Optimalf 139
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
2.00 1.82 1.67 1.54 1.43 1.33 1.25 1.18 1.11 1.05 1.00
4.00 3.82 3.67 3.54 3.43 3.33 3.25 3.18 3.11 3.05 3.00
2.00 1.72 1.47 1.24 1.03 0.83 0.65 0.48 0.31 0.15 0.00
4.00 3.61 3.23 2.85 2.47 2.08 1.69 1.29 0.87 0.44 0.00
4.002.00 3.411.62 2.841.29 2.291.00 1.780.74 1.300.52 0.880.34 0.190.52 0.090.24 0.020.06 0.000.00
2.00 1.53 1.14 0.80 0.53 0.33 0.18 0.08 0.02 .00 0.00
4.00 3.22 2.50 1.85 1.28 0.81 0.46 0.21 0.07 0.01 0.00
4.002.00 3.041.45 1.00.20 1.490.65 0.920.38 0.200.51 0.090.24 0.030.09 0.010.02 .00.00 0.000.00
2.00 1.37 0.88 0.52 0.28 0.13 0.05 0.01 .00 .00 0.00
4.00 2.88 1.94 1.20 0.66 0.32 0.12 0.03 0.01 .00 0.00
2.00 1.29 0.77 0.42 0.20 0.08 0.02 0.01 .00 .00 0.00
2.004.00 1.222.72 0.681.70 0.340.96 0.140.48 0.050.20 0.010.06 0.01.00 .00.00 .00.00 0.000.00
4.00 2.57 1.50 0.78 0.34 0.12 0.03 0.01 .00 .00 0.00
2.00 1.16 0.60 0.25 0.08 0.03 0.01 .00 .00 .00 0.00
2.004.00 1.092.43 0.531.32 0.220.62 0.070.25 0.020.08 0.020.02 .00.00 .00.00 .00.00 0.000.00
4.00 2.29 1.16 0.50 0.18 0.05 0.01 .00 .00 .00 0.00
2.00 1.03 0.46 0.18 0.05 0.01 .00 .00 .00 .00 0.00
4.00 2.17 1.02 0.40 0.13 0.03 .00 .00 .00 .00 0.00
4.002.00 2.050.98 0.900.41 0.330.14 0.090.04 0.020.01 .00.00 .00.00 .00.00 .00.00 0.000.00
2.00 0.92 0.36 0.11 0.03 .00 .00 .00 .00 .00 0.00
4.00 1.94 0.79 0.26 0.07 0.01 .00 .00 .00 .00 0.00
4.002.00 0.871.83 0.700.32 0.090.21 0.050.02 0.01.00 .00.00 .00.00 .00.00 .00.00 0.000.00
2.00 0.82 0.21 0.07 0.01 .00 .00 .00 .00 .00 0.00
4.00 1.73 0.61 0.17 0.03 .00 .00 .00 .00 .00 0.00
2.00 0.78 0.24 0.06 0.01 .00 .00 .00 .00 .00 0.00
2.004.00 0.741.63 0.220.54 0.050.14 0.010.02 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
4.00 1.54 0.47 0.11 0.02 .00 .00 .00 .00 .00 0.00
2.00 0.69 0.19 0.04 0.01 .00 .00 .00 .00 .00 0.00
2.004.00 0.661.46 0.170.42 0.030.09 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
4.00 1.38 0.37 0.07 0.01 .00 .00 .00 .00 .00 0.00
2.00 0.62 0.19 0.02 .00 .00 .00 .00 .00 .00 0.00
4.00 1.30 0.32 0.06 0.01 .00 .00 .00 .00 .00 0.00
2.00 0.59 0.13 0.02 .00 .00 .00 .00 .00 .00 0.00
1.00 0.32 0.08 0.01 .00 .00 .00 .00 .00 .00 0.00
$2.50 stake has grown to $127,482, thanks to optimal f. Now look what
happens in this extremely favorable situation if you miss the optimalfby
20%. Atfvalues of .6 and .2 you don’t make one-tenth as much as you do
at .4 in this case! This particular situation, a 50/50 bet paying 5 to 1, has a
mathematical expectation of (5.5)+(1(−.5))=2. Yet if you bet using
an fvalue greater than .8, you lose money in this situation. Clearly, the
question of what is the correct quantity to bet or trade has been terribly
underrated.
The graphs bear out a few more interesting points. The first is that
at no other fixed fraction will you make more money than optimalf. In
other words, it does not pay to bet $1 for every $2 in your stake in the
above example of+5,−1. In such a case, you would make less money
than if you bet $1 for every $2.50 in your stake.It does not pay to risk
more than the optimalf—in fact, you pay a price to do so!Notice in
Figure 4.7 that you make less atf=.55 than atf=.5. The second interesting
point to notice is how important the biggest loss is in the calculations.
Traders may be incorrectly inclined to use maximum drawdown rather than
biggest loss.