120 THE HANDBOOK OF PORTFOLIO MATHEMATICS
The Kelly criterion states that we should bet that fixed fraction of our stake
(f) which maximizes the growth function G(f):
G(f)=P*ln(1+B*f)+(1−P)*ln(1−f) (4.02)where: f=The optimal fixed fraction.
P=The probability of a winning bet/trade.
B=The ratio of amount won on a winning bet to amount
lost on a losing bet.
ln( )=The natural logarithm function to the base
e=2.71828....As it turns out, for an event with two possible outcomes, this optimal
fcan be found quite easily with the Kelly formulas.Kelly
Beginning around the late 1940s, Bell System engineers were working on
the problem of data transmission over long distance lines. The problem fac-
ing them was that the lines were subject to seemingly random, unavoidable
“noise” that would interfere with the transmission. Some rather ingenious
solutions were proposed by engineers at Bell Labs. Oddly enough, there are
great similarities between this data communications problem and the prob-
lem of geometric growth as it pertains to gambling money management (as
both problems are the product of an environment of favorable uncertainty).
The Kelly formula is one of the outgrowths of these solutions.
The first equation here is:
f= (^2) P− 1 (4.03)
where: f=The optimal fixed fraction.
P=The probability of a winning bet/trade.
This formula will yield the correct answer for optimal fprovided the
sizes of wins and losses are the same. As an example, consider the following
stream of bets:
−1,+1,+1,−1,−1,+1,+1,+1,+1,− 1
There are 10 bets, 6 winners, hence:
f=(. (^6) 2)− 1
= 1. 2 − 1
=. 2