Doubling Time Calculator

t = ln(2) ÷ ln(1 + r) t ≈ 72 ÷ R

r

The growth rate for one period, written as a decimal rather than a percentage: a rate of 5 percent per period enters the formula as 0.05. It must be positive, since a quantity that is not growing never doubles at all, and it is entered as a percentage in the field. Compound growth is what this describes — each period's growth is applied to the total including everything grown before it, which is what makes the doubling time finite instead of impossible.

ln 2

The natural logarithm of two, about 0.693147, and the only constant in the exact formula. It appears because doubling means the growth factor has reached exactly two, so the question is how many multiplications by (1 + r) it takes to turn one into two — and the logarithm is the operation that answers that.

t

The doubling time, in periods, printed to three decimal places. It carries no unit of its own: if the rate you entered was a monthly rate, t is a number of months; if it was annual, t is a number of years. The last doubling is partial, which is why t is not a whole number — after 14 periods at 5 percent the quantity is a little short of double, and it crosses over partway through the fifteenth.

R

The same rate, but as a percentage, which is the number typed into the field. The rule of 72 is the approximation that divides 72 by R, and it is the second reading on the panel. It is accurate near 8 percent, overshoots at low rates and undershoots at high ones, and it exists because a division by a round number was much easier to do in your head than a logarithm.

Reach for this page when the growth rate is given and the time is what you want. The exponential growth calculator runs the same model in the other direction, starting from an initial quantity and a number of periods; the half-life calculator is the mirror image, with a factor of one half instead of two; and the rule of 72 calculator is the same arithmetic in a money setting, where the rule of thumb rather than the exact answer is the headline.

1. Five percent per period

   1. The rate as a decimal is 0.05, so each period multiplies the quantity by 1.05
   2. The exact answer is ln 2 ÷ ln 1.05 = 0.693147 ÷ 0.048790 = 14.207 periods
   3. The rule of 72 gives 72 ÷ 5 = 14.4 periods
   4. The two agree to a little over a week's worth of a period, which is the usual gap between the rule and the truth

   Double-checking the answer is easy: 1.05 raised to the 14th power is 1.98, just under two, and to the 15th is 2.08, just over. So the crossing happens between the fourteenth and fifteenth period, exactly where 14.207 says. That check is worth doing once, because it shows the decimal is a position on a timeline rather than an accounting artefact.

2. One percent per period

   1. The exact answer is ln 2 ÷ ln 1.01 = 0.693147 ÷ 0.009950 = 69.661 periods
   2. The rule of 72 gives 72 ÷ 1 = 72 periods
   3. The rule is out by 2.3 periods here, which at one percent means more than two extra periods of waiting
   4. The error is systematic rather than random: the rule of 72 is tuned for rates around 8 percent and drifts at the low end

   The gap at low rates is the reason the two readings are shown side by side rather than one replacing the other. A rule of thumb that is two periods out over a lifetime of compounding is a real difference, and seeing both numbers is what tells you which one to trust for the rate you actually have.

3. Twenty percent per period

   1. The exact answer is ln 2 ÷ ln 1.2 = 0.693147 ÷ 0.182322 = 3.802 periods
   2. The rule of 72 gives 72 ÷ 20 = 3.6 periods
   3. This time the rule comes in low rather than high, which is what happens past 8 percent
   4. The check: 1.2 cubed is 1.728 and 1.2 to the fourth is 2.074, so the crossing sits between period three and period four

   Between this example and the one percent one, the error of the rule changes sign — overestimating at low rates, underestimating at high ones — and that is the shape of the approximation rather than a mistake in it. The rate where the two readings come closest is around 8 percent, where the rule gives 9 and the exact answer is 9.006.

The doubling time has no unit of its own. It is measured in whatever period the rate was measured in, so 14.207 is years if the rate was annual, months if it was monthly, and minutes if you were watching bacteria. The page cannot tell which, and it will not guess. The rate must be positive and lies between 0.1 and 1000 percent per period; zero is refused because neither formula has an answer for it, and a negative rate describes a quantity that shrinks, which never doubles and belongs to the half-life calculator instead. The constant 2 is fixed: this page finds the time to double and cannot be asked for the time to triple or to grow by any other multiple. Growth is assumed to be smooth and constant, applied continuously rather than in yearly steps, which is the usual model but not what a bank actually does. The rule of 72 reading is an approximation by construction and is shown for comparison rather than as an answer, and no error analysis between the two readings is given here.

How do you calculate doubling time?

Divide the natural log of 2 by the natural log of one plus the rate. At 5 percent per period that is 0.693147 ÷ 0.048790, which comes to 14.207 periods. The logarithm is doing the work because doubling is a multiplication repeated an unknown number of times, and the logarithm is what turns that into a division.

What is the rule of 72 and is it accurate?

It is the shortcut that divides 72 by the percentage rate, so 5 percent doubles in about 14.4 periods. It is accurate to within a few percent near 8 percent, overestimates the time at low rates — at 1 percent it says 72 against a true 69.661 — and underestimates it at high rates, where 20 percent gives 3.6 against a true 3.802. This page shows both numbers so the gap is visible at the rate you actually have.

Is the doubling time measured in years?

Only if the growth rate you entered was annual. The rate is per period and the period is whatever you decide it is: a monthly rate gives an answer in months, a daily one in days, and a bacterial culture doubling every twenty minutes gives an answer in twenty-minute units. The page deliberately does not label the output, because it cannot know which you meant.

Why is a zero or negative growth rate refused?

Because nothing doubles. At zero the quantity stays where it is forever and the rule of 72 divides by zero; at a negative rate the quantity shrinks and the honest question becomes the half-life rather than the doubling time. Rates below 0.1 percent per period are refused for the same practical reason, and the half-life calculator takes the shrinking case.

What is the difference between this and exponential growth?

Direction. This page starts from a growth rate and returns a time; the exponential growth calculator starts from a rate and a number of periods and returns the resulting quantity. The model underneath is the same one, so a doubling time of 14.207 periods at 5 percent is exactly the statement that the growth factor reaches 2 at that point on the exponential curve.

• Exponential Growth — growth at a rate proportional to the current quantity, and the doubling time that follows from it — Wolfram MathWorld (United States)
• Exponential Function — the base e, the natural logarithm that undoes it, and why ln 2 appears in every doubling formula — Wolfram MathWorld (United States)
• Compound Interest — the same growth factor applied period after period, which is where the doubling time is usually met first — Wolfram MathWorld (United States)