Rule of 72 Calculator

Rule of 72 Calculator

to Double my Money
at an Annual Rate of
%
Answer:
Double Your Money in
13.71 years
at 5.25%

(actual years = 13.55)

Calculator Use

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.

Interest Rate
the annual nominal interest rate of your investment in percent
Time Period in Years
number of years for this investment.  (you can interchange months or any period as long as your interest rate entered is in the same time period)
Compounding
the frequency of compounding coincides with periods.  Once per period.

Rule of 72 Formula

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

where
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.

Derivation of the Rule of 72 Formula

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,

tln(1+r) = ln(2)

and isolating t on the left;

t = {ln(2)/ln(1+r)}

it should be easy to show that 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)} ={0.69/r} (1.0395)

Solving this equation for r times t

 rt = 0.69 \times 1.0395 \approx 0.72

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%.

Example Calculations in Years

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%

Example Calculation in Months

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

References

Vaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition) Washington DC: The Mathematical Association of Amercia, 2009, page 75.

Weisstein, Eric W. "Rule of 72." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Ruleof72.html

 

Cite this content, page or calculator as:

Furey, Edward "Rule of 72 Calculator" From http://www.CalculatorSoup.com - Online Calculator Resource.