule Of 70. At 5%, there is only 0.001117 difference between the Rule Of 70 Approx. Doubling Factor and the Rule Of 72 Approx Doubling factor

X and Y are calculated as for Table 2 (and the Rule Of 70), though of course P is calculated by dividing the growth rate r into 72 rather than 70.

Variable Rate Compound Interest

Both the Rule of 70 and the Rule of 72 are explained in terms of constant rates of exponential growth, which is the equivalent of fixed rate compound interest.

To understand what happens when a population (or loan / investment) experiences variable rates of compound interest, including a mix of positive and negative rates, please read my articles the Scales of 70 and the Scales of e

Conclusion

As rules of thumb, both the Rule Of 70 and the Rule Of 72 have their uses. Do a Google search and you will find millions of hits for each, demonstrating the enduring popularity of each. Often, adherents of one of these rules of thumb are to surprised to learn of the existence of the other.

It has proved useful to compare negative and positive growth at the same rate, revealing a useful scientific law of regarding Natural Logarithms and the imbalance between positive and negative growth  for a given growth rate.

Given the superior accuracy of the Rule Of 70 to the Rule Of 72 at negative rates of growth the Exponentialist approach is to use the Rule Of 70 in preference to the Rule Of 72 when an approximate answer will suffice.

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Copyright © 2001 David A. Coutts
Last modified: 04 January, 2010