356 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 10.8 Geometric mean HPR two-to-one coin toss (=TWR atT=1)
For a positive arithmetic expectation, the geometric mean does not
have any points of inflection. However, the TWR, ifT>1, has two points of
inflection, one to the left of the peak and one to the right. The one which
concerns us is, of course, the one to the left of the peak.
The left point of inflection is nonexistent atT=1 and, asTincreases,
it approaches the optimalffrom the left (Figures 10.9 and 10.10). WhenT
is infinite, the point of inflection converges upon optimalf.
Unfortunately, the left point of inflection migrates toward optimalfas
Tapproaches infinity, just like with the GRR. Again, just like EACG, if you
FIGURE 10.9 dTWR/dffor 40 plays (T=40) of the two-to-one coin toss. The peak
to the left and the trough to the right are the points of inflection