Ralph Vince - Portfolio Mathematics

The Leverage Space Model 305

The Objective Function

The objective function we wish to maximize is the geometric mean HPR,
simply calledG:

G(f 1 ...fn)=

(m

∏

k= 1

HPRk

)

(

1

/∑m

k= 1

Probk

)

(9.01)

where: n=The number of scenario spectrums (market systems or

portfolio components).

m=The possible number of combinations of outcomes

between the various scenario spectrums (market

systems) based on how many scenarios are in each set.

m=The number of scenarios in the first spectrum*the

number of scenarios in the second spectrum*...*the

number of scenarios in thenth spectrum.

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

set offvalues. Probkis the sum of the values in

brackets{}in Equation (9.02) for allmvalues of a given

set offvalues.

HPR=The holding period return of eachk. This is given as:

HPRk=

(

1 +

(∑n

i= 1

(fi*(−PLk,i/BLi))

))Probk

(9.02)

where: n=The number of components (scenario spectrums, i.e.,

market systems) in the portfolio.

fi=Thefvalue being used for componenti.

fimust be>0, and can be infinitely high (i.e., can

be greater than 1.0).

PLk,i=The outcome profit or loss for theith component (i.e.,

scenario spectrum or market system) associated with the

kth combination of scenarios.

BLi=The worst outcome of scenario spectrum (market

system)i.

We can estimate Probkin the earlier equation forGas:

Probk=

(n− 1

∏

i= 1

( n

∏

j=i+ 1

P(ik|jk)

))(1/(n−l))

(9.03)