A composition between risk and deviation measures
aa r X i v : . [ q -f i n . R M ] M a y A composition between risk and deviation measures
Marcelo Brutti Righi ∗ Federal University of Rio Grande do Sul [email protected]
Abstract
The intuition of risk is based on two main concepts: loss and variability. Inthis paper, we present a composition of risk and deviation measures, which con-template these two concepts. Based on the proposed Limitedness axiom, we provethat this resulting composition, based on properties of the two components, is acoherent risk measure. Similar results for the cases of convex and co-monotonerisk measures are exposed. We also provide examples of known and new risk mea-sures constructed under this framework in order to highlight the importance of ourapproach, especially the role of the Limitedness axiom.
Keywords : Coherent risk measures, Generalized deviation measures, Convex risk mea-sures, Co-monotone coherent risk measures, Limitedness.
The intuition of risk is based on two main concepts: the possibility of a negative outcome,i.e. a loss; and the variability in terms of an expected result, i.e. a deviation. Since thetime when the modern theory of finance was accepted, the role of risk measurementhas attracted attention. Initially, it was predominantly used as a dispersion measure,such as variance, which contemplates the second pillar of the intuition. More recently,the occurrence of critical events has turned the attention to tail-risk measurement, asis the case of well-known Value at Risk (VaR) and Expected Shortfall (ES) measures,which contemplate the first pillar. Theoretical and mathematical discussions have gainedattention in the literature, giving importance to distinct axiomatic structures for classesof risk measures and their properties. See F¨ollmer and Weber (2015) for a recent review.Despite their fundamental importance, such classes present a very wide range for thoserisk measures that can be understood as valid or useful. Thus, they can be considered asa first step, in which measures with poor theoretical properties are discarded. The nextstep would be to consider, inside a class, those measures more suited to practical use.Thus, to ensure a more complete measurement, it is reasonable to consider contemplatingboth pillars of intuition on risk. These pillars include the possibility of negative resultsand variability over an expected result, as a single measure. ∗ We would like to thank two anonymous reviewers for their comments, which have helped to improveour manuscript. We are also grateful for the financial support of CNPq (Brazilian Research Council)and FAPERGS (Rio Grande do Sul State Research Council). ρ + D in order to maintaindesired theoretical properties, which are central to the theory of risk measures. In ourmain context, ρ is a coherent risk measure in the sense of Artzner et al. (1999), whereas D is a generalized deviation measure, as proposed by Rockafellar et al. (2006). Thefinancial interpretation of ρ + D is the same as any coherent risk measure, but it servesas a more conservative protection once it yields higher values due o the penalty resultingfrom dispersion, while keeping the desired properties. Nevertheless, instead of makingthis protection arbitrary, our approach contemplates a deviation term and leads to desiredtheoretical properties.We prove a useful result that relates Limitedness; an axiom we propose of the form ρ ( X ) ≤ − inf( X ), with Monotonicity and Lower Range Dominance. The milestone isthat, in these cases, we always obtain D ( X ) ≤ − ρ ( X ) − inf X , i.e. the dispersion termconsiders ’financial information’ from the interval between the loss represented by ρ andthe maximum loss − inf X = sup − X . Thus, we can state that this combination is againa coherent risk measure. Under Translation Invariance, one can think in ρ ( X ) + D ( X )as ρ ( X ′ ), where X ′ = X − D ( X ), i.e. a real valued penalization on the initial position X . Moreover, this can be extended to acceptance sets, which are composed by positions X with non-positive risk, of the form A ρ + D := { X : ρ ( X ) + D ( X ) ≤ } = { X : ρ ( X ) ≤−D ( X ) } . In this sense, it is possible to explicitly observe the penalization reasoning interms of the deviation term. A position must have risk, in terms of the loss measure ρ ,at most of −D ( X ) ≤ X ′ = X − D ( X ) works as a penalization, A ρ + D isnot an acceptance set without Limitedness for ρ + D because Monotonicity plays a keyrole.Moreover, we also discuss issues regarding Law Invariance and representations intro-duced in Kusuoka (2001). Our results can be extended to the case of convex measuresin the sense of F¨ollmer and Schied (2002), Frittelli and Rosazza Gianin (2002) and Pflug(2006), or co-monotone coherent measures, as for the spectral or distortion classes pro-posed by Acerbi (2002) and Grechuk et al. (2009). We also provide some examples ofknown and new proposed functionals composed by risk and deviation measures in order2o illustrate our results, especially the role of our Limitedness axiom. In these examples,it is possible to generate the deviation term from a chosen risk measure, which easesthe financial meaning. It is valid to point out that, for practical matters, both ρ and D will be in the same monetary unit, but our results are valid even if this is not the case.Moreover, we are concerned over how to make a composition between risk and deviationmeasures rather than to claim it as new classes of risk measures. We highlight that,beyond the specific examples we expose, any combination of risk and deviation measuresleading to Limitedness can be taken into consideration under the results we present inthis paper. Moreover, our approach is static and univariate, which is standard in riskmeasures theory. Extensions to dynamic and multivariate cases are beyond our scope.Furthermore, extensions to a robust framework induced by uncertainty on models, forrisk forecasting as in Righi and Ceretta (2015), linked to probability measures are alsobeyond the present scope.We contribute to existing literature because, to the best of our knowledge, no suchresult as that proposed by us has been considered in previous studies. Rockafellar et al.(2006) presented an interplay between coherent risk measures and generalized deviationmeasures, and Rockafellar and Uryasev (2013) proposed a risk quadrangle, where thisrelationship is extended by adding intersections with concepts of error and regret undera generator statistic. In fact, these authors prove that any given generalized deviation D with D ≤ E [ X ] − inf X , one can obtain the coherent risk measure E [ − X ] + D ( X ).However, these studies are centered on an interplay of concepts, rather than a combinationthat joins both pillars of the intuition on risk, since their formulation is only valid, in ournotation, for the case ρ ( X ) = E [ − X ]. Filipovi´c and Kupper (2007) presented results inwhich convex functions possess Monotonicity and Translation Invariance, both of whichare convex risk measures. Nonetheless, their result is based on the supremum of functionson a vector space, and not on a relation of axioms for risk measures such as in ourapproach. Furthermore, we also present and prove results about some new examples ofrisk measures that rely on our approach.The reader should notice that the goal is to compose a new functional from risk anddeviation terms, instead of decomposing a given functional into these two components.The key point is to simultaneously consider both concepts (risk and deviation) in a singlefunctional and to guarantee the presence of theoretical properties. The approach for ac-tuarial science, a sum of expectation plus a risk loading, does not necessarily guaranteethe theoretical properties, as is the case for mean plus standard deviation, for instance.Other possibilities beyond the linear sum of risk and deviation, such as a risk measurewith more risk aversion to mean-preserving spreads, may be understood as another mea-surement of the risk term and do not explicit the dispersion term and do not guaranteetheoretical properties. It is more a dominance stochastic approach, which is related toprobability distributions.The remainder of this paper is structured as follows: Section 2 presents the notation,definitions and preliminaries from the literature; Section 3 contains our main resultsregarding the proposed composition of risk and deviation measures under the Limitednessaxiom; Section 4 exposes examples and results of known and new proposed compositionsin order to illustrate our approach, especially the role for Limitedness axiom; and Section5 summarizes and concludes the paper. 3 Preliminaries
Unless otherwise stated, the content is based on the following notation. Consider therandom result X of any asset ( X ≥ X < , F , P ). In addition, P = { Q : Q ≪ P } is the non-empty set of probability measures Q defined in (Ω , F ), which are absolutely continuousin relation to P . We have that d Q d P is the density of Q relative to P , which is known as theRadon-Nikodym derivative. P (0 , is the set of probability measures defined in (0 , P . E P [ X ] is the expectedvalue of X under P . F X is the probability function of X and its inverse is F − X , defined as F − X ( α ) = inf { x : F X ( x ) ≥ α } . We define X + = max( X,
0) and X − = max( − X, L p = L p (Ω , F , P ), with 1 ≤ p ≤ ∞ , be the space of equivalence classes of random variablesdefined by the norm k X k p = ( E P [ | X | p ]) p with finite p and k X k ∞ = inf { k : | X | ≤ k } . X ∈ L p indicates that k X k p < ∞ . We have that L q , p + q = 1, is the dual space of L p .In this section, we present some definitions and results from the literature that serveas a background to our main results. In this sense, we begin by defining the axioms forrisk and deviation measures. There is a large number of possible properties. We focuson those that are most prominent in the literature and that are used in this paper. Eachclass of risk measures is based on a specific set of axioms. We also define the classes ofrisk measures that are representative in this paper. Definition 2.1.
A functional ρ : L p → R ∪ {∞} is a risk measure, which may fulfill thefollowing properties: • Monotonicity: if X ≤ Y , then ρ ( X ) ≥ ρ ( Y ) , ∀ X, Y ∈ L p . • Translation Invariance: ρ ( X + C ) = ρ ( X ) − C, ∀ X ∈ L p , C ∈ R . • Sub-additivity: ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ) , ∀ X, Y ∈ L p . • Positive Homogeneity: ρ ( λX ) = λρ ( X ) , ∀ X ∈ L p , λ ≥ . • Convexity: ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) , ∀ X, Y ∈ L p , ≤ λ ≤ . • Fatou Continuity: if | X n | ≤ Y, { X n } ∞ n =1 , Y ∈ L p , and X n → X , then ρ ( X ) ≤ lim inf ρ ( X n ) . • Law Invariance: if F X = F Y , then ρ ( X ) = ρ ( Y ) , ∀ X, Y ∈ L p . • Co-monotonic Additivity: ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) , ∀ X, Y ∈ L p with X, Y co-monotone, i.e. (cid:0) X ( w ) − X ( w ′ ) (cid:1) (cid:0) Y ( w ) − Y ( w ′ ) (cid:1) ≥ , ∀ w, w ′ ∈ Ω . • Limitedness: ρ ( X ) ≤ − inf X = sup − X, ∀ X ∈ L p .Remark . Monotonicity requires that, if one position generates worse results to an-other, its risk shall be greater. Translation Invariance ensures that, if a certain gain isadded to a position, its risk shall decrease by the same amount. Risk measures that sat-isfy both Monotonicity and Translation Invariance are called monetary and are Lipschitzcontinuous in L ∞ . Sub-additivity, which is based on the principle of diversification, im-plies that the risk of a combined position is less than the sum of individual risks. PositiveHomogeneity is related to the position size, i.e. the risk proportionally increases with4osition size. These two axioms are together known as sub-linearity. Convexity is a well-known property of functions that can be understood as a relaxed version of sub-linearity.Any two axioms among Positive Homogeneity, Sub-Additiviy and Convexity implies thethird one. Fatou continuity is a well-established property for functions, directly linked tolower semi-continuity and continuity from above. Law invariance ensures that two posi-tions with the same probability function have equal risks. Co-monotonic Additivity is anextreme case where there is no diversification, because the positions have perfect positiveassociation. Co-monotonic Additivity implies Positive Homogeneity. Limitedness ensuresthat the risk of a position is never greater than the maximum loss. In this paper, we arealways working with normalized risk measures in the sense of ρ (0) = 0, since this is easilyobtained through a translation. Definition 2.3.
A functional D : L p → R + ∪ {∞} is a deviation measure, which mayfulfill the following properties: • Non-Negativity: For all X ∈ L p , D ( X ) = 0 for constant X , and D ( X ) > fornon-constant X. • Translation Insensitivity: D ( X + C ) = D ( X ) , ∀ X ∈ L p , C ∈ R . • Sub-additivity: D ( X + Y ) ≤ D ( X ) + D ( Y ) , ∀ X, Y ∈ L p . • Positive Homogeneity: D ( λX ) = λ D ( X ) , ∀ X ∈ L p , λ ≥ . • Lower Range Dominance: D ( X ) ≤ E P [ X ] − inf X, ∀ X ∈ L p . • Fatou Continuity: if | X n | ≤ Y, { X n } ∞ n =1 , Y ∈ L p , and X n → X , then D ( X ) ≤ lim inf D ( X n ) . • Law Invariance: if F X = F Y , then D ( X ) = D ( Y ) , ∀ X, Y ∈ L p . • Co-monotonic Additivity: D ( X + Y ) = D ( X ) + D ( Y ) , ∀ X, Y ∈ L p with X, Y co-monotone.Remark . Non-negativity assures that there is dispersion only for non-constant posi-tions. Translation Insensitivity indicates that the deviation does not change if a constantvalue is added. Lower Range Dominance restricts the measure to a range that is lowerthan the range between expectation and the minimum value. These axioms are relatedto the concept of norm, as explored in Righi and Borenstein (2017).
Definition 2.5.
Let ρ : L p → R ∪ {∞} and D : L p → R + ∪ {∞} .(i) ρ is a coherent risk measure in the sense of Artzner et al. (1999) if it fulfills theaxioms of Monotonicity, Translation Invariance, Sub-additivity, and Positive Ho-mogeneity.(ii) ρ is a convex risk measure in the sense of F¨ollmer and Schied (2002) and Frittelli and Rosazza Gianin(2002) if it fulfills the axioms of Monotonicity, Translation Invariance, and Con-vexity.(iii) D is a generalized deviation measure in the sense of Rockafellar et al. (2006) if itfulfills the axioms of Non-negativity, Translation Insensitivity, Sub-additivity, andPositive Homogeneity. iv) D is a convex deviation measure in the sense of Pflug (2006) if it fulfills the axiomsof Non-negativity, Translation Insensitivity, and Convexity.(v) A risk or deviation measure is said to be law invariant, lower-range dominated,limited, co-monotone, or Fatou continuous if it fulfills the axioms of Law Invari-ance, Lower Range Dominance, Limitedness, Co-monotonic Additivity, or FatouContinuity, respectively.Remark . Given a coherent risk measure ρ , it is possible to define an acceptance setof positions that do not have positive risk as A ρ = { X ∈ L p : ρ ( X ) ≤ } . Let L p + be thecone of the non-negative elements of L p and L p − its negative counterpart. This acceptanceset contains L p + , has no intersection with L p − , and is a convex cone. The risk measureassociated with this set is ρ ( X ) = inf { m : X + m ∈ A ρ } , i.e. the minimum capital thatneeds to be added to X to ensure it becomes acceptable. For convex risk measures, A ρ need not be a cone.A coherent risk measure can be represented as the worst possible expectation fromscenarios generated by probability measures Q ∈ P , known as dual sets. Artzner et al.(1999) presented this result for finite L ∞ spaces. Delbaen (2002) generalized for all L ∞ spaces, whereas Inoue (2003) considered the spaces L p , ≤ p ≤ ∞ . F¨ollmer and Schied(2002), Frittelli and Rosazza Gianin (2002) and Kaina and R¨uschendorf (2009) presenteda similar result for convex risk measures based on a penalty function. It is also possible torepresent generalized deviation measures in a similar approach, with the due adjustments,as demonstrated by Rockafellar et al. (2006) and Grechuk et al. (2009). Ang et al. (2018)adapted this framework for coherent risk measures. Pflug (2006) proved similar resultsfor convex deviation measures also based on a penalty function. In this sense, the dualrepresentations we consider in this paper are formally guaranteed by the following results. Theorem 2.7.
Let ρ : L p → R ∪ {∞} and D : L p → R + ∪ {∞} . Then:(i) ρ is a Fatou continuous coherent risk measure if, and only if, it can be represented as ρ ( X ) = sup Q ∈P ρ E Q [ − X ] , where P ρ = { Q ∈ P : d Q d P ∈ L q , ρ ( X ) ≥ E Q [ − X ] , ∀ X ∈ L p } is a closed and convex dual set.(ii) ρ is a Fatou continuous convex risk measure if, and only if, it can be represented as ρ ( X ) = sup Q ∈P { E Q [ − X ] − γ ρ ( Q ) } , where γ ρ : P → R ∪{∞} is a lower semi-continuousconvex penalty function conform γ ρ ( Q ) = sup X ∈A ρ E Q [ − X ] , with γ ρ ( Q ) ≥ − ρ (0) .(iii) D is a lower-range dominated Fatou continuous generalized deviation measure if,and only if, it can be represented as D ( X ) = E P [ X ] − inf Q ∈P D E Q [ X ] , where P D = { Q ∈ P : d Q d P ∈ L q , D ( X ) ≥ E P [ X ] − E Q [ X ] , ∀ X ∈ L p } is a closed and convex dualset.(iv) D is a lower-range dominated Fatou continuous convex deviation measure if, andonly if, it can be represented as D ( X ) = E P [ X ] − inf Q ∈P { E Q [ X ] + γ D ( Q ) } , where γ D is similar to γ ρ . Main results
We now turn the focus to our main contribution, the proposed approach for combinationof risk and deviation measures. We initially prove interesting results that relate Mono-tonicity and Lower Range Dominance axioms to Limitedness. Based on these and thepreviously exposed results, we are able to prove our main theorem. The results can beextended to the convex and co-monotone coherent cases.
Proposition 3.1.
Let ρ : L p → R ∪ {∞} and D : L p → R + ∪ {∞} .Then:(i) If ρ fulfills Sub-additivity (Convexity) and Limitedness, then it possesses Mono-tonicity.(ii) If ρ fulfills Translation Invariance and Monotonicity, then it possesses Limitedness.(iii) if ρ + D is a coherent (convex) risk measure, then D possesses Lower Range Dom-inance.Proof. For (i) , remember that because L p spaces are composed by equivalence classes ofrandom variables, we have that if X = Y , then ρ ( X ) = ρ ( Y ). We begin by supposing theSub-additivity of ρ . Let X, Y ∈ L p , X ≤ Y . There is Z ∈ L p , Z ≥ Y = X + Z .From Limitedness, we must have ρ ( Z ) ≤ − inf Z ≤
0. Thus, by Sub-additivity we obtain ρ ( Y ) = ρ ( X + Z ) ≤ ρ ( X ) + ρ ( Z ) ≤ ρ ( X ), as required. By the same logic, let ρ haveConvexity. Thus, for any λ ∈ (0 ,
1) there is some Z ∈ L p , Z ≥ Y = λX + (1 − λ ) Z . This leads to ρ ( Y ) = ρ ( λX + (1 − λ ) Z ) ≤ λρ ( X ) + (1 − λ ) ρ ( Z ) ≤ λρ ( X ).As λ is an arbitrary value in (0 , ρ ( Y ) ≤ ρ ( X ), as desired.For (ii) , note that because X ≥ inf X , Monotonicity and Translation Invarianceimplies ρ ( X ) ≤ ρ (inf X ) = − inf X , which is Limitedness.For (iii) , note that for a coherent (convex) risk measure ρ , due to its dual representa-tion, we have that E P [ − X ] ≤ ρ ( X ) ≤ sup − X = − inf X with extreme situations where P ρ equals a singleton { P } or the whole P . Thus, if ρ + D is coherent (convex), hence lim-ited, then D is lower-range dominated because D ( X ) ≤ − ρ ( X ) − inf X ≤ E P [ X ] − inf X .This concludes the proof. Remark . As proved by B¨auerle and M¨uller (2006), in the presence of Law Invariance,Convexity and Monotonicity are equivalent to second-order stochastic dominance foratom-less spaces. As Limitedness implies Monotonicity, in the presence of Convexity andLaw Invariance, it also implies second-order stochastic dominance.
Theorem 3.3.
Let ρ : L p → R ∪{∞} be a coherent risk measure and D : L p → R + ∪{∞} a generalized deviation measure. Then:(i) ρ + D is a coherent risk measure if and only if it fulfills Limitedness.(ii) ρ and D are Fatou continuous and ρ + D is limited if, and only if, ρ + D can berepresented as ρ ( X ) + D ( X ) = sup Q ∈P ρ + D E Q [ − X ] , where P ρ + D = { Q ∈ P : d Q d P = d Q ρ d P + d Q D d P − , Q ρ ∈ P ρ , Q D ∈ P D } . iii) ρ and D are law invariant and ρ + D is limited if, and only if, ρ + D can be representedas ρ ( X ) + D ( X ) = sup m ∈M R ρ α ( X ) md ( α ) , where ρ α ( X ) = − α R α F − X ( u ) du and M = { m ∈ P (0 , : R (0 ,
1] 1 α dm ( α ) = d Q d P , Q ∈ P ρ + D } .Proof. We begin with (i) . According to Proposition 3.1, if ρ + D is a coherent riskmeasure then it fulfills Limitedness. For the converse part, the Translation Invariance,Sub-additivity, and Positive Homogeneity of ρ + D is a consequence of the individualaxioms fulfilled by ρ and D individually by definition. As there is Limitedness by as-sumption, ρ + D respects Monotonicity due to Proposition 3.1. Hence, it is a coherentrisk measure.For (ii) , ρ + D being limited implies it is a coherent risk measure, by the previousresult. As ρ and D are Fatou continuous, by Theorem 2.7 they have representations withdual sets P ρ and P D . Thus, ρ + D is also Fatou continuous and has dual representation.We then obtain that: ρ ( X ) + D ( X ) = sup Q ρ ∈P ρ E Q ρ [ − X ] + E P [ X ] − inf Q D ∈P D E Q D [ X ]= sup Q ρ ∈P ρ , Q D ∈P D { E Q ρ [ − X ] − E P [ − X ] + E Q D [ − X ] } = sup Q ρ ∈P ρ , Q D ∈P D (cid:26) E P (cid:20) − X (cid:18) d Q ρ d P + d Q D d P − (cid:19)(cid:21)(cid:27) = sup Q ∈P ρ + D E Q [ − X ] , where P ρ + D = { Q ∈ P : d Q d P = d Q ρ d P + d Q D d P − , Q ρ ∈ P ρ , Q D ∈ P D } . To show that P ρ + D is composed by valid probability measures, we verify that for Q ∈ P ρ + D , E P (cid:2) d Q d P (cid:3) = E P h d Q ρ d P i + E P (cid:2) d Q D d P (cid:3) − E P [1] = 1. In addition, Monotonicity of ρ + D implies that d Q d P ≥
0, because for
X, Y ∈ L p with X ≤ Y we have from the dual representation thatsup Q ∈P ρ + D E P [ − X d Q d P ] ≥ sup Q ∈P ρ + D E P [ − Y d Q d P ]. This inequality could not be guaranteed if d Q d P assume negative values. Now, we assume that ρ + D has such dual representation. Then ρ + D is a Fatou continuous coherent risk measure that respects Limitedness. Reversingthe deduction steps, one recovers the individual dual representations of both ρ and D .By Theorem 2.7 these two measures possess Fatou continuity.Regarding (iii) , Kusuoka (2001) showed that coherent risk measures that fulfill LawInvariance and Fatou continuity axioms can be represented as sup m ∈M R ρ α ( X ) md ( α ) forsome M ⊂ P (0 , . Moreover, Jouini et al. (2006) and Svindland (2010) have proved thatlaw-invariant convex risk measures defined in atom-less spaces are Fatou continuous.Since we are considering an atom-less probability space, we get that ρ + D can have thiskind of representation because it is limited, then coherent. We can define a continuousvariable u ∼ U (0 ,
1) uniformly distributed between 0 and 1, such that F − X ( u ) = X .For Q ∈ P ρ + D , we can obtain d Q d P = H ( u ) = R ( u,
1] 1 α dm ( α ), where H is a monotonicallydecreasing function and m ∈ P (0 , . As H is anti-monotonic in relation to X, one can8each the supremum in a dual representation. Then we obtain ρ ( X ) + D ( X ) = sup Q ∈P ρ + D E Q [ − X ]= sup Q ∈P ρ + D E P (cid:20) − X d Q d P (cid:21) = sup m ∈M (cid:26)Z − F − X ( u ) (cid:20)Z ( u, α dm ( α ) (cid:21) du (cid:27) = sup m ∈M (cid:26)Z (0 , (cid:20) α Z α − F − X ( u ) du (cid:21) dm ( α ) (cid:27) = sup m ∈M (cid:26)Z (0 , ρ α dm ( α ) (cid:27) , where M = n m ∈ P (0 , : R ( u,
1] 1 α dm ( α ) = d Q d P , Q ∈ P ρ + D o . We now assume that ρ + D has such representation. Then it is a law-invariant coherent risk measure. This is onlypossible if both ρ and D are law invariant. By (i) , it is also limited. This concludes theproof.Assertions of Theorem 3.3 can be extended to the case where ρ is a convex risk measureand D a convex deviation measure. For the law invariant case, Frittelli and Rosazza Gianin(2005) and Noyan and Rudolf (2015) proved representations similar to those of Kusuoka(2001) for convex risk measures. The results of Theorem 3.3 can also be extended to thecase where ρ and D are co-monotone. In this scenario, M becomes a singleton, as is thecase with the spectral risk measures proposed by Acerbi (2002) and concave distortionfunctions, which are widely used in insurance. Grechuk et al. (2009), Wang et al. (2017)and Furman et al. (2017) prove results linking these classes and axioms for generalizeddeviation measures. We state these two extensions without proof, because the deductionsare quite similar to the coherent case. Theorem 3.4.
Let ρ : L p → R ∪ {∞} be a convex risk measure and D : L p → R + ∪ {∞} a convex deviation measure. Then:(i) ρ + D is a convex risk measure if and only if it fulfills Limitedness.(ii) ρ and D are Fatou continuous and ρ + D is limited if, and only if, ρ + D can berepresented as ρ ( X ) + D ( X ) = sup Q ∈P { E Q [ − X ] − γ ρ + D ( Q ) } , where γ ρ + D = γ ρ + γ D .(iii) ρ and D are law invariant and ρ + D is limited if, and only if, ρ + D can berepresented as ρ ( X ) + D ( X ) = sup m ∈P (0 , nR ρ α ( X ) dm ( α ) − γ ρ + D ( m ) o . Theorem 3.5.
Let ρ : L p → R ∪ {∞} be a co-monotone coherent risk measure and D : L p → R + ∪ {∞} a co-monotone generalized deviation measure. Then:(i) ρ + D is a co-monotone coherent risk measure if, and only if, it fulfills Limitedness.(ii) ρ and D are Fatou continuous and ρ + D is limited if, and only if, ρ + D can berepresented as ρ ( X ) + D ( X ) = sup Q ∈P ρ + D E Q [ − X ] .(iii) ρ and D are law invariant and ρ + D is limited if, and only if, ρ + D can berepresented as ρ ( X ) + D ( X ) = R ρ α ( X ) dm ( α ) , where m ∈ P (0 , . Examples
In this section, we provide examples of functionals composed by risk and deviation mea-sures in order to illustrate the importance of Limitedness, since it is central to our results.In practical situations, typically the idea is to consider ρ + β D , where β assumes the roleof some penalty coefficient indicating that the proportion of deviation that must be in-cluded. Thus, it works similarly to an aversion term. Note that, if D is a generalizeddeviation measure, then so is β D for β >
0. The same is true if D is convex or co-monotone generalized. It is valid to point out that in this situation the acceptance set isdefined as A ρ + β D := { X : ρ ( X ) + β D ( X ) ≤ } = { X : − ρ ( X ) D ( X ) ≥ β } , which is related toa performance criteria similar to a Sharpe ratio. The sing of minus is due to ρ representlosses. When β = 0, D lacks the Non-Negativity axiom. Nonetheless, in this case, ourcomposition is trivially the initial risk measure ρ . We explore results with a main focuson the class of coherent risk measures, especially dual representations. Representationsregarding Convexity, Law Invariance and Co-monotonic Additivity can be obtained inthe same spirit as in the previous theorems.Our first example is the intuitive mean plus (p-norm) standard deviation, directlylinked to the variance premium and mean-variance Markowitz portfolio theory. Thenegatives of mean and standard deviation are canonical examples of coherent risk andgeneralized deviation measures, respectively. We now define this risk measure. Definition 4.1.
The mean plus standard deviation is a functional
M SD β : L p → R ∪{∞} defined conform: M SD β ( X ) = − E P [ X ] + β k X − E P [ X ] k p , ≤ β ≤ . This risk measure fulfills Translation Invariance, Convexity, Positive Homogeneity andLaw Invariance. However, it does not possess Monotonicity. This fact is due to the lack ofLimitedness, since it is easy to obtain β k X − E P [ X ] k p > E P [ X ] − inf X for some randomvariable with skewed F X . Indeed, by considering the whole distribution of X makesthis risk measure flawed in its financial meaning because it penalizes profit and loss inthe same way. Its tail counterpart, when X is restricted to values below its α -quantile F − X ( α ), is proposed and studied by Furman and Landsman (2006) and inherits its mainproperties. In order to circumvent such drawbacks, it becomes necessary to consider themean plus (p-norm) semi-deviation. We give a formal definition. Definition 4.2.
The mean plus semi-deviation is a functional
M SD β − : L p → R ∪ {∞} defined conform: M SD β − ( X ) = − E P [ X ] + β k ( X − E P [ X ]) − k p , ≤ β ≤ . It is clear that semi-deviation is a lower range dominated generalized deviation mea-sure. This risk measure is studied in detail by Ogryczak and Ruszczy´nski (1999) andFischer (2003). It is well known that this functional is a law invariant coherent risk mea-sure. We now provide an alternative proof based on our setting in order to explicit therole of Limitedness axiom.
Proposition 4.3.
The mean plus semi-deviation is a law invariant coherent risk measurewith dual set P MSD β − = (cid:8) Q ∈ P : d Q d P = 1 + β ( W − E P [ W ]) , W ≤ , k W k q ≤ (cid:9) . roof. From the properties of both components, which are law invariant coherent risk andgeneralized deviation measures respectively, we have from Theorem 3.3 the combinationis a coherent risk measure if and only if it is limited. This comes from the fact that( X − E P [ X ]) − ≤ E P [ X ] − inf X, ∀ X ∈ L p . Thus, we have E P [ X ] − inf X ≥ k ( X − E P [ X ]) − k ∞ ≥ k ( X − E P [ X ]) − k p ≥ β k ( X − E P [ X ]) − k p . Hence, M SD β − ≤ − inf X .Regarding the structure of P MSD β , one must note that the dual set of − E P [ X ]is a singleton, while for the semi-deviation multiplied by β it is composed, conformRockafellar et al. (2006), by relative densities of the form d Q d P = β (1 + E P [ W ] − W ) +(1 − β ) , W ≤ , k V k q ≤
1. From Theorem 3.3 we have that the representation is givenby P MSD β − = (cid:8) Q ∈ P : d Q d P = 1 + β ( E P [ W ] − W ) , W ≤ , k W k q ≤ (cid:9) . This concludes theproof.From the previous proposition, we can see the importance of Limitedness for Mono-tonicity of the mean plus semi-deviation risk measure. This notion of penalization overa risk measure by the deviation of results worst than this value can be extended whenthe negative expectation is replaced by alternative risk measures. An advantage of thisapproach is that the agent chooses a risk measure and the deviation is directly generatedfrom it. This is explored when ρ is the ES, by Righi and Ceretta (2016). Moreover,Righi and Borenstein (2017) explores for other risk measures beyond ES, such as expec-tiles and entropic ones, calling the approach a loss-deviation for portfolio optimization.Their results point out the advantages of these risk measures, but no theoretical resultsare presented. We thus present a formal definition and explore theoretical properties.In order to ease notation, we define ρ ∗ ( X ) = − ρ ( X ). The minus sign is simply anadjustment to ease the notation. Definition 4.4.
Let ρ : L p → R ∪ {∞} be a risk measure. Then its loss-deviation is afunctional LD βρ : L p → R ∪ {∞} defined conform: LD βρ ( X ) = ρ ( X ) + β k ( X − ρ ∗ ( X )) − k p , ≤ β ≤ . Despite this very interesting intuitive meaning, the penalization term k ( X − ρ ∗ ( X )) − k p is not sub-additive for any convex risk measure, with the exception of the negative expec-tation. To see this fact, note that ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − ≤ ( X + Y − ρ ∗ ( X + Y )) − ,with equality if and only if ρ ( X ) = − E P [ X ]. Thus, it is not a generalized, even convex,deviation measure. Nonetheless, this penalization term composed with a coherent (con-vex) risk measure ρ results in a sub-additive (convex) loss-deviation. We now expose theformalization of such facts. Proposition 4.5.
Let ρ : L p → R ∪ {∞} be a coherent (convex) risk measure and LD βρ : L p → R ∪ {∞} its loss-deviation. Then:(i) LD βρ is a coherent (convex) risk measure. If ρ is law invariant then LD βρ also is.Moreover, if ρ is co-monotone, then LD βρ is sub-additive for any co-monotone pair X, Y ∈ L p .(ii) If ρ is Fatou continuous coherent, then LD βρ has, for W = { W : W ≤ , k W k q ≤ } ,dual set P LD βρ = n Q ∈ P : d Q d P = d Q ρ d P (1 + βE P [ W ]) − βW, d Q ρ d P ∈ P ρ , W ∈ W o .Proof. When β = 0 all claims are obvious from the assumptions on ρ . We thus focuson the case 0 < β ≤
1. Regarding (i), Translation Invariance and Positive Homogeneity11re easily obtained from the fact that ρ fulfills such properties. Let ∆ X + Y = k ( X + Y − ρ ∗ ( X + Y )) − k p − ( k ( X − ρ ∗ ( X )) − k p + k ( Y − ρ ∗ ( Y )) − k p ). From the Sub-additivity of ρ ,we get for any X, Y ∈ L p that:∆ X + Y ≤ k ( X + Y − ρ ∗ ( X + Y )) − k p − k ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − k p ≤ k ( X + Y − ρ ∗ ( X + Y )) − − ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − k p ≤ k ( X + Y − ρ ∗ ( X + Y )) − − ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − k ∞ = ρ ( X ) + ρ ( Y ) − ρ ( X + Y ) ≤ β ( ρ ( X ) + ρ ( Y ) − ρ ( X + Y )) . Thus, we obtain LD βρ ( X + Y ) = ρ ( X + Y ) + β k ( X + Y − ρ ∗ ( X + Y )) − k p ≤ ρ ( X ) + β k ( X − ρ ∗ ( X )) − k p + ρ ( Y ) + β k ( Y − ρ ∗ ( Y )) − k p = LD βρ ( X ) + LD βρ ( Y ), as desired. The first andsecond inequalities are due to both the p-norm and negative part satisfying Sub-additivityproperty, while the last one is because ρ fulfills Sub-additivity and β ≤
1. Moreover, since ρ ∗ ( X + Y ) ≥ ρ ∗ ( X )+ ρ ∗ ( Y ), we have that: ( X + Y − ρ ∗ ( X + Y )) − − ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − assumes value 0 if X + Y ≥ ρ ∗ ( X + Y ), ρ ∗ ( X + Y ) − ρ ∗ ( X ) − ρ ∗ ( Y ) if X + Y ≤ ρ ∗ ( X )+ ρ ∗ ( Y ),and some scalar C ≤ ρ ∗ ( X + Y ) − ρ ∗ ( X ) − ρ ∗ ( Y ) otherwise. This explains the equality k ( X + Y − ρ ∗ ( X + Y )) − − ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − k ∞ = ρ ( X )+ ρ ( Y ) − ρ ( X + Y ). When ρ isa convex risk measure, the deduction is quite similar, beginning with ∆ X + Yλ = k ( λX +(1 − λ ) Y − ρ ∗ ( λX + (1 − λ ) Y )) − k p − ( λ k ( X − ρ ∗ ( X )) − k p + (1 − λ )( k ( Y − ρ ∗ ( Y )) − k p ) , ≤ λ ≤ LD βρ fulfills Sub-additivity (Convexity) when ρ does. This fact, together withthe Limitedness axiom implies, from Proposition 3.1, that LD βρ satisfies Monotonicity.Limitedness comes from the fact that ( X − ρ ∗ ( X )) − ≤ ρ ∗ ( X ) − inf X, ∀ X ∈ L p . Thus,we have ρ ∗ ( X ) − inf X ≥ k ( X − ρ ∗ ( X )) − k ∞ ≥ k ( X − ρ ∗ ( X )) − k p ≥ β k ( X − ρ ∗ ( X )) − k p .Hence, LD βρ ( X ) ≤ − inf X . Hence LD βρ is a coherent or convex risk measure when ρ lies in these same classes. Moreover, it is direct that the Law Invariance of ρ impliesthe same axiom for LD βρ because the p-norm is based on expectation. Moreover, if ρ isco-monotone, we have for co-monotonic X, Y ∈ L p that: LD βρ ( X + Y ) = ρ ( X + Y ) + β k ( X + Y − ρ ∗ ( X + Y )) − k p = ρ ( X ) + ρ ( Y ) + β k ( X + Y − ρ ∗ ( X ) − ρ ∗ ( Y )) − k p ≤ ρ ( X ) + ρ ( Y ) + β ( k ( X − ρ ∗ ( X )) − k p + k ( Y − ρ ∗ ( Y )) − k p )= LD βρ ( X ) + LD βρ ( Y ) , which is Sub-additivity for this case, as claimed.Concerning (ii), let W = { W : W ≤ , k W k q ≤ } . It is well known, see Pflug (2006),12hat k X − k p = sup W ∈W E P [ XW ]. From that, we can obtain for any X ∈ L p that: LD βρ ( X ) = ρ ( X ) + β sup W ∈W E P [( X − ρ ∗ ( X )) W ]= β sup W ∈W (cid:26) E P [ XW ] + ρ ( X ) (cid:18) β + E P [ W ] (cid:19)(cid:27) = β sup W ∈W ( E P [ XW ] + sup Q ∈P ρ E Q [ − X ] ! (cid:18) β + E P [ W ] (cid:19)) = sup Q ∈P ρ ,W ∈W { E Q [ − X ](1 + βE P [ W ]) + βE P [ XW ] } = sup Q ∈P ρ ,W ∈W (cid:26) E P (cid:20) − X (cid:18) d Q ρ d P (1 + βE P [ W ]) − βW (cid:19)(cid:21) , d Q ρ d P ∈ P ρ (cid:27) = sup Q ∈P LDβρ E Q [ − X ] , where P LD βρ = (cid:26) Q ∈ P : d Q d P = d Q ρ d P (1 + βE P [ W ]) − βW, d Q ρ d P ∈ P ρ , W ∈ W (cid:27) . In the third equality, we use the assumption that ρ is Fatou continuous and, by Theorem2.7, possesses a dual representation. In this same equality, it is valid to note that β − > E P [ W ] ≥ −
1, which implies that β − + E P [ W ] ≥
0. It remains to show that P LD βρ is composed by valid probability measures, i.e. ∀ Q ∈ P LD βρ it is true that d Q d P ≥ E P (cid:2) d Q d P (cid:3) = 1, and d Q d P ∈ L q . In this sense, since βE P [ W ] ≥ E P [ W ] ≥ −
1, we get that d Q d P = d Q ρ d P (1 + βE P [ W ]) − βW ≥ , ∀ Q ∈ P LD βρ . Moreover, we have that E P (cid:2) d Q d P (cid:3) = E P h d Q ρ d P (1 + βE P [ W ]) − βW i = E P h d Q ρ d P i (1 + βE P [ W ]) − βE P [ W ] = 1 , ∀ Q ∈ P LD βρ .Finally, we also have that (cid:13)(cid:13) d Q d P (cid:13)(cid:13) q = (cid:13)(cid:13)(cid:13) d Q ρ d P (1 + βE P [ W ]) − βW (cid:13)(cid:13)(cid:13) q ≤ (1 + βE P [ W ]) (cid:13)(cid:13)(cid:13) d Q ρ d P (cid:13)(cid:13)(cid:13) q + β k W k q < ∞ . This concludes the proof. Remark . When ρ ( X ) = − E [ X ], we obtain the mean plus semi-deviation as a par-ticular case, where relative densities have the form d Q d P = d Q ρ d P (1 + βE P [ W ]) − βW =1 + β ( E P [ W ] − W ) , W ∈ W , the same as in Proposition 4.3.Despite the fact that this kind of risk measure is not contemplated by our mainTheorems 3.3, 3.4 and 3.5, its coherence (convexity) is guaranteed by Proposition 3.1.This reinforces the role of Limitedness when one combines risk and deviation measures.If ρ fulfills Law Invariance and Co-monotonic Additivity, then LD βρ lies in the moreflexible class of Natural risk measures proposed by Kou et al. (2013), which must satisfyMonotonicity, Translation Invariance, Positive Homogeneity, Law Invariance and Co-monotonic Sub-Additivity. There are other examples in the literature of functionalscomposed by a coherent risk measure and a non-convex deviation that is again a coherentrisk measure. This is exactly what happens for the Tail Gini Shortfall, proposed byFurman et al. (2017), and its extension introduced in Berkhouch et al. (2017). The ideaof such risk measures is to have a composition of the form ρ + β D between ES and a Ginifunctional restricted to the distribution tail. In both cases, it is a necessary restrictionon the range of values for β . In general, it is possible to ’force’ Limitedness over ρ + D
13y replacing D for β D , under some restriction on the range of β , despite the propertiesof D . We now provide such results in a formal way. Proposition 4.7.
Let ρ : L p → R ∪ {∞} be limited and D : L p → R + ∪ {∞} . Then, ρ + β D fulfills Limitedness if and only if β ≤ inf n ρ ∗ ( X ) − inf X D ( X ) : X ∈ L p , D ( X ) > o .Proof. When D ( X ) = 0, it is straightforward that Limitedness is achieved by the assump-tion on ρ . Thus, we focus on the cases when D ( X ) > X . Let K = inf n ρ ∗ ( X ) − inf X D ( X ) : X ∈ L p , D ( X ) > o . For β ≤ K , we thus get the following: ρ ( X ) + β D ( X ) ≤ ρ ( X ) + K D ( X ) ≤ ρ ( X ) + (cid:18) ρ ∗ ( X ) − inf X D ( X ) (cid:19) D ( X )= − inf X. For the converse relation, we now assume that ρ ( X ) + β D ( X ) ≤ − inf X, ∀ X ∈ L p . Inthis case, we obtain that: β ≤ inf (cid:26) ρ ∗ ( X ) − inf X D ( X ) : X ∈ L p (cid:27) ≤ inf (cid:26) ρ ∗ ( X ) − inf X D ( X ) : X ∈ L p , D ( X ) > (cid:27) . This concludes the proof.
Remark . In the proposition, we have not used the typical practical constraint β ≥ ρ + β D ≤ ρ . Due toProposition 3.1, we have that ρ is limited for all frameworks of Theorems 3.3, 3.4 and3.5. Thus, this last result is not restricted. Regarding practical interpretation, if ρ ∗ isa parsimonious risk measure that is far from inf X , then β can assume larger values,i.e. more protection from D can be added without losing Limitedness. The contraryreasoning is also valid.Even with this constraint for the value of β , Monotonicity is not necessarily achievedbecause, under Limitedness, Sub-Additivity or Convexity is demanded by Proposition3.1. At the same time, it would be nice to obtain the deviation term in our compositiondirectly from the risk measure we choose, as in the loss-deviation approach. Under allthese perspectives, we consider, as our next example, deviations defined as risk measuresapplied over a demeaned financial position. We now define such types of deviation. Definition 4.9.
Let ρ : L p → R ∪ {∞} be a risk measure. Then the deviation inducedby ρ is a functional D ρ : L p → R ∪ {∞} defined conform: D ρ ( X ) = ρ ( X − E P [ X ]) . This characterization is explored in Rockafellar et al. (2006) and Rockafellar and Uryasev(2013), which prove that D ρ is indeed a lower range dominated generalized (convex) de-viation when ρ is a coherent (convex) risk measure. For the case of coherent ρ it isnot hard to realize that P D ρ = P ρ . It is interesting to consider risk measures strictlylarger than the negative expectation in order to have a well-defined situation since E P [ X − E P [ X ]] = 0 , ∀ X ∈ L p . We now investigate the properties of a compositiongiven by ρ + β D ρ . 14 roposition 4.10. Let ρ : L p → R ∪ {∞} be a coherent (convex) risk measure such as ρ ( X ) > − E P [ X ] , ∀ X ∈ L p . Then:(i) The composition ρ + β D ρ is a coherent (convex) risk measure if and only if ≤ β ≤ inf n ρ ∗ ( X ) − inf XE P ( X ) − ρ ∗ ( X ) : X ∈ L p o . Moreover, if ρ fulfills Law Invariance (Co-monotonicAdditivity), then ρ + β D ρ is law invariant (co-monotone).(ii) If ρ is Fatou continuous coherent and ≤ β ≤ inf n ρ ∗ ( X ) − inf XE P ( X ) − ρ ∗ ( X ) : X ∈ L p o , then thecombination ρ + β D ρ has dual set P ρ + β D ρ = (cid:8) Q ∈ P : d Q d P = (1 + β ) d Q ρ d P − β, Q ρ ∈ P ρ (cid:9) .Proof. For (i), Translation Invariance, Sub-Additivity, Convexity, Positive Homogeneity,Law Invariance and Co-monotonic Additivity are direct when ρ possess these propertiesand β ≥
0. Regarding Monotonicity, from Proposition 3.1 we just need Limitednessbecause Sub-additivity (Convexity) is present. Moreover, by Proposition 4.7 we havethat ρ + β D ρ is limited if and only if β ≤ inf n ρ ∗ ( X ) − inf Xρ ( X − E P [ X ]) : X ∈ L p , ρ ( X − E P [ X ]) > o = n ρ ∗ ( X ) − inf XE P [ X ] − ρ ∗ ( X ) : X ∈ L p o , as claimed.Concerning (ii), since ρ is Fatou continuous coherent, it has a representation underdual set P ρ . The same is true for β D ρ under relative densities β d Q ρ d P − (1 − β ). The role of β is according to Rockafellar et al. (2006). Moreover, from (i), ρ + β D ρ is also coherent for0 ≤ β ≤ inf n ρ ∗ ( X ) − inf XE P ( X ) − ρ ∗ ( X ) : X ∈ L p o . Its Fatou continuity is direct from that of ρ . FromTheorem 3.3 its dual set is P ρ + β D ρ = n Q ∈ P : d Q d P = d Q ρ d P + β d Q ρ d P − (1 − β ) − , Q ρ ∈ P ρ o .After some simple manipulation, the claim is achieved. This concludes the proof. In this paper, we present a composition of risk and deviation measures, which considersthe concepts of loss and variability, in order to keep desired theoretical properties. Moststudies are so far only concerned with specific examples, while we present a generalapproach. Our results are based on the proposed Limitedness axiom, which indicates thatthe composition value must not be over a certain limit – the supremum of possible losses.In this context, we prove that this composition is a coherent, convex or co-monotonerisk measure, conforming to properties of the two components. In a second contribution,we provide results about specific examples of known and new risk measures constructedunder this framework. In such results, the importance of our approach becomes clear,especially the role of Limitedness axiom.
References
Acerbi, C., 2002. Spectral measures of risk: A coherent representation of subjective riskaversion. Journal of Banking & Finance 26, 1505–1518.Ang, M., Sun, J., Yao, Q., 2018. On the dual representation of coherent risk measures.Annals of Operations Research 262, 29–46.Artzner, P., Delbaen, F., Eber, J., Heath, D., 1999. Coherent measures of risk. Mathe-matical Finance 9, 203–228. 15¨auerle, N., M¨uller, A., 2006. Stochastic orders and risk measures: Consistency andbounds. Insurance: Mathematics and Economics 38, 132–148.Berkhouch, M., Lakhnati, G., Righi, M., 2017. Extended gini-type measures of risk andvariability. ArXiv working paper.Chen, Z., Wang, Y., 2008. Two-sided coherent risk measures and their application inrealistic portfolio optimization. Journal of Banking & Finance 32, 2667–2673.Delbaen, F., 2002. Coherent risk measures on general probability spaces, in: Sandmann,K., Sch¨onbucher, P.J. (Eds.), Advances in Finance and Stochastics: Essays in Honourof Dieter Sondermann. Springer Berlin Heidelberg, pp. 1–37.Dentcheva, D., Penev, S., Ruszczy´nski, A., 2010. Kusuoka representation of higher orderdual risk measures. Annals of Operations Research 181, 325–335.Filipovi´c, D., Kupper, M., 2007. Monotone and cash-invariant convex functions and hulls.Insurance: Mathematics and Economics 41, 1–16.Fischer, T., 2003. Risk capital allocation by coherent risk measures based on one-sidedmoments. insurance: Mathematics and Economics 32, 135–146.F¨ollmer, H., Schied, A., 2002. Convex measures of risk and trading constraints. Financeand Stochastics 6, 429–447.F¨ollmer, H., Weber, S., 2015. The axiomatic approach to risk measures for capitaldetermination. Annual Review of Financial Economics 7, 301–337.Frittelli, M., Rosazza Gianin, E., 2002. Putting order in risk measures. Journal of Banking& Finance 26, 1473–1486.Frittelli, M., Rosazza Gianin, E., 2005. Law invariant convex risk measures. Advances inMathematical Economics 7, 33–46.Furman, E., Landsman, Z., 2006. Tail Variance Premium with Applications for EllipticalPortfolio of Risks. ASTIN Bulletin 36, 433–462.Furman, E., Wang, R., Zitikis, R., 2017. Gini-type measures of risk and variability: Ginishortfall, capital allocations, and heavy-tailed risks. Journal of Banking & Finance 83,70–84.Grechuk, B., Molyboha, A., Zabarankin, M., 2009. Maximum Entropy Principle withGeneral Deviation Measures. Mathematics of Operations Research 34, 445–467.Inoue, A., 2003. On the worst conditional expectation. Journal of Mathematical Analysisand Applications 286, 237–247.Jouini, E., Schachermayer, W., Touzi, N., 2006. Law invariant risk measures have theFatou property. Advances in Mathematical Economics 9, 49–71.Kaina, M., R¨uschendorf, L., 2009. On convex risk measures on lp-spaces. MathematicalMethods of Operations Research 69, 475–495.16ou, S., Peng, X., Heyde, C.C., 2013. External risk measures and basel accords. Math-ematics of Operations Research 38, 393–417.Krokhmal, P., 2007. Higher moment coherent risk measures. Quantitative Finance 7,373–387.Kusuoka, S., 2001. On law invariant coherent risk measures. Advances in MathematicalEconomics 3, 158–168.Noyan, N., Rudolf, G., 2015. Kusuoka representations of coherent risk measures in generalprobability spaces. Annals of Operations Research 229, 591–605.Ogryczak, W., Ruszczy´nski, A., 1999. From stochastic dominance to mean-risk models:Semideviations as risk measures. European Journal of Operational Research 116, 33–50.Pflug, G., 2006. Subdifferential representations of risk measures. Mathematical Program-ming 108, 339–354.Righi, M., Borenstein, D., 2017. A simulation comparison of risk measures for portfoliooptimization. Finance Research Letters in press.Righi, M., Ceretta, P., 2015. A comparison of expected shortfall estimation models.Journal of Economics and Business 78, 14 – 47.Righi, M., Ceretta, P., 2016. Shortfall Deviation Risk: an alternative to risk measurement.Journal of Risk 19, 81–116.Rockafellar, R., Uryasev, S., 2013. The fundamental risk quadrangle in risk manage-ment, optimization and statistical estimation. Surveys in Operations Research andManagement Science 18, 33–53.Rockafellar, R., Uryasev, S., Zabarankin, M., 2006. Generalized deviations in risk analy-sis. Finance and Stochastics 10, 51–74.Svindland, G., 2010. Continuity properties of law-invariant (quasi-)convex risk functionson L ∞∞