118 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 4.1 The curve of optimalf
have an edge (i.e., a positive mathematical expectation), there exists an
optimal fixed fraction(f)between 0 and 1 as a divisor of your biggest loss
to bet on each and every event.
Most people think that the optimal fixed fraction is the percentage of
your total stake to bet. Optimalfis not in itself the percentage of our total
stake to bet; it is the divisor of our biggest loss, the result of which we divide
our total stake by to know how many bets to make or contracts to have on.
You will also notice thatmargin has nothing to do whatsoever with
what is the mathematically optimal number of contracts to have on.
As you can see in Figure 4.1,fis a curve cupped downward from 0 to
- The highest point forfis that fraction of your stake to bet on each and
every event (bet) to maximize your winnings.
Most people incorrectly believe thatfis a straight-line function rising
up and to the right. They believe this because they think it would mean that
the more you are willing to risk, the more you stand to make. People reason
this way because they think that a positive mathematical expectation is just
the mirror image of a negative expectancy. They mistakenly believe that if
increasing your total action in a negative expectancy game results in losing
faster, then increasing your total action in a positive expectancy game will
result in winning faster. This is not true. At some point in a positive ex-
pectancy situation, to increase your total action further works against you.
That point is a function of both the system’s profitability and its consistency
(i.e., its geometric mean), since you are reinvesting the returns.
Asymmetrical Leverage
Recall that the amount required to recoup a loss increases geometrically
with the loss.We can show that the percentage gain to recoup a loss is:
Required Gain=(1/(1−loss in percent))− 1 (4.01)