362 THE HANDBOOK OF PORTFOLIO MATHEMATICS
this problem can be found by the fundamental equation for trading. Since
growth—that is, TWR—is the geometric mean holding period return to the
powerT, the number of plays is given by:
TWR=GT (10.19)By hiding out in the corner, we have a much smallerG. However, by
increasingT, that is, the number of trades, the effect of an exponential
decrease in growth is countered, by itself an exponential function.
In short, if a trader must minimize drawdowns, he or she is far better
off to trade at a very lowfvalue and get off many more holding periods in
the same span of time.
For example, consider playing only one of the two-to-one coin-toss
games. After 40 holding periods, at the optimalfvalue of .25, the geometric
mean HPR is 1.060660172, and the TWR is 10.55. If we were to play this same
game with anfvalue of .01, our geometric mean HPR would be 1.004888053,
which crosses 10.55 when raised to the power of 484. Thus, if you can get
off 484 plays (holding periods) in the same time it would take you to get
off 40 plays, you would see equivalent growth, with a dramatic reduction in
drawdowns. Further, you would have insulated yourself tremendously from
changes in the landscape. That is, you would also have insulated yourself a
great deal from changing market conditions.
It may appear that you want to trade more than one component (i.e., sce-
nario spectrum) simultaneously. That is, to increaseT, you want to trade
many more components simultaneously. This is counter to the idea pre-
sented earlier in discussing the points of inflection that you may be better
off to trade only one component. However, by increasing the number of
components traded simultaneously, you increase the compositefof the
portfolio. For example, if you were to trade 20 scenario spectrums simulta-
neously, each with a .005 value off, you would have a compositefof the
entire portfolio of 0.1. At such a level, if the worstcase scenarios were to
manifest simultaneously, you would experience a 10% drawdown on equity.
By contrast, you are better off to trade only one scenario spectrum whereby
you can get off the equivalent of 20 holding periods in the same spame of
time. This may not be possible, but it is the direction you want to be working
in to minimize drawdowns.
Finally, when a trader seeks to approach drawdown minimization, he
or she can use the continuous dominance notion in doing so. Continuous
dominance is great in the theoretical ideal model. However, it is extremely
sensitive to changes in the landscape. That is, as the scenarios used as input
change to accommodate changing market characteristics, continuous dom-
inance begins to run into trouble. In a gambling game where the conditions
do not change from one period to the next, continuous dominance is ideal.