# Modeling the Dependence of Losses of a Financial Portfolio Using Nested Archimedean Copulas

Authors: Wendkouni Yaméogo; Diakarya Barro
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In financial analysis, stochastic models are more and more used to estimate potential outcomes in a risky framework. This paper proposes an approach of modeling the dependence of losses on securities, and the potential loss of the portfolio is divided into sectors each including two subsectors. The Weibull model is used to describe the stochastic behavior of the default time while a nested class of Archimedean copulas at three levels is used to model the maximum of the value at risk of the portfolio.


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## Body

## 1. Introduction

Stochastic models are more and more used in financial analysis, and they allow institutions to include uncertainties in their estimates, which account for situations where outcomes are risky. In particular, the copula functions are used to capture the dependence of the price movement of financial instruments. The word “copulas” was first introduced by Nelsen [1] in a pioneering work, under the well-known theorem which bears his name and which clarifies that for any n-dimensional distribution F=F1,…,Fn with continuous marginal, there exists a unique copula CF mapping ℝn to 0,1n such that x1,…,xnT⟶F1x1,…,FnxnT with(1)Fx1,…,xn=CFF1x1;…;Fnxn.Archimedean copulas form a family of copulas used in the construction of multivariate distributions involving one-dimensional generator functions.Well defined by Savu and Trede [2], the hierarchical Archimedean copulas (HACs) offer more flexibility than the classic Archimedean family. In financial modeling, the HACs allow, at the first level, the modeling of the joint dependence of firms including similar sectors. At the second level, the Archimedean copulas, subcopulas of the inners, can beused to combine the sectors. HACs therefore make it possible to model the dependence between a large number of assets in a flexible and intuitive way. Particularly, Gumbel copula, which belongs both to the Archimedean class and the extremal one, is a suitable model used in financial applications (according to Hofert [3]), and it is easy to simulate in hierarchical version (according to McNeil [4]).The HACs are used in the management of default risk. This default is triggered by a formal credit specified in the contract. This event can be the bankruptcy of the company, a default, or the restructuring of its debt. For the evaluation of these, products derived from credits of intensity models are used. The existing intensity models are constant intensityλt=λ by pieces such as(2)λt=∑k=1qλk1Tk−1<t≤Tk.In this case, the default time follows the exponential distribution with parameterλ intensity governed by an extended CIR process. In the domain of quantitative finance, large portfolio managersneed stochastics models whichcombine both nested Archimedean copulas and extreme valueslaws.In our study, we consider the valuation of credit derivatives that the default intensity also called the hazard rate is given by(3)hθx,k,λ=kλx−θλk−1,that is, the default time follows Weibull’s law (with λ>0 and k≥1 ), whose law covers both a whole family of law.The major contribution of this paper is the study of the dependence of losses of securities in a large portfolio and the maximum of values at risk of the same portfolio. We show that thhe default time of eachasset follows theWeibull law by using a partially nested Archimedean copula at three levels. The rest of our paper is organized as follows. Section2 is devoted to the preliminaries of the study where the main properties of both the classical and nested Archimedean copulas and an overview of Weibull negative distribution are given. In Section 3, we study the portfolio losses and default time modeling while Section 4 deals with the nested Archimedean copulas and losses. In Section 5, sampling the hightest values at risk is clarified.


## 2. Materials and Methods

In this section, we give important definitions and useful properties on Archimedean copulas of extreme values and the value at risk. These results are necessary for our approach. We refer the readers to Schonbucher [5], Capéraà et al. [6], and Cherubini et al. [7] for the applications of Archimedean copulas to different degrees of quantitative finance, Savu and Trede [2] or Hofert and Scherer [8] for hierarchical Archimedean copulas, Veysseyre [9] for details on using Weibull law, and Tyrrell Rockafellar and Uryasev [10] and Roncalli [11] for the value at risk.


### 2.1. An Overview of Archimedean and Extremal Copulas

Archimedean copulas play an important role in practical applications.Definition 1.
Ann-dimensional copula C is an Archimedean one if there exists a continuous, strictly decreasing, and convex function φ:0,1⟶0,+∞, completely monotone on 0;∞, and n−2 times derivativable such as(4)−1kφ−1kt≥0,for allk=1,2,…,n−2,called the Archimedean generator function of C, such that, for all u1,…,un∈0,1n,(5)Cu1,…,un=φφ−1u1+⋯+φ−1un,ui∈0,1nsince∑i=1nφui≤φ0,0,else,where the generalized inverse of the Archimedean generator is given by φ−1y=inft∈0,1:φt≤y. The generator must be chosen from class C2 (a C2 function has both a continuous first derivative and a continuous second derivative) so that φ1=0,φ′u≤0 et φ″u>0.
Even if Archimedean copulas simulate tails of distributions, their use becomes restrictive for copulas of high-dimensional analysis. Moreover, copula depends on only one parameter which applies to all the steps. The use of hierarchical copulas allows to bypass the restrictions caused by Archimedean copulas. There are partially nested Archimedean copulas and fully nested Archimedean copulas. Here, we use partially nested Archimedean copulas. For example, by coupling margins of the extreme value theory (EVT) with the multivariate Gumbel copula, we obtain the distribution of multidimensional extreme value according to Sklar theorem.(6)Gx1,…,xn=exp−∑i=1n−lnG∗xiθ1/θ,where G∗ is a suitable normalized limit of law of maxima such that(7)G∗xi=limn⟶+∞PMn−bnan≤xi,with an>0 and bn∈ℝ being normalizing coefficients and Mn being the statistic of maxima. An extreme value copula verifies the following max-stability relation:(8)Cu1k,…,unk=Cku1,…,un,for all ui∈0,1 for i=1,…,n.


### 2.2. An Overview of Weibull Negative Distribution

In stochastic analysis, the Weibull law is a continuous probabilitydistribution within the extreme values family as well as the Gumbel and the Frechet laws. With three parameters (forx>0 ), this law is characterized as follows:(9)fθx;k;λ=kλx−θλk−1exp−x−θλk,the density function,Fθx;k;λ=1−exp−x−θλk,the distribution function,Sθx;k;λ=exp−x−θλk,the survival distribution function,where θ=minx, k>0 is the shape parameter, and λ>0 is the scale parameter.We have below graphical representations of the functions of density and distribution of Weibull law for same values of parameters. Figure1is taken from [12].Figure 1
Density function (a) and distribution function (b) of Weibull.
(a)(b)One of the uses of Weibull distribution is the extreme value theory where the asymptotic behavior of distributions, mainly for component-wise maxima of laws, is appropriately normalized. Particularly, in one-dimensional extreme framework, the well-known Fisher–Tippett–Gnedenko theorem [13] shows that the three following distributions can characterize these asymptotic behaviors:(10)Λx=exp−exp−x;∞<x<∞,Gumbel distribution,Φθx=exp−x−θ;x>0;θ>0,Fréchet distribution,Ψθx=exp−−xθ;x>0,θ>0,Weibull distribution.These one-dimensional extreme values distributions can be unified by the three real-valued parametricdistribution, the so-called generalized extreme values (GEV) model, defined by(11)GEVμ,σ,ξy=exp−1+ξy−μσ+−1/ξ,ifξ≠0,where u+=maxu,0 and μ∈ℝ, σ>0, and ξ∈ℝ are, respectively, the parameters of location.Notice that instead of the generalized type of marginal, it is not restrictive to consider any of the three types given by the relation (10). Indeed, these three models are linked each to other by the following functional transformations, even in spatial context:(12)Y∼Φθ⇔lnYθ∼Λ⇔−1Y∼Ψθ⇔Z=μx+σxξxYξx−1∼GEVσx,μx,ξx.


### 2.3. An Overview of the Value at Risk

In actuarial and financial sciences, the value at risk (VaR) quantifies numerically the size of the loss for which there is a low probability of being exceeded. It is characterized by the confidence level, the time horizon chosen, and the distribution of profit or loss.LetX1,…,Xn be n random independent variables, identically distributed with common distribution function F. A simple way to study the behavior of extreme events is to consider the random variable Mn=maxX1,…,Xn representing the largest observed loss out of the n observed losses X1,…,Xn (we agree that the loss is a positive number). So, for all realization x,(13)PMn≤x=P∩i=1nXi≤x=Fx,…,x=CF1x,…,F1x.The value at risk (VaR) is a risk measure mainly used to measure the large portfolio market risk. We call value at risk with a confidence levelα∈0;1 the lower α -quantile:(14)VaRX;α=infx,FXx≥α=FX−1α,where FX−1 is the right continuous inverse of FX.Loyara and Barro [14] showed that the VaR is also intrinsically linked in its expression to this function, making it possible to bridge the copula function with the VaR. More generally, in multivariate studies, for a random vector X satisfying the regularity conditions, one defines the multidimensional VaR at probability level α by(15)VaRαX=EX|X∈∂Lα,where ∂Lα is the boundary of the α−level set of Ft, the univariate component of the vector.The three following sections provide the main results of our study.


## 2.1. An Overview of Archimedean and Extremal Copulas

Archimedean copulas play an important role in practical applications.Definition 1.
Ann-dimensional copula C is an Archimedean one if there exists a continuous, strictly decreasing, and convex function φ:0,1⟶0,+∞, completely monotone on 0;∞, and n−2 times derivativable such as(4)−1kφ−1kt≥0,for allk=1,2,…,n−2,called the Archimedean generator function of C, such that, for all u1,…,un∈0,1n,(5)Cu1,…,un=φφ−1u1+⋯+φ−1un,ui∈0,1nsince∑i=1nφui≤φ0,0,else,where the generalized inverse of the Archimedean generator is given by φ−1y=inft∈0,1:φt≤y. The generator must be chosen from class C2 (a C2 function has both a continuous first derivative and a continuous second derivative) so that φ1=0,φ′u≤0 et φ″u>0.
Even if Archimedean copulas simulate tails of distributions, their use becomes restrictive for copulas of high-dimensional analysis. Moreover, copula depends on only one parameter which applies to all the steps. The use of hierarchical copulas allows to bypass the restrictions caused by Archimedean copulas. There are partially nested Archimedean copulas and fully nested Archimedean copulas. Here, we use partially nested Archimedean copulas. For example, by coupling margins of the extreme value theory (EVT) with the multivariate Gumbel copula, we obtain the distribution of multidimensional extreme value according to Sklar theorem.(6)Gx1,…,xn=exp−∑i=1n−lnG∗xiθ1/θ,where G∗ is a suitable normalized limit of law of maxima such that(7)G∗xi=limn⟶+∞PMn−bnan≤xi,with an>0 and bn∈ℝ being normalizing coefficients and Mn being the statistic of maxima. An extreme value copula verifies the following max-stability relation:(8)Cu1k,…,unk=Cku1,…,un,for all ui∈0,1 for i=1,…,n.


## 2.2. An Overview of Weibull Negative Distribution

In stochastic analysis, the Weibull law is a continuous probabilitydistribution within the extreme values family as well as the Gumbel and the Frechet laws. With three parameters (forx>0 ), this law is characterized as follows:(9)fθx;k;λ=kλx−θλk−1exp−x−θλk,the density function,Fθx;k;λ=1−exp−x−θλk,the distribution function,Sθx;k;λ=exp−x−θλk,the survival distribution function,where θ=minx, k>0 is the shape parameter, and λ>0 is the scale parameter.We have below graphical representations of the functions of density and distribution of Weibull law for same values of parameters. Figure1is taken from [12].Figure 1
Density function (a) and distribution function (b) of Weibull.
(a)(b)One of the uses of Weibull distribution is the extreme value theory where the asymptotic behavior of distributions, mainly for component-wise maxima of laws, is appropriately normalized. Particularly, in one-dimensional extreme framework, the well-known Fisher–Tippett–Gnedenko theorem [13] shows that the three following distributions can characterize these asymptotic behaviors:(10)Λx=exp−exp−x;∞<x<∞,Gumbel distribution,Φθx=exp−x−θ;x>0;θ>0,Fréchet distribution,Ψθx=exp−−xθ;x>0,θ>0,Weibull distribution.These one-dimensional extreme values distributions can be unified by the three real-valued parametricdistribution, the so-called generalized extreme values (GEV) model, defined by(11)GEVμ,σ,ξy=exp−1+ξy−μσ+−1/ξ,ifξ≠0,where u+=maxu,0 and μ∈ℝ, σ>0, and ξ∈ℝ are, respectively, the parameters of location.Notice that instead of the generalized type of marginal, it is not restrictive to consider any of the three types given by the relation (10). Indeed, these three models are linked each to other by the following functional transformations, even in spatial context:(12)Y∼Φθ⇔lnYθ∼Λ⇔−1Y∼Ψθ⇔Z=μx+σxξxYξx−1∼GEVσx,μx,ξx.


## 2.3. An Overview of the Value at Risk

In actuarial and financial sciences, the value at risk (VaR) quantifies numerically the size of the loss for which there is a low probability of being exceeded. It is characterized by the confidence level, the time horizon chosen, and the distribution of profit or loss.LetX1,…,Xn be n random independent variables, identically distributed with common distribution function F. A simple way to study the behavior of extreme events is to consider the random variable Mn=maxX1,…,Xn representing the largest observed loss out of the n observed losses X1,…,Xn (we agree that the loss is a positive number). So, for all realization x,(13)PMn≤x=P∩i=1nXi≤x=Fx,…,x=CF1x,…,F1x.The value at risk (VaR) is a risk measure mainly used to measure the large portfolio market risk. We call value at risk with a confidence levelα∈0;1 the lower α -quantile:(14)VaRX;α=infx,FXx≥α=FX−1α,where FX−1 is the right continuous inverse of FX.Loyara and Barro [14] showed that the VaR is also intrinsically linked in its expression to this function, making it possible to bridge the copula function with the VaR. More generally, in multivariate studies, for a random vector X satisfying the regularity conditions, one defines the multidimensional VaR at probability level α by(15)VaRαX=EX|X∈∂Lα,where ∂Lα is the boundary of the α−level set of Ft, the univariate component of the vector.The three following sections provide the main results of our study.


## 3. Portfolio Losses and Default Time Modeling

Note that in this study, the modeled portfolio contains assets with credit risk. These companies are assigned to one of the industry sectors according to a certain attribute. Sector is further divided into two subsectors according to an attribute. The default times of these companies are denoted asτi and are modeled using Weibull law for the parameter θ=0. The default time is modeled as a downtime, with a distribution function(16)Ft=Pτ≤t=1−St,where t≥0 and St is a survival function used to calculate the probability that a time default τ occurs beyond a certain time interval t. The hazard rate ht is defined by hk,λt=ft/1−Ft.(17)hk,λt=kλtλk−1is increasing k≥1. The default time for an element i of the portfolio is defined as the first time where the expression exp−∫0thisds falls below the stochastic trigger threshold represented by the variable Ui.(18)τi=inft:exp−∫0thisds≤Ui=S−1Ui=F−11−Ui,where Ui is uniformly distributed over 0;1. In what follows, by coupling Weibull marginal distribution with Gumbel one which is also an extremal copula, we obtain this result.Proposition 1.
LetC be a Gumbel copula which models the dependence of the default times of the assets of a portfolio distributed marginally by the Weibull law. Then, the associated distribution function is(19)Ft1,…,tn=exp−∑i=1n−ln1−exp−tiλikiθ1/θ,for all t1,…,tn∈ℝn where θ is the parameter of Gumbel copula θ≥1, λi and ki are the parameters of Weibull law, and n is the number of the elements of the portfolio.Proof.
According to the relation (1), for all t1,…,tn∈ℝn, it follows that(20)Ft1,…,tn=CFt1,…,Ftn,where(21)Cu1,…,un=exp−∑1=in−lnuiθ1/θ.
So, it follows that(22)Ft1,…,tn=exp−∑i=1n−ln1−exp−tiλikiθ1/θ.
While dealing with the distribution of losses of the portfolio, notice that by defining an indicator functionIit=1τi<t at time t and an individual loss Li∗, the cumulative loss of the portfolio is linked such that(23)L∗t=∑i=1nLi∗Iit,where for i=1,…,n,(24)Li∗=1−RciAi,where Ai is the nominal value of element i of the portfolio and Rci is its recovery rate.
Note that for the distribution of portfolio losses, only the copula function ofτi is relevant. So, we have the following proposition.Corollary 1.
The copula of losses is given, for allui∈0,1 for i=1,…,n.(25)Cu1,…,un=exp−∑i=1nu˜θ1/θ.where u˜i=−lnui with θ≥1.Proof.
With the aim to construct a class of Archimedean copulas belonging to a given domain of attraction, Capéràa et al. [6] have introduced, in bivariate context, the family of Archimax copulas by combining the extreme values and Archimedean copula families into a single class. A member of this class with generator φ has the representation, for all u,v∈0,12,(26)Cϕ,Au,v=ϕ−1minϕ0,ϕu+ϕvAϕuϕu+ϕv,where A is a Pickands dependence function mapping the bivariate unit simplex S2 to 1/2,1, verifying: maxt;1−t≤At≤1, with Sn being defined by Sn=x∈ℝ+n;x1=∑i=1nxi=1.
By noting that only Gumbel copula models the default time, the parametric Archimedean generator is given byφtx=−lnx. So, relation (1) allows to obtain(27)Cϕ,Au,v=ϕ−1ϕu+ϕvAϕuϕu+ϕv.It comes that, in two dimensional study, one has the Gumbel model:(28)Cu,v=exp−∑i=1n−lnuθ+−lnvθ1/θ,which gives by extending to n -dimensional cases:(29)Cu1,…,un=exp−∑i=1n−lnuiθ1/θ=exp−∑i=1nu˜θ1/θ,where u˜i=−lnui with θ≥1.


## 4. Nested Archimedean Copulas and Losses

The class of Archimedean copulas founded a wide range of applications for a number of properties like their computational tractability and their flexibility to model dependencies between random variables, their exchangeability due to the analytic symmetry given by the generator, and the quadrant dependence of the underlying multivariate random vectors see [1] and [15]. However, in multivariate analysis applied in portfolio management, for example, we need nonexchangeable models. To face this drawback, some authors use Laplace transforms to derive more flexible extension of this family of copulas by nesting of generators allowing different degrees of positive dependence in bivariate margins. Denote an m-dimensional construction defined interactively for all u1,…,um∈0,1m by(30)Cψu1,…,um=ψ0ψ0−1u1+ψ1−1CΨ1u1,…,um.The inverseψ0−1 of the generator is completely monotone, i.e., for all t≥0 and ∀k≥0, −1kψ−1kt≥0 (see [8]).We present a model based on nested Archimedean copulas (NACs) able to capture hierarchically the structures of dependence between losses in a credit portfolio. The dependence between the losses of all the companies in the same subsector is described by an internal copulaCij of generator Φij, and the two subsectors of each sector are then coupled via a copula Ci of generator Φi and the sectors are then coupled via a copula C of generator Φ0. The companies in a credit portfolio have been classified according to an attribute which is the sector of activity of the industry. Each sector of activity is divided into two subsectors. It yields the following result under the two following assumptions.Hypothesis 1.
Suppose the dependence between the losses of all the companies in the same subsector is described by an internal copulaCli with generator Φli; the subsectors are then coupled via a copula Ci with generator Φl and the sectors via a copula C with generator Φ0. Suppose that the copulas Cli, Cl, and C are Gumbel copulas.Hypothesis 2.
Suppose the dependence between the losses of all the companies in the same subsector is described by an internal copulaCli with generator Φli; the subsectors are then coupled via a copula Ci with generator Φl and the sectors via a copula C with generator Φ0.Proposition 2.
Let us assume that the dependence between the losses of all the companies in the same subsector is described by an internal copulaCli with generator Φli; the subsectors are then coupled via a copula Cl with generator Φl and the sectors via a copula C with generator Φ0. Then, the partially nested Archimedean copula C is(31)Cu=Φ0−1∑k=1sΦ0Φl−1∑i=12ΦlΦli−1∑j=1dliΦkiujli,where u=u111,…,uds2s2, s is the number of business sectors, and dli is the number of businesses in subsector i of sector l, such that(32)∑i=12∑l=1sdli=n,with n being the number of companies and dli being the number of companies belonging to subsector i of sector l.Proof.
LetCl and Cli be Archimedean copulas with generators Φl and Φli, respectively. One has(33)C1u1=Φ1−1Φ1C11u1+Φ1C12u1.which gives(34)C1u1=Φ1−1Φ1Φ11−1Φ11u111+⋯+Φ11ud1111+Φ1Φ12−1Φ12u112+⋯+Φ12ud1212.
Furthermore, for the first sector, it follows that(35)C1u1=Φ1−1∑i=12Φ1Φ1i−1∑j=1d1iΦ1iuj1i.which provides, for the second member,(36)C2u2=Φ2−1∑i=12Φ2Φ2i−1∑j=1d2iΦ2iuj2i.
So, more generally, for alll∈1,2,…,s,(37)Clu1l1,…,udlil2=Φl−1∑i=12ΦlΦli−1∑j=1dliΦliujli.
Furthermore, the copulaC is given by(38)Cu111,…,uds2s2=CC1,…,Cs=Φ0−1∑l=1sΦ0Cl,which gives(39)Cu111,…,uds2s2=Φ0−1∑l=1sΦ0Φl−1∑i=12ΦlΦli−1∑j=1dliΦliujli,where s is the number of companies sectors ∑i=12∑l=1sdli=n, while n represents the number of companies.Proposition 3.
Suppose that the copulasCli, Cl, and C are Gumbel copulas. Then,(i)
The dependence of losses on companies in the same sector is given by(40)Clu1l1,…,udl1l1,u1l2,…,udl2l2=exp−∑i=12∑j=1dli−lnujliθliθl/θli1/θl.(ii)
The dependence of losses on portfolio companies is given by(41)CC1,…,Cs=Cu111,…,uds2s2=exp−∑l=1s−lnexp−∑i=12∑j=1dli−lnujliθliθl/θli1/θlθo1/θo.
The parametersθ0, θl, and θli of copulas C, Cl, and Clj satisfy the condition θ0<θl<θli with l∈1,…,s and i∈1,2.Proof.
(i)
LetCli and Cl be Gumbel copulas of parameters θli and θl, respectively; one has(42)Cliu1li,…,udlili=exp−∑j=1dli−lnujliθli1/θli.Therefore, one has(43)ClCl1,Cl2=Clu1l1,…,udl1l1,u1l2,…,udl2l2=exp−∑i=12∑j=1dli−lnujliθliθl/θli1/θl.Finally, it follows that(44)Clu1l1,…,udl1l1,u1l2,…,udl2l2=exp−∑i=12∑j=1dli−lnujliθliθl/θli1/θl.(ii)
LetCli and Cl be Gumbel copulas of parameters θli and θl, respectively; one has(45)CC1,…,Cs=Cu111,…,uds2s2=exp−∑l=1s−lnexp−∑i=12∑j=1dli−lnujliθliθl/θli1/θlθo1/θo.The parametersθ0, θl, and θli of copulas C, Cl, and Cli satisfy the condition θ0<θl<θli with l∈1,…,s and i∈1,2 (see [15]).


## 5. Sampling the Maximum Values at Risk in a Portfolio

### 5.1. On the Maximum of the VaR in Each Subsector

Here we determine the maximum of the value at risk in each subsector.Proposition 4.
Suppose the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli of parameter θli and that the default times are identically distributed by the law of Weibull. Then, the maximum of value at risk in each subsector is given by(46)VaRlimaxα=λ−ln1−explnαdli1/θli1/k,where λ and k are the parameters of Weibull law and dli is the number of elements in the subsector li.Proof.
According to the relation (1), one has(47)Ft1,…,tn=CFF1t1,…,Fntn,withFiti=1−exp−tiλk,where if t1li,…,tdlili=t,…,t,(48)Flit1li,…,tdlili=CliF1lit1li,…,Fdlilitdlili=expdli1/θliln1−exp−tλk.
One hasVaRlimaxα=Fli−1α, where(49)Flit1li,…,tdlili=α.
That is,(50)expdli1/θliln1−exp−tλk=α.
Equivalently, it follows that(51)dli1/θliln1−exp−tλk=lnα.
Thus, it follows that(52)1−exp−tλk=explnαdli1/θli,
Finally, one has(53)t=−λkln1−explnαdli1/θli1/k.
So, one obtains easily(54)VaRlimaxα=λ−ln1−explnαdli1/θli1/k,which proved the result as disserted.
Figure2 is a graphic representation of VaRlimax where some parameters are fixed.
The graph shows that the maximum of the value at risk is an increasing function of alpha for the parameters which are fixed. With other parameters, we obtain the same property.Figure 2
Maximum of the value at risk as a function of confidence levelα for dli=20, θli=8 for all l∈1,…,s and i∈1,2 in a subsector.


### 5.2. Computation of the Maximum Potential Loss in Each Sector

In this subsection, we determine the maximum of value at risk in each sector.Proposition 5.
Let us presume that the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli with parameter θli; the two subsectors of each sector are then coupled via a copula Cl with parameter θl. And if the default times are identically distributed by Weibull law, then the maximum of value at risk in each sector is(55)VaRlmaxα=λ−ln1−explnαdl1θl/θl1+dl2θl/θl2−1/θl1/k,where λ and kk≥1 are the parameters of Weibull distribution and dl1 and dl1 are, respectively, the number of companies of subsectors l1 and l2.Proof.
According to the canonical parametric relation (1), Ft1,…,tn=CF1t1,…,Fntn with Fiti=1−e−ti/λk. So, it follows that(56)Clu1l1,…,udl1l1,u1l2,…,udl2l2=exp−∑i=12∑j=1dli−lnujliθliθl/θli1/θl.
Moreover, ift1l1,…,tdl1l1,t1l2,…,tdl2l2=t,…,t, one has(57)Flt1l1,…,tdl1l1,t1l2,…,tdl2l2=exp−dl1−ln1−exp−1λtkθl1θl/θl1+dl2−ln1−exp−1λtkθl2θl/θl21/θl,which is equivalent to(58)Flt1l1,…,tdl1l1,t1l2,…,tdl2l2=exp−dl1θl/θl1−ln1−exp−1λtkθl+dl2θl/θl2−ln1−exp−tλkθ1/θl,which provides(59)VaRlα=F−1α,andFlt1l1,…,tdl1l1,…,tdl2l2=α,which means that(60)exp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θl=α.
Therefore, it follows that(61)−ln1−exp−tλkθl=−lnαθldl1θl/θl1+dl2θl/θl2.
From the above formula, one can have(62)t=−λkln1−explnαdl1θl/θl1+dl2θl/θl21/θl1/k,and finally(63)VaRlmaxα=λ−ln1−explnαdl1θl/θl1+dl2θl/θl2−1/θl1/k.
So, the result is proved. Figure3 is a graphic representation of VaRlmax where some parameters are fixed.
The graph shows that the maximum of the value at risk is an increasing function of alpha for the parameters which are fixed. With other parameters, we obtain the same property.Figure 3
Maximum of the value at risk as a function of confidence levelα for dli=20, θl=5, θl1=8, θl2=7,5 for all l∈1,…,s and i∈1,2 in a sector.


### 5.3. On the Maximum of the VaR in the Portfolio

In this section, we compute the maximum of value at risk in the portfolio made up of securities belonging to the different sectors and its variation as a function ofλ and k.Proposition 6.
Suppose the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli with parameter θli; the two subsectors of each sector are then coupled via a copula Cl with parameter θl and the sectors are coupled via a copula C with parameter θ0. And if the default times are identically distributed by Weibull law, then the maximum potential loss of the portfolio is(64)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k,where λ and kk≥1 are the parameters of Weibull distribution with dl1 and dl1 being, respectively, the number of companies of subsectors l1 and l2.Proof.
Using the parametric form of Sklar theorem, it follows that(65)Ft111,…,tds2s2=CF111t111,…,Fds2s2tds2s2.
So, ift111,…,tds2s2=t,…,t, one has(66)Ft111,…,tds2s2=exp−∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ01/θ0.
Since(67)Ft111,…,tds2s2=α,equivalently, one has(68)∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ01/θ0=−lnα.
It follows that(69)∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ0=−lnαθ0.
Moreover,∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−ln1−exp−t/λkθ0=−lnαθ0. Therefore, it follows that(70)−ln1−exp−tλkθ0=−lnαθ0∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1.
We obtain(71)−ln1−exp−tλk=−lnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ0.
It follows that(72)exp−tλk=1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ0.
Finally, we have(73)t=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k.
So,(74)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k.
Thus, the result is proved as disserted.
VaRmax depends on λ, k and on correlations between sectors, subsectors, and companies in the same sector. The following theorem is a key result of our study.Proposition 7.
The maximum potential loss of the portfolio satisfies the following properties:(i)
VaRmax is an increasing function of λ.(ii)
It is an increasing function ofk if(75)∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl1/θ0≤lnαln1−e−1,and decreasing otherwise.Proof.
(i)
From the previous results, one can write that(76)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl11/θ01/k.So, we obtain easily(77)∂VaRmaxα∂λ=−ln1−explnα∑l=1sdl1θlθl1+dl2θlθl2θ0θl−1θ01k>0.(ii)
Let us denotefk=x1/k, a function f defined on 1,+∞ for x>0. It is obvious that f is increasing in k if 0<x≤1 and decreasing if x>1. So, one obtains the result as disserted.
Figure4 is a graphic representation of VaRmax where some parameters are fixed.
The graph shows that maximum of the value at risk of the portfolio is an increasing function of alpha for the parameters which are fixed. This result is demonstrated and generalized by the previous proposition.Figure 4
Maximum of the value at risk as a function of confidence levelα for dli=20, θ1=5, θ2=6, θ3=7,θ11=8, θ12=7,5, θ21=8, θ22=9, θ31=8,5, θ32=9,5 for all l∈1,…,s and i∈1,2 in the portfolio made up of 3 sectors.


## 5.1. On the Maximum of the VaR in Each Subsector

Here we determine the maximum of the value at risk in each subsector.Proposition 4.
Suppose the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli of parameter θli and that the default times are identically distributed by the law of Weibull. Then, the maximum of value at risk in each subsector is given by(46)VaRlimaxα=λ−ln1−explnαdli1/θli1/k,where λ and k are the parameters of Weibull law and dli is the number of elements in the subsector li.Proof.
According to the relation (1), one has(47)Ft1,…,tn=CFF1t1,…,Fntn,withFiti=1−exp−tiλk,where if t1li,…,tdlili=t,…,t,(48)Flit1li,…,tdlili=CliF1lit1li,…,Fdlilitdlili=expdli1/θliln1−exp−tλk.
One hasVaRlimaxα=Fli−1α, where(49)Flit1li,…,tdlili=α.
That is,(50)expdli1/θliln1−exp−tλk=α.
Equivalently, it follows that(51)dli1/θliln1−exp−tλk=lnα.
Thus, it follows that(52)1−exp−tλk=explnαdli1/θli,
Finally, one has(53)t=−λkln1−explnαdli1/θli1/k.
So, one obtains easily(54)VaRlimaxα=λ−ln1−explnαdli1/θli1/k,which proved the result as disserted.
Figure2 is a graphic representation of VaRlimax where some parameters are fixed.
The graph shows that the maximum of the value at risk is an increasing function of alpha for the parameters which are fixed. With other parameters, we obtain the same property.Figure 2
Maximum of the value at risk as a function of confidence levelα for dli=20, θli=8 for all l∈1,…,s and i∈1,2 in a subsector.


## 5.2. Computation of the Maximum Potential Loss in Each Sector

In this subsection, we determine the maximum of value at risk in each sector.Proposition 5.
Let us presume that the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli with parameter θli; the two subsectors of each sector are then coupled via a copula Cl with parameter θl. And if the default times are identically distributed by Weibull law, then the maximum of value at risk in each sector is(55)VaRlmaxα=λ−ln1−explnαdl1θl/θl1+dl2θl/θl2−1/θl1/k,where λ and kk≥1 are the parameters of Weibull distribution and dl1 and dl1 are, respectively, the number of companies of subsectors l1 and l2.Proof.
According to the canonical parametric relation (1), Ft1,…,tn=CF1t1,…,Fntn with Fiti=1−e−ti/λk. So, it follows that(56)Clu1l1,…,udl1l1,u1l2,…,udl2l2=exp−∑i=12∑j=1dli−lnujliθliθl/θli1/θl.
Moreover, ift1l1,…,tdl1l1,t1l2,…,tdl2l2=t,…,t, one has(57)Flt1l1,…,tdl1l1,t1l2,…,tdl2l2=exp−dl1−ln1−exp−1λtkθl1θl/θl1+dl2−ln1−exp−1λtkθl2θl/θl21/θl,which is equivalent to(58)Flt1l1,…,tdl1l1,t1l2,…,tdl2l2=exp−dl1θl/θl1−ln1−exp−1λtkθl+dl2θl/θl2−ln1−exp−tλkθ1/θl,which provides(59)VaRlα=F−1α,andFlt1l1,…,tdl1l1,…,tdl2l2=α,which means that(60)exp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θl=α.
Therefore, it follows that(61)−ln1−exp−tλkθl=−lnαθldl1θl/θl1+dl2θl/θl2.
From the above formula, one can have(62)t=−λkln1−explnαdl1θl/θl1+dl2θl/θl21/θl1/k,and finally(63)VaRlmaxα=λ−ln1−explnαdl1θl/θl1+dl2θl/θl2−1/θl1/k.
So, the result is proved. Figure3 is a graphic representation of VaRlmax where some parameters are fixed.
The graph shows that the maximum of the value at risk is an increasing function of alpha for the parameters which are fixed. With other parameters, we obtain the same property.Figure 3
Maximum of the value at risk as a function of confidence levelα for dli=20, θl=5, θl1=8, θl2=7,5 for all l∈1,…,s and i∈1,2 in a sector.


## 5.3. On the Maximum of the VaR in the Portfolio

In this section, we compute the maximum of value at risk in the portfolio made up of securities belonging to the different sectors and its variation as a function ofλ and k.Proposition 6.
Suppose the dependence between the losses of all the companies in the same subsector is described by a Gumbel copulaCli with parameter θli; the two subsectors of each sector are then coupled via a copula Cl with parameter θl and the sectors are coupled via a copula C with parameter θ0. And if the default times are identically distributed by Weibull law, then the maximum potential loss of the portfolio is(64)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k,where λ and kk≥1 are the parameters of Weibull distribution with dl1 and dl1 being, respectively, the number of companies of subsectors l1 and l2.Proof.
Using the parametric form of Sklar theorem, it follows that(65)Ft111,…,tds2s2=CF111t111,…,Fds2s2tds2s2.
So, ift111,…,tds2s2=t,…,t, one has(66)Ft111,…,tds2s2=exp−∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ01/θ0.
Since(67)Ft111,…,tds2s2=α,equivalently, one has(68)∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ01/θ0=−lnα.
It follows that(69)∑l=1s−lnexp−dl1θl/θl1−ln1−exp−tλkθl+dl2θl/θl2−ln1−exp−tλkθl1/θlθ0=−lnαθ0.
Moreover,∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−ln1−exp−t/λkθ0=−lnαθ0. Therefore, it follows that(70)−ln1−exp−tλkθ0=−lnαθ0∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1.
We obtain(71)−ln1−exp−tλk=−lnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ0.
It follows that(72)exp−tλk=1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ0.
Finally, we have(73)t=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k.
So,(74)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl−1/θ01/k.
Thus, the result is proved as disserted.
VaRmax depends on λ, k and on correlations between sectors, subsectors, and companies in the same sector. The following theorem is a key result of our study.Proposition 7.
The maximum potential loss of the portfolio satisfies the following properties:(i)
VaRmax is an increasing function of λ.(ii)
It is an increasing function ofk if(75)∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl1/θ0≤lnαln1−e−1,and decreasing otherwise.Proof.
(i)
From the previous results, one can write that(76)VaRmaxα=λ−ln1−explnα∑l=1sdl1θl/θl1+dl2θl/θl2θ0/θl11/θ01/k.So, we obtain easily(77)∂VaRmaxα∂λ=−ln1−explnα∑l=1sdl1θlθl1+dl2θlθl2θ0θl−1θ01k>0.(ii)
Let us denotefk=x1/k, a function f defined on 1,+∞ for x>0. It is obvious that f is increasing in k if 0<x≤1 and decreasing if x>1. So, one obtains the result as disserted.
Figure4 is a graphic representation of VaRmax where some parameters are fixed.
The graph shows that maximum of the value at risk of the portfolio is an increasing function of alpha for the parameters which are fixed. This result is demonstrated and generalized by the previous proposition.Figure 4
Maximum of the value at risk as a function of confidence levelα for dli=20, θ1=5, θ2=6, θ3=7,θ11=8, θ12=7,5, θ21=8, θ22=9, θ31=8,5, θ32=9,5 for all l∈1,…,s and i∈1,2 in the portfolio made up of 3 sectors.


## 6. Conclusion

In this study, we contribute to the modeling of the dependence of losses of a large portfolio. Mainly, using a negative Weibull-distributed law and a family of nested Archimedean copulas, we estimate the highest value of the VaR. The use of the three-level nested Archimedean copula with Gumbel’s copulas provides interesting results on the maximum of value at risk of the credit portfolio divided into subsectors and sectors when the default time follows Weibull law. It has been shown that this maximum of value at risk of the portfolio is an increasing function ofλ. But it is an increasing or decreasing of k according to the conditions given by Proposition 7.



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