How Does the Rule of 72 Relate to Doubling Time in Investments?

[Muon12]

I've been studying the rule of 72 lately, which gives the time in years that it takes for an investment to double based on the annual percentage rate of growth.I have tried to compare it's formula t=72/x (x being the number in front of the percent sign) with the exact value for the doubling time (in years) of an investment based on interest. The exact formula is t=(ln2)/ln(1+(x/100)). This is also where x is the number in front of the % sign. What I don't yet understand is how 72/x is connected to the exact value formula. Iv'e seen the graphs and understand how close they are... but why are they so close?

 

[kebz33]

we want to double p, our initial investment. then (pardon my crappy notation):
p(1+i)^t = 2p
where i is our rate of interest. we can solve for t and get
t=[ln(2)/i]*[i/ln(1+i)]
so with this formula, it is a little easier to see what is going on:
(1/i)ln(2)*[i/ln(1+i)] =appx .72
so basically i/ln(1+i) will give you more or less the same number (slighty greater than 1) for all choices of i (i would be expressed as a decimal, IE 8% = .08).

 

[Muon12]

Rule of 72

"which equals .72/i no matter the choice of i, so long as i is expressed as a decimal, IE 8% = .08" -kebz33



Yes, I thought it had something to do with the ln2 value. What I'm still unclear about is how .72/i will come out to .6931 no matter what i is. How does that work?

PS, the indicator says I'm not logged on... but I am. Maybe I'm not doing something right here...

 

[kebz33]

i edited my post somewhat to make things more clear

actually its ln(2) which equals .6931 apprx. which is already close to .72 as it is.

 

[HallsofIvy]

It's not all that close to 72. I wonder why they don't use 70 instead (oh, as I type this it occurs to me that 72= 6*12 so if you are thinking in terms of months, that's simpler).

Another way to see the result is to approximate ln(1+ (x/100) by its "Taylor polynomial": x/100- (1/2)(x/100)²+ (1/3)(x/100)³+...

If we take just the "linear part", x/100, then
ln(2)/ln(1+(x/100)) becomes ln(2)/(x/100))= 100*ln(2)/x= 69/x.

Again, replacing 69 by 72 (just because it is divisible by 12!) we get 72/x.

I might point out that this is not so much mathematics as "business" and there is no telling what "business" people may do!

 

[Muon12]

Rule of 72, fake?

So, Halls of Ivy, are you saying that the Rule of 72 is really just molded around the needs of buisnesses, as opposed to being a legitimate formula? I have to say that I am surprised by that. Here's why: first of all, the graph of 72/x actually interects the graph of (ln2)/(ln(1+x/100), while staying incredibly close to matching the values between the 4% and 15% range, the range where most interest rates usually fall. I guess it's close to being the secant line of the ln equation (I think that's what it's called.) Also, wasn't it Albert einstein who discovered the formula and it's potential usefullness? I have doubts that Einstein would alter a formula just so that it could be useful for buisnesses, though I may be wrong. Although I do see how 69/x would work, I don't see why the Rule of 72 is less true, since in some cases, it's actually MORE accurate than 69/x. Ex. at point (7.85,9.17): 72/7.85= 9.17 (almost exact same value as ln rule), 69/7.85= 8.79 which is far less accurate.

 

[HallsofIvy]

[quote[So, Halls of Ivy, are you saying that the Rule of 72 is really just molded around the needs of buisnesses, as opposed to being a
legitimate formula?[/quote]
No, I simply pointed out, like several other people, that 69/i would be more accurate than 72/i. Although the choice of 72 over 69 was probably based on the needs of business, I don't see why that would make it not a "legitimate formula". There are many "legitimate formula" that are designed for business use.

Also, wasn't it Albert einstein who discovered the formula and it's potential usefullness?

Not likely. The "rule of 72" was used long before Einstein was born.