The rule of 72 is a mathematical shortcut used to predict when a population, investment or other growing category will double in size for a given rate of growth. It is also used as a heuristic device to demonstrate the nature of compound interest. It has been recommended by many statisticians that the number 69 be used, rather than 72, to estimate the results of continuous compounding rates of growth. Calculate how quickly continuous compounding will double the value of your investment by dividing 69 by its rate of growth.

The rule of 72 was actually based on the rule of 69, not the other way around. For non-continuous compounding, the number 72 is more popular because it has more factors and is easier to calculate returns quickly.

Continuous Compounding

In finance, continuous compounding refers to a growth rate with compounding periods that are infinitesimally small; the interest generated is calculated and compounded more than once per second, for example.

Because an investment with continuous compounding grows faster than an investment with simple or discrete compounding, standard time value of money calculations are ill-equipped to handle them.

Rule of 72 and Compounding

The rule of 72 comes from a standard compound interest formula:

 V F u t u r e = P V ( 1 + r ) n where: V F u t u r e = Future value P V = Present value r = Interest rate \begin{aligned} &V_{Future} = PV * \left(1 + r \right)^n\\ &\textbf{where:}\\ &V_{Future} = \text{Future value}\\ &PV = \text{Present value}\\ &r = \text{Interest rate}\\ &n = \text{Number of compounding periods} \end{aligned} VFuture=PV(1+r)nwhere:VFuture=Future valuePV=Present valuer=Interest rate

This formula makes it possible to find a future value that is exactly twice the present value. Do this by substituting FV = 2 and PV = 1:

 2 = ( 1 r ) n 2 = \left(1- r \right)^n 2=(1r)n

Now, take the logarithm of both sides of the equation, and use the power rule to simplify the equation further:

 2 = ( 1 r ) n ln 2 = ln ( 1 r ) n = n ln ( 1 r ) 0 . 6 9 3 n r \begin{aligned} 2 &= \left(1- r \right)^n\\ &\therefore\\ \ln{2} &= \ln{\left(1- r \right)^n} \\ &= n*\ln{\left(1- r \right)}\\ &\therefore\\ 0.693 &\approx n*r \end{aligned} 2ln20.693=(1r)n=ln(1r)n=nln(1r)nr

Since 0.693 is the natural logarithm of 2. This simplification takes advantage of the fact that, for small values of r, the following approximation holds true:

 ln ( 1 + r ) r \ln{\left(1+r\right)}\approx r ln(1+r)r

The equation can be further rewritten to isolate the number of time periods: 0.693 / interest rate = n. To make the interest rate an integer, multiply both sides by 100. The last formula is then 69.3 / interest rate (percentage) = number of periods.

It isn't very easy to calculate some numbers divided by 69.3, so statisticians and investors settled on the nearest integer with many factors: 72. This created the rule of 72 for quick future value and compounding estimations.

Continuous Compounding and the Rule of 69(.3)

The assumption that the natural log of (1 + interest rate) equals the interest rate is only true as the interest rate approaches zero in infinitesimally small steps. In other words, it is only under continuous compounding that an investment will double in value under the rule of 69.

If you really want to calculate how quickly an investment will double for a given interest rate, use the rule of 69. More specifically, use the rule of 69.3.

Suppose a fixed-rate investment guarantees 4% continuously compounding growth. By applying the rule of 69.3 formula and dividing 69.3 by 4, you can find that the initial investment should double in value in 17.325 years.