204 THE HANDBOOK OF PORTFOLIO MATHEMATICS
first derivative of the fundamental equation of trading (i.e., the estimated
TWR) is continuous for allfwithin the interval, sincefresults in AHPRs
and variances in those HPRs, within the interval, which are differentiable in
the function in that interval; thus, the function, the estimated TWR, is con-
tinuous within the interval. Per Rolle’s Theorem, it must, therefore, have at
least one relative extremum in the interval, and since the interval is positive,
that is, above the X-axis, the interval must contain at least one maximum.
In fact, there can be only one maximum in the interval given that the
change in the geometric mean HPR (a transformation of the TWR, given that
the geometric mean HPR is theTth root of the TWR) is a direct function of
the change in the AHPR and the variance, both of which vary inopposite
directions to each other asfvaries, per the Pythagorean theorem. This
guarantees that there can be only one peak. Thus, there must be a peak in
the interval, and there can be only one peak. There is anfthat is optimal
at only one value forf, where the first derivative of the TWR with respect
tofequals zero.
Let us go back to Equation (4.07). Now, we again consider our two-to-
one coin toss. There are two trades, two possible scenarios. If we take the
first derivative of (4.07) with respect tof, we obtain:
dTWR
df=
((
1 +f*(
−trade 1
biggest loss))
*
(
−trade 2
biggest loss))
+
((
−trade 1
biggest loss)
*
(
1 +f*(
−trade 2
biggest loss)))
(5.11)
If there were more than two trades, the same basic form could be used,
only it would grow monstrously large in short order, so we’ll use only two
trades for the sake of simplicity. Thus, for the sequence+2,−1atf=.25:
dTWR
df=
((
1 +. (^25) *
(
− 2
− 1
))
*
(
− 1
− 1
))
+
((
− 2
− 1
)
*
(
1 +. (^25) *
(
− 1
− 1
)) )
dTWR
df=(( 1 +. (^25) 2 )− 1 )+( (^2) ( 1 +. (^25) − 1 ))
dTWR
df
=(( 1 +. 5 )− 1 )+( (^2) ( 1 −. 25 ))
dTWR
df
=(1. (^5) −1)+(2.75)
dTWR
df