The Leverage Space Portfolio Model in the Real World 383
Notice that fornthings takenqat a time, the total number of permuta-
tions is thereforenq.
We can take the sum of theseβvalues for all permutations (ofnthings
takenqat a time, and again here,n=qfor the moment), and divide by
the number of permutations to obtain a real probability of ruin, withruin
defined as dropping tobof our starting stake, asRR(b):
RR(b,q)=∀nPq∑nq
k= 1βknq(12.05)
This is what we are doing in discerning the probability of ruin to a
givenb,RR(b). If there are two HPRs. There are 2× 2 =4 permutations,
from which we are going to determine aβvalue for each [usingRR(.6)].
Summing theseβvalues and dividing by the number of permutations, 4,
gives us our probability of ruin.
Note the input parameters. We have a value forbinRR(b)—that is,
the percentage of our starting stake left. Various values forb, of course,
will yield various results. Additionally, we are using HPRs, implying we
have anfvalue here. Differentfvalues will give different HPRs will give
different values forβ. Thus, what we are ultimately concerned with here—
and the reader is advised at this point not to lose sight of this—is that we
are essentially looking to holdbconstant in our analysis and are concerned
with those f values that yield an acceptableRR(b). That is, we want to
find thosefvalues that give us an acceptable probability for a given risk of
ruin.
Again we digress now for purposes of clarifying. For the moment, let us
suspend the notion of each play’s being a multiple on our stake. That is, let
us suspend thinking of these streams in terms of HPRs and TWRs. Rather,
let us simply contemplate the case of being presented with the prospect
of three consecutive coin tosses. We can therefore say that there are eight
separate streams, eight permutations, that the sequence which H and T may
comprise (∀ 2 P 3 ).
HHH
HHT
HTH
H T T (ruin)*
THH
THT
T T H (ruin)
T T T (ruin)