ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0
32 THE HANDBOOK OF PORTFOLIO MATHEMATICS
All of the aboveis calculated with$50 commissions and slippage taken
off of each trade.As you can see, this was a terrific system before this rule.
Sogood,in fact, thatit was difficult toimprove uponitin any way.Yet,
once the dependency was found and exploited, the system was materially
improved.It was with a confidence limitofslightly over 95%.Itis rare to
find a confidence limitthishighin futures tradingsystems.However, from a
statistical point of view,itis hardly high enough to assume that dependency
exists.Ideally, yet rarely you will find systems that have confidence limits
in the highnineties.
So far we have only looked at dependency from the point of view of
whether the last trade was a winner or a loser.We are tryingto deter-
mineif the sequence of wins and losses exhibit dependency or not.The
runs test for dependency automatically takes the percentageofwins and
lossesinto account.However,in performingthe runs test on runs of wins
and losses, we have accounted for the sequence of wins and losses but
not theirsize.For the system to be trulyindependent, not only must the
sequence of wins and losses beindependent;the sizes of the wins and
losses within the sequence must also beindependent.Itis possible for the
wins and losses to beindependent, while theirsizes are dependent (or vice
versa).
One possible solutionis to run the runs test on only the winningtrades,
segregatingthe runsin some way (e.g., those that aregreater than the me-
dian win versus those that are less).Then look for dependency amongthe
size of the winningtrades;then do the same for the losingtrades.The Linear Correlation Coefficient
Thereis, however, a different, possibly better way to quantify this possible
dependency between the size of the wins and losses.The technique to be
discussed next looks at the sizes of wins and losses from an entirely dif-
ferent mathematical perspective than does the runs test, and when usedin
conjunction with the latter, measures the relationship of trades with more
depth than the runs test alone could provide.This technique utilizes the
linear correlation coefficient, r, sometimes called Pearson’s r, to quantify
the dependency/independency relationship.
Look at Figure 1. 5 .It depicts two sequences that are perfectly corre-
lated with each other.We call this effect“positive”correlation.
Now look at Figure 1. 6 .It shows two sequences that are perfectly un-
correlated with each other.When one lineiszigging, the otheriszagging.
We call this effect“negative”correlation.
The formula for findingthe linear correlation coefficient (r) between
two sequences, X and Y, follows.(A bar over the variable means the mean