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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 Ω within a probability space. This concept is inherently an objective matter, tied to the nature of the world.

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 X. In this context, uncertain outcomes are modeled as random variables, meaning we work with a vector space X of random variables (primarily bounded for mathematical convenience).

Definition: Preference Order and Numerical Representation

A preference order on X is a binary relation xy indicating that choice y is preferred to choice x.
We assume this relation satisfies the following normative properties:

  • Transitivity: xy and yz imply xz;
  • Completeness: For any two possible choices x and y, either xy or yx.

A function U:XR is called a numerical representation (or utility) of a preference order if:

xyif and only ifU(x)U(y)

Preference orders are a generic way to represent subjective views on outcomes. The first property, transitivity, ensures consistency: if y is preferred to x, and z is preferred to y, then z must also be preferred to x. This property appears quite intuitive. The second property, completeness, requires that, for any two elements, one must always express a preference between them. This is a strong assumption, as it mandates the ability to decide between any two elements in a potentially infinite set X.

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 R, providing a quantitative measure of preferences.

Note

Note first that if we have a numerical representation U for a preference order , it is not unique. Any strictly increasing function ϕ:RR defines another numerical representation U~=ϕU. Indeed, xy is equivalent to U(x)U(y), which is equivalent to ϕ(U(x))=U~(x)U~(y)=ϕ(U(y)).

Second, starting directly with a function U:XR, it defines a preference order by:

xyU(x)U(y)

As an exercise, show that , so defined through a function U, is a preference order, satisfying transitivity and completeness.

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 U exists. However, under reasonable assumptions, this is often the case.

Proposition

If the set X is countable

, then any preference order on X admits a numerical representation.

Proof

Without loss of generality, assume X={x1,x2,}. On N, define a probability measure P[{n}]=pn=1/2n, since pn=1. For each xn, define An={k:xkxn}, the set of indices k for elements in X that are less preferred than xn. By definition, xnxm if and only if AnAm. The function:

U:XRxnU(xn)=P[An]={k:xkxn}pk

defines a numerical representation of . Indeed, xnxm if and only if AnAm. Since P assigns a unique probability to each element of N, it follows that AnAm if and only if U(xn)=P[An]P[Am]=U(xm). This completes the proof.

This proposition uses probability measures to define a numerical representation. The argument extends to more general sets, provided you can relate sublevel sets {x~:x~x} using topological arguments like smoothness. If such smoothness is absent, preference orders may exist without a numerical representation.

The Lexicographical Order Does Not Admit a Numerical Representation

Consider X=[0,1]×[0,1] and define the lexicographical order as:

x=(x1,x2)y=(y1,y2)if and only if{either x1<y1,or x1=y1 and x2y2.

This is a preference order (similar to library book ordering). However, since X is uncountable and the preference order lacks smoothness, it can be shown that no numerical representation exists. Try this as an exercise: assume a numerical representation exists and derive a contradiction.

Decision theory typically frames preferences and utilities (where higher values are better). However, when discussing risk, we consider possible loss profiles L—random variables representing losses. For simplicity, we consider complete binary relations , where L1L2 means "L1 is less risky than L2." Thus, loss profiles are ranked by based on perceived risk. However, the basic properties of a preference order on L do not inherently convey insights about risk perception.

Definition: Risk Order and Risk Measures

A preference order on L is called a risk order if the following two additional assumptions are satisfied:

  • Diversification: If L1 is more risky than L2, then any diversified position between the two is less risky than the worse one:

    if L1L2,thenL1λL1+(1λ)L2,for every 0λ1.
  • Monotonicity (worse for sure is more risky): If the losses of L1 are worse than those of L2 in every state of the world, then L1 is more risky than L2:

    if L1(ω)L2(ω) for every ω,then L1L2.

A numerical representation R:LR of a risk order is called a risk measure.

These two additional properties express reasonable and intuitive features of risk perception. They also have implications for the properties of risk measures.

Proposition

Let R be a numerical representation of a preference order on L. Then the following assertions are equivalent:

  • is a risk order;
  • R satisfies:
    • Quasi-convexity: max{R(L1),R(L2)}R(λL1+(1λ)L2) for every 0λ1;
    • Monotonicity: If L1(ω)L2(ω) for every ω, then R(L1)R(L2).
Proof

Let L1 and L2 be two loss profiles. Assume that is a risk order.

For quasi-convexity, due to the completeness of the relation, assume without loss of generality that L1L2, which is equivalent to R(L1)=max{R(L1),R(L2)}.
For any 0λ1, the diversification property implies L1λL1+(1λ)L2, which gives:

max{R(L1),R(L2)}=R(L1)R(λL1+(1λ)L2),

showing quasi-convexity of R.

For monotonicity, assume L1(ω)L2(ω) for every ω. By the monotonicity assumption, L1L2, which implies R(L1)R(L2).

The reverse implication—that a numerical representation being quasi-convex and monotone implies is a risk order—is straightforward to verify.

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 R is called:

  • Cash-Invariant: if R(Lm)=R(L)m for every mR;
  • Positive-Homogeneous: if R(λL)=λR(L) for every λ>0;
  • Law-Invariant: if R(L)=R(L~) whenever the CDFs of L and L~ 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 X. The question is how much liquidity m is needed in the bank account to ensure the overall position (cash plus risky assets) has acceptable risk.
The threshold is that the total risk must be below zero. The total loss profile is Lm, where L=X. By cash-invariance:

0R(Lm)=R(L)mmR(L).

Thus, the minimal liquidity required to make the risky position acceptable is m=R(L). This interpretation ties the risk measure to capital requirements.

Moreover, cash-invariance, combined with quasi-convexity, implies convexity.

Lemma

If R is a cash-invariant risk measure, then R is convex.

Proof

Let R be a cash-invariant risk measure, 0λ1, and L1,L2 be loss profiles. To prove R(λL1+(1λ)L2)λR(L1)+(1λ)R(L2), define m1=R(L1) and m2=R(L2). By cash-invariance and quasi-convexity:

R(λL1+(1λ)L2)λm1(1λ)m2=R(λL1+(1λ)L2λm1(1λ)m2)=R(λ(L1m1)+(1λ)(L2m2))max{R(L1m1),R(L2m2)}=max{0,0}=0.

Positive Homogeneity

Positive homogeneity has a financial interpretation: if L represents the loss exposure of an investment with risk R(L), scaling the investment by λ>0 scales the corresponding risk by λ. While desirable for mathematical reasons, super-linear scaling might be expected in some contexts. Positive homogeneity also implies sub-additivity, ensuring that risk is not exacerbated by combining positions.

Lemma

Let R be a cash-invariant risk measure. If R is positive homogeneous, then:

R(L1+L2)R(L1)+R(L2).
Proof

Let R be a cash-invariant and positive-homogeneous risk measure. By convexity:

R(L1+L2)=R(212(L1+L2))=2R(12L1+12L2)(Positive Homogeneity)2(12R(L1)+12R(L2))(Convexity)=R(L1)+R(L2).