The rule of 72 states that, if you want to find the number of years it would take to double your money at a given interest rate (r), you divide 72 by r. For example at 8% compound interest it would take 9 years (72 / 8) to double your money. I have been using this rule for a very long time without actually understanding why it worked. Recently I was explaining this rule to my son.
Me: At 24% we can double the money in 3 years.
Son: How do you know?
Me: I used rule of 72 and got 3 years (72 / 24).
Son: If I compound $1 at 24% for 3 years I get $1.91 [1.24 * 1.24 * 1.24]. Do you know why I did not get $2. Is rule of 72 incorrect?
I could not answer his question as I never understood why the formula worked. In order to answer his question, I wanted to rediscover the formula myself.
Let pv = present value fv = future value r = rate of interest ln = natural logarithm n = number of years From compound interest we know that fv = pv * (1 + r) n 2 * pv = pv * (1 + r)n -- [fv is double of pv] 2 = (1 + r)n -- [cancel pv on both sides] ln(2) = n * ln (1 + r) -- [taking log on both sides] n = ln(2) / ln (1 + r) n = ln(2) / r -- [for low values of r we get ln(1 + r) = r] n = 0.693 / r -- [ln(2) is 0.693]
Using the above formula it takes 69.3 years to double your money at 1% annual interest [0.693 / 0.01]. But the compound interest formula tells that it will take 69.5 years.
At 4% annual interest I can double the money in 17.325 years [0.693 / 0.04]. But the compound interest formula tells that it will take 17.7 years.
At 10% annual interest I can double the money in 6.93 years [0.693 / 0.10]. But the compound interest formula tells that it will take 7.25 years.
I captured the results in an excel sheet which is given below. What do you see?
I see that (1) Formula derived by me tells that I can double the money in shorter period compared to the actual compound interest (2) As the interest rate increases the difference between the two formulas goes up. Why does this happen? The reason for this difference is because of the assumption ln(1 + r) = r that we used in deriving 0.693 / r.
From the table and chart given below you can see that when r goes up the difference between r and ln(1 + r) goes up. This is because log of a number grows at a much smaller rate compared to the number. If you want a refresher on logarithms read here.
Since r is in the denominator (n = 0.693 / r ) and r is higher than ln (r) the derived formula tells that we can double the money in shorter period. Also as the interest rate goes up the error rate gets magnified.
The formula we derived only has 0.693 where did 72 come from? The number 69.3 (0.693 * 100) is hard to remember compared to 72. Also 72 is divisible by several numbers 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 compared to 69.3 and hence it is easy to do mental math on it. This was the reason why 72 was chosen.
From all this I concluded that (1) Rule of 72 is a reasonable approximation (2) It works well on low interest rates and it is accurate for a range of interest rate from 6% to 10% (3) As the interest rate goes up the error rate increases. I found this document really useful in understanding this rule. Here is an excerpt from the document
So, for very small rates, 69.3 would be more accurate than 72. For higher rates, a bigger numerator would be better (e.g. for 20%, using 76 to get 3.8 years would be only about 0.002 off, where using to 72 get 3.6 would be about 2.002 off). 72 is a reasonable approximation across this range and is easily divisible by many numbers. The rule of 72 is only an approximation that is accurate for a range of interest rate (from 6% to 10%). Outside that range the error will vary from 2.4% to 14.0%.
As Richard Feynman says: There is a huge difference between knowing the name of something and knowing something. By rediscovering the formula I really learned something.