How Long Till You Double Your Money? Is the Rule Of 72 Good Enough?

[Do let me know if you have seen the calculations behind the "Rule of 72" before this :)]

I am sure almost everyone on the personal finance blogging scene must be knowing about this rule of thumb…sort of. Here, I am just revisiting the idea behind this deceptively simple concept from ground up.

This is the “Rule of 72″: to quickly estimate the number of years it will take your initial invested amount to double, just divide 72 by the interest rate. For example, if your interest rate (APY APR) is 12%, the time it will take to double your investment is 72 divided by 12 equal to 6 years.

So is this rule applicable to all interest rates?

For the mathematically inclined, here is a check on how accurate this rule is (If you have an HSBC Direct or Emigrant Direct or ING Direct account and are not interested in the math just scroll towards the end of the article:) )

We start with the compound interest formula:

Where, P is the initial investment, A is compounded amount after t years, r is the rate of interest in decimal form, n is the number of times the interest is compounded in a year.

For simplicity, we will assume n=1 (I will discuss what happens when n varies from 1 to 365 in a later post).

Now, we are interested in doubling the amount so we substitue A=2P in the above formula to get:

We now take logarithm of both sides of the equation to get:

Then, t can be written as:

Now, the rule of 72 says:

(Note: Here r is in %, not in decimal form; in decimal form it will be 0.72/r). We have to check if the expression we derived for t above evaluates to 72/r. In other words, we can check if t multiplied by r gives us 72.
There is no easy way to do this, so I will explain it graphically. From our expression for t we can write:

We are going to check if this “Factor” turns out to be 72 (Note: in this last logarithmic formula, r must be in decimal form. We will then multiply the result by 100). Below is a graph drawn using the above equation and shows the relationship between the “Factor” and interest rate %.

We can easily observe from the graph that the factor 72 is not constant ! that means the rule is valid ONLY for one particular interest rate !! The factor of 72 is valid only for a rate of 7.8% !

However, since most of the investments are within the shaded region shown on the graph, the number 72 works for us as a rough estimate.

And now it’s show-me-the-money-already-damn-it time
For those of us who have accounts in HSBC Direct and Emigrant Direct, the rule is more like “Rule of 71″ and you will double you initial investment in about 14.3 years and 14.1 years respectively (now they will be same, read about it here on Blueprint for Financial Prosperity), at their current rates of interest. If you bank with ING Direct, the rule is more like “Rule of 70.8″ and you will double your money in about 16.4 years. Keep in mind that these are estimates and that we made some assumptions above (like n = 1) so the real numbers will vary a bit.

Now hopefully, after the above calculations, some of you will look at the “Rule of 72″ with a little more respect in future :).

Feel free to point out any mistakes or suggest improvements to the calculations above.