Optimalf 135
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
3.20 3.02 2.87 2.74 2.63 2.53 2.45 2.38 2.31 2.25 2.20
1.60 1.36 1.15 0.96 0.79 0.63 0.49 0.36 0.23 0.11 0.00
2.56 2.25 1.97 1.71 1.45 1.20 0.96 0.72 0.48 0.24 0.00
2.051.28 1.681.01 1.360.79 1.060.60 0.800.44 0.570.30 0.380.19 0.110.22 0.050.10 0.010.03 0.000.00
1.02 0.76 0.54 0.37 0.24 0.14 0.08 0.03 0.01 .00 0.00
1.64 1.26 0.93 0.66 0.44 0.27 0.15 0.07 0.02 .00 0.00
1.310.82 0.940.57 0.640.37 0.410.23 0.240.13 0.130.07 0.060.03 0.010.02 .00.00 .00.00 0.000.00
0.66 0.42 0.26 0.14 0.07 0.03 0.01 .00 .00 .00 0.00
1.05 0.70 0.44 0.26 0.13 0.06 0.02 0.01 .00 .00 0.00
0.52 0.32 0.18 0.09 0.04 0.02 .00 .00 .00 .00 0.00
0.420.84 0.240.52 0.120.30 0.060.16 0.020.07 0.010.03 0.01.00 .00.00 .00.00 .00.00 0.000.00
0.67 0.39 0.21 0.10 0.04 0.01 .00 .00 .00 .00 0.00
0.34 0.18 0.08 0.03 0.01 .00 .00 .00 .00 .00 0.00
0.270.54 0.290.13 0.060.14 0.060.02 0.010.02 0.01.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.43 0.22 0.10 0.04 0.01 .00 .00 .00 .00 .00 0.00
0.21 0.10 0.04 0.01 .00 .00 .00 .00 .00 .00 0.00
0.34 0.16 0.07 0.02 0.01 .00 .00 .00 .00 .00 0.00
0.270.17 0.120.07 0.050.03 0.020.01 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.14 0.05 0.02 0.01 .00 .00 .00 .00 .00 .00 0.00
0.22 0.09 0.03 0.01 .00 .00 .00 .00 .00 .00 0.00
0.180.11 0.070.04 0.020.01 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.09 0.03 0.01 .00 .00 .00 .00 .00 .00 .00 0.00
0.14 0.05 0.02 .00 .00 .00 .00 .00 .00 .00 0.00
0.07 0.02 0.01 .00 .00 .00 .00 .00 .00 .00 0.00
0.060.11 0.020.04 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.09 0.03 0.01 .00 .00 .00 .00 .00 .00 .00 0.00
0.05 0.01 .00 .00 .00 .00 .00 .00 .00 .00 0.00
0.040.07 0.010.02 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.06 0.02 .00 .00 .00 .00 .00 .00 .00 .00 0.00
0.03 0.01 .00 .00 .00 .00 .00 .00 .00 .00 0.00
0.05 0.01 .00 .00 .00 .00 .00 .00 .00 .00 0.00
0.02 0.01 .00 .00 .00 .00 .00 .00 .00 .00 0.00
0.01 .00 .00 .00 .00 .00 .00 .00 .00 .00 0.00
This is not a contention that the fractional bet situation is the same as the
real-life integer-bet situation. Rather, the contention is that for the purposes
of studying these functions we are better off considering the fractional bet,
since it represents the universe of integer bets. The fractional bet situation
is what we can expect in real life in an asymptotic sense (i.e., in the long
run).
This discussion leads to another interesting point that is true in a fixed
fractional betting situation where fractional bets are allowed (think of frac-
tional bets as the average outcome of all integer bets at different initial
bankroll values, since that is what fractional betting represents here). This
point is thatthe TWR is the same regardless of the starting value.In the
examples just cited, if we have an initial stake of one starting value, 20 units,
our TWR (ending stake divided by initial stake) is 1.15. If we have an initial
stake of 400 units, 20 starting values, our TWR is still 1.15.
Figure 4.4 shows thefcurve for 20 sequences of the+1.5,−1.
Refer now to the+2,−1 graph (Figure 4.5). Notice that here the optimal
fis .25 where the TWR is 10.55 after 40 bets (20 sequences of+2,−1). Now
look what happens if you bet only 15% away from the optimal .25f.Atanf