Ralph Vince - Portfolio Mathematics

Optimalf 161

Note that here, for a local minimum, the first operator is a plus (+) sign, not
a minus (−) sign as when we were looking for a local maximum.

The Next Step

The real problem with the formula to this point is that it makes the assump-
tion that all HPRs have an equal probability of occurrence. What is needed
is a new formula that allows for different probabilities associated with dif-
ferent HPRs. Such a formula would allow you to find an optimal fgiven a
description of a probability distribution of HPRs. To accommodate this, we
need to rework (4.06) to:

HPR=

⎛

⎝ 1 +

⎛

⎝(A

W

f

)

⎞

⎠

⎞

⎠

P

(4.16)

where A=outcome of the scenario

P=probability of the scenario

W=worst outcome of allnscenarios

f=value forfwhich we are testing

Now, we obtain the terminal wealth relative, or TWR^4 , originally given
by (3.03) and (4.07) to:

TWR=

∏T

i= 1

HPRi

or

TWR=

∏T

i= 1

⎛

⎝ 1 +

⎛

⎝(Ai

W

f

)

⎞

⎠

⎞

⎠

Pi

(4.17)

Finally, if we take Equation (4.18) to thepiroot, we can find our
average compound growth per play, also called the geometric mean HPR,
and replace that given in (4.08) which will become more important later on:

G=TWR^1 /pi (4.18)

(^4) In this formulation, unlike the 1990 formulations, the TWR has no special meaning.
In this instance, it is simply an interim value used to findG, and it doesnotrepresent
the multiple made on our starting stake. The variable named “TWR” is maintained
solely for consistency’s sake.