Rule of 72
This is the most important calculation any investor needs to know.
It is simply a shortcut to estimate the number of years required to double your money for a given rate of return / interest / yield.
How the rule applies
The rule states that if divide the rate of return / interest rate / yield into the number 72, the result is the number of years to double your money.
It also works the other way round: if you divide the number of years to double your money into 72, the answer is the growth rate you will need.
The Formula:
(1+R/100)/t=2(1+R/100)t=X
The formula is (1+R/100)t=2(1+R/100)t=X, where X is the multiple of the money you need, to find t being the amount of time for the compounding.
Double your money
If you have a return of 6%, 6 into 72 gives 12, so at 6% per year it takes 12 years to double your money.
Return required
If you want to double your money in 8 years, 72 divided by 8 gives 9, so to double your money in 8 years you need 9% annual return.
Effect of inflation
If you want to know how long it takes for 3% inflation to halve the purchasing power of your money, 72 divided by 3 gives 24, so at 3% annual inflation the buying power of your money drops in half in 24 years.
This is simply the basis of compounding returns; if you have a 4% return and that return itself grows at 5% per year then (72 ÷ 5) means that after 15 years your annual return has doubled from 4% to 8%. (If you have a pension this should start looking valuable.)
If you take that annual yield/growth as annual income then you can see how the income grows over time. If you don’t take the income out, but leave it rolled up in your investment, then you will see the full value of compounding over time, so starting with £100, at a 4% yield, that yield growing at 5% per year, by the time the yield has doubled to 8% the investment of £100 has grown to £231, in a predictable way.
This means the capital has grown at 5.75% per year through income alone, and the income/yield at year 15 is now 8% of £231, not the £100.
If you kept the capital the same by taking out the income / yield / growth each year you will have an income of 8% x £100 = £8 each year. This is an 8% yield to cost.
If you have left the income to compound — perhaps while you have been growing your pension — then you will have £231 x 8% = £18 each year. This is an 18% yield to cost.
This is why we are income investors.
Rule of 72 Chart
When dealing with low rates of return, the Rule of 72 is fairly accurate. This chart compares the numbers given by the rule of 72 and the actual number of years it takes an investment to double.
Notice that, although it gives a quick rough estimate, the rule of 72 gets less precise as rates of return become higher. Therefore, when dealing with higher rates, it's best to calculate the precise number of years algebraically by means of the future value formula.
To get a little more nerdy (as we do from time to time) we look at why this works, outlined for us by moneychimp.com.
| Rate of return | Rule of 72 | Actual no. years |
|---|---|---|
| 2% | 36.0 | 35.00 |
| 3% | 24.0 | 23.45 |
| 5% | 14.4 | 14.21 |
| 7% | 10.3 | 10.24 |
| 9% | 8.0 | 8.04 |
| 12% | 6.0 | 6.12 |
| 25% | 2.9 | 3.11 |
| 50% | 1.4 | 1.71 |
| 72% | 1.0 | 1.28 |
| 100% | 0.7 | 1.00 |
A man called Euler, 1707–83
If you have an initial investment principal of P, and an annual interest rate R, then after T years your investment will have a value of PERT, where E is Euler's number, an irrational number that shows up all over the place in mathematics.
The Rule of 72 — why it works
The Rule of 72 applies to annually compounded interest, but it's easiest to understand by looking at the case of continuously compounded interest first. We'll write P for the starting principal and R for the return rate (as a decimal); we're looking for Y to double P:
2P = PeYr
Solve for Y: Y = ln(2) / r
The log of 2 is about equal to .69, so. Y = .69 / r
You can think of this as The Rule of 69 (multiplying the .69 by one hundred, so that the interest rate can be expressed as a percent instead of a decimal). It isn't an estimate — it's the exact answer for doubling your money, assuming that the interest is compounded continuously. It's valid for any value of R.
Solving the formula for annually compounded interest is messier:
2P = P(1 + r)Y
Y = ln(2) / ln(1 + r)
Approximated as a neat fraction again Y = K / r
…where K is some number that will make the approximation pretty good for some ranges of r (and pretty lousy for others). We'll choose K to make the approximation work for a return rate of ten percent:
ln(2) / ln(1 + r) = K / r
ln(2) / ln(1 + .1) = K / 0.1
K = [ln(2) / ln(1.1)] x 0.1 K = .727
Now 72.7 is really closer to 73 than 72, so why isn't it The Rule of 73? Well... uh, the continuously compounded case gives you 69, so you want to round the 72.7 down instead of up... um, plus, 72 is easier to work with, since it's divisible by 12... anyway, it's only an estimate. So, 72 it is!