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ln2/r gives the doubling time for continuously compounded interest and
exponential growth.
ln2/ln(1+r) gives the doubling time for interest compounded annually.
[Check by solving 2A = A(1+r)^t for t.] The slightly larger resulting
time is to be expected.
ln2/(n ln(1+r/n) gives the doubling time for interest (rate r) compounded
n times per year. [Check by solving 2A = A(1+r/n)^(n t) for t.] For an
interest rate of 5.1% = .051, compounded monthly (n=12), this actually
gives 13.62 years.
If one is content with sufficiently approximate results, the "rule of 70"
can be thought of as a rough approximation of any of these. It's
systematically high as an approximation for true exponential growth (but
getting better for larger r), for the others the error varies depending on
r and n, crossing from negative to positive near 2% for annually
compounded interest, and somewhere around 25% for monthly compounded
interest. For interest rates between 2% and 25% one can assume that the
rule of 70 will give answers that are too small for annual interest, too
large for monthly interest; it is always too large for continuously
compounded interest (exponential growth).
I just graphed the difference between the correct formulas and the rule of
70 approximation for the 3 cases: exponential, annual and monthly, the
pdf generated is here:
http://www.southern.edu/~caviness/RuleOf70Errors.pdf Basically this is
the following Mathematica command:
Plot[Evaluate[{Log[2]/r , Log[2]/Log[1 + r], Log[2]/(12 Log[1 + r/12])} -
70/(100 r)], {r, .01, .5}]
The original comment related to exponential growth, not finance at all, so
the "Pe^(rt)" formula holds, and ln2/r is the most appropriate formula for
doubling time. As an approximation for this, the rule of 70 gives answers
that are too large and really doesn't save much in the way of calculator
button clicks. It may be easier to remember, but thinking of "doubling"
might jog the memory to remember the "ln2".
Ken
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of Robert
Cohen
Sent: Tuesday, 21 September 2010 6:10 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
It seems to me that the ln2/r rule is a little worse than the 70/r rule.
Using ln2/ln(1+r), I get ln2/ln(1+.051) ~ 13.9 years.
----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@po-box.esu.edu http://www.esu.edu/~bbq
-----Original Message-----_______________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of Ken
Caviness
Sent: Thursday, September 16, 2010 8:57 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
Ah, the "Rule of 70". That's sometimes taught in finance, business or
economics classes. I'm not sure why. Most people can't divide x into
70 without a calculator anyway, and most calculators have an "ln x"
button, so it would be basically as quick to use the more exact
formula, ln2/r. And this has the advantage of immediate extensibility
to questions such as, "How long does it take for a population
increasing at rate x to triple?" -- (ln3/r)
This is just exponential growth, also known as continuously compounded
interest, which I was taught to remember using the "pert" formula: A
= P e^(rt), where P is the principal or original number/amount, A is
the future amount, t is the time in some convenient units, r is the
interest or growth rate (per time unit). The more standard
exponential growth/decay formula is N = N_0 e^(+-r t), sometimes using
lambda instead of rate r, but it's the same thing.
To get the rule of 70, I just plugged in P = 1, A = 2:
A = P e^(rt) ==> 2 = e^(rt) ==> ln2 = rt ==> t = ln2/r ==> t = ~
0.693/r = 69.3/(100r) = ~70/x.
Of course, you have to use the decimal for the rate instead of the
percentage. :-)
So doubling time with a rate of 5.1% per year is ln2 / .051 = ~ 13.6
years. Using 70/5.1 gives ~ 13.7 years, so the rule of 70 is not bad,
just (in my opinion) unnecessary.
Cheers,
Ken
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of curtis
osterhoudt
Sent: Thursday, 16 September 2010 1:23 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
Remember the simple rule-of-thumb: If something is growing at x% per
time y, the doubling time is roughly 70/x to double in y units. That
is, a percentage growth rate of (say) 1% leads to a doubling of
population in about 70 years. That's _scary_ to me, for _any_
population.
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the
unknown.
Some do bow in that final direction. Others advance upon it.
To bow before the one is to lose sight of the three. I may submit to
the unknown, but never to the unknowable." ~~Roger Zelazny, in "Lord
of Light"
***************************************/
________________________________
From: Bernard Cleyet <bernardcleyet@redshift.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Thu, September 16, 2010 11:15:34 AM
Subject: Re: [Phys-l] H. Sapiens
Contrary to what I presume many think, Mexico is less adding to that
problem:
"Throughout most of the twentieth century Mexico's population was
characterized by rapid growth. Even though this tendency has been
reverted and average annual population growth over the last five years
was less than 1%, the demographic transition is still in progress, and
Mexico still has a large cohort of youths."
http://en.wikipedia.org/wiki/Demographics_of_Mexico
bc
Demographic transition:
http://anthrocivitas.net/forum/showthread.php?t=1539
On 2010, Sep 11, , at 20:15, ludwik kowalski wrote:
> Actually, I was wrong about the "less than 20 years, as you
can see at:
>
> http://en.wikipedia.org/wiki/World_population
>
> Ludwik
>
> = = = = = = = = = = = = = = = = = = = = = = = = = = =
>
> On Sep 11, 2010, at 11:05 PM, ludwik kowalski wrote:
>
>> On Sep 11, 2010, at 10:49 PM, brian whatcott wrote:
>>
>>> There are less than 8 billion people presently on Earth. . . .
>>
>> This is about four times more than when I was a kid. It
will probably
>>double again in less than 20 years. How can one be optimistic about
>>the future of sapients?
>>
>>
>> Ludwik
>>
>> http://csam.montclair.edu/~kowalski/life/intro.html
>>
>>
>>
>>
>> _______________________________________________
>> Forum for Physics Educators
>> Phys-l@carnot.physics.buffalo.edu
>> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
>
> Ludwik
>
> http://csam.montclair.edu/~kowalski/life/intro.html
>
>
>
>
> _______________________________________________
> Forum for Physics Educators
> Phys-l@carnot.physics.buffalo.edu
> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
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