Measures of Contribution for Portfolio Risk

2 CHAPTER 1. INTRODUCTION
1.1 Nature of Risk
What is risk? A well-known encyclopedia1 defines it in this way:
“Risk is a concept connected to the eventuality to suffer a
damage due to foreseeable circumstances. In medicine, the risk
owner, on which hangs a high statistical evidence of particular
pathological events; nuclear risk, connected to the possibility of
exposure to radiations irradiate from plants, apparatus or nu-
clear materials; in economy, insurance, bank and corporate in-
dustry, the probability to get a suffer pursuant to non predictable
facts.”
When we speak about risk concepts typically we refer to events, deci-
sions, consequences and uncertainty. Mainly the risk of a negative event
(downside risk) is mentioned rather than a positive one, so a potential gain.
For financial risks, the subject of this thesis, we can propose the definition
in [18]: “any event or action that may adversely affect an organization’s
ability to achieve its objectives and execute its strategies” or, alternatively,
“the quantifiable likelihood of loss or less-than-expected returns”. Although
these capture some of the elements of risk, no one is thorough in all the
contexts.
Independently of any context, risk is strongly related with uncertainty
and for this reason to the notion of randomness. Smith et al. [52] and
Metelli [41] use for this intention to recall the distinction between certainty,
risk and uncertainty:
• a choice is taken in condition of certainty when is related to an envi-
ronmental situation perfectly known a priori. The agent acts in way
purely deterministic.
1
Italian Treccani.
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
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.
1.2. FINANCIAL RISK 5
The realization of such an unfavorable event can happen for different
reasons, each of these identify various kind of risk. Most of common investors
know only this generic definition of risk, completely ignoring different sorts
and characters of it that in practice will help to move in a easier way in
financial markets and in personal investments. As a matter of fact in reality
there exist a lot of types of risk, each with own quality and features.
We can start giving a brief summary of different risks that are present
in financial industry.
1. Market risk: Is the risk of a changing of the value of the financial
position due to variations in the value of underlying instruments as
stocks, bonds, currencies, commodities, etc.
2. Credit risk: Is the risk that in a credit operation, the debtor cannot
pay, even for a part, to his reimbursement obligation of capital and
interests.
3. Operational risk: Is the risk of losses for processes aspects, errors due
to human factors, to fraud and employee unfaithfulness, but also due
to technological factor (system failure). Near to these internal causes
there are internal proveniences for operational risks such events that
can afflict damages to the functionality of processes or more general
that can cause losses. In this field we include legal, reputation and
strategic risks.
Limits to these three categories are not well defined nor they build up
an exhaustive list of every possible risk that a financial institution meet.
There are notions of risk that are included in every block as model risk
and liquidity risk: the former is associated to the use of mis-specified mod-
els (inappropriate) for the measurement of risk. We can think about the
Black-Scholes model for pricing an exotic option in circumstances where the
6 CHAPTER 1. INTRODUCTION
assumptions on underlying components are violated (for example hypothe-
sis of returns normally distributed). It can be observed that model risk is
always present at some degree. Liquidity risk comes when an investment
has negotiability difficulties in the sense that it cannot be bought or sold
quickly to prevent or minimize a loss. We can think about liquidity [18] as
“oxygen for a healthy market”. One need liquidity to survive even one does
not feel its present. Its absent, however, is perceived immediately, often
with disastrous consequences.
Concepts, techniques and instruments that will be introduced will be
principally applied to the three groups of risk such as market, credit and
operational. Is good to notice that the only way to control financial risk
is to use an integrated approach that includes every single typology of risk
and their interactions. Although this is a clear aim, current models cannot
fully satisfy yet.
1.3 Measurement and Management
This thesis will be more concentrated on the techniques of risk measure-
ment, a field that is part of the risk management. We will now clarify the
two concepts.
Risk Measurement
We assume to keep a portfolio of d investments with related weights
u1, . . . , ud so a change in the value of portfolio in a specified time horizon
(called P&L, from profit and loss) can be described as X = ∑d
i=1 uiXi,
where Xi is the variation in the value of the i-th investment. To measure
the risk of such portfolio is essentially to determine the distribution function
FX(x) = P (X ≤ x), or functionals that describe this function as the mean,
the variance or the 99-th percentile.