158 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 4.12 A function with two local extremes
previous parabola’s abscissa. Convergence is determined when the absolute
value of the difference between two abscissas is less than a prescribed
amount called the tolerance, or TOL for short. This amount should be chosen
with respect to how accurate you want yourfto be. Generally, I use a value
of .005 for TOL. This gives the same accuracy in searching for the optimal
fas the brute force techniques described earlier.
We can start with two of the three coordinate points as (0, 0), (1.0, 0).
The third coordinate point must be a point that lies on the actualfcurve
itself. Let us choose the X value here to be 1 – TOL, or .995. To make
sure that the coordinate lies on the f curve, we determine our Y value
by finding what the TWR is atf =.995. Assume we are looking for the
optimalf for the four-trade example−1,−3, 3, 5. For these four trades
the TWR atf=.995 is .017722. Now we have the three coordinates: (0, 0),
(.995, .017722), (1.0, 0). We plug them into the above described equation
to find the abscissa of a parabola that contains these three points, and our
result is .5.
Now we compute the TWR corresponding to this abscissa; this equals
1.145833. Since the X value here now (.5) is to the left of the value for X2 pre-
viously (.995), we move our three points over to the left, and compute a new
abscissa to the parabola that contains the three points (0, 0), (.5, 1.145833),
(.995, .017722).