ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0
26 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Assume the followingstream of coinflips where a plus (+) stands for
awin and a minus (−) stands for a loss:
++−−−−−−−+−+−+−−−+++−+++−+++
There are 28 trades, 14 wins and 14 losses.Say thereis$1 won on a win
and$1 lost on a losingflip.Hence, the net for this seriesis$ 0.
Now assume you possess theinfant’smind.You do not knowif there
is dependency or notin the coin-toss situation (although thereisn’t).Upon
seeingsuch a stream of outcomes you deduce the followingrule, which
says,“Don’t bet after two losers;go to the sidelines and wait for a winner
to resume betting.”With this new rule, the previous sequence would have
been:
++−−−+−+−−++−+++−+++
So, with this new rule the old sequence would have produced 12 winners
and 8 losers for a net of$ 4 .You’re quite confident of your new rule.You
haven’t learned to differentiate an exact sequence (whichis all that this
stream of tradesis) from an end result (the end result beingthat thisisa
break-evengame).
Thereisamajor problem here, though, and thatis that you do not
knowif thereis dependencyin the sequence of flips.Unless dependency
is proven, no attempt to improve performance based on the stream of
profits and losses alone is of any value, and quite possibly you may do
more harm than good.^2 Let us continue with theillustration and we will
see why.(^2) Adistinction must be drawn between a stationary and a nonstationary distribu-
tion.A stationary distributionis one where the probability distribution does not
change.An example would be a casinogame such as roulette, where you are al-
ways at a.0526 disadvantage.A nonstationary distributionis one where the expec-
tation changes over time (in fact, the entire probability distribution may change
over time).Tradingisjust such a case.Tradingis analogousinthis respect to a
drunk wanderingthrough a casino,goingfromgame togame.First he plays roulette
with$5chips (for a−.0526 mathematical expectation), then he wanders to a black-
jack table, where the deck happens to be runningfavorable to the player by 2%.
Hisdistribution of outcomes curve moves around as he does;the mathematical
expectation and distribution of outcomesis dynamic.Contrast this to stayingat
one table, at onegame.In such a case the distribution of outcomesis static.We
sayitisstationary.The outcomes of systems tradingappear to be a nonstation-
ary distribution, which wouldimply that thereis perhaps some technique that may
be employed to allow the trader to advantageously“trade his equity curve.”Such
techniques are, however, beyond the mathematical scope of this book and will not
be treated here.Therefore, we will not treat nonstationary distributions any differ-
ently than stationary onesin the text, but be advised that the two are profoundly
different.