Set-Valued Haezendonck-Goovaerts Risk Measure and Its Properties

We propose a new set-valued riskmeasure, which is called set-valuedHaezendonck-Goovaerts riskmeasure. First, we construct the set-valued Haezendonck-Goovaerts risk measure and then provide an equivalent representation. The properties of the set-valued Haezendonck-Goovaerts risk measure are investigated, which show that the set-valued Haezendonck-Goovaerts risk measure is coherent. Finally, an example of set-valuedHaezendonck-Goovaerts riskmeasure is given, which exhibits the fact that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.


Introduction
Premium principles are important and basic issues in insurance and actuarial science.Among various premium principles, the so-called Haezendonck-Goovaerts premium principle (also called Haezendonck-Goovaerts risk measure) has especially been attracting the attention of the insurance community and the financial community, not only because the Haezendonck-Goovaerts risk measure is a coherent risk measure introduced by Artzner et al. [1] but also because it has good properties and a wide range of applications in the finance field and the insurance field.Therefore, it plays an important role in both financial community and insurance community.Haezendonck and Goovaerts [2] introduced a new premium principle based on Orlicz norms.Goovaerts et al. [3] further studied the premium principle by Haezendonck and Goovaerts [2] and called it the Haezendonck risk measure in honor of the late Haezendonck.Haezendonck-Goovaerts risk measure describes a class of concrete risk measures which derived by different parameters.Bellini and Rosazza Gianin [4] further studied Haezendonck risk measure.Bellini and Rosazza Gianin [5] continued the study of Haezendonck and used the terminology of Haezendonck-Goovaerts risk measure in order to better acknowledge both authors' contribution in the seminal papers by Haezendonck and Goovaerts [2] and Goovaerts et al. [3].All the above premium principles are for single risks (i.e., single claims).For more studies about insurance in recent years, see [6][7][8][9][10][11] and the references cited therein.
In some sense, risk measures can be considered as the counterpart of premium principles; see Bellini and Rosazza Gianin [4].Meanwhile, in finance, in order to evaluate the risk of multivariate risks, with possible dependence between separate individual risks, the so-called scalar multivariate risk measures have been studied.For scalar multivariate coherent and convex risk measures, see Burgert and Rüschendorf [12], Rüschendorf [13], and the references cited therein.Moreover, besides the possible dependence between the separate individual risks, if the assets in different security markets are also involved in the multivariate risks, the so-called set-valued risk measures have been suggested and investigated.To be specific, the set-valued risk measure was firstly introduced to allow for random portfolios valued in R  and relate each component of this portfolio to a specific security market.The motivation is that most of the investors are not able to aggregate their portfolio because of transaction costs between different security markets [14].For the set-valued multivariate coherent and convex risk measures, see Jouini et al. [14], Hamel et al. [15], and Ararat et al. [16].For more study about risk measure and set-valued risk measure in recent years, see [3,[17][18][19][20] and the references cited therein.
The existing Haezendonck-Goovaerts risk measure only takes into account a single risk position.When investors tend to consider a portfolio which contains different currencies or securities in different markets, the problems of conveniences and transaction costs can not be solved by the existing Haezendonck-Goovaerts risk measure.In the present paper, we propose the set-valued Haezendonck-Goovaerts risk measures for multivariate risks (i.e., risk portfolio vectors) to measure the risk of financial portfolios taking into account the transaction costs.Also, the set-valued Haezendonck-Goovaerts risk measure provides a convenient and effective way to compare different portfolios and their margins.By construction, we will define the set-valued Haezendonck-Goovaerts risk measure.Then an equivalent representation is provided, whose properties are investigated.It turns out that the set-valued Haezendonck-Goovaerts risk measures are coherent.Finally, an example is given, which shows that the set-valued average valued at risk by Hamel et al. [15] is a particular case of the set-valued Haezendonck-Goovaerts risk measures.
The rest of this paper is organized as follows.In Section 2, we briefly introduce the preliminaries.Section 3 is the study of the regular set-valued Haezendonck-Goovaerts risk measures.Section 4 falls on an extended form of setvalued Haezendonck-Goovaerts risk measures.Section 5 is an example of set-valued Haezendonck-Goovaerts risk measures, which also shows that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.
The prototype of Haezendonck-Goovaerts risk measure is called the Orlicz premium principle studied by Haezendonck and Goovaerts [2].The Orlicz premium principle is defined on Orlicz space.Definition 1.Let Φ be a Young Function; the set  Φ of random variables X defined as follows is called an Orlicz space: For more information about Orlicz space as well as Orlicz heart, see Haezendonck and Goovaerts [2] and Bellini and Rosazza Gianin [5].
The following definition of Orlicz premium principle is from Haezendonck and Goovaerts [2].
∞ with X not being equal to zero a.s. and (X) be the solution of the equation Note that the solution (X) exists uniquely and that it is exactly the Orlicz norm ‖X‖ Φ as in Definition 1.Moreover, (⋅) is a coherent risk measure by Artzner et al. [1].For more details, see Haezendonck and Goovaerts [2].
For more details about the definition above, see Jouini et al. [14].
Hamel et al. [15] introduced the so-called set-valued average value at risk, which is coherent.Definition 5. Let  ∈ (0, 1]  and X ∈  ∞  ; the average value at risk of X is defined as It has an equivalent representation as follows: AV@  (X) = [inf (4)
Remark 7. The intersection with M in Definition 1 has an interpretation as follows.When measuring the risk of portfolio X, we should work out the set of all the margins that can cancel the risk of portfolio X.After intersecting the set M, HG  not only shows the valid margins but also aggregates the margins.It aggregates the valid margins from -dimension to -dimension valid margins and the other ( − )-dimension should be zero.Aggregating margins has plenty of financial explanations.For example, different elements of the vector represent the amount in different currencies.Suppose that there are  kinds of currencies in the vector margin.It is unnecessary for the regulator to ask investor for -dimension vector margin.They could aggregate some elements, which are in the same currency, into one new element of the vector margin.This idea also makes sense when considering the margins needed by different departments in a company.The leader of this company may just deliver the sum of margins of different departments.For more details, see Jouini et al. [14].
Proof.The condition of Definition 6 equals Z ≥ (X −) + ,  ∈ R  .Then we have According to the monotonicity of   (⋅), we have Hence, we obtain Proposition 9.The function X  →   (X) satisfies the following properties: (a) It is positive homogeneous; that is, for any X ∈  0  and any  > 0, we have   (X) =   (X).
Remark 10.Properties (a)-(d) ensure that set-valued Haezendonck-Goovaerts risk measure defined in Definition 6 is a coherent set-valued risk measure.
(e) It is obvious.
The proof of Proposition 9 is complete.
Before ending this section, we will introduce the concept of scalarization of set-valued Haezendonck-Goovaerts risk measure.The relationship between ( 5) and (6) will also be further illustrated by scalarization of set-valued Haezendonck-Goovaerts risk measure.For simplicity, we denote the equivalent representation of set-valued Haezendonck-Goovaerts risk measure in (6) by HG sca  (X).In the following, we assume that HG sca  (X) is a nonempty closed set.Besides, it is also convex.So it is the intersection of all closed half-space including it.Such a half-space has an element  ∈ (R  + ∩ M) + \ {0} = (R  + + M ⊥ ) \ {0}.Then we have the following relationship: Now we consider a real-valued function extended by HG  (X): Let any V ∈ M ⊥ ; we can get that  HG  (X),+V =  HG  (X), since HG  (X) ⊆ M for all X ∈  0  .Then we can restrict the function  HG  (X), to those with  ∈ R  + and obtain the following relationship: Furthermore, it is not difficult to restrict the functions  HG  (X), to those with  ∈  (1), where (1) is defined as , and we obtain Finally, it is easy to prove the next proposition which will show that the functions  HG  (X), are composed of the scalar type HG  (X) (i.e., HG sca  (X)) functions for the components of X.
Remark 12. Proposition 11 together with the previous discussion, by the scalarization, provides the following representation of the set-valued Haezendonck-Goovaerts risk measure.
If HG  (X) is closed, then Remark 13.Proposition 11 means that the scalarization functions  HG  (X), are convex combinations of the scalar Haezendonck-Goovaerts risk measure applied to the components of the portfolio X.Moreover,  ∈ HG  (X) if and only if the convex combinations    cancel the risk of the convex combinations   X for all  ∈ (1).That explains why the multidimensional confidential level  may have different components.
The regulators and investors may have different attitudes towards the risk of different components of the portfolio X.On the other hand,  ∈ (1) can be understood as weighting coefficients of the  components of assets.When investors evaluate the risk of portfolio, they may have more and less favorable assets.Proposition 11 shows that the regulator does not care about the weighting which investors have chosen.The regulator only needs to be the safe side no matter the weighting that the investors choose.

The Extended Set-Valued Haezendonck-Goovaerts Risk Measure
In this section, we will give an extended set-valued Haezendonck-Goovaerts risk measure, which can match more complex market situation.
Remark 15.Different closed convex cones represent different attitudes towards risks.( 0  ) + is a specific closed convex cone in ( 0  ).When choosing  = ( 0  ) + , the regulators (investors) consider −X ⪰  0 (equivalently −X ≥ 0) as that −X is component-wise greater than 0. They demand that each position of the portfolio −X can not be lost.When choosing a general closed convex cone , −X ⪰  0 means that −X ∈ .In this case, the regulators (or investors) are risk appetite.They allow several positions to be negative and this portfolio is also acceptable to them.
Remark 16.There are pros and cons of the set-valued Haezendonck-Goovaerts risk measure compared with current risk measures like value at risk, scalar Haezendonck-Goovaerts risk measure, and other coherent risk measures.The pros are that it provides a more convenient and effective way to compare different portfolios.For example, there are two-dimensional portfolios like (5,6) and (6,5).Two elements of these portfolios represent Dollar and Euro, HG ext (X)fl {  (Z) +  | Z ∈ K, −X + Z +  ∈ ,  ∈ R  } ∩ M.