398 THE HANDBOOK OF PORTFOLIO MATHEMATICS
particular horizontal run ofkfrom 1 tonq, we have a probabilityProbk,i.
Now multiplying theseProbk,i’s together in the horizontal run forifrom 1
toqwill give thepkfor thisk.
Prob1, 1*Prob1, 2*···*Prob1,q=p 1
...
Probnq,1*Probnq,2*···*Probnq,q=pnqn.b. Now, when dependency is present in the stream
of outcomes, the pkvalues are necessarily affected.For example, in the simplistic binomial outcome case of a coin toss,
where I have two possible outcomes (n=2), heads and tails, with outcomes
+2 and−1, respectively, and I look at flipping the coin two times (q=2), I
have the following four (nq) possible outcomes:
pk
Outcome 1 (k=1) H H .25
Outcome 2 (k=2) H T .25
Outcome 3 (k=3) T H .25
Outcome 4 (k=4) T T .25
Now let us assume there is perfect negative correlation involved—that
is, winners always beget losers, and vice versa. In this idealized case, we
then have the following:
pk
Outcome 1 (k=1)HH0
Outcome 2 (k=2) H T .5
Outcome 3 (k=3) T H .5
Outcome 4 (k=4)TT0
Unfortunately, when serial dependency seems to exist, it is never at
such an idealized value as 1.0, as shown here. Fortunately, however, serial
dependency rarely exists, and its appearance of existence in small amounts
is usually, and typically, incidental, and can thus be worked with as being
zero. However, if thepkvalues are deemed to be more than merely “inci-
dental,” then they can, and in fact, must, be accounted for as they are used
in the equations given in this chapter.
Additionally, the incorporation of rules to address dependency when it
seems present, of the type like, “Don’t trade after two consecutive losers,
etc,” could in this analysis be turned into the familiar tails, or T in the
following stream: