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.
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 Section 2, 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. Let be a fixed positive integer. The space represents the set of financial risk positions. With positive values of , we denote the gains while the negative denotes the losses. Let be the sample size of in the scenario, . Let . More precisely, suppose that the behavior of is represented by a collection of data , where and is the data subset that corresponds to the scenario with respect to . For each , , is the data subset that corresponds to the observation of in the scenario.
In this paper, an element of is denoted by . An element of is denoted by . Let be a closed convex polyhedral cone of where and where . Let be the positive dual cone of , that is, , where means the transpose of . For any , stands for and stands for for . Denote and where . By , and we denote the positive dual cone of in , i.e., . The partial order with respect to is defined as , which means where , and means where .
Let be the linear subspace of for . The introduction of was considered in [6, 9]. Denote , where and . Therefore, a regulator can only accept security deposits in the first reference instruments. Denoting by the closed convex polyhedral cone in , the positive dual cone of in and is the interior of in . We denote and , where the represents the closed convex hull of .
By [18], a set-valued risk statistic is any map ,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: and [A1] Monotonicity: for any , implies that [A2] M-translative invariance: for any and , [A3] Convexity: for any and , [A4] Positive homogeneity: for any and [A5] Subadditivity: for any
We end this section with more notations. A function is said to be proper if and for all . is said to be closed if 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 any , is defined as follows:
Therefore, the position that belongs to 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 function that satisfies the following properties: [] Normalization: and [] Cash cover: for any , [] Monotonicity: for any , implies that [] Regulator-dependence: for any , [] Convexity: for any and ,
Remark 1. The property of means any fixed negative risk position can be canceled by its positive quality ; says that if is bigger than for the partial order in , then the need less capital requirement than , so contains ; 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 regarded as 0 position.
We now construct an example for regulator-based risk statistics.
Example 1. The coherent risk measure AV@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 and , we define asIt is clear that satisfies the cash cover, monotonicity, regulator dependence properties, and convexity, so is a regulator-based risk statistic.
Definition 2. Let , . Define a function asIn fact, the is the support function of . 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). Let be a set-valued closed convex function. Then, the Legendre–Fenchel conjugate and the biconjugate of can be defined, respectively, as
Definition 3. (indicator function). For any , the -valued indicator function is defined as
Remark 2. The conjugate of -valued indicator function is
Remark 3. It is easy to see that the regulator-based risk statistic does not have cash additivity, see [9]. However, has cash subadditivity introduced in [23, 24]. Indeed, from Theorem 2 of [10], satisfies the Fatou property. Then, considering any and , for any , we havewhere the last inclusion is due to the property . Using the arbitrariness of , we have the following lemma.
Lemma 2. Assume that is a regulator-based risk statistic. For any , ,which also implies
Proposition 1. Let be a proper closed convex regulator-based risk statistic with . Then,
Now, we state the main result of this paper, the dual representations of regulator-based risk statistics.
Theorem 1. If is a proper closed convex regulator-based risk statistic, then there is a , which is not identically of the setsuch that for any ,
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 replace by that is a linear subspace of . We also replace by that is a is a closed convex polyhedral cone where . The partial order with respect to is defined as , which means . Let . Denoting by the closed convex polyhedral cone in , is the positive dual cone of in and is the interior of in . We denote and . We still start from the viewpoint of regulators which only care about the positions that need to pay capital requirements. Therefore, for any , we define as
Then, we state the axioms related to regulator-based risk statistics.
Definition 4. A regulator-based risk statistic is a function that satisfies the following properties: [] Normalization: and [] Cash cover: for any , [] Monotonicity: for any , implies that [] Regulator-dependence: for any , [] Convexity: for any , , We need more notations. Let , . Define a function aslet be a set-valued closed convex function. Then, the Legendre–Fenchel conjugate and the biconjugate of can be defined, respectively, asFor any , the -valued indicator function is defined asThe conjugate of -valued indicator function isAssume that is a regulator-based risk statistic. For any , ,which also impliesNext, we state the dual representations of regulator-based risk statistics.
Proposition 2. Let be a proper closed convex regulator-based risk statistic with . Then,
Theorem 2. If is a proper closed convex regulator-based risk statistic, then there is a , that is not identically of the setsuch that for any ,
5. Proofs of Main Results
Proof of Lemma 2. the proof of Lemma 2 is straightforward from Remark 3.
Proof of Proposition 1. if , there exits an such that . Using the definition of , we have for . Therefore,The last equality is due to when . Using the definition of , we conclude that . Hence,It is easy to check that for any and ,when , and we have . However, . Therefore, , and we can find such that for any ,Therefore, we haveTherefore,Therefore,From the definition of , the inverse inclusion is always true. So, we conclude thatIt is also easy to check thatwhere the last equality comes from the fact that is a linear space and . We now derive that . In this context, from , we derive it in two cases.
Case 1. When , using the definition, we have . Hence,
Case 2. When , we can always find an such that . Then,where . It is relatively simple to check that . Therefore,that is,Consequently, we haveWe now need only to derive that . In fact, for any and , . Therefore,Using the arbitrariness of , we haveTherefore,
Proof of Theorem 1. the proof is straightforward from Lemma 1 and Proposition 1.
Data Availability
No data and code were generated or used during the study.
Disclosure
This manuscript has been released as a preprint at arXiv: 1904.08829v4.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by funds from Education Department of Guangdong (2019KQNCX156).