196 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Thus, if we were to toss a coin 10 times (N=10) we would have the
following probabilities of being in positive territory for K of the tosses:
K Probability^40 .14795
1 .1061
2 .0796
3 .0695
4 .065
5 .0637
6 .065
7 .0695
8 .0796
9 .1061
10 .14795You would expect to be in positive territory for 5 of the 10 tosses, yet
that is the least likely outcome! In fact, the most likely outcome is that you
will be in positive territory for all of the tosses or for none of them!
This principle is formally detailed in thefirst arc sine law, which states:
For a Fixed A (0<A<1) and as N approaches infinity, the probability that
K/N spent on the positive side is<A tends to:
Prob{(K/N) < A}= 2 /π*sin−^1√
A (5.09)
Even with N as small as 20, you obtain a very close approximation for the
probability.
Equation (5.09), the first arc sine law, tells us that with probability .1,
we can expect to see 99.4% of the time spent on one side of the origin, and
with probability .2, the equity curve will spend 97.6% of the time on the same
side of the origin! With a probability of .5, we can expect the equity curve
to spend in excess of 85.35% of the time on the same side of the origin. That
is just how perverse the equity curve of a fair coin is!
(^4) Note that since neither K nor N may equal 0 in Equation (5.08) (as you would then
be dividing by 0), we can discern the probabilities corresponding to K=0 and K=
N by summing the probabilities from K=1toK=N – 1 and subtracting this sum
from 1. Dividing this difference by 2 will give us the probabilities associated with
K=0 and K=N.