ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0
14 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 1.2 Normal probability function: Center line and 1 standard deviation in
either directionThe mathematical expectationis what we expect togain or lose, on
average, each bet.However,it does not explain the fluctuations from bet to
bet.In our coin-toss example we know that thereis a 50/50 probability of a
toss’s comingup heads or tails.We expect that after N trials approximately(^1) / (^2) N of the tosses will be heads, and (^1) / (^2) N of the tosses will be tails.
Assumingthat we lose the same amount when we lose as we make when
we win, we can say we have a mathematical expectation of 0, regardless of
how largeNis.
We also know that approximately 68% of the time we will be+or−
1 standard deviation away from our expected value.For 10 trials (N=10)
this means our standard deviationis1. 58 .For 100 trials (N=100) this
means we have a standard deviation size of 5.At 1,000 (N=1,000) trials the
standard deviationis approximately 15. 81 .For 10,000 trials (N=10,000)
the standard deviationis50.
N Std Dev Std Dev/N as%
10 1.58 15.8%
100 5 5.0%
1,000 15.81 1.581%
10,000 50 0.5%
Notice that as Nincreases, the standard deviationincreases as well.
This means that contrary to popular belief,the longer you play, the