Monetary Risk Measures
aa r X i v : . [ q -f i n . M F ] D ec Monetary Risk Measures ∗ Guangyan Jia † Jianming Xia ‡ Rongjie Zhao § Abstract
In this paper, we study general monetary risk measures (without any convexity orweak convexity). A monetary (respectively, positively homogeneous) risk measure canbe characterized as the lower envelope of a family of convex (respectively, coherent)risk measures. The proof does not depend on but easily leads to the classical repre-sentation theorems for convex and coherent risk measures. When the law-invarianceand the SSD (second-order stochastic dominance)-consistency are involved, it is notthe convexity (respectively, coherence) but the comonotonic convexity (respectively,comonotonic coherence) of risk measures that can be used for such kind of lower en-velope characterizations in a unified form. The representation of a law-invariant riskmeasure in terms of VaR is provided.
Key words:
Monetary risk measure, (comonotonic) convex risk measure, (comono-tonic) coherent risk measure, law-invariance
It is an important subject to measure the risk of a financial position. VaR (value at risk)has long been a standard risk measure in industry, whether by choice or by regulation.However, VaR has been criticized in both academia and industry, mainly for two short-comings: (1) VaR focuses on the probability of loss, regardless of the magnitude, andtherefore, fails to capture “tail risk”; (2) VaR is not sub-additive, and therefore, violatesthe principle of diversification.Recognizing the shortcomings of VaR, in their seminal work, Artzner, Delbaen, Eberand Heath (1999) argued that a good risk measure should satisfy a set of reasonableaxioms (monotonicity, translation-invariance, sub-additivity and positive homogeneity),leading to the so-called coherent risk measures. F¨ollmer and Schied (2002), as well as ∗ Supported by National Key R&D Program of China (NO. 2018YFA0703900) and the Major Project ofNational Social Science Foundation of China (NO.19ZDA091): Research on Dynamic Monitoring of LocalFinancial Operation and Early Warning of Systemic Risk. † Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China; Email:[email protected]. ‡ RCSDS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China; Email: [email protected]. § School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;Email: [email protected]. For representations of coherent risk measures on general probability spaces, see Delbaen (2002). For discussions on law-invariance of risk measures, see among others Kusuoka(2001), Frittelli and Rosazza Gianin (2005), Jouini, Schachermayer and Touzi (2006), andSong and Yan (2009).All kinds of risk measures above satisfy two basic axioms: monotonicity and translation-invariance. A risk measure satisfying the two basic axioms is usually called a monetaryrisk measure; see, e.g., F¨ollmer and Schied (2016). In this paper, our focus is on thecharacterizations and representations of a general monetary risk measure (without anyconvexity or weak convexity).A closely related work to ours is Mao and Wang (2020), which is fundamental and in-spiring. They argued that a risk measure should be consistent with risk aversion, which canbe described by the consistency with respect to SSD (second-order stochastic dominance),leading to the so-called SSD-consistent risk measures. They provided characterizationsand representations, as well as some applications, of SSD-consistent risk measures. Al-though no convexity is imposed a priori on an SSD-consistent risk measure, we will seethat an SSD-consistent risk measure still satisfies a kind of weak convexity, which is calledID-convexity (identical-distribution convexity) in this paper; see Theorem 5.5 below.The most famous non convex risk measure is VaR, which is excluded by Mao andWang (2020) since it is not SSD-consistent. Every risk measure has its advantages as wellas disadvantages. No one can do everything better than another. We do not argue forone risk measure against another, but investigate general monetary risk measures, whichinclude VaR as a special case.The contribution and the structure of our paper are as follows (the results for positivelyhomogeneous risk measures are similar and hence not list here for the brevity).Section 2 collects some preliminaries about risk measures and acceptance sets.In Section 3, we show that a monetary risk measure ρ is the lower envelope of a familyof convex risk measures and has the following representation ρ ( X ) = min λ ∈ Λ sup Q ∈ M ( P ) ( E Q [ − X ] − α λ ( Q )) , where M ( P ) denotes all probability measures which are absolutely continuous with re-spect to P and { α λ | λ ∈ Λ } is a family of convex functionals α λ : M ( P ) → ( −∞ , ∞ ].As a consequence, we have, for every monetary (similarly, positively homogeneous) risk Some other relaxations, from the translation-invariance and convexity to the quasi-convexity (andcash-subadditivity), were carried out by El Karoui and Ravanelli (2009) and Cerreia-Vioglio et al. (2011). ρ , ρ ( X ) = inf { h ( X ) | h convex (coherent) and h ≥ ρ } , which extends the following equality in Artzner, Delbaen, Eber and Heath (1999, Propo-sition 5.2): VaR t ( X ) = inf { h ( X ) | h coherent and h ≥ VaR t } . (1.1)It is interesting that the proof of the representation theorem for general monetary riskmeasures does not depend on but easily leads to the classical representation theorem forconvex and coherent risk measures, as Corollary 3.3 and Remark 3.5 show.In Section 4, we show that a law-invariant monetary risk measure has the followingrepresentation ρ ( X ) = min λ ∈ Λ sup g ∈G (cid:18)Z VaR t ( X ) g ( t ) dt − α λ ( g ) (cid:19) , where G denotes all probability density functions on (0 ,
1) and { α λ | λ ∈ Λ } is a family ofconvex functionals α λ : G → ( −∞ , ∞ ].In Section 5, we investigate SSD-consistent risk measures. Some results of Mao andWang (2020) are recovered and some other representations of SSD-consistent risk mea-sures are provided. We provide another equivalent formulation of SSD-consistency: SSD-consistency is equivalent to the combination of law-invariance, the Fatou property andID-convexity. As far as we know, the ID-convexity has not yet been introduced in theliterature to study risk measures.From Sections 3–5, we know each monetary risk measure is the lower envelope of afamily of convex risk measures. When the law-invariance is involved, it is a “natural”expectation that each law-invariant monetary risk measure is the lower envelope of afamily of law-invariant convex risk measures. This “natural” expectation, however, is notcorrect since law-invariant convex risk measures are SSD-consistent and hence so is thelower envelope of some of them. For example,VaR t ( X ) = inf { h ( X ) | h law-invariant and coherent and h ≥ VaR t } . (1.2)A natural question arises: What is the “good” property such that general (respectively, law-invariant,SSD-consistent) monetary risk measures can be represented in a unified form:they are the lower envelopes of a family of general (respectively, law-invariant,SSD-consistent) monetary risk measures having the “good” property?
Section 6 replies to this question. It turns out that the comonotonic convexity is thedesired “good” property. 3
Preliminaries: Risk Measures and Acceptance Sets
Let (Ω , F , P ) be a complete probability space. We use L ∞ ( P ) to denote L ∞ (Ω , F , P ) forbrevity. We study risk measures on L ∞ ( P ). Some basic definitions and facts are recalledin this section. Definition 2.1.
A mapping ρ : L ∞ ( P ) → R is called a monetary risk measure if it satisfiesthe following two conditions for all X, Y ∈ L ∞ ( P ) . • Monotonicity: If X ≤ Y P -a.s., then ρ ( X ) ≥ ρ ( Y ) . • Translation-Invariance: If m ∈ R , then ρ ( X + m ) = ρ ( X ) − m . Definition 2.2.
A monetary risk measure ρ : L ∞ ( P ) → R is called a positively homoge-neous risk measure if it satisfies • Positive Homogeneity: ρ ( αX ) = αρ ( X ) for X ∈ L ∞ ( P ) and α ≥ . Definition 2.3.
A monetary risk measure ρ : L ∞ ( P ) → R is called a convex risk measureif it satisfies • Convexity: ρ ( αX + (1 − α ) Y ) ≤ αρ ( X ) + (1 − α ) ρ ( Y ) for X, Y ∈ L ∞ ( P ) and α ∈ [0 , . Definition 2.4.
A monetary risk measure ρ : L ∞ ( P ) → R is called a coherent riskmeasure if it is convex and positively homogeneous. Given a monetary risk measure ρ , its acceptance set A ρ is given by A ρ , { X ∈ L ∞ ( P ) | ρ ( X ) ≤ } . Given a subset A ⊆ L ∞ ( P ), let the mapping ρ A be given by ρ A ( X ) , inf { m ∈ R | X + m ∈ A } , X ∈ L ∞ ( P ) . The following proposition summarizes the relation between monetary risk measures andtheir acceptance sets; see, e.g., F¨ollmer and Schied (2016, Propositions 4.6–4.7).
Proposition 2.5.
A mapping ρ : L ∞ ( P ) → R is a monetary risk measure if and only if ρ = ρ A for a nonempty subset A ⊆ L ∞ ( P ) satisfying the following two conditions. (i) inf { m ∈ R | m ∈ A } > −∞ ; (ii) X ∈ A , Y ∈ L ∞ ( P ) , Y ≥ X P -a.s. ⇒ Y ∈ A .If it is the case, the set A can be chosen as the acceptance set A ρ of ρ . Moreover, we have • ρ is a positively homogeneous risk measure if and only if A ρ is a cone; ρ is a convex risk measure if and only if A ρ is a convex set; • ρ is a coherent risk measure if and only if A ρ is a convex cone. The following easy lemma will be frequently used, whose proof is omitted.
Lemma 2.6.
Assume { A λ | λ ∈ Λ } is a class of subsets of L ∞ ( P ) and each A λ satisfiesconditions (i)–(ii) of Proposition 2.5. Let A = S λ ∈ Λ A λ . Then ρ A ( X ) = inf λ ∈ Λ ρ A λ ( X ) for X ∈ L ∞ ( P ) . Let M ( P ) = M (Ω , F , P )denote the set of all probability measures on (Ω , F ) which are absolutely continuous withrespect to P . Let M ,f ( P ) = M ,f (Ω , F , P )denote the set of all finitely additive probability measures on (Ω , F ) which are absolutelycontinuous with respect to P . It is well known that M ,f ( P ) is compact under the weak*-topology, which is induced by L ∞ ( P ).The following theorem shows that a monetary risk measure is the lower envelope of afamily of convex risk measures. Theorem 3.1.
For a mapping ρ : L ∞ ( P ) → R , the following assertions are equivalent. (a) ρ is a monetary risk measure. (b) There exists a family { α λ | λ ∈ Λ } of convex functionals α λ : M ( P ) → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ sup Q ∈ M ( P ) ( E Q [ − X ] − α λ ( Q )) for X ∈ L ∞ ( P ) . (c) There exists a family { α λ | λ ∈ Λ } of lower semi-continuous (under the weak* topol-ogy), convex functionals α λ : M ,f ( P ) → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ max Q ∈ M ,f ( P ) ( E Q [ − X ] − α λ ( Q )) for X ∈ L ∞ ( P ) . (d) There exists a family { ρ λ | λ ∈ Λ } of convex risk measures on L ∞ ( P ) such that ρ ( X ) = min λ ∈ Λ ρ λ ( X ) for X ∈ L ∞ ( P ) . (e) For each X ∈ L ∞ ( P ) , ρ ( X ) = inf { h ( X ) | h is a convex risk measure and h ≥ ρ } . (3.1)5 roof. We only prove “(a) ⇒ (b)” and “(a) ⇒ (c)”, since “(b) ⇒ (d) ⇒ (e) ⇒ (a)” and “(c) ⇒ (d)”are obvious. (a) ⇒ (b): Assume ρ is a monetary risk measure. For any Z ∈ A ρ , let A ( Z ) = { Y ∈ L ∞ ( P ) | Y ≥ Z P -a.s. } . Firstly, each A ( Z ) is obviously a convex subset of L ∞ ( P ) satisfying conditions (i)–(ii) of Proposition 2.5. Then by Proposition 2.5, each ρ A ( Z ) is a convex risk measure.Next, A ρ = S Z ∈ A ρ A ( Z ). Then by Lemma 2.6, ρ ( X ) = ρ A ρ ( X ) = inf Z ∈ A ρ ρ A ( Z ) ( X ) , ∀ X ∈ L ∞ ( P ) . (3.2)Moreover, for each X ∈ L ∞ ( P ), we have Z = X + ρ ( X ) ∈ A ρ and ρ A ( Z ) ( X ) = inf { m ∈ R | X + m ≥ X + ρ ( X ) } = ρ ( X ) , which implies that the infimum in (3.2) can be attained and hence ρ ( X ) = min Z ∈ A ρ ρ A ( Z ) ( X ) . Finally, for each X ∈ L ∞ ( P ) and Z ∈ A ρ , we have ρ A ( Z ) ( X ) = inf { m ∈ R | X + m ≥ Z P -a.s. } = ess sup ( Z − X )= sup Q ∈ M ( P ) E Q [ Z − X ]= sup Q ∈ M ( P ) ( E Q [ − X ] − α Z ( Q )) , (3.3)where α Z ( Q ) = E Q [ − Z ] for each Q ∈ M ( P ). Therefore, { α Z | Z ∈ A ρ } is a desiredfamily of convex functionals on M ( P ). (a) ⇒ (c): According the proof of “(a) ⇒ (b)”, we have ρ A ( Z ) ( X ) = ess sup ( Z − X )= max Q ∈ M ,f ( P ) E Q [ Z − X ]= max Q ∈ M ,f ( P ) ( E Q [ − X ] − α Z ( Q )) , where α Z ( Q ) = E Q [ − Z ] for each Q ∈ M ,f ( P ). Therefore, { α Z | Z ∈ A ρ } is adesired family of convex functionals on M ,f ( P ). (cid:3) Remark 3.2. (a)
From the proof of Theorem 3.1, where a similar argument of Mao andWang (2020) is used, we can see that a monetary risk measure ρ : L ∞ ( P ) → R hasthe following representations ρ ( X ) = min Z ∈ A ρ sup Q ∈ M ( P ) ( E Q [ − X ] − E Q [ − Z ])= min Z ∈ A ρ max Q ∈ M ,f ( P ) ( E Q [ − X ] − E Q [ − Z ]) , ∀ X ∈ L ∞ ( P ) . b) The characterizations (3.1) and (3.4) below are extensions of equality (1.1) , which isfrom Artzner, Delbaen, Eber and Heath (1999). (c)
For some related applications in decision theory under uncertainty, see Xia (2020).
As the next corollary shows, the proof of Theorem 3.1 does not depend on but leadsto the classical representation theorem for convex risk measures of F¨ollmer and Schied(2002) and Frittelli and Rosazza Gianin (2002).
Corollary 3.3.
Assume ρ : L ∞ ( P ) → R is a convex risk measure. Then there exists alower-semicontinuous convex functional α : M ,f ( P ) → ( −∞ , ∞ ] such that ρ ( X ) = max Q ∈ M ,f ( P ) ( E Q [ − X ] − α ( Q )) for X ∈ L ∞ ( P ) . Proof.
The convexity of ρ implies that its acceptance set A ρ is convex. Moreover, M ,f ( P ) is a weak*-compact and convex set. From the representations in Remark 3.2 andby the minmax theorem , we have for each X ∈ L ∞ ( P ) that ρ ( X ) = min Z ∈ A ρ max Q ∈ M ,f ( P ) ( E Q [ − X ] − E Q [ − Z ])= max Q ∈ M ,f ( P ) inf Z ∈ A ρ ( E Q [ − X ] − E Q [ − Z ])= max Q ∈ M ,f ( P ) ( E Q [ − X ] − α ( Q )) , where α ( Q ) = sup Z ∈ A ρ E Q [ − Z ] defines the desired lower-semicontinuous convex functionalon M ,f ( P ). (cid:3) The following theorem shows that a positively homogeneous risk measure is the lowerenvelope of a family of coherent risk measures.
Theorem 3.4.
For a mapping ρ : L ∞ ( P ) → R , the following assertions are equivalent. (a) ρ is a positively homogeneous risk measure. (b) There exists a family { Q λ | λ ∈ Λ } of nonempty, weak*-compact, convex subsets of M ,f ( P ) such that ρ ( X ) = min λ ∈ Λ max Q ∈ Q λ E Q [ − X ] for X ∈ L ∞ ( P ) . (c) There exists a family { ρ λ | λ ∈ Λ } of coherent risk measures on L ∞ ( P ) such that ρ ( X ) = min λ ∈ Λ ρ λ ( X ) for X ∈ L ∞ ( P ) . The minmax theorem is frequently used in this paper, for which see, e.g., Mertens, Sorin and Zamir(2015, Theorem I.1.1). d) For each X ∈ L ∞ ( P ) , ρ ( X ) = inf { h ( X ) | h is a coherent risk measure and h ≥ ρ } . (3.4) Proof.
We only prove “(a) ⇒ (b)”, since “(b) ⇒ (c) ⇒ (d) ⇒ (a)” is obvious.Assume ρ is a positively homogeneous risk measure. For any Z ∈ A ρ , let A ( Z ) begiven as in the proof of Theorem 3.1. By Proposition 2.5, we know that A ρ is a cone,which combined with Remark 3.2 imply that, for each X ∈ L ∞ ( P ), ρ ( X ) = min Z ∈ A ρ max Q ∈ M ,f ( P ) ( E Q [ − X ] + E Q [ Z ])= min Z ∈ A ρ , α ≥ max Q ∈ M ,f ( P ) ( E Q [ − X ] + E Q [ αZ ])= min Z ∈ A ρ inf α ≥ max Q ∈ M ,f ( P ) ( E Q [ − X ] + E Q [ αZ ])= min Z ∈ A ρ max Q ∈ M ,f ( P ) inf α ≥ ( E Q [ − X ] + E Q [ αZ ]) (by the minmax theorem)= min Z ∈ A ρ max Q ∈ M ,f ( P ) (cid:18) E Q [ − X ] + inf α ≥ E Q [ tZ ] (cid:19) = min Z ∈ A ρ max Q ∈ M ,f ( P ) E Q [ Z ] ≥ E Q [ − X ] , where we can apply the minmax theorem since M ,f ( P ) is weak*-compact. Let Q Z = { Q ∈ M ,f ( P ) | E Q [ Z ] ≥ } for each Z ∈ A ρ . Then { Q Z | Z ∈ A ρ } is a desired family ofsubsets of M ,f ( P ). (cid:3) Remark 3.5.
From the proof of Theorem 3.4, we can see that a positively homogeneousrisk measure ρ : L ∞ ( P ) → R has the following representation ρ ( X ) = min Z ∈ A ρ max Q ∈ M ,f ( P ) E Q [ Z ] ≥ E Q [ − X ] , ∀ X ∈ L ∞ ( P ) . Similar to Corollary 3.3, we can see that the proof of Theorem 3.4 does not depend onbut leads to the classical representation theorem for coherent risk measures of Artzner,Delbaen, Eber and Heath (1999).
From now on, we assume the probability space (Ω , F , P ) is nonatomic, that is, it supportsa random variable with a continuous distribution. Now we study the class of all riskmeasures which are law-invariant. Definition 4.1.
A monetary risk measure ρ : L ∞ ( P ) → R is called law-invariant if ρ ( X ) = ρ ( Y ) whenever X and Y have the same distribution under P . efinition 4.2. Given two random variables
X, Y ∈ L ∞ ( P ) . We say that X first-orderstochastically dominates (FSD, in short) Y and write it X (cid:23) Y , if E P [ f ( X )] ≥ E P [ f ( Y )] for all increasing functions f . It is easy to see that a monetary risk measure ρ : L ∞ ( P ) → R is law-invariant if andonly if it is FSD-consistent in the following sense: • FSD-Consistency: ρ ( X ) ≤ ρ ( Y ) whenever X (cid:23) Y .For a random variable X ∈ L ∞ ( P ), its right-continuous probability distribution func-tion is denoted by F X and its (upper) quantile function q X : [0 , → R is given by q X ( t ) , inf { x ∈ R | F X ( x ) > t } , t ∈ [0 , ,q X (1) , q X (1 − ) = lim t ↑ q X ( t ) . For more details about quantile functions, see Appendix A.3 of F¨ollmer and Schied (2016).One of the most well known law-invariant risk measures is VaR. The VaR of X at level t ∈ [0 ,
1] is given by VaR t ( X ) , − q X ( t ) . It is well known that, for any
X, Y ∈ L ∞ ( P ), X (cid:23) Y ⇔ VaR t ( X ) ≤ VaR t ( Y ) , ∀ t ∈ [0 , . (4.1)Now we introduce some notations. • G denotes all nonnegative Borel functions g on (0 ,
1) such that R g ( t ) dt = 1. Thatis, G is the set of all probability density functions on (0 , • B [0 , denotes all Borel subsets of [0 , • ℓ denotes the Lebesgue measure on [0 , • M ,f ( ℓ ) denotes all finitely additive probability measures on ([0 , , B [0 , ) which areabsolutely continuous with respect to ℓ .It is well known that M ,f ( ℓ ) is compact under the weak*-topology, which is inducedby L ∞ ([0 , , B [0 , ) , ℓ ).The following theorem characterizes law-invariant monetary risk measures on L ∞ ( P ). Theorem 4.3.
Assume the probability space (Ω , F , P ) is nonatomic. For a mapping ρ : L ∞ ( P ) → R , the following two assertions are equivalent. Throughout the paper “increasing” means “non-decreasing” and “decreasing” means “non-increasing.” a) ρ is a law-invariant monetary risk measure. (b) There exists a family { α λ | λ ∈ Λ } of convex functionals α λ : G → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ sup g ∈G (cid:18)Z VaR t ( X ) g ( t ) dt − α λ ( g ) (cid:19) for X ∈ L ∞ ( P ) . (4.2) (c) There exists a family { α λ | λ ∈ Λ } of lower semi-continuous (under the weak*-topology), convex functionals α λ : M ,f ( ℓ ) → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ max µ ∈ M ,f ( ℓ ) (cid:18)Z VaR t ( X ) µ ( dt ) − α λ ( µ ) (cid:19) for X ∈ L ∞ ( P ) . (4.3) Proof. “(b) ⇒ (a)” and “(c) ⇒ (a)” are obvious.Now we show “(a) ⇒ (b)”. Assume ρ is a law-invariant monetary risk measure. For any Z ∈ A ρ , let D ( Z ) = { Y ∈ L ∞ ( P ) | Y (cid:23) Z } . By a similar discussion as in the proof of Theorem 3.1, we can see that each ρ D ( Z ) is alaw-invariant monetary risk measure and ρ ( X ) = min Z ∈ A ρ ρ D ( Z ) ( X ) . Moreover, for each X ∈ L ∞ ( P ) and Z ∈ A ρ , we have ρ D ( Z ) ( X ) = inf { m ∈ R | X + m (cid:23) Z } = inf { m ∈ R | VaR t ( X + m ) ≤ VaR t ( Z ) , ∀ t ∈ (0 , } = inf { m ∈ R | VaR t ( X ) − m ≤ VaR t ( Z ) , ∀ t ∈ (0 , } = sup t ∈ (0 , (VaR t ( X ) − VaR t ( Z ))= sup g ∈G Z (VaR t ( X ) − VaR t ( Z )) g ( t ) dt, = sup g ∈G (cid:18)Z VaR t ( X ) g ( t ) dt − α Z ( g ) (cid:19) , where α Z ( g ) = R VaR t ( Z ) g ( t ) dt for each g ∈ G . Therefore, { α Z | Z ∈ A ρ } is a desiredfamily of convex functionals on G .The proof of “(a) ⇒ (c)” is similar to the corresponding part of Theorem 3.1. (cid:3) Remark 4.4.
From the proof of Theorem 4.3, we can see that a law-invariant monetaryrisk measure ρ : L ∞ ( P ) → R has the following representation ρ ( X ) = min Z ∈ A ρ sup g ∈G (cid:18)Z VaR t ( X ) g ( t ) dt − Z VaR t ( Z ) g ( t ) dt (cid:19) = min Z ∈ A ρ max µ ∈ M ,f ( ℓ ) (cid:18)Z VaR t ( X ) µ ( dt ) − Z VaR t ( Z ) µ ( dt ) (cid:19) , for all X ∈ L ∞ ( P ) . L ∞ ( P ), whose proof, based on Theorem 4.3, is similar to Theorem 3.4 and henceomitted. Theorem 4.5.
Assume the probability space (Ω , F , P ) is nonatomic. For a mapping ρ : L ∞ ( P ) → R , the following two assertions are equivalent. (a) ρ is a law-invariant, positively homogeneous risk measure. (b) There exists a family { M λ | λ ∈ Λ } of nonempty, weak*-compact, convex subsets of M ,f ( ℓ ) such that ρ ( X ) = min λ ∈ Λ max µ ∈ M λ Z VaR t ( X ) µ ( dt ) for X ∈ L ∞ ( P ) . Remark 4.6.
Similarly to Remark 3.5, we can see that a law-invariant, positively homo-geneous risk measure ρ : L ∞ ( P ) → R has the following representation ρ ( X ) = min Z ∈ A ρ max µ ∈ M ,f ( ℓ ) R VaR t ( Z ) µ ( dt ) ≤ Z VaR t ( X ) µ ( dt ) , ∀ X ∈ L ∞ ( P ) . Now we study SSD-consistent risk measures in this section.
Definition 5.1.
Given two random variables
X, Y ∈ L ∞ ( P ) . We say that X second-orderstochastically dominates Y and write it X (cid:23) Y , if E P [ f ( X )] ≥ E P [ f ( Y )] for all increasing and concave functions f . Definition 5.2.
A monetary risk measure ρ : L ∞ ( P ) → R is called SSD-consistent if ρ ( X ) ≤ ρ ( Y ) whenever X (cid:23) Y . Mao and Wang (2020) comprehensively investigated SSD-consistent risk measures andprovided four equivalent conditions of SSD-consistency. Theorem 5.5 below gives anotherone: SSD-consistency is equivalent to the combination of law-invariance, the Fatou prop-erty and a kind of weak convexity, called ID-convexity. As far as we know, the ID-convexityhas not yet been introduced in the literature to study risk measures.
Definition 5.3.
A monetary risk measure ρ : L ∞ ( P ) → R is called ID-convex (identical-distribution convex) if it satisfies • ID-Convexity: ρ ( P ni =1 α i X i ) ≤ P ni =1 α i ρ ( X i ) whenever X X , . . . , X n are identi-cally distributed under P , each α i ≥ and P ni =1 α i = 1 . efinition 5.4. We say a monetary risk measure ρ : L ∞ ( P ) → R has the Fatou propertyif ( X n ) is a bounded sequence in L ∞ ( P ) X n → X P -a.s. ) ⇒ lim inf n →∞ ρ ( X n ) ≥ ρ ( X ) . Theorem 5.5.
Assume the probability space (Ω , F , P ) is nonatomic. For a monetary riskmeasure ρ : L ∞ ( P ) → R , the following two assertions are equivalent. (a) ρ is SSD-consistent. (b) ρ is law-invariant and ID-convex and has the Fatou property. Proof.“(a) ⇒ (b)”: Assume ρ is SSD-consistent. The law-invariance of ρ is clear. The Fatouproperty is from Mao and Wang (2020, Theorem 3.5). It remains to show the ID-convexity. Actually, let X X , . . . , X n be identically distributed under P , each α i ≥ P ni =1 α i = 1. Obviously, we have P ni =1 α i X i (cid:23) X and hence ρ n X i =1 α i X i ! ≤ ρ ( X ) = n X i =1 α i ρ ( X i ) . Therefore, ρ is ID-convex. “(b) ⇒ (a)”: Assume ρ is law-invariant and ID-convex and has the Fatou property. Forany X, Y ∈ L ∞ ( P ) with X (cid:23) Y , we need to show ρ ( X ) ≤ ρ ( Y ). Actually, by aresult of Ryff (1967), see also Carlier and Lachapelle (2011, Lemma 2.3), there existsa sequence Z n of the form Z n = P N n i =1 α ni Y ni with α ni ≥ P N n i =1 α ni = 1 and each Y ni has the same distribution of Y , such that Z n converges to X P -a.s.. Then by thelaw-invariance, ID-convexity and the Fatou property of ρ , we have ρ ( X ) ≤ lim inf n →∞ ρ ( Z n ) ≤ lim inf n →∞ N n X i =1 α ni ρ ( Y ni ) = ρ ( Y ) . (cid:3) The most well known SSD-consistent risk measure is AVaR (average value at risk ).The AVaR of X at level t ∈ [0 ,
1] is given byAVaR t ( X ) , t Z t VaR s ( X ) ds, t ∈ (0 , , AVaR ( X ) , VaR ( X ) . It is well known that, for any
X, Y ∈ L ∞ ( P ), X (cid:23) Y ⇔ AVaR t ( X ) ≤ AVaR t ( Y ) , ∀ t ∈ [0 , . (5.1) Also sometimes termed “expected shortfall” or “conditional value at risk” in the literature. W , { w : [0 , → [0 , | ≤ w (0) ≤ w (1) = 1 , w is increasing and right-continuous } . Then W is the set of all probability distribution functions on [0 , w ∈ W , the probability of { } is w (0) and the probability of { } is1 − w (1 − ). It is well known that W is compact under the weak topology, which is inducedby all continuous functions on [0 , Theorem 5.6.
Assume the probability space (Ω , F , P ) is nonatomic. For a mapping ρ : L ∞ ( P ) → R , the following assertions are equivalent. (a) ρ is an SSD-consistent monetary risk measure. (b) There exists a family { α λ | λ ∈ Λ } of convex functionals α λ : G → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ sup g ∈G (cid:18)Z AVaR t ( X ) g ( t ) dt − α λ ( g ) (cid:19) for X ∈ L ∞ ( P ) . (5.2) (c) There exists a family { α λ | λ ∈ Λ } of lower semi-continuous (under the weak topology),convex functionals α λ : W → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ max w ∈W Z [0 , AVaR t ( X ) dw ( t ) − α λ ( w ) ! for X ∈ L ∞ ( P ) . (5.3) (d) There exists a family { ρ λ | λ ∈ Λ } of law-invariant, convex risk measures on L ∞ ( P ) such that ρ ( X ) = min λ ∈ Λ ρ λ ( X ) for X ∈ L ∞ ( P ) . (e) For each X ∈ L ∞ ( P ) , ρ ( X ) = inf { h ( X ) | h is a law-invariant, convex risk measure and h ≥ ρ } . Proof.
The “(b) ⇒ (d) ⇒ (e) ⇒ (a)” and “(c) ⇒ (d)” parts are obvious. The proof of “(a) ⇒ (b)”and “(a) ⇒ (c)” parts is similar to Theorem 4.3. (cid:3) Theorem 5.7.
Assume the probability space (Ω , F , P ) is nonatomic. For a mapping ρ : L ∞ ( P ) → R , the following assertions are equivalent. (a) ρ is an SSD-consistent, positively homogeneous risk measure. (b) There exists a family {W λ | λ ∈ Λ } of nonempty, weakly compact, convex subsets of W such that ρ ( X ) = min λ ∈ Λ max w ∈W λ Z [0 , AVaR t ( X ) dw ( t ) for X ∈ L ∞ ( P ) . c) There exists a family { ρ λ | λ ∈ Λ } of law-invariant, coherent risk measures on L ∞ ( P ) such that ρ ( X ) = min λ ∈ Λ ρ λ ( X ) for X ∈ L ∞ ( P ) . (d) For each X ∈ L ∞ ( P ) , ρ ( X ) = inf { h ( X ) | h is a law-invariant, coherent risk measure and h ≥ ρ } . Proof.
It is similar to Theorem 3.4, based on Theorem 5.6. (cid:3)
Remark 5.8.
Similar to Remarks 4.4 and 4.6, we have the following two assertions. (a)
An SSD-consistent monetary risk measure ρ : L ∞ ( P ) → R has the following repre-sentations ρ ( X ) = min Z ∈ A ρ sup g ∈G (cid:18)Z AVaR t ( X ) g ( t ) dt − Z AVaR t ( Z ) g ( t ) dt (cid:19) = min Z ∈ A ρ max w ∈W Z [0 , AVaR t ( X ) dw ( t ) − Z [0 , AVaR t ( Z ) dw ( t ) ! , ∀ X ∈ L ∞ ( P ) . (b) An SSD-consistent, positively homogeneous risk measure ρ : L ∞ ( P ) → R has thefollowing representation ρ ( X ) = min Z ∈ A ρ max w ∈W R [0 , AVaR t ( Z ) dw ( t ) ≤ Z [0 , AVaR t ( X ) dw ( t ) , ∀ X ∈ L ∞ ( P ) . The SSD-consistency obviously implies the law-invariance. Then a question arises:what is the representation in terms of VaR for an SSD-consistent monetary risk measure,when it is regarded as a law-invariant monetary risk measure? The same question arisesfor SSD-consistent, positively homogeneous risk measures as well. They are positivelyreplied by the next corollary, where the following notations will be used. G ↓ , { g ∈ G | g is decreasing on (0 , } , Ψ , { ψ ∈ W | ψ is concave on [0 , } . By F¨ollmer and Schied (2016, Lemma 4.69), the identity ( ψ ′ ( t ) = R ( t, s − dw ( s ) , t ∈ [0 , ψ (0) = w (0) (5.4)defines a bijection J : W → Ψ , ψ ′ denotes the right derivative. Under identity (5.4), an application of Fubinitheorem implies that Z [0 , VaR t ( X ) dψ ( t ) = Z [0 , AVaR t ( X ) dw ( t ) , ∀ X ∈ L ∞ ( P ) . (5.5)Let Ψ be endowed with the topology induced by the bijection J . Then W and Ψ arehomeomorphic and J is a homeomorphism. Therefore, Ψ is a compact space and, forevery X ∈ L ∞ ( P ), the functional ψ Z [0 , VaR t ( X ) dψ ( t )is continuous on Ψ. Corollary 5.9.
Assume the probability space (Ω , F , P ) is nonatomic. For a mapping ρ : L ∞ ( P ) → R , we have the following two assertions. (a) ρ is an SSD-consistent monetary risk measure if and only if one of the followingconditions holds. (i) There exists a family { α λ | λ ∈ Λ } of convex functionals α λ : G ↓ → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ sup g ∈G ↓ (cid:18)Z VaR t ( X ) g ( t ) dt − α λ ( g ) (cid:19) for X ∈ L ∞ ( P ) . (5.6) (ii) There exists a family { α λ | λ ∈ Λ } of lower semi-continuous, convex functionals α λ : Ψ → ( −∞ , ∞ ] such that ρ ( X ) = min λ ∈ Λ max ψ ∈ Ψ Z [0 , VaR t ( X ) dψ ( t ) − α λ ( ψ ) ! for X ∈ L ∞ ( P ) . (b) ρ is an SSD-consistent, positively homogeneous risk measure if and only if there existsa family { Ψ λ | λ ∈ Λ } of nonempty, compact, convex subsets of Ψ such that ρ ( X ) = min λ ∈ Λ max ψ ∈ Ψ λ Z [0 , VaR t ( X ) dψ ( t ) for X ∈ L ∞ ( P ) . Proof.
It is an obvious consequence of Theorems 5.6 and 5.7. (cid:3)
Remark 5.10.
Similar to Remark 5.8, an SSD-consistent monetary risk measure ρ : L ∞ ( P ) → R has the following representations ρ ( X ) = min Z ∈ A ρ sup g ∈G ↓ (cid:18)Z VaR t ( X ) g ( t ) dt − Z VaR t ( Z )) g ( t ) dt (cid:19) = min Z ∈ A ρ max ψ ∈ Ψ Z [0 , VaR t ( X ) dψ ( t ) − Z [0 , VaR t ( Z ) dψ ( t ) ! , for all X ∈ L ∞ ( P ) . Characterizations in a Unified Form
We have shown that each monetary risk measure ρ is the lower envelope of a family ofconvex risk measures; particularly, ρ ( X ) = inf { h ( X ) | h convex and h ≥ ρ } . When the law-invariance is involved, it is a “natural” expectation that each law-invariantmonetary risk measure is the lower envelope of a family of law-invariant convex risk mea-sures. But each law-invariant convex risk measure is SSD-consistent. Therefore, the lowerenvelope of a family of law-invariant convex risk measures must be SSD-consistent as well.As a consequence, the previous “natural” expectation is not correct. For example,VaR t ( X ) = inf { h ( X ) | h law-invariant, convex and h ≥ VaR t } . Furthermore, as Theorem 5.6 shows, each SSD-consistent monetary risk measure ρ is thelower envelope of a family of law-invariant convex risk measures; particularly, ρ ( X ) = inf { h ( X ) | h law-invariant, convex and h ≥ ρ } . A natural question arises: what is a “good” property such that general/law-invariant/SSD-consistent monetary risk measures can be characterized in a unified form as follows? • A monetary risk measure is the lower envelope of a family of “good” risk measures. • A law-invariant monetary risk measure is the lower envelope of a family of law-invariant, “good” risk measures. • An SSD-consistent monetary risk measure is the lower envelope of a family of SSD-consistent, “good” risk measures.The convexity is two strict to unify the characterizations, as we have seen. It turns out thatthe “comonotonic convexity” (CoM-convexity, for short) is the desired “good” property,which is weaker than the convexity and were introduced by Song and Yan (2006, 2009),as well as by Kou, Peng and Heyde (2013), to investigate risk measures. Some basicdefinitions and facts about CoM-convex (CoM-coherent) risk measures are summarized inAppendix A.The following theorem replies to the previous question.
Theorem 6.1.
For a mapping ρ : L ∞ ( P ) → R , we have the following assertions. (a) ρ is a monetary risk measure if and only if it is the lower envelope of a family ofCoM-convex risk measures. (b) Assume (Ω , F , P ) is nonatomic, then ρ is a law-invariant monetary risk measure ifand only if it is the lower envelope of a family of law-invariant, CoM-convex riskmeasures. c) Assume (Ω , F , P ) is nonatomic, then ρ is an SSD-consistent monetary risk measureif and only if it is the lower envelope of a family of SSD-consistent, CoM-convex riskmeasures. Proof. (a) is an obvious consequence of Theorem 3.1. (c) is a consequence of Theorems5.6 and A.8. The “if” part of (b) is obvious. Now we show the “only if” part of (b).Assume ρ is a law-invariant monetary risk measure. Recalling the proof of Theorem4.3, the set D ( Z ) is given by D ( Z ) = { Y ∈ L ∞ ( P ) | Y (cid:23) Z } . It is easy to see that D ( Z ) is CoM-convex in the following sense: αX + (1 − α ) Y ∈ D ( Z ) whenever X, Y ∈ D ( Z ) are comonotonic and α ∈ [0 , . Then it is easy to verify that ρ D ( Z ) is CoM-convex. Therefore, { ρ D ( Z ) | Z ∈ A ρ } is adesired family of law-invariant, CoM-convex risk measures, whose lower envelope is ρ . (cid:3) Similarly, we have the following theorem.
Theorem 6.2.
For a mapping ρ : L ∞ ( P ) → R , we have the following assertions. (a) ρ is a positively homogeneous risk measure if and only if it is the lower envelope of afamily of CoM-coherent risk measures. (b) Assume (Ω , F , P ) is nonatomic, then ρ is a law-invariant, positively homogeneousrisk measure if and only if it is the lower envelope of a family of law-invariant,CoM-coherent risk measures. (c) Assume (Ω , F , P ) is nonatomic, then ρ is an SSD-consistent, positively homogeneousrisk measure if and only if it is the lower envelope of a family of SSD-consistent,CoM-coherent risk measures. The next two corollaries show that the proofs of Theorem 4.3 and Corollary 5.9lead to another proof of the representation theorems of Song and Yan (2009) for law-invariant/SSD-consistent and CoM-convex/CoM-coherent risk measures.
Corollary 6.3.
Assume (Ω , F , P ) is nonatomic and ρ : L ∞ ( P ) → R is a law-invariantCoM-convex risk measure. Then there exists a lower-semicontinuous convex functional α : M ,f ( ℓ ) → ( −∞ , ∞ ] such that ρ ( X ) = max µ ∈ M ,f ( ℓ ) (cid:18)Z VaR t ( X ) µ ( dt ) − α ( µ ) (cid:19) , for X ∈ L ∞ ( P ) . Furthermore, assume ρ : L ∞ ( P ) → R is law-invariant and CoM-coherent. Then thereexists a nonempty, weak* compact and convex subset M ⊆ M ,f ( ℓ ) such that ρ ( X ) = max µ ∈ M Z VaR t ( X ) µ ( dt ) , for X ∈ L ∞ ( P ) . roof. Assume ρ : L ∞ ( P ) → R is a law-invariant CoM-convex risk measure. Let Q denote the set of quantile functions q X of all random variables X ∈ L ∞ ( P ). Consider asubset Q ρ of Q given by Q ρ = { q ∈ Q | q = q Z for some Z ∈ A ρ } . Then Q ρ is a convex set. Actually, for any q , q ∈ Q ρ , we have q i = q Z i for some Z i ∈ A ρ , i = 1 ,
2. Consider random variables Y i = q i ( U ), i = 1 ,
2, where U is a (0 , i , Z i and Y i have the samedistribution. Moreover, Y and Y are comonotonic. Then we have for each given α ∈ (0 , αq + (1 − α ) q = q Y , where Y = αY + (1 − α ) Y . Moreover, ρ ( Y ) ≤ αρ ( Y ) + (1 − α ) ρ ( Y ) = αρ ( Z ) + (1 − α ) ρ ( Z ) ≤ , which implies that Y ∈ A ρ and hence αq + (1 − α ) q ∈ Q ρ . Therefore, Q ρ is convex.The convexity of Q ρ and the weak*-compactness and convexity of M ,f ( ℓ ) allows foran application of the minmax theorem to the representations in Remark 4.4, which leadsto, for each X ∈ L ∞ ( P ), ρ ( X ) = min q ∈Q ρ max µ ∈ M ,f ( ℓ ) (cid:18)Z VaR t ( X ) µ ( dt ) + Z q ( t ) µ ( dt ) (cid:19) = max µ ∈ M ,f ( ℓ ) inf q ∈Q ρ (cid:18)Z VaR t ( X ) µ ( dt ) + Z q ( t ) µ ( dt ) (cid:19) = max µ ∈ M ,f ( ℓ ) (cid:18)Z VaR t ( X ) µ ( dt ) − α ( µ ) (cid:19) , where α ( µ ) = sup q ∈Q ρ (cid:16) − R q ( t ) µ ( dt ) (cid:17) defines the desired lower-semicontinuous convexfunctional on M ,f ( ℓ ).Furthetrmore, if ρ is law-invariant and CoM-coherent, then it is easy to see that A ρ and Q ρ are cones, which implies that α ( µ ) = sup q ∈Q ρ (cid:18) − Z q ( t ) µ ( dt ) (cid:19) = sup q ∈Q ρ ,α> (cid:18) − Z αq ( t ) µ ( dt ) (cid:19) = ( q ∈Q ρ R q ( t ) µ ( dt ) ≥ , ∞ otherwise . Let M = ( µ ∈ M ,f ( ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup q ∈Q ρ Z q ( t ) µ ( dt ) ≥ ) . Then M is the desired subset of M ,f ( ℓ ). (cid:3) Corollary 6.4.
Assume (Ω , F , P ) is nonatomic and ρ : L ∞ ( P ) → R is an SSD-consistentCoM-convex risk measure. Then there exists a lower-semicontinuous convex functional : Ψ → ( −∞ , ∞ ] such that ρ ( X ) = max ψ ∈ Ψ Z [0 , VaR t ( X ) ψ ( dt ) − α ( ψ ) ! , for X ∈ L ∞ ( P ) . Furthermore, assume ρ : L ∞ ( P ) → R is SSD-consistent and CoM-coherent. Then thereexists a nonempty, compact and convex subset Ψ ⊆ Ψ such that ρ ( X ) = max ψ ∈ Ψ Z [0 , VaR t ( X ) ψ ( dt ) , for X ∈ L ∞ ( P ) . Proof.
It is similar to Corollary 6.3, based on Remarks 5.10, and hence omitted. (cid:3)
AppendixA Comonotonic Convex Risk Measures
Definition A.1.
Two random variables
X, Y ∈ L ∞ ( P ) are called comonotonic if thereexists some Ω ∈ F such that P (Ω ) = 1 and ( X ( ω ) − X ( ω ′ ))( Y ( ω ) − Y ( ω ′ )) ≥ for ω, ω ′ ∈ Ω . Definition A.2.
A monetary risk measure ρ : L ∞ ( P ) → R is called comonotonic additive(CoM-additive, in short) if ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) whenever X, Y ∈ L ∞ ( P ) are comonotonic. Definition A.3.
A monetary risk measure ρ : L ∞ ( P ) → R is called comonotonic convex(CoM-convex, in short) if it satisfies • CoM-Convexity: ρ ( αX + (1 − α ) Y ) ≤ αρ ( X ) + (1 − α ) ρ ( Y ) whenever X, Y ∈ L ∞ ( P ) are comonotonic and α ∈ [0 , . Definition A.4.
A monetary risk measure ρ : L ∞ ( P ) → R is called a comonotoniccoherent (CoM-coherent, in short) if it is CoM-convex and positively homogeneous. Definition A.5.
A set function c : F → [0 , is called a capacity on (Ω , F ) if c ( ∅ ) = 0 , c (Ω) = 1 , and c ( A ) ≤ c ( B ) whenever A ⊂ B . A capacity c on (Ω , F ) is called absolutelycontinuous with respect to P if c ( A ) = c ( B ) whenever P ( A △ B ) = 0 . Let C ( P ) denote all capacities on (Ω , F ) that are absolutely continuous with respectto P . It is well known that a monetary risk measure ρ : L ∞ ( P ) → R is CoM-additive ifand only if there exists some capacity c ∈ C ( P ) such that ρ ( X ) = Z ( − X ) dc for X ∈ L ∞ ( P ) , R ( − X ) dc is the Choquet integral of − X with respect to c ; see, e.g., Schmeidler(1986), and also F¨ollmer and Schied (2016).Obviously, any convex (coherent) risk measure is CoM-convex (CoM-coherent). For thefollowing two theorems, which characterize CoM-convex (CoM-coherent) risk measures, seeSong and Yan (2006) as well as Kou, Peng and Heyde (2013). Theorem A.6.
A mapping ρ : L ∞ ( P ) → R is a CoM-convex risk measure if and only ifthere exists some convex functional α : C ( P ) → ( −∞ , ∞ ] such that ρ ( X ) = sup c ∈ C ( P ) (cid:26)Z ( − X ) dc − α ( c ) (cid:27) for X ∈ L ∞ ( P ) . Theorem A.7.
A mapping ρ : L ∞ ( P ) → R is a CoM-coherent risk measure if and onlyif there exists some nonempty subset C ⊆ C ( P ) such that ρ ( X ) = sup c ∈ C (cid:26)Z ( − X ) dc (cid:27) for X ∈ L ∞ ( P ) . For the following theorem, which characterizes SSD-consistent, CoM-convex (CoM-coherent) risk measures, see Song and Yan (2009, Theorems 3.2 and 3.6).
Theorem A.8.
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