Measures of Contribution for Portfolio Risk

1.1. NATURE OF RISK 3
• a choice is taken in a position of risk when is related to more environ-
mental situations, each of them rules out the others. Event probabil-
ities are known and also the related payoff characteristics; it derives
a probabilistic model. Results coming out from scenario/probability
couples are definable in a probabilistic way: they are different on a
quality level (event likelihood) and quantitative (dimension of the re-
sult). Everything is based on the consistency of the likelihood assigned.
• when we act in state of uncertainty we have a huge variety of environ-
mental cases, but we are not able to know objectively a priori neither
the related likelihood to every situation, nor the dimension of poten-
tial payoffs; we can only propose subjective probabilities and results,
making use of prediction ability and experience.
In both second and third point decision comes through a probabilistic
model that determines the expected value of every alternative, i.e. the
likelihood weighted result; in fact, to make risk the variable to take correctly
management choices, it needs to be measurable
Coming back to the notion of randomness, this is backed out of a clear
and profitable explanation for many centuries; we have to wait the year
1933, when the russian mathematician A. N. Kolmogorov gave an axiomatic
definition of randomness and probability [30].
He developed a research that was by now crystallized in the debate be-
tween who was considering probability as the limit of relative frequencies
(frequentist school) and who was looking for a logical foundation of that.
The mathematizing that gave to this concept was really suitable notwith-
standing the acceptance of one or the other school of thinking and became
the lingua franca in the speeches about risk and uncertainty.
In the language of Kolmogorov a probabilistic model is described by a
triplet (Ω,F , P ). An element ω of Ω represents a realization of an experi-
ment, in economy it is often referred to a state of nature. The statement