# Acceptability Indexes for Portfolio Vectors

Authors: Xianfu Zeng; Yanhong Chen; Yijun Hu
Journal: Mathematical Problems in Engineering
 (2019)

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## 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.


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## Body

## 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 Section2, 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. LetΩ,F be a fixed measurable space and Ω,F,P a fixed probability space. We denote by G=L∞Ω,F the space of bounded F-measurable random variables. Denote H=LpΩ,F,P for 1≤p≤+∞, where, when p=+∞, H denotes the space of essentially bounded F-measurable random variables and, when 1≤p<+∞, H denotes the space of random variables with finite p-order moment. For X,Y∈H, we will identify X with Y if PX=Y=1. The space G (or H) represents financial risk positions. Positive values of X∈G or H correspond to gains, while negative values correspond to losses. For X∈G, define X∞≔supω∈ΩXω, and then G,⋅∞ is a Banach space. For H∈H, define Hp≔esssupH, if p=+∞; EHp1/p, if 1≤p<+∞, where EHp means the integral of Hp with respect to the probability P, and then H,⋅p is a Banach space.A mapμ:F⟶R is called a finitely additive set function if for any finite collection A1,…,An∈F of mutually disjoint sets, μ∪i=1nAi=∑i=1nμAi, and if μ∅=0. The total variation of a finitely additive set function μ is defined as μvar≔sup∑i=1nμAi:A1,…,Andisjoint sets inF,n∈ℕ. By baΩ,F, we denote the space of all finitely additive measures μ with μvar<+∞. EμX denotes the integral of X∈G with respect to μ∈baΩ,F. We denote by Ms,fΩ,F≔μ:F⟶0,1:μ∈baΩ,F,μΩ≤1, by M1,fΩ,F≔μ∈Ms,fΩ,F:μΩ=1, and by MsΩ,F≔μ∈Ms,fΩ,F:μisσ−additive. We also denote by baΩ,F,P (Ms,fΩ,F,P,M1,fΩ,F,P,MsΩ,F,P, resp.) the subclass of baΩ,F (Ms,fΩ,F, M1,fΩ,F, MsΩ,F, resp.), which are absolutely continuous with respect to P.LetN≥1 be a fixed positive integer, which represents the number of assets. For 1≤i≤N, let Ωi,Fi be a fixed measurable space and Ωi,Fi,Pi a fixed probability space. We denote by Gi≔L∞Ωi,Fi and Hi≔LpiΩi,Fi,Pi, 1≤pi≤+∞, 1≤i≤N. Let XN be the product space X1×⋯×XN, where Xi is either Gi or Hi for 1≤i≤N. Define a norm in the space XN by M≔X1+⋯+XN for M=X1,…,XN∈XN, where Xi equals Xi∞ if Xi is Gi, and Xipi if Xi is Hi for 1≤i≤N. Then, XN,⋅ is a Banach space.Next, we introduce a notationMs,fXi for 1≤i≤N. When Xi is Gi, Ms,fXi stands for Ms,fΩi,Fi. When Xi is ℋi,(1)
Ifpi=+∞, then Ms,fXi means the set Ms,fΩi,Fi,Pi(2)
Ifpi=1, then Ms,fXi stands for the set Q∈MsΩi,Fi,Pi:dQ/dPi∈L∞Ωi,Fi,Pi(3)
If1<pi<+∞, then Ms,fXi means the set Q∈MsΩi,Fi,Pi:dQ/dPi∈LqiΩi,Fi,Pi, where qi is the conjugate index of piForM1=X1,…,XN, M2=Y1,…,YN∈XN, M1+M2 means X1+Y1,…,XN+YN, and M1≤M2 means X1≤Y1,…,XN≤YN. aM stands for aX1,…,aXN for M=X1,…,XN∈XN and a∈R. We set 0≔0,…,0, 1≔1,…,1, and for any fixed i between 1 and N, ei≔0,…,0,1,0,…,0, where 1 occupies the ith position. For 1≤i≤N, Xei means 0,…,0,X,0,…,0, where X occupies the ith position for any random variable X.On a general level, a multivariate acceptability index (or a performance measure for portfolio vectors) is any mapα:XN⟶0,+∞. Given a portfolio vector M∈XN, αM measures the degree of quality of M.Definition 1.
A quasi-concave acceptability index is a mapα:XN⟶0,+∞ satisfying the following two axioms:(A1) Quasi-concavity: for any realc∈0,1 and any M1,M2∈XN, then(1)αcM1+1−cM2≥minαM1,αM2.(A2) Monotonicity:M1≥M2 implies αM1≥αM2 for any M1,M2∈XN.Furthermore, a quasi-concave acceptability indexα is called a coherent acceptability index if it also satisfies:(A3) Scaling invariance:αcM=αM for any M∈XN and c>0.
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 ifM1 dominates M2, then M1 is acceptable at least at the same level as M2 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 mapρ:XN⟶R is called a convex risk measure, if it satisfies the following three axioms:(R1) Monotonicity:M1≥M2 implies ρM1≤ρM2 for any M1,M2∈XN(R2) Translation invariance:ρM+m1=ρM−m for any M∈XN and m∈R(R3) Convexity:ρcM1+1−cM2≤cρM1+1−cρM2 for any c∈0,1 and M1,M2∈XNFurthermore, a convex risk measureρ is called a coherent risk measure if it also satisfies(R4) Positive homogeneity:ρcM=cρM for any M∈XN and c>0Definition 3.
We call a mapf:XN⟶R has the Fatou property, if Mn;n≥1 is a bounded sequence in XN which converges to M∈XN, then(2)fM≤liminfn⟶∞fMn.Definition 4.
A family of risk measuresρxx∈0,+∞ is called increasing if ρxM≥ρyM for all x≥y>0 and M∈XN.


## 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 thatα:XN⟶0,+∞ is a quasi-concave (coherent, resp.) acceptability index. Then, the set of functions ρx, x∈0,+∞, defined by(3)ρxM≔infm∈R:αM+m1≥x,M∈XN,is an increasing family of convex (coherent, resp.) risk measures.(2)
Assume thatρxx∈0,+∞ is an increasing family of convex (coherent, resp.) risk measures. Then, the function α defined by(4)αM≔supx∈0,+∞:ρxM≤0,M∈XN,is a quasi-concave (coherent, resp.) acceptability index.Remark 1.
Ifα is a quasi-concave (coherent, resp.) acceptability index with the Fatou property, then ρxx∈0,+∞ defined by (3) also has the Fatou property. And vise versa, if ρxx∈0,+∞ 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)
Ifα is a quasi-concave (coherent, resp.) acceptability index, then there exists an increasing family of convex (coherent, resp.) risk measures ρxx∈0,+∞, such that(5)αM=supx∈0,+∞:ρxM≤0.(2)
Ifρxx∈0,+∞ is an increasing family of convex (coherent, resp.) risk measures, then there exists a quasi-concave (coherent, resp.) acceptability index α, such that(6)ρxM=infm∈R:αM+m1≥x,here we assume that infø=∞ and supø=0.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 ρ:XN⟶R is a convex risk measure if and only if there exists a function F:∏i=1NMs,fXi⟶R∪+∞ with(7)infQ1,…,QN∈Ms,fX1×⋯×Ms,fXNΣi=1NQi1=1FQ1,…,QN∈R,such that(8)ρM=supQ1,…,QN∈Ms,fXi×⋯×Ms,fXNΣi=1NQi1=1EQ1−X1+⋯+EQN−XN−FQ1,…,QN,for any M=X1,…,XN∈XN, where F can be chosen as (9)FminQ1,…,QN≔supX1,…,XN∈AρEQ1−X1+⋯+EQN−XN,where(10)Aρ≔M∈XN:ρM≤0.Remark 2.
A functionρ:XN⟶R is a convex risk measure with the Fatou property if and only if there exists a function F:∏i=1NMsXi⟶R∪+∞ with(11)infQ1,…,QN∈Ms,fX1×⋯×Ms,fXNΣi=1NQi1=1FQ1,…,QN∈R,such that(12)ρM=supQ1,…,QN∈Ms,fXi×⋯×Ms,fXNΣi=1NQi1=1EQ1−X1+⋯+EQN−XN−FQ1,…,QN,for any M=X1,…,XN∈XN, where F can be chosen as(13)FminQ1,…,QN≔supX1,…,XN∈AρEQ1−X1+⋯+EQN−XN,where(14)Aρ≔M∈XN:ρM≤0.Corollary 1.
(representation result for coherent risk measures). A function ρ:XN⟶R is a coherent risk measure if and only if there exists a subset(15)D⊂Q≔Q1,…,QN∈∏i=1NMs,fXi:∑i=1NQi1=1,such that(16)ρM=supQ1,…,QN∈DEQ1−X1+⋯+EQN−XN,for any M=X1,…,XN∈XN.Remark 3.
A functionρ:XN⟶R is a coherent risk measure with the Fatou property if and only if there exists a subset(17)D⊂Q≔Q1,…,QN∈∏i=1NMsXi:∑i=1NQi1=1,such that(18)ρM=supQ1,…,QN∈DEQ1−X1+⋯+EQN−XN,for any M=X1,…,XN∈XN.
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 α:XN⟶0,+∞ is a quasi-concave acceptability index if and only if there exists an decreasing family Fxx∈0,+∞ of functionals such that(19)αM=supx∈0,+∞:infQ1,…,QN∈Ms,fX1×⋯×Ms,fXNΣi=1NQi1=1EQ1X1+⋯+EQNXN+FxQ1,…,QN≥0,for any M∈XN, where F can be chosen as(20)FxQ1,…,QN≔supM∈XN,αM≥x−EQ1X1+⋯+EQNXN,and here we assume that inf∅=∞ and sup∅=0.Corollary 2.
(representation result for coherent acceptability indexes). A map α:XN⟶0,+∞ is a coherent acceptability index if and only if there exists a family of subsets Dxx∈0,+∞ of Q≔Q1,…,QN∈∏i=1NMs,fXi:∑i=1NQi1=1 with Dx⊆Dy for 0<x≤y, such that(21)αM=supx∈0,+∞:infQ1,…,QN∈DxEQ1X1+⋯+EQNXN≥0,and here we assume that inf∅=∞ and sup∅=0.Remark 4.
(a)
A mapα:XN⟶0,+∞ is a quasi-concave acceptability index with the Fatou property if and only if there exists an decreasing family Fxx∈0,+∞ of functionals such that(22)αM=supx∈0,+∞:infQ1,…,QN∈MsX1×⋯×MsXNΣi=1NQi1=1EQ1X1+⋯+EQNXN+FxQ1,…,QN≥0,for anyM∈XN, where F can be chosen as(23)FxQ1,…,QN≔supM∈XN,αM≥x−EQ1X1+⋯+EQNXN,and here we assume thatinf∅=∞ and sup∅=0.(b)
A mapα:XN⟶0,+∞ is a coherent acceptability index with the Fatou property if and only if there exists a family of subsets Dxx∈0,+∞ of Q≔Q1,…,QN∈∏i=1NMsXi:∑i=1NQi1=1 with Dx⊆Dy for 0<x≤y, such that(24)αM=supx∈0,+∞:infQ1,…,QN∈DxEQ1X1+⋯+EQNXN≥0,and here we assume thatinf∅=∞ and sup∅=0.(c)
LetN=1 and X=L∞Ω,F,P, 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.



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*Source: 2196563-2019-09-18.xml*