Calculate the number of years to double your investment at a given interest rate. Or, calculate the interest rate required to double your money within a certain time frame.
The Rule of 72 is a simple mental or pencil and paper method to estimate time or interest required to double your money. It is a simplified compound interest calculation.
The rule of 72 is a simple way to estimate a compound interest calculation that doubles an investment without applying the actual formula. This allows you to do an estimate, pretty well, in your head or just using a pencil and paper.
The rule says that in order to double an amount of money at a given rate and time, the interest percentage per period times the number of periods is equal to 72. We can write it as
R * t = 72
R = interest rate per period as a percentage, not a decimal
r = interest rate in decimal form, r = R/100
t = number of periods
Commonly, periods are years so R would be the rate per year and t is the number of years. Now we can calculate the number of years to double our investment at some known interest rate by solving for t; t=72/R. And, we can calculate the interest rate required to double our money within a known time frame by solving for R; R=72/t.
Generally, periods is in years but you can use any time period as long as the interest rate you use is the corresponding periodic rate.
The basic compound interest formula is
A = P(1 + r)t (equation 1)
where A is the accrued amount, P is the principal investment, r is the interest rate per period t in decimal, and t is the number of periods. If we want to reformat this formula to show that the accrued amount is double of our principal investment, P, then we have A = 2P. Rewriting;
2P = P(1 + r)t , and cancelling P from both sides we have;
(1 + r)t = 2
We can solve this equation for t by taking the natural log, ln(), of both sides,
and isolating t on the left;
it should be easy to show that we can rewrite this to an equivalent form
Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8% (r=0.08)*
Solving this equation for r times t
finally putting r the decimal rate into R the percent rate by multiplying both sides by 100,
R*t = 72
*8% is used as a common average and makes this formula most accurate for interest rates from 6% to 10%.
If you invest money at 6% interest per year, how long will it take you to double your investment?
t=72/R = 72/6 = 12 years
What compounding interest rate do you need to double your money in 10 years?
R = 72/t = 72/10 = 7.2%
If you invest 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 our answer is in months not years)
144 months = 144 months / 12 months per years = 12 years
Vaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition) Washington DC: The Mathematical Association of Amercia, 2009, page 75.