ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0
76 HANDBOOK OF PORTFOLIO MATHEMATICS
where: L=The lower boundary for P to be at Z standard deviations.
P=The variable of interest representing the probability of be-
ing in one of two mutually exclusive groups.
Z=The selected number of standard deviations.
N=The total number of events in the sample.Suppose our sample consisted of 100 plays. Thus:L=. 51 − 3 ∗√
(. 51 ∗(1−.51))/(100−1)
=. 51 − 3 ∗
√
(. 51 ∗.49)/ 99
=. 51 − 3 ∗
√
. 2499 / 99
=. 51 − 3 ∗
√
. 0025242424
=. 51 − 3 ∗. 05024183938
=. 51 −. 1507255181
=. 3592744819
Based on our history of 100 plays which generated a 51% win rate, we
can state that it would take a three-sigma event for the population of plays
(the future if we play an infinite number of times into the future) to have
less than 35.92744819% winners.
What kind of a confidence level does this represent? That is a function
of N, the total number of plays in the sample. We can determine the con-
fidence level of achieving 35 or 36 wins in 100 tosses by Equation (2.35).
However, (2.35) is clumsy to work with as N gets large because of all
of the factorial functions in (2.35). Fortunately, the Normal distribution,
Equation (2.21) for one-tailed probabilities, can be used as a very close
approximation for the Binomial probabilities. In the case of our example,
using Equation (2.21), 3 standard deviations translates into a 99.865% con-
fidence. Thus, if we were to play this gambling system over an infinite num-
ber of times, we could be 99.865% sure that the percentage of wins would
be greater than or equal to 35.92744819%.
This technique can also be used for statistical validation of trading sys-
tems. However, this method is only valid when the following assumptions
are true. First, the N events (trades) are all independent and randomly se-
lected. This can easily be verified for any trading system. Second, the N
events (trades) can all be classified into two mutually exclusive groups
(wins and losses, trades greater than or less than the median trade, etc.).
This assumption, too, can easily be satisfied. The third assumption is that
the probability of an event being classified into one of the two mutually
exclusive groups is constant from one event to the next. This is not neces-
sarily true in trading, and the technique becomes inaccurate to the degree