172 THE HANDBOOK OF PORTFOLIO MATHEMATICS
the money risked on the decision at hand. Since, in almost every case, the
money risked on an event today will be risked again on a different event in
the future, and money made or lost in the past affects what we have avail-
able to risk today, we should decide, based on geometric mean, to maximize
the long-run growth of our money. Even though the scenarios that present
themselves tomorrow won’t be the same as those today, by always deciding
based on greatest geometric mean, we are maximizing our decisions. It is
analogous to a dependent trials process, like a game of blackjack. In each
hand, the probabilities change and, therefore, the optimal fraction to bet
changes as well. By always betting what is optimal for that hand, however,
we maximize our long-run growth. Remember that, to maximize long-run
growth, we must look at the current contest as one that expands infinitely
into the future. In other words, we must look at each individual event as
though we were to play it an infinite number of times if we wanted to max-
imize growth over many plays of different contests.
As a generalization, whenever the outcome of an event has an effect
on the outcome(s) of subsequent event(s), we are better off to maximize
for greatest geometric expectation. In the rare cases where the outcome
of an event has no effect on subsequent events, we are then better off to
maximize for greatest arithmetic expectation.
Mathematical expectation (arithmetic) does not take the dispersion
between the outcomes of the different scenarios into account and, therefore,
can lead to incorrect decisions when reinvestment is considered.
Using this method of scenario planning gets you quantitatively posi-
tioned with respect to the possible scenarios, their outcomes, and the like-
lihood of their occurrence. The method is inherently more conservative than
positioning yourself per the greatest arithmetic mathematical expectation.
The geometric mean of a data set is never greater than the arithmetic mean.
Likewise, this method can never have you position yourself (have a greater
commitment) otherwise than selecting by the greatest arithmetic mathe-
matical expectation would. In the asymptotic sense (the long-run sense),
this is not only the superior method of positioning yourself as it achieves
greatest geometric growth; it is also a more conservative one than position-
ing yourself per the greatest arithmetic mathematical expectation.
Since reinvestment is almost always a fact of life (except on the day
before you retire)—that is, you reuse the money that you are using today—
we must make today’s decision under the assumption that the same decision
will present itself a thousand times over, in order to maximize the results of
our decision. We must make our decisions and position ourselves in order
to maximize geometric expectation. Further, since the outcomes of most
events do, in fact, have an effect on the outcomes of subsequent events,
we should make our decisions and position ourselves based on maximum
geometric expectation. This tends to lead to decisions and positions that
are not always obvious.