Ralph Vince - Portfolio Mathematics

The Geometry of Leverage Space Portfolios 329

To make 400% profit (i.e., a goal or TWR, of 5) requires of the .2 static
technique:

ln(5)

ln(1.005)

= 322. 6902

Which compares to its dynamic counterpart:

ln(21)

ln(1.01933)

= 1590201

It takes the dynamic almost half the time it takes the static to reach the
goal of 400% in this example. However, if you look out in time 322.6902 days
to where the static technique doubled, the dynamic technique would be at
a TWR of:

=. 8 + 1. 01933322.^6902 ∗. 2

=. 8 + 482. 0659576 ∗. 2

= 97. 21319

This represents making over 9,600% in the time it took the static to make
400%.
We can now amend Equation (5.07) to accommodate both the static and
fractional dynamicfstrategies to determine the expected length required
to achieve a specific goal as a TWR. To begin with, for the static fractional
f, we can create Equation (5.07b):

T=

ln(goal)

ln(FGHPR)

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.

FGHPR=The adjusted geometric mean. This is the geometric

mean, run through Equation (5.06) to determine

the geometric mean for a given static fractionalf.

ln( )=The natural logarithm function.

For a dynamic fractionalf, we have Equation (5.07c):

T=

ln

((

(goal−1)

FRAC

)

+ 1

)

ln(geometric mean)