We invest $1000 and, after a certain length of time, our investment has doubled in value.
If our annual return was 6%, how long did it take to double?
>I'd say twelve years.
How'd you get that?
>Easy. The Rule of 72. I divide 6 into 72 and I get 12, because N x R= 72, eh?
I assume N is the number of years and R% is the return.
Okay, at 6% annual return, how long would it take to triple?
>Triple? I have no idea ... but I have a funny feeling you're going to tell me.
Okay.
We start with $A and, with an annual return of R%, we'd have
 A(1+R/100) after one year, and
 A(1+R/100)^{2} after two years, and
 A(1+R/100)^{3} after three years, and
 ... and
 A(1+R/100)^{N} after N years.
In order for our $A to double, we'll need
A(1+R/100)^{N} = 2A
or cancelling the A
(1+R/100)^{N} = 2
or taking logarithms
N log(1+R/100) = log(2)
or dividing by log(1+R/100)
N = log(2)/log(1+R/100)
so multiplying by R
NxR = {log(2)R}/log(1+R/100)
So, is N x R = 72?
In Fig. 1 we've plotted this expression for NxR
... namely {log(2)R}/log(1+R/100)
versus R and find
(surprise!) that it's roughly 72. That means that:
NxR = 72
approximately ... to multiply by 2
so NumberofYears x AnnualPercentageReturn = 72 (approximately)
For example, using R = 7%, we'd get
(using logarithms to the base 10, tho' any base will do):
log(2)*7/log(1.07) = (0.301)(7)/0.0294 = 71.7
Neat, eh?
>I assume that, at the red dot, it's exact.
Yes, and ...
>And to triple ... to multiply our investment by 3?


If we wanted our $A to be multiplied by a factor
F (instead of 2) we'd need
NxR = {log(F)R}/log(1+R/100)
and I've plotted these, too, for 3
and 5 so we get ...
>Wait! To multiply by 3 we have the Rule of 114, since
NxR = 114 ...
Approximately!
>... and if we wanted our R = 6% per year investment to triple, it'd take N = 114/6 = 19 years.
Approximately, remember.
>... and to multiply by 5 we have the Rule of 166, since
NxR = 166, and ...
Remember, it's approximately!
>... so it'd take N = 166/6 = ... uh, how many years is that?
Suppose you want your R = 6% investment to increase by a factor of 10.
Since 10 = 2x5 we can use N = 72/6 = 12 years to get the
multiply by 2 then
another N = 166/6 = 27.7 years to get the additional multiple by 5 so it'd take
12 + 27.7 = 39.7 years. Neat, eh?
>You forgot to say approximately!
By the way, what's the exact answer?
Roughly 39.51653063577150 years
You can play, here (to get the exact answer ... for comparison):
