Characteristics of Optimalf 197
Now here is thesecond arc sine law, which also uses Equation (5.09)
and hence has the same probabilities as the first arc sine law, but applies
to an altogether different incident, the maximum or minimum of the equity
curve. The second arc sine law states that the maximum (or minimum) point
of an equity curve will most likely occur at the endpoints, and least likely at
the center. The distribution is exactly the same as the amount of time spent
on one side of the origin!
If you were to toss the coin N times, your probability of achieving the
maximum (or minimum) at point K in the equity curve is also given by
Equation (5.08):
Prob∼ 1 /π*√
K*
√
(N−K)
Thus, if you were to toss a coin 10 times (N=10), you would have the
following probabilities of the maximum (or minimum) occurring on the Kth
toss:
K Probability0 .14795
1 .1061
2 .0796
3 .0695
4 .065
5 .0637
6 .065
7 .0695
8 .0796
9 .1061
10 .14795In a nutshell, the second arc sine law states that the maximum or mini-
mum is most likely to occur near the endpoints of the equity curve and least
likely to occur in the center.
Time Spent in a Drawdown
Recall the caveats involved with the arc sine laws. That is, the arc sine laws
assume a 50% chance of winning and a 50% chance of losing. Further, they
assume that you win or lose the exact same amounts and that the generating
stream is purely random. Trading is considerably more complicated than
this. Thus, the arc sine laws don’t apply in a pure sense, but they do apply
in spirit.