Ralph Vince - Portfolio Mathematics

Characteristics of Optimalf 203

changes are the result of inefficiencies in the way we are carrying out our
trading, as well as inefficiencies in our trading program or methodology.

Why IsfOptimal?

To see thatfis optimal in the sense of maximizing wealth:

sinceG=

(

∏T

i= 1

HPRi

) 1 /T

and

(T

∏

i= 1

HPRi

) 1 /T

=exp

⎛

⎜

⎜

⎜

⎝

∑T

i= 1

In(HPRi)

T

⎞

⎟

⎟

⎟

⎠

Then, if one acts to maximize the geometric mean at every holding period,
if the trial is sufficiently long, by applying either the weaker law of large
numbers or the Central Limit Theorem to the sum ofindependentvariables
(i.e., the numerator on the right side of this equation), almost certainly
higher terminal wealth will result than from using any other decision rule.
Furthermore, we can also apply Rolle’s Theorem to the problem of the
proof offs optimality. Recall that we are definingoptimalhere as meaning
that which will result in the greatest geometric growth as the number of trials
increases. The TWR is the measure of average geometric growth; thus, we
wish to prove that there is a value forfthat results in the greatest TWR.
Rolle’s Theorem states that if acontinuousfunction crosses a line par-
allel to the X-axis at two points,aandb, and the function is continuous
throughout the intervala,b, then there exists at least one point in the interval
where the first derivative equals zero (i.e., at least one relative extremum).
Given that all functions with a positive arithmetic mathematical expec-
tation cross the X-axis twice^6 (the X being thefaxis), atf=0 and at that
point to the right wherefresults in computed HPRs where the variance in
those HPRs exceeds the difference of the arithmetic mean of those HPRs
minus one, we have oura,binterval on X, respectively. Furthermore, the

(^6) Actually, atf=0, the TWR=0, and thus we cannot say that it crosses 0 to the
upside here. Instead, we can say that at anfvalue which is an infinitesimally small
amount beyond 0, the TWR crosses a line an infinitesimally small amount above 0.
Likewise to the right but in reverse, the line, thefcurve, the TWR, crosses this line
which is an infinitesimally small amount above the X-axis as it comes back down to
the X-axis.