122 THE HANDBOOK OF PORTFOLIO MATHEMATICS
when wins are not all for the same amount and/or losses are not all for the
same amount is a mistake. It will not yield the optimalf.
Notice that the numerator in this formula equals the mathematical ex-
pectation for an event with two possible outcomes as defined in Chapter 1.
Therefore, we can say that as long as all wins are for the same amount and
all losses are for the same amount (regardless of whether the amount that
can be won equals the amount that can be lost), the optimalfis:
f=Mathematical Expectation/B (4.05)where: f=The optimal fixed fraction.
B=The ratio of amount won on a winning bet to amount
lost on a losing bet.Finding the Optimalfby the Geometric Mean
In trading we can count on our wins being for various amounts and our
losses being for various amounts. Therefore, the Kelly formula cannot give
us the correct optimalf. How then can we find the optimalfto tell us how
many contracts to have on and have it be mathematically correct?
As you will see later in this chapter, trading the correct quantities of
contracts/shares is a far bigger problem than was previously thought. Quan-
tity can mean the difference between winning and losing. All systems expe-
rience losing trades. All systems experience drawdown. These are givens,
facts of life. Yet if you can always have the right amount of contracts on (i.e.,
the mathematically correct amount), then there is consolation in the losses.
Now here is the solution. To begin with, we must amend our formula
for finding HPRs to incorporatef.
HPR= 1 +f*(−trade/biggest loss) (4.06)And again, TWR is simply the geometric product of the HPRs and work-
ing from (3.03), geometric mean is simply the Nth root of the TWR.
TWR=
∏N
i= 1(1+f*(−tradei/biggest loss)) (4.07)Geo. Mean=(N
∏
i= 1(1+f*(−tradei/biggest loss)))l/N
(4.08)The geometric mean can also be calculated here by the procedure for
finding the estimated geometric mean by using the HPRs as formulated