Upper risk bounds in internal factor models with constrained specification sets

For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.


Introduction
In order to reduce the standard upper risk bounds for a portfolio S = d i=1 X i based on marginal information, a promising approach to include structural and dependence information are partially specified risk factor models, see Bernard et al. (2017b). In this approach, the risk vector X = (X i ) 1≤i≤d is described by a factor model with functions f i , systemic risk factor Z , and individual risk factors ε i . It is assumed that the distributions H i of (X i , Z ) , 1 ≤ i ≤ d , and thus also the marginal distributions F i of X i and G of Z are known. The joint distribution of (ε i ) 1≤i≤d and Z , however, is not specified in contrast to the usual independence assumptions in factor models. It has been shown in Bernard et al. (2017b) that in the partially specified risk factor model a sharp upper bound in convex order of the joint portfolio is given by the conditionally comonotonic sum, i.e., it holds for some U ∼ U (0, 1) independent of Z . Furthermore, S c Z is an improvement over the comonotonic sum, i.e, For a law-invariant convex risk measure : L 1 ( , A, P) → R that has the Fatouproperty it holds that is consistent with respect to the convex order which yields that assuming generally that X i ∈ L 1 (P) are integrable and defined on a non-atomic probability space ( , A, P) , see (Bäuerle and Müller (2006), Theorem 4.3). We assume that Z is real-valued. Then, the improved upper risk bound depends only on the marginals F i , the distribution G of Z , and on the bivariate copulas C i = C X i ,Z specifying the dependence structure of (X i , Z ) . An interesting question is how the worst case dependence structure and the corresponding risk bounds depend on the specifications C i , 1 ≤ i ≤ d . More generally, for some subclasses S i ⊂ C 2 of the class of two-dimensional copulas C 2 , the problem arises how to obtain (sharp) risk bounds given the information that C i ∈ S i , 1 ≤ i ≤ d . More precisely, for univariate distribution functions F i , G , we aim to solve the constrained maximization problem for some suitable dependence specification sets S i . As an extension of (5), we also determine solutions of the constrained maximization problem with dependence specification sets S i and marginal specification sets F i ⊂ F 1 , where F 1 denotes the set of univariate distribution functions.
A main aim of this paper is to solve the constrained supermodular maximization problem max (X 1 , . . . , for F i ∈ F i and G ∈ F 0 . A solution of this stronger maximization problem allows more general applications. In particular, it holds that and thus a solution of (7) also yields a solution of (5). Note that solutions of the maximization problems do not necessarily exist because both the convex ordering of the constrained sums and the supermodular ordering are partial orders on the underlying classes of distributions that do not form a lattice, see Müller and Scarsini (2006). In general, the existence of solutions also depends on the marginal constraints F i and G . In this paper, we determine solutions of the maximization problems for large classes F i ⊂ F 1 of marginal constraints under some specific dependence constraints S i .
In Ansari and Rüschendorf (2016), some results on the supermodular maximization problem are given for normal and Kotz-type distributional models for the risk vector X . Some general supermodular ordering results for conditionally comonotonic random vectors are established in Ansari and Rüschendorf (2018). Therein, as a useful tool, the upper product d i=1 D i of bivariate copulas D i ∈ C 2 is introduced by (Note that we superscribe copulas with upper indices in this paper which should not be confused with exponents.) If the risk factor distribution G is continuous, then d i=1 C i is the copula of the conditionally comonotonic risk vector F −1 X i |Z (U ) 1≤i≤d with specifications C X i ,Z = C i , see Ansari and Rüschendorf (2018), Proposition 2.4. Thus, ordering the dependencies of conditionally comonotonic random vectors is based on ordering the corresponding upper products. In particular, a strong dependence ordering condition on the copulas A i , B i ∈ C 2 (based on the sign sequence ordering) allows us to infer inequalities of the form see (Ansari and Rüschendorf 2018), Theorem 3.10. In this paper, we characterize upper product inequalities of the type for copulas D 2 , . . . D d , E ∈ C 2 , where M 2 denotes the upper Fréchet copula in the bivariate case. These inequalities are based on simple lower orthant ordering conditions on the sets S i such that solutions of the maximization problems (5) -(7) exist and can be determined.
The problem to find risk bounds for the Value-at-Risk (VaR) or other risk measures of a portfolio under the assumption of partial knowledge of the marginals and the dependence structure is a central problem in risk management. Bounds for the VaR (or the closely related distributional bounds), resp., for the Tail-Value-at-Risk (TVaR) based on some moment information have been studied extensively in the insurance literature by authors such as Kaas and Goovaerts (1986), Denuit et al. (1999), de Schepper andHeijnen (2010), Hürlimann (2002); Hürlimann (2008), Goovaerts et al. (2011), Bernard et al. (2017a; Bernard et al. (2018), Tian (2008), and Cornilly et al. (2018). Hürlimann (2002) derived analytical bounds for VaR and TVaR under knowledge of the mean, variance, skewness, and kurtosis.
The more recent literature has focused on the problem of finding risk bounds under the assumption that all marginal distributions are known but the dependence structure of the portfolio is either unknown or only partially known. Risk bounds with pure marginal information were intensively studied but were often found to be too wide in order to be useful in practice (see Embrechts and Puccetti (2006); Embrechts et al. (2013); Embrechts et al. (2014)). Related aggregation-robustness and model uncertainty for risk measures are also investigated in Embrechts et al. (2015). Several approaches to add some dependence information to marginal information have been discussed in ample literature (see Puccetti and Rüschendorf (2012a); Puccetti and Rüschendorf (2012b); Puccetti and Rüschendorf (2013); Bernard and Vanduffel (2015), Bernard et al. (2017a); Bernard et al. (2017b), Bignozzi et al. (2015); Rüschendorf and Witting (2017); Puccetti et al. (2017) ). For some surveys on these developments, see Rueschendorf (2017aRueschendorf ( , 2017b. Apparently, a relevant dependence information and structural information leading to a considerable reduction of the risk bounds is given by the partially specified risk factor model as introduced in Bernard et al. (2017b). In this paper, we show that for a large relevant class of partially specified risk factor models-the internal risk factor models-more simple sufficient conditions for the supermodular ordering of the upper products-and thus for the conditionally comonotonic risk vectors-can be obtained by simple lower orthant ordering conditions on the dependence specifications. These simplified conditions allow easy applications to ordering results for risk bounds with subset specification sets S i described above. We give an illuminating application to real market data which clearly shows the potential usefulness of the comparison results. For some further details, we refer to the dissertation of Ansari (2019).

Internal risk factor models
A simplified supermodular ordering result for conditionally comonotonic random vectors can be obtained in the case that the risk factor Z is itself a component of these risk vectors. As a slight generalization, we define the notion of an internal risk factor model.
Definition 1 (Internal risk factor model) A (partially specified) internal risk factor model with internal risk factor Z is a (partially specified) risk factor model such that for some j ∈ {1, . . . , d} and a non-decreasing function g j holds X j = g j (Z ) .
Without loss of generality, the distribution function of the internal risk factor can be chosen continuous, i.e., Z ∼ G ∈ F 1 c . Thus, the not necessarily uniquely determined copula of (X j , Z ) can be chosen as the upper Fréchet copula M 2 . This means that X j and Z are comonotonic and Z can be considered as a component of the risk vector X which explains the denomination of Z as an internal risk factor.
In partially specified risk factor models, the dependence structure of the worst case conditionally comonotonic vector is represented by the upper product of the dependence specifications if G ∈ F 1 c , i.e., Thus, assuming w.l.o.g. that j = 1 , our aim is to derive supermodular ordering results for the upper product M 2 ∨ D 2 ∨ · · · ∨ D d with respect to the dependence specifications D i .
For a function f : be the difference operator, where ε > 0 and e i denotes the unit vector w.r.t. the canonical base in R d .
The following theorem is a main result of this paper. It characterizes the upper product inequality (9) concerning partially specified internal risk factor models.
Theorem 1 (Supermodular ordering of upper products) Let D 2 . . . , D d , E ∈ C 2 . Then, the following statements are equivalent: The proof of the equivalence of (i) and (ii) is not difficult, whereas the equivalence w.r.t. the supermodular ordering in (iii) which we derive in Section 3 requires some effort. Its proof is based on the mass transfer theory for discrete approximations of the upper products and, further, on a conditioning argument using extensions of the standard orderings ≤ lo , ≤ uo , ≤ sm as well as of the comonotonicity notion to the frame of signed measures.
The reverse direction follows from the closures of the upper product (see Ansari and Rüschendorf (2018), Proposition 2.4.(iv)) and of the lower orthant ordering under marginalization.
As a consequence of the above supermodular ordering theorem for upper products, we obtain improved bounds in partially specified internal risk factor models in comparison to the standard bounds based on marginal information.
Theorem 2 (Improved bounds in internal risk factor models) for U ∼ U (0, 1) independent of Y 1 . In particular, this implies Proof Without loss of generality, let X i ∼ U (0, 1) . Then, (X 1 , . . . , X d ) follows a partially specified internal risk factor model with internal risk factor Z = X 1 and dependence constraints C X i ,Z = C i , 2 ≤ i ≤ d . We obtain where the first inequality follows from Ansari and Rüschendorf (2018), Proposition 2.4.(i) and the second inequality holds due to Theorem 1. Thus, (12) follows from the representation in (10). The statement in (13) is a consequence of (8) and (12). Ansari and Rüschendorf (2018), (v, u) . In this case, the statements of Theorems 1 and 2 are known from the literature, see, e.g., (Müller (1997), Theorem 2.7). Further, for d > 2 , the result in Theorem 2 cannot be obtained by a simple supermodular mixing argument because, in the general case, a supermodular ordering of all conditional distributions is not possible, i.e., there exists a z outside a null set such that (12), resp., (13) solves the constrained maximization problem (7), resp., (5) for the dependence specification sets

Remark 1 (a) The upper bound in
As a consequence of Theorem 2, we also obtain improved upper bounds under some correlation information. For a bivariate random vector Then, C r is a bivariate copula and is given by Then, D t is a bivariate copula and given by see Nelsen et al. (2001) [Theorem 2].
The risk bounds can be improved under correlation bounds as follows.

Remark 2 For r
only an improved upper bound in supermodular ordering for the constrained risk vectors but not a solution of maximization problem (5), resp., (7) can be achieved.
To also allow a comparison of the univariate marginal distributions, remember that a bivariate copula D is conditionally increasing (CI) if there exists a bivariate random vector (U 1 , U 2 ) ∼ D such that U 1 |U 2 = u 2 is stochastically increasing in u 2 and U 2 |U 1 = u 1 is stochastically increasing in u 1 . Equivalently, If the upper bound E in Theorem 2 is conditionally increasing, then the case of increasing marginals in convex order can also be handled.
is easily shown to be CI, the statement follows from Müller and Scarsini (2001), (14).

Let the sets S i of dependence specifications be given as in
For a generalization of Theorem 2, we need an extension of (8) as follows.
for all increasing convex functions ψ i and increasing functions f i k .
Proof By straightforward calculations, it can be shown that the function h : is supermodular for all increasing convex functions ϕ . Then, the invariance under increasing transformations and the concatenation property of the supermodular order (see, e.g., Shaked and Shantikumar (2007) Hence, the assertion follows from Shaked and Shantikumar (2007) The application to improved portfolio TVaR bounds in Section 4 is based on the following generalization of Theorem 2.
Proof Statement (17) follows from Theorem 2 with the concatenation property of the supermodular ordering. Statement (18) is a consequence of Theorem 2 and Lemma 1.

Remark 4
Under the assumptions of Theorem 4, the right hand side in (18) solves maximization problem (5) for

Proof of the supermodular ordering in Theorem 1
In this section, we prove the equivalence of (i) and (iii) in Theorem 1. This requires some preparations. We approximate the upper products by discrete upper products based on grid copula approximations. Then, we show that these discrete upper products can be supermodularly ordered using a conditioning argument and mass transfer theory from Müller (2013). However, requires an extension of the orderings ≤ lo , ≤ uo , ≤ sm , and of comonotonicity to the frame of signed measures.

Extensions of ≤ lo , ≤ uo , and ≤ sm to signed measures
We generalize the concept of comonotonicity to signed measures as follows.

Quasi-comonotonicity
We say that a probability distribution Q , resp., a distribution function F is comonotonic if there exists a comonotonic random vector ξ such that ξ ∼ Q , resp., For a signed measure P ∈ M 1 d , we define the associated measure generating function F = F P by F(x) = P((−∞, x]) and its univariate marginal measure generating functions F i by F i ( We define the notion of quasi-comonotonicity as follows. Definition 3 (Quasi-comonotonicity) We denote P , resp., F as quasi- Obviously, if P ∈ M 1 d , then the quasi-comonotonicity and comonotonicity of P are equivalent.
The following lemma characterizes the lower orthant ordering of (quasi-) comonotonic distributions in terms of the upper orthant order.
, . . . , d} such that a 1 ≥ . . . ≥ a d . Then, the survival function F corresponding to F is calculated by where the fourth equality holds true because P is quasi-comonotonic, F i ≤ 1 and F i (∞) = 1 for all i . The fifth equality follows since there are k−1 k− j subsets of {1, . . . , k} with k − j + 1 elements such that k is the maximum element. The sixth equality holds due to the symmetry of the binomial coefficient.
Let G be the distribution function corresponding to Q with univariate margins G i . Then, it holds analogously that G( where we use for the second equivalence that The third equivalence holds true because G i ≥ 0 and G i (−∞) = 0 for all i .

Grid copula approximation
In this subsection, we consider the approximation of the upper product by grid copulas. In the proof of the supermodular ordering in Theorem 1, we make essential use of the property that this approximation is done by distributions with finite support. For n ∈ N and d ≥ 1 , denote by Denote by C d,n (resp., ∈ C s 2,n ) the set of all (signed) n-grid d-copulas.
An 1 n -scaled doubly stochastic matrix or, if the dimension of the matrix is clear, a mass matrix is defined as an n × n-matrix with non-negative entries and row, resp., column sums equal to 1 n . By an signed 1 n -scaled doubly stochastic matrix or also signed mass matrix, we mean an 1 n -scaled doubly stochastic matrix where negative entries are also allowed.
Obviously, there is a one-to-one correspondence between the set of (signed) n-grid 2-copulas and the set of (signed) 1 n -scaled doubly stochastic matrices. For a bivariate (signed) n-grid copula E ∈ C 2,n (∈ C s 2,n ), the associated (signed) probability mass function e is defined by denotes the difference operator of length 1 n with respect to the i-th variable, i.e., i n g(u) := g(u) − g((u − 1 n e i ) ∨ 0) for u ∈ G d n,0 and the i-th unit vector e i . Further, define its associated (signed) mass matrix (e kl ) 1≤k,l≤n by For every d-copula D ∈ C d , denote by G n (D) its canonical n-grid d-copula given by Define the upper product : (C 2,n ) d → C d,n for grid copulas D 1 n , . . . , D d n ∈ C 2,n by for (u 1 , . . . , u d ) ∈ [0, 1] d . A version for signed grid copulas is defined analogously.
The following result gives a sufficient supermodular ordering criterion for the upper product based on the approximations by grid copulas, see Ansari and Rüschendorf (2018), Proposition 3.7.
We make use of the above ordering criterion because the approximation is done by distributions with finite support. But the supermodular ordering of distributions with finite support enjoys a dual characterization by mass transfers as follows.

Mass transfer theory
This section and the notation herein is based on the mass transfer theory as developed in Müller (2013).
For signed measures P, Q ∈ M 1 d with finite support, denote the signed measure Q − P a transfer from P to Q . To indicate this transfer, write where We focus on the set of -monotone, resp., -antitone, resp., supermodular transfers. These sets induce the classes F of -monotone, resp., F − of -antitone, resp., F sm of supermodular functions on S .
Definition 5 Let η > 0 . Let x ≤ y with strict inequality x i < y i for k indices i 1 , . . . , i k for some k ∈ {1, . . . , d} . Denote by V o (x, y) , resp., V e (x, y) the set of all vertices z of the k-dimensional hyperbox [x, y] such that the number of components with z i = x i , i ∈ {i 1 , . . . , i k } is odd, resp., even.

(i) A transfer indicated by
is called a supermodular transfer, where ∧ , resp., ∨ denotes the componentwise minimum, resp., maximum.
The characterizations of the orderings ≤ uo , ≤ lo , resp., ≤ sm by mass transfers due to (Müller (2013), Theorems 2.5.7 and 2.5.4) also hold in the case of signed measures because the proof makes only a statement on transfers, i.e., on the difference of measures. Remark 5 From Definition 5, we obtain that exactly the one-dimensionalmonotone, resp., -antitone transfers affect the univariate marginal distributions. Hence, for measures P, Q ∈ M 1 d ( ) with equal univariate distributions, i.e., P π i = Q π i , π i the i-th projection, for all 1 ≤ i ≤ d , holds that P ≤ uo Q , resp., P ≤ lo Q if and only if Q can be obtained from P by a finite number of at least 2-dimensional -monotone, resp., -antitone transfers. But note that also the onedimensional -monotone, resp., -antitone transfers can affect the copula, resp., dependence structure. Now, we are able to give the proof of the main ordering result of this paper.

Proof of '(i) ⇐⇒ (iii)' in Theorem 1
Assume that (iii) holds. Then, the closures of the upper product and the supermodular ordering under marginalization imply For the reverse direction, assume that D i ≤ lo E for all 2 ≤ i ≤ d . Consider the discretized grid copulas D i n := G n (D i ), M 2 n := G n (M 2 ), and E n := G n (E) , 2 ≤ i ≤ d, and denote by d i n , resp., e n the associated mass matrices of D i n , resp., E n . We prove for the upper products of grid copulas, defined in (20), that showing that there exists a finite number of supermodular transfers that transfer C n to B n . This yields (iii) applying Propositions 2 (iii) and 1.
To show (22), consider for 2 ≤ i ≤ d the signed grid copulas (D i n,k ) 1≤k≤n on G 2 n defined through the signed mass matrices (d i n,k ) 1≤k≤n given by d i n,1 : = d i n , and For all 2 ≤ i ≤ d and for all n ∈ N , the sequence (d i n,k ) 1≤k≤n of signed mass matrices adjusts the signed mass matrix d i n column by column to the signed mass matrix e n . It holds that d i n,n = e n for all i and n .
for all 1 ≤ k ≤ n − 1 . Then, transitivity of the supermodular ordering implies (22) because C n,1 = C n and C n,n = B n . We observe that D i ≤ lo E yields D i n ≤ lo E n and also for all 1 ≤ k ≤ n − 1 . Further, we observe that C n,k and C n,k+1 are (signed) grid copulas with uniform univariate marginals, i.e., for all u j ∈ G 1 n,0 and 1 ≤ j ≤ d . This holds because 2 n,0 and for all i and k , even if d i n,k can get negative for t = k n and some u i < 1 .
To show (29), let us fix u 1 = k n . Then, we calculate where the first equality follows from (27), the first inequality is Jensen's inequality, the second inequality is due to (25). Equality (33) holds because E n is a grid copula and does not depend on i , the third equality holds by definition of 2 n , and the last equality is true because E n is a grid copula, thus 2-increasing, and hence 2 n E n (·, t) is increasing for all t ∈ G 1 n .
Then, from (32) and (34) it follows for the k-th columns of the matrices that where the equality holds true due to (27). This means that holds. Further, C u 1 n,k corresponds to a quasi-comonotonic signed measure in M 1 d with univariate marginals given by n 2 n D i n,k (·, u 1 ) ≤ 1 , and C u 1 n,k+1 corresponds to a comonotonic probability distribution. Thus, we obtain from Lemma 2 that (29) holds.
Next, we show (30). Due to (29) and Proposition 2, there exists a finite number of reverse -monotone transfers that transfer C u 1 n,k to C u 1 n,k+1 , i.e., there exist m ∈ N and a finite sequence P u 1 l 1≤l≤m of signed measures on G d−1 n such that , and Since the univariate margins of C u 1 n,k and C u 1 n,k+1 do not coincide, some of the transfers μ u 1 l l must be one-dimensional, see Remark 5. Each one-dimensional transfer μ u 1 l transports mass from one point for some η l > 0 . Since applying mass transfers is commutative, we first choose to apply all of these one-dimensional reverse -monotone transfers. Because δdimensional -monotone transfers leave the univariate marginals unchanged for δ ≥ 2 , see Remark 5, the univariate margins of C u 1 n,k must be adjusted to the univariate margins of C u 1 n,k+1 having applied all of these one-dimensional reverse -monotone transfers. Then, since the grid copula of C u 1 n,k+1 is the upper Fréchet bound and hence the greatest element in the ≤ uo -ordering, no further reverse -monotone transfer is possible. Thus, C u 1 n,k+1 is reached from above having applied only one-dimensional -monotone transfers μ u 1 l , 1 ≤ l ≤ m − 1 , on P C u 1 n,k , i.e., For all reverse -monotone transfers μ u 1 l , consider its corresponding reverse transfer μ The transfers μ u 1 +1/n l l are one-dimensional -monotone transfers. Then, it holds true that they adjust the univariate marginals of P C u 1 +1/n n,k to the univariate marginals of P C u 1 +1/n n,k+1 . This can be seen because only two entries (in column k) of matrix ι are changed by the mass transfer μ u 1 l . All other columns and matrices j = ι are unaffected by this transfer. From (28) follows that exactly the reverse transfers μ u 1 +1/n l applied simultaneously on the corresponding entries in column k +1 of mass matrix ι guarantee the uniform margin condition (26) to stay fulfilled. Having applied all transfers μ l , then each column j = k + 1 of the mass matrix d i n,k is adjusted to column j of the mass matrix d i n,k+1 for all 2 ≤ i ≤ d . But this also means that column k + 1 of the mass matrix d i n,k must be adjusted to column k + 1 of d i n,k+1 due to the uniform margin condition.
Since applying the one-dimensional transfers μ u 1 +1/n l on P C u 1 +1/n n,k (which is comonotonic) can change the dependence structure, the signed measure P u 1 +1/n m is not necessarily quasi-comonotonic, i.e., P u 1 +1/n m does not necessarily coincide with P C u 1 +1/n n,k+1 (which is quasi-comonotonic). We show that Since C u 1 n,k ≤ lo C u 1 n,k+1 , see (35), it also holds that 2 n D i n,k u i , k n ≤ 2 n D i n,k+1 u i , k n ∀u i ∈ G 1 n ∀i ∈ {2, . . . , d} , where we use that 2 n D i n,k ·, k n , 2 n D i n,k+1 ·, k n ≤ 1 n and 2 n D i n,k 1, k n = 2 n D i n,k+1 1, k n = 1 n for all 2 ≤ i ≤ d . By construction of d i n,l 1≤l≤n , it follows that 2 n D i n,k u i , k+1 n ≥ 2 n D i n,k+1 u i , k+1 n ∀u i ∈ G 1 n ∀i ∈ {2, . . . , d} .
Further, (30) and Proposition 2 imply that there exist m ∈ N and a finite number of reverse -monotone transfers (γ l ) 1≤l≤m that adjust P C u 1 +1/n n,k+1 to P C u 1 +1/n n,k . With the same argument as above, these transfers are one-dimensional. Further, the reverse transfers (γ r l ) 1≤l≤m , where γ r l = −γ l , correspond to the -monotone transfers , which proves (39). Hence, (38) yields It remains to show (31). Each transfer μ u 1 l , , resp.,, μ can be extended to a reverse -monotone, resp., -monotone transfer μ l,r , resp., μ l on Then, for each l ∈ {1, . . . , m − 1} , applying the transfers μ l,r and μ l in (41) simultaneously yields exactly a transfer ν l on {u 1 , u 1 + 1 n } × G d−1 n between u 1 , u l and u 1 + 1 n , v l , indicated by Each transfer ν l is a supermodular transfer. Denote by ε {x} the one-point probability measure in x . Then, finally, we obtain which implies (31) using Proposition 2. The first and last equality hold due to the definition of the measures. The second equality is given by (37) and (40), the third equality holds by the definition of μ l,r , resp., μ l , and the fourth equality holds true by the definition of ν l .
Remark 6 (a) The proof is based on an approximation by finite sequences of signed grid copulas that fulfill the conditioning argument in (28) Assumptions (I) -(III) are consistent. Assumption (I) is a standard assumption on the log-increments of (S i t ) t≥0 while Assumption (II) generalizes the dependence assumptions for multivariate Lévy models because neither multivariate stationarity nor independence for all increments is assumed. Assumption (III) reduces the dependence structure between the k-th log-increment of the i-th component and the k-th log-increment of the first component (which is the internal risk factor) by a subclass S i k = {C ∈ C 2 |C ≤ lo E k } of bivariate copulas. Then, Theorem 4 yields improved bounds in convex order for the portfolio T if the claims Y i T are of the form For the estimation of the distribution of S i T , we make the following specification of Assumption (4): (1) Each S i t t≥0 , i = 1, . . . , 8 , follows an exponential NIG process, i.e., where S i 0 > 0 and where each L i t t≥0 is an NIG process with parameters For the estimation of upper bounds in supermodular order for the increments ξ 1 k , . . . , ξ 8 k , we specify Assumption (I) as follows: (3) For fixed ν ∈ (2, ∞] , the copula E k in Assumption (III) is given by a t-copula with some correlation parameter ρ k ∈ [−1, 1] (which we specify later) and ν degrees of freedom, i.e., We make use of the relation between the (pseudo-)correlation parameter ρ of elliptical copulas and Kendall's τ given by ρ(τ ) = sin π 2 τ , see McNeil et al. (2015) [Proposition 5.37], because Kendall's rank correlation does not depend on the specified univariate marginal distributions in contrast to Pearson's correlation. Thus, in order to determine a reasonable value for ρ k , we estimate an upper bound for Since it is not possible to determine the dependence structure of each increment from a single observation, we estimate τ i k from a sample of past observations. To do so, we assume that the dependence structure of ξ i k , ξ 1 k does not jump too rapidly to strong positive dependence in a short period of time as follows: historical estimatesτ k,30 = max 2≤i≤8 {τ i k,30 } (solid graph) as an estimator for τ k , and it also shows the estimated historical upper boundρ k := max 2≤i≤8 {ρ τ i k,n + k } (dotted graph) with error k for ρ k , 31 ≤ k ≤ 2540 , see Assumption 4.
As we observe from Fig. 2 there is no strong correlation between the log-returns x 1 k k of "AUDI" and the log-returns x i k k , i = 1 , of the other assets. We use this property to apply Theorem 4 as follows.
For the prediction of an improved worst-case upper bound for T w.r.t. convex order for T = 1 year, resp., T = 2 years, we choose the worst-case period of the historical estimatesρ k for ρ k with a length of m = 254 trading days, resp., m = 508 trading days. We identify visually that (ρ k ) k takes the historically largest values in a period of length m = 254 , resp., m = 508 for 1797 ≤ k ≤ 2050 , resp., 1543 ≤ k ≤ 2050 , see the plot at the bottom right in Fig. 2. Thus, we decide on (ρ k ) 1797≤k≤2050 , resp., (ρ k ) 1543≤k≤2050 as the worst-case estimate for (ρ k ) 1≤k≤254 , resp., (ρ k ) 1≤k≤508 with error k = 0.05 in (42).
Then, we obtain from Theorem 4 that ν for ρ k =ρ 2050−m+k+1 and 2 ≤ i ≤ 8 , U k ∼ U (0, 1) and U k , ζ 1 l independent for all 1 ≤ k, l ≤ m . Denote by τ ζ 1 k the distributional transform of ζ 1 k , see Rüschendorf (2009), and let t ν be the distribution function of the t-distribution with ν degrees of freedom. Then, it holds that where f is given by Note that the distribution function of ( f (r, ν, Z , ε), Z ) , Z , ε ∼ U (0, 1) independent, is the t-copula with correlation r and ν degrees of freedom, see Aas et al. (2009).
The Tail-Value-at-Risk at level λ (also known as Expected Shortfall) is defined by for a real-valued random variable ζ . If ζ is integrable, then TVaR λ is a convex law-invariant risk measure, see, e.g., Föllmer and Schied (2010), which satisfies the Fatou-property. As a consequence of (4) and (43)

Empirical results and conclusion
The improved risk bounds TVaR λ c T,(ρ k ),ν for TVaR λ ( T ) are compared in Table 1 with the standard comonotonic risk bound TVaR λ c T (5 million simulated points) for different values of λ and ν and for T = 1 year (= 254 trading days), resp., T = 2 years (= 508 trading days).
As observed from Table 1, there is a substantial improvement of the risk bounds up to 20% for T = 1 year and about 20% for T = 2 years for all degrees of freedom ν of the t-copulas C ρ k ν and high levels of λ . For T = 2 years, the improvement is even better because the two-year worst-case period forρ k also contains the one-year worst-case period forρ k where in the latter one attains higher values.
We see that the improvement is larger for higher values of ν. This can be explained by the fact that C ρ ν has a higher tail-dependence for smaller values of ν, see, e.g., Demarta and McNeil (2005). Thus, for small ν, more extreme events (= realizations of the log-increments) occur more often simultaneously which sums up to a higher risk.
The results of this application clearly indicate the potential usefulness and flexibility of the comparison results for the supermodular ordering to an improvement of the standard risk bounds.