350 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 10.5 Continuous dominance vs. dynamicf
to dynamicf. If we were to do this, Figure 10.5 shows how much better
we would have fared, on average, over the first 20 plays or holding periods,
than by simply trading a dynamic fractionalfstrategy:
Notice that, at every period, an account traded this way has a higher
expected value than even the dynamic fractionalf. Further, from period 17
on, where we switched from static to dynamic, both lines are forevermore
on the same gradient. That is, the dynamic line will never be able to catch up
to the continuous dominance line. Thus, the principle of always trading the
highest gradient to achieve continuous dominance helps a money manager
maximize where an account will be at any point in the future, not just in an
asymptotic sense.
To clarify by carrying the example further, suppose we play this two-to-
one coin-toss game, and we start out with an account of $200. Our optimal
fis .25, and a .2f, one-fifth of this, means we are trading anf value of
.05, or we bet $1 for every $20 in our stake. Therefore, on the first play we
bet $10. Since we are trading constant contract, regardless of where the
account equity is thereafter, we bet $10 on each subsequent play until we
switch over to staticf. This occurs on the third play. So, on the third bet,
we take where our stake is and bet $1 for every $20 we have in equity. We
proceed as such through play 16, where, going into the seventeenth play,
we will switch over to dynamic. Thus, as we go into every play, from play 3
through play 16, we divide our total equity by $20 and bet that many dollars,
thus performing a static fractionalf.
So, assume that after the second play we have $210 in our stake. We
would now bet $10 on the next play (since 210/20=10.5, and we must round
down to the integer). We keep doing this going into each play through the
sixteenth play.