Ralph Vince - Portfolio Mathematics

ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0

The Random Process and Gambling Theory 29

Let’s construct an example to follow alongwith.Assume the following

trades:

− 3 + 2 + 7 − 4 + 1 − 1 + 1 + 6 − 10 − 2 + 1

The net profitis+ 7 .The total number of tradesis12;therefore, N= 12

(we are violatingthe rule that there must be at least 30 trades only to

keep the example simple).Now we are not concerned here with how

bigthe wins and losses are, but rather how many wins and losses there

are and how many streaks.Therefore, we can reduce our run of trades

to a simple sequence of pluses and minuses.Note that a trade with a

profit and loss (P&L) of 0isregarded as a loss.We now have:

−++−+−++−−−+

As can be seen, there are six profits and six losses.Therefore, X= 2

*^6 *^6 =^72 .As can also be seen, there are eight runsinthis sequence, so

R= 8 .We will define arunas any time we encounter a sign change when

readingthe sequence as shown above from left to right (i.e., chronolog-

ically).Assume also that we start at 1.Therefore, we would count this

sequence as follows:

−++−+−++−−−+

12 3456 7 8

2.Solve for the equation:

N∗(R−.5)−X

For our example this would be:

12 ∗(8−.5)− 72

12 ∗ 7. 5 − 72

90 − 72

18

3.Solve for the equation:

X∗(X−N)/(N−1)

So for our example this would be:

72 ∗(72−12)/(12−1)

72 ∗ 60 / 11

4 , 320 / 11

392. 727272