Monetary Risk Measures∗

Dr. Robert Philipowski

Recommended book:

H. Follmer, A. Schied, Stochastic Finance (3rd edition). Walter de Gruyter, Berlin, 2011.

Contents

1 Monetary risk measures and their acceptance sets 2

2 Simple examples 5

3 Coherent and convex risk measures 6

4 Robust representation of convex risk measures 104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Existence of a robust representation . . . . . . . . . . . . . . . . . . . . . . 12

5 Convex risk measures on L∞(Ω,F) 14

6 Convex risk measures on L∞(Ω,F ,P) 17

7 Average value at risk 21

8 Law-invariant risk measures and Kusuoaka’s representation theorem 25

∗Notes of the second part of a course on mathematical finance given at the University of Bonn duringthe winter semester 2015/2016

1

1 Monetary risk measures and their acceptance sets

Given a risky financial position we want to quantify its risk in the sense of determining theminimal amount of money that has to be added in order to make the position “acceptable”.

Definition 1.1. Let Ω be a non-empty set (interpreted as the set of possible scenarios).A financial position is a function X : Ω→ R. Here X(ω) is interpreted as the (discounted)net worth of the position in scenario ω at the end of the period under consideration.

In the sequel we fix a set G of “admissible” financial positions. (From the economicpoint of view G should be as large as possible, ideally the space of all functions X : Ω→ R,but for mathematical reasons it can be convenient to impose some restrictions, e.g. mea-surability if Ω is equipped with a σ-field, or boundedness.)

Assumption 1.2. G is a linear space containing the constants.

Given a subset A of G (interpreted as the set of acceptable positions or acceptance set)we will define the risk of a position X ∈ G by

ρA(X) := infm ∈ R : X +m ∈ A

(with the convention inf ∅ = ∞). Conversely, given a “risk measure”, i.e. a functionρ : G → R := R ∪ ±∞, we will define the acceptance set of ρ by

Aρ := X ∈ G : ρ(X) ≤ 0.

Of course not all subsets A of G are reasonable acceptance sets. We therefore define:

Definition 1.3. A subset A of G is called an acceptance set if:

1. A∩constant functions 6= ∅. That is, there exists an amount of money m such thathaving m for sure is acceptable.

2. For all X ∈ G there exists m ∈ R such that X + m /∈ A. That is, no financialposition is “infinitely good”. (This assumption is not evident; if E(X) = ∞, risk-neutral people would pay arbitrary amounts of money to get the position X.)

3. A is monotone in the following sense: For all X ∈ A and all Y ∈ G satisfying Y ≥ X(i.e. Y (ω) ≥ X(ω) for all ω ∈ Ω) we have Y ∈ A. That is, if some position isacceptable then all better positions are acceptable as well.

Definition 1.4. An acceptance set A is normalized if infm ∈ R |m ∈ A = 0.

Hence ifA is normalized positive amounts of money are accepted, and negative amountsare not.

Remark 1.5. Suppose that A is not normalized, and let x0 := infm ∈ R |m ∈ A andA′ := A− x0. Then A′ is normalized, and X ∈ A if and only if X − x0 ∈ A′. Hence onecan restrict one’s attention to normalized acceptance sets without any loss of generality.Nevertheless it is sometimes convenient not to insist on normalization.

Proposition 1.6. Let A ⊂ G be an acceptance set and define the function ρA : G → R by

ρA(X) := infm ∈ R |X +m ∈ A

(that is, ρA(X) is the minimal amount of money which, when added to X, makes Xacceptable). Then ρA has the following properties:

2

1. ρA does not take the value −∞.

2. For each constant m ∈ R, ρA(m) is finite.

3. ρA is monotone, i.e.Y ≥ X ⇒ ρA(Y ) ≤ ρA(X)

4. ρA is cash-invariant, i.e. for all positions X and all constants m we have

ρA(X +m) = ρA(X)−m.

5. A is normalized if and only if ρA(0) = 0.

Proof. These properties are immediate consequences of the definition of an acceptanceset.

Definition 1.7. A function ρ : G → R with the first four of the above properties is calleda monetary risk measure. It is called normalized if ρ(0) = 0.

Remark 1.8. Again we can always pass to a normalized monetary risk measure by re-placing ρ with ρ− ρ(0).

Lemma 1.9. Let ρ : G → R be a monetary risk measure. Then

ρ(X) ≤ ρ(Y ) + ‖X − Y ‖∞

for all X,Y ∈ G.

Proof. We haveY ≤ X + ‖X − Y ‖∞

so that, using monotonicity and cash invariance,

ρ(Y ) ≥ ρ(X + ‖X − Y ‖∞) = ρ(X)− ‖X − Y ‖∞.

Corollary 1.10. Let ρ : G → R be a monetary risk measure.

1. ρ is continuous with respect to the supremum norm on G.

2. It is even 1-Lipschitz in the following sense: Whenever X,Y ∈ G are such that ρ(X)or ρ(Y ) is finite, we have

|ρ(X)− ρ(Y )| ≤ ‖X − Y ‖∞.

Proof. To prove the first statement let (Xn)n∈N be a sequence in G such that‖Xn −X‖∞ → 0 for some X ∈ G. Then on the one hand

ρ(Xn) ≤ ρ(X) + ‖Xn −X‖∞,

so thatlim supn→∞

ρ(Xn) ≤ ρ(X),

and on the other handρ(X) ≤ ρ(Xn) + ‖Xn −X‖∞,

so thatlim infn→∞

ρ(Xn) ≥ ρ(X),

Hencelimn→∞

ρ(Xn) = ρ(X).

The second statement follows immediately from the lemma.

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Proposition 1.11. Let ρ : G → R be a monetary risk measure. Then the set

Aρ := X ∈ G : ρ(X) ≤ 0

is an acceptance set (called the acceptance set of ρ). Moreover, Aρ is closed with respectto the supremum norm on G.

Proof. The first statement follows immediately from the definition of a monetary riskmeasure. The second statement follows from the fact that Aρ is the pre-image of theclosed set R− under the continuous map ρ.

Proposition 1.12.

1. For each monetary risk measure ρ : G → R we have

ρAρ = ρ.

2. For each acceptance set A ⊂ G we have

AρA ⊇ A,

with equality if and only if A is closed with respect to ‖ · ‖∞.

Proof.

1. We have

ρAρ(X) = infm ∈ R |X +m ∈ Aρ= infm ∈ R | ρ(X +m) ≤ 0= infm ∈ R | ρ(X) ≤ m= ρ(X).

2. a) To prove that AρA ⊇ A let X ∈ G \ AρA . Then ρA(X) > 0, i.e.

infm ∈ R |X +m ∈ A > 0.

Consequently X /∈ A.

b) If A is not closed with respect to ‖ · ‖∞, equality cannot hold because AρA isclosed.

c) Suppose that A is closed, and let X ∈ AρA . Then ρA(X) ≤ 0, i.e.

infm ∈ R |X +m ∈ A ≤ 0.

Consequently X + m ∈ A for each m > 0. Since A is closed, it follows thatX ∈ A.

Conclusion. There is a one-to-one correspondence between monetary risk measures andclosed acceptance sets:

Monetary risk measures ↔ Closed acceptance setsρ 7→ Aρ = X ∈ G | ρ(X) ≤ 0

ρA(X) = infm ∈ R |X +m ∈ A ←[ A

4

2 Simple examples

Which monetary risk measure (or, equivalently, which closed acceptance set) should oneuse in practice?

Example 2.1 (Worst case risk measure). Extremely pessimistic people would choose theworst case risk measure defined by

ρ(X) := − inf X.

This is indeed a normalized monetary risk measure. Its acceptance set is

Aρ = X ∈ G |X(ω) ≥ 0 for all ω ∈ Ω.

For practical purposes, however, the worst case risk measure is too pessimistic.

Example 2.2 (Value at risk). Idea: Very unlikely events should be neglected. To makethis idea precise we assume that Ω is equipped with a σ-field F and a probability measure Pand that all elements of G are measurable with respect to F . Given a number α ∈ [0, 1)(e.g. α = 0.01) we regard a financial position as acceptable if the probability to obtain anegative value is ≤ α. That is, we choose

A := X ∈ G |P[X < 0] ≤ α.

A is indeed a normalized acceptance set. (Exercise: Is A closed?) The correspondingmonetary risk measure is called value at risk at level α and denoted VaRα. We have

VaRα(X) = infm ∈ R : X +m ∈ A= infm ∈ R : P[X < −m] ≤ α.

Question. Is value at risk a good risk measure?

• It is easy to understand.

• It is easy to calculate as soon as P is specified.

For these reasons value at risk has been the standard so far. However, it can penalizediversification: Consider two risky financial positions X1 and X2. Diversification meanstaking a convex combination of them, i.e.

X = λX1 + (1− λ)X2

for some λ ∈ [0, 1]. Diversification should not increase the risk, i.e. the risk of X should beat most as high as the maximum of the risks of X1 and X2. Hence a reasonable monetaryrisk measure ρ should satisfy

ρ(λX1 + (1− λ)X2) ≤ max(ρ(X1), ρ(X2))

for all X1, X2 ∈ G and all λ ∈ [0, 1]. In terms of acceptability, if two positions X1 and X2

are acceptable, then each convex combination of them should be acceptable as well. Inother words, the acceptance set should be convex.

Does value at risk have this property?

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Example 2.3. Let α = 0.01, and let X1 and X2 be two independent random variableswith

P[Xi = 1, 000] = 0.992, P[Xi = −10, 000] = 0.008.

(Think of Xi as investing 10,000 e in a bond with interest rate 10% and default probabil-ity 0.8%.) We have E(Xi) = 992− 80 = 912, so that the expected return of each positionequals 9.12%. Moreover, the probability of a loss is less than α for each position, so thatboth of them are “acceptable”. (We obtain Var0.01(Xi) = −1, 000 for both positions.)

Instead of choosing X1 or X2 it should be even less risky to diversify, i.e. to chooseX := 1

2X1 + 12X2. However, the probability that at least one of the two variables is

negative equals 2 · 0.008− 0.0082 ≈ 0.016 > α. Hence X is “not acceptable”. (We obtainVar0.01(X) = 4, 500.)

Conclusion. Value at risk can penalize diversification.

What should we do?

a) Try to modify value at risk in such a way that this problem does not occur any more.

b) Systematic approach initiated by Artzner, Delbaen, Eber and Heath (“Coherentmeasures of risk”, 1999): Formulate a set of axioms that a good risk measure shouldsatisfy, and then investigate the structure of these risk measures.

3 Coherent and convex risk measures

Following Artzner et al. (1999) we define:

Definition 3.1 (Coherent risk measure). A monetary risk measure ρ : G → R is coherentif it is

1. positively homogeneous, i.e.ρ(λX) = λρ(X)

for all X ∈ G and all λ ≥ 0 (with the convention 0 · ∞ := 0),

2. subadditive, i.e.ρ(X + Y ) ≤ ρ(X) + ρ(Y )

for all X,Y ∈ G.

Remark 3.2.

• Positive homogeneity seems very reasonable: if we double a financial position, thenthe risk should also double. Note that value at risk is positively homogeneous andthat each positively homogeneous risk measure is normalized.

• Subadditivity allows to decentralize the task of managing the risk arising from acollection 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 risklimits. Note that positive homogeneity and subadditivity together imply that

ρ(λX1 + (1− λ)X2) ≤ λρ(X1) + (1− λ)ρ(X2)

≤ max(ρ(X1), ρ(X2)),

so that a coherent risk measure does not penalize diversification. In particular, valueat risk is not coherent (unless α = 0).

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Remark 3.3. A monetary risk measure ρ is positively homogeneous if and only if itsacceptance set Aρ is a cone, i.e. if and only if for all X ∈ Aρ and all λ ≥ 0 we haveλX ∈ Aρ. Consequently, if X is acceptable, positive homogeneity implies that for eachλ ≥ 0 the position λX is acceptable as well. Follmer and Schied (“Convex measures of riskand trading constraints”, 2002), however, argued that this property is questionable andthat consequently the axioms of positive homogeneity and subadditivity are too strong.Instead, one should only make sure that diversification is not penalized. This propertycan be formulated in several equivalent ways:

Lemma 3.4. For a monetary risk measure ρ : G → R the following properties are equi-valent:

1. ρ is quasi-convex, i.e.

ρ(λX1 + (1− λ)X2) ≤ max(ρ(X1), ρ(X2))

for all X1, X2 ∈ G and all λ ∈ [0, 1].

2. The acceptance set of ρ is convex.

3. ρ is convex, i.e.

ρ(λX1 + (1− λ)X2) ≤ λρ(X1) + (1− λ)ρ(X2)

for all X1, X2 ∈ G and all λ ∈ [0, 1].

Proof. 1. ⇒ 2. trivial.3. ⇒ 1. trivial.2. ⇒ 3. Suppose that Aρ is convex and that ρ(X1) and ρ(X2) are both finite. Then bycash invariance

ρ(λX1 + (1− λ)X2

)− λρ(X1)− (1− λ)ρ(X2)

= ρ(λX1 + (1− λ)X2 + λρ(X1) + (1− λ)ρ(X2)

)= ρ

(λ (X1 + ρ(X1))︸ ︷︷ ︸

∈Aρ

+(1− λ) (X2 + ρ(X2))︸ ︷︷ ︸∈Aρ︸ ︷︷ ︸

∈Aρ because Aρ is convex

)

≤ 0.

Definition 3.5. A monetary risk measure with these properties is called convex.

Remark 3.6.

1. Every coherent risk measure is convex.

2. A convex risk measure is coherent if and only if it is positively homogeneous.

3. Let ρ be a normalized convex risk measure. Then for each X ∈ G and each λ ≥ 0,

ρ(λX)

≤ λρ(X) if λ ∈ [0, 1],

≥ λρ(X) if λ ≥ 1.

7

Proof. The first two statements are trivial. If λ ∈ [0, 1] we obtain

ρ(λX) = ρ(λX + (1− λ) · 0)

≤ λρ(X) + (1− λ)ρ(0)

= λρ(X).

If λ ≥ 1, then 1/λ ≤ 1, so that

ρ(X) = ρ

(1

λλX

)≤ 1

λρ(λX).

Example 3.7 (Acceptance in terms of expected utility). Let Ω be equipped with a σ-fieldF and a probability measure P, and assume that all elements of G are measurable withrespect to F and bounded. Let moreover u : R → R be strictly increasing and concave(“the utility function of a risk-averse or risk-neutral decision maker”). Fix x0 ∈ R, and let

A := X ∈ G |E[u(X)] ≥ u(x0).

That is, X is accepted if and only if the certainty equivalent of X – i.e. u−1(E[u(X)]) –is greater than or equal to x0. A is clearly an acceptance set, which is normalized if andonly if x0 = 0. Moreover, concavity of u implies that for X,Y ∈ A and λ ∈ [0, 1],

E[u(λX + (1− λ)Y )] ≥ E[λu(X) + (1− λ)u(Y )]

= λE[u(X)] + (1− λ)E[u(Y )]

≥ u(x0),

so that A is convex. The corresponding convex risk measure is given by

ρA(X) = infm ∈ R |E[u(X +m)] ≥ u(x0).

Let us consider two important special cases:

a) Letu(x) = x.

Then

ρA(x) = infm ∈ R |E(X) +m ≥ x0= x0 − E(X)

= E(−X) + x0.

b) Letu(x) = 1− e−βx (β > 0)

and x0 = 0. We have

u′(x) = βe−βx > 0,

u′′(x) = −β2e−βx < 0,

so that u is indeed strictly increasing and concave. The condition E[u(X + m)] ≥u(x0) now reads

1− e−βm E[e−βX

]≥ 0

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or, equivalently,

m ≥ 1

βlogE

[e−βX

].

Hence

ρA(X) =1

βlogE

[e−βX

].

(“entropic risk measure”).

Proposition 3.8. The entropic risk measure can be represented as follows:

1

βlogE

[e−βX

]= sup

Q∈M1(Ω,F)

(EQ(−X)− 1

βH(Q |P)

).

Here H(Q |P) is the relative entropy of Q with respect to P, defined by

H(Q |P) :=

+∞ if Q is not absolutely continuous with respect to PE(dQdP log dQ

dP

)= EQ

(log dQ

dP

)if Q is absolutely continuous with respect to P.

Lemma 3.9. For all probability measures P and Q,

H(Q |P) ≥ 0,

with equality if and only if Q = P.

Proof. We may assume that Q is absolutely continuous with respect to P; we denote thedensity by ρ. Since the function f(x) := x log x is strictly convex, Jensen’s inequalityimplies that

H(Q |P) =

∫Ωf(ρ) dP

≥ f

(∫Ωρ dP

)= f(1) = 0,

with equality if and only if ρ is constant P-a.s, i.e. if and only if Q = P.

Proposition 3.8 follows immediately from the following lemma:

Lemma 3.10. For any probability measure Q on (Ω,F),

H(Q |P) ≥ βEQ(−X)− logE[e−βX

],

with equality if and only if

Q = PX :=e−βX

E[e−βX ]P.

Proof. We havedQdP

=dQdPX

dPX

dP=

dQdPX

e−βX

E[e−βX ]

and consequently

logdQdP

= logdQdPX

− βX − logE[e−βX ].

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Integration with respect to Q then yields

H(Q |P) = EQ

(log

dQdP

)= EQ

(log

dQdPX

)− βEQ(X)− logE[e−βX ]

= H(Q |PX)︸ ︷︷ ︸≥0

−βEQ(X)− logE[e−βX ]

≥ βEQ(−X)− logE[e−βX

],

with equality if and only if Q = PX .

Example 3.11. We work in the setting of the last example, but instead of just oneprobability measure P we are given a whole non-empty set S of probability measures.Moreover, for each Q ∈ S we are given a number x0(Q) ∈ R. Now let

A :=⋂Q∈SX ∈ G |EQ[u(X)] ≥ u(x0(Q)).

That is, X is accepted if and only if for each probability measure Q ∈ S the certaintyequivalent of X with respect to Q is greater than or equal to x0(Q). A is not necessarily anacceptance set because it can be empty. However, if the function Q 7→ x0(Q) is bounded,A is a convex acceptance set.

In the special case u(x) = x we get

ρA(X) = infm ∈ R |X +m ∈ A= infm ∈ R | ∀Q ∈ S,EQ(X +m) ≥ x0(Q)

= infm ∈ R | infQ∈S

(EQ(X) +m− x0(Q)

)≥ 0

= − infQ∈S

(EQ(X)− x0(Q)

)= sup

Q∈S

(EQ(−X) + x0(Q)

).

Note that this expression looks quite similar to the formula that we obtained for theentropic risk measure. Remarkably, “many” convex risk measures can be represented inthis way.

4 Robust representation of convex risk measures

4.1 Introduction

In this section we suppose that Ω is equipped with a σ-field F and that all elements of Gare F-measurable. Our goal is to show that if G is “nice”, then all “nice” convex riskmeasures ρ : G → R can be represented in the form

ρ(X) = supQ∈S

(EQ(−X)− α(Q)

)where

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• S is a non-empty set of probability measures on (Ω,F) such that all elements of Gare integrable with respect to all Q ∈ S,

• the function α : S → R ∪ +∞ is bounded from below and not identically +∞.

Proposition 4.1. A function ρ : G → R of that form is a convex risk measure.

Proof. Since α is not identically +∞, ρ does not take the value −∞. Since α is boundedfrom below the constants have finite risk. Monotonicity, cash-invariance and convexity areclear.

Remark 4.2. A representation of that form is called a robust representation. The idea isas follows:

1. For various probabilistic models Q one computes the expected loss with respect to Q.

2. These models are taken more or less seriously, as specified by the penalty function α.

3. Finally one computes ρ(X) as the worst case of the penalized expected loss over allprobabilistic models in the class S.

Remark 4.3. Probability measures Q ∈ S for which α(Q) =∞ do not contribute to thesupremum so that the corresponding terms can be dropped. On the other hand, one canartificially add probability measures Q to the supremum by choosing α(Q) := ∞. Onemay therefore assume that S is the set of all probability measures on (Ω,F) with respectto which all elements of G are integrable.

Proposition 4.4.

1. For each Q ∈ S we have

α(Q) ≥ αmin(Q)

:= supX∈G

(EQ(−X)− ρ(X)

)= sup

X∈AρEQ(−X).

2. The robust representation also holds with αmin instead of α.

Proof. 1. The robust representation immediately implies that for all Q ∈ S and all X ∈ Gwe have

α(Q) ≥ EQ(−X)− ρ(X)

and consequentlyα(Q) ≥ αmin(Q).

Moreover,

supX∈G

(EQ(−X)− ρ(X)

)= sup

X∈GEQ(−[X + ρ(X)])

≤ supX′∈Aρ

EQ(−X ′)

and

supX∈Aρ

EQ(−X) ≤ supX∈Aρ

EQ(−[X + ρ(X)])

≤ supX∈G

EQ(−[X + ρ(X)]).

11

2. On the one hand, since αmin(Q) ≤ α(Q) the robust representation immediately impliesthat

ρ(X) ≤ supQ∈S

(EQ(−X)− αmin(Q)

).

On the other hand, for each X ∈ G we have αmin(Q) ≥ EQ(−X)− ρ(X) and consequently

supQ∈S

(EQ(−X)− αmin(Q)

)≤ sup

Q∈S

(EQ(−X)− EQ(X) + ρ(X)

)= ρ(X).

Proposition 4.5. For a convex risk measure ρ admitting a robust representation thefollowing statements are equivalent:

1. ρ is coherent.

2. There exists a robust representation whose penalty function α only takes the values 0and ∞.

3. αmin only takes the values 0 and ∞.

Proof. 3.⇒ 2. Trivial.2. ⇒ 1. If α only takes the values and ∞, then ρ is clearly positively homogeneous andhence coherent.1.⇒ 3. Let Q ∈ S and λ > 0. Recall (Remark 3.3) that the acceptance set of a coherentrisk measure is a cone, so that X ∈ Aρ if and only if λX ∈ Aρ. Consequently

αmin(Q) = supX∈Aρ

EQ(−X) = supX∈Aρ

EQ(−λX) = λαmin(Q).

4.2 Existence of a robust representation

We now suppose that G is equipped with a norm ‖ ·‖ such that (G, ‖ ·‖) is a Banach space.Our main example will be L∞(Ω,F), the space of all bounded F-measurable functionson Ω equipped with the supremum norm.

The idea is to consider the Legendre transform of ρ, defined by

ρ∗(`) := supX∈G

(`(X)− ρ(X)

), ` ∈ G′ := dual space of G.

Applying the Legendre transform a second time (together with the canonical embeddingof G into its double dual G′′) we get

ρ∗∗(X) := sup`∈G′

(`(X)− ρ∗(`)

), X ∈ G.

If ρ∗∗ = ρ, we are on the way to obtain a robust representation.

Remark 4.6. For each ` ∈ G′ the function X 7→ `(X) − ρ∗(`) is affine and continuous.Consequently, ρ∗∗ is convex and lower semicontinuous. Hence ρ∗∗ = ρ can only hold if ρitself is convex and lower semicontinuous.

Theorem 4.7 (Fenchel-Moreau). A function ρ : G → R satisfies ρ∗∗ = ρ if and only ifρ is convex and lower semicontinuous.

Corollary 4.8. Each convex and lower semicontinuous risk measure ρ : G → R has therepresentation

ρ(X) = sup`∈G′

(`(X)− ρ∗(`)

).

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Example 4.9. Every monetary risk measure is even Lipschitz continuous with respect tothe supremum norm. Hence on L∞(Ω,F) every convex risk measure has such a represen-tation.

Proposition 4.10. Whenever ρ∗(`) <∞, we have

1. ` ≤ 0, i.e. `(X) ≤ 0 for all X ≥ 0,

2. `(1) = −1.

Hence for each ` ∈ G′ which contributes to the supremum, −` behaves as integrationwith respect to a probability measure.

Proof of Proposition 4.10. Let ` ∈ G′ be such that ρ∗(`) <∞.

1. Take X ≥ 0. Then

+∞ > ρ∗(`)

= supY ∈G

(`(Y )− ρ(Y )

)≥ sup

λ≥0

(`(λX)− ρ(λX)

)≥ sup

λ≥0λ`(X)− ρ(0).

This supremum can only be less than +∞ if `(X) ≤ 0.

2. Similarly,

+∞ > supY ∈G

(`(Y )− ρ(Y )

)≥ sup

λ∈R

(`(λ)− ρ(λ)

)= sup

λ∈R

(λ`(1)− ρ(0) + λ

)= sup

λ∈Rλ(`(1) + 1

)− ρ(0).

This supremum can only be less than +∞ if `(1) = −1.

Corollary 4.11. Every convex and lower semicontinuous risk measure ρ : G → R has therepresentation

ρ(X) = sup`∈G′, `≤0, `(1)=−1

(`(X)− ρ∗(`)

).

Under appropriate additional conditions each ` ∈ G′ with ` ≤ 0 and `(1) = −1 is of theform `(X) = −EQ(X) for some probability measure Q on (Ω,F). Defining α(Q) := ρ∗(`)then gives the representation

ρ(X) = supQ

(EQ(−X)− α(Q)

)where the supremum is over a certain class of probability measures.

13

5 Convex risk measures on L∞(Ω,F)

We now suppose that G = L∞(Ω,F). Assuming boundedness of all financial positions Xis not very restrictive from the economic point of view, but quite convenient because itimplies that

1. ρ(X) is finite for every monetary risk measure ρ,

2. EQ(−X) is well-defined and finite for every probability measure Q on (Ω,F).

Let ρ be a convex risk measure on L∞(Ω,F). Since ρ is even Lipschitz continuous withrespect to ‖ · ‖ we have

ρ(X) = sup`(X)− ρ∗(`)

∣∣∣ ` ∈ L∞(Ω,F)′, ` ≤ 0, `(1) = −1.

How do elements of L∞(Ω,F)′ look like?

Definition 5.1.

1. A function µ : F → R is called a finitely additive set function if

(a) µ(∅) = 0,

(b) µ(A ∪B) = µ(A) + µ(B) for all disjoint A,B ∈ F .

2. The total variation of a finitely additive set function µ is

‖µ‖var := sup

n∑i=1

|µ(Ai)|∣∣∣∣n ∈ N, A1, . . . , An ∈ F are pairwise disjoint

.

3. We denote the set of all finitely additive set functions of finite total variation byMf (Ω,F).

Definition 5.2. The integral of a bounded measurable function f : Ω → R with respectto a finitely additive set function with finite total variation is defined as follows:

1. Let G0 be the space of elementary functions of the form

X =

n∑i=1

αi1Ai

with αi ∈ R and Ai ∈ F . Note that this space is dense in L∞(Ω,F) with respect tothe supremum norm. For such a function X ∈ G0 we set

Iµ(X) :=

n∑i=1

αiµ(Ai).

One has to check that this definition is independent of the choice of the representationof X, but this can be done in the same way as when µ is a measure.

2. Choose a representation of X with pairwise disjoint sets Ai. Then

|Iµ(X)| ≤n∑i=1

|αi| |µ(Ai)| ≤ ‖X‖∞‖µ‖var.

Hence there is a unique extension of Iµ to a continuous linear functional on L∞(Ω,F).We write ∫

ΩXdµ

for this extension applied to a function X ∈ L∞(Ω,F).

14

Theorem 5.3. For each continuous linear functional ` on G there exists a finitely additiveset function µ ∈Mf (Ω,F) such that

`(X) =

∫ΩXdµ

for each X ∈ L∞(Ω,F).

Proof. We defineµ(A) := `(1A).

Finite additivity of µ is clear. To prove that µ is of finite variation let A1, . . . , An bepairwise disjoint. Let A+ be the union of those sets Ai for which µ(Ai) ≥ 0, and A− theunion of those sets Ai for which µ(Ai) < 0. Then

n∑i=1

|µ(Ai)| = µ(A+)− µ(A−)

= `(1A+ − 1A−)

≤ ‖`‖ ‖1A+ − 1A−‖∞≤ ‖`‖.

Hence ‖µ‖var ≤ ‖`‖.

Corollary 5.4. Every convex risk measure ρ : L∞(Ω,F) → R can be represented in theform

ρ(X) = supQ∈M1,f

(EQ(−X)− α(Q)

),

where M1,f denotes the space of all finitely additive probability measures on (Ω,F), i.e.the set of all non-negative finitely additive set functions Q satisfying Q(Ω) = 1.

Question. Under which conditions is there a representation in terms of true (i.e.σ-additive) probability measures? In other words, which convex risk measures ρ canbe represented in the form

ρ(X) = supQ∈M1

(EQ(−X)− α(Q)

),

where M1 denotes the space of all true (i.e. σ-additive) probability measures on (Ω,F)?

We start with a necessary condition:

Proposition 5.5. Let ρ : L∞(Ω,F)→ R be a convex risk measure that can be representedin terms of true probability measures. Then it has the following “Fatou property”: Foreach bounded and pointwise convergent sequence (Xn)n∈N we have

ρ( limn→∞

Xn) ≤ lim infn→∞

ρ(Xn).

Proof. Using Lebesgue’s dominated convergence theorem (which holds for true probabilitymeasures, but not for finitely additive ones) we obtain

ρ( limn→∞

Xn) = supQ∈M1

(EQ(− lim

n→∞Xn)− α(Q)

)= sup

Q∈M1

limn→∞

(EQ(−Xn)− α(Q)

)≤ lim inf

n→∞sup

Q∈M1

(EQ(−Xn)− α(Q)

)= lim inf

n→∞ρ(Xn).

15

Proposition 5.6. For a monetary risk measure ρ : L∞(Ω,F)→ R the Fatou property isequivalent to continuity from above, i.e.

Xn X ⇒ ρ(Xn) X.

Proof. Suppose first that ρ has the Fatou property, and let Xn X. Note that thisimplies that the sequence (Xn)n∈N is bounded (from below by inf X and from above bysupX1). Hence the Fatou property implies that

ρ(X) ≤ lim infn→∞

ρ(Xn).

Moreover, monotonicity of ρ implies that the sequence (ρ(Xn))n∈N is convergent, and

limn→∞

ρ(Xn) ≤ ρ(X).

Hence ρ(Xn) ρ(X).Conversely, assume that ρ is continuous from above, and let (Xn)n∈N be a bounded and

pointwise convergent sequence in L∞(Ω,F). Let X := limn→∞Xn and Yn := supm≥nXm.Then Yn X, so that by continuity from above

ρ(Yn) ρ(X).

Moreover, Xn ≤ Yn, so that

lim infn→∞

ρ(Xn) ≥ limn→∞

ρ(Yn) = ρ(X).

A sufficient condition for the existence of a robust representation in terms of trueprobability measures is continuity from below:

Proposition 5.7. Let ρ be a convex risk measure on L∞(Ω,F) → R that is continuousfrom below, i.e.

Xn X ⇒ ρ(Xn) X,

and has the representation

ρ(X) = supQ∈M1,f

(EQ(−X)− α(Q)

).

Then α(Q) =∞ for all Q ∈M1,f which are not σ-additive, and consequently

ρ(X) = supQ∈M1

(EQ(−X)− α(Q)

).

Proof. Let Q ∈ M1,f be such that α(Q) < ∞. In order to show that Q is σ-additivewe show that for every sequence (An)n∈N in F that increases to Ω we have Q(An) 1.To this end let X = λ1An for some λ > 0. We have

ρ(λ1An) ≥ EQ(−λ1An)− α(Q)

and consequentlyλQ(An) ≥ −ρ(λ1An)− α(Q).

As n→∞, λ1An λ, hence continuity from below implies that

ρ(λ1An) ρ(λ) = ρ(0)− λ.

16

It follows thatλ lim inf

n→∞Q(An) ≥ λ− ρ(0)− α(Q)

and consequently

lim infn→∞

Q(An) ≥ 1− ρ(0) + α(Q)

λ.

Letting λ→∞ gives the result.

Summary. For a convex risk measure ρ on L∞(Ω,F) we have the following implications:

ρ continuous from below

⇒ ρ has a robust representation in terms of true probability measures

⇒ ρ continuous from above

⇔ ρ has the Fatou property

6 Convex risk measures on L∞(Ω,F ,P)

We now fix a probability measure P on (Ω,F) and consider risk measures ρ on L∞(Ω,F)with the property

X = Y P-a.s. ⇒ ρ(X) = ρ(Y ).

Such risk measures can be regarded as functionals on the space L∞(Ω,F ,P) of equivalenceclasses (w.r.t. to P-a.s. equality) of elements of L∞(Ω,F). We therefore call them riskmeasures on L∞(Ω,F ,P).

Proposition 6.1. Let ρ be a convex risk measure on L∞(Ω,F ,P) that has the represen-tation

ρ(X) = supQ∈M1,f

(EQ(−X)− α(Q)

).

Then α(Q) = ∞ for all Q ∈ M1,f that are not absolutely continuous with respect to P.Consequently, ρ has the representation

ρ(X) = supQ∈M1,f (P)

(EQ(−X)− α(Q)

),

where M1,f (P) denotes the set of all finitely additive probability measures on (Ω,F) thatare absolutely continuous with respect to P.

Proof. Let Q ∈ M1,f be not absolutely continuous with respect to P. Then there existsA ∈ F with P(A) = 0 and Q(A) > 0. Let Xn := −n1A. Then Xn = 0 P-a.s, andconsequently

α(Q) ≥ EQ(n1A)− ρ(−n1A)

= nQ(A)− ρ(0)

→ ∞ (n→∞).

For convex risk measures on L∞(Ω,F ,P) existence of a representation in terms of trueprobability measures can be characterized in a number of ways. To formulate and provethe next result we have to recall some definitions and results from functional analysis:

17

1. The space L∞(Ω,F ,P) can be identified with the dual space of L1(Ω,F ,P). Namely,every continuous linear functional ` on L1(Ω,F ,P) is of the form

`(X) =

∫ΩXY dP

for some Y ∈ L∞(Ω,F ,P).

2. Given a Banach space G, the weak∗ topology on G′ is the coarsest topology for whichall functionals of the form

` 7→ `(X), X ∈ G,

are continuous. For example, the weak∗ topology on L∞(Ω,F ,P) (regarded as dualspace of L1(Ω,F ,P)) is the coarsest topology for which all functionals of the form

Y 7→∫

ΩXY dP, X ∈ L1(Ω,F ,P),

are continuous. One can show that every linear functional on G′ which is continuouswith respect to the weak∗ topology is of the form

` 7→ `(X)

for some X ∈ G.

3. Krein-Shmulyan theorem: A convex subset K of G′ is weak∗ closed if and only if foreach r > 0 the set

Kr := K ∩ x ∈ G′ | ‖x‖G′ ≤ r

is weak∗ closed.

Theorem 6.2. For a convex risk measure ρ on L∞(Ω,F ,P) the following properties areequivalent:

1. ρ can be represented in the form

ρ(X) = supQ∈M1

(EQ(−X)− α(Q)

).

2. ρ can be represented in the form

ρ(X) = supQ∈M1(P)

(EQ(−X)− α(Q)

),

where M1(P) denotes the set of all true probability measures on (Ω,F) that areabsolutely continuous with respect to P.

3. ρ is continuous from above, i.e.

Xn X ⇒ ρ(Xn) ρ(X).

4. ρ has the Fatou property, i.e. for every bounded and pointwise convergent sequence(Xn)n∈N,

ρ( limn→∞

Xn) ≤ lim infn→∞

ρ(Xn).

18

5. ρ is lower semicontinuous with respect to the weak∗ topology on L∞(Ω,F ,P) (regardedas dual pace of L1(Ω,F ,P)).

6. The acceptance set Aρ is closed in the weak∗ topology of L∞(Ω,F ,P).

Remark 6.3.

1. In 3. and 4. one can replace pointwise convergence with P-almost sure covergence.

2. Every convex risk measure on L∞(Ω,F) is Lipschitz continuous and consequentlylower semicontinuous with respect to ‖ · ‖∞. However, the weak∗ topology is coarserthan the norm topology, so that lower semicontinuity with respect to the weak∗

topology is a stronger property than with respect to the norm topology. Note how-ever that for a convex function on a Banach space G lower semicontinuity withrespect to the weak topology is equivalent to lower semicontinuity with respect to thenorm topology. (Suppose that f : G → R is lower semicontinuous with respect tothe norm topology. Then for each c ∈ R the set f ≤ c is strongly closed. Sincemoreover it is convex it is weakly closed. The last fact (every strongly closed andconvex subset K of a Banach space G is weakly closed) can be proved as follows: Letx ∈ G \K. Then by the Hahn-Banach separation theorem there exists a continuouslinear functional ` such that `|K ≤ 0 and `(x) > 0. Hence the set ` > 0 is aweakly open neighborhood of x contained in the complement of K. Since x ∈ G \Kis arbitrary it follows that G \K is weakly open.)

Proof of Theorem 6.2.

1. The equivalence 1. ⇔ 2. follows from the last proposition, and the implications1. ⇒ 3. ⇔ 4. have already been proved (Propositions 5.5 and 5.6). Since Aρ =X ∈ L∞ | ρ(X) ≤ 0, the implication 5. ⇒ 6. is trivial. In the sequel we will prove4. ⇒ 5. and 6. ⇒ 2.

2. We now prove 4. ⇒ 5. We have to show that for each c ∈ R the set

K := X ∈ L∞(Ω,F ,P) | ρ(X) ≤ c

is weak∗ closed. Convexity of ρ implies that K is convex, so that by the Krein-Shmulyan theorem it suffices to prove that for each r > 0 the set

Kr = X ∈ L∞(Ω,F ,P) | ‖X‖∞ ≤ r and ρ(X) ≤ c

is weak∗ closed.

(a) We first show that Kr is closed with respect to the L1-norm (note that Kr ⊂L∞ ⊆ L1). To this end let (Xn)n∈N be a sequence in Kr converging in L1 tosome X ∈ L1. Then there exists a subsequence (Xnk)k∈N converging P-almostsurely to X, so that

‖X‖∞ ≤ r.

Moreover, by the Fatou property

ρ(X) ≤ lim infk→∞

ρ(Xnk) ≤ c.

Hence X ∈ Kr, and Kr is closed with respect to the L1-norm.

(b) Since Kr is convex it follows that it is also weakly closed in L1.

19

(c) Weak∗ convergence in L∞ implies weak convergence in L1 because the space oftest functions is larger (L1 instead of L∞). Consequently weak closedness in L1

implies weak∗ closedness in L∞. Hence Kr is weak∗ closed in L∞.

3. We now prove 6. ⇒ 2. To this end we will show that for all X ∈ L∞,

ρ(X) = supQ∈M1(P)

(EQ(−X)− αmin(Q)

),

where we recall that

αmin(Q) = supX∈L∞

(EQ(−X)− ρ(X)

)= sup

X∈AρEQ(−X).

Letm := sup

Q∈M1(P)

(EQ(−X)− αmin(Q)

).

Clearly m ≤ ρ(X), so that it suffices to show that m ≥ ρ(X), i.e.

X +m ∈ Aρ.

Suppose instead that X + m /∈ Aρ. Under this assumption we will construct aprobability measure Q0 on (Ω,F) such that

EQ0(−X)− αmin(Q0) > m,

in contradiction to the definition of m.

(a) We apply the Hahn-Banach separation theorem to the convex and weak∗-closedset Aρ and the point X+m, and obtain a linear functional ` on L∞(Ω,F ,P)that is continuous with respect to the weak∗ topology and separates Aρ andX +m in the sense that

infY ∈Aρ

`(Y ) > `(X +m).

(b) Since ` is continuous with respect to the weak∗ topology of L∞, it is of the form

`(Y ) =

∫ΩY ZdP

for some Z ∈ L1(Ω,F ,P).

(c) Now we show that Z ≥ 0: For each Y ≥ 0 and each λ > 0 we have ρ(λY ) ≤ ρ(0),so that

λY + ρ(0) ∈ Aρand consequently

λ`(Y ) + `(ρ(0)) = `(λY + ρ(0)) > `(X +m).

It follows that

`(Y ) >`(X +m− ρ(0))

λ.

By letting λ → ∞ we see that `(Y ) ≥ 0 and consequently Z ≥ 0. Moreover,` 6≡ 0 and consequently Z 6≡ 0.

20

(d) It follows that

Q0 :=Z

E(Z)P

defines a probability measure on (Ω,F). Moreover,

EQ0(X) +m =E(XZ)

E(Z)+m

=E((X +m)Z)

E(Z)

=`(X +m)

E(Z)

<infY ∈Aρ `(Y )

E(Z)

=infY ∈Aρ E(Y Z)

E(Z)

= infY ∈Aρ

EQ0(Y )

= − supY ∈Aρ

EQ0(−Y )

= −αmin(Q0).

7 Average value at risk

In this section we suppose that Ω is equipped with a σ-field F and a probability measure Pand that all elements of G are measurable with respect to F . For λ ∈ (0, 1] we define theaverage value at risk (or expected shortfall or conditional value at risk) of X at level λ by

AVaRλ(X) :=1

λ

∫ λ

0VaRα(X)dα.

Is this integral well-defined? We have to recall some facts about quantiles.

Definition 7.1. Let µ be a probability measure on R and α ∈ (0, 1).

1. A number q ∈ R is called an α-quantile of µ if

µ((−∞, q]) ≥ α and µ([q,∞)) ≥ 1− α.

2. A function qµ : (0, 1) → R is called a quantile function of µ if for each α ∈ (0, 1),qµ(α) is an α-quantile of µ.

Remark 7.2.

1. For each α ∈ (0, 1) the set of α-quantiles of µ is a non-empty bounded closed interval.We denote its endpoints by q−µ (α) and q+

µ (α).

2. Pictures (nice case, discontinuity, interval of constancy)!!!

3. Numbers α for which q−µ (α) < q+µ (α) correspond to intervals of constancy

of the cumulative distribution function of µ. It follows that the setα ∈ (0, 1) | q−µ (α) < q+

µ (α) is countable.

21

4. Let X be a random variable with distribution µ. Then

−VaRα(X) = supc ∈ R |µ((−∞, c]) ≤ α.

It follows that −VaRα(X) is the largest α-quantile of µ,

−VaRα(X) = q+µ (α).

5. Let U be a standard uniform random variable (i.e. a random variable which is isuniformly distributed on (0, 1)), and let qµ be any quantile function of µ. Thenqµ(U) has distribution µ. (Standard method to simulate random variables with agiven distribution µ)

Proposition 7.3. The integral ∫ 1

0VaRα(X)dα

exists in R if and only if X is quasi-integrable (i.e. has a well-defined expectation). In thiscase ∫ 1

0VaRα(X)dα = E(−X).

Corollary 7.4. If X is integrable, AVaRλ(X) exists and is finite for all λ ∈ (0, 1].

Proof of Proposition 7.3. Let µ be the distribution of X, and let U be a standard uniformrandom variable. Supposing that all integrals exist we obtain∫ 1

0VaRα(X)dα = −

∫ 1

0q+µ (α)dα

= −E[q+µ (U)]

= E(−X).

Moreover, this reasoning shows that∫ 1

0 VaRα(X)dα exists if and only if X is quasi-integrable.

From now on we consider only integrable random variables, i.e. we assume thatG ⊆ L1(Ω,F ,P).

Proposition 7.5. Let qX be any quantile function of the distribution of X. Then

AVaRλ(X) =1

λE[(qX(λ)−X)+]− qX(λ).

Proof. Let U be a standard uniform random variable. Then X has the same distributionas qX(U), and consequently

1

λE[(qX(λ)−X)+]− qX(λ) =

1

λE[(qX(λ)− qX(U))+]− qX(λ)

=1

λ

∫ 1

0(qX(λ)− qX(α))+dα− qX(λ)

=1

λ

∫ λ

0(qX(λ)− qX(α))dα− qX(λ)

= − 1

λ

∫ λ

0qX(α)dα

=1

λ

∫ λ

0VaRα(X)dα

= AVaRλ(X).

22

Proposition 7.6. We have

AVaRλ(X) = supEQ(−X)

∣∣∣Q P,dQ

dP≤ 1

λ

.

Corollary 7.7. Average value at risk is a coherent risk measure.

Proof of Proposition 7.6. The supremum on the right-hand side is equal to

supE(−ϕX)

∣∣∣ϕ ∈ L1(Ω,F ,P), E(ϕ) = 1, 0 ≤ ϕ ≤ 1

λ

.

E(−ϕX) is large if ϕ takes large values at points where X takes small values and viceversa. Hence the supremum is attained for

ϕ :=

1/λ on X < qX(λ)0 on X > qX(λ)κ on X = qX(λ),

where qX is any quantile function of the distribution of X and κ is such that E(ϕ) = 1,i.e.

1

λP(X < qX(λ)) + κP(X = qX(λ)) = 1.

(Note that κ necessarily satifies κ ∈ [0, 1/λ] because P(X < qX(λ)) ≤ λ ≤ P(X ≤ qX(λ)).)It follows that

supE(−ϕX)

∣∣∣ϕ ∈ L1(Ω,F ,P)E(ϕ) = 1, 0 ≤ ϕ ≤ 1

λ

= − 1

λE(X · 1X<qX(λ))− κE(X · 1X=q+(λ))

= − 1

λE(X · 1X<qX(λ))− κqX(λ)P(X = q+(λ))

= − 1

λE(X · 1X<qX(λ))− qX(λ) +

1

λq+(λ)P(X < qX(λ))

=1

λE((q+(λ)−X) · 1X<qX(λ))− qX(λ)

=1

λE((qX(λ)−X)+)− qX(λ)

= AVaRλ(X).

Proposition 7.8. For λ ∈ (0, 1) let

WCEλ(X) := supE(−X|A) |A ∈ F , P(A) > λ

be the worst conditional expectation at level λ and

TCEλ(X) := E(−X | −X ≥ VaRλ(X))

be the tail conditional expectation at level λ. Then

AVaRλ(X) ≥ WCEλ(X)

≥ TCEλ(X)

≥ VaRλ(X).

Moreover, the first two inequalities are equalities if P(−X ≥ VaRλ(X)) = λ. (By thedefinition of −VaRλ(X) as a λ-quantile of X this probability is always greater than orequal to λ, and it is equal to λ if X has a continuous distribution.)

23

Proof. The third inequality is trivial.To prove the first inequality let A ∈ F with P(A) ≥ λ and define Q := P(· |A). Then

Q P with dQ/dP ≤ 1/λ, so that

E(−X |A) ≤ AVaRλ(X).

To prove the second inequality note that for every ε > 0,

P−X ≥ VaRλ(X)− ε > λ.

(This follows immediately from the fact that −VaRλ(X) = supc ∈ R |P[X ≤ c] ≤ λ.)Consequently, for every ε > 0,

WCEλ(X) ≥ E(−X | −X ≥ VaRλ(X)− ε).

Letting ε→ 0 gives the second inequality (using Lebesgue’s dominated convergence theo-rem).

Finally, suppose that P(−X ≥ VaRλ(X)) = λ. Then on the one hand

TCEλ(X) =E(−X · 1−X≥VaRλ(X))

P−X ≥ VaRλ(X)

=1

λE(−X · 1−X≥VaRλ(X)),

and on the other hand

AVaRλ(X) =1

λE[(−VaRλ(X)−X)+] + VaRλ(X)

=1

λE[(−VaRλ(X)−X) · 1−X≥VaRλ(X)] + VaRλ(X)

=1

λE(−X · 1−X≥VaRλ(X))

as well.

Remark 7.9.

1. The inequalities in the proposition can be strict.

2. WCEλ is coherent.

3. TCEλ is not even convex.

4. In contrast to the other risk measures appearing in the proposition, WCEλ(X)depends not only on the distribution of X under P, but also on the structure ofthe underlying probability space.

Example 7.10. Let Ω = ω1, ω2, ω3, F = P(Ω), P(ω1) = P(ω2) = 0.1 andP(ω3) = 0.8. Define X1, X2 : Ω → R by X1(ω1) = −100, X1(ω2) = X1(ω3) = 0,X2(ω2) = −100 and X2(ω1) = X2(ω3) = 0. We obtain

VaR0.15(Xi) = 0,

AVaR0.15(Xi) =1

0.15· 0.1 · 100 =

200

3≈ 67,

WCEλ(Xi) = 50,

TCEλ(Xi) = 10.

24

Moreover, letting Y := (X1 + X2)/2 we have Y (ω1) = Y (ω2) = −50 and Y (ω3) = 0, andconsequently

VaR0.15(Y ) = 50,

TCE0.15(Y ) = 50,

which shows that TCE is not convex.

8 Law-invariant risk measures and Kusuoaka’s representa-tion theorem

We still suppose that Ω is equipped with a σ-field F and a probability measure P and thatall elements of G are measurable with respect to F . If we regard P as the “true probabilitymeasure”, it is natural to concentrate on law-invariant risk measures.

Definition 8.1. A monetary risk measure ρ : G → R is called law-invariant or distribution-based if ρ(X) = ρ(Y ) whenever X and Y have the same distribution under P.

Example 8.2.

• Value at risk (not convex),

• Average value at risk (convex).

Law-invariance clearly implies that ρ(X) = ρ(Y ) whenever X = Y P-a.s. Hence everylaw-invariant convex risk measure on L∞(Ω,F) that is continuous from above has therepresentation

ρ(X) = supQ∈M1(Ω,F ,P)

(EQ(−X)− αmin(Q)

)with

αmin(Q) = supX∈L∞(Ω,F)

(EQ(−X)− ρ(X)

)= sup

X∈AρEQ(−X).

If ρ is law-invariant it will turn out that the only relevant aspect of the probability mea-sures Q is the distribution of the density dQ/dP under P. Consequently, the idea ofKusuoka’s representation theorem is to write ρ(X) as a supremum over distributions ofprobability densities with respect to P.

Assumption 8.3. The probability space (Ω,F ,P) is atomless, i.e. for every A ∈ F withP(A) > 0 there exists A′ ∈ F with A′ ⊂ A and 0 < P(A′) < P(A).

Lemma 8.4. For a probability space (Ω,F ,P) the following properties are equivalent:

1. (Ω,F ,P) is atomless.

2. There exists a standard uniform random variable U on (Ω,F ,P).

3. For every real-valued random variable X on (Ω,F ,P) there exists a standard uniformrandom variable U on (Ω,F ,P) such that

X = qX(U) P-a.s.,

where qX is any quantile function of the distribution of X.

25

Proof. The implications 3. ⇒ 2. ⇒ 1. are trivial. The implication 1. ⇒ 3. can be foundin the book by Follmer and Schied.

We will encounter maximization problems of the following form: Given two randomvariables X and Y on a probability space (Ω,F ,P),

maximize E(XY )

over all random variables Y on (Ω,F ,P) with the same distribution as Y . On an atomlessprobability space this problem can be easily solved:

Lemma 8.5. Let X and Y be two random variables on an atomless probability space(Ω,F ,P), and assume that one of them is integrable and the other one is bounded (thislast assumption is made to ensure that E(XY ) exists and is finite). Let U be a standarduniform random variable on (Ω,F ,P) such that X = qX(U) P-a.s. (U exists by theprevious lemma.) Then the maximum is attained by Y = qY (U), and consequently

supY∼Y

E(XY ) = E(qX(U)qY (U)) =

∫ 1

0qX(t)qY (t)dt.

Proof (heuristic argument). E(XY ) becomes large if Y takes large values when X doesso, and small values when X does so. Since quantile functions are non-decreasing, thisrequirement is achieved by the choice Y = qY (U).

Proposition 8.6. Let ρ be a law-invariant convex risk measure ρ on L∞(Ω,F) that is con-tinuous from above (and hence admits a robust representation in terms of true probabilitymeasures). Then its minimal penalty function is given by

αmin(Q) = supX∈Aρ

∫ 1

0VaRt(X)qdQ/dP(1− t)dt.

Corollary 8.7. αmin(Q) depends only on the distribution of dQ/dP under P.

Proof of Proposition 8.6. Using the law-invariance of ρ and the last lemma we obtain

αmin(Q) = supX∈Aρ

EQ(−X)

= supX∈Aρ

supX∼X

EQ(−X)

= supX∈Aρ

supX∼X

E(−X dQ

dP

)= sup

X∈Aρ

∫ 1

0q−X(t)qdQ/dP(t)dt.

Moreover,q−X(t) = −qX(1− t) = VaR1−t(X).

for all but countably many t. Hence the claim follows.

The last proposition suggests that

supQ∈M1(P)

(EQ(−X)− αmin(Q)

)

26

might be formulated as a supremum over quantile functions q of probability densitieswith respect to P. Such functions are non-decreasing and non-negative and satisfy∫ 1

0 q(t)dt = E(q(U)) = 1. Conversely, if (Ω,F ,P) is atomless, every non-decreasing func-

tion q : (0, 1) → R+ with∫ 1

0 q(t)dt = 1 is a quantile function of a probability density,namely of q(U) for some standard uniform random variable U .

Proposition 8.8 (Preliminary version of Kusuoka’s representation theorem). Let ρ be alaw-invariant convex risk measure on L∞(Ω,F) that is continuous from above. Then

ρ(X) = supq

(∫ 1

0VaRt(X) q(1− t)dt− β(q)

),

where the supremum is over all non-decreasing functions q : (0, 1)→ R+ with∫ 1

0 q(t)dt = 1,and

β(q) = supX∈Aρ

∫ 1

0VaRt(X)q(1− t)dt.

Proof. We have

ρ(X) = supQ∈M1(P)

(EQ(−X)− αmin(Q)

)= sup

Q∈M1(P)

(E(−XdQ

dP

)− αmin(Q)

)

= supQ∈M1(P)

supQ∼Q

(E

(−XdQ

dP

)− αmin(Q)

)

= supQ∈M1(P)

(supQ∼Q

E

(−XdQ

dP

)− αmin(Q)

)

= supQ∈M1(P)

(∫ 1

0q−X(t)qdQ/dP(t)dt− αmin(Q)

)= sup

Q∈M1(P)

(∫ 1

0VaRt(X) qdQ/dP(1− t)dt− β(qdQ/dP)

).

Writing this as a supremum over all quantile functions of probability densities (with respectto P) we get the result.

Lemma 8.9. There is a one-to-one correspondence between left-continuous non-decreasingfunctions q : (0, 1)→ R+ with

∫ 10 q(t)dt = 1, and probability measures µ on (0, 1], which is

given as follows: With a probability measure µ on (0, 1] we associate the function q definedby

q(t) =

∫(1−t,1]

1

sµ(ds).

Proof.

1. For a given probability measure µ on (0, 1] define the measure ν on (0, 1] by

ν(ds) =1

sµ(ds).

Thenq(t) = ν((1− t, 1]),

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from which it is clear that q is non-negative, non-decreasing and left-continuous.Moreover, ∫ 1

0q(t)dt =

∫ 1

0q(1− t)dt

=

∫ 1

0ν((t, 1])dt

=

∫ 1

0

∫(0,1]

1s∈(t,1]ν(ds)dt

=

∫(0,1]

∫ 1

01t∈(0,s]dt ν(ds)

=

∫(0,1]

sν(ds)

= µ((0, 1])

= 1.

2. Conversely, let q : (0, 1) → R+ be non-negative, non-decreasing and left-continuousand such that

∫ 10 q(t)dt = 1. We extend q to a function on [0, 1) by setting q(0) := 0,

and define f : (0, 1]→ R+ by

f(t) = −q(1− t).

f is non-decreasing and right-continuous and therefore induces a measure ν (theStieltjes measure) on (0, 1] by

ν((a, b]) := f(b)− f(a) = q(1− a)− q(1− b).

In particular, for b = 1 we get

ν((a, 1]) = q(1− a)

(because q(0) = 0). Let nowµ(ds) := sν(ds).

By essentially the same computation as above we obtain that µ is a probabilitymeasure:

µ((0, 1]) =

∫(0,1]

sν(ds)

=

∫(0,1]

∫ 1

01t∈(0,s)dt ν(ds)

=

∫ 1

0

∫(0,1]

1s∈(t,1]ν(ds) dt

=

∫ 1

0ν((t, 1]) dt

=

∫ 1

0q(1− t) dt

=

∫ 1

0q(t) dt

= 1.

28

Finally, the two procedures (µ → ν → q and q → ν → µ) are clearly inverse to eachother.

Theorem 8.10 (Kusuoaka’s representation theorem). Let ρ be a law-invariant convexrisk measure ρ on L∞(Ω,F) that is continuous from above. Then

ρ(X) = supµ∈M1((0,1])

(∫(0,1]

AVaRλ(X)µ(dλ)− γ(µ)

),

where

γ(µ) = supX∈Aρ

∫(0,1]

AVaRλ(X)µ(dλ).

Proof. By the preliminary version of Kusuoka’s representation theorem,

ρ(X) = supq

(∫ 1

0VaRt(X) q(1− t)dt− β(q)

),

where the supremum is over all non-decreasing functions q : (0, 1)→ R+ with∫ 1

0 q(t)dt = 1,and

β(q) = supX∈Aρ

∫ 1

0VaRt(X)q(1− t)dt.

For

q(t) =

∫(1−t,1]

1

sµ(ds)

we obtain ∫ 1

0VaRt(X) q(1− t)dt =

∫ 1

0VaRt(X)

∫(t,1]

1

sµ(ds)dt

=

∫ 1

0

∫(0,1]

VaRt(X)1s∈(t,1]1

sµ(ds)dt

=

∫(0,1]

∫ 1

0VaRt(X)1t∈(0,s]

1

sdtµ(ds)

=

∫(0,1]

1

s

∫ s

0VaRt(X)dt µ(ds)

=

∫(0,1]

AVaRs(X)µ(ds).

In view of the one-to-one correspondence between functions q and probability measures µthis proves the result.

29