The Rule of 72 is a simple way to estimate a compound interest calculation for doubling
an investment. The formula is interest rate multiplied by the number of time periods = 72:
r x t = 72
where
•
•
r = interest rate per period as a percentage
t = number of periods
Commonly, periods are years so r is the interest rate per year and t is the number of
years. You can calculate the number of years to double your investment at some known
interest rate by solving for t: t = 72 / r. You can also calculate the interest rate required to
double your money within a known time frame by solving for r: r = 72 / t.
Derivation of the Rule of 72 Formula
The basic compound interest formula is:
F = P(1 + r)t,
where F is the future amount, P is the present or principal investment, r is the interest
rate per period in decimal form, and t is the number of periods. If we change this formula
to show that the future amount is twice the principal investment, P, then we have F = 2 x
P. Rewriting the formula:
2 x P = P(1 + r)t , and dividing by P on both sides gives us
(1 + r)t = 2
We can solve this equation for t by taking the natural log, ln(), of both sides,
t × ln(1+r) = ln(2)
and isolating t on the left:
t = ln(2) / ln(1+r)
We can rewrite this to an equivalent form:
t =(ln(2) / r) × (r / ln(1+r))
Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8%
(r=0.08)*:
t = (0.69/r) × (0.08 / ln(1.08))
t = (0.69 / r) x (1.0395)
Solving this equation for r times t:
r x t = 0.69 × 1.0395≈0.72
Finally, multiply both sides by 100 to put the decimal rate r into the percentage rate r:
r x t = 72
*8% is used as a common average and makes this formula most accurate for interest
rates from 6% to 10%.
Example Calculations in Years
If you invest a sum of money at 6% interest per year, how long will it take you to double
your investment?
t = 72 / r = 72 / 6 = 12 years
What interest rate do you need to double your money in 10 years?
r = 72 / t = 72 / 10 = 7.2%
Example Calculation in Months
If you invest a sum of money at 0.5% interest per month, how long will it take you to
double your investment?
t = 72 / r = 72 / 0.5 = 144 months (since r is a monthly rate the answer is in months
rather than years)
144 months = 144 months / 12 months per years = 12 years
References
Vaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second
Edition), Washington DC: The Mathematical Association of America, 2009, page 75.
Weisstein, Eric W. "Rule of 72." From MathWorld--A Wolfram Web Resource, Rule of
72.