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Posted by Robert11 on May 11, 2009, 4:41 pm
Hello,
Figure that this must be the right and best group for this question.
Have read the Google links on it, but still unsure.
Dividing 72 by the interest rate gives the number of years to double an
investments value (approx.)..
Fine.
Questions:
I guess the assumption is that when they say, e.g., 6%, they mean at the end
of every year, 6% is added in, and left in for the subsequent years. This
is certainly compounding, but not very real world.
Or, do they mean a "certain amount" is added daily, but at the end of the
year it amounts to the Principal + 6% ?
I guess I am asking what the compunding "Period" implied in the formula
actually is ?
**And also, how to adjust the formula for different periods ? e.g.,
compunded every 3 months, monthly,
or even daily perhaps ? **
Thanks,
Bob
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Posted by Douglas Johnson on May 11, 2009, 8:54 pm
>**And also, how to adjust the formula for different periods ? e.g.,
>compunded every 3 months, monthly,
>or even daily perhaps ? **
See http://en.wikipedia.org/wiki/Rule_of_72 for a very good answer.
-- Doug
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Posted by Xho Jingleheimerschmidt on May 12, 2009, 5:02 am
Robert11 wrote:
...
> Dividing 72 by the interest rate gives the number of years to double an
> investments value (approx.)..
> Fine.
>
> Questions:
>
> I guess the assumption is that when they say, e.g., 6%, they mean at the end
> of every year, 6% is added in, and left in for the subsequent years. This
> is certainly compounding, but not very real world.
As you said, the rule of 72 is an approximation. Is it worth agonizing
over the exact compounding schedule of something that is only an
approximation in the first place? The approximation remains
approximately correct whether compounding is done yearly, quarterly, or
daily.
> Or, do they mean a "certain amount" is added daily, but at the end of the
> year it amounts to the Principal + 6% ?
Over the course of the year, you have 365 different principals. Which
one of those are you referring to? If the principal at the start of the
year, then ending the year at that principal plus 6% is functionally
equivalent to annual compounding at 6%.
Xho
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Posted by Bill Woessner on May 12, 2009, 11:16 am
> I guess I am asking what the compunding "Period" implied in the formula
> actually is ?
The Rule of 72 is only an approximation. It isn't exact for any given
compounding period (or interest rate, for that matter). That being
said, the approximation is closest to annual compounding. However, it
works reasonably well for shorter compounding periods.
> **And also, how to adjust the formula for different periods ? e.g.,
> compunded every 3 months, monthly,
> or even daily perhaps ? **
If only there were a mathematician in the house...
Honestly, at this point, I would just throw away the approximation and
solve for the doubling time exactly. Let r be the interest rate, n be
the number of compounding periods per year and t be time (in years).
The equation you need to solve is:
(1 + r / n)^(nt) = 2
Solving for t, we get:
t = ln(2) / (n * ln(1 + r / n))
If you really wanted a rule like the Rule of 72, you could use a
Maclaurin series to approximate the natural log in the denominator:
t ~= ln(2) / (n * (r / n - (r / n)^2 / 2))
~= ln(2) / (r * (1 - r / (2 * n)))
>From here, we can reproduce the rule of 72 by letting n=1 and the
SECOND r=7%:
t ~= ln(2) / (r * (1 - .07 / (2 * 1)))
~= ln(2) / (.965r)
~= .72 / r
Ta da! If you wanted a rule that's more appropriate for, say, monthly
compounding and interest rates around 6.3%, you'd get:
t ~= ln(2) / (r * (1 - .063 / (2 * 12)))
~= ln(2) / (.997r)
~= .695 / r
Sadly, the Rule of 69.5 just doesn't have the same ring to it. That
alludes to another reason why the Rule of 72 is so popular. 72 is a
highly composite number. It's evenly divisible by 2, 3, 4, 6, 8, 9,
12, 18, 24 and 36. So if you use the Rule of 72 to approximate the
doubling time for 6%, it comes out an even 12 years. For 8%, you get
9 years. It's handy that way.
I hope none of my students read this newsgroup. This would make an
excellent exam question for next semester...
--Bill
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Posted by on May 12, 2009, 2:44 pm
> **And also, how to adjust the formula for different periods ? e.g.,
> compunded every 3 months, monthly,
> or even daily perhaps ? **
I hope my high school algebra teacher doesn't take away my diploma if
I get this wrong.
R = rate (simple or same as if you compound yearly)
C = number of times you compound a year
Y= number of years to double money
^ = exponential 2^3 is 2 to the 3 or 8
* = multiply 2*3 is 6
log = log base 10, log100=2
doubling formula is:
(1+(R/C))^(Y*C) =2
crunch around
Y = log(2)/(C*log(1+R/C))
At 6%, monthly compounding, it takes 11.5813 years to double your
money.
At 6%, daily compounding, it takes 11.5534 years to double your money.
Thats a difference of 10 days interest over 11 years.
Don't even think about asking about leap year :)
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