Re: Bill Reid, Kelly Criterion


<GoldenGemNetwork@xxxxxxxxx> wrote in message
news:838ccfa9-f118-417d-8dfd-21e2237d6aa5@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Hi Bill, I wish to apologize for my lack of clarity.

No, it's probably me, I admitted I'm not a mathematician...but then
I also averred that could both work against AND for me...it's a mindset
"thang", doncha know?

It's particularly true of the market(s)...a mathematician wants a
"closed-form" equation to explain economics, and all such attempts
have met with futility, because the system is multi-variate, relatively
"open", with cross-dependancies between the variables, and thus
is not soluble using linear equations...in other words, it's "chaos"
out there...

....but I haven't given up totally, I keep working with what DOES
work (or DID work for hundreds of years)...

Below, you
discuss two very relevant examples,
one theoretical and one from actual examples. I'll re-present these
and then try to comment briefly and clearly.

By the way I liked your joke when I said I hoped you hadn't lost
interest in the group, and you said it is impossible to lose something
you never had.

What makes you think it was a joke?

Uh, sure, but I'm not sure what the point is...if it's because of this:

Yes, at the limit in theory your portfolio can become
infinitesemally
small, shrinking to $0.00000000000000001 from $1,000,000.00, and
you'll have to claw your way back over the next billion years or so.

But the chances of this are also infinitesamally small, equivalent
to
like a 30,000-sigma event in a normal curve, or maybe 1 chance out
of a number that is bigger than all the atoms in the universe.

Well, there's better ways to calculate this than what I alluded to...but
I'm really NOT sure what your point is...

For example, say the standard deviation of logs of price changes
year to year is .1

Here's some actual numbers that I just noticed laying around:

DJIA 1990 - 2000 Daily Change Distribution

-7.18% --> -5.97% = 2 0.07%
-5.97% --> -4.75% = 1 0.04%
-4.75% --> -3.53% = 4 0.14%
-3.53% --> -2.32% = 34 1.22%
-2.32% --> -1.10% = 198 7.12%
-1.10% --> +0.11% = 1230 44.26%
+0.11% --> +1.33% = 1121 40.34%
+1.33% --> +2.55% = 167 6.01%
+2.55% --> +3.76% = 16 0.58%
+3.76% --> +4.98% = 6 0.22%

The average daily change was +0.0528% (+0.1050% compounded),
with a standard deviation of 0.6491%. So you're saying that there is
some calculational inconsistency if we take the "logs" of something
there and project it out another 10 years in the future?

OK

OK, does that mean I've summarized your point?

Let me try to work from your actual data.

You have daily change +.0538% and standard deviation 0.6491

Yup, given the limits of IEEE floating-point math using C-standard
"doubles" all-around and turning off my compiler's optimizer...and of
course, my own stupidity...

Now, I am going to say, that it is a not sensitlbe to take standard
deviation of percentage change.

You may be right, except of course that "percent" is just
"per 100" (per the Latin)...so you could divide the "change"
numbers by 100 if THAT'S the problem...but if you want
the actual DIFFERENCE (price_today-price_yesterday),
then we have a problem as time goes by because those
numbers become less significant as time goes by and
prices (hopefully) drift upward...same is true of just
calculating a "mean" by adding up the closing DJIA for about
2300 trading days (10 years worth) and dividing by about
2300...in the example decade, the Dow increased something
like five-fold, so the "average" Dow doesn't mean much...

Your use of logarithms in the Kelly criterion is what that is all
about.

Well, OK, doesn't mean we have to use logarithms everywhere, though
the "trick" of showing a log-adjusted DJIA over the last 100 years on a
chart makes the rise look a lot less terrifyingly steep over the last few
years,
if your goal is not to scare people...

A percentage change like in the last column above of +4.98%
means you have multiplied by 1.0498 .

OK, the DJIA on that day was 1.0498*DJIA_the_day_before, or
a 0.0498 increase...

That is great, and you have
deleted the 1 and divided by 100.

Yeah, or conversely (more conventionally) multiplied 0.0498*100
(again, per the Latin)...

Instead of deleting the 1 -- which
is a bit crude -- you really wanted to take the logarithm.

Well, no I didn't, or I would have...when I run those numbers
I want to determine the distribution of historical changes for
a certain time periods or "fractals"...THEN I want to take the
"logs", if that is the BEST determinant of my FUTURE results
that I can come up with...

And you're right that at the point that we "take the logs" as
you put it, we take the "log" of 1.0498 for that particular result
in the distribution (since we are interested in the logarithmic
increase or decrease of our beginning "wealth", which is
1.0).

The natural
logarithm of 1.0498 is 0.0486 which is a bit smaller.

OK, now my actual next step is to multiply that by 1/~2300 (or
whatever the actual distribution of that result is) and take the log of
THAT...now THAT'S a small number! (IEEE floating-point under-flow?)

This hardly
makes any difference, but it is the difference between the Kelly
approach you favour versus the 'ordinary' approach.

I'm not really sure the "approaches" are exclusive...but if we don't
"take the logs", we wind up with a "expectation-only" average gain
and wind up endlessly (and pointlessly) debating how much "risk"
we should take for that "gain"...

Note that despite the fact the 90s had one of the all-time great
rises in the DJIA, on one day we would have LOST 7.18% of our
investment from the previous day (equivalent to almost a 900-point
loss next Monday)...does that sound "risky" to you?

It DOES "sound" risky to me, just stated baldly that way...but
WOW, you could also MAKE almost 5% in ONE DAY (625 points
next Monday)...I'm so confused, what should I do?

But, to a very close approximation, the distribution I am talking
about is the same as the one you are talking about. Though if one were
talking about year-on-year data, the difference between using the
logarithm, versus just deleting the leading 1, would become more
significant.

Well, yeah, if you take the "log" of the wrong thing, it's just gonna
get "wronger" the bigger the numbers...

I am just going to approximate the natural log by deleting the leading
1, as you have.

Now, I am saying, if you iterate that over 365 years, if you believe
it is a normal distribution, you get then the mean of 365 * .0538 =
19.637 and the standard deviation is (square root of 365) * .6491=
12.401

"In the long run, we are all dead"

Maybe you should work with the variance instead of the standard
deviation to avoid having to take the square root...

The standard deviation used to be large compared to the mean, now they
are of the same order of magnitude.

Remember, our goal in taking the "logs" is to provide a method of
growing our wealth more quickly than all other possible methods
AT THE LIMIT (infinity years) with probability 1. The longer you
"play" the strategy, the more insignificant any short-term portfolio
swings seem in comparison to your original amount of wealth...

If you go over ten years the standard deviation becomes small compared
to the mean.

Sounds good to me! Again, our goal is put all that "deviation" out of
our minds and focus on making the most amount of money...

Now that the numbers are so large, adding 1 is not the same as undoing
the natural log. The full exponential series is not 1+x it is 1+x
+x^2/2 +x^3/6 +...

So the mean change corresponds to an expected earning ratio of more
than 3 milion, in one year (!!!). which is what I get when I raise e
to the 19.637 power.

You should spam the Internet with this amazing discovery! Better
than $100 a day by a long shot!

Reminds me of all the "day-traders" here who were claiming to
or attempting to make like 25% a week...

So assuming the changes day to day are random with a normal
distribution and a drift is not what happens!

It kind of looks like a "random" normal distribution, though a
little skewed...but looks can be deceiving...

Well, Mandelbrot claims they are a generalization of normal, instead
of using the square root which is the 1/2 power you use the 1/alpha
power where alpha is 1.7 in place of 2.

Man, that guy could lay it on thick...I just accept the fact that I
can't calculate the size of a shoreline to his satisfaction and plow
ahead anyway...

But, the fact we cannot just assume the mean daily gain continues for
this particular example is more striking.

Nope, but absent any other information we can't assume it will
be any different! The great thing about the markets is that virtually
ALL market strategies, to the extent they are based on anything
at all other than a "guess", use cherry-picked time frames to
"prove" that a particular strategy "works" as advertised...

My original question was that with these sorts of calculation,
transforming daily data into yearly (and yearly into decadely etc to
try to come up with a plan for the future),

The "plan" for the "logs" calculation is that as every "trade" (or
"bet") becomes available, we take the ones that have the highest
growth factor. We are talking about hundreds, thousands, of
taken trades out of the possibility of millions over the course
of a trading lifetime, not just a single trade that has a certain
projected distribution or an unvarying portfolio of investments.
A single trade, despite the "logs", will either win or lose as
always, and the nature of dynamic market means that if you
truly have a method for projecting future results you will also
find that your original projection changes every day, so you
MUST "trade" for this "system" to work.

the statistics say if we
believe the distribution is normal we multiply the mean of the
percentaage changes (more correctly the log changes)

I do kinda believe you're off-base here taking the "logs"
of the MEAN and standard deviation just because we used
a logarithmic utility function for trade selection, but whatever,
I'm not sure you're coming to a startling conclusion as
a result below...

by the number of
days, and multiply the standard dev by the 1/2 power of the number of
days.
And if we believe it is alpha stable we use 1/alpha in place of 1/2
but hte same rule is used.

But the fact the standard deviation grows more slowly leads to the
advice, invest invest invest. The probabiliyt of even gteting down to
where you are now, becomes zero in the limit....

Well, yeah, if you "invest invest invest" (non-stupidly) you WILL
make money, "at the limit", with a 100% probability (or screw
the Latin, probability 1), barring meteor collisions or the like that
really renders the whole thing moot in the first place.

Now the next question is: why is this true? We look at the longest-term
history we can find, and it IS true, but WHY?

I've only answered this question about 8 billion times here, but it
is crucial to the TRUE next step, which is to construct a "bullet-proof"
model of the market/economy that you can run as a simulation
backwards and forwards, which forms the basis of "predicting"
future results over shorter time-frames, and thus dramatically
increasing our logarithmic portfolio growth rather than basing it
solely on 200-year averages...

---
William Ernest Reid
Post count: 902



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