128 THE HANDBOOK OF PORTFOLIO MATHEMATICS
profits are always reinvested and fractional contracts can be purchased.
In effect, this is the mathematical expectation when you are trading on
a fixed fractional basis.This figure shows you what effect there is from
losers occurring when you have many contracts on and winners occur-
ring when you have fewer contracts on. In effect, this approximates how
a system would have fared per contract per trade doing fixed fraction.
(Actually, the geometric average trade is your mathematical expectation
in dollars per contract per trade. The geometric mean minus 1 is your
percent mathematical expectation per trade—e.g., a geometric mean of,
say, 1.025 represents a mathematical expectation of 2.5% per trade, irre-
spective of size.) So many traders look simply at the average trade of a
market system to see if it is high enough to justify trading the system. How-
ever, in making their decision, they should be looking at the geometric
average trade (which is never greater than the average trade) as well as at
the PRR.Geo. Avg. Trade=G*(biggest loss/−f) (4.09)where: G=Geometric mean−1.
f=Optimal fixed fraction.
(And, of course, our biggest loss is always a negative number.)For example, suppose a system has a geometric mean of 1.017238, the
biggest loss is $8,000, and the optimalfis .31. Our geometric average trade
would equal:Geo. Avg. Trade=(1. 017238 −1)*(−8,000/−.31)=. (^017238) *25, 806. 45
=$444. 85
A Simpler Method for Finding the Optimalf CONTENTS ix
The Virtues of the Optimalf
There are numerous ways to arrive at the optimal value forf. The tech-
nique for finding the optimal fthat has been presented thus far in this
chapter is perhaps the most mathematically logical. That is to say, it is ob-
vious upon inspection that this technique will yield the optimalf. It makes
more intuitive sense when you can see the HPRs laid out than does the
next and somewhat easier method. So here is another way for calculating