#### Abstract

In this paper, we introduce two new classes of acceptability indexes, named quasi-concave acceptability indexes and coherent acceptability indexes, for portfolio vectors. We establish the one-to-one correspondence between quasi-concave (coherent, resp.) acceptability indexes and convex (coherent, resp.) risk measures for portfolio vectors. We derive the representation results for coherent and convex risk measures. Finally, based on these results, we derive the representation results for quasi-concave acceptability indexes and coherent acceptability indexes for portfolio vectors. These new acceptability indexes can be considered as a kind of multivariate extension of univariate coherent and quasi-concave acceptability indexes introduced by Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013), respectively.

#### 1. Introduction

Recently, several authors have focused their attention on acceptability indexes. Coherent acceptability indexes have been defined by Cherny and Madan [1] as performance measures of terminal cash flows seen as random variables, by proposing some basic axioms to be satisfied by every sound financial acceptability indexes. Further, Rosazza Gianin and Sgarra [2] introduced the broader class, named quasi-concave acceptability indexes, by dropping one of coherency axioms. More recently, Bielecki et al. [3] and Biagini and Bion-Nadal [4] extended the definitions of coherent and quasi-concave acceptability indexes to a dynamic setting, respectively. For more works about acceptability indexes, see Bielecki et al. [5] and the references therein.

In all the above-mentioned works, a financial position is described by a random variable or a one-dimensional stochastic process. However, in order to evaluate the degree of quality of a multivariate portfolio, among which there may exist some possible dependence between the components, it is natural to consider a random vector rather than random variables composed by all components. This observation motivates us to study acceptability indexes for portfolio vectors.

In the present paper, we will study multivariate coherent and quasi-concave acceptability indexes. Meanwhile, in order to investigate the correspondence between multivariate acceptability indexes and multivariate risk measures, we will make some adaptation to one of the axioms of multivariate risk measures introduced by Burgert and Rüschendorf [6] and will establish the one-to-one correspondence between quasi-concave (resp. coherent) acceptability indexes and convex (resp. coherent) risk measures for portfolio vectors. Representation results for (the adapted) convex and coherent risk measures will be given. Finally, we will provide representation results for these new acceptability indexes.

It should be mentioned that there also exist many papers about multidimensional risk measures. For scalar multivariate risk measures, see Burgert and Rüschendorf [6], Rüschendorf [7], Ekeland and Schachermayer [8], Ekeland et al. [9], Rüschendorf [10], Wei and Hu [11], Chen et al. [12], and the references therein. For set-valued multivariate risk measures, see Jouini et al. [13], Hamel [14], Hamel and Heyde [15], Hamel et al. [16], Hamel et al. [17], Labuschagne and Offwood-Le Roux [18], Farkas et al. [19], Molchanov and Cascos [20], Chen and Hu [21], and the references therein.

The rest of the paper is organized as follows. In Section 2, we briefly introduce some preliminaries. The main results are stated in Section 3, and their proofs are postponed to Appendix.

#### 2. Preliminaries

In this section, we will briefly introduce the preliminaries. Let be a fixed measurable space and a fixed probability space. We denote by the space of bounded -measurable random variables. Denote for , where, when , denotes the space of essentially bounded -measurable random variables and, when , denotes the space of random variables with finite *p*-order moment. For , we will identify *X* with *Y* if . The space (or ) represents financial risk positions. Positive values of or correspond to gains, while negative values correspond to losses. For , define , and then is a Banach space. For , define , if ; , if , where means the integral of with respect to the probability *P*, and then is a Banach space.

A map is called a finitely additive set function if for any finite collection of mutually disjoint sets, , and if . The total variation of a finitely additive set function *μ* is defined as . By , we denote the space of all finitely additive measures *μ* with . denotes the integral of with respect to . We denote by , by , and by . We also denote by (, resp.) the subclass of (, , , resp.), which are absolutely continuous with respect to *P*.

Let be a fixed positive integer, which represents the number of assets. For , let be a fixed measurable space and a fixed probability space. We denote by and , , . Let be the product space , where is either or for . Define a norm in the space by for , where equals if is , and if is for . Then, is a Banach space.

Next, we introduce a notation for When is , stands for . When is ,(1)If , then means the set (2)If , then stands for the set (3)If , then means the set where is the conjugate index of

For , , means , and means . stands for for and . We set , , and for any fixed *i* between 1 and *N*, , where 1 occupies the *i*th position. For , means , where *X* occupies the *i*th position for any random variable *X*.

On a general level, a multivariate acceptability index (or a performance measure for portfolio vectors) is any map . Given a portfolio vector , measures the degree of quality of .

*Definition 1. *A quasi-concave acceptability index is a map satisfying the following two axioms: (A1) Quasi-concavity: for any real and any , then (A2) Monotonicity: implies for any . Furthermore, a quasi-concave acceptability index *α* is called a coherent acceptability index if it also satisfies: (A3) Scaling invariance: for any and .The above axioms have natural financial interpretation. (A1), quasi-concavity, means that a diversified portfolio performs at higher level than its components. (A2), monotonicity, states that if dominates , then is acceptable at least at the same level as is. (A3), scaling invariance, means that cash flows with the same direction of trade have the same level of acceptance. For more explanations about these axioms, see Cherny and Madan [1], Rosazza Gianin and Sgarra [2], Biagini and Bion-Nadal [4], Bielecki, Cialenco, and Zhang [3], and Bielecki et al. [5].

As mentioned in Cherny and Madan [1] and Rosazza Gianin and Sgarra [2], there is a strong relationship between univariate acceptability indexes and univariate risk measures. For multivariate acceptability indexes, it is natural to link them with multivariate risk measures. Theorem 1 shows that this is indeed the case. The delicate issue however is what family of multivariate risk measures should be used. It turns out that to produce a quasi-concave or coherent acceptability index for portfolio vectors, one needs to make an adaptation to the definition of coherent and convex risk measures for portfolio vectors introduced by Burgert and Rüschendorf [6]; see also Rüschendorf [10] and Wei and Hu [11]. Definition 2 is such an adaptation.

*Definition 2. *A map is called a convex risk measure, if it satisfies the following three axioms: (R1) Monotonicity: implies for any (R2) Translation invariance: for any and (R3) Convexity: for any and Furthermore, a convex risk measure *ρ* is called a coherent risk measure if it also satisfies (R4) Positive homogeneity: for any and

*Definition 3. *We call a map has the Fatou property, if is a bounded sequence in which converges to , then

*Definition 4. *A family of risk measures is called increasing if for all and .

#### 3. Main Results

In this section, we will state the representation results for quasi-concave and coherent acceptability indexes for portfolio vectors defined in Definition 1. We need first to give some propositions and theorems, whose proofs will be given in the Appendix.

The following proposition shows that a quasi-concave (coherent, resp.) acceptability index can induce an increasing family of convex (coherent, resp.) risk measures, and vice versa, an increasing family of convex (coherent, resp.) risk measures can also induce a quasi-concave (coherent, resp.) acceptability index.

Proposition 1. (1)*Assume that is a quasi-concave (coherent, resp.) acceptability index. Then, the set of functions , , defined by** **is an increasing family of convex (coherent, resp.) risk measures.*(2)*Assume that is an increasing family of convex (coherent, resp.) risk measures. Then, the function α defined by*

*is a quasi-concave (coherent, resp.) acceptability index.*

*Remark 1. *If *α* is a quasi-concave (coherent, resp.) acceptability index with the Fatou property, then defined by (3) also has the Fatou property. And vise versa, if is an increasing family of convex (coherent, resp.) risk measures with the Fatou property, then *α* defined by (4) also has the Fatou property.

The following theorem shows that a quasi-concave (coherent, resp.) acceptability index can be represented by a family of convex (coherent, resp.) risk measures, and vise versa, convex (coherent, resp.) risk measures can be represented by a quasi-concave (coherent, resp.) acceptability index.

Theorem 1. (1)*If α is a quasi-concave (coherent, resp.) acceptability index, then there exists an increasing family of convex (coherent, resp.) risk measures , such that*

*(2)*

*If is an increasing family of convex (coherent, resp.) risk measures, then there exists a quasi-concave (coherent, resp.) acceptability index*

*α*, such that*here we assume that and .*

Next, we will state the representation results for coherent and convex risk measures defined in Definition 2.

Theorem 2. **(representation result for convex risk measures)**. A function is a convex risk measure if and only if there exists a function withsuch thatfor any , where *F* can be chosen as where

*Remark 2. *A function is a convex risk measure with the Fatou property if and only if there exists a function withsuch thatfor any , where *F* can be chosen aswhere

Corollary 1. **(representation result for coherent risk measures)**. A function is a coherent risk measure if and only if there exists a subsetsuch thatfor any .

*Remark 3. *A function is a coherent risk measure with the Fatou property if and only if there exists a subsetsuch thatfor any .

Now, we are in a position to state the main results of this paper, whose proofs will be provided in the next section.

Theorem 3. **(representation result for quasi-concave acceptability indexes)**. A map is a quasi-concave acceptability index if and only if there exists an decreasing family of functionals such thatfor any , where *F* can be chosen asand here we assume that and .

Corollary 2. **(representation result for coherent acceptability indexes)**. A map is a coherent acceptability index if and only if there exists a family of subsets of with for , such thatand here we assume that and .

*Remark 4. *(a)A map is a quasi-concave acceptability index with the Fatou property if and only if there exists an decreasing family of functionals such that for any , where *F* can be chosen as and here we assume that and .(b)A map is a coherent acceptability index with the Fatou property if and only if there exists a family of subsets of with for , such that and here we assume that and .(c)Let and , and then representation results (22) and (24) are reduced to the one-dimensional case which coincides with the representation results of Rosazza Gianin and Sgarra (2013, Proposition 3) and Cherny and Madan (2009, Theorem 1), respectively.

#### Appendix

In the appendix, we will provide all the proofs of the results stated in Section 3.

*Proof of Proposition 1. *(1)Assume that is a quasi-concave acceptability index, and then we will show that defined by (3) is an increasing family of convex risk measures. Assume that with , and then for any , , which implies Next we will show that satisfies (R1), monotonicity. Let , with , and then from the monotonicity of *α*, we have Now we will show satisfies (R2), translation invariance. For any and , we have that Finally, we will show that satisfies (R3), the convexity. For any , , and , by the quasi-concavity of *α*, we have which implies i.e., . Hence, By taking infimum in (A.4), first with respect to and then with respect to , we have . Hence, is an increasing family of convex risk measures. In particular, if *α* additionally satisfies scale invariance, then by the scaling invariant of *α*, we have for any and , which yields that satisfies (R4), positive homogeneity.(2)Assume that is an increasing family of convex risk measures. We will check that *α* defined by (4) is a quasi-concave acceptability index. If are such that and , then, by the definition (4) of *α*, and the monotonicity of in *x*, we conclude that for any , By convexity of , for any and , we have Hence, , and thus, by the definition (4) of *α*, we have . This yields the quasi-concavity of *α.* The monotonicity is clear. Hence, *α* defined by (4) is a quasi-concave acceptability index. In particular, if is an increasing family of coherent risk measures, then it is easy to check that *α* defined by (4) is scale invariant, which implies that it is a coherent acceptability index.

*Proof of Theorem 1. *(1)Assume that *α* is a quasi-concave (coherent, resp.) acceptability index. For every define as follows: for any . By Proposition 1, is an increasing family of convex (coherent, resp.) risk measures. We will show that for any . It is easy to see that, for any and , if and only if Hence,(2)Assume that is an increasing family of convex (coherent, resp.) risk measures. Define the function *α* as follows: for all . By Proposition 1, *α* is a quasi-concave (coherent, resp.) acceptability index. Finally, it is easy to see that for any , and , if and only if . In fact, if , then , i.e., , which, together with the property that is increasing with respect to *x*, implies that On the other hand, if , then . Thus, , i.e., .Hence,The proof of Theorem 1 is completed.

*Proof of Theorem 2. *The sufficiency is obvious. Next, we prove the necessity. First, we show thatfor any .

To this end, given , note that . Thus, for any with , we have thatwhich implies (A.13).

For given, we will now construct with , such thatwhich, together with (A.13), proves representation (8).

By translation invariance, it suffices to prove (A.15) for with Moreover, we may assume, without loss of generality, that . Then, is not contained in the nonempty convex setSince is a convex set, by the Hahn-Banach theorem, there exists a nonzero continuous linear functional *λ* (depending on ) on such thatFor , defineThen, obviously are continuous linear functionals on , respectively, and satisfyfor any . We further claim that *λ* and , , have the following two properties:(1) for any . Particularly, for , for any .(2).First, we prove (1). For any and any , . Hence,which could not be true if .

Next, we prove (2). Since *λ* is nonzero, there exists an such that . Taking the truncation argument into account, with no loss of generality, we assume that . Thus, , , where means that if , and if . Thus, by property (1). Hence, Therefore, we can choose *λ* such that .

Consequently, are continuous linear functionals on respectively, and satisfy .

By the preceding discussion and Theorem A.51 of Föllmer and Schied [22], we conclude that there exist , such thatNote that we know thatOn the other hand, for any and each . This shows thatwhich, together with (A.17), implies thatThus, are desired, and the proof of (8) is completed. Theorem 2 is proved.

*Proof of Corollary 1. *Assume that *ρ* is a coherent risk measure, and then is a cone. Thus,for all and . Hence, can take only the values 0 and . Setand then, by Theorem 2, we haveThe proof of Corollary 1 is completed.

*Proof of Theorem 3. *Assume that *α* is a quasi-concave acceptability index, and then by Theorem 1, we know that there exists an increasing family of convex risk measures , such that (5) holds. By Theorem 2, for each , , there exists a function such that (8) holds. Since is increasing, is decreasing, which, together with (5) and (8) yields (19). The proof of Theorem 3 is completed.

*Proof of Corollary 2. *By the same argument as in the proof of Theorem 3, and using Theorem 1 and Corollary 1, one can steadily show Corollary 2.

#### Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11771343) and the Fundamental Research Funds for the Central Universities (no. 531107051210).