What is the "rule of 72" (sometimes called the "rule of 70") with regards to investment, and how do I apply it?
The Rule of 72 is a rough guide for calculating how long it would take to double your investment through compound interest, given a fixed yearly rate of return.
It means that the time taken (in years) to double your investment value is approximately equal to:
72 / return of investment (%) per year.
Example: Assuming you have invested an amount, X, in an investment returning 6% per year, and you reinvest all the returns from the investment. Then, using the Rule of 72, the time taken to double your investment value to 2X would be approximately
72 / 6 = 12 years.
It could also be used to calculate how long it takes for inflation to reduce the value of your money by half .
The Rule of 72 demonstrates the principle behind why it's never too soon to start investing – it could be the difference between $2 million and $4 million.
Here another example on how to apply it:
If I invest 1$ at 5% for 30 years, what will I have at the end?
5% means we double every 72/5 years, so about every 14 years. Thus, in 30 years, I double it a bit more than twice, so I expect to have a bit more than $1 x 2 x 2 = $4.
If you want to be more precise, you could figure that we double twice in about 28 years, then have 2 years left, with every year giving 5%, so we slap 2*5% or 10% on top of the $4, to get $4.40.
If you want to be really precise, you get a calculator and compute
- 1.05^30 = 4.32 for annual compounding, or
- exp(0.05*30) = 4.48 for continuous compounding
and we see that the rule performs quite well :-)
With continuous compounding at a rate that yields 100x% per annum, in y years (y need not be an integer) an investment has increased by the factor (1 + x)^y. For any given x value, this factor has value 2 exactly when y years have elapsed where (1 + x)^y = 2, which, upon taking natural logarithms and remembering that log(a^b) = b*log(a), gives
y = ln(2)/ln(1 + x) = 0.693/ln(1 + x)
Here, ln denotes the natural logarithm, and ln(2) happents to have value 0.693... Now, to a first-order approximation, ln(1 + x) equals x (actually a little bit less) when x is small, and so an investment yielding 100x% per annum should double approximately 69.3/x years. But if we take the "little bit less" into account, 70/x or even 72/x yields a closer approximation.
At 5% annual yield, an investment will double in a little over 14.2 years. The 70/x and 72/x formulas give 14 and 14.4 years as the answer.
At 2% annual yield, an investment will double in a little over 35 years. The 70/x and 72/x formulas give 35 and 36 years as the answer.
At 0.1% annual yield (which I think what my bank is paying on my checking account), I will have twice as much money in 693.5 years -- too long to wait -- but the 69.3/x formula is looking pretty good as compared to the 70/x and 72/x formulas. Thus, as might be expected, that 69.3 is working better for small x.
At 10% annual yield, the investment doubles in 72.7 years instead of the 70 or 72 years estimated with the rule of 70 or rule of 72.
At 100% annual yield, the investment doubles in 1 year which is a lot longer than the 0.7 or 0.72 years estimated by the rule of 70 or 72.
In summary, the rule of 70 or 72 works well for annual yields in the typical range of a few percentage points but not that well for trivial yields (rule of 69.3 is better) or the pie-in-the-sky yields that beginning investors plan on getting so as to retire by age 35.