Conditional tail risk expectations for location-scale mixture of elliptical distributions
aa r X i v : . [ m a t h . S T ] J u l Conditional tail risk expectations for location-scale mixture ofelliptical distributions
Baishuai Zuo, Chuancun Yin*
School of Statistics, Qufu Normal University, Qufu, Shandong 273165, P. R. China *Corresponding author. Email address: [email protected] (Chuancun Yin)July 21, 2020
Abstract
We present general results on the univariate tail conditional expectation (TCE) and mul-tivariate tail conditional expectation for location-scale mixture of elliptical distributions. Examples in-clude the location-scale mixture of normal distributions, location-scale mixture of Student- t distributions,location-scale mixture of Logistic distributions and location-scale mixture of Laplace distributions. Wealso consider portfolio risk decomposition with TCE for location-scale mixture of elliptical distributions. Keyword
Tail conditional expectations; Portfolio allocations; Multivariate risk measures; Location-scalemixture; Elliptical distributions
Tail conditional expectation (TCE), one of important risk measures, is common and practical. TCEof a random variable X is defined as T CE X ( x q ) = E ( X | X > x q ) , where x q is a particular value, generallyreferred to as the q -th quantile with F X ( x q ) = 1 − q. Here F X ( x ) = 1 − F X ( x ) is tail distribution functionof X . The TCE has been discussed in many literatures ( see Landsman and Valdez (2003), Ignatieva andLandsman (2015, 2019)). Recently, a new type of multivariate tail conditional expectation (MTCE) wasdefined by Landsman et al. (2016). It is the following special case when q = ( q, q, · · · , q ). M T CE q ( X ) = E [ X | X > V aR q ( X )]= E [ X | X > V aR q ( X ) , · · · , X n > V aR q n ( X n )] , q = ( q , · · · , q n ) ∈ (0 , n , X = ( X , X , · · · , X n ) T is an n × F X ( x )and tail function F X ( x ), V aR q ( X ) = ( V aR q ( X ) , V aR q ( X ) , · · · , V aR q n ( X n )) T , and V aR q k ( X k ) , k = 1 , , · · · , n is the value at risk (VaR) measure of the random variable X k , beingthe q k -th quantile of X k (see Landsman et al. (2016)). On the basis of it, Mousavi et al. (2019) studymultivariate tail conditional expectation for scale mixtures of skew-normal distribution.Closely related to tail conditional expectation is portfolio risk decomposition with TCE, it’s researchhas experienced a rapid growth in the literature. Portfolio risk decomposition based on TCE for theelliptical distribution was studied in Landsman and Valdez (2003) and extended to the multivariate skew-normal distribution in Vernic (2006). The phase-type distributions and multivariate Gamma distributionwere researched in Cai and Li (2005) and Furman and Landsman (2007), respectively. Furthermore,Hashorva and Ratovomirija (2015) considered the capital allocation with TCE for mixed Erlang dis-tributed risks joined by the Sarmanov distribution, and Ignatieva and Landsman (2019) has given theexpression of TCE-based allocation for the generalised hyperbolic distribution. Recently, Zuo and Yin(2020) deals with the tail conditional expectation for univariate generalized skew-elliptical distributionsand multivariate tail conditional expectation for generalized skew-elliptical distributions.The rest of the paper is organized as follows. In Section 2 we define the location-scale mixture ofelliptical distributions and establish some properties of it. Furthermore, we give several examples asspecial cases of it. In Section 3 we derive TCE for univariate cases of mixture of elliptical distributions,and in Section 4, we provide expression of MTCE for mixture of elliptical distributions. Section 5 offersexpression of portfolio risk decomposition with TCE for mixture of elliptical distributions. Section 6 givesconcluding remarks. In this section we introduce the location-scale mixture of elliptical (LSME) distributions and some itsproperties. Let Y ∼ LSM E n ( µ , Σ , β , Θ , g n ) be an n -dimensional LSME distribution with locationparameter µ and positive definite scale matrix Σ = ( σ i,j ) ni,j =1 , if Y = µ + Θ β + Θ Σ X , (2.1)2here β ∈ R n , and X ∼ E n ( , I n , g n ) . Assume that X is independent of non-negative scalar randomvariable Θ. We have Y | Θ = θ ∼ E n ( µ + θ β , θ Σ , g n ) . (2.2)Here X is an n -dimensional elliptical random vector, and denoted by X ∼ E n ( µ , Σ , g n ). If it’s proba-bility density function exists, the form will be f X ( x ) := 1 p | Σ | g n (cid:26)
12 ( x − µ ) T Σ − ( x − µ ) (cid:27) , x ∈ R n , (2.3)where µ is an n × Σ is an n × n scale matrix and g n ( u ), u ≥
0, is the density generatorof X . This density generator satisfies condition: (see Fang et al. (1990)) Z ∞ u n − g n ( u )d u < ∞ . (2.4)The characteristic function of X takes the form ϕ X ( t ) = exp (cid:8) i t T µ (cid:9) ψ (cid:0) t T Σ t (cid:1) , t ∈ R n , with function ψ ( t ) : [0 , ∞ ) → R , called the characteristic generator. Furthermore, the condition | ψ ′ (0) | < ∞ , (2.5)guarantees the existence of the covariance matrix of X (see Fang et al. (1990)). Suppose A is a k × n matrix, and b is a k × AX + b ∼ E k (cid:0) Aµ + b , A Σ A T , g k (cid:1) . (2.6)To express conditional tail risk measures for n -dimensional mixture of elliptical distributions weintroduce the cumulative generator G n ( u ) (see Landsman et al.(2018)): G n ( u ) = Z ∞ u g n ( v )d v. (2.7)Let X ∗ ∼ E n ( µ , Σ , G n ) be an elliptical random vector with generator G n ( u ), whose the densityfunction (if it exists) f X ∗ ( x ) = − ψ ′ (0) p | Σ | G n (cid:26)
12 ( x − µ ) T Σ − ( x − µ ) (cid:27) , x ∈ R n . (2.8)We list some examples of the mixture of elliptical family, including location-scale mixture of normal(LSMN) distributions, location-scale mixture of Student- t (LSMSt) distributions, location-scale mixtureof Logistic (LSMLo) distributions and location-scale mixture of Laplace (LSMLa) distributions.3 xample 2 . (Mixture of normal distribution). An n -dimensional normal random vector X with locationparameter µ and scale matrix Σ has density function f X ( x ) = (2 π ) − n p | Σ | exp (cid:26) −
12 ( x − µ ) T Σ − ( x − µ ) (cid:27) , x ∈ R n , and denoted by X ∼ N n ( µ , Σ ). Therefore, the location-scale mixture of normal random vector Y ∼ LSM N n ( µ , Σ , β , Θ) is defined as Y = µ + Θ β + Θ Σ X , (2.9)and µ , Σ , Θ and β are the same as in (2 . Example 2 . (Mixture of student- t distribution). An n -dimensional student- t random vector X withlocation parameter µ , scale matrix Σ and m > f X ( x ) = c n p | Σ | (cid:20) x − µ ) T Σ − ( x − µ ) m (cid:21) − m + n , x ∈ R n , where c n = Γ ( m + n ) Γ( m/ mπ ) n , and denoted by X ∼ St n ( µ , Σ , m ). So that the location-scale mixture ofstudent- t random vector Y ∼ LSM St n ( µ , Σ , β , Θ) satisfies Y = µ + Θ β + Θ Σ X , (2.10)and µ , Σ , Θ and β are the same as in (2 . Example 2 . (Mixture of Logistic distribution). Density function of an n -dimension Logistic randomvector X with location parameter µ and scale matrix Σ can be expressed as f X ( x ) = c n p | Σ | exp (cid:8) − ( x − µ ) T Σ − ( x − µ ) (cid:9)(cid:2) (cid:8) − ( x − µ ) T Σ − ( x − µ ) (cid:9)(cid:3) , x ∈ R n , where c n = (2 π ) − n/ (cid:2)P ∞ i =0 ( − i − i − n/ (cid:3) − , and denoted by X ∼ Lo n ( µ , Σ ). The location-scalemixture of Logistic random vector Y ∼ LSM Lo n ( µ , Σ , β , Θ) satisfies Y = µ + Θ β + Θ Σ X , (2.11)and µ , Σ , Θ and β are the same as in (2 . Example 2 . . (Mixture of Laplace distribution). Density of Laplace random vector X with locationparameter µ and scale matrix Σ is given by f X ( x ) = c n p | Σ | exp n − [( x − µ ) T Σ − ( x − µ )] / o , x ∈ R n , where c n = Γ( n/ π n/ Γ( n ) , and denoted by X ∼ La n ( µ , Σ ). Hence, the location-scale mixture of Laplacerandom vector Y ∼ LSM La n ( µ , Σ , β , Θ) is defined as Y = µ + Θ β + Θ Σ X , (2.12)and µ , Σ , Θ and β are the same as in (2 . Univariate cases
Theorem 3.1.
Let Y ∼ LSM E ( µ, σ , β, Θ , g ) be an univariate location-scale mixture of ellipticalrandom variable defined as (2 . . We suppose Z ∞ g ( u )d u < ∞ . (3.13) Then
T CE Y ( y q ) = µ + E θ h θβ + δ θ ( √ θσ ) i , (3.14) where δ θ = √ θσ G ( z q ) F Z ( z q ) , with Z ∼ E (0 , , g ) and z q = y q − µ − θβ √ θσ . Proof. Using definition and tower property of expectations, we obtain
T CE Y ( y q ) = E [ Y | Y > y q ]= E Θ [ E ( Y | Y > y q , Θ)] . Since E [ Y | Y > y q , Θ = θ ] = E [( Y | Θ = θ ) | ( Y | Θ = θ ) > y q ]= E [ M | M > y q ]= T CE M ( y q ) , where M ∼ E ( µ + θβ, θσ , g ), and the second equality we have used (2.2).Using Theorem 1 in Landsman and Valdez (2003), we obtain (3 . Remark 3 . . We find that
T CE Y | Θ ( y q ) is a special case of Theorem 1 in Landsman and Valdez (2003). Corollary 3.1.
Let Y ∼ LSM N ( µ, σ , β, Θ) be an univariate location-scale mixture of normal randomvariable defined as (2 . . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixtureof normal distributions. Its’ form is the same as (3 . , where δ θ = √ θσ φ ( z q )1 − Φ ( z q ) . Additionally, φ ( · ) and Φ ( · ) denote the density and distribution functions of normal distributions, re-spectively. G ( u ) = g ( u ) = φ ( u ) = (2 π ) − e − u in Theorem 3.1, we directlyobtain our result. This completes the proof of Corollary 3.1. Corollary 3.2.
Let Y ∼ LSM St ( µ, σ , β, Θ) be an univariate location-scale mixture of student- t random variable defined as (2 . . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Student- t distributions. Its’ form is the same as (3 . , where δ θ = √ θσ G ( z q ) F Z ( z q ) = √ θσ c mm − (1 + z q m ) − ( m − / F Z ( z q ) = √ θσ t m, ( z q ; 0 , T m, ( z q ; 0 , . In addition, t m, ( z q ; 0 , and T m, ( z q ; 0 , are the density and distribution functions of Student- t distri-butions, respectively (see Landsman et al. (2016)). Proof. Letting g ( u ) = c (1 + um ) − ( m +1) / , G ( u ) = c mm − (1 + um ) − ( m − / and c = Γ(( m +1) / m/ mπ ) (seeLandsman et al. (2016)) in Theorem 3.1, we immediately obtain our result. This completes the proof ofCorollary 3.2. Corollary 3.3.
Let Y ∼ LSM Lo ( µ, σ , β, Θ) be an univariate location-scale mixture of Logisticrandom variable defined as (2 . . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Logistic distributions. Its’ form is the same as (3 . , where δ θ = √ θσ G ( z q ) F Z ( z q ) = √ θσ c − z q )1+exp( − z q ) F Z ( z q ) = " √ π ) − + φ ( z q )] √ θσ φ ( z q ) F z ( z q ) . In addition, φ ( · ) is the density functions of normal distributions (see Landsman and Valdez (2003)). Proof. Letting g ( u ) = c − u )[1+exp( − u )] , G ( u ) = c − u )1+exp( − u ) and c = (see Landsman and Valdez(2003)) in Theorem 3.1, we directly obtain our result. This completes the proof of Corollary 3.3. Corollary 3.4.
Let Y ∼ LSM La ( µ, σ , β, Θ) be an univariate location-scale mixture of Laplacerandom variable defined as (2 . . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Laplace distributions. Its’ form is the same as (3 . , where δ θ = √ θσ G ( z q ) F Z ( z q ) = √ θσ c (1 + q z q ) exp( − q z q ) F Z ( z q ) = √ (cid:16) q z q (cid:17) √ θσ e ( z q ) F z ( z q ) . Additionally, e ( · ) is the density functions of exponential power distributions with a density generator ofthe form g ( u ) = c exp( −√ u ) and c = √ (see Landsman and Valdez (2003)). Proof. Letting g ( u ) = c exp( −√ u ), G ( u ) = c (1 + √ u ) exp( −√ u ) and c = (see Landsman et al.(2016)) in Theorem 3.1, we immediately obtain our result. This completes the proof of Corollary 3.4.6 Multivariate cases
In this section, we consider the multivariate TCE for mixture of elliptical distributions. To calculateit we definite shifted cumulative generator (see Landsman et al. (2016)) G ∗ n − ( u ) = Z ∞ u g n ( v + a )d v, a ≥ , n > , (4.15)with G ∗ n − ( u ) < ∞ . (4.16)Here we consider g ∗ n − ( u ) = g n ( u + a ) as a density generator, if it satisfies the condition: Z ∞ u n − g n ( u + a )d u < ∞ , ∀ a ≥ . (4.17)Let Y ∼ LSM E n ( µ , Σ , β , Θ , g n ) and M = Y | Θ = θ ∼ E n ( µ + θ β , θ Σ , g n ). Then Z = ( θ Σ ) − ( M − µ − θ β ) ∼ E n ( , I n , g n ) . Writing ξ q = ( ξ q , , ξ q , , · · · , ξ q ,n ) T = ( θ Σ ) − ( y q − µ − θ β ) , where y q = V aR q ( Y ), and ξ q , − k = ( ξ q , , ξ q , , · · · , ξ q ,k − , ξ q ,k +1 , · · · , ξ q ,n ) T .To derive formula for MTCE we introduce tail function F Z − k ( t ) of ( n − Z − k = ( Z , Z , · · · , Z k − , Z k +1 , · · · , Z n ) T , F Z − k ( t ) = Z ∞ t f Z − k ( v )d v , v , t ∈ R n − , d v = d v d v · · · d v n , with the pdf f Z − k ( z − k ) = − ψ ∗ ′ (0) G ∗ n − (cid:26) z T − k z − k (cid:27) = − ψ ∗ ′ (0) G n (cid:26) z T − k z − k + 12 ξ q ,k (cid:27) , k = 1 , , · · · , n, where ψ ∗ ( · ) is the characteristic generator corresponding to G ∗ n − , and G ∗ n − as formula (4 . Theorem 4.1.
Let Y ∼ LSM E n ( µ , Σ , β , Θ , g n ) be an n -dimensional location-scale mixture of ellip-tical random variable defined as (2 . . We suppose satisfy conditions (2 . , (4 . and (4 . .Then M T CE q ( Y ) = µ + E θ h θ β + √ θ Σ δ q i , (4.18)7 here δ q = ( δ , q , δ , q , · · · , δ n, q ) T , with δ k, q = − c n ψ ∗ ′ (0) F z − k ( ξ q , − k ) F z ( ξ q ) and c n = Γ( n/ π ) n/ (cid:2)R ∞ u n − g n ( u )d u (cid:3) − . Proof. Using the tower property of expectations, we obtain
M T CE q ( Y ) = E [ Y | Y > y q ]= E Θ [ E ( Y | Y > y q , Θ)] . Since E [ Y | Y > y q , Θ = θ ] = E [( Y | Θ = θ ) | ( Y | Θ = θ ) > y q ]= E [ M | M > y q ]= M T CE q ( M ) , where M ∼ E n ( µ + θ β , θ Σ , g n ), and the second equality we have used (2.2). Using Theorem 1 inLandsman et al. (2016), we obtain (4 . Remark 4 . . We remark that Theorem 1 in Landsman et al. (2016), which corresponding the result of
M T CE q ( Y | Θ) with q = ( q, q, · · · , q ) T , is a special case of our Theorem 4.1. Corollary 4.1.
Suppose Y ∼ LSM N n ( µ , Σ , β , Θ) is an n -variate location-scale mixture of normalrandom variable defined as (2 . . Under conditions in Theorem 4.1, we obtain the MTCE for location-scale mixture of normal distributions. Its’ form is the same as (4 . , where δ k, q = − c n ψ ∗ ′ (0) F z − k ( ξ q , − k ) F z ( ξ q ) = φ ( ξ k, q ) Φ z − k ( ξ q , − k )Φ z ( ξ q ) . Additionally, φ n ( · ) and Φ n ( · ) denote the density and distribution functions of normal distributions, re-spectively. Proof. Letting the density generator G n ( u ) = g n ( u ) = φ n ( u ) = c n e − u , c n = (2 π ) − n and ψ ∗ ′ (0) = − (2 π ) n φ ( ξ q ,k )in Theorem 4.1, we directly obtain our result. This completes the proof of Corollary 4.1. Corollary 4.2.
Suppose that Y ∼ LSM St n ( µ , Σ , β , Θ) is an n -variate location-scale mixture ofstudent- t random vector defined as (2 . . Under conditions in Theorem 4.1, we obtain the MTCE for ocation-scale mixture of Student- t distributions. Its’ form is the same as (4 . , where δ k, q = − c n ψ ∗ ′ (0) F z − k ( ξ q , − k ) F z ( ξ q )= Γ( m − ) m m ) p π ( m − (cid:18) m − m (cid:19) n ξ q ,k m ! − ( m + n − T m − ,n − ( ξ q , − k ; , ∆ k ) T m,n ( ξ q ; , I n ) , and ∆ k = m (1 + ξ q ,k m ) m − I n − , I k ( k = n − or n ) is a k -dimensional identity matrix. In addition, T m − ,n − ( ξ q , − k ; 0 , ∆ k ) and T m,n ( ξ q ; 0 , I n ) are distribution functions of Student- t distributions (see Landsman et al. 2016). Proof. Letting g n ( u ) = c n (1 + um ) − ( m + n ) / , G n ( u ) = c n mm + n − (1 + um ) − ( m + n − / , c n = Γ(( m + n ) / m/ mπ ) n (see Landsman et al. (2016)) and ψ ∗ ′ (0) = − Γ( m − ) π ( n − / ( m − ( n − / m Γ( m + n − )( m + n − ξ q ,k m ! − ( m + n − / in Theorem 4.1, we immediately obtain our result. This completes the proof of Corollary 4.2. Corollary 4.3.
Assume Y ∼ LSM Lo n ( µ , Σ , β , Θ) is an n -variate location-scale mixture of Logisticrandom vector defined as (2 . . Under conditions in Theorem 4.1, we obtain the MTCE for location-scalemixture of Logistic distributions. Its’ form is the same as (4 . , where δ k, q = − c n ψ ∗ ′ (0) F z − k ( ξ q , − k ) F z ( ξ q )= L ( − exp( − ξ q ,k ) , n − ,
1) exp( − ξ q ,k ) √ π (cid:2)P ∞ i =0 ( − i − i − n/ (cid:3) F z − k ( ξ q , − k ) F z ( ξ q ) , and pdf of Z − k : f Z − k ( t ) = − ψ ∗ ′ (0) exp( − t T t − ξ q ,k )1 + exp( − t T t − ξ q ,k ) , k = 1 , , · · · , n, t ∈ R n − , and ψ ∗ ′ (0) = − (2 π ) n − Γ( n − ) Z ∞ t ( n − / exp( − t − ξ q ,k )1 + exp( − t − ξ q ,k ) d t = − (2 π ) ( n − / L ( − exp( − ξ q ,k ) , n − , ξ q ,k ) . (4.19) Additionally, L ( · ) is the well known Lerch zeta function (see Lin and Srivastava (2004)). g n ( u ) = c n exp( − u )[1+exp( − u )] , G n ( u ) = c n exp( − u )1+exp( − u ) , c n = (2 π ) − n/ " ∞ X i =0 ( − i − i − n/ − and formula (4 .
19) in Theorem 4.1, we directly obtain our result. This completes the proof of Corollary4.3.
Corollary 4.4.
Assume that Y ∼ LSM La n ( µ , Σ , β , Θ) be an n -variate location-scale mixture ofLaplace random vector defined as (2 . . Under conditions in Theorem 4.1, we obtain the MTCE forlocation-scale mixture of Laplace distributions. Its’ form is the same as (4 . , where δ k, q = − c n ψ ∗ ′ (0) F z − k ( ξ q , − k ) F z ( ξ q )= − Γ( n/ ψ ∗ ′ (0)2 π n/ Γ( n ) F z − k ( ξ q , − k ) F z ( ξ q ) , and pdf of Z − k : f Z − k ( t ) = − ψ ∗ ′ (0) (cid:16) q t T t + ξ q ,k (cid:17) exp n − q t T t + ξ q ,k o , k = 1 , , · · · , n, t ∈ R n − , and ψ ∗ ′ (0) = − (2 π ) ( n − / Γ (cid:0) n − (cid:1) (cid:20)Z ∞ t n − (cid:16) q t + ξ q ,k (cid:17) exp n − q t + ξ q ,k o d t (cid:21) . (4.20)Proof. Letting g n ( u ) = c n exp( −√ u ), G n ( u ) = c n (1 + √ u ) exp( −√ u ), c n = Γ( n/ π n/ Γ( n ) (see Landsmanet al. 2016) and formula (4 .
20) in Theorem 4.1, we immediately obtain our result. This completes theproof of Corollary 4.4.
Let Y = ( Y , Y , · · · , Y n ) T ∼ LSM E n ( µ , Σ , β , Θ , g n ), e = (1 , , · · · , T is an n × S = n X j =1 Y j = e T Y = e T µ + Θ e T β + Θ e T Σ X , (5.21)which is the sum of mixture of elliptical risks. Proposition 5 . . Under the conditions (3 .
13) and (5 . S can be expressed as T CE S ( s q ) = µ S + E θ h θβ S + δ S ( √ θσ S ) i , (5.22)10here δ S = √ θσ S G ( z q ) F Z ( z q ) , with Z ∼ E (0 , , g ) and z q = s q − µ S − θβ S √ θσ S .Proof. Let L = e T Σ X . Due to (2 . L ∼ E (0 , σ S , g ) with σ S = e T Σ e . So that L ′ ∼ E (0 , , g ) with L ′ = Lσ S . Therefore, S = µ S + Θ β S + Θ σ L L ′ ∼ LSM E ( µ S , σ S , β S , Θ , g ) , (5.23)with µ S = e T µ and β S = e T β .By using Theorem 3.1, we obtain (5 . Lemma 5.1.
Let Y = ( Y , Y , · · · , Y n ) T ∼ LSM E n ( µ , Σ , β , Θ , g n ) as (2 . . Then the vector Y k,S = ( Y k , S ) T , (1 ≤ k ≤ n ) has a mixture of elliptical distribution, namely, Y k,S ∼ LSM E ( µ k,S , Σ k,S , β k,S , Θ , g ) , where µ k,S = ( µ k , e T µ ) T = ( µ k , P ni =1 µ i ) T , Σ k,S = σ k σ k,S Σ k,S σ S ! and β k,S = ( β k , β S ) T , and σ k = σ k,k , σ k,S = P ni =1 σ k,i , β S = e T β = P ni =1 β i , σ S = e T Σ e = P ni,j =1 σ i,j . Proof. Write P = ( P , P , · · · , P n ) T = Σ X . Due to (2 . P ∼ E n ( , Σ , g n ) . From (2 . Y k = µ k +Θ β k +Θ P k , 1 ≤ k ≤ n . According to (5 . S = µ S +Θ β S +Θ L with L ∼ E (0 , σ S , g ). So that Y k,S = µ k,S + Θ β k,S + Θ ( P k , L ) T . By Lemma 1 in Landsman and Valdez (2003), we obtain ( P k , L ) T ∼ E ( µ k,S , Σ k,S , g ). Therefore, Y k,S ∼ LSM E ( µ k,S , Σ k,S , β k,S , Θ , g ). This completes the proof of Lemma 5.1. Lemma 5.2.
Let Y = ( Y , Y ) T ∼ LSM E ( µ , Σ , β , Θ , g ) . Assume that condition (3 . holds.Then T CE Y | Y ( y q ) = µ + E θ [ θβ + δ θσ σ ρ , ] , (5.24) where δ = √ θσ G ( z ,q ) F z ( z ,q ) ,ρ , = σ , σ σ , σ = √ σ , , σ = √ σ , and z ,q = y q − µ − θβ √ θσ . T CE Y | Y ( y q ) = E [ Y | Y > y q ]= E θ [ E [ Y | Y > y q , Θ = θ ]]= E θ [ E [ Q | Q > y q ]] , where ( Q , Q ) T = Y | Θ = θ ∼ E ( µ + θ β , θ Σ , g ), and the third equality we have used (2.2).By Lemma 2 in Landsman and Valdez (2003), we can obtain (5 . Theorem 5.1.
Let Y = ( Y , Y , · · · , Y n ) T ∼ LSM E n ( µ , Σ , β , Θ , g n ) be an n -dimensinal location-scale mixture of elliptical random vector defined as (2 . . We suppose condition (3 . holds, and let S = P ni =1 Y i . Then the contribution of risk Y k , ≤ k ≤ n , to the total TCE can be given by T CE Y k | S ( s q ) = µ k + E θ [ θβ k + δ S θσ k σ S ρ k,S ] , (5.25) where ρ k,S = σ k,S σ k σ S and δ S is the same as in Proposition 5.1. Proof. By Lemma 5.1, we know Y k,S = ( Y k , S ) T ∼ LSM E ( µ k,S , Σ k,S , β k,S , Θ , g ), (1 ≤ k ≤ n ).Let Y subject to Y k,S in Lemma 5.2, we can immediately obtain (5 . Remark 5 . . Letting the density generator G ( u ) = g ( u ) = φ ( u ) = (2 π ) − e − u in Theorem 5.1, weobtain the portfolio risk decomposition with TCE for location-scale mixture of normal distributions. Its’form is the same as (5 . δ S = √ θσ S φ ( z q )1 − Φ ( z q ) . Additionally, φ ( · ) and Φ ( · ) denote the density and distribution functions of normal distributions. Remark 5 . . Letting g ( u ) = c (1+ um ) − ( m +1) / , G ( u ) = c mm − (1+ um ) − ( m − / and c = Γ(( m +1) / m/ mπ ) (see Landsman et al. (2016)) in Theorem 5.1, we obtain the portfolio risk decomposition with TCE forlocation-scale mixture of Student- t distributions. Its’ form is the same as (5 . δ S = √ θσ S G ( z q ) F Z ( z q ) = √ θσ S c mm − (1 + z q m ) − ( m − / F Z ( z q ) = √ θσ S t m, ( z q ; 0 , T m, ( z q ; 0 , . In addition, t m, ( z q ; 0 ,
1) and T m, ( z q ; 0 ,
1) are the density and distribution functions of Student- t distri-butions, respectively (see Landsman et al. (2016)).12 emark 5 . . Letting g ( u ) = c − u )[1+exp( − u )] , G ( u ) = c − u )1+exp( − u ) and c = (see Landsman and Valdez(2003)) in Theorem 5.1, we obtain the portfolio risk decomposition with TCE for location-scale mixtureof Logistic distributions. Its’ form is the same as (5 . δ S = √ θσ S G ( z q ) F Z ( z q ) = √ θσ S c − z q )1+exp( − z q ) F Z ( z q ) = " √ π ) − + φ ( z q )] √ θσ S φ ( z q ) F z ( z q ) . In addition, φ ( · ) is the density functions of normal distributions (see Landsman and Valdez (2003)). Remark 5 . . Letting g ( u ) = c exp( −√ u ), G ( u ) = c (1+ √ u ) exp( −√ u ) and c = (see Landsmanet al. (2016)) in Theorem 5.1, we obtain the portfolio risk decomposition with TCE for location-scalemixture of Laplace distributions. Its’ form is the same as (5 . δ S = √ θσ S G ( z q ) F Z ( z q ) = √ θσ S c (1 + q z q ) exp( − q z q ) F Z ( z q ) = √ (cid:16) q z q (cid:17) √ θσ S e ( z q ) F z ( z q ) . Additionally, e ( · ) is the density functions of exponential power distributions with a density generator ofthe form g ( u ) = c exp( −√ u ) and c = √ (see Landsman and Valdez (2003)). In this paper we consider the univariate and multivariate location-scale mixture of elliptical distribu-tion, which is ( A = Σ ) generalization of normal mean-variance mixture distribution in Kim and Kim(2019). It has received much attention in finance and insurance applications, since this distribution notonly include location-scale mixture of normal (LSMN) distributions, location-scale mixture of Student- t (LSMSt) distributions, location-scale mixture of Logistic (LSMLo) distributions and location-scale mix-ture of Laplace (LSMLa) distributions, but also include the generalized hyperbolic distribution (GHD)and the slash distribution. The GHD is a special case of this mixture random variable with X ∼ N n ( , I n )and the distribution of Θ given by a generalized inverse gaussian N − ( λ, χ, ψ ) (see Kim and Kim (2019)for details). The GHD is an important distribution, and has a lot of applications (see Kim (2010) andIgnatieva and Landsman (2015)). Slash distribution also is a special case of this mixture random variablewith X ∼ N n ( , I n ) and Θ ∼ BP ( η = 1 , α = 1 , β = q/ BP ( · ) is the 3-parameter beta prime(BP) or inverted beta distribution (see Kim and Kim (2019) for details). This distribution has beendiscussed in many literatures (see Gneiting (1997), Gen¸c (2007) and Wang and Genton (2006)). We alsoconsider univariate TCE, multivariate TCE and portfolio risk decomposition with TCE for location-scalemixture of elliptical distribution. As special cases, we provided univariate TCE, multivariate TCE andportfolio risk decomposition with TCE for LSMN, LSMSt, LSMLo and LSMLa distributions.13 cknowledgments The research was supported by the National Natural Science Foundation of China (No.11571198, 11701319)