352 THE HANDBOOK OF PORTFOLIO MATHEMATICS
managers would restrict themselves to that real estate which is to the left
of the peak when looking from south to north at the landscape, and left of
the peak when looking from east to west at the landscape. We could carry
the thought into more dimensions, but the termto the left, is irrespective
of the number of dimensions; it simply means at less than full optimal with
respect to each axis (scenario spectrum).
Money managers arenotwealth maximizers. That is, their utility func-
tion or, rather, the utility functions imposed on them by their clients and
their industry, theirU′′(x), is less than zero. They are, therefore, to the left
of the peaks of their optimalfs.
Thus, given the real-world constraints of smoother equity curves than
full optimal calls for, as well as the realization that a not-so-typical draw-
down at the optimal level will surely cause a money manager’s clients to flee,
we are faced with the prospect of where, to the left, is an opportune point
(to satisfy theirU′′(x))? Once this opportune point is found, we can then
exercise continuous dominance. In so doing, we will be ensuring that by
trading at this opportune point to the left, we will have the highest expected
value for the account at any point thereafter. It does not mean, however,
that it will outpace an account traded at the full optimalfset. It will not.
Now we actually begin to work with this new framework. Hence, the
point of this section is twofold: first, to point out that there are possible
advantageous points to the left, and, second, but more importantly, to show
you, by way of examples, how the new framework can be used.
There are a number of advantageous points to the left of the peak, and
what follows is not exhaustive. Rather, it is a starting place for you.
The first point of interest to the left pertains to constant contract trading,
that is, always trading in the same unit size regardless of where equity runs
up or shrinks. This should not be dismissed as overly simplistic by candidate
money managers for the following reason:Increasing your bet size after
a loss maximizes the probability of an account being profitable at any
arbitrary future point. Varying the trading quantity relative to account
equity attempts to maximize the profitability (yet it does not maximize
the probability of being profitable).
The problem with trading the same constant quantity is that it not only
puts you to the left of the peak, but, as the account equity grows, you are
actually migrating toward zero on the variousfaxes.
For instance, let’s assume we are playing the two-to-one coin toss game.
The peak is atf=.25, or making one bet for every $4 in account equity.
Let’s say we have a twenty dollar account, and we plan to always make two
bets, that is, to always bet $2 regardless of where the equity goes. Thus,
we start out (fortunately, this is a two-dimensional case since we are only
discussing one scenario spectrum) trading at anf$ of $10, which is anfof
.1, sincef$=−BL/f, it follows thatf=−BL/f$. Now, let us assume that