190 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FSD= SD*FRAC
=. (^3750004853) *. 5
=. 1875002427
FGHPR=
√
FAHPR^2 −FSD^2
=
√
1. 0652 −. 18750024272
=
√
1. 12890625 −. 03515634101
=
√
1. 093749909
= 1. 04582499
Thus, for an optimalfof .25, or making one bet for every $4 in equity, we
have values of 1.125, 1.06066, and .3750004853 for the arithmetic average,
geometric average, and SD of HPRs, respectively. Now we have solved for a
fractional (.5)fof .125 or making one bet for every $8 in our stake, yielding
values of 1.0625, 1.04582499, and .1875002427 for the arithmetic average,
geometric average, and SD of HPRs, respectively.
We can now take a look at what happens when we practice a fractional
fstrategy. We have already determined that under fractionalfwe will make
geometrically less money than under optimalf. Further, we have determined
that the drawdowns and variance in returns will be less with fractionalf.
What about time required to reach a specific goal?
We can quantify the expected number of trades required to reach a
specific goal. This is not the same thing as the expected time required to
reach a specific goal, but since our measurement is in trades we will use
the two notions of time and trades elapsed interchangeably here:
T=ln(Goal) / ln(Geometric Mean) (5.07)where: T=The expected number of trades to reach a
specific goal.
Goal=The goal in terms of a multiple on our starting stake,
a TWR.
ln()=The natural logarithm function.or:
T=LogGeometric MeanGoal
(i.e. The ‘Log base Geoemetric Mean’ of the Goal) (5.07a)Returning to our 2:1 coin-toss example, at optimalf we have a geomet-
ric mean of 1.06066, and at halffthis is 1.04582499. Now let’s calculate