# Regulator-Based Risk Statistics for Portfolios

Authors: Xiaochuan Deng; Fei Sun
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Risk statistic is a critical factor not only for risk analysis but also for financial application. However, the traditional risk statistics may fail to describe the characteristics of regulator-based risk. In this paper, we consider the regulator-based risk statistics for portfolios. By further developing the properties related to regulator-based risk statistics, we are able to derive dual representation for such risk.


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## Body

## 1. Introduction

Risk measure is a popular topic in both financial application and theoretical research. The quantitative calculation of risk involves two problems: choosing an appropriate risk model and allocating risk to individual institutions. This has led to further research on risk statistics. In a seminal paper, Sun et al. and Liu et al. [1, 2] first introduced the class of natural risk statistics with representation results. Furthermore, Ahmed et al. [3] derived an alternative proof for the natural risk statistics. Later, Tian and Jiang and Tian and Suo [4, 5] obtained the representation results for convex risk statistics and quasiconvex risk statistics, respectively.However, traditional risk statistics may fail to describe the characteristics of regulator-based risk. Therefore, the study of regulator-based risk statistics is particularly interesting. On the other hand, in the abovementioned research on risk statistics, the set-valued risk was never studied. Jouini et al. [6] pointed out that a set-valued risk measure is more appropriate than a scalar risk measure especially in the case where several different kinds of currencies are involved when one is determining capital requirements for the portfolio. Indeed, a natural set-valued risk statistic can be considered as an empirical (or a data-based) version of a set-valued risk measure. More recent studies of set-valued risk measures include those of [7–14] and the references therein.The main focus of this paper is regulator-based risk statistics for portfolios. In this context, both empirical versions and data-based versions of regulator-based risk measures are discussed. By further developing the properties related to regulator-based risk statistics, we are able to derive their dual representations. Indeed, this class of risk statistics can be considered as an extension of those introduced in [15–17].The remainder of this paper is organized as follows. In Section2, we briefly introduce some preliminaries. In Section 3, we state the main results of regulator-based risk statistics, including the dual representations. Section 4 investigates the data-based versions of regulator-based risk measures. Finally, in Section 5, the main proofs in this paper are discussed.


## 2. Preliminary Information

In this section, we briefly introduce some preliminaries that are used throughout this paper. Letd≥1 be a fixed positive integer. The space ℝd×n represents the set of financial risk positions. With positive values of X∈ℝd×n, we denote the gains while the negative denotes the losses. Let nj be the sample size of D=X1,…,Xd in the jth scenario, j=1,…,l. Let n≔n1+⋯+nl. More precisely, suppose that the behavior of D is represented by a collection of data X=X1,…,Xd∈ℝn×⋯×ℝn, where Xi=Xi,1,…,Xi,l∈ℝn and Xi,j=x1i,j,…,xnji,j∈ℝnj is the data subset that corresponds to the jth scenario with respect to Xi. For each j=1,…,l, h=1,…,nj, Xhj≔xh1,j,xh2,j,…,xhd,j is the data subset that corresponds to the hth observation of D in the jth scenario.In this paper, an elementz of ℝd is denoted by z≔z1,…,zd. An element X of ℝd×n is denoted by X≔X1,…,Xd≔x11,1,…,xn11,1,…,x11,l,…,xnl1,l,…,x1d,1,⋯xn1d,1,…,x1d,l,…,xnld,l. Let K be a closed convex polyhedral cone of ℝd where K⊇ℝ++d≔x1,…,xd∈ℝd;xi>0,1≤i≤d and K∩ℝ−d=∅ where ℝ−d≔x1,…,xd∈ℝd;xi≤0,1≤i≤d. Let K+ be the positive dual cone of K, that is, K+≔u∈ℝd:utrv≥0for anyv∈K, where utr means the transpose of u. For any X=X1,…,Xd,Y=Y1,…,Yd∈ℝd×n, X+Y stands for X1+Y1,…,Xd+Yd and aX stands for aX1,…,aXd for a∈ℝ. Denote K1n≔z11n,z21n,…,zd1n:z∈K and z1n≔z,z,…,z:z∈ℝ∈ℝn where 1n≔1,…,1∈ℝn. By K1n+, and we denote the positive dual cone of K1n in ℝd×n, i.e., K1n+≔w∈ℝd×n:wztr≥0for anyz∈K. The partial order with respect to K is defined as a≤Kb, which means b−a∈K where a,b∈ℝd, and X≤K1nY means Y−X∈K1n where X,Y∈ℝd×n.LetM≔ℝm×0d−m be the linear subspace of ℝd for 1≤m≤d. The introduction of M was considered in [6, 9]. Denote M+≔M∩ℝ+d, where ℝ+d≔x1,…,xd∈ℝd;xi≥0,1≤i≤d and M⊥≔0m×ℝd−m. Therefore, a regulator can only accept security deposits in the first m reference instruments. Denoting KM≔K∩M by the closed convex polyhedral cone in M, KM+≔u∈M:utrz≥0for anyz∈KM the positive dual cone of KM in M and intKM is the interior of KM in M. We denote QMt≔A⊂M:A=clcoA+KM and QM+t≔A⊂KM:A=clcoA+KM, where the clcoA represents the closed convex hull of A.By [18], a set-valued risk statistic is any map ρ,(1)ρ:ℝd×n⟶2M,which can be considered as an empirical (or a data-based) version of a set-valued risk measure. The axioms related to this set-valued risk statistic are organized as follows:[A0] Normalization:KM⊆ρ0 and ρ0∩−intKM=ϕ[A1] Monotonicity: for anyX,Y∈ℝd×n, X−Y∈K1n implies that ρX⊇ρY[A2] M-translative invariance: for anyX∈ℝd×n and z∈ℝd, ρX−z1n=ρX+z[A3] Convexity: for anyX,Y∈ℝd×n and λ∈0,1, ρλX+1−λY⊇λρX+1−λρY[A4] Positive homogeneity:ρλX=λρX for any X∈ℝd×n and λ>0[A5] Subadditivity:ρX+Y⊇ρX+ρY for any X,Y∈ℝd×nWe end this section with more notations. A functionρ:ℝd×n⟶2M is said to be proper if domρ≔X∈ℝd×n:ρX≠∅≠∅ and ρX≠M for all X∈domρ. ρ is said to be closed if graphρ is a closed set. For the properties of the graphs, see [19–21].


## 3. Empirical Versions of Regulator-Based Risk Measures

In this section, we state the dual representations of regulator-based risk statistics, which are the empirical versions of regulator-based risk measures. Firstly, for anyX∈ℝd×n, X∧K1n0 is defined as follows:(2)X∧K1n0≔X,X∉K1n,0,X∈K1n.Therefore, the position that belongs toK regarded is as 0 position. Next, we derive the properties related to regulator-based risk statistics.Definition 1.
A regulator-based risk statistic is a functionϱ:ℝd×n⟶QM+t that satisfies the following properties:[P0] Normalization: KM⊆ϱ0 and ϱ0∩−intKM=ϕ[P1] Cash cover: for any z∈KM, z∈ϱ−z1n[P2] Monotonicity: for any X,Y∈ℝd×n, X−Y∈ℝd×n∩K1n implies that ϱX⊇ϱY[P3] Regulator-dependence: for any X∈ℝd×n, ϱX=ϱX∧K1n0[P4] Convexity: for any X,Y∈ℝd×n and λ∈0,1, ϱλX+1−λY⊇λϱX+1−λϱYRemark 1.
The property ofP1 means any fixed negative risk position −z can be canceled by its positive quality z; P2 says that if X1 is bigger than X2 for the partial order in K, then the X1 need less capital requirement than X2, so ϱX1 contains ϱX2; P3 means the regulator-based risk statistics start only from the viewpoint of regulators which only care about the positions that need to pay capital requirements, while the positions that belong to K regarded as 0 position.
We now construct an example for regulator-based risk statistics.Example 1.
The coherent risk measureAV@R was studied by Föllmer and Schied [22] in detail. They have given several representations and many properties such as law invariance and the Fatou property [12]. First, they introduced set-valued AV@R, where the representation result is derived. Moveover, they also proved that it is a set-valued coherent risk measure. We now define the regulator-based average value at risk. For any X∈ℝd×n and 0<α<1, we define ϱX as(3)ϱX≔AV@RαlossX,≔infz∈ℝd1α−X∧K1n0M+z+−z+ℝ+m,
It is clear thatϱ satisfies the cash cover, monotonicity, regulator dependence properties, and convexity, so ϱ is a regulator-based risk statistic.Definition 2.
LetY∈ℝd×n, u∈M. Define a function SY,uX:ℝd×n⟶2M as(4)SY,uX≔z∈M:XtrY≤utrz.
In fact, theSY,uX is the support function of X. Before we derive the dual representations of regulator-based risk statistics, the Legendre–Fenchel conjugate theory ([9]) should be recalled.Lemma 1 (see [9], Theorem 2).
LetR:ℝd×n⟶QMt be a set-valued closed convex function. Then, the Legendre–Fenchel conjugate and the biconjugate of R can be defined, respectively, as(5)−R∗Y,u≔cl∪X∈ℝd×nRX+SY,u−X,Y∈ℝd×n,u∈ℝd,(6)RX=R∗∗X≔∩Y,u∈ℝd×n×KM+\0−R∗Y,u+SY,uX,X∈ℝd×n.Definition 3.
(indicator function). For anyZ⊆ℝd×n, the QMt-valued indicator function IZ:ℝd×n⟶QMt is defined as(7)IZX≔clKM,X∈Z,ϕ,X∉Z.Remark 2.
The conjugate ofQMt-valued indicator function IZ is(8)−IZ∗Y,u≔cl∪X∈ZSY,u−X.Remark 3.
It is easy to see that the regulator-based risk statisticϱ does not have cash additivity, see [9]. However, ϱ has cash subadditivity introduced in [23, 24]. Indeed, from Theorem 2 of [10], ϱ satisfies the Fatou property. Then, considering any X∈ℝd×n and z∈KM, for any ε∈0,1, we have(9)ϱ1−εX−z1n=ϱ1−εX+ε−zε1n,⊇1−εϱX+εϱ−zε1n,⊇1−εϱX+z,where the last inclusion is due to the property P1. Using the arbitrariness of ε, we have the following lemma.Lemma 2.
Assume thatϱ is a regulator-based risk statistic. For any z∈ℝ+d, X∈ℝd×n,(10)ϱX−z1n⊇ϱX+z,which also implies(11)ϱX+z1n⊆ϱX−z.Proposition 1.
Letϱ:ℝd×n⟶QM+t be a proper closed convex regulator-based risk statistic with u∈−∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+M⊥∩KM+\0. Then,(12)−ϱ∗Y,u=cl∪X∈ℝd×nSY,u−X,Y∈−ℝ+d×n∩K+1n,M,elsewhere.Now, we state the main result of this paper, the dual representations of regulator-based risk statistics.Theorem 1.
Ifϱ:ℝd×n⟶QM+t is a proper closed convex regulator-based risk statistic, then there is a −α:−ℝ+d×n∩K+1n×KM+0⟶QM+t, which is not identically M of the set(13)W=Y,u∈−ℝ+d×n∩K+1n×KM+\0u∈−∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+M⊥,such that for any X∈ℝd×n,(14)ϱX=∩Y,u∈W−αY,u+SY,uX∧K1n0.


## 4. Alternative Data-Based Versions of Regulator-Based Risk Measures

In this section, we develop another framework, the data-based versions of regulator-based risk measures. This framework is a little different from the previous one. However, almost all the arguments are the same as those in the previous section. Therefore, we only state the corresponding notations and results and omit all the proofs and relevant explanations.We replaceM by M˜∈ℝd×n that is a linear subspace of ℝd×n. We also replace K by K˜∈ℝd×n that is a is a closed convex polyhedral cone where K˜⊇ℝ++d×n. The partial order with respect to K˜ is defined as X≤K˜Y, which means Y−X∈K˜. Let M˜+≔M˜∩ℝ+d×n. Denoting K˜M˜≔K˜∩M˜ by the closed convex polyhedral cone in M˜, K˜M˜+≔u˜∈M:u˜trz˜≥0for anyz˜∈K˜M˜ is the positive dual cone of K˜M˜ in M˜ and intK˜M˜ is the interior of K˜M˜ in M˜. We denote QM˜t≔A˜⊂M˜:A˜=clcoA˜+K˜M˜ and QM˜+t≔A˜⊂K˜M˜:A˜=clcoA˜+K˜M˜. We still start from the viewpoint of regulators which only care about the positions that need to pay capital requirements. Therefore, for any X∈ℝd×n, we define X∧K˜0 as(15)X∧K˜0≔X,X∉K˜,0,X∈K˜,Then, we state the axioms related to regulator-based risk statistics.Definition 4.
A regulator-based risk statistic is a functionϱ˜:ℝd×n⟶QM˜+t that satisfies the following properties:[Q0] Normalization: K˜M˜⊆ϱ˜0 and ϱ˜0∩−intK˜M˜=ϕ[Q1] Cash cover: for any z˜∈K˜M˜, z˜∈ϱ˜−z˜[Q2] Monotonicity: for any X1,X2∈ℝd×n, X1−X2∈ℝd×n∩K˜ implies that ϱ˜X1⊇ϱ˜X2[Q3] Regulator-dependence: for any X∈ℝd×n, ϱ˜X=ϱ˜X∧K˜0[Q4] Convexity: for any X,Y∈ℝd×n, λ∈0,1, ϱ˜λX+1−λY⊇λϱ˜X+1−λϱ˜Y
We need more notations. LetY∈ℝd×n, u˜∈M˜. Define a function SY,u˜X:ℝd×n⟶2M˜ as(16)SY,u˜X≔z˜∈M˜:XtrY≤u˜trz˜,let R˜:ℝd×n⟶QM˜t be a set-valued closed convex function. Then, the Legendre–Fenchel conjugate and the biconjugate of R˜ can be defined, respectively, as(17)−R˜∗Y,u≔cl∪X∈ℝd×nR˜X+SY,u˜−X,Y∈ℝd×n,u˜∈ℝd×n,(18)R˜X=R˜∗∗X≔∩Y,u˜∈ℝd×n×K˜M˜+\0−R˜∗Y,u˜+SY,u˜X,X∈ℝd×n.
For anyZ˜⊆ℝd×n, the QM˜t-valued indicator function IZ˜:ℝd×n⟶QM˜t is defined as(19)IZ˜X≔clK˜M˜,X∈Z˜,ϕ,X∉Z˜.
The conjugate ofQM˜t-valued indicator function IZ˜ is(20)−IZ˜∗Y,u˜≔cl∪X∈Z˜SY,u˜−X.
Assume thatϱ˜ is a regulator-based risk statistic. For any z˜∈ℝ+d×n, X∈ℝd×n,(21)ϱ˜X−z˜⊇ϱ˜X+z˜,which also implies(22)ϱ˜X+z˜⊆ϱ˜X−z˜.
Next, we state the dual representations of regulator-based risk statistics.Proposition 2.
Letϱ˜:ℝd×n⟶QM˜+t be a proper closed convex regulator-based risk statistic with u˜∈−∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+M˜⊥∩K˜M˜+\0. Then,(23)−ϱ˜∗Y,u˜=cl∪X∈ℝd×nSY,u˜−X,Y∈−ℝ+d×n∩K˜+,M˜,elsewhere.Theorem 2.
Ifϱ˜:ℝd×n⟶QM˜+t is a proper closed convex regulator-based risk statistic, then there is a −α:−ℝ+d×n∩K˜+×K˜M˜+\0⟶QM˜+t, that is not identically M˜ of the set(24)W˜=Y,u˜∈−ℝ+d×n∩K˜+×K˜M˜+\0:u˜∈−∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+M˜⊥such that for any X∈ℝd×n,(25)ϱ˜X=∩Y,u˜∈W−αY,u˜+SY,u˜X∧K˜0.


## 5. Proofs of Main Results

Proof of Lemma2.
the proof of Lemma2 is straightforward from Remark 3.Proof of Proposition1.
ifY∉−ℝ+d×n∩K+1n, there exits an X¯∈ℝd×n∩K1n such that X¯trY>0. Using the definition of SY,u, we have SY,u−tX¯=z∈M:−tX¯trY≤utrz for t>0. Therefore,(26)cl∪X∈ℝd×nSY,u−X⊇∪t>0SY,u−tX¯=M.
The last equality is due to−tX¯trY⟶−∞ when t⟶+∞. Using the definition of SY,u, we conclude that cl∪X∈ℝd×nSY,u−X⊆M. Hence,(27)cl∪X∈ℝd×nSY,u−X=M,Y∉−ℝd×n∩K+1n.
It is easy to check that for anyX∈ℝd×n and v∈M,(28)SY,u−X−v1n=z∈M:−XtrY≤utrz+Ytrv1n,=z−v∈M:−XtrY≤utrz−v+Y+u1ntrv1n+v,=z∈M:−XtrY≤utrz+Y+u1ntrv1n+v,when −∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+u∈M⊥, and we have SY,u−X−v1n=SY,u−X+v. However, u∉−∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+M⊥. Therefore, −∑j=1l∑h=1njYh1,j,…,−∑j=1l∑h=1njYhd,j+u∉M⊥, and we can find v∈M such that for any z∈M,(29)−XtrY≤utrz+Y+u1ntrv1n.
Therefore, we have(30)z+v∈SY,u−X−v1n.
Therefore,(31)∪z,v∈Mz+v⊂∪v∈MSY,u−X−v1n.
Therefore,(32)M⊂∪v∈MSY,u−X−v1n.
From the definition ofSY,u, the inverse inclusion is always true. So, we conclude that(33)M=∪v∈MSY,u−X−v1n.
It is also easy to check that(34)−ϱ∗Y,u=cl∪X∈ℝd×n,v∈MϱX+v1n+SY,u−X−v1n,=cl∪X∈ℝd×n,v∈MϱX+v1n+M,=M,where the last equality comes from the fact that M is a linear space and ϱX⊆M. We now derive that −ϱ∗Y,u=cl∪X∈ℝd×nSY,u−X. In this context, from −ϱ∗Y,u=cl∪X∈ℝd×nϱX+SY,u−X, we derive it in two cases.Case 1.
WhenX∧K1n0=0, using the definition, we have ϱX=ϱ0∋0. Hence,(35)cl∪X∈ℝd×nϱX+SY,u−X⊃cl∪X∈ℝd×nSY,u−X.Case 2.
WhenX∧K1n0=X, we can always find an α∈KM such that α∈ϱX. Then,(36)ϱX+SY,u−X⊇α+SY,u−X=SY,u−X−α1n=SY,u−β,where β=X+α1n. It is relatively simple to check that β∈ℝd×n. Therefore,(37)cl∪X∈ℝd×nϱX+SY,u−X⊇cl∪z∈ℝd×nSY,u−z,that is,(38)−ϱ∗Y,u⊇cl∪X∈ℝd×nSY,u−X.
Consequently, we have(39)−ϱ∗Y,u⊇cl∪X∈ℝd×nSY,u−X.
We now need only to derive that−ϱ∗Y,u⊆cl∪X∈ℝd×nSY,u−X. In fact, for any z∈ϱX and X∈ℝd×n, X+z1n∈ℝd×n. Therefore,(40)cl∪X∈ℝd×nSY,u−X=cl∪X∈ℝd×nSY,u−X⊇SY,u−X−z1n=SY,u−X+z.
Using the arbitrariness ofz, we have(41)ϱX+SY,u−X⊆cl∪X∈ℝd×nSY,u−X.
Therefore,(42)−ϱ∗Y,u⊆cl∪X∈ℝd×nSY,u−X.Proof of Theorem1.
the proof is straightforward from Lemma1 and Proposition 1.



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