Here’s a question that has been floating around the internet for a while, to demonstrate how the seemingly intuitive response is not always the right one. Try this – read the following question and come up with the answer as quickly as possible.
Suppose a businessman runs a very successful business and found the secrets of wealth by being able to double his money daily. After 50 days, he reached his milestone of $1 million. How many days did he take to make $500,000?
Most people would instinctively say 25 days but the correct answer is actually 49. This simple exercise illustrates how counter-intuitive thinking about exponential growth can be. In this post, I will attempt to de-mystify a little bit of that, although I will have to appeal to some secondary school A Maths at some point. We will look into the Rule of 72, and other similar “Rules of X”, and a general way of computing how fast it takes to double, triple or multiply your money.
Rule of 72
Let me explain the Rule of 72 first. It is a simple and approximate way to determine how long it takes to double your money based on a constant rate of return. For example, if you deposit $1,000 into a bank account that offers a fixed annual rate of 2%, it will take about 72/2 = 36 years for that amount to grow to $2,000. This is extremely useful to re-frame our thinking about rate of returns to the length of time needed to observe an effect on our pot of money. Of course, the inherent assumption is that both the principal sum and the rate of return are constant, which is rarely ever the case in reality.
Note: 36 years is an approximate figure. The actual figure is 35 years, but 36 is a pretty good approximate for a such a short-cut computation compared to the algebra that will be coming later.
The table below shows how this works for other rate of returns, but it gets increasingly inaccurate with larger rate of returns (>15%).
How do we derive the Rule of 72? We will have to do some algebraic manipulations with the compound interest formula , where
is the final amount,
is the initial principal amount,
is the interest rate, and
is the time, usually measured in years because interest rates are usually on a per annum basis.
Since we want the final amount to be double that of the principal, then we substitute in to get .
Dividing both sides by gives
Taking logarithms on both sides and making the subject gives
Let and we realise that seems like a better approximation, but another consideration is how easy it is to apply the formula. 72 is better because it has more factors than 70 and thus easier to work with without using a calculator. To understand this better, try working out 70 divided by 3 or 6.
We can also do something similar to generate the table for tripling our money, except that this time, we will use instead to get the formula . And what’s the rule for tripling your money? Using the same method above, a Rule of 110 might work. But quite unlike the case for doubling, it seems that there is no consensus of a rule for tripling – a quick search show that there are several such rules used out there: Rule of 110, Rule of 114, Rule of 115 or Rule of 120.
Personally, I would go with the Rule of 114 because it balances between a reasonable margin of error and the ease of use as 114 is a multiple of 3. From the above tables, the most accurate for rates below 10% and above 10% are the Rule of 110 and Rule of 120 respectively.
One should be able to make the astute observation that only the numerator changes according to the multiple we are interested in. So for some factor that we wish our money to be multiplied by, we will substitute to get the general formula . This then gives the Rule of 144 for quadrupling, which is also intuitive because that 144 is just double of 72. For small rate of returns, computing for larger factors to multiply one’s money by is less meaningful as it will take a pretty long time, in the region of ~100 years or more.