Risk Preferences and Measures
So far, we have introduced potential examples of risk measures and discussed their shortcomings regarding properties deemed sound for risk assessment. Additionally, the selection of these measures might appear arbitrary. In the following, we aim to formalize the concepts of risk and uncertainty.
On the one hand, uncertainty refers to the possibility of multiple outcomes.
In other words, it considers the set
On the other hand, risk represents a subjective or personal perception of uncertainty.
It depends on an individual's viewpoint and can be understood as a cautious response to uncertainty.
To model this consistently within a mathematical framework, we rely on decision theory, which captures preferences among various choices.
The set of possible choices is denoted by
Definition: Preference Order and Numerical Representation
A preference order
We assume this relation satisfies the following normative properties:
- Transitivity:
and imply ; - Completeness: For any two possible choices
and , either or .
A function
Preference orders are a generic way to represent subjective views on outcomes.
The first property, transitivity, ensures consistency: if
These two rational (or normative, as decision theorists would say) assumptions often fail in empirical decision-making. However, they are intended to model fully rational behavior in decision-making processes involving prospective outcomes.
A numerical representation maps the preference ranking into
Note
Note first that if we have a numerical representation
Second, starting directly with a function
As an exercise, show that
Third, even if a numerical function defines a preference order, the reciprocal is not necessarily true.
Additional assumptions are required to ensure that, for a given preference order, a numerical representation
Proposition
If the set
Proof
Without loss of generality, assume
defines a numerical representation of
This proposition uses probability measures to define a numerical representation.
The argument extends to more general sets, provided you can relate sublevel sets
The Lexicographical Order Does Not Admit a Numerical Representation
Consider
This is a preference order (similar to library book ordering). However, since
Decision theory typically frames preferences and utilities (where higher values are better). However, when discussing risk, we consider possible loss profiles
Definition: Risk Order and Risk Measures
A preference order
-
Diversification: If
is more risky than , then any diversified position between the two is less risky than the worse one: -
Monotonicity (worse for sure is more risky): If the losses of
are worse than those of in every state of the world, then is more risky than :
A numerical representation
These two additional properties express reasonable and intuitive features of risk perception. They also have implications for the properties of risk measures.
Proposition
Let
is a risk order; satisfies:- Quasi-convexity:
for every ; - Monotonicity: If
for every , then .
- Quasi-convexity:
Proof
Let
For quasi-convexity, due to the completeness of the relation, assume without loss of generality that
For any
showing quasi-convexity of
For monotonicity, assume
The reverse implication—that a numerical representation being quasi-convex and monotone implies
This proposition shows that neither the mean-variance risk measure nor the Value at Risk represents a risk order. Additional properties may be required of a risk measure, but they might not always align with the underlying risk order.
Definition
A risk measure
- Cash-Invariant: if
for every ; - Positive-Homogeneous: if
for every ; - Law-Invariant: if
whenever the CDFs of and coincide.
Aside from law invariance, the other two properties do not hold if the risk measure is transformed by a strictly increasing function. Nevertheless, they are commonly used and practical.
Cash-Invariance
Cash-invariance is typically required from regulatory or financial perspectives. For instance, consider a financial institution with a risky position
The threshold is that the total risk must be below zero. The total loss profile is
Thus, the minimal liquidity required to make the risky position acceptable is
Moreover, cash-invariance, combined with quasi-convexity, implies convexity.
Lemma
If
Proof
Let
Positive Homogeneity
Positive homogeneity has a financial interpretation: if
Lemma
Let
Proof
Let