146 THE HANDBOOK OF PORTFOLIO MATHEMATICS
game (the action). If you were to bet, say, $100, on the situation described
above, you would be at anffactor way out to the right of the optimalf.You
wouldn’t stand a chance over time—no matter how good a card counter
you were! If you are too far to the right of the optimalf, even if you know
exactly what cards remain in the deck, you are in a losing situation!
Next are four more charts, which, if you still do not see, drive home
the importance of being near the optimalf. These are equity curve charts.
An equity curve is simply the total equity of an account (plotted on the Y
axis) over a period of time or series of trades (the X axis). On these four
charts, we assume an account starts out with 10 units. Then the following
sequence of 21 trades/bets is encountered:
1, 2, 1,−1, 3, 2,−1,−2,−3, 1,−2, 3, 1, 1, 2, 3, 3,−1, 2,−1, 3If you have done the calculations yourself, you will find that the optimalf
is .6, or bet one unit for every five in your stake (since the biggest losing
trade is for three units).
The first equity curve (Figure 4.8) shows this sequence on a constant
one-contract basis. Nice consistency. No roller-coaster drawdowns. No ge-
ometric growth, either.
Next comes an equity curve withfat .3, or bet one unit for every 10
units in your stake (Figure 4.9). Makes a little more than constant contract.
On the third equity curve graph you see the sequence at the optimalf
value of .6, or one bet for every five in your stake (Figure 4.10). Notice how
much more it has made than atf=.3.
The final equity curve shows the sequence of bets at f=.9, or one
bet for every 3^1 / 3 units in your stake (Figure 4.11). Notice how quickly the
equity took off until it hit the drawdown periods (7 through 12). Whenfis
too high, the market systems get beaten down so low during a drawdown
that it takes far longer to come out of them, if ever, than at the optimal
values.
Even at the optimal values, the drawdowns can be quite severe for any
market/system. It is not unusual for a market system trading one contract
under optimalfto see 80 to 95% of its equity erased in the bad drawdowns.
But notice how at the optimal values the equity curve is able to recover in
short order and go on to higher ground. These four charts have all traded
the same sequence of trades, yet look at how using the optimalfaffects
performance, particularly after drawdowns.
Obviously, the greater an account’s capitalization, the more accurately
its traders can stick to optimalf, as the dollars per single contract required
are a smaller percentage of the total equity. For example, suppose optimal
ffor a given market system dictates we trade one contract for every $5,000
in an account. If an account starts out with $10,000 equity, then it can gain