330 THE HANDBOOK OF PORTFOLIO MATHEMATICS
where: T=The expected number of holding periods
to reach a specific goal.
goal=The goal in terms of a multiple on our starting
stake, a TWR.
FRAC=The initial active equity percentage.
geometric mean=the raw geometric mean HPR at the
optimalf; there is no adjustment performed
on it as there is in Equation (5.07b)
ln( )=The natural logarithm function.Thus, to illustrate the use of Equation (5.07c), suppose we want to
determine how long it will take an account to double (i.e., TWR=2) at .1
active equity and a geometric mean of 1.01933:
T=
ln((
(goal−1)
FRAC)
+ 1
)
ln(geometric mean)=
ln((
(2−1). 1
)
+ 1
)
ln(1.01933)=
ln(
(1). 1 +^1
)
ln(1.01933)=ln( 10 + 1 )
ln(1.01933)=ln(11)
ln(1.01933)=2. 397895273
. 01914554872
= 125. 2455758
Thus, if our geometric means are determined off scenarios which have
a daily holding period basis, we can expect about 125^1 / 4 days to double.
If our scenarios used months as holding period lengths, we would have to
expect about 125^1 / 4 months to double.
As long as you are dealing with aTlarge enough that Equation (5.07c)
is greater than Equation (5.07b), then you are benefiting from dynamic frac-
tionalftrading. This can, likewise, be expressed as Equation (10.05):
FGHPRT<=geometric meanT ∗FRAC+ 1 −FRAC (10.05)Thus, you must iterate to that value ofTwhere the right side of the
equation exceeds the left side—that is, the value forT(the number of holding