126 THE HANDBOOK OF PORTFOLIO MATHEMATICS
mean to make apples-to-apples comparisons with other market systems,
as well as use thefto know how many contracts to trade for that par-
ticular market system.Once the highestfis found, it can readily be turned into a dollar
amount by dividing the biggest loss by the negative optimalf.For example,
if our biggest loss is $100 and our optimalfis .25, then−$100/−.25=$400.
In other words, we should bet one unit for every $400 we have in our stake.
In the sequence of bets that our coin-toss example would generate,
we find that the optimal f value for the sequence+2,−1 is .25. Since
our biggest loss is $1, 1/.25=$4. In other words, we should bet $1 for
every $4 we have in our stake in order to make the most money out of this
game. To bet a higher number or a lower number will not result in a greater
gain! After 10 bets, for every $4 we started out with in our stake, we will
have $9.
This approach to finding the optimalfwill yield the same result as:
f=((B+1)*P−1)/BYou obtain the same result, of course, when losses are all for the same
amount and wins are all for the same amount. In such a case, either technique
is correct. When both wins and losses are for the same amount, you can
use any of the three methods—the Kelly formula just shown, the fthat
corresponds to the highest TWR, or:
f= (^2) *P− 1
Any of the three methods will give you the same answer when all wins and
losses are for the same amount.
All three methods for finding the optimal fmeet the four desirable
properties of a money-management strategy outlined earlier, given the con-
straints of the two formulas (i.e., all wins being for the same amount and
all losses being for the same amount, or all wins and losses being for the
same amount). Regardless of constraints, the optimalfvia the highest TWR
will always meet the four desirable properties of a money-management
strategy.
If you’re having trouble with some of these concepts, try thinking in
terms of betting in units, not dollars (e.g., one $5 chip or one futures con-
tract or one 100-share unit of stock). The amount of dollars you allocate to
each unit is calculated by figuring your largest loss divided by the negative
optimalf.
The optimalfis a result of the balance between a system’s profit-making
ability (on a constant one-unit basis) and its risk (on a constant one-unit
basis). Notice that margin doesn’t matter, because the size of individual