Ralph Vince - Portfolio Mathematics

Optimalf 131

Recall that we determined earlier in this chapter that the Kelly formula did
not apply to this sequence, since wins were not all for the same amount and
losses were not all for the same amount. We also decided to average the
wins and average the losses and take these averages as our values into the
Kelly formula (as many traders mistakenly do). Doing this we arrived at an
fvalue of .16. It was stated that this is an incorrect application of Kelly, that
it would not yield the optimal f. The Kelly formula must be specific to a
single bet. We cannot average our wins and losses from trading and obtain
the true optimalfusing the Kelly formula.
Our highest TWR on this sequence of bets/trades is obtained at .24, or
betting $1 for every $71 in our stake. That is the optimal geometric growth we
can squeeze out of this sequence of bets/trades trading fixed fraction. Let’s
look at the TWRs at different points along 100 loops through this sequence
of bets.
At one loop through, nine bets/trades, the TWR for f=.16 is 1.085;
forf=.24 it is 1.096. This means that for one pass through this sequence
of bets an f=.16 made 99% of what anf =.24 would have made. To
continue:

Passes Total TWR for TWR for Percentage

Through Bets/Trades f=. 24 f=.16 Difference

1 9 1.096 1.085 1%

10 90 2.494 2.261 9.4%

40 360 38.694 26.132 32.5%

100 900 9313.312 3490.761 62.5%

As can be seen, using anfvalue that we mistakenly figured from Kelly
made only 37.5% as much as our optimal f of .24 after 900 bets/trades
(100 cycles through the series of nine outcomes). In other words, our optimal
fof .24 (which is only .08 more than .16) made almost 267% the profit that
f=.16 did after 900 bets!
Let’s go another 11 cycles through this sequence of trades, so we have
a total of 999 trades. Now our TWR forf=.16 is 8563.302 (not even what it
was forf=.24 at 900 trades) and our TWR forf=.24 is 25,451.045. At 999
tradesf=.16 is only 33.6% off=.24, orf=.24 is 297% off=.16! Here
you can see that using the Kelly formula does not yield the true optimalf
for trading.
As can be seen from the above, using the optimalfdoes not appear to
offer much advantage over the short run, but over the long run it becomes