# Conditional tail risk expectations for location-scale mixture of elliptical distributions

We present general results on the univariate tail conditional expectation (TCE) and multivariate tail conditional expectation for location-scale mixture of elliptical distributions. Examples include 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. We also consider portfolio risk decomposition with TCE for location-scale mixture of elliptical distributions.

## Authors

• 6 publications
• 15 publications
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## 1 Introduction and Motivation

Tail conditional expectation (TCE), one of important risk measures, is common and practical. TCE of a random variable

is defined as where is a particular value, generally referred to as the

-th quantile with

Here is tail distribution function of . The TCE has been discussed in many literatures ( see Landsman and Valdez (2003), Ignatieva and Landsman (2015, 2019)). Recently, a new type of multivariate tail conditional expectation (MTCE) was defined by Landsman et al. (2016). It is the following special case when .

 MTCEq(X)=E[X|X>VaRq(X)] =E[X|X1>VaRq1(X1),⋯,Xn>VaRqn(Xn)], q=(q1,⋯,qn)∈(0, 1)n,

where is an vector of risks with cumulative distribution function and tail function ,

 VaRq(X)=(VaRq1(X1), VaRq2(X2),⋯,VaRqn(Xn))T,

and is the value at risk (VaR) measure of the random variable , being the -th quantile of

(see Landsman et al. (2016)). On the basis of it, Mousavi et al. (2019) study multivariate tail conditional expectation for scale mixtures of skew-normal distribution.

Closely related to tail conditional expectation is portfolio risk decomposition with TCE, it’s research has experienced a rapid growth in the literature. Portfolio risk decomposition based on TCE for the elliptical 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 distribution were 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 distributed risks joined by the Sarmanov distribution, and Ignatieva and Landsman (2019) has given the expression 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 distributions and 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 of elliptical distributions and establish some properties of it. Furthermore, we give several examples as special 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 offers expression of portfolio risk decomposition with TCE for mixture of elliptical distributions. Section 6 gives concluding remarks.

## 2 Mixture of elliptical distributions

In this section we introduce the location-scale mixture of elliptical (LSME) distributions and some its properties. Let be an -dimensional LSME distribution with location parameter and positive definite scale matrix , if

 Y=μ+Θβ+Θ12Σ12X, (2.1)

where , and Assume that is independent of non-negative scalar random variable . We have

 Y|Θ=θ∼En(μ+θβ, θΣ, gn). (2.2)

Here is an -dimensional elliptical random vector, and denoted by

. If it’s probability density function exists, the form will be

 fX(x):=1√|Σ|gn{12(x−μ)TΣ−1(x−μ)}, x∈Rn, (2.3)

where is an location vector, is an scale matrix and , , is the density generator of . This density generator satisfies condition: (see Fang et al. (1990))

 ∫∞0un2−1gn(u)du<∞. (2.4)

takes the form , with function called the characteristic generator. Furthermore, the condition

 |ψ′(0)|<∞, (2.5)

guarantees the existence of the covariance matrix of (see Fang et al. (1990)). Suppose is a matrix, and is a vector. Then

 AX+b∼Ek(Aμ+b, AΣAT, gk). (2.6)

To express conditional tail risk measures for -dimensional mixture of elliptical distributions we introduce the cumulative generator (see Landsman et al.(2018)):

 ¯¯¯¯Gn(u)=∫∞ugn(v)dv. (2.7)

Let be an elliptical random vector with generator , whose the density function (if it exists)

 fX∗(x)=−1ψ′(0)√|Σ|¯¯¯¯Gn{12(x−μ)TΣ−1(x−μ)}, x∈Rn. (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- (LSMSt) distributions, location-scale mixture of Logistic (LSMLo) distributions and location-scale mixture of Laplace (LSMLa) distributions.
(Mixture of normal distribution). An -dimensional normal random vector with location parameter and scale matrix has density function

 fX(x)=(2π)−n2√|Σ|exp{−12(x−μ)TΣ−1(x−μ)}, x∈Rn,

and denoted by . Therefore, the location-scale mixture of normal random vector
is defined as

 Y=μ+Θβ+Θ12Σ12X, (2.9)

and , ,  and  are the same as in .
(Mixture of student- distribution). An -dimensional student- random vector with location parameter , scale matrix and degrees of freedom has density function

 fX(x)= cn√|Σ|[1+(x−μ)TΣ−1(x−μ)m]−m+n2, x∈Rn,

where , and denoted by . So that the location-scale mixture of student- random vector satisfies

 Y=μ+Θβ+Θ12Σ12X, (2.10)

and , ,  and  are the same as in .
(Mixture of Logistic distribution). Density function of an -dimension Logistic random vector with location parameter and scale matrix can be expressed as

 fX(x)=cn√|Σ|exp{−12(x−μ)TΣ−1(x−μ)}[1+exp{−12(x−μ)TΣ−1(x−μ)}]2, x∈Rn,

where , and denoted by . The location-scale mixture of Logistic random vector satisfies

 Y=μ+Θβ+Θ12Σ12X, (2.11)

and , ,  and  are the same as in .
(Mixture of Laplace distribution). Density of Laplace random vector with location parameter and scale matrix is given by

 fX(x)= cn√|Σ|exp{−[(x−μ)TΣ−1(x−μ)]1/2}, x∈Rn,

where , and denoted by . Hence, the location-scale mixture of Laplace random vector is defined as

 Y=μ+Θβ+Θ12Σ12X, (2.12)

and , ,  and  are the same as in .

## 3 Univariate cases

###### Theorem 3.1.

Let be an univariate location-scale mixture of elliptical random variable defined as . We suppose

 ∫∞0g1(u)du<∞. (3.13)

Then

 TCEY(yq)=μ+Eθ[θβ+δθ(√θσ)2], (3.14)

where

 δθ=1√θσ¯¯¯¯G1(12z2q)¯¯¯¯FZ(zq),

with and .

Proof. Using definition and tower property of expectations, we obtain

 TCEY(yq) =E[Y|Y>yq] =EΘ[E(Y|Y>yq,Θ)].

Since

 E[Y|Y>yq,Θ=θ] =E[(Y|Θ=θ)|(Y|Θ=θ)>yq] =E[M|M>yq] =TCEM(yq),

where , and the second equality we have used (2.2).
Using Theorem 1 in Landsman and Valdez (2003), we obtain , which completes the proof of Theorem 3.1.
We find that is a special case of Theorem 1 in Landsman and Valdez (2003).

###### Corollary 3.1.

Let be an univariate location-scale mixture of normal random variable defined as . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of normal distributions. Its’ form is the same as , where

 δθ=1√θσϕ1(12z2q)1−Φ1(zq).

Additionally, and denote the density and distribution functions of normal distributions, respectively.

Proof. Letting the density generator in Theorem 3.1, we directly obtain our result. This completes the proof of Corollary 3.1.

###### Corollary 3.2.

Let be an univariate location-scale mixture of student- random variable defined as . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Student- distributions. Its’ form is the same as , where

 δθ=1√θσ¯¯¯¯G1(12z2q)¯¯¯¯FZ(zq)=1√θσc1mm−1(1+z2qm)−(m−1)/2¯¯¯¯FZ(zq)=1√θσtm,1(zq;0,1)¯¯¯¯Tm,1(zq;0,1).

In addition, and are the density and distribution functions of Student- distributions, respectively (see Landsman et al. (2016)).

Proof. Letting , and (see Landsman et al. (2016)) in Theorem 3.1, we immediately obtain our result. This completes the proof of Corollary 3.2.

###### Corollary 3.3.

Let be an univariate location-scale mixture of Logistic random variable defined as . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Logistic distributions. Its’ form is the same as , where

 δθ=1√θσ¯¯¯¯G1(12z2q)¯¯¯¯FZ(zq)=1√θσc1exp(−12z2q)1+exp(−12z2q)¯¯¯¯FZ(zq)=[12[(√2π)−1+ϕ(zq)]]1√θσϕ(zq)¯¯¯¯Fz(zq).

In addition, is the density functions of normal distributions (see Landsman and Valdez (2003)).

Proof. Letting , and (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 be an univariate location-scale mixture of Laplace random variable defined as . Under conditions in Theorem 3.1, we obtain the TCE for location-scale mixture of Laplace distributions. Its’ form is the same as , where

 δθ=1√θσ¯¯¯¯G1(12z2q)¯¯¯¯FZ(zq)=1√θσc1(1+√z2q)exp(−√z2q)¯¯¯¯FZ(zq)=√2(1+√z2q)1√θσe(z2q)¯¯¯¯Fz(zq).

Additionally, is the density functions of exponential power distributions with a density generator of the form and (see Landsman and Valdez (2003)).

Proof. Letting , and (see Landsman et al. (2016)) in Theorem 3.1, we immediately obtain our result. This completes the proof of Corollary 3.4.

## 4 Multivariate cases

In this section, we consider the multivariate TCE for mixture of elliptical distributions. To calculate it we definite shifted cumulative generator (see Landsman et al. (2016))

 (4.15)

with

 ¯¯¯¯G∗n−1(u)<∞. (4.16)

Here we consider as a density generator, if it satisfies the condition:

 ∫∞0un2−1gn(u+a)du<∞, ∀a≥0. (4.17)

Let and . Then

 Z=(θΣ)−12(M−μ−θβ)∼En(0, In, gn).

Writing

 ξq=(ξq,1, ξq,2,⋯,ξq,n)T=(θΣ)−12(yq−μ−θβ),

where , and .

To derive formula for MTCE we introduce tail function of -dimensional random vector ,

 ¯¯¯¯FZ−k(t)=∫∞tfZ−k(v)dv,   v,t∈Rn−1,   dv=dv1dv2⋯dvn,

with the pdf

 fZ−k(z−k) =−1ψ∗′(0)¯¯¯¯Gn{12zT−kz−k+12ξ2q,k}, k=1, 2,⋯,n,

where is the characteristic generator corresponding to , and as formula .

###### Theorem 4.1.

Let be an -dimensional location-scale mixture of elliptical random variable defined as . We suppose satisfy conditions , and .

Then

 MTCEq(Y)=μ+Eθ[θβ+√θΣ12δq], (4.18)

where

 δq=(δ1,q, δ2,q,⋯,δn,q)T,

with and .

Proof. Using the tower property of expectations, we obtain

 MTCEq(Y) =E[Y|Y>yq] =EΘ[E(Y|Y>yq,Θ)].

Since

 E[Y|Y>yq,Θ=θ] =E[(Y|Θ=θ)|(Y|Θ=θ)>yq] =E[M|M>yq] =MTCEq(M),

where , and the second equality we have used (2.2). Using Theorem 1 in Landsman et al. (2016), we obtain , which completes the proof of Theorem 4.1.
We remark that Theorem 1 in Landsman et al. (2016), which corresponding the result of with , is a special case of our Theorem 4.1.

###### Corollary 4.1.

Suppose is an -variate location-scale mixture of normal random variable defined as . Under conditions in Theorem 4.1, we obtain the MTCE for location-scale mixture of normal distributions. Its’ form is the same as , where

 δk,q=−cnψ∗′(0)¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq)=ϕ1(ξk,q)¯¯¯¯Φz−k(ξq,−k)¯¯¯¯Φz(ξq).

Additionally, and denote the density and distribution functions of normal distributions, respectively.

Proof. Letting the density generator , and

 ψ∗′(0)=−(2π)n2ϕ1(ξq,k)

in Theorem 4.1, we directly obtain our result. This completes the proof of Corollary 4.1.

###### Corollary 4.2.

Suppose that is an -variate location-scale mixture of student- random vector defined as . Under conditions in Theorem 4.1, we obtain the MTCE for location-scale mixture of Student- distributions. Its’ form is the same as , where

 δk,q=−cnψ∗′(0)¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq) =Γ(m−12)m2Γ(m2)√π(m−1)(m−1m)n2(1+ξ2q,km)−(m+n−2)2¯¯¯¯Tm−1,n−1(ξq,−k;0,Δk)¯¯¯¯Tm,n(ξq;0,In),

and

 Δk=⎛⎜ ⎜⎝m(1+ξ2q,km)m−1⎞⎟ ⎟⎠In−1,

is a

and are distribution functions of Student- distributions (see Landsman et al. 2016).

Proof. Letting , , (see Landsman et al. (2016)) and

 ψ∗′(0)=−Γ(m−12)π(n−1)/2(m−1)(n−1)/2mΓ(m+n−22)(m+n−2)(1+ξ2q,km)−(m+n−2)/2

in Theorem 4.1, we immediately obtain our result. This completes the proof of Corollary 4.2.

###### Corollary 4.3.

Assume is an -variate location-scale mixture of Logistic random vector defined as . Under conditions in Theorem 4.1, we obtain the MTCE for location-scale mixture of Logistic distributions. Its’ form is the same as , where

 δk,q=−cnψ∗′(0)¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq) =L(−exp(−ξ2q,k2), n−12, 1)exp(−ξ2q,k2)√2π[∑∞i=0(−1)i−1i1−n/2]¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq),

and pdf of :

 fZ−k(t)=−1ψ∗′(0)exp(−tTt2−ξ2q,k2)1+exp(−tTt2−ξ2q,k2), k=1, 2,⋯,n, t∈Rn−1,

and

 ψ∗′(0) =−(2π)n−12Γ(n−12)⎡⎢ ⎢⎣∫∞0t(n−3)/2exp(−t−ξ2q,k2)1+exp(−t−ξ2q,k2)dt⎤⎥ ⎥⎦ =−(2π)(n−1)/2L(−exp(−ξ2q,k2), n−12, 1)exp(ξ2q,k2). (4.19)

Additionally, is the well known Lerch zeta function (see Lin and Srivastava (2004)).

Proof. Letting , ,

 cn=(2π)−n/2[∞∑i=0(−1)i−1i1−n/2]−1

and formula in Theorem 4.1, we directly obtain our result. This completes the proof of Corollary 4.3.

###### Corollary 4.4.

Assume that be an -variate location-scale mixture of Laplace random vector defined as . Under conditions in Theorem 4.1, we obtain the MTCE for location-scale mixture of Laplace distributions. Its’ form is the same as , where

 δk,q=−cnψ∗′(0)¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq) =−Γ(n/2)ψ∗′(0)2πn/2Γ(n)¯¯¯¯Fz−k(ξq,−k)¯¯¯¯Fz(ξq),

and pdf of :

 fZ−k(t)=−1ψ∗′(0)(1+√tTt+ξ2q,k)exp{−√tTt+ξ2q,k}, k=1, 2,⋯,n, t∈Rn−1,

and

 ψ∗′(0)=−(2π)(n−1)/2Γ(n−12)[∫∞0tn−32(1+√t+ξ2q,k)exp{−√t+ξ2q,k}dt]. (4.20)

Proof. Letting ,