348 THE HANDBOOK OF PORTFOLIO MATHEMATICS
The growth functions are taken from Equation (10.05). Thus, the static
fgrowth function is the left side of (10.05) and the dynamicfis the right
side. Thus, the growth function for staticfis:
Y =FGHPRT (10.12)And for dynamicf,itis:
Y=geometric meanT∗ FRAC+ 1 −FRAC (10.13)Equations (10.11) through (10.13) give us the growth function as a mul-
tiple of our starting stake, at a given number of elapsed holding periods,T.
Thus, by subtracting 1 from Equations (10.11) through (10.13), we obtain
the percent growth as depicted in Figure 10.3.
The gradients, depicted in Figure 10.4, are simply the first derivatives
ofYwith respect toT, for Equations (10.11) through (10.13). Thus, the
gradients are given by the following.
For constant contract trading:
dY
dT=
((AHPR−1)∗FRAC)
(1+AHPR−)∗FRAC∗T
(10.14)
For static fractionalf:dY
dT=FGHPRT∗ln(FGHPR) (10.15)And finally for dynamic fractionalf:dY
dT=geometric meanT∗ln(geometric mean)∗FRAC (10.16)where: T=The number of holding periods.
FRAC=The initial active equity percentage.
geometric mean=The raw geometric mean HPR at the optimalf.
AHPR=The arithmetic average HPR at full optimalf.
FGHPR=The fractionalfgeometric mean HPR given by
Equation (5.06).
ln( )=The natural logarithm function.
The way to implement these equations, especially as your scenarios
(scenario spectrums) and joint probabilities change from holding period to
holding period, is as follows. Recall that just before each holding period we
must determine the optimal allocations. In the exercise of doing that, we
derive all of the necessary information to get the values for the variables
listed above (for FRAC, geometric mean, AHPR, and the inputs to Equation