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Regarding some postings late last week pertaining to fixed fractional
trading, if I may put my two cents in, there is NO single fraction of your
capital to invest which maximizes profits asymptotically or otherwise,
regardless of whether the stream of profits and losses are independent or
not.
In fact, I wrote about this in books published in 1992 and 1995. On p. 180
of latter, it is shown that by always trading that money management
strategy with the steepest gradient not only maximizes expected gain at any
point in time, but, by virtue of this, also maximizes gain asymptotically.
If we consider a simple two-outcome case, wherein we can either win 2
units with a probability of .5 or lose 1 unit with a probability of .5, we
find that the optimal fraction to wager is .25 of our bankroll on each and
every wager.
*However*, as pointed out in the 1995 book, if we wager .25 of our
INITIAL bankroll, regardless of the size of our current bankroll (a
strategy I refer to as "consant contract," as it is akin to trading, say,
10 bond contracts always on each and every trade, regardles of current
equity) we find that it's gradient is steeper than trading the optimal
fixed fraction method of .25 of our current equity on each and every play,
for the first 24 plays of this game, on average.
Thus, only after 24 plays at constant contract in this game are yuo better
off to trade fixed fractionally. For the first 24 plays, we would expect
the constant contract player to have a greater stake than the optimal fixed
fraction player. This then leads to the fact that from play 25 onward, the
constant contract player, now having a greater stake than the optimal fixed
fractional player, should now switch to trading using optimal fixed
fraction, and will always have greater wealth than his counterpart who
traded optimal fixed fraction all the way along. This is referred to in the
1995 book as continuous dominance, because it is not only asymptotically
optimal, it is also optimal at ANY point in time, including T=1.
To disregard or disparage notions like this is a big mistake I feel.
Although this case uses a simple 2 possible outcome case, and in real
trading we can have many more outcomes, the math is exact given the
outcomes and their probabilities of occurence. Considering how much we can
benefit from systematic money mangement, I very strongly believe a great
deal of our reseach effort should be directed towards discerning possible
outcomes and their probabilities. This an be solidified greatly by
1. Generating a model of the distribution of price changes
2. Transforming that distribution by
A. The rules of our trading system or metodology
B. Selecting instruments and strategies which clearly define the scenarios
(e.g. a long option clearly truncates the tansformed dstribution on the
left - the adverse - tail)
In so doing, we can create scenarios for the next holding period, and
formlate their probabilitites of occurence. With this input, we can apply
systematic money management. To simply disregard this as too complicated,
or worse yet, too unreliable, because we have not performed the exercises
listed above, is to deny ourselves a great deal of power available to us.
-Ralph Vince
"When you're out of Schlitz, you're out of beer."
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