306 THE HANDBOOK OF PORTFOLIO MATHEMATICS
The expressionP(ik|jk) is simply the joint probability of the scenario in
theith spectrum and thejth spectrum, corresponding to thekth combination
of scenarios. For example, if we have three coins, each coin represents
a scenario spectrum, represented by the variablen, and each spectrum
contains two scenarios: heads and tails. Thus, there are eight (2 (^2) 2)
possible combinations, represented by the variablem.
In Equation (9.01), the variablekproceeds from 1 tom,inodometric
fashion:
Coin 1 Coin 2 Coin 3 k
ttt1
tth2
tht3
thh4
ht t 5
ht h6
hht 7
hhh8
That is, initially all spectrums are set to their worst (leftmost) values. Then,
the rightmost scenario spectrum cycles through all of its values, after which
the second rightmost scenario spectrum increments to the next (next right)
scenario. You proceed as such again, with the rightmost scenario spectrum
cycling through all of its scenarios, and when the second rightmost scenario
spectrum has cycled through all of its values, the third rightmost scenario
spectrum increments to its next scenario. The process is exactly the same
as an odometer on a car, hence the termodometrically.
So in the expressionP(ik|jk), ifkwere at the value 3 above (i.e.,k=3),
andiwas 1 andjwas 3, we would be looking for the joint probability of coin
1 (coming up tails and coin 3) coming up tails. Equation (9.03) helps us in es-
timating the joint probabilities of particular individual scenarios occurring
innspectrums simultaneously. To put it simply, if I have two scenario spec-
trums, at any givenkI will have only one joint probability to incorporate. If
I have three scenario spectrums, I will have three joint probabilities to in-
corporate (spectrums 1 and 2, spectrums 1 and 3, and spectrums 2 and 3). If
four scenario spectrums, I will have six joint probabilities to compute using
(9.03); if five scenario spectrums, then I have 10 joint probabilities to com-
pute using (9.03). Quite simply, in (9.03) the number of joint probabilities
you will have to incorporate at anyP(i) is:
n!/(nā2)!/ 2 =number of joint probabilities required as input
to (9.03)