300 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 9.3 Two-to-one coin toss—one play
on each game, you will now go broke, with a probability that approaches
certainty as the length of the game increases.
When you begin trading more than one market system, you no longer
reside on a line that has a peak; instead, you reside in ann+1 (where
n=the number of market systems you are trading) dimensional terrain
that has a single peak! In our single-coin-toss example, we had a peak on
the line at 25%. Here we have one game (n=1) and thus a two (i.e.,n+1)
dimensional landscape (the line) with a single peak. When we play two of
these games simultaneously, we now have a three-dimensional landscape
(i.e.,n+1) within leverage space with a single peak. If the correlation
coefficient between the coins is zero, then the peak is at 23% for the first
game and 23% for the second as well. Notice that there is still only one peak,
even though the dimensions of the landscape have increased!
When we are playing two games simultaneously, we are faced with a
three-dimensional landscape, where we must find the highest point. If we
were playing three games simultaneously, we would be looking for the peak
in a four-dimensional landscape. The dimensions of the topography within
which we must find a peak are equal to the number of games (markets and
systems) we are playing plus one.