Ralph Vince - Portfolio Mathematics

The Leverage Space Portfolio Model in the Real World 381

Expressed mathematically, we can say that at anyiin (12.02) if the
interim value for (12.02)<=0, then ruin has occurred:

∑q

i= 1

((

i∏− 1

t= 0

HPRt

)

*HPRi−b

)

(12.02)

where: HPR 0 =1.0

q=The number of scenarios in multiplicative sequence

(in this case 2, the same asn).^4

b=That multiple on our stake, as a lower barrier, where

we determine ruin to occur (0<=b<=1).

Again, if at any arbitraryq, we have a value<=0, we can conclude that
ruin has occurred.
One way of expressing this mathematically would be:

int

⎛

⎜

⎜

⎜

⎝

∑q

i= 1

((i− 1

∏

t= 0

HPRt

)

*HPRi−b

)

∑q

i= 1

∣

∣

∣

∣

((i− 1

∏

t= 0

HPRt

)

*HPRi−b

)∣

∣

∣

∣

⎞

⎟

⎟

⎟

⎠

=β (12.03)

where: HPR 0 =1.0

q=The number of scenarios in multiplicative sequence.

∑q

i= 1

∣

∣

∣

∣

∣

((

i∏− 1

t= 0

HPRt

)

*HPRi−b

)∣

∣

∣

∣

∣

= 0

In (12.03) note thatβcan take only one of two values, either 1 (ruin has
not occurred) or 0 (ruin has occurred).
There is the possibility that the denominator in (12.03) equals 0, in which
caseβshould be set to 0.
We digress for purpose of clarity now. Suppose we have a stream of
HPRs. Let us suppose we have the five separate HPRs of:

.9

1.05

.7

.85

1.4

(^4) For the moment, considerqthe same asn. Later in this chapter, they become two
distinct variables.