4Monetary measures of risk

In this chapter, we discuss the problem of quantifying the risk of a financial position. As in Chapter 2, such a position will be described by the corresponding payoff profile, that is, by a real-valued function X on some set of possible scenarios. In a probabilistic model, specified by a probability measure on scenarios, we could focus on the resulting distribution of X and try to measure the risk in terms of moments or quantiles. Note that a classical measure of risk such as the variance does not capture a basic asymmetry in the financial interpretation of X: Here it is the downside risk that matters. This asymmetry is taken into account by measures such as Value at Risk which are based on quantiles for the lower tail of the distribution, see Section 4.4 below. Value at Risk, however, fails to satisfy some natural consistency requirements. Such observations have motivated the systematic investigation of measures of risk that satisfy certain basic axioms.

From the point of view of an investor, we could simply turn around the discussion of Chapter 2 and measure the risk of a position X in terms of the loss functional

Here U is a utility functional representing a given preference relation on financial positions. Assuming robust preferences, we are led to the notion of robust shortfall risk defined by

where (x) := −u(−x) is a convex increasing loss function and Q is a class of probability measures. The results of Section 2.5 show how such loss functionals can be characterized in terms of convexity and monotonicity properties of the preference relation.

From the point of view of a supervising agency, however, a specific monetary purpose comes into play. In this perspective a risk measure is viewed as a capital requirement: We are looking for the minimal amount of capital which, if added to the position and invested in a risk-free manner, makes the position acceptable. For example, a financial position could be viewed as being acceptable if the robust shortfall risk of X does not exceed a given bound. This monetary interpretation is captured by an additional axiom of cash invariance. Together with convexity and monotonicity, it singles out the class of convex risk measures. These measures can be represented in the form

where α is a penalty function defined on probability measures on Ω. Under the additional condition of positive homogeneity, we obtain the class of coherent risk measures. Herewe are back to the situation in Proposition 2.84, and the representation takes the form

where Q is some class of probability measures on Ω.

The axiomatic approach to such monetary risk measures was initiated by P. Artzner, F. Delbaen, J. Eber, and D. Heath [12], and it will be developed in the first three sections. Our presentation will be self-contained and can thus be read without relying on the results of the preceding chapters. In Section 4.4 we discuss some coherent risk measures related to Value at Risk. These risk measures only involve the distribution of a position under a given probability measure. In Section 4.5 we characterize the class of convex risk measures which share this property of law-invariance. Section 4.6 discusses the role of concave distortions, and in Section 4.7 the resulting risk measures are characterized by a property of comonotonicity. In Section 4.8 we discuss risk measures which arise naturally in the context of a financial market model, if acceptability is defined in terms of hedging opportunities. In Section 4.9 we analyze the structure of utility-based risk measures which are induced by our notion of robust shortfall risk.

4.1Risk measures and their acceptance sets

Let Ω be a fixed set of scenarios. A financial position is described by a mapping where X(ω) is the discounted net worth of the position at the end of the trading period if the scenario ω ∈ Ω is realized. The discounted networth corresponds to the profits and losses of the position and is also called the P&L. Our aim is to quantify the risk of X by some number ρ(X), where X belongs to a given class X of financial positions. Throughout this section, X will be a linear space of bounded functions containing the constants. We do not assume that a probability measure is given on Ω.

Definition 4.1. A mapping ρ : X → ℝ is called a monetary risk measure if it satisfies the following conditions for all X, Y ∈ X .

–Monotonicity: If X ≤ Y, then ρ(X) ≥ ρ(Y).

–Cash invariance: If m ∈ ℝ, then ρ(X + m) = ρ(X) − m.

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The financial meaning of monotonicity is clear: The downside risk of a position is reduced if the payoff profile is increased. Cash invariance is also called translation invariance or translation property. It is motivated by the interpretation of ρ(X) as a capital requirement, i.e., ρ(X) is the amount which should be added to the position X in order to make it acceptable from the point of view of a supervising agency. Thus, if the amount m is added to the position and invested in a risk-free manner, the capital requirement is reduced by the same amount. In particular, cash invariance implies

and

For most purposes it would be no loss of generality to assume that a given monetary risk measure satisfies the condition of

–Normalization: ρ(0) = 0.

For a normalized risk measure, cash invariance is equivalent to cash additivity, i.e., to ρ(X + m) = ρ(X)+ ρ(m). In some situations, however, it will be convenient not to insist on normalization.

Remark 4.2. We are using the convention that X describes the worth of a financial position after discounting. For instance, the discounting factor can be chosen as 1/(1 + r) where r is the return of a risk-free investment. Instead of measuring the risk of the discounted position X, one could consider directly the nominal worth

The corresponding risk measure () := ρ(X) is again monotone. Cash invariance is replaced by the following property:

i.e., the risk is reduced by mif an additional amount mis invested in a risk-free manner. Conversely, any : X → ℝ which is monotone and satisfies (4.2) defines a monetary measure of risk via ρ(X) := (1 + r)X.

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Lemma 4.3. Any monetary risk measure ρ is Lipschitz continuous with respect to the supremum norm · :

Proof. Clearly, by monotonicity and cash invariance. Reversing the roles of X and Y yields the assertion.

From now on we concentrate on monetary risk measures which have an additional convexity property.

Definition 4.4. A monetary risk measure ρ : X → ℝ is called a convex risk measure if it satisfies

–Convexity:

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Consider the collection of possible future outcomes that can be generated with the resources available to an investor: One investment strategy leads to X, while a second strategy leads to Y. If one diversifies, spending only the fraction λ of the resources on the first possibility and using the remaining part for the second alternative, one obtains λX + (1 − λ)Y. Thus, the axiom of convexity gives a precise meaning to the idea that diversification should not increase the risk. This idea becomes even clearer when we note that, for a monetary risk measure, convexity is in fact equivalent to the weaker requirement of

–Quasi Convexity:

Exercise 4.1.1. Prove that a monetary risk measure is quasi-convex if and only if it is convex.

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Exercise 4.1.2. Show that if ρ is convex and normalized, then

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Definition 4.5. A convex risk measure ρ is called a coherent risk measure if it satisfies

–Positive Homogeneity: If λ ≥ 0, then ρ(λX) = λρ(X).

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When a monetary risk measure ρ is positively homogeneous, then it is normalized, i.e., ρ(0) = 0. Under the assumption of positive homogeneity, convexity is equivalent to

–Subadditivity:

This property allows to decentralize the task of managing the risk arising from a collection of different positions: If separate risk limits are given to different “desks”, then the risk of the aggregate position is bounded by the sum of the individual risk limits.

In many situations, however, risk may grow in a nonlinear way as the size of the position increases. For this reason we will not insist on positive homogeneity. Instead, our focus will be mainly on convex measures of risk.

Exercise 4.1.3. Let ρ be a normalized monetary risk measure on X . Show that any two of the following properties imply the remaining third.

–Convexity.

–Positive homogeneity.

–Subadditivity.

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A monetary risk measure ρ induces the class

of positions which are acceptable in the sense that they do not require additional capital. The class Aρ will be called the acceptance set of ρ. The following two propositions summarize the relations between monetary risk measures and their acceptance sets.

Proposition 4.6. Suppose that ρ is a monetary risk measure with acceptance set A := Aρ.

(a) A is nonempty, closed in X with respect to the supremum norm · , and satisfies the following two conditions:

(b) ρ can be recovered from A :

(c) ρ is a convex risk measure if and only if A is convex.

(d) ρ is positively homogeneous if and only if A is a cone. In particular, ρ is coherent if and only if A is a convex cone.

Proof. (a): Properties (4.3) and (4.4) are straightforward, and closedness follows from Lemma 4.3.

(b): Cash invariance implies that for X ∈ X ,

(c): A is clearly convex if ρ is a convex measure of risk. The converse will follow from Proposition 4.7 together with (4.7).

(d): Clearly, positive homogeneity of ρ implies that A is a cone. The converse follows as in (c).

Conversely, one can take a given class A ⊂ X of acceptable positions as the primary object. For a position X ∈ X , we can then define the capital requirement as the minimal amount m for which m + X becomes acceptable:

Note that, with this notation, (4.5) takes the form

Proposition 4.7. Assume that A is a nonempty subset of X which satisfies (4.3) and (4.4). Then the functional ρA has the following properties:

(a) ρA is a monetary risk measure.

(b) If A is a convex set, then ρA is a convex risk measure.

(c) If A is a cone, then ρA is positively homogeneous. In particular, ρA is a coherent risk measure if A is a convex cone.

(d) AρA is equal to the closure of A with respect to the supremum norm · . In particular, AρA = A holds if and only if A is · -closed.

Proof. (a): It is straightforward to verify that ρA satisfies cash invariance and monotonicity. We show next that ρA takes only finite values. To this end, fix some Y in the nonempty set A . For X ∈ X given, there exists a finite number m with m + X > Y, because X and Y are both bounded. Then

and hence ρA (X) ≤ m < ∞. Note that (4.3) is equivalent to ρA (0) > −∞. To show that ρA (X) > −∞ for arbitrary X ∈ X , we take mʹ such that X + mʹ ≤ 0 and conclude by monotonicity and cash invariance that ρA (X) ≥ ρA (0) + mʹ > −∞.

(b): Suppose that X1, X2 ∈ X and that m1, m2 ∈ ℝ are such that mi + Xi ∈ A . If λ ∈ [0, 1], then the convexity of A implies that λ(m1 + X1) + (1 − λ)(m2 + X2) ∈ A . Thus, by the cash invariance of ρA ,

and the convexity of ρA follows.

(c): As in the proof of convexity, we obtain that ρA (λX) ≤ λρA (X) for λ ≥ 0 if A is a cone. To prove the converse inequality, let m < ρA (X). Then m + X / ∈ A and hence λm + λX / ∈ A for λ > 0. Thus λm < ρA (λX), and (c) follows.

(d): Let A denote the closure of A with respect to ·.We first show A ⊂ AρA . If X ∈ A , there exists a sequence (Xn) in A such that Xn − X → 0. It follows from Lemma 4.3 that ρA (Xn) → ρA (X), and so ρA (X) ≤ 0. Hence X ∈ AρA and in turn A ⊂ AρA .

For the proof of the converse inclusion, take X ∈ AρA . Then

and so there exists a sequence (mn) such that mn ↓ ρA (X) and mn + X ∈ A . By (4.4), we also have Xn := mn + X − ρA (X) ∈ A . Since Xn − X= mn − ρA (X) ↓ 0, we obtain X ∈ A and in turn AρA ⊂ A .

Exercise 4.1.4. Let A be a nonempty subset of X which satisfies (4.3) and (4.4). Show that A is · -closed if and only if A satisfies the following property: if X +m ∈ A for all m > 0 then X ∈ A.

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In the following examples, we take X as the linear space of all bounded measurable functions on some measurable space (Ω,F), and we denote by M1 = M1(Ω,F) the class of all probability measures on (Ω,F).

Example 4.8. Consider the worst-case risk measure ρmax defined by

The value ρmax(X) is the least upper bound for the potential loss which can occur in any scenario. The corresponding acceptance set A is given by the convex cone of all nonnegative functions in X . Thus, ρmax is a coherent risk measure. It is the most conservative measure of risk in the sense that any normalized monetary risk measure ρ on X satisfies

Note that ρmax can be represented in the form

where Q is the class M1 of all probability measures on (Ω,F).

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Example 4.9. Let Q be a set of probability measures on (Ω,F), and consider a mapping γ : Q → ℝ with supQ γ(Q) < ∞,which specifies for each Q ∈ Qsome “floor” γ(Q). Suppose that a position X is acceptable if

The set A of such positions satisfies (4.3) and (4.4), and it is convex. Thus, the associated monetary risk measure ρ = ρA is convex, and it takes the form

Alternatively, we can write

where the penalty function α : M1 → (−∞, ∞] is defined by α(Q) = −γ(Q) for Q ∈ Q and α(Q) = +∞ otherwise. Note that ρ is a coherent risk measure if γ(Q) = 0 for all Q ∈ Q.

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Example 4.10. Consider a utility function u on ℝ, a probability measure Q ∈ M1, and fix some threshold c ∈ ℝ. Let us call a position X acceptable if its certainty equivalent is at least c, i.e., if its expected utility EQ[ u(X) ] is bounded from below by u(c). Clearly, the set

is nonempty, convex, and satisfies (4.3) and (4.4). Thus, ρA is a convex risk measure. As an obvious robust extension, we can define acceptability in terms of a whole class Q of probability measures on (Ω,F), i.e.,

with constants cQ such that supQ∈Q cQ < ∞. The corresponding risk measures are called utility-based shortfall risk measures. They will be studied in more detail in Section 4.9.

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Example 4.11. Suppose now that we have specified a probabilistic model, i.e., a probability measure P on (Ω,F). In this context, a position X is often considered to be acceptable if the probability of a loss is bounded by a given level λ ∈ (0, 1), i.e., if

The corresponding monetary risk measure V@Rλ, defined by

is called Value at Risk at level λ. Note that it is well defined on the space L0(Ω,F, P) of all random variables which are P-a.s. finite, and that

if X is a Gaussian random variable with variance σ2(X) and Φ−1 denotes the inverse of the distribution function Φ of N (0, 1). Clearly, V@Rλ is positively homogeneous, but in general it is not convex, as shown by Example 4.46 below. In Section 4.4, Value at Risk will be discussed in detail. In particular, we will study some closely related coherent and convex risk measures.

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Exercise 4.1.5. Compute V@Rλ(−X) if X is

(a) uniform,

(b) log-normally distributed, i.e., X = eσZ+μ with Z ∼ N(0, 1) and μ, σ ∈ ℝ.

(c) an indicator function, i.e., X = A for some event A ∈ F.

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Example 4.12. As in Example 4.11, we fix a probability measure P on (Ω,F). For an asset with payoff∈ L2 = L2(Ω,F, P), price π(), and variance σ2() ≠ 0, the Sharpe ratio is defined as

where X := (1 + r)−1 − π() is the corresponding discounted net worth. Suppose that we find the position X acceptable if the Sharpe ratio is bounded from below by some constant c > 0. The resulting functional ρc on L2 defined by (4.6) for the class

is given by

It is sometimes called the mean-standard deviation risk measure. It is cash invariant and positively homogeneous, and it is convex since σ( · ) is a convex functional on L2. But ρc is not a monetary risk measure, because it is not monotone. Indeed, if X = eZ and Z is a random variable with normal distribution N (0, σ2), then X ≥ 0, and monotonicity would imply ρc(X) ≤ ρc(0) = 0. But

becomes positive for sufficiently large σ. Note, however, that (4.10) shows that ρc(X) coincides with V@Rλ(X) if X is Gaussian and if c = Φ−1(1 − λ) with 0 < λ ≤ 1/2. Thus, both ρc and V@Rλ have all the properties of a coherent risk measure if restricted to a Gaussian subspace of L2, i.e, any linear space consisting of normally distributed random variables. But neither ρc nor V@Rλ can be coherent on X , and hence on the full space L2, since the existence of normal random variables on (Ω,F, P) implies that X will also contain random variables as considered in Example 4.46.

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Example 4.13. Let u : ℝ → ℝ be a strictly increasing continuous function. For X ∈ X := L∞(Ω,F, P) we consider the certainty equivalent of the law of X under P as a functional of X by setting

Then ρ(X) := −c(X) is monotone,

If ρ is also cash invariant, and hence a monetary risk measure, then Proposition 2.46 shows that u is either linear or a function with exponential form: u(x) = a + beαx or u(x) = a − be−αx for constants a ∈ ℝ and b, α > 0. In the linear case we have

In the first exponential case ρ is of the form

In the second exponential case ρ is given by

and called the entropic risk measure for reasons that will become clear in Example 4.34. There (and in Exercise 4.1.6 below) we will also see that ρ is a convex risk measure.

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Exercise 4.1.6. Let u : ℝ → ℝ be a strictly increasing continuous function and let ρ be defined as in Example 4.13. Show that ρ is quasi-convex,

if u is concave. Conclude that the entropic risk measure in (4.12) is a convex risk measure.

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Exercise 4.1.7. For any X ∈ L1 define

where is an independent copy of X .Note that Δ(·), often called Gini’s mean difference, satisfies Δ(X) ∈ [0, 2E[ |X| ]], with Δ(X) = 0 if and only if X is P-a.s. constant. In analogy to the mean-standard deviation risk measure in (4.11), consider the functional on L1 defined by

for some constant λ ≥ 0.

(a) Show that ρλ(X) is a coherent risk measure for any Note that a non-constant position X is acceptable if and only if E [ X ] > 0 and the Gini coefficient of X, defined as

satisfies G(X) ≥ (2λ)−1.

(b) Show that for there are nonnegative random variables X such that ρλ(X) > 0.

In particular, ρλ cannot be monotone for

(c) Show that

where F denotes the distribution function of X, and that

whenever X has a continuous distribution.

(d) Use (4.13) to show that for all X ∈ L2 and conclude that ρλ is dominated by the mean-standard deviation risk measure ρc in (4.11) if As in Example 4.12, we denote here by σ(X) the square root of the variance of X.

(e) Compute Δ(X) and G(X) if X has a Pareto distribution with shape parameter α > 1 and minimum 1, that is, log X is exponentially distributed with parameter α.

(f) Consider a log-normally distributed random variable X = exp(m + σZ), where Z has a standard normal law N (0, 1). Show that G(X) = erf(σ/2), where erf(·) is the Gaussian error function, that is,

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Example 4.14. Let c : F → [0, 1] be any set function which is normalized and monotone in the sense that c(∅) = 0, c(Ω) = 1, and c(A) ≤ c(B) if A ⊂ B. For instance, c can be given by c(A) := ψ(P[ A ]) for some probability measure P and an increasing function ψ : [0,1] → [0, 1] such that ψ(0) = 0 and ψ(1) = 1. The Choquet integral of a bounded measurable function X ≥ 0 with respect to c is defined as

If c is a probability measure, Fubini’s theorem implies that ∫ X dc coincides with the usual integral. In the general case, the Choquet integral is a nonlinear functional of X, but we still have for constants λ, m ≥ 0. If X ∈ X is arbitrary, we take m ∈ ℝ such that X + m ≥ 0 and get

The right-hand side is independent of m ≥ − inf X, and so it makes sense to extend the definition of the Choquet integral by putting

for all X ∈ X . It follows that

for all λ ≥ 0 and m ∈ ℝ. Moreover, we have

Thus, the Choquet integral of the loss,

is a positively homogeneous monetary risk measure on X . In Section 4.7, we will characterize these risk measures in terms of a property called “comonotonicity”. We will also show that ρ is convex, and hence coherent, if and only if c is submodular or strongly subadditive, i.e.,

In this case, ρ admits the representation

where Qc is the core of c, defined as the class of all finitely additive and normalized set functions Q : F → [0, 1] such that Q[ A ]≤ c(A) for all A ∈ F; see Theorem 4.94.

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Exercise 4.1.8. Let P be a probability measure on (Ω,F) and fix n ∈ ℕ. For X ∈ X let X1, . . . , X n be independent copies of X and set

The functional ρ : X → ℝ is sometimes called MINVAR. Show that ρ is a coherent risk measure on X . Show next that ρ fits into the framework of Example 4.14. More precisely, show that ρ can be represented as a Choquet integral,

where the set function c is of the form c(A) = ψ(P[A]) for a concave increasing function ψ : [0,1] → [0, 1] such that ψ(0) = 0 and ψ(1) = 1.

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In the next two sections, we are going to show how representations of the form (4.8), (4.15), (4.9), or (4.14) for coherent or convex risk measures arise in a systematic manner.