188 THE HANDBOOK OF PORTFOLIO MATHEMATICS
since you have not been able to trim back size for each loss as you would
have had the trades occurred sequentially.
Therefore, we can say that there is a gain from adding each new market
system to the portfolio provided that the market system has a correlation
coefficient less than one and a positive mathematical expectation, or a neg-
ative expectation but a low enough correlation to the other components in
the portfolio to more than compensate for the negative expectation. There
is a marginally decreasing benefit to the geometric mean for each market
system added. That is, each new market system benefits the geometric mean
to a lesser and lesser degree. Further, as you add each new market system,
there is a greater and greater efficiency loss caused as a result of simultane-
ous rather than sequential outcomes. At some point, to add another market
system may do more harm then good.
TIME REQUIRED TO REACH A SPECIFIED
GOAL AND THE TROUBLE WITH
FRACTIONALf
Suppose we are given the arithmetic average HPR and the geometric average
HPR for a given system. We can determine the standard deviation (SD) in
HPRs from the formula for estimated geometric mean:
EGM=√
AHPR^2 −SD^2
where: AHPR=The arithmetic mean HPR.
SD=The population standard deviation in HPRs.Therefore, we can estimate the SD as:SD^2 =AHPR^2 −EGM^2Returning to our 2:1 coin-toss game, we have a mathematical expecta-
tion of $.50, and an optimalfof betting $1 for every $4 in equity, which yields
a geometric mean of 1.06066. We can use Equation (5.03) to determine our
arithmetic average HPR:
AHPR= 1 +(ME/f$ ) (5.03)where: AHPR=The arithmetic average HPR.
ME=The arithmetic mathematical expectation in units.
f$=The biggest loss/−f.
f=The optimalf(0 to 1).