Ralph Vince - Portfolio Mathematics

Optimalf 121

If the winners and losers were not all the same size, this formula would
not yield the correct answer. Such a case would be our two-to-one coin-toss
example, where all of the winners were for 2 units and all of the losers for
1 unit. For this situation the Kelly formula is:

f=((B+1)*P−1)/B (4.04)

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.

For the two-to-one coin toss:

f=((2+1)*. 5 −1)/ 2

=(3*. 5 −1)/ 2

=(1. 5 −1)/ 2

=. 5 / 2

=. 25

This formula will yield the correct answer for optimal f, provided all
wins are always for the same amount and all losses are always for the same
amount. If this condition is not met, the formula will not yield the correct
answer.
Consider the following sequence of bets/trades:

+9,+18,+7,+1,+10,−5,−3,−17,− 7

Since all wins and all losses are of different amounts, the previous formula
does not apply. However, let’s try it anyway and see what we get.
Since five of the nine events are profitable, P=.555. Now let’s take
averages of the wins and losses to calculate B (here is where so many
traders go wrong). The average win is 9 and the average loss is 8. Therefore,
we will say that B=1.125. Plugging in the values we obtain:

f=((1. 125 +1)*. 555 −1)/ 1. 125

=(2. (^125) *. 555 −1)/ 1. 125
=(1. 179375 −1)/ 1. 125
=. 179375 / 1. 125
=. 159444444
So we sayf=.16. We will see later in this chapter that this is not the
optimalf. The optimalffor this sequence of trades is .24. Applying Kelly