150 THE HANDBOOK OF PORTFOLIO MATHEMATICS
size? Conversely, would someone trading 10 contracts on every trade who
was suddenly cut down to trading less than 10 contracts inject enough
capital into the account to margin 10 contracts again on the next trade?
It’s quite unlikely. Any time a trader trading on a constant contract basis
deviates from always trading the same constant contract size, the problem
of what quantities to trade in arises. This is so whether the trader recognizes
this problem or not. As you have seen demonstrated in this chapter, this is a
problem for the trader. Constant contract trading is not the solution, because
you can never experience geometric growth trading constant contract. So,
like it or not, the question of what quantity to take on the next trade is
inevitable for everyone. To simply select an arbitrary quantity is a costly
mistake. Optimalfis factual; it is mathematically correct.
Are there traders out there who aren’t planning on reinvesting their
profits? Unless we’re looking at optimalfvia the highest TWR, we wouldn’t
know a good market system.
If a system is good enough, it is often possible to have a value forfthat
implies applying a dollar amount per contract that is less than the initial
margin. Remember thatfgives us the peak of the curve; to go off to the
right of the peak (take on more contracts) provides no benefit. But the trader
need not use that value forfthat puts him at the peak; he may want to go
to the left of the peak (i.e., apply more dollars in equity to each contract he
puts on). You could, for instance, divide your account into two equal parts
and resolve to keep one part cash and one part as dollars to apply to trading
positions and usefon that half. This in effect would amount to a halffor
fractionalfstrategy.
By now it should be obvious that we have a working range for usable
values off, that range being from zero to the optimal value. The higher
you go within this range, the greater the return (up to but not beyond the
optimalf) and the greater the risk (the greater the expected drawdowns in
size—not, however, in frequency). The lower you go in this range, the less the
risk (again in terms of extent but not frequency of drawdowns), and the less
the potential returns. However, as you move down this range toward zero,
the greater the probability is that an account will be profitable (remember
that a constant-contract-based account has a greater probability of being
profitable than a fractionalfone). Ziemba’sGambling Timesarticles on
Kelly demonstrated thatat smaller profit targets the half Kelly was more
apt to reach these levels before halving than was a full Kelly bet. In other
words, the fractional Kelly (fractionalf) bet is safer—it has less variance
in the final outcome after X bets. This ability to choose a fraction of the
optimalf(choosing a value between 0 and the optimalf) allows you to
have any desired risk/return trade-off that you like.
Referring back to our four equity curve charts where the optimalf=.60,
notice how nice and smooth the halffchart off=.30 is. Halffmakes for