An article on the old financial rule of 70. It explains how it works and introduces some new "rules" of my own:)

One day I was talking to a friend of mine about how house prices have appreciated in our region. He mentioned casually that prices had doubled in about 10 years. The conversation went something like this:
Him: I wonder how much of an annual percentage growth this is...
Me: Ooh..., about 7%.
He then pulledout his scientific calculator and made the actual calculation.
Him: Man.., you are right!... Are you some sort of human calculator?
Now, before you start thinking that I'm some sort of genius that can do logarithms in his head, I'll tell you my little secret: I just used the "rule of 70". This is nothing new. People in the finance field have been using this ruleofthumb for a long time. But as the Engineer that I still am in the heart, I wondered where this "rule' came from, and hence, this article.
Simply stated, the "rule of 70" says that the number of years it takes for an amount growing at x % per year to double is roughly equal to 70/x.
So, in the example above if 70/x = 10 years, (it took ten years for house prices to double) then x = 7%.
As I said, a nobrainer to calculate using the rule of 70. Now the question for me was: where did this rule come from? And what are its limitations? And could there be other useful rules like this?
So, let's first see what is the exact formula for calculating the growth rate x. If a is the amount growing at x % and n the number of years needed to double the amount then, the following relation must be met:
(1+x/100)^n . a = 2 . a <=> (1+x/100)^n = 2
and solving for n we get:
n . ln (1+x/100) = ln(2) <=>
n = ln(2) / ln(1+x/100)
This is why I mentioned in the beginning that, if it wasn't for the rule of 70, I would have to know how to do logarithms in my head... So we have to simplify the expression above. Fortunately, for small values of x we know that ln(1+x) ~= x. Trust me on this one... simple Taylor series expansion ( ln(1+x) = x x^2/2+x^3/3x^4/4... so for small x you can approximate ln(1+x)~=x ). So, for small values of x/100 we can approximate ln(1+x/100) with x/100 and the equation above becomes:
n ~= ln(2) / (x/100) = 100.ln(2)/x
It so happens that 100.ln(2) is something like 69.315 which people further approximated to 70. And voila'! The rule of 70s is born:
n ~= 70/x
Remember that the key to this rule is the approximation we made of: ln(1+x/100) ~= x/100. This approximation is only good for small values of x/100. In other words, the rule of 70 works best for small growth rates. Let me give you an example to clarify this.
Say your 401K money is growing at roughly 10% a year. How long will it take to double? Well, according to the rule of 70:
n ~= 70/10 = 7 years
The exact calculation using logarithms is actually 7.27 years. The "rule" gives a pretty good match. Now suppose you hired Warren Buffet to manage your money and you are getting a whopping 90% annual return :). In this very hypothetical scenario, the rule would tell you that your money would double in:
n ~= 70/90 = 0.77 years or roughly 9 months
In reality, your money will only double in about 13 months. The "rule" gives you a very poor, overly optimistic, approximation in this case. So keep this in mind when using the rule. The following plot shows how the rule (line in red) and the exact calculation (green line) compare for a range of growth rates up to 100%. As expected, the curves start to diverge as the growth rate increases.
figure 1  Years to Double vs. Growth Rate
Man I always wanted to have a "rule" named after me, this is so cool. Hopefully nobody beat me to it, but if someone did, my apologies in advance:) So, suppose you want to know how long before your money triples. How about using
Paulo's rule of 110?
"The number of years it takes for an amount growing at x % per year to triple is roughly equal to 110/x."
n ~= 110/x
As an example, if your money is growing at 11%, it will take roughly 10 years to triple.
And what about Paulo's rule of 230?
"The number of years it takes for an amount growing at x % per year to increase tenfold is roughly equal to 230/x."
As an example, if your money is growing at 10%, it will take roughly 23 years for it to increase tenfold. And I could go on with this all night... But don't worry, I won't.
The cool thing about combining these three rules (rule of 70, 110 and 230) is that now you can also calculate other numbers that are multiples. Say someone asks how many years is it going to take for their house to increase 6 times in price assuming it increases 10% a year. Well, since 6 = 2 x 3 you can use the rule of 70 for the doubling (70/10 = 7 years ) and then Paulo's rule of 110 for the tripling (110/10 = 11 years) to get a total of 7+11 = 18 years! The exact number is 18.8 but this is close enough.
Now you could say, why not just get a calculator and calculate these things... well what fun is that?
Hopefully you learned something of value with this article. And remember, these rules work best for small growth rates and they will always give you a slightly "optimistic" result. It usually takes a little longer for the money to grow than the rule would have you believe.
Comments, questions, suggestions? You can reach me at: contact (at sign) paulorenato (dot) com