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Muon12

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- Thread starter Muon12
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- #1

Muon12

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- #2

kebz33

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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).

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).

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- #3

Muon12

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"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...

- #4

kebz33

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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.

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

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- #5

HallsofIvy

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

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!

- #6

Muon12

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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

- #7

HallsofIvy

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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.

Not likely. The "rule of 72" was used long before Einstein was born.Also, wasn't it Albert Einstien who discovered the formula and it's potential usefullness?

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