Abstract
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 \(S=\sum _{i=1}^{d} X_{i}\) 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, \(S_{Z}^{c}\) is an improvement over the comonotonic sum, i.e,
For a lawinvariant convex risk measure \(\Psi \colon L^{1}(\Omega,\mathcal {A},P) \to \mathbb {R}\) that has the Fatouproperty it holds that Ψ 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 nonatomic probability space \((\Omega,\mathcal {A},P)\,,\) see (Bäuerle and Müller (2006), Theorem 4.3).
We assume that Z is realvalued. 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 \({\mathcal {S}}^{i}\subset \mathcal {C}_{2}\) of the class of twodimensional copulas \(\mathcal {C}_{2}\,,\) the problem arises how to obtain (sharp) risk bounds given the information that \(C^{i}\in {\mathcal {S}}^{i}\,,\) 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 \({\mathcal {S}}^{i}\,.\) As an extension of (5), we also determine solutions of the constrained maximization problem
with dependence specification sets \({\mathcal {S}}^{i}\) and marginal specification sets \({\mathcal {F}}_{i}\subset {\mathcal {F}}^{1}\,,\) where \({\mathcal {F}}^{1}\) denotes the set of univariate distribution functions.
A main aim of this paper is to solve the constrained supermodular maximization problem
for \(F_{i}\in {\mathcal {F}}_{i}\) and \(G\in {\mathcal {F}}_{0}\,.\) 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 \({\mathcal {F}}_{i}\subset {\mathcal {F}}^{1}\) of marginal constraints under some specific dependence constraints \({\mathcal {S}}^{i}\,.\)
In Ansari and Rüschendorf (2016), some results on the supermodular maximization problem are given for normal and Kotztype 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 \(\bigvee _{i=1}^{d} D^{i}\) of bivariate copulas \(D^{i}\in \mathcal {C}_{2}\) is introduced by
for u=(u_{1},…,u_{d})∈[0,1]^{d}, where ∂_{2} denotes the partial derivative operator w.r.t. the second variable. (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 \(\bigvee _{i=1}^{d} C^{i}\) is the copula of the conditionally comonotonic risk vector \(\left (F_{X_{i}Z}^{1}(U)\right)_{1\leq i \leq d}\) 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 \(A^{i},B^{i}\in \mathcal {C}_{2}\) (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 \(D^{2},\ldots D^{d},E\in \mathcal {C}_{2}\,,\) 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 \({\mathcal {S}}^{i}\) such that solutions of the maximization problems (5) – (7) exist and can be determined.
The problem to find risk bounds for the ValueatRisk (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 TailValueatRisk (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 and Heijnen (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 aggregationrobustness 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 (2017a, b).
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 \({\mathcal {S}}^{i}\) 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 factorZ is a (partially specified) risk factor model (X_{i})_{1≤i≤d}, X_{i}=f_{i}(Z,ε_{i}), such that for some j∈{1,…,d} and a nondecreasing 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., \(Z\sim G\in {\mathcal {F}}_{c}^{1}\,.\) 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 \(G\in {\mathcal {F}}_{c}^{1}\,,\) 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 \(f\colon \mathbb {R}^{d}\to \mathbb {R}\,,\) let \(\triangle _{i}^{\varepsilon } f(x):=f(x+\varepsilon e_{i})g(x)\) be the difference operator, where ε>0 and e_{i} denotes the unit vector w.r.t. the canonical base in \(\mathbb {R}^{d}\,.\) Then, f is said to be supermodular, resp., directionally convex if \(\triangle _{i}^{\varepsilon _{i}} \triangle _{j}^{\varepsilon _{j}} f\geq 0\) for all 1≤i<j≤d, resp., 1≤i≤j≤d. For ddimensional random vectors ξ,ξ^{′}, the supermodular ordering ξ≤_{sm}ξ^{′}, resp., the directionally convex ordering ξ≤_{dcx}ξ^{′} is defined via \({\mathbb {E}} f(\xi) \leq {\mathbb {E}} f(\xi ^{\prime })\) for all supermodular, resp., directionally convex functions f for which the expectations exist. The lower, resp., upper orthant ordering ξ≤_{lo}ξ^{′}, resp., ξ≤_{uo}ξ^{′} is defined by the pointwise comparison of the corresponding distribution, resp., survival functions, i.e., \(F_{\xi }(x)\leq F_{\xi ^{\prime }}(x)\,,\) resp., \(\overline {F}_{\xi }(x)\leq \overline {F}_{\xi ^{\prime }}(x)\) for all \(x\in \mathbb {R}^{d}\,.\) Remember that the convex ordering ζ≤_{cx}ζ^{′} for realvalued random variables ζ,ζ^{′} is defined via \({\mathbb {E}} \varphi (\zeta)\leq {\mathbb {E}} \varphi (\zeta ^{\prime })\) for all convex functions φ for which the expectation exists. Note that these orderings depend only on the distributions and, thus, are also defined for the corresponding distribution functions. For an overview of stochastic orderings, see Müller and Stoyan (2002), Shaked and Shantikumar (2007), and Rüschendorf (2013).
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 \(D^{2}\ldots,D^{d},E\in \mathcal {C}_{2}\,.\) Then, the following statements are equivalent:
 (i)
D^{i}≤_{lo}E for all 2≤i≤d.
 (ii)
\(M^{2}\vee D^{2}\vee \cdots \vee D^{d} \leq _{lo} M^{2}\vee \underbrace {E \vee \cdots \vee E}_{(d1)\text {times}}\,.\)
 (iii)
\(M^{2}\vee D^{2}\vee \cdots \vee D^{d} \leq _{sm} M^{2}\vee \underbrace {E \vee \cdots \vee E}_{(d1)\text {times}}\,.\)
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.
Proof
(Proof of ‘(i) ⇔ (ii)’:) Assume that D^{i}≤_{lo}E. Then, for u=(u_{1},…,u_{d})∈[0,1]^{d}, we obtain from the definition of the upper product that
using that \(\phantom {\dot {i}\!}\partial _{2} M^{2}(u_{1},t)={\mathbb{1}}_{\{u_{1}\geq t\}}\) almost surely.
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.
The proof of ‘(i) ⇔ (iii)’ is given in Section 3. □
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 \(F_{j}\in {\mathcal {F}}^{1}\,,\) let X_{j}∼F_{j}, 1≤j≤d, be realvalued random variables such that \(C^{i}=C_{X_{i},X_{1}}\leq _{lo}E\) for all 2≤i≤d. Then, for Y_{1},…,Y_{d} with X_{j}=dY_{j} for all 1≤j≤d and \(C_{Y_{i},Y_{1}}=E\) for all 2≤i≤d holds
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). □
Remark 1

(a)
The upper bound in (12) is comonotonic conditionally on Y_{1}. Further, the vector \(\left (F_{Y_{2}Y_{1}}^{1}(U),\ldots,F_{Y_{d}Y_{1}}^{1}(U)\right)\) is comonotonic because all copulas \(C_{Y_{i},Y_{1}}=E\) coincide, 2≤i≤d, cf. Ansari and Rüschendorf (2018), Proposition 2.4(v).

(b)
For d=2, (12) reduces to (X_{1},X_{2})≤_{sm}(Y_{1},Y_{2}) and the upper product in Theorem 1 simplifies to M^{2}∨D^{2}=(D^{2})^{T}, resp., M^{2}∨E=E^{T}, where the copula C^{T} is the transposed copula of \(C\in \mathcal {C}_{2}\,,\) i.e., C^{T}(u,v)=C(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
$$(X_{1},\ldots,X_{d})X_{1}=z~\not \leq_{sm} ~(Y_{1},F_{Y_{2}Y_{1}}^{1}(U),\ldots,F_{Y_{d}Y_{1}}^{1}(U))Y_{1}=z\,,$$unless \(C_{X_{i},X_{1}}=E\) for all i, see (Ansari (2019), Proposition 3.18).

(c)
If (X_{i},X_{1}) are negatively lower orthant dependent for all 2≤i≤d, i.e., \(C_{X_{i},X_{1}}(u,v)\leq \Pi ^{2}(u,v)=uv\) for all (u,v)∈[0,1]^{2}, then Theorem 2 simplifies to
$$\begin{aligned} (X_{1},\ldots,X_{d})&\leq_{sm} \left(X_{1},F_{X_{2}}^{1}(U),\ldots,F_{X_{d}}^{1}(U)\right)\\ \text{and}~~~~\sum_{i=1}^{d} X_{i} &\leq_{cx} X_{1}+\sum\limits_{i=2}^{d} F_{X_{i}}^{1}(U)\,, \end{aligned} $$where U∼U(0,1) is independent of X_{1}.

(d)
For \(G\in {\mathcal {F}}^{1}_{c},\) the right side in (12), resp., (13) solves the constrained maximization problem (7), resp., (5) for the dependence specification sets
$$\begin{array}{@{}rcl@{}} {\mathcal{S}}^{1}&=&\{M^{2}\}\,,~~~\text{and}\\ {\mathcal{S}}^{i}&=&\{C\in \mathcal{C}^{2}\,\,C\leq_{lo} E\}\,,~~2\leq i \leq d\,. \end{array} $$(14)
As a consequence of Theorem 2, we also obtain improved upper bounds under some correlation information. For a bivariate random vector \((V_{1},V_{2})\sim C\in \mathcal {C}_{2}\,,\) denote Spearman’s ρ, resp., Kendall’s τ of (V_{1},V_{2}) by ρ_{S}(V_{1},V_{2})=ρ_{S}(C), resp., τ(V_{1},V_{2})=τ(C).
For r∈[−1,1], define \(C^{r}(u,v):=\sup \{C(u,v)~~C\in \mathcal {C}_{2}\,,~\rho _{S}(C)=r\}\,,\) (u,v)∈[0,1]^{2}. Then, C^{r} is a bivariate copula and is given by
where \(\phi (a,b)=\frac 1 6 \left [(9b+3\sqrt {9b^{2}3a^{6}})^{1/3}+(9b3\sqrt {9b^{2}3a^{6}})^{1/3}\right ]\,,\) see Nelsen et al. (2001) [Theorem 4].
For t∈[−1,1], define \(D^{t}(u,v):=\sup \{C(u,v)~~C\in \mathcal {C}_{2}\,,~\tau (C)=t\}\,,\) (u,v)∈[0,1]^{2}. 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.
Corollary 1
(Improved bounds based on correlations)
Let X_{1},…,X_{d} be realvalued random variables such that either
 (i)
ρ_{S}(X_{1},X_{i})<0.5 for all 2≤i≤d, or
 (ii)
τ(X_{1},X_{i})<0 for all 2≤i≤d.
Let r:= max2≤i≤d{ρ_{S}(X_{1},X_{i})}, resp., t:= max2≤i≤d{τ(X_{1},X_{i})}. Then, for Y_{1},…,Y_{d} with Y_{j}=dX_{j}, 1≤j≤d, and \(C_{Y_{i},Y_{1}}=C^{r}\,,\) resp., \(C_{Y_{i},Y_{1}}=D^{t}\) for all 2≤i≤d, it holds true that
where U∼U(0,1) is independent of Y_{1}.
Proof
The result follows from Theorem 2 using the monotonicity of the distributional bound C^{r} in r, resp., D^{t} in t w.r.t. the lower orthant ordering, see Nelsen et al. (2001) [Corollary 5 (a),(b),(e), resp., Corollary 3 (a),(b),(e)]. □
Remark 2
For r∈(−1,0.5) and t∈(−1,0), it holds that ρ_{S}(C^{r})>r and τ(D^{t})>t, see Nelsen et al. (2001) [Corollary 3(h), resp., 5(h)]. Thus, for \(F_{i}\in {\mathcal {F}}^{1},\) 1≤i≤d,\(G\in {\mathcal {F}}_{c}^{1}\) and
2≤i≤d, 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, ∂_{2}D(u,v) is almost surely decreasing in v for all u∈[0,1] and ∂_{1}D(u,v) is almost surely decreasing in u for all v∈[0,1].
If the upper bound E in Theorem 2 is conditionally increasing, then the case of increasing marginals in convex order can also be handled.
Theorem 3
(Improved bounds in ≤_{dcx}order) Let X_{1},…,X_{d} be realvalued random variables with \(C_{X_{i},X_{1}}\leq _{lo}E\) for all 2≤i≤d. Assume that E is conditionally increasing. Then, for Y_{1},…,Y_{d} with X_{j}≤_{cx}Y_{j} for all 1≤j≤d and \(C_{Y_{i},Y_{1}}=E\) for all 2≤i≤d holds
where U∼U(0,1) is independent of Y_{1}. This implies
Proof
Let Y1′,…,Yd′ with X_{j}=dYj′ for all 1≤j≤d and \(C_{Y_{i}',Y_{1}'}=E\) for all 2≤i≤d. Then, we obtain from (12) that
for V∼U(0,1) independent of Y1′. Since both \(\left (Y_{1}',F_{Y_{2}',Y_{1}'}^{1}(V),\ldots,F_{Y_{d}',Y_{1}'}^{1}(V)\right)\) and \(\left (Y_{1},F_{Y_{2}Y_{1}}^{1}(U),\ldots,F_{Y_{d}Y_{1}}^{1}(U)\right)\) have the same copula \(M^{2}\vee \underbrace {E\vee \cdots \vee E}_{(d1)\text {times}}\,,\) which is easily shown to be CI, the statement follows from Müller and Scarsini (2001), Theorem 4.5 using Yi′≤_{cx}Y_{i}. □
Remark 3
For \(F_{1},\ldots,F_{d}\in {\mathcal {F}}^{1},\) consider the sets \({\mathcal {F}}_{i}':=\{F\in {\mathcal {F}}^{1}F\leq _{cx} F_{i}\}\,.\) Let the sets \({\mathcal {S}}^{i}\) of dependence specifications be given as in (14). If \(E=C_{Y_{i},Y_{1}}\) is CI and Y_{i}∼F_{i}, then the upper bound in (16) solves maximization problem (6) with marginal specification sets \({\mathcal {F}}_{0}={\mathcal {F}}_{c}^{1}\) and \({\mathcal {F}}_{i}={\mathcal {F}}_{i}'\) for 1≤i≤d.
For a generalization of Theorem 2, we need an extension of (8) as follows.
Lemma 1
Let \(X=\left (X_{k}^{i}\right)_{1\leq i \leq d, 1\leq k \leq m}\) and \(Y=\left (Y_{k}^{i}\right)_{1\leq i \leq d, 1\leq k \leq m}\) be (d×m)matrices of real random variables with independent columns.
If \(\left (X_{k}^{i}\right)_{1\leq i \leq d}\leq _{sm} \left (Y_{k}^{i}\right)_{1\leq i \leq d}\) for all 1≤k≤m, then it holds true that
for all increasing convex functions ψ_{i} and increasing functions \(f_{k}^{i}\,.\)
Proof
By straightforward calculations, it can be shown that the function \(h\colon (\mathbb {R}^{m})^{d} \to \mathbb {R}\) given by
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) [Theorem 9.A.9(a),(b)]) imply that
where ≤_{icx} denotes the increasing convex order. Since it holds for 1≤i≤d that \(\sum _{k=1}^{m} f_{k}^{i}\left (X_{k}^{i}\right) {\stackrel {\mathrm {d}}=} \sum _{k=1}^{m} f_{k}^{i}\left (Y_{k}^{i}\right)\,,\) we obtain
Hence, the assertion follows from Shaked and Shantikumar (2007) [Theorem 4.A.35]. □
The application to improved portfolio TVaR bounds in Section 4 is based on the following generalization of Theorem 2.
Theorem 4
(Concatenation of upper bounds)
For \(F_{i}^{k}\in {\mathcal {F}}^{1}\,,\) let \(\left (X_{1}^{k},\ldots,X_{d}^{k}\right)\,,\) 1≤k≤m, be independent random vectors with \(X_{i}^{k}\sim F_{i}^{k}\,.\) Assume that \(C_{X_{i}^{k},X_{1}^{k}}\leq _{lo} E^{k}\) for \(E^{k}\in \mathcal {C}_{2}\) for all 2≤i≤d, 1≤k≤m. Then, for independent vectors \(\left (Y_{1}^{k},\ldots,Y_{d}^{k}\right)\) with \(Y_{i}^{k}{\stackrel {\mathrm {d}}=} X_{i}^{k}\) and \(C_{Y_{i}^{k},Y_{1}^{k}}=E^{k}\) for all 2≤i≤d, 1≤k≤m holds
where U^{1},…,U^{m}∼U(0,1)are i.i.d. and independent of \(Y_{1}^{k}\) for all k. This implies
for all increasing convex functions φ_{1},…,φ_{d}.
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
where \(F_{i}=F_{\varphi _{i}\left (\sum _{k} X_{i}^{k}\right)}\,,\) 1≤i≤d and \(G\in {\mathcal {F}}_{c}^{1}\,.\)
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
For a Borelmeasurable subset \(\Xi \subset \mathbb {R}^{d}\,,\) denote by \({\mathcal {B}}(\Xi)\) the Borel σalgebra on Ξ. Denote by \({\mathcal {M}}_{d}^{1}\) the set of probability measures on \({\mathcal {B}}(\Xi)\,.\) A signed measure on \({\mathcal {B}}(\Xi)\) is a σadditive mapping \(\mu \colon {\mathcal {B}}(\Xi) \to \mathbb {R}\) such that μ(∅)=0. Let \({\mathbb {M}}^{0}_{d}={\mathbb {M}}_{d}^{0}(\Xi)\,,\) resp., \({\mathbb {M}}^{1}_{d}={\mathbb {M}}_{d}^{1}(\Xi)\) be the set of all signed measures μ on \({\mathcal {B}}(\Xi)\) with μ(Ξ)=0, resp., μ(Ξ)=1 and finite variation norm ∥μ∥=μ^{+}(Ξ)+μ^{−}(Ξ)<∞, where μ^{+},μ^{−} are the unique measures obtained from the Hahn–Jordan decomposition of μ=μ^{+}−μ^{−}. Then, the definition of the orderings ≤_{lo}, ≤_{uo}, and ≤_{sm} can be extended to signed distributions using this decomposition.
Definition 2
Let \(P,Q\in {\mathbb {M}}^{1}_{d}\) be signed measures. Then, define
 (i)
the lower orthant orderP≤_{lo}Q if P((−∞,x])≤Q((−∞,x]) holds for all \(x\in \mathbb {R}^{d}\,,\)
 (ii)
the upper orthant orderP≤_{uo}Q if P((x,∞))≤Q((x,∞)) holds for all \(x\in \mathbb {R}^{d}\,,\)
 (iii)
the supermodular orderP≤_{sm}Q if \(\int f(x)\, {\mathrm {d}} P(x)\leq \int f(x) \, {\mathrm {d}} Q(x)\) holds for all supermodular integrable functions f.
We generalize the concept of comonotonicity to signed measures as follows.
Quasicomonotonicity
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., F_{ξ}=F.
For a signed measure \(P\in {\mathbb {M}}_{d}^{1}\,,\) we define the associated measure generating functionF=F_{P} by F(x)=P((−∞,x]) and its univariate marginal measure generating functionsF_{i} by \(F_{i}(x_{i})=P(\mathbb {R}\times \cdots \mathbb {R}\times (\infty,x_{i}]\times \mathbb {R}\times \cdots \times \mathbb {R})\) for \(x=(x_{1},\ldots,x_{d})\in \mathbb {R}^{d}\) and 1≤i≤d. We define the notion of quasicomonotonicity as follows.
Definition 3
(Quasicomonotonicity) We denote P, resp., F as quasicomonotonic if \(F(x)=\min \limits _{1\leq i \leq d}\left \{F_{i}(x_{i})\right \}\) for all \(x=(x_{1},\ldots,x_{d})\in \mathbb {R}^{d}.\)
Obviously, if \(P\in {\mathcal {M}}^{1}_{d}\,,\) then the quasicomonotonicity 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.
Lemma 2
Let \(P\in {\mathbb {M}}_{d}^{1}\) be a signed distribution with univariate marginal distribution functions F_{i}, 1≤i≤d. Let \(Q\in {\mathcal {M}}_{d}^{1}\) be a probability distribution. Assume that F_{i}(t)≤1 for all \(t\in \mathbb {R}\,,\) 1≤i≤d. If P is quasicomonotonic and Q is comonotonic, then it holds that
Proof
Let \(A_{i}=\{(y_{1},\ldots,y_{d})\in \mathbb {R}^{d}\,\, y_{i}\in (x_{i},\infty ]\},\phantom {\dot {i}\!}\) 1≤i≤d, and let \(a_{j}:=F_{i_{j}}(x_{i_{j}})\phantom {\dot {i}\!}\) for i_{1},…,i_{d}∈{1,…,d} such that a_{1}≥…≥a_{d}. Then, the survival function \(\overline {F}\) corresponding to F is calculated by
where the fourth equality holds true because P is quasicomonotonic, F_{i}≤1 and F_{i}(∞)=1 for all i. The fifth equality follows since there are \(\binom {k1}{kj}\) 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 \(\overline {G}(x)=Q((x,\infty))=1\max _{i}\left \{G_{i}(x_{i})\right \}\) for \(x=(x_{1},\ldots,x_{d})\in \mathbb {R}^{d}\,.\) We obtain that
where we use for the second equivalence that F_{i},G_{i}≤1 and F_{i}(∞)=G_{i}(∞)=1 for all 1≤i≤d. 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 \(n\in {\mathbb {N}}\) and d≥1, denote by
the (extended) uniform unit ngrid of dimension d with edge length \(\tfrac 1 n\,.\)
The following notion of an ngrid dcopula is related to an dsubcopula with domain \({\mathbb {G}}_{n}^{d}\,,\) see, e.g., Nelsen (2006), Definition 2.10.5. For our purpose, we also need a signed version. Denote by ⌊·⌋ the componentwise floor function.
Definition 4
(Grid copula) For \(d\in {\mathbb {N}}\,,\) a (signed) ngrid dcopula (briefly grid copula) \(D\colon [0,1]^{d} \to \mathbb {R}\) is the (signed) measure generating function of a (signed) measure \(\mu \in {\mathbb {M}}_{d}^{1}({\mathbb {G}}_{n,0}^{d})\) with uniform univariate margins, i.e., it holds that
 (i)
\(D(u)=D\left (\frac {\lfloor n u \rfloor }{n}\right)=\mu \left (\left [0,\frac {\lfloor n u \rfloor }{n}\right ]\right)\) for all u∈[0,1]^{d}, and
 (ii)
for all i=1,…,d holds \(D(u)=\tfrac k n\) for all k=0,…,n, if \(u_{i}=\tfrac k n\) and u_{j}=1 for all j≠i.
Denote by \(\mathcal {C}_{d,n}\) (resp., \(\in \mathcal {C}_{2,n}^{s}\)) the set of all (signed) ngrid dcopulas.
An \(\tfrac 1 n\)scaled doubly stochastic matrix or, if the dimension of the matrix is clear, a mass matrix is defined as an n×nmatrix with nonnegative entries and row, resp., column sums equal to \(\tfrac 1 n\,.\) By an signed \(\tfrac 1 n\)scaled doubly stochastic matrix or also signed mass matrix, we mean an \(\tfrac 1 n\)scaled doubly stochastic matrix where negative entries are also allowed.
Obviously, there is a onetoone correspondence between the set of (signed) ngrid 2copulas and the set of (signed) \(\tfrac 1 n\)scaled doubly stochastic matrices.
For a bivariate (signed) ngrid copula \(E\in \mathcal {C}_{2,n}\) (\(\in \mathcal {C}_{2,n}^{s}\)), the associated (signed) probability mass function e is defined by
where \(\Delta _{n}^{i}\) (distinct from \(\triangle _{i}^{\varepsilon _{i}}\)) denotes the difference operator of length \(\tfrac 1 n\) with respect to the ith variable, i.e.,
for \(u\in {\mathbb {G}}_{n,0}^{d}\) and the ith unit vector e_{i}. Further, define its associated (signed) mass matrix (e_{kl})_{1≤k,l≤n} by
For every dcopula \(D\in \mathcal {C}_{d}\,,\) denote by \({\mathbb {G}}_{n}(D)\) its canonical ngrid dcopula given by
Define the upper product \(\bigvee \colon (\mathcal {C}_{2,n})^{d} \to \mathcal {C}_{d,n}\) for grid copulas \(D_{n}^{1},\ldots,D_{n}^{d}\in \mathcal {C}_{2,n}\) 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.
Proposition 1
Let \(D^{i},E^{i}\in \mathcal {C}_{2}\) be bivariate copulas for 1≤i≤d. Then, it holds true that
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 \(P,Q\in {\mathbb {M}}^{1}_{d}\) with finite support, denote the signed measure Q−P a transfer from P to Q. To indicate this transfer, write
where \((QP)^{}=\sum _{i=1}^{n} \alpha _{i} \delta _{x_{i}}\) and \((QP)^{+}=\sum _{i=1}^{m} \beta _{i} \delta _{y_{i}}\) for α_{i},β_{j}>0 and \(x_{i},y_{j}\in \mathbb {R}^{d}\,,\) 1≤i≤n, 1≤j≤m. A reverse transfer from P to Q is a transfer from Q to P.
Since \(Q=P+(QP)=P\sum _{i=1}^{n} \alpha _{i} \delta _{x_{i}} + \sum _{i=1}^{m} \beta _{i} \delta _{y_{i}}\,,\) the mapping in (21) illustrates the mass that is transferred from P to Q. By definition, it holds that \(QP\in {\mathbb {M}}_{d}^{0}\,.\) Thus, mass is only shifted and, in total, neither created nor lost.
For a set \(M\subset {\mathbb {M}}^{0}_{d}\) of transfers, one is interested in the class \({\mathcal {F}}\) of continuous functions \(f\colon S\to \mathbb {R}\) such that
whenever μ∈M, where \(\mu :=\sum _{j=1}^{m} \beta _{j} \delta _{y_{j}}\sum _{i=1}^{n} \alpha _{i} \delta _{x_{i}}\,,\)α_{i},β_{j}>0. Then, \({\mathcal {F}}\) is said to be induced from M.
We focus on the set of Δmonotone, resp., Δantitone, resp., supermodular transfers. These sets induce the classes \({\mathcal {F}}_{\Delta }\) of Δmonotone, resp., \({\mathcal {F}}_{\Delta }^{}\) of Δantitone, resp., \({\mathcal {F}}_{sm}\) 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 \({\mathcal {V}}_{o}(x,y)\,,\) resp., \({\mathcal {V}}_{e}(x,y)\) the set of all vertices z of the kdimensional 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
$$\begin{array}{@{}rcl@{}} \eta\left(\sum_{z\in {\mathcal{V}}_{0}(x,y)} \delta_{z} \right)\to \eta\left(\sum_{z\in {\mathcal{V}}_{e}(x,y)} \delta_{z} \right) \end{array} $$is called a (kdimensional) Δmonotone transfer.
 (ii)
A transfer indicated by
$$\begin{aligned} \eta \left(\sum_{z\in {\mathcal{V}}_{o}(x,y)} \delta_{z} \right) &\to \eta\left(\sum_{z\in {\mathcal{V}}_{e}(x,y)} \delta_{z}\right)~~~\text{if }k \text{ is even, and}\\ \eta \left(\sum_{z\in {\mathcal{V}}_{e}(x,y)} \delta_{z} \right) &\to \eta\left(\sum_{z\in {\mathcal{V}}_{o}(x,y)} \delta_{z}\right)~~~\text{if }k \text{ is odd} \end{aligned} $$is called a (kdimensional) Δantitone transfer.
 (iii)
For \(v,w\in \mathbb {R}^{d}\,,\) a transfer indicated by
$$\eta(\delta_{v} + \delta_{w}) \to \eta (\delta_{v\wedge w}+\delta_{v\vee w})$$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.
Proposition 2
For signed measures \(P,Q\in {\mathbb {M}}^{1}_{d}\) with finite support holds:
 (i)
P≤_{uo}Q if and only if Q can be obtained from P by a finite number of Δmonotone transfers.
 (ii)
P≤_{lo}Q if and only if Q can be obtained from P by a finite number of Δantitone transfers.
 (iii)
P≤_{sm}Q if and only if Q can be obtained from P by a finite number of supermodular transfers.
Remark 5
From Definition 5, we obtain that exactly the onedimensional Δmonotone, resp., Δantitone transfers affect the univariate marginal distributions. Hence, for measures \(P,Q\in {\mathcal {M}}^{1}_{d}(\Xi)\) with equal univariate distributions, i.e., \(\phantom {\dot {i}\!}P^{\pi _{i}}=Q^{\pi _{i}}\,,\)π_{i} the ith 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 2dimensional Δmonotone, resp., Δantitone transfers. But note that also the onedimensional Δmonotone, resp., Δantitone 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 (D^{i})^{T}=M^{2}∨D^{i}≤_{sm}M^{2}∨E=E^{T}. But this means that D^{i}≤_{lo}E.
For the reverse direction, assume that D^{i}≤_{lo}E for all 2≤i≤d. Consider the discretized grid copulas \(D_{n}^{i}:={\mathbb {G}}_{n}(D^{i}), M_{n}^{2}:={\mathbb {G}}_{n}(M^{2}),\) and \(E_{n}:={\mathbb {G}}_{n}(E)\,,\) 2≤i≤d, and denote by \(d_{n}^{i}\,,\) resp., e_{n} the associated mass matrices of \(d_{n}^{i}\,,\) 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 \((D_{n,k}^{i})_{1\leq k \leq n}\) on \({\mathbb {G}}_{n}^{2}\) defined through the signed mass matrices \((D_{n,k}^{i})_{1\leq k \leq n}\) given by
for 1≤k≤n−1.
For all 2≤i≤d and for all \(n\in {\mathbb {N}}\,,\) the sequence \((D_{n,k}^{i})_{1\leq k \leq n}\) of signed mass matrices adjusts the signed mass matrix \(d_{n}^{i}\) column by column to the signed mass matrix e_{n}. It holds that \(d_{n,n}^{i}=e_{n}\) for all i and n.
For \(C_{n,k}:=M^{2}_{n}\vee D_{n,k}^{2} \vee \cdots \vee D_{n,k}^{d}\,,\) 1≤k≤n, we show that
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 \(D_{n}^{i}\leq _{lo} E_{n}\) 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 \(u_{j}\in {\mathbb {G}}_{n,0}^{1}\) and 1≤j≤d. This holds because \(\Delta _{n}^{2} D_{n,k}^{i}(u_{i},t)\leq \tfrac 1 n\) for all \((u_{i},t)\in {\mathbb {G}}_{n,0}^{2}\) and for all i and k, even if \(d_{n,k}^{i}\) can get negative for \(t=\tfrac k n\) and some u_{i}<1.
By construction of \((D_{n,k}^{i})_{1\leq k \leq n}\,,\) it holds that
for all 1≤k≤n−1 and for all \(u_{i}\in {\mathbb {G}}_{n,0}^{1}\,,\) 2≤i≤d.
To show (24), fix column k∈{1,…,n−1} of the signed mass matrices. Conditioning under \(u_{1}\in {\mathbb {G}}_{n}^{1}\,,\) consider the conditional (signed) measure generating functions
for l=1,…,n, where
is the upper product of the (signed) grid copulas \(M_{n},D_{n,l}^{2},\ldots,D_{n,l}^{d}\) for \(u\in {\mathbb {G}}_{n,0}^{d}\,.\) Hence, it holds for the conditional (signed) measure generating function that
and for its corresponding (signed) survival function that
where \(u_{1}=(u_{2},\ldots,u_{d})\in {\mathbb {G}}_{n,0}^{d1}\,.\)
By the construction of \((D_{n,l}^{i})_{1\leq l \leq n}\,,\) it holds that
We show that
where \(P^{U_{1}}(\cdot)\times P_{C_{n,k}^{\cdot }}\,,\) resp., \(P^{U_{1}}(\cdot)\times P_{C_{n,k+1}^{\cdot }}\) is the conditional measure generating function of \(P_{C_{n,k}}\,,\) resp., \(P_{C_{n,k+1}}\) given the set \(\{\tfrac k n,\tfrac {k+1} n\}\times {\mathbb {G}}_{n,0}^{d1}\,.\) Then, (28) and (31) imply (24) using (a slightly generalized version of) the closure of the supermodular ordering under mixtures given by Shaked and Shantikumar (2007) [Theorem 9.A.9.(d)].
To show (29), let us fix \(u_{1}=\tfrac k n\,.\) 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 \(\Delta _{n}^{2}\,,\) and the last equality is true because E_{n} is a grid copula, thus 2increasing, and hence \(\Delta _{n}^{2} E_{n}(\cdot,t)\) is increasing for all \(t\in {\mathbb {G}}_{n}^{1}\,.\)
Then, from (32) and (34) it follows for the kth columns of the matrices that
where the equality holds true due to (27). This means that
holds. Further, \(C_{n,k}^{u_{1}}\) corresponds to a quasicomonotonic signed measure in \({\mathbb {M}}_{d}^{1}\) with univariate marginals given by \(n\Delta _{n}^{2} D_{n,k}^{i}(\cdot,u_{1})\leq 1\,,\) and \(C_{n,k+1}^{u_{1}}\) 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 Δmonotone transfers that transfer \(C_{n,k}^{u_{1}}\) to \(C_{n,k+1}^{u_{1}}\,,\) i.e., there exist \(m\in {\mathbb {N}}\) and a finite sequence \(\left (P_{l}^{u_{1}}\right)_{1\leq l \leq m}\) of signed measures on \({\mathbb {G}}_{n}^{d1}\) such that
Since the univariate margins of \(C_{n,k}^{u_{1}}\) and \(C_{n,k+1}^{u_{1}}\) do not coincide, some of the transfers \(\left (\mu _{l}^{u_{1}}\right)_{l}\) must be onedimensional, see Remark 5. Each onedimensional transfer \(\mu _{l}^{u_{1}}\) transports mass from one point \(u^{l}=\left (u_{2}^{l},\ldots,u_{d}^{l}\right)\in {\mathbb {G}}_{n}^{d1}\) to another point \(v^{l}=\left (v_{2}^{l},\ldots,v_{d}^{l}\right)\in {\mathbb {G}}_{n}^{d1}\) such that \(v_{\iota }^{l}< u_{\iota }^{l}\) for an ι∈{2,…,d} and \(u_{j}^{l}=v_{j}^{l}\) for all j≠ι, i.e., \(\mu _{l}^{u_{1}}=\eta ^{l} \left (\delta _{v^{l}}\delta _{u^{l}}\right)\) is indicated by
for some η^{l}>0. Since applying mass transfers is commutative, we first choose to apply all of these onedimensional reverse Δmonotone transfers. Because δdimensional Δmonotone transfers leave the univariate marginals unchanged for δ≥2, see Remark 5, the univariate margins of \(C_{n,k}^{u_{1}}\) must be adjusted to the univariate margins of \(C_{n,k+1}^{u_{1}}\) having applied all of these onedimensional reverse Δmonotone transfers.
Then, since the grid copula of \(C_{n,k+1}^{u_{1}}\) is the upper Fréchet bound and hence the greatest element in the ≤_{uo}ordering, no further reverse Δmonotone transfer is possible. Thus, \(C_{n,k+1}^{u_{1}}\) is reached from above having applied only onedimensional Δmonotone transfers \(\mu _{l}^{u_{1}}\,,\) 1≤l≤m−1, on \(P_{C_{n,k}^{u_{1}}}\,,\) i.e.,
For all reverse Δmonotone transfers \(\mu _{l}^{u_{1}},\phantom {\dot {i}\!}\) consider its corresponding reverse transfer \(\mu _{l}^{u_{1}+1/n}:=\mu _{l}^{u_{1}}\phantom {\dot {i}\!}\) on \({\mathbb {G}}_{n}^{d1}\) indicated by \(\phantom {\dot {i}\!}\eta ^{l} \delta _{v^{l}}\to \eta ^{l} \delta _{u^{l}}\,.\) Define
The transfers \(\phantom {\dot {i}\!}\left (\mu _{l}^{u_{1}+1/n}\right)_{l}\) are onedimensional Δmonotone transfers. Then, it holds true that they adjust the univariate marginals of \(\phantom {\dot {i}\!}P_{C_{n,k}^{u_{1}+1/n}}\) to the univariate marginals of \(P_{C_{n,k+1}^{u_{1}+1/n}}.\phantom {\dot {i}\!}\) This can be seen because only two entries (in column k) of matrix ι are changed by the mass transfer \(\mu _{l}^{u_{1}}.\phantom {\dot {i}\!}\) All other columns and matrices j≠ι are unaffected by this transfer. From (28) follows that exactly the reverse transfers \(\mu _{l}^{u_{1}+1/n}\phantom {\dot {i}\!}\) 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 \(d_{n,k}^{i}\) is adjusted to column j of the mass matrix \(d_{n,k+1}^{i}\) for all 2≤i≤d. But this also means that column k+1 of the mass matrix \(d_{n,k}^{i}\) must be adjusted to column k+1 of \(d_{n,k+1}^{i}\) due to the uniform margin condition.
Since applying the onedimensional transfers \(\phantom {\dot {i}\!}\mu _{l}^{u_{1}+1/n}\) on \(\phantom {\dot {i}\!}P_{C_{n,k}^{u_{1}+1/n}}\) (which is comonotonic) can change the dependence structure, the signed measure \(P_{m}^{u_{1}+1/n}\) is not necessarily quasicomonotonic, i.e., \(P_{m}^{u_{1}+1/n}\phantom {\dot {i}\!}\) does not necessarily coincide with \(P_{C_{n,k+1}^{u_{1}+1/n}}\) (which is quasicomonotonic). We show that
Since \(C_{n,k}^{u_{1}}\leq _{lo} C_{n,k+1}^{u_{1}}\,,\) see (35), it also holds that
where we use that
for all 2≤i≤d. By construction of \(\left (d^{i}_{n,l}\right)_{1\leq l\leq n}\,,\) it follows that
This implies
But this means that \(C_{n,k}^{u_{1}+1/n}\geq _{lo} C_{n,k+1}^{u_{1}+1/n}\,.\) Due to (23), it holds that \(C_{n,k}^{u_{1}+1/n}\) is comonotonic and \(C_{n,k+1}^{u_{1}+1/n}\) is quasicomonotonic with univariate marginal measure generating functions \(n\Delta _{n}^{2} D_{n,k+1}^{i}\left (\cdot,\tfrac {k+1}n\right)\leq 1\,.\) Thus, Proposition 2 yields (30).
Further, (30) and Proposition 2 imply that there exist \(m'\in {\mathbb {N}}\) and a finite number of reverse Δmonotone transfers \(\phantom {\dot {i}\!}(\gamma _{l})_{1\leq l \leq m'}\) that adjust \(P_{C_{n,k+1}^{u_{1}+1/n}}\phantom {\dot {i}\!}\) to \(\phantom {\dot {i}\!}P_{C_{n,k}^{u_{1}+1/n}}\,.\) With the same argument as above, these transfers are onedimensional. Further, the reverse transfers \(\phantom {\dot {i}\!}(\gamma _{l}^{r})_{1\leq l \leq m'}\,,\) where \(\gamma _{l}^{r}=\gamma _{l}\,,\) correspond to the Δmonotone transfers \(\phantom {\dot {i}\!}\left (\mu _{l}^{u_{1}+1/n}\right)_{1\leq l \leq m}\) that adjust the margins of \(C_{n,k}^{u_{1}+1/n}\phantom {\dot {i}\!}\) to the margins of \(\phantom {\dot {i}\!}C_{n,k+1}^{u_{1}+1/n}\,.\) This yields m=m^{′},\(\sum _{l=1}^{m1} \mu _{l}^{u_{1}+1/n} =\sum _{l=1}^{m'1} \gamma _{l}^{r}\phantom {\dot {i}\!}\) and thus \(P_{m}^{u_{1}+1/n}=P_{C_{n,k+1}^{u_{1}+1/n}}\,,\) which proves (39). Hence, (38) yields
It remains to show (31). Each transfer \(\mu _{l}^{u_{1}}\,,\,,\) resp.„ \(\mu _{l}^{u_{1}+1/n}\) on \({\mathbb {G}}_{n}^{d1}\) can be extended to a reverse Δmonotone, resp., Δmonotone transfer μ_{l,r}, resp., μ_{l} on \(\{u_{1}\}\times {\mathbb {G}}_{n}^{d1}\,,\) resp., \(\{u_{1}+\tfrac 1 n\}\times {\mathbb {G}}_{n}^{d1}\,,\) indicated by
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 \(\{u_{1},u_{1}+\tfrac 1 n\}\times {\mathbb {G}}_{n}^{d1}\) between (u_{1},u^{l}) and \(\left (u_{1}+\tfrac 1 n,v^{l}\right)\,,\) indicated by
Each transfer ν_{l} is a supermodular transfer. Denote by ε_{{x}} the onepoint 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}.\(\square \)
Remark 6

(a)
The proof is based on an approximation by finite sequences of signed grid copulas that fulfill the conditioning argument in (28)–(31). Further, we use the necessary condition that the lower orthant ordering holds true–indeed, (32) and (34) yield C_{n,k}≤_{lo}C_{n,k+1}–in order to show that the supermodular ordering is also fulfilled.

(b)
The condition that the upper bound E for D^{i} is a joint upper bound, i.e., it does not depend on i, is crucial for the proof. Otherwise, Eq. (33) can fail, see also (11). In general, it holds that
$$\begin{array}{@{}rcl@{}} D^{i}\leq_{lo} E^{i} ~\forall i ~\not \Longrightarrow ~M^{2} \vee D^{1} \vee \cdots \vee D^{d} \leq_{lo} M^{2} \vee E^{1} \vee \cdots \vee E^{d}\,. \end{array} $$For a counterexample assume that D^{1}=D^{2}<_{lo}E^{1}<_{lo}E^{2}. Then, it holds
$$M^{2}\vee D^{1} \vee D^{2} (1,\cdot,\cdot)=M^{2} >_{lo} E^{1}\vee E^{2} = M^{2}\vee E^{1}\vee E^{2} (1,\cdot,\cdot)\,,$$using the marginalization and the maximality property of the upper product, see Ansari and Rüschendorf (2018), Proposition 2.4, which yields a contradiction to M^{2}∨D^{1}∨D^{2}≤_{lo}M^{2}∨E^{1}∨E^{2}.

(c)
While in the proof the \(D_{n,k}^{i}\,,\) 1≤k≤n, can be signed grid copulas with \(\Delta _{n}^{2} D_{n,k}^{i}(u_{i},t)\leq \tfrac 1 n\) for all \((u_{i},t)\in {\mathbb {G}}_{n,0}^{2}\,,\) it is necessary that E_{n} is a grid copula and not only a signed grid copula. Otherwise, both monotonicity properties in (33) and (34) can fail.
We illustrate the idea of the proof with an example for n=4 and d=3 :
Example 1
Let \(D^{2}_{4},D^{3}_{4},E_{4}\) be 4grid copulas given through the mass matrices \(d^{2}_{4},d^{3}_{4}\,,\) resp., e_{4} by
Then, we observe that \(D^{2}_{4},D^{3}_{4}\leq _{lo} E_{4}\,.\) Consider the signed 4grid copulas \(D_{4,l}^{i}\,,\) 1≤l≤4, i=2,3, in Fig. 1 constructed by (23). The conditional distribution of \(M^{2}_{4}\vee D^{2}_{4,1}\vee D^{3}_{4,1}\) under \(u_{1}=\tfrac 1 4\) is given by \(4\, \Delta _{4}^{2} \, M^{2}_{4}\vee D^{2}_{4,1}\vee D^{3}_{4,1}\left (\tfrac 1 4,\cdot,\cdot \right)=4 \min \limits _{i}\{\Delta _{4}^{2} D^{i}\left (\cdot,\tfrac 1 4\right)\}\,,\) where the arguments of the minfunction correspond to the distributions given through the first columns of \(d_{4}^{2}\,,\) resp., \(d_{4}^{3}\,.\)
Since
the solidmarked reverse Δmonotone transfers can be applied to adjust the first column of \(d^{2}_{4,1}\,,\) resp., \(d^{3}_{4,1}\) to the first column of e_{4}. These transfers are balanced by the dashedmarked Δmonotone transfers in the second columns which guarantee that the new matrices \(d_{4,2}^{2}\,,\) resp., \(d_{4,2}^{3}\) are still (signed) copula mass matrices. This procedure is repeated column by column until \(d_{4,4}^{2}=d_{4,4}^{3}=e_{4}\,.\)
Application to improved portfolio TVaR bounds
In this section, we determine improved TailValueatRisk bounds for a portfolio \(\Sigma _{t}=\sum _{i=1}^{8} Y_{t}^{i}\,,\)t≥0, t in trading days, of d=8 (derivatives on) assets \(S_{t}^{i}\) applying Theorem 4 about internal risk factor models. More specifically, let \(Y_{t}^{i}=S_{t}^{i}\) for i=1,…,6 and \(Y_{t}^{i}=(S_{t}^{i}K^{i})_{+}\) for i=7,8, where K^{7}=70 and K^{8}=10. In this application, \((S_{t}^{i})_{t\geq 0}\) denotes the asset price process of Audi (i=1), Allianz (i=2), Daimler (i=3), Siemens (i=4), Adidas (i=5), Volkswagen (i=6), SAP (i=7), resp., Deutsche Bank (i=8).
We aim to determine improved TVaR bounds for Σ_{T} for T=1 year=254 trading days, resp., T=2 years=508 trading days. The underlying process \(S_{t}=(S_{t}^{1},\ldots,S_{t}^{8})\) is modeled by an integrable exponential process S_{t}=S_{0} exp(L_{t}) under the following assumptions:
Let \(m\in {\mathbb {N}}\) and 0=t_{0}<t_{1}<⋯<t_{m}=T with \(t_{i}t_{i1}=\tfrac T m\) for 1≤i≤m.
 (I)
The component processes \((L_{t}^{i})_{t\geq 0}\) are Lévy processes for all i.
 (II)
The increments \((\xi _{k}^{1},\ldots,\xi _{k}^{d}):=(L_{t_{k}}^{1}L_{t_{k1}}^{1},\ldots,L_{t_{k}}^{d}L_{t_{k1}}^{d})\,,\) 1≤k≤m, are independent in k (but not necessarily stationary).
 (III)
For all k, there exists a bivariate copula \(E^{k}\in \mathcal {C}_{2}\) such that \(C_{\xi _{k}^{i},\xi _{k}^{1}}\leq _{lo} E^{k}\) for all 2≤i≤d.
Assumptions (I)–(III) are consistent. Assumption (I) is a standard assumption on the logincrements of \((S_{t}^{i})_{t\geq 0}\) 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 kth logincrement of the ith component and the kth logincrement of the first component (which is the internal risk factor) by a subclass \({\mathcal {S}}_{k}^{i}=\{C\in \mathcal {C}_{2}C\leq _{lo} E^{k}\}\) of bivariate copulas.
Then, Theorem 4 yields improved bounds in convex order for the portfolio Σ_{T} if the claims \(Y_{T}^{i}\) are of the form \(Y_{T}^{i}=\psi _{i}(S_{T}^{i})=\psi _{i}(\exp (L_{T}^{i}))=\psi _{i}\left (\exp \left (\sum _{k=1}^{m} \xi _{k}^{i}\right)\right)\) with ψ_{i} increasing convex.
For the estimation of the distribution of \(S_{T}^{i}\,,\) we make the following specification of Assumption (4):
 (1)
Each \(\left (S_{t}^{i}\right)_{t\geq 0}\,,\)i=1,…,8, follows an exponential NIG process, i.e.,
$$\begin{array}{@{}rcl@{}} S_{t}^{i}=S_{0}^{i}\exp\left(L_{t}^{i}\right)\,~~t\geq 0\,, \end{array} $$where \(S_{0}^{i}>0\) and where each \(\left (L_{t}^{i}\right)_{t\geq 0}\) is an NIG process with parameters α_{i},β_{i},δ_{i},ν_{i}.
For the estimation of upper bounds in supermodular order for the increments \(\left (\xi _{k}^{1},\ldots,\xi _{k}^{8}\right)\), we specify Assumption (I) as follows:
 (3)
For fixed ν∈(2,∞], the copula E^{k} in Assumption (III) is given by a tcopula with some correlation parameter ρ_{k}∈[−1,1] (which we specify later) and ν degrees of freedom, i.e., \(E^{k}=C_{\nu }^{\rho _{k}}\,.\)
We make use of the relation between the (pseudo)correlation parameter ρ of elliptical copulas and Kendall’s τ given by \(\rho (\tau)=\sin \left (\tfrac \pi 2 \tau \right)\,,\) 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 \(\tau _{k}:=\max _{2\leq i \leq 8}\{\tau _{k}^{i}\}\,,\) where \(\tau _{k}^{i}:=\tau \left (C_{\xi _{k}^{i},\xi _{k}^{1}}\right)\,.\) Since it is not possible to determine the dependence structure of each increment from a single observation, we estimate \(\tau _{k}^{i}\) from a sample of past observations. To do so, we assume that the dependence structure of \(\left (\xi _{k}^{i},\xi _{k}^{1}\right)\) does not jump too rapidly to strong positive dependence in a short period of time as follows:
 (2)
For \(n\in {\mathbb {N}}\,,\) define the averaged correlations over the past n time points at time k by \(\tau _{k,n}^{i}:=\tfrac 1 n \sum _{j=0}^{n1}\tau _{kn+j}^{i}\,,\) for k>n. Then, we assume that
$$\begin{array}{@{}rcl@{}} \tau_{k}=\max_{2\leq i \leq d}\{\tau_{k}^{i}\}\leq \max_{2\leq i \leq d}\{\tau_{k,n}^{i}\}+\epsilon_{k} \end{array} $$(42)for some error ε_{k}≥0 (which we fix later).
The above assumptions include the basic assumptions of multivariate exponential Lévy models because the stationarity condition in Assumption (II) yields (42). Further, ρ_{k}=1 yields E_{k}=M^{2} which means that Assumption (III) is trivially fulfilled in this case. Note that in this application the dependence constraints are allowed to come from quite a big subclass of copulas (see Remark 4).
Under the Assumptions (I)–(III), the dependence structure of \(\left (Y_{T}^{i},Y_{T}^{1}\right)\) is not uniquely determined for i=2,…,8. Thus, we need to solve the constrained maximization problem (5) to obtain improved upper bounds compared to applying (2) for partially specified risk factor models.
We mention that the structure of this section and the underlying data are similar to Ansari and Rüschendorf (2018), Section 4. But, there, the risk factor (which is the “DAX”) is an external risk factor which is not part of the portfolio, whereas in our application the internal risk factor “AUDI” is part of the underlying portfolio. This allows use of the simplified ordering conditions established in this paper. Further, the improved TVaRbounds in this application are based on large sets of dependence specifications of the daily logreturns (see Assumption (III) and Remark 4), whereas in Ansari and Rüschendorf (2018) all the dependence constraints on the time T logreturns are assumed to come from a oneparametric family of copulas.
Application to real market data
As data set, we take the daily adjusted close data from “Yahoo! Finance” from 23/04/2008 to 20/04/2018. It contains the values of 2540 trading days for 8 assets (with some missing data) which we denote by \(\left (s_{k}^{1},\ldots,s_{k}^{8}\right)_{1\leq k \leq 2540}\,.\) More precisely, \(\left (s_{k}^{1}\right)_{k}\) are the adjusted close data of “AUDI AG (NSU.DE)”, \(\left (s_{k}^{2}\right)_{k}\) of “Allianz SE (ALV.DE)”, \(\left (s_{k}^{3}\right)_{k}\) of “Daimler AG (DAI.DE)”, \(\left (s_{k}^{4}\right)_{k}\) of “Siemens Aktiengesellschaft (SIE.DE)”, \(\left (s_{k}^{5}\right)_{k}\) of “adidas AG (ADS.DE)”, \(\left (s_{k}^{6}\right)_{k}\) “Volkswagen AG (VOW.DE)”, \(\left (s_{k}^{7}\right)_{k}\) of “SAP SE (SAP.DE)” and \(\left (s_{k}^{8}\right)_{k}\) of “Deutsche Bank Aktiengesellschaft (DBK.DE)”.
We choose \(\hat {\tau }_{k,n}^{i}:=\hat {\tau }\left (\left (x_{kn+j}^{i},x_{kn+j}^{1}\right)_{0\leq j< n}\right)\) as an estimator for \(\tau _{k,n}^{i}\) in Assumption (4), where \(\hat {\tau }\) denotes Kendall’s rank correlation coefficient (see, e.g., (McNeil et al. (2015), equation (5.50))) and \(\left (x_{k}^{i}\right)_{2\leq k \leq 2540}\) are the historical logreturns of the ith component, i.e., \(x_{k}^{i}:=\log s_{k}^{i}\log s_{k1}^{i}\) for 2≤k≤2540. Further, we choose n=30 and ε_{k}=ε=0.05 in (42).
In Fig. 2, the historical estimates \(\hat {\tau }_{k,30}\) are illustrated for 31≤k≤2540 and for i=2,…,8. Further, the plot at the bottomright shows the maximum of the historical estimates \(\hat {\tau }_{k,30}=\max _{2\leq i \leq 8}\{\hat {\tau }_{k,30}^{i}\}\) (solid graph) as an estimator for τ_{k}, and it also shows the estimated historical upper bound \(\overline {\hat {\rho }_{k}}:= \max _{2\leq i\leq 8}\{\rho \left (\hat {\tau }_{k,n}^{i}+\epsilon _{k}\right)\}\) (dotted graph) with error ε_{k} for ρ_{k}, 31≤k≤2540, see Assumption 4.
As we observe from Fig. 2 there is no strong correlation between the logreturns \(\left (x_{k}^{1}\right)_{k}\) of “AUDI” and the logreturns \(\left (x_{k}^{i}\right)_{k}\,,\)i≠1, of the other assets. We use this property to apply Theorem 4 as follows.
For the prediction of an improved worstcase upper bound for Σ_{T} w.r.t. convex order for T=1 year, resp., T=2 years, we choose the worstcase period of the historical estimates \(\overline {\hat {\rho }}_{k}\) for ρ_{k} with a length of m=254 trading days, resp., m=508 trading days. We identify visually that \((\overline {\hat {\rho }}_{k})_{k}\) 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 \((\overline {\hat {\rho }}_{k})_{1797\leq k \leq 2050}\,,\) resp., \((\overline {\hat {\rho }}_{k})_{1543\leq k \leq 2050}\) as the worstcase estimate for (ρ_{k})_{1≤k≤254}, resp., (ρ_{k})_{1≤k≤508} with error ε_{k}=0.05 in (42).
Then, we obtain from Theorem 4 that
where \(\zeta _{k}^{i}\sim \xi _{k}^{i}\) for 1≤i≤8,\(C_{\zeta _{k}^{i},\zeta _{k}^{1}}=E^{k}=C_{\nu }^{\rho _{k}}\) for \(\rho _{k}=\overline {\hat {\rho }}_{2050m+k+1}\) and 2≤i≤8, U^{k}∼U(0,1) and \(U^{k},\zeta _{l}^{1}\) independent for all 1≤k,l≤m. Denote by \(\tau _{\zeta _{k}^{1}}\) the distributional transform of \(\zeta _{k}^{1}\,,\) see Rüschendorf (2009), and let t_{ν} be the distribution function of the tdistribution 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 tcopula with correlation r and ν degrees of freedom, see Aas et al. (2009).
The TailValueatRisk at level λ (also known as Expected Shortfall) is defined by
for a realvalued random variable ζ. If ζ is integrable, then TVaR_{λ} is a convex lawinvariant risk measure, see, e.g., Föllmer and Schied (2010), which satisfies the Fatouproperty. As a consequence of (4) and (43) we obtain
Empirical results and conclusion
The improved risk bounds \({\text {TVaR}}_{\lambda }\left (\Sigma _{T,(\rho _{k}),\nu }^{c}\right)\) for TVaR_{λ}(Σ_{T}) are compared in Table 1 with the standard comonotonic risk bound \({\text {TVaR}}_{\lambda }\left (\Sigma _{T}^{c}\right)\) (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 tcopulas \(C_{\nu }^{\rho _{k}}\) and high levels of λ. For T=2 years, the improvement is even better because the twoyear worstcase period for \(\overline {\hat {\rho _{k}}}\) also contains the oneyear worstcase period for \(\overline {\hat {\rho _{k}}}\) 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 \(C_{\nu }^{\rho }\) has a higher taildependence for smaller values of ν, see, e.g., Demarta and McNeil (2005). Thus, for small ν, more extreme events (= realizations of the logincrements) 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.
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Ansari, J., Rüschendorf, L. Upper risk bounds in internal factor models with constrained specification sets. Probab Uncertain Quant Risk 5, 3 (2020). https://doi.org/10.1186/s4154602000045y
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DOI: https://doi.org/10.1186/s4154602000045y
Keywords
 Risk bounds
 Risk factor model
 Supermodular order
 Convex order
 Convex risk measure
 Upper product of bivariate copulas
 Comonotonicity