vector-valued tail value-at-risk and capital allocation · 2017. 9. 28. · vector-valued tail...

Methodol Comput Appl Probab (2016) 18:653–674DOI 10.1007/s11009-015-9444-9

Vector-Valued Tail Value-at-Risk and Capital Allocation

Helene Cossette1 ·Melina Mailhot2 · Etienne Marceau1 ·Mhamed Mesfioui3

Received: 18 August 2014 / Revised: 10 February 2015 /Accepted: 13 March 2015 / Published online: 16 April 2015© Springer Science+Business Media New York 2015

Abstract Enterprise risk management, actuarial science or finance are practice areas inwhich risk measures are important to evaluate for heterogeneous classes of homogeneousrisks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. Theyare based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al.(Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived.Notably, they are shown to be positive homogeneous, invariant under translation and subad-ditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sumof random pairs are presented, allowing to use a more accurate model for dependent classesof homogeneous risks.

Keywords Bivariate Tail Value-at-Risk · Multivariate risk measures · Capital allocation ·Copulas · Bounds

Mathematics Subject Classification (2010) 62P05 · 91B30

1 Introduction

Recently, attention has been given in the literature to the improvement of risk measures, andparticularly, to their extension to a multivariate setting. Regulators and investors are inter-ested in measuring risks, in order to allocate capital, elaborate investment strategies and

� Melina [email protected]

1 Ecole d’Actuariat, Universite Laval, Quebec City, Quebec, Canada

2 Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada

3 Departement de Mathematiques et Informatique, Universite du Quebec a Trois-Riviares,Trois-Rivieres, Quebec, Canada

mailto:[email protected]

654 Methodol Comput Appl Probab (2016) 18:653–674

control the probability of meeting risky commitments or objectives. Many researchers havebeen interested in univariate risk measures. See e.g. McNeil et al. (2005) for a review of uni-variate VaR and TVaR risk measures. Risk measures are used for multiple purposes, notablyto study scenarios for capital allocation under different dependence risk levels and to pricefinancial products. Ignoring the dependence between risks often results in over-estimatingor under-estimating them. In many cases, external risk factors and multiple business activi-ties make the use of a univariate risk measure insufficient, due to undesirable compensationsresulting from the aggregation. Moreover, due to recent financial crises, considering eachclass of risks to be covered at level α in addition to a protection of risks, considering theirdependence, would provide additional security (see BCBS 2003, 2004 for discussions onthis subject), which is encouraged by regulatory organizations. In addition to providing newrisk measures with interesting characteristics, the bivariate risk measure framework is moti-vated by applications related to capital requirements. It allows to account for the dependencerelationship between risks, or various lines of business, that cannot be aggregated.

Different multivariate risk measures have been proposed in the last decade. The main dif-ficulty lies in establishing couple orders because of partial ordering. Barnett (1976) presentsfour different ways of defining couple ordering, and discusses different measures based onthe latter. Serfling (2002) reviews and compares different multivariate quantile curve rep-resentations. Bedoui and Dbabis (2009) apply the bivariate VaR using quantile curves todetermine the marginal rate of substitution between the two univariate VaRs of hedge fundsand share indices, claiming that it represents the dependence between each fund or indicemore accurately than the correlation coefficient. Cossette et al. (2012) reformulated thebivariate lower and upper orthant Value-at-Risk (VaR), introduced by Embrechts and Puc-cetti (2006). These two risk measures use boundary sets whose distribution and survivalfunction are accumulating to a fixed level. With analogous characteristics as in the univari-ate case, the multivariate VaR does not give any information about the thickness of the tailof the cumulative distribution function (cdf). In practice, we are not only concerned withthe value at default but also with the expected loss.

Jouini et al. (2004) defines the multivariate Worst Conditional Expectation (WCE), froman axiomatic representation of multivariate risk measures, as an extension to Artzner et al.(1999). Tahar and Lepinette (2012, 2014) also studies the WCE and introduces the General-ized Worst Conditional Expectation, based on the representation introduced in Jouini et al.(2004). Furthermore, Tahar and Lepinette (2012, 2014) investigates multivariate Tail Con-ditional Expectation, which consists in the selection of vectors that respect a conditionalexpectation from an acceptance region. Their approach substantially differs from the onepresented in this article. Here, we consider the conditional expectation of a set instead of theexpected conditional quantiles. Di Bernardino et al. (2015) present two conditional Value-at-Risk, using multivariate distribution functions, conditioning on different events than thesuggested risk measures in this paper. Cousin and Di Bernardino (2013, 2014) considera multivariate VaR, conditioning on a fixed threshold, and a Conditional Tail Expecta-tion (CTE), based on their definition of the VaR. Mainik and Schaanning (2012) present avector-valued Conditional Expected Shortfall (CoES), based on the univariate VaR of theconditioning random variable (rv), differing from conditioning on the bivariate lower andupper orthant VaR introduced in Cossette et al. (2012), as presented in this paper. Cousinand Di Bernardino (2013) and Mainik and Schaanning (2012) methods provide sets withthe same dimension as the number of rv’s, but does not fulfill as many properties as the onepresented in this article. Our measures consider joint thresholds, instead of conditional uni-variate ones. Their representations of CTE and CoES is similar to the univariate top-downTVaR-based capital allocation rule (see Cossette et al. 2012 for more details), except that it

Methodol Comput Appl Probab (2016) 18:653–674 655

is conditional to the joint distribution. Sordo et al. (2015) compare stochastic orderings ofdependent vectors, conditioning on marginal behaviors. Also, see Balbas et al. (2012) forrepresentations and applications of multivariate risk measures within dynamic models.

In this paper, we define the bivariate lower and upper orthant Tail Value-at-Risk (TVaR)at level α, denoted TVaRα(X) and TVaRα(X) respectively, and describe their behavior.These measures allow flexibility for choosing capital allocation sets. The approach of inte-grating the inversion of multivariate cdf’s and survival functions is very similar to univariaterv’s and makes them attractive in practice.The approach proposed significantly simplifiesprevious ones and allows to retrieve suitable and natural properties. Limits and propertiesare established and bivariate lower and upper orthant TVaR’s of sums of random pairs arediscussed. This article also contributes to capital allocation techniques, by providing twodifferent approaches to calculate capital allocation couples from TVaRα(X) and TVaRα(X).None of the previously discussed methods include an alternative representation for the upperorthant bivariate TVaR, which considers a joint survival threshold, that can be used in amultivariate ruin probability or dynamic risk measures setting.

The paper is organized as follows. Section 2.1 introduces the concept of TVaR and setsnotation. Section 2 defines bivariate lower and upper orthant TVaR’s and studies the lim-its of the curves obtained by TVaRα(X) and TVaRα(X). It also establishes properties ofTVaRα(X) and TVaRα(X). Section 3 presents two approaches to compute capital alloca-tion couples from these bivariate curves. Section 4 concerns the contributions of lower andupper orthant TVaR for sums of random pairs and applications of bivariate lower and upperorthant TVaR with investment products and for bounds of sums of random pairs.

2 Bivariate TVaR and Resulting Properties

2.1 Preliminaries

Let us denote by X = (X1, X2), a vector of rv’s with joint cdf FX, survival function FX andmarginal cdf’s FXi

, for Xi , i = 1, 2. Also, denote the conditional cdf’s of (X2 | X1 ≤ x1)

and (X2 | X1 ≥ x1) by FX2|X1≤x1(x2) = FX(x1,x2)FX1 (x1)

and FX2|X1≥x1(x2) = FX2 (x2)−FX(x1,x2)

1−FX1 (x1),

respectively. Analogously, we can obtain the conditional cdf’s of (X1 | X2 ≤ x2) and(X1 | X2 ≥ x2). For a fixed x1, let us define the functions x2 �→ Fx1(x2) = FX(x1, x2),and x2 �→ Fx1(x2) = FX(x1, x2). Analogously, for a fixed x2, let us define the functionsx1 �→ Fx2(x1) = FX(x1, x2), and x1 �→ Fx2(x1) = FX(x1, x2). As further discussedin Section 2.2, for a level α, such that FX(x1, x2) = α, and a fixed xi , the functionFxi

(xj ) ensures that FXj |Xi≤xi(xj ) = α

FXi(xi )

. Also, as discussed in Section 2.3, for a

level α, such that FX(x1, x2) = 1 − α, and a fixed xi , the function Fxi(xj ) ensures that

FXj |Xi≥xi(xj ) = α−FXi

(xi )

1−FXi(xi )

.

Let F−1xi

(α) and F−1xi

(α) be their corresponding generalized inverse functions given by

F−1xi

(α) = inf{t ∈ R : Fxi

(t) ≥ α}

and F−1xi

(α) = inf{t ∈ R : Fxi

(t) ≤ α}

respectively, for i = 1, 2. For continuous rv’s, Cossette et al. (2012) denote the bivariatelower orthant VaR at a fixed probability level α by either the set

VaRα(X) ={(

x1, F−1x1

(α))

, x1 ≥ F−1X1

(α)}

, (2.1)


orVaRα(X) =

{(F−1

x2(α) , x2

), x2 ≥ F−1

X2(α)

}. (2.2)

Therefore, we have VaRα,xi(X) = F−1

xi(α), i = 1, 2 of Eq. 2.1 such that

FX(x1, VaRα,x1

(X)) = FX

(VaRα,x2

(X), x2) = α. (2.3)

The bivariate upper orthant VaR at probability level α is denoted either by the set

VaRα(X) ={(

x1, F−1x1

(1 − α))

, x1 ≤ F−1X1

(α)}

, (2.4)

orVaRα(X) =

{(F−1

x2(1 − α) , x2

), x2 ≤ F−1

X2(α)

}. (2.5)

These curves, studied in Cossette et al. (2012), are bivariate reformulations of the multi-variate lower and upper orthant VaR defined by Embrechts and Puccetti (2006), as well asquantile curves considered by Tibiletti (1993), Fernandez-Ponce and Suarez-Llorens (2002)and Belzunce et al. (2007).

To add information to the bivariate VaR regarding the tail of the distribution, we proposein this paper the bivariate TVaR. We aim to obtain two curves from the joint cdf to charac-terize the bivariate lower orthant TVaR and two curves from the joint survival function tocharacterize the bivariate upper orthant TVaR. These risk measures will provide more infor-mation than the bivariate lower and upper orthant VaR on the bivariate tail distribution. Thisis an extension of univariate TVaR which provides more information on the tail distributioncompared to the univariate VaR.

2.2 Bivariate Lower Orthant TVaR

Definition 2.1 Bivariate lower orthant TVaR is defined by

TVaRα,x(X) = ((x1, TVaRα,x1

(X)),(TVaRα,x2

(X), x2))

,

where

TVaRα,x1(X) = E[X2 | X2 > VaRα,x1

(X),X1 ≤ x1], x1 ≥ F−1X1

(α)

andTVaRα,x2

(X) = E[X1 | X1 > VaRα,x2(X),X2 ≤ x2], x2 ≥ F−1

X2(α).

Then, the bivariate lower orthant TVaR of X at a fixed level α is represented by the set oftwo curves

TVaRα(X) ={

TVaRα,x(X), xi ≥ F−1Xi

(α), i = 1, 2}

, (2.6)

such that x1 ≥ F−1X1

(α) and x2 ≥ F−1X2

(α) for all x = (x1, x2).

Bivariate lower orthant TVaR can be interpreted as a set composed of two curves, one foreach rv Xi (i = 1, 2). TVaRα,xj

(X) is the expectation of a rv under study Xi , if the related

rv Xj exceeds its lower orthant VaR curve at level α, but remains below xj ∈ [F−1Xj

(α),∞[,which provides an expression in terms of Xi . The latter condition ensures that both Xi andXj are at least protected up to the level α. The measure is taken for all possible values ofXj exceeding its α-level. Hence, for Xi (i = 1, 2), we obtain two curves for the bivariatelower orthant TVaR. This measure is particularly relevant in risk management.

The next result provides an interesting expression for the components of the lower orthantTVaR in terms of the components of Eq. 2.1.


Proposition 2.2 For all xi ≥ F−1Xi

(α), i = 1, 2, we have

TVaRα,xi(X) = 1

FXi(xi) − α

∫ FXi(xi )

α

VaRu,xi(X)du, i = 1, 2. (2.7)

Proof Note that VaRα,xj(X) represents the univariate VaR of

(Xi |Xj ≤ xj

)at level

αFXj

(xj ), namely

VaRα,xj(X) = VaR α

FXj(xj )

(Xi |Xj ≤ xj ), xj ≥ F−1Xj

(α).

Then, one has

TVaRα,xi(X) = E[Xj | Xj > VaRα,xi

(X),Xi ≤ xi]=

∫ ∞

VaRα,xi(X)

xj dFxi(xj )

1 − αFXi

(xi )

,

and substituting xj = F−1xi

(u),

TVaRα,xi(X) = 1

1 − αFXi

(xi )

∫ 1

αFXi

(xi )

F−1xi

(u) du

= 1

FXi(xi) − α

∫ FXi(xi )

α

F−1xi

(u

FXi(xi)

)du

= 1

FXi(xi) − α

∫ FXi(xi )

α

VaRu,xi(X)du.

One could easily expand the representation to higher dimensions. This new approachis as easy to understand as the univariate TVaR, since it consists in the integration of theunivariate VaR. Also, it allows to retrieve suitable and natural properties, as presented inSection 2.4.

We use �(FX1 , FX2) to represent the Frechet class, that is the class of joint cdf’s FX withfixed marginal cdf’s FX1 and FX2 . Let us denote by M(x1, x2) = min(FX1(x1), FX2(x2))

and W(x1, x2) = sup(FX1(x1) + FX2(x2) − 1, 0) to denote the Frechet upper and lowerbounds respectively. It is well known that W(x1, x2) ≤ FX(x1, x2) ≤ M(x1, x2) for allx1, x2 ∈ R. Also, we will use �(x1, x2) = FX1(x1)FX2(x2) to represent the bivariatecdf under the assumption of independence. We use the notation M(x1, x2), W(x1, x2) and�(x1, x2) to represent the bivariate cdf’s of XM = (X1, X2), XW = (X1, X2) and X� =(X1, X2), respectively.

Example 1 We obtain the following special cases when the bivariate random vector(X1, X2) is either comonototic, counter-comonotonic or independent respectively:

TVaRα,xi(XM) = 1

FXi(xi) − α

∫ FXi(xi )

α

VaRu(Xj )du, i, j = 1, 2 (i �= j), (2.8)

TVaRα,xi(XW ) = 1

FXi(xi) − α

∫ FXi(xi )

α

VaRu−FXi(xi )+1(Xj )du, i, j = 1, 2 (i �= j),

(2.9)


TVaRα,xi(X�) = 1

FXi(xi) − α

∫ FXi(xi )

α

VaR uFXi

(xi )(Xj )du, i, j = 1, 2 (i �= j). (2.10)

We apply the results above with the random vector (X1, X2) whose joint cdf is defined withindependent exponential marginals, i.e. Xi ∼Exponential(λi), i = 1, 2. Using Eq. 2.10, weobtain

TVaRα,xi(X�) = 1

FXi(xi) − α

[1

λj

(FXi(xi) − α)

{1 − ln

(FXi

(xi) − α

FXi(xi)

)}],

and

TVaRα,xi(XM) = 1

FXi(xi ) − α

(1

λj

[(1 − FXi

(xi )) ln(1 − FXi(xi )) − (1 − α) ln(1 − α) + (FXi

(xi ) − α)])

,

where FXi(xi) = (

1 − e−λixi), i, j = 1, 2 (i �= j).

The following results on the behavior of the α-curves xj �→ TVaRα,xj(X) when xj gets

near the upper support of the rv Xj , denoted uXj, and near F−1

Xj(α) = VaRα(Xj ), j = 1, 2

can be observed in Fig. 1.

Proposition 2.3 Let X = (X1, X2) be a pair of rv’s with cdf FX and marginal distributionsFX1 and FX2 . Assume that FX is continuous and strictly increasing. Then,

limxj →uXj

TVaRα,xj(X) = TVaRα(Xi), lim

xj →F−1Xj

(α)

TVaRα,xj(X) = uXi

, (2.11)

where uXirepresents the upper support of the rv Xi , i, j = 1, 2 (i �= j).

Proof As shown in Cossette et al. (2012), one has that limxj →uXj

VaRα,xj(X) = VaRα(Xi).

Combined with FXj(uXj

) = 1 and (2.7), one gets the result for the first part of Eq. 2.11.Also, lim

xj →F−1Xj

(α)

VaRα,xj(X) = uXi

so that integrating this constant on [α, FXi(xi)] also

results in uXi.

Example 2 Consider the random vector (X1, X2) whose joint cdf is defined with the Gum-bel copula with dependence parameter θ = 5 and marginals X1 ∼Exponential(λ = 1/100)

and X2 ∼Pareto(αP = 1.1, λP = 10). Let the confidence level be α = 99%. One canappreciate the limits of the lower orthant TVaR given in Proposition 2.3 in Fig. 1, and theshape of the lower orthant VaR and TVaR level curves. When x1 gets close to VaRα(X2),then TVaRα,x1

(X) approaches infinity, the upper support of X2. Also, when x1 gets closeto its upper support (infinity), TVaRα,x1

(X) approaches TVaRα(X2). The convexity of thiscurve is studied in Cossette et al. (2012), since it is expressed in terms of the bivariate lowerorthant VaR. The results have been obtained using numerical integration tools in MatLab.

2.3 Bivariate Upper Orthant TVaR

Definition 2.4 Bivariate upper othant TVaR is defined by

TVaRα,x(X) = ((x1, TVaRα,x1(X)

),(TVaRα,x2(X), x2

)),


Fig. 1 Graphical representationof VaRα,x1

(X), TVaRα,x1(X) and

TVaRα,x2(X)


withTVaRα,x1(X) = E[X2 | X2 > VaRα,x1(X),X1 ≥ x1], x1 ≥ F−1

X1(α),

andTVaRα,x2(X) = E[X1 | X1 > VaRα,x2(X),X2 ≥ x2], x2 ≥ F−1

X2(α).

Then, the bivariate upper orthant TVaR of X at level α is represented by the set

TVaRα(X) ={

TVaRα,x(X), xi ≤ F−1Xi

(α), i = 1, 2}

, (2.12)

such that xi ≤ F−1Xi

(α), i = 1, 2 for all x = (x1.x2).

Bivariate upper orthant TVaR is a set of two curves. Each curve represents the expectationof a rv under study Xi , if the related random variable Xj is below its upper orthant VaRcurve at level α. The measure is taken for all possible values of Xj smaller than its α-level.Hence, for Xi (i = 1, 2), we obtain two curves for the bivariate upper orthant TVaR. Thismeasure is particularly relevant in finance and actuarial science. It could represent jointthresholds of investments and pension funds.

An alternative expression for the components of Eq. 2.12 is derived next.

Proposition 2.5 For all xi ≤ F−1Xi

(α), i = 1, 2, we have

TVaRα,xi(X) = 1

1 − α

∫ 1

α

VaRu,xi(X)du, i = 1, 2.

Proof Based on the probabilities P(Xi ≥ VaRα,xj

(X),Xj ≥ xj

) = 1 − α and

P(Xi ≤ VaRα,xj

(X) | Xj ≥ xj

) = 1 − 1−α1−FXj

(xj ), one observes that VaRα,xj

(X) coincides

with the VaR of (Xi |Xj ≥ xj ) at levelα−FXj

(xj )

1−FXj(xj )

, that is

VaRα,xj(X) = VaR α−FXj

(xj )

1−FXj(xj )

(Xi |Xj ≥ xj ), xj ≤ F−1Xj

(α).

Let Fxi(xj ) be the cdf of the rv

(Xj |Xi ≥ xi

), i, j = 1, 2, i �= j . Then, one has

TVaRα,xj(X) = E[Xi | Xi > VaRα,xj

(X),Xj ≥ xj ]= 1 − FXj

(xj )

1 − α

∫ ∞

VaRα,xj(X)

xidFxj(xi), i, j = 1, 2 (i �= j).

Letting u = Fxj(xi) and v = (1 − FXj

(xj ))u + FXj(xj ), we get

TVaRα,xj(X) = 1 − FXj

(xj )

1 − α

∫ 1

α−FXj(xj )

1−FXj(xj )

F−1xj

(u)du = 1

1 − α

∫ 1

α

F−1xj

(v − FXj

(xj )

1 − FXj(xj )

)

dv

= 1

1 − α

∫ 1

α

VaRv,xj(X)dv, i, j = 1, 2 (i �= j).

Example 3 We obtain the following special cases for XM , XW and X�, respectively:

TVaRα,xi(XM) = TVaRα(Xj ), i, j = 1, 2 (i �= j), (2.13)


TVaRα,xi(XW ) = 1

1 − α

∫ 1

α

VaRu−FXi(xi )(Xj )du, i, j = 1, 2 (i �= j), (2.14)

and,

TVaRα,xj(X�) = 1

1 − α

∫ 1

α

VaR u−FXj(xj )

1−FXj(xj )

(Xi)du, i, j = 1, 2 (i �= j). (2.15)

In the same context as for Example 1, we obtain with Eq. 2.15 the following closed-formexpressions

TVaRα,xi(X�) = 1

1 − α

[α − 1

λj

{ln

(α − 1

FXi(xi) − 1

)− 1

}], i, j = 1, 2 (i �= j),

and

TVaRα,xi(XM) = VaRα(Xj ) + E[Xj ], i, j = 1, 2 (i �= j),

where VaRα(Xj ) = − 1λj

ln(1 − α) and E[Xj ] = 1λj

, FXi(xi) = (

1 − e−λixi), i, j =

1, 2 i �= j .

Proposition 2.6 Let X = (X1, X2) be a pair of rv’s with joint cdf FX and marginal cdf’sFX1 and FX2 . Assume that FX is continuous and strictly increasing. Then,

limxj →lXj

TVaRα,xj(X) = TVaRα(Xi), lim

xj →F−1Xj

(α)

TVaRα,xj(X) = lXi

, (2.16)

where lXirepresents the lower support of the rv Xi .

Proof The results are obtained similarly as for TVaRα,xi(X), from the limits of bivariate

lower and upper orthant VaR.

Example 4 Consider the random vector (X1, X2) defined in Example 2. Let the confidencelevel be α = 99 %. One can observe the limits given in Proposition 2.6 in Fig. 2, and theshape of the upper orthant level curve. From Fig. 2 C, one sees that when x1 gets close toVaRα(X2), then TVaRα,x1(X) approaches lX2 , the upper support of X2. Also, when x1 getsclose to its lower support, then TVaRα,x1(X) approaches TVaRα(X2). Note that the resultshave been obtained using numerical integration in MatLab.

2.4 Properties of Bivariate TVaR

2.4.1 Homogeneity and Translation Property

In this section, we discuss homogeneous and translation properties of TVaRα(X) andTVaRα(X). As for the univariate TVaR, TVaRα(X) and TVaRα(X) are homogeneous andinvariant under translation measure, as shown in the following proposition.

Proposition 2.7 Let X = (X1, X2) be a continuous random vector. Let φ1 and φ2 be realfunctions defined on the supports of X1 and X2 respectively.


Fig. 2 Graphical representation of VaRα,x1 (X), TVaRα,x1 (X) and TVaRα,x2 (X)


1. (Translation) For all c ∈ R2,

TVaRα,x+c(X + c) = TVaRα,x(X) + c, TVaRα,x+c(X + c) = TVaRα,x(X) + c.

2. (Homogeneity) If a ≥ 0, then

TVaRα,ax(aX) = aTVaRα,x(X), TVaRα,ax(aX) = aTVaRα,x(X).

Proof Results from expectation properties and corollaries 2.6 and 2.7 from Cossette et al.(2012).

2.4.2 Concordance Order

In this section, we show that the curves TVaRα(X) and TVaRα(X) are monotone withrespect to the concordance order.

Definition 2.8 Consider two pairs of risks X1 = (X1,1, X1,2) and X2 = (X2,1, X2,2) withjoint cdf’s FX1 and FX2 , respectively. The ordering between two bivariate lower and upperorthant TVaR’s of two pairs of risks is defined as follows :

TVaRα(X1) ≺ TVaRα(X2) ⇔ TVaRα,xi(X1) ≺ TVaRα,xi

(X2), xi ≥ F−1Xi

(α),

TVaRα(X1) ≺ TVaRα(X2) ⇔ TVaRα,xi(X1) ≺ TVaRα,xi

(X2), xi ≤ F−1Xi

(α),

for i = 1, 2.

The following corollary compares the impact of the dependence between the rv’s onthe bivariate lower and upper orthant TVaR. The concordance order allows to compare theimpact of the dependence between rv’s, without changing their marginal cdf’s. See e.g. Joe(1997), Muller and Stoyan (2002), and Denuit et al. (2005) or Shaked and Shanthikumar(2007) for details on the concordance order.

Definition 2.9 Given two random vectors X1 = (X1,1, X1,2) and X2 = (X2,1, X2,2), withjoint cdf’s FX1 and FX2 respectively, X1 is said to be more concordant than X2, denoted asX1 ≺co X2, if FX1(x, y) ≤ FX2(x, y), for all x1, x2 ∈ R.

Note that it is well known that XW ≺co X� ≺co XM and that XW ≺co X ≺co XM

(see references mentioned above). The following corollaries compare the impact of thedependence between the rv’s on the bivariate lower and upper orthant TVaR.

Corollary 2.10 Let X1 = (X1,1, X1,2) and X2 = (X2,1, X2,2) be two pairs of risks withthe same marginal cdf’s. Then, for X1 ≺co X2,

TVaRα(X2) ≺ TVaRα(X1), and TVaRα(X1) ≺ TVaRα(X2).

Proof The results are consequences of Lemma 2.17 of Cossette et al. (2012).

Remark 2.11 A random couple X1 being less concordant that another random couple X2will have higher weights in the tail distribution, leading to higher TVaRα(X1) curves.

Let us now derive bounds on TVaRα(X) as stated in the next result.


Corollary 2.12 Let X = (X1, X2) be a pair of risks with bivariate cdf FX ∈ �(FX1 , FX2).Then,

TVaRα(XM) ≺ TVaRα(X) ≺ TVaRα(XW ),

andTVaRα(XW ) ≺ TVaRα(X) ≺ TVaRα(XM).

Proof Since XW ≺co X ≺co XM , then by Corollary 2.10, we obtain the desired results.

Remark 2.13 If the random vector X is positive quadrant dependent (PQD), than theupper bound TVaRα(XW ) of TVaRα(X) and (respectively the lower bound TVaRα(X) ofTVaRα(XW )) is improved by TVaRα(X�) (respectively TVaRα(X�)).

3 Bivariate TVaR and Allocation

The objective of this section is to propose two different approaches to select couples fromthe bivariate lower and upper orthant TVaR curves that could be suitable for different pur-poses, such as capital requirements. In actuarial science or finance, one might be interestedin minimizing the aggregation of the components of a couple from the quantile curves. Mostdefinitions of multivariate risk measures suggest multivariate risk measures that result in aset of the same dimension as the number of rv’s. This couple does not minimize the amountsthat would be allocated to each risk, preserving a protection at level α for each component,which might not be desirable from an investor’s or manager’s point of view. The methodsproposed here consist in finding a set from an orthogonal projection of the curves on the uni-variate VaR of the components, or to the couples found in Proposition 2.3 and Proposition2.6, representing the limits of bivariate lower and upper orthant TVaR.

3.1 Approach 1

This method consists in selecting the capital allocation couple from the orthogonal projec-tion of the bivariate lower and upper orthant VaR and then computing the TVaR with thesevalues. As explained in Cossette et al. (2012), the allocation couple based on the orthogonalprojection allocation is obtained with the set that minimizes the distance between the curveVaRα,xi

(X), i = 1, 2 and the couple (VaRα(X1), VaRα(X2)). One obtains (x1, VaRα,x1(X))

for x1 ≥ F−1X1

(α), and equivalently (VaRα,x2(X), x2) for x2 ≥ F−1

X2(α), for which the sum

of the components of the bivariate lower orthant curve is the smallest.The first step of this approach is to obtain the values x∗

i , solutions to the followingoptimization problems

minxi≥F−1

Xi(α)

{(xi − VaRα(Xi))

2 + (VaRα,xi

(X) − VaRα(Xj ))2

}i, j = 1, 2, (i �= j).

(3.1)The second step is to compute the components of TVaRα(X) by

TVaRα,x∗i(X) = 1

α − FXi(x∗

i )

∫ FXi(x∗

i )

α

VaRu,x∗i(X)du, i = 1, 2.

Hence,

TVaRα,x∗(X) =(

TVaRα,x∗2(X), TVaRα,x∗

1(X)

).


One obtains the components of TVaRα,x∗(X) =(


1(X)

)similarly.

An illustration of this approach is provided in the following example.

Example 5 Consider the random vector (X1, X2) whose joint cdf is defined with the Frankcopula with dependence parameter θ and two marginals such that X1 ∼ Gamma(α1 =15, β1 = 0.5) and X2 ∼ Weibull(α2 = 2, β = 0.0295). We give results in Table 1 forthe couples obtained from Approach 1, using different dependence levels θ and risk lev-els α. These results have been obtained with optimization tools and numerical integrationtechniques in MatLab.

The lower orthant TVaR couples represent the optimal amounts to protect against theexpectations of worst losses, with respect to this approach, that should be kept aside topreserve a level α for both rv’s. It is higher than for a comonotonic relation, because wedo not assume that X1 can compensate for X2. It means that even though there is a pos-itive or negative dependence between the rv’s, the capital allocated for each rv has to behigher than when the risks are aggregated. Also, for the bivariate lower orthant TVaR, theallocation couples decrease as θ increases much more slowly than the upper orthant TVaRincreases, exhibiting greater dependence in the positive tail than in the negative tail of thisbivariate distribution. One sees that the allocation couples increase (decrease) as θ increasesfor the bivariate upper (lower) orthant TVaR. The upper orthant TVaR couples representthe optimal amounts to protect against the expectations of worst losses,that should be keptaside to preserve a survival level (1 − α) for X1 and X2, considering that each risk hasto preserve this survival level, with no possibility of aggregation. Also, one observes thatvalues from the lower orthant TVaR are always higher than those from the upper orthantTVaR.

3.2 Approach 2

This second method is an extension of the orthogonal projection allocation of the bivari-ate VaR presented in Cossette et al. (2012). Instead of considering the quantile curves, weare now interested in the minimization constraint applied to the bivariate lower and upperorthant TVaR.

Table 1 Allocation couples based on an orthogonal projection of bivariate VaR

θ α (TVaRα,x∗2(X), TVaRα,x∗

1(X)) (TVaRα,x∗

2(X), TVaRα,x∗

1(X))

−20 0.95 (22.2949, 55.9469) (52.5346, 70.9545)

0.99 (21.8323, 70.5542) (58.5444, 83.2069)

0.995 (22.9236, 75.9581) (60.9951, 87.9394)

0 0.95 (37.3730, 59.3089) (52.4537, 70.8971)

0.99 (40.6514, 68.4332) (58.5306, 83.1966)

0.995 (41.8736, 73.8770) (60.9885, 87.9343)

20 0.95 (46.1574, 65.5968) (50.9454, 69.4871)

0.99 (49.5760, 76.0230) (58.2668, 82.9881)

0.995 (51.2390, 80.1028) (60.8634, 87.8379)


Definition 3.1 Let Ci,α represent the α-curve obtained from TVaRα,xj(X), that is

Ci,α ={(

xj , TVaRα,xj(X)

), xj ≥ F−1

Xj(α)

}i, j = 1, 2, (i �= j).

For a fixed α, the method consists in finding the closest couples (x∗j , TVaRα,x∗

j(X))

from Ci,α to the couples (VaRα(Xj ), TVaRα(Xi)), for i, j = 1, 2, (i �= j).(VaRα(Xj ), TVaRα(Xi)) represent the limits of bivariate lower orthant TVaR. The criterionconsists in choosing the values x∗

i , i = 1, 2, solution to the minimization problems

minxi≥F−1

Xi(α)

{(xi − VaRα(Xi))

2 + (TVaRα,xi

(X) − TVaRα(Xj ))2

}i, j = 1, 2, (i �= j).

(3.2)The bivariate lower orthant TVaR optimal couple is given by

TVaRα,x∗(X) =(


1(X)

).

Definition 3.2 Let us define the α-curves obtained from TVaRα,xj(X), with respective values

xj , xj ≥ F−1Xj

(α), that is

Ci,α ={(

xj , TVaRα,xj(X)

), xj ≤ F−1

Xj(α)

}i, j = 1, 2, (i �= j).

Analogously, one can derive the bivariate upper orthant TVaR optimal couple. Hence, thecapital allocation couple (x∗

i , x∗j ) is obtained by solving the next optimization problems

minxj ≤F−1

Xj(α)

{(xj − VaRα(Xj )

)2 + (TVaRα(Xi) − TVaRα,xj

(X))2

}i, j = 1, 2, (i �= j).

Thus, the bivariate upper orthant TVaR optimal couple is given by

TVaRα,x∗(X) =(


1(X)

).

The Approach 2 is illustrated in the following example.

Example 6 With the same random couple as in Example 5, we obtain the capital allocationcouples displayed in Table 2.

The same conclusions as for the Approach 1 can be drawn concerning the increasing anddecreasing values of the allocation couples depending on the level α and the dependence level θ .

Table 2 Allocation couples from orthogonal projection of bivariate TVaR

θ α (TVaRα,x∗2(X), TVaRα,x∗

1(X)) (TVaRα,x∗

2(X), TVaRα,x∗

1(X))

−20 0.95 (21.9603, 58.4596) (52.7070, 71.2561)

0.99 (21.6741, 70.3947) (58.6475, 83.3926)

0.995 (22.5260, 75.1945) (61.0831, 88.0969)

0 0.95 (36.2199, 58.0645) (52.6213, 71.1950)

0.99 (39.5128, 68.3097) (58.6352, 83.3813)

0.995 (41.0402, 72.4334) (61.0828, 88.0863)

20 0.95 (45.5256, 64.7971) (51.1100, 69.8482)

0.99 (48.9504, 75.3375) (58.3644, 83.1665)

0.995 (50.5651, 78.9820) (60.9504, 87.9907)


Moreover, the values of (TVaRα,x∗2(X), TVaRα,x∗

1(X)) from Approach 2 are always higher than

those from Approach 1, because the latter selects the couple providing the higher sum of thecomponents from the threshold curves.

4 Bivariate TVaR of Sums of Random Pairs and Illustrations

4.1 Context

In this section, we investigate the behavior of the bivariate lower and upper orthant TVaR of sumsof random pairs. Such sums may be encountered in several situations in actuarial science, suchas dependent classes of pooled risks in an insurance company. Here, we consider the bivariaterv’s (S1, S2), representing aggregate risks, that are defined as follows:

S =(

S1

S2

)=

n∑

i=1

(Xi

Yi

). (4.1)

Note that S1 and S2 can represent the aggregate amount of claims for lines of business 1 and 2respectively, and Xj and Yj , j = 1, ..., n can represent risks within each line of business. In whatfollows, we establish desirable properties of the upper and lower orthant TVaR of S.

For the remaining of this section, X1, . . . , Xn (respectively Y1, . . . , Yn) are continuous rv’swith cdf’s FX1 , . . . , FXn (respectively GY1 , . . . , GYn ). Let us denote by FS1 and FS2 the cdf’s ofthe univariate aggregated sums S1 = ∑n

i=1 Xi and S2 = ∑nj=1 Yj .

4.2 Subadditivity in Distributions

An interesting characteristic of the univariate TVaR is its subadditivity, i.e. for a fixed level α,we have

TVaRα(S1) ≤n∑

i=1

TVaRα(Xi) and TVaRα(S2) ≤n∑

j=1

TVaRα(Yj ).

We will show next the subadditivity in distribution for the bivariate lower and upper orthantTVaR. By subadditivity in distribution, we mean that the random couples are considered to becomposed of a component X (respectively Y ) and a replica of Y (respectively X), denoted Y

(respectively X), that has the same distribution as Y (respectively X), denoted Xi =d Xi andYi =d Yi respectively, i = 1, . . . , n. Also note that the random vectors (Xi , Yi) and (S1, Yi) havethe same dependence structure.

In what follows, we need the next lemma.

Lemma 4.1 Let X be a rv with survival function FX(x) = P(X > x), and let A be an eventsuch that P(A) = FX(x). Then,

E (X|A) ≤ E (X|X > x) .

Proof See Ruschendorf (1982).

Proposition 4.2 Define the rv’s Xi such that Xi = F−1Xi

◦ FS1(S1) and Yi such that Yi =G−1

Yi◦FS2(S2), where (F ◦G)(x) = F(G(x)) is the composite function. Note that Xi =d Xi and


Yi =d Yi , i = 1, . . . , n, and the random vectors (Xi, Yi ) and (Xi, S2) have the same dependencestructure. The upper and lower orthant TVaR are subadditive in distribution.

TVaRα,s2(S) ≤

n∑

i=1

TVaRα,yi(Xi, Yi ), TVaRα,s2(S) ≤

n∑

i=1

TVaRα,yi(Xi, Yi ), (4.2)

where s2 = ∑ni=1 yi = ∑n

i=1 G−1Yi

◦ FS2(s2), and

TVaRα,s1(S) ≤

n∑

i=1

TVaRα,xi(Xi , Yi), TVaRα,s1(S) ≤

n∑

i=1

TVaRα,xi(Xi , Yi), (4.3)

where s1 = ∑ni=1 xi = ∑n

i=1 F−1Xi

◦ FS1(s1).

Proof Because the random vectors (Xi, Yi ) and (Xi, S2) have the same dependence structure,we have

TVaRα,s2(S) = E

(S1|S2 > VaRα,s2

(S), S2 ≤ s2)

=n∑

i=1

E

(Xi |S2 > VaRα,s2

(X), Yi ≤ yi

).

Also,

P(S2 > VaRα,s2

(S)|S2 ≤ s2) =

P

(Xi > VaRα,yi

(Xi, Yi ), Yi ≤ yi

)

FS2(s2)

= (1 − α)/FS2(s2).

Thus, from Lemma 4.1, one deduces

E

(Xi |S2 > VaRα,s2

(S), Yi ≤ yi

)≤ E

(Xi |Xi > VaRα,yi

(Xi, Yi ), Yi ≤ yi

)

= TVaRα,yi(Xi, Yi ).

Analogously, one can get the second part of inequalities (4.2) and (4.3).

Proposition 4.2 states that without any assumption on the marginals and joint cdf’s, the bivari-ate lower and upper orthant TVaR curves of sums of n risks are always smaller than the sum ofthe curves for each random couple of risks i, i = 1, ..., n.

4.3 Bivariate TVaR and Comonotonicity

The distribution of aggregate risks is not a trivial issue. When explicit formulas are not available,it is possible to use simulation or numerical algorithms. Nevertheless, very interesting results,simplifying the calculation steps, are available when the rv’s are comonotonic, meaning a perfectdependence structure.

Proposition 4.3 For any dependence relation between (X1, ..., Xn) and (Y1, ..., Yn), we havethe following results:

1. If (X1, . . . , Xn) is comonotonic and if no assumption is made on the dependence within(Y1, . . . , Yn), then

TVaRα,s1(S) ≤

n∑

i=1

TVaRα,xi(Xi, Yi), and TVaRα,s1(S) ≤

n∑

i=1

TVaRα,xi(Xi, Yi),

where s1 = ∑ni=1 xi = ∑n

i=1 F−1Xi

◦ FS1(s1).


2. If (Y1, . . . , Yn) is comonotonic and if no assumption is made on the dependence within(X1, . . . , Xn), then

TVaRα,s2(S) ≤

n∑

i=1

TVaRα,yi(Xi, Yi), and TVaRα,s2(S) ≤

n∑

i=1

TVaRα,yi(Xi, Yi),


i=1 G−1Yi

◦ FS2(s2).

Proof One easily sees that if (X1, . . . , Xn) (respectively Y1, . . . , Yn) are comonotonic, thenXi = Xi (respectively Yi = Yi ), which leads to the desired result.

By combining items 1. and 2. of Proposition 4.3, we obtain the following proposition.

Proposition 4.4 Let (X1, . . . , Xn) be comonotonic and (Y1, . . . , Yn) be also comonotonic.Moreover, no assumption is made on the dependence between (X1, . . . , Xn) and (Y1, . . . , Yn).Then,

TVaRα,s2(S) =

n∑

i=1

TVaRα,yi(Xi, Yi) and TVaRα,s2(S) =

n∑

i=1

TVaRα,yi(Xi, Yi), (4.4)

TVaRα,s1(S) =

n∑

i=1

TVaRα,xi(Xi, Yi) and TVaRα,s1(S) =

n∑

i=1

TVaRα,xi(Xi, Yi), (4.5)


i=1 G−1Yi

◦ FS2(s2), s1 = ∑ni=1 xi = F−1

Xi◦ FS1(s1).

Proof Let us define F−1S1

(u) = ∑ni=1 F−1

Xi(u) and F−1

S2(u) = ∑n

i=1 G−1Yi

(u). Since(X1, . . . , Xn) is comonotonic and (Y1, . . . , Yn) is comonotonic, then there exists a uniform ran-dom vector (U1, U2) such that S1 = F−1

S1(U1) and S2 = F−1

S1(U2). Given that FXi

and GYiare

increasing functions and given Lemma 4.1, one obtains

TVaRα,s2(S) = 1

FS2(s2) − α

∫ FS2 (s2)

α

VaRu,s2

(F−1

S1(U1), F

−1S2

(U2))

du

= 1

FS2(s2) − α

∫ FS2 (s2)

α

F−1S1

(VaRu,FS2 (s2)(U1, U2)

)du

=n∑

i=1

1

GYi(yi) − α

∫ GYi(yi )

α

VaRu,yi

(F−1

Xi(U1), G

−1Yi

(U2))

du

=n∑

i=1

1

GYi(yi) − α

∫ GYi(yi )

α

VaRu,yi(Xi, Yi)du

=n∑

i=1

TVaRα,yi(Xi, Yi).

We obtain proofs of the second equality of Eqs. 4.4 and 4.5 using analogous arguments.

Proposition 4.4 states that for a portfolio composed of two classes of comonotonic risks, thesum of all bivariate lower and upper orthant TVaR curves for each risk provides the bivariatelower and upper orthant TVaR curves for each class of risks.

Combining Proposition 4.2 with Corollary 2.12, we get an upper bound on TVaRα,s(S)

(TVaRα,s(S)) that only depends on the marginal cdf’s as shown next.


Corollary 4.5 For S1 and S2, we have that

TVaRα,si(S) ≤

n∑

i=1

TVaRα,si(XW,i , YW,i),

TVaRα,si (S) ≤n∑

i=1

TVaRα,si (XM,i , YM,i), i = 1, 2,

where

TVaRα,si(XW,i , YW,i) = 1

FSi(si) − α

∫ FSi(si )

α

VaRu−FSi(si )+1(Xj )du, i, j = 1, 2 (i �= j),

TVaRα,s2(XM,i, YM,i) = 1

1 − α

∫ 1

α

VaRu(Xi)du, i = 1, 2.

Proof Since Yi and Yi are identically distributed, then from Proposition 4.2 and Corollary 2.12,one has

TVaRα,s2(S1, S2) ≤

n∑

i=1

TVaRα,yi(Xi, Yi )

=n∑

i=1

1

GYi(yi ) − α

∫ GYi(yi )

α

VaRu−GYi(yi )+1(Xi)du

= 1

FS2(s2) − α

∫ FS2 (s2)

α

VaRu−FS2 (s2)+1(Xi)du.

One can proceed analogously for TVaRα,s2(S1, S2) with the second part of Corollary 2.12.

Corollary 4.5 is interesting because it allows to obtain upper bounds for the bivariate lowerand upper orthant TVaR of S1 and S2 when the dependence structure between the random couples(X1, Y1), ..., (Xn, Yn) is unknown.

4.4 Capital Allocation Couples

In many practical situations, it may be useful to consider the capital allocation set of the bivariateTVaR of (S1, S2) described in Section 3, instead of the curves. When the dependence structure ofthe random vector (S1, S2) is not available, one can try to find allocation couples from the upperbounds for the bivariate lower and upper TVaR of (S1, S2).

Here, we propose a procedure to obtain allocation couples for the upper bounds given inCorollary 4.5. For that purpose, let m(1) and m(2) be functions defined by

m(1)(u) =n∑

i=1

1

u − α

∫ u

α

VaRv−u+1(Xi)dv,

m(2)(u) =n∑

i=1

1

u − α

∫ u

α

VaRv−u+1(Yi)dv,

for all u ∈ [α, 1].Clearly, one sees that the upper bounds for TVaRα,s2

(S1, S2) and TVaRα,s1(S1, S2) are

respectively given by m(1)(FS2(s2)) and m(2)(FS1(s1)). This allows us to obtain the capital


allocation points of these upper bounds without knowing the values of FS2(s2) and FS1(s1).Since

limu→1

m(1)(u) =n∑

i=1

TVaRα(Xi), limu→1

m(2)(u) =n∑

i=1

TVaRα(Yi),

the capital allocation set of the upper bounds of(TVaRα,s2

(S1, S2), TVaRα,s1(S1, S2)

)is given

by the pair(m(1)(u∗

1), m(2)(u∗

2)). The latter can be used to appreciate the bivariate lower orthant

TVaR. The values of u∗1 and u∗

2 are found as follows:

u∗1 = min

u≥α

⎧⎨

⎩(u − α)2 +

(

m(2)(u) −n∑

i=1

TVaRα(Yi)

)2⎫⎬

⎭, (4.6)

and

u∗2 = min

u≥α

⎧⎨

⎩(u − α)2 +

(

m(1)(u) −n∑

i=1

TVaRα(Xi)

)2⎫⎬

⎭. (4.7)

It is also possible to obtain an allocation couple from the lower bound of the upper bivariateTVaR, using the following functions and considering the minimization equations with

m(1)(u) =n∑

i=1

1

1 − α

∫ 1

α

VaRv−u(Xi)dv, m(2)(u) =n∑

i=1

1

1 − α

∫ 1

α

VaRv−u(Yi)dv,

and

limu→0

m(1)(u) =n∑

i=1

TVaRα(Xi), limu→0

m(2)(u) =n∑

i=1

TVaRα(Yi).

The values u∗1 and u∗

2 of these lower bounds may be obtained by solving respectively theminimization problems

u∗1 = min

u≤α

⎧⎨

⎩(u − α)2 +

(

m(2)(u) −n∑

i=1

TVaRα(Yi)

)2⎫⎬

⎭,

and

u∗2 = min

u≤α

⎧⎨

⎩(u − α)2 +

(

m(1)(u) −n∑

i=1

TVaRα(Xi)

)2⎫⎬

⎭.

This method is very useful considering that only the marginal cdf’s are necessary. Oftenfor each aggregate class, explicit distributions are not available. In such a case, one can useapproximation techniques.

4.5 Applications

The following section presents two applications of the bivariate lower orthant TVaR. Both situa-tions are from an insurer’s point of view and aim to establish capital allocation requirements fortwo different classes of aggregated risks. Moreover, these examples motivate the use of multivari-ate risk measures by protecting each class individually, and considering the dependence betweenthem when allocating capital. The first application considers sums of comonotonic rv’s withintwo dependent classes. The second one illustrates allocation sets obtained from the upper boundof the lower orthant TVaR. This bound, presented in Section 4, can be used when the dependencestructure of the overall portfolio is unknown.


In both illustrations, we want to highlight the fact that each class is protected at level α.The suggested bivariate lower orthant TVaR provides information on the tail of the joint cdf ofthe risks, and on tails of the distributions of each business line or classes of risks of a globalportfolio in addition to considering the dependence between them. One can also clearly interpretthe allocation couples resulting from Section 3, in addition to the information provided by theoverall level curves.

The second example provides useful bounds that only rely on the marginal cdf’s, which isvery convenient for computation. This bound is relevant in the case where the dependence withinand between each risk of a bivariate portfolio is unknown.

Note that the applications could easily be extended to higher dimensions.

4.5.1 Capital Allocation for Bivariate Sums of Couples of rv’s

We consider two pairs of rv’s (X1, Y1) and (X2, Y2), where each set (Xi, Yi), i = 1, 2 couldrepresent two different investment products offered by a financial institution. Each product offersthe maximum between a preset return, and a variable return, in terms of two dependent assets RA

and RB . In this case, X1 offers a smaller guaranteed rate than X2. Let ϕ1 = 0.89 and ϕ2 = 0.93be preset participation rates and

X1 = 100 × max(e0.01×5, ϕ1e

RA

), X2 = 100 × max

(e0.02×5, ϕ2e

RA

),

Y1 = 100 × max(e0.01×5, ϕ1e

RB

), Y2 = 100 × max

(e0.02×5, ϕ2e

RB

),

The joint cdf of (RA, RB) is defined with the Frank copula

Cθ (u1, u2) = − 1

θln

{

1 +(e−θu1 − 1

) (e−θu2 − 1

)

(e−θ − 1

)

}

with a dependence parameter θ = 10.79855 and normal marginals where

RA ∼ Normal(μRA

= 5 × 0.0459, σ 2RA

= 5 × 0.18952)

andRB ∼ Normal

(μRB

= 5 × 0.0534, σ 2RB

= 5 × 0.26482)

.

Clearly, X1 and X2 are comonotonic as well as Y1 and Y2. We define the pair of rv’s (S1, S2) withS1 = X1 + X2 and S2 = Y1 + Y2. Note that VaR0.99(S1) = 613.4366, VaR0.99(S2) = 942.5839TVaR0.99(S1) = 714.8063 and TVaR0.99(S2) = 1173.9345.

The allocation couple obtained from the approach presented in Section 3.1 corresponds tothe couple obtained for Approach 1 in Table 3. The orthogonal projection sets given in Eq. 3.2correspond to the allocated values for Approach 2 of Table 3. One sees that the allocation valuesfor each class based on Approach 1 (Section 3.1) has lower, but similar, capital allocation valuesthan for Approach 2 (Section 3.2), which should always be the case.

4.5.2 Capital Allocation Bounds for Sums of Random Pairs

In this application, two business lines of an insurance portfolio are considered. It is assumedthat each business line is composed of two risks and their allocated loss adjustment expenses

Table 3 Bivariate allocation setsApproach

(TVaR0.99,s∗

2(S), TVaR0.99,s∗

1(S)

)

1 (818.8619, 1270.7121)

2 (819.6650, 1271.1062)


Table 4 Allocation couples atlevels 95 %, 99 % and 99.5 % α (TVaRα,x∗

2(X), TVaRα,x∗

1(X))

95 % (780.5106, 1115.7060)

99 % (945.4633, 1537.7790)

995 % (1034.7970, 1734.9880)

(ALAE). The objective is to illustrate the bivariate lower orthant curves and allocation sets fromSection 4.4. Let S = (S1, S2)

t , be defined as in Eq. 4.1. Let us assume X1 ∼Lognormal(μ11 =4.7179, σ 2

11 = 0.1795), X2 ∼ Lognormal(μ12 = 4.7619, σ 212 = 0.1795), Y1 ∼

Lognormal(μ21 = 0.2670, σ 221 = 0.3507) and Y2 ∼ Lognormal(μ22 = 4.800, σ 2

22 = 0.3507).Also, for the ALAE variable, we assume that X3 and Y3 are exponentially distributed with param-eters λ1 = 0.1 and λ2 = 0.15 respectively. In this application we suppose that the dependencewithin and between each class is unknown.

The objective is to establish the capital allocation for S1 and S2, at probability level 99 % foreach business line, using the upper bound of the lower orthant TVaR, as provided by Eqs. 4.6 and4.7.

The upper bound for the lower orthant TVaR is obtained by assuming a comonotonic depen-dence structure within each class and counter-comonotonicity between the classes. As mentionedin Section 4.4, in such a case, the solution only depends on the marginal cdf’s of every single riskof the overall portfolio.

Results obtained with Eqs. 4.6 and 4.7 are presented in Table 4.

Ackowledgments and Compliance with Ethical Statement We would like to thank the two anonymousreferees for their useful comments. This work was partially supported by the Natural Sciences and Engineer-ing Research Council of Canada (Cossette 054993, Marceau 053934, Mesfioui 261968), the Fonds quebecoisde la recherche sur la nature et les technologies (Mailhot 138773), the Chaire en actuariat de l’UniversiteLaval (Cossette, Mailhot and Marceau; FO502323) and Concordia University (Mailhot V00654). No humanparticipants and/or animals were involved in this research. Conflict of Interest: The authors declare that theyhave no conflict of interest. The authors (Cossette, Mailhot, Marceau and Mesfioui) consent to the submissionof this research paper and completed the consent form.

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