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>"If you are increasing your trade size as your account increases,
>then your ideal account size would grow exponentially and you would
>calculate the returns on the logarithm of the account value."
>
>Logarithm? What logarithm? Can you refer me to a formula, web site,
>whatever?
I did a Google search of "logarithm" and found lots of references.
For example see:
<http://taipan.nmsu.edu/aght/soils/soil_physics/tutorials/log/log_home.html>
<http://www.math.utah.edu/~alfeld/math/log.html>
It is a common math function. The Excel spreadsheet function is "LN()".
The log function is used when you would like to replace
multiplications by additions.
So:
a = x * y
can be replace by:
LN(a) = LN(x) + LN(y)
For our purposes compounding in an investment requires multiplying:
1+R = (1+r1) * (1+r2) * (1+r3) * ....
where R is the overall return and r1, r2, r3, ... are the periodic
(such as monthly) returns. Compounding requires that you figure the
return on the value at the beginning of each month, not from the
original value.
This tends to be messy so we convert this to additions using logarithms.
LN(1+R) = LN(1+r1) + LN(1+r2) + LN(1+r3) + ...
Now we can use existing Excel functions such as SUM, AVERAGE, STDEV,
etc., for all the calculations.
Logarithms are also used in "ratio charts" such as the attached 30
year chart of the Dow Jones Industrial Average. The scale on the
right is logarithmic and so a fixed distance on the vertical scale
represents a constant percentage increase. The red line shows a
constant rate of increase of about 10% over 20 years.
Another benefit is that the Central Limit Theorem of statistics says
that the probability distribution function of a sum of random
variables tends to be "normal" (like the bell-shaped curve)
regardless of the probability distribution functions of the
individual terms so our assumptions of a "normal distribution" tends
to be pretty accurate at this level. This is usually called a
"Log-normal" distribution.
Bob Fulks
Attachment:
Description: "DJIA5.gif"
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