Ralph Vince - Portfolio Mathematics

326 THE HANDBOOK OF PORTFOLIO MATHEMATICS

Prob=The sum of probabilities of allmof the HPRs for a given

set offvalues. Probkis the sum of the values in

brackets{}in the numerator, for allmvalues of a

given set offvalues.

fi=Thefvalue being used for componenti.fimust be greater

than 0, and can be infinitely high (i.e., it can be greater

than 1.0).

PLk,j=The outcome profit or loss for theith component

(i.e., scenario spectrum or market system) associated

with thekth combination of scenarios.

BLi=The worst outcome of scenario spectrum

(market system)i.

Thus, Probkin the equation is equal to Equation (9.03)
Equation (10.01) simply takes the coefficient of each HPRtimesits
probability and sums these. The resultant sum is then divided by the sum
of the probabilities.
The variance in the HPRs for a given set of multiple simultaneous sce-
nario spectrums being traded at givenfvalues can be determined by first
taking theraw coefficientof the HPRs, the rawcoef:

rawcoefk= 1 +

∑n

i= 1

(

fi∗

(

−PLk,i

BLi

))

(10.02)

Then, these raw coefficients are averaged for all values ofkbetween 1 and
m, to obtain arimeanrawcoef:

arimeanrawcoef=

(m

∑

k= 1

rawcoefk

)

m

(10.03)

Now, the varianceVcan be determined as:

V=

∑m

k= 1

(rawcoefk−arimeanrawcoef)^2 ∗Probk

∑m

k= 1

Probk

(10.04)

Where again, Probkis determined by Equation (9.03).
If we know what the AHPR is, and the variance at a givenflevel (say
the optimalflevel for argument’s sake), we can convert these numbers into
what they would be trading at a level of dilution we’ll call FRAC. And, since
we are able to figure out the two legs of the right triangle, we can also
figure the estimated geometric mean HPR at the diluted level. The formulas