Ralph Vince - Portfolio Mathematics

310 THE HANDBOOK OF PORTFOLIO MATHEMATICS

simply take the sum of the joint probabilities of the scenarios in spectrum 1
and 2 times the joint probability of the scenarios in spectrum 1 and 3, times
the joint probabilities of the scenarios in spectrums 2 and 3. Expressed
differently:

ij

12

13

23

If there were four spectrums, we would take the product of all the joint
probabilities as:

ij

12

13

14

23

24

34

Since all of our joint probabilities are .25, we get forx:

x=

(n− 1

∏

i= 1

( n

∏

j=i+ 1

P(ik|jk)

))(1/(n−1))

x=(. (^25) *.25)(1/(n−1))
x=(.015625)^1 /(3−1)
x=(.015625)^1 /^2
x=. 125
Thus,xequals .125, which represents the joint probability of thekth
combination of scenarios. (Note that we are going to determine a joint
probability of three random variables by using joint probabilities of two
random variables!)
Thus, HPRk=.7.^125 =.9563949076 whenk=1. Per Equation (9.02),
we must figure this for all values ofkfrom 1 throughm(in this case,m
equals 8). Doing this, we obtain: