Measures of Contribution for Portfolio Risk

4 CHAPTER 1. INTRODUCTION
“the probability that an event A occurs” is indicated with P (A), where A is
an element of F , set of all the events. P denotes the probability measure.2
Let us consider the following example: an investor who bought stocks of
a particular company; an insurance firm that has sold an insurance policy;
a person who decides to convert a fixed-rate mortgage into a floating one.
To model these assumptions a mathematician should now define a one
period risky position (or simply risk) X that is function of the probability
space (Ω,F , P ); this function is called a random variable. We leave for
the moment the set of all the possible values X not specified. Most part of
modeling of the risky position is related to its distribution function FX(x) =
P (X ≤ x), the probability that for the end of the period in consideration
the value of the risk X is less or equal to a given number x. Most risky
opportunities are indicated by a random vector (X1, . . . , Xd), defined by X;
the time can be introduced in a way to reach the notion of random process
(or stochastic), often indicated with (Xt).
1.2 Financial Risk
In this dissertation we discuss financial risk (even if it is applied in in-
surance too). Most of the dissatisfactions linked to financial investments are
due to a limited understanding of financial risk. Normal people, as a mat-
ter of fact, refer to “risk” as something more general instead professionals
describe it as an element very specific and with different typologies. In fi-
nance risk expresses in general the probability to get a different return than
expected. If one invests 100 euro in a risky opportunity wishing to obtain a
return for example of 6%, the risk is to get a lower return or a negative one.
2
(Ω,F .P ) is called measure space. (Ω,F) is called measurable space. The set Ω is often
called sample space and F is called sigma-algebra (or σ-algebra or field of events). Subsets
of Ω that belong to F are known as measurable sets of Ω with respect to F , or more briefly
F-measurable sets.