Ralph Vince - Portfolio Mathematics

The Geometry of Leverage Space Portfolios 327

are now given for the diluted AHPR, called FAHPR, the diluted standard
deviation (which is simply the square root of variance), called FSD, and the
diluted geometric mean HPR, called FGHPR here:

FAHPR=(AHPR−1) * FRAC+ 1

FSD=SD * FRAC

FGHPR=

√

FAHPR^2 −FSD^2

where: FRAC=The fraction of optimalfwe are solving for.

AHPR=The arithmetic average HPR at the optimalf.

SD=The standard deviation in HPRs at the optimalf.

FAHPR=The arithmetic average HPR at the fractionalf.

FSD=The standard deviation in HPRs at the fractionalf.

FGHPR=The geometric average HPR at the fractionalf.

Let’s assume we have a system where the AHPR is 1.0265. The standard
deviation in these HPRs is .1211 (i.e., this is the square root of the variance
given by Equation (10.04)); therefore, the estimated geometric mean is 1.019.
Now, we will look at the numbers for a .2 static fractionalfand a .1 static
fractionalf.The results, then, are:

Fullf .2f .1f

AHPR 1.0265 1.0053 1.00265
SD .1211 .02422 .01211
GHPR 1.01933 1.005 1.002577
Here is what will also prove to be a useful equation, the time expected
to reach a specific goal:

T=

ln(goal)

ln(geometric mean)

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.

ln ( )=The natural logarithm function.

Now, we will compare trading at the .2 static fractionalfstrategy, with
a geometric mean of 1.005, to the .2 dynamic fractionalf strategy (20%
as initial active equity) with a daily geometric mean of 1.01933. The time