Ralph Vince - Portfolio Mathematics

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

The Random Process and Gambling Theory 27

Since thiswasacoin toss, there wasin fact no dependencyin the

trials—thatis, the outcome of each successive flipwasindependent of (un-

affected by) the previous flips.Therefore, this exact sequence of 28 flips

was totally random.(Remember, each exact sequence has an equal proba-

bility of occurring.Itis the end results that follow the Normal Distribution,

with the peak of the distribution occurringat the mathematical expecta-

tion.The end resultinthis case, the mathematical expectation,is a net

profit/loss of zero.) The next exact sequence of 28 flipsisgoingto appear

randomly, and thereis an equal probability of the followingsequence ap-

pearingas any other:

−−+−−+−−+−−+−−+−−+−−++++++++

Once again, the net of this sequenceis nothingwon and nothinglost.Ap-

plyingyour rule here, the outcomeis:

−−−−−−−−−−−−−−+++++++

Fourteen losses and seven wins for a net loss of$ 7.

As you can see, unless dependencyis proven (inastationary process),

no attempt toimprove performance based on the stream of profits and

losses aloneis of any value, and you may do more harm thangood.

The Runs Test, Z Scores, and Confidence Limits

For certain events, such as the profit and loss stream of a system’s trades,

where dependency cannot be determined uponinspection, we have the

runs test.The runs testis essentially a matter of obtainingthe Z scores for

the win and loss streaks of a system’s trades.Here’s how to doit.First,

you will need a minimum of 30 closed trades.Thereis a very valid statisti-

cal reason for this.Z scores assume a Normal Probability Distribution (of

streaks of wins and lossesinthisinstance).Certain characteristics of the

Normal Distribution are no longer valid when the number of trialsis less

than 30.Thisis because a minimum of 30 trials are necessaryin order to

resolve the shape of the Normal Probability Distribution clearly enoughto

make certain statistical measures valid.

The Z scoreissimply the number of standard deviations the datais

from the mean of the Normal Probability Distribution.For example, a Z

score of 1.00 would mean that the data you are testingiswithin 1 stan-

dard deviation from the mean.(Incidentally, thisis perfectly normal.) The

Z scoreis then convertedinto a confidence limit, sometimes also called a