312 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Thus, Equation (9.01) gives us:
G(.21,.21,.21)=
(
∏mk= 1HPRk)
(
1/∑m
k= 1
Probk)G(.21,.21,.21)=(. (^883131) (^1) . (^1) 1. (^062976) (^1) 1. (^062976) 1. 062976
*^1 .107296)
(1/(. 125 +. 125 +. 125 +. 125 +. 125 +. 125 +. 125 +.125))
G(.21,.21,.21)= 1. 174516 (1/1)
G(.21,.21,.21)= 1. 174516
This is thefvalue combination that results in the greatestGfor these
scenario spectrums. Since this is a very simplified case, that is, all scenario
spectrums were identical, and all had correlation of zero between them,
we ended up with the samefvalue for all three scenario spectrums of .21.
Usually, this will not be the case, and you will have a differentfvalue for
each scenario spectrum.
Now that we know the optimalfvalues for each scenario spectrum,
we can determine how much those decimalfvalues are, in currency, by
dividing the largest loss scenario in each of the spectrums by the negative
optimalffor each of those spectrums. For example, for the first scenario
spectrum, Coin 1, we had a largest loss of−1. Dividing−1 by the negative
optimalf,−.21, we obtain 4.761904762 asf$ for Coin 1.
To summarize the procedure, then:
1.Start with anfvalue set forf 1 ...fnwherenis the number of components
in the portfolio, that is, market systems or scenario spectrums. This initial
fvalue set is given by the optimization technique selected.
2.Go through the combinations of scenario setskfrom 1 tom, odomet-
rically, and calculate an HPR for eachk, multiplying them all together.
While doing so, keep a running sum of the exponents of the HPRs.
3.Whenkequalsm, and you have computed the last HPR, the final product
must be taken to the power of 1, divided by the sum of the exponents
(probabilities) of all the HPRs, to getG, the geometric mean HPR.
4.This geometric mean HPR gives us onealtitudeinn+1 space. We wish
to find the peak in this space, so we must now select a new set offvalues
to test to help us find the peak. This is the mathematical optimization
process.
Mathematical Optimization versus Root Finding
ROOT FINDING
Equations have a left and a right side. Subtracting the two makes the equa-
tion equal to 0. Inroot finding, you want to know what values of the