EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 2074-2085
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

An Approach of Estimating the Value at Risk of
Heavy-tailed Distribution using Copulas

Korotimi Ouédraogo1,∗, Diakarya Barro2

1 LANIBIO, UJKZ, BP: 7023 Ouagadougou 03, Burkina Faso
2 Université Thomas SANKARA, Burkina Faso

Abstract. The value at risk (VaR) plays a fundamental role in modeling risk in financial studies.
We propose a approach in estimating the VaR for heavy-tailed distribution by taking into account
the effects of certain covariates on the variable of interest. This method, involves estimating the
extreme conditional quantiles by using the assciated copula. Morever, we use Bernstein copulas to
estimate the intermediate conditional quantile in a non-parametric approach of the direct method.
Then, the extreme conditional quantile is also estimated and we study the asymptotic properties
of this new estimator.

2020 Mathematics Subject Classifications: 91G70, 60G65, 62G32, 62H05

Key Words and Phrases: Copulas, Value at risk, extreme values, conditional quantiles, heavy-
tailed distribution

1. Introduction

The Value at Risk (VaR) is one of the best known risk measures in many fields such as
finance and insurance. This measure quantifies the maximum loss that a portfolio manager
can incur during a certain time horizon at a given confidence level. For a given level of
confidence, the corresponding VaR for a random variable Y with distribution function F0

is given by
V aRY (α) = QY (α) = inf {y ∈ R : F0(y) ≥ α} . (1)

However, in some domains such as finance or actuarial science, the usual methods of
estimating the VaR can be affected by factors such as interest rates or inflation. In the
same vein, in many applications, the tail quantiles of the variable of interest Y depend on
some covariate X from whom the VaR depends on. But, in practice its estimation does
not take into account the effect of covariates on the variable of interest. So, it turns out to
be necessary in the estimation of certain risk measures such as VaR, to take into account
the influence of covariates.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4280

Email addresses: korotimioued@gmail.com. (K. Ouédraogol Wahid),
dbarro2@gmail.com (D. Barro)

https://www.ejpam.com 2074 © 2022 EJPAM All rights reserved.



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2075

A VaR of level α being a quantile of order α, the estimation of a value at risk in the
presence of covariates consists in the estimation of a conditional quantile. Let (Y,X) be a
pair of random variables such that Y is the univariate response or interest variable, with
marginal distribution function F0 and X the covariate is a random vector with marginal
distribution function F ; the conditional distribution function will be noted FY (• | X).
The VaR under the effects of covariates of level α is defined by

CoV aRY (α) = QY (α | X = x) = inf {y : FY (y | x) ≥ α} , (2)

where x is a prespecified covariate vector.
Several methods of estimating extreme conditional quantiles based on extreme values

theory (EVT) exist. More particularly, J. Beirlant et al.[1] used univariate extreme theory
and quantile regression to estimate conditional quantiles. Based on the nearest neighbours
method, Gardes and al. [2] have proposed a method for estimating extreme conditional
quantiles. However, this method suffers from the lack of data in the local neighbourhood.
Based also on EVT and quantile regression, Wang and al. [3] proposed an estimation
method assuming the linear dependence model. Furthermore, in 2013, Wang and al.[4]
proposed another estimation method by relaxing the linearity assumption while Noh et
al. [5] focused on a semi-parametric method for estimating conditional quantiles based on
quantile regression and copulas. In 2017, Nasri & Bouezmarni [6] proposed two parametric
and semi-parametric estimation methods based on copulas.

In this study, we consider a random vector X = (X1;X2; ...;Xm)T of dimension m with
joint distribution function F and marginal distribution function F1; ...;Fm. According to
Sklar’s theorem [7] we can find a copula C such that

F (x1; ...;xm) = C(F1(x1); ...;Fm(xm)). (3)

If the marginals distributions functions F1, ..., Fm are continuous, then the copula C is
unique. Conversly, if C is a m-dimensional copula an F1, ..., Fm univariate distributions
functions, then the function F defined by (3) is a distribution function with margins
F1, ..., Fm and

C(u1, ..., um) = F (F−1
1 (u1), ..., F

−1
m (um)). (4)

More generaly, the copula C and the marginal distribution functions are not known
and are estimated. Several methods of estimating copulas have been proposed. Parametric
methods estimate the copula and the marginal distribution functions in a parametric way
such as in Oakes [13], Romano and Joe [17]. Semi-parametric methods assume instead a
parametric model for the copula and a non-parametric model for the marginal functions,
see [10] , [11] , [14] and [23] . Finally there are the non-parametric estimation methods. It
assumes a non-parametric model for the copulas and the marginal distributions. We can
cite the work of Deuhevels [9], Gijbels and Mielniczuk [12].

In this work we use a non-parametric copula estimation method based on Bernstein
polynomials to estimate the intermediate conditional quantile. Sancetta and Satchell
[18] proposed in 2004 a method for estimating the copula function based on Bernstein



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2076

polynomials called Bernstein copula. In the following, we consider the univariate case, i.e.
m = 1, and we will use a non-parametric estimation of copulas for the estimation of our
intermediate quantile.

Our method of estimating extreme conditional quantiles is based on univariate EVT
and the relationship between copulas and marginal distribution functions. The method
consists in to use the copulas to estimate the intermediate conditional quantile and then
using the EVT to estimate the extreme conditional quantile. EVT provides a elegant
mathematicals tool for analyzing rares events. Our method of estimating the extreme
conditional quantile differs from existing methods in that by using copulas namely Bern-
stein copulas, the dependence relationship between the variable of interest and the covari-
ate is general. In particular, Bernstein copulas provide a more adequate estimate of the
underlying dependence structure.

The paper is structured as follows. In the second section, we will discuss the various
preliminaries that concern extreme values theory and copulas. In the section 3 we present
our estimation method and study the asymptotic behaviour of our estimator. The last
section consists of the conclusion followed by a discussion.

2. Materials and methods

In the following sections, we estimate the intermediate conditional quantile based on
a copulas and we introduce the notions of copulas with a focus on Bernstein ones.

2.1. An overview of extreme conditional quantiles.

We suppose in the remainder of this work that FY (• | x) belongs to the domain of
attraction of a distribution of extreme values Gγ (γ ∈ R). Let Z1; ...;Zn be a sample of
random variables with distribution F and let Mn = max1≤i≤n Zi. The law of Mn suitably
normalised converges to

Gγ(x) = exp
{
−(1 + γz)

−1
γ

}
; (5)

when n → ∞ and for 1 + γz > 0
According to the Fisher-Tippet-Gnedenko theorem (see [8]), Gγ belongs to one of the

three following types of distributions:

Gγ(x) =



exp
[
−(x)

−1
γ

]
for x ≥ 0 and γ > 0 (Fréchet model)

exp [− exp(−x)] for x ∈ R and γ = 0 (Gumbel model)

exp
[
−(−x)

−1
γ

]
for x < 0 (Weibull Negative model)

.

We are particularly interested in the case where γ > 0 i.e. the conditional distribution
function FY (• | x) belongs to the Fréchet attraction domain. In this case the extreme



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2077

conditional quantile is defined by:

qY (αn | X = x) =

(
1− βn
1− αn

)γ(x)

qY (βn | x) ; (6)

with βn such that βn → 1 and n(1− βn) → 0 when n → ∞ and the index of extreme
values γ depends on the covariate X.

2.2. On the concept of Bernstein copulas

The use of Bernstein Copulas in our estimation have many reasons.:

• Firstly, it can approximate any behavior in the tail, and it recently has attracted
attention in insurance modeling and is thus a natural candidate for further analysis.
The Bernstein copulas are therefore relevant for the estimation of the CoVaR.

• Secondly, the Bernstein copula is attractive from a modeling perspective.

• Thirdly, the Bernstein copula is also suitable in higher dimensions, which is a major
advantage compared to other parametric and non-parametric estimators.

• Fourthly, its mathematical properties are interesting as the Bernstein estimator con-
verges to the underlying dependence structure, provides a higher rate of consistency
than other common nonparametric estimators [22].

In what follows, we first deal with some important result on copulas, the notion of
Bernstein copulas and the relationship between the conditional distribution function and
copulas. Consider (Y,X) a pair of random variables with marginal distribution functions
F0 and F and joint distribution function H. According to relation (3) if F0 and F are

continuous, then C is unique and we have C(u, v) = H(F−1
0 (u), F−1(v)). Given a sample

of random variables (Y1, X1), ..., (Yn, Xn), the empirical estimator of the copula C is defined
by, for all u, v ∈ [0, 1] by

Cn(u, v) = Hn(F
−1
0n (u), F−1

n (v))

with

Hn(x, y) = n−1
n∑

i=1

I(Xi ≤ x, Yi ≤ y);

Fn(x) = Hn(x,∞) = n−1
n∑

i=1

I(Xi ≤ x);

and

F0n(y) = Hn(∞, y) = n−1
n∑

i=1

I(Yi ≤ y).



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2078

While modeling insurance tools, Sancetta & Satchelle [18] proposed the Bernstein
copula function defined below as an estimator of the copula C

Bp(u, v) =

p∑
k=0

p∑
l=0

C

(
k

p
,
l

p

)
B(p, k, u)B(p, l, v) (7)

where B(p, k, u) = Ck
pu

k(1−u)p−k is a binomial probability called Beirnstein’s polynomial.

Note that limp→∞Bp(u, v) = C(u, v) is uniform over [0, 1]2 since C is continuous over
[0, 1]2. However, this estimator depends on the copula C which is always unknown. They
thus proposed in the above estimator to replace the copula C by its empirical estimator
Cn [18]. This estimator is given by

Cp,n(u, v) =

p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p, k, u)B(p, l, v). (8)

In the same vein, in the following paper [19] it has been showed that the almost certain
convergence and the asymptotic normality of this estimator, see [22] and [23].

2.2.1. Copulas and conditional distribution function

The conditional quantile estimation based on copulas is based on quantile regression and
the asymptotic distribution of this estimator has been proven to be Gaussian (see Noh et al
[5]). Authors such as Kraus and Czado [20], Nasri and Bouezmarni [21] have used copulas
based on the plug-in method. More recently, Remillard, Nasri and Bouezmarni [6] have
used copulas and the direct estimation method of conditional quantiles to estimate the
conditional quantile. The asymptotic normality of their estimator has been studied. We
consider Y the one-dimensional random variable corresponding to the response variable of
marginal distribution F0 and X = (X1, X2; ...;Xm)T a random vector of dimension m and
of marginal distribution function F (x) = (F1(x1); ...;Fm(xm)). We note the conditional
distribution of Y | X by

FY (y | x) = P(Y ≤ y | X = x), y ∈ R, x = (x1, ..., xm) ∈ Rm. (9)

We can establish a relation between the conditional distribution function, the copula
function and the marginal distributions. This relation was established in its first version,
i.e. in the case m = 1 in Bouyé & Salomon 14. In the case of dimension m ≥ 2 [6], the
relation is given for all real x and y by:

FY (y | x) = C̃(F0(y), F (x)); (10)

where C̃ is the copula such that [16]

C̃(F0(y), F (x)) =
∂F1(x1)...∂Fm(xm)C(F0(y), F1(x1), ..., Fm(xm))

∂x1...∂xmC(1, F1(x1), ..., Fm(xm))
. (11)



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2079

Now by definition of the copula C(1, F (x)) = F (x). So, equation(11) becomes

C̃(F0(y), F (x)) =
∂F1(x1)...∂Fm(xm)C(F0(y), F1(x1), ..., Fm(xm))

f(x1, ..., xm)
(12)

In the rest of our work, we consider that m = 1, i.e. that X is a one-dimensional random
variable. We therefore consider the pair of random variables (Y,X) of marginal distribution
resp F0 and F and of conditional distribution FY (• | X). The relation between the copulas
and the conditional distribution is now established as follows:

FY (y | x) = C̃(F0(y), F (x)) =
∂C(F0(y), F (x))

∂F (x)
. (13)

The conditional quantile function can be defined as the inverse of the conditional distri-
bution function. For a conditional quantile of order α we have

QY (α | x) = F−1
0 (Γ(α, F (x))) (14)

where Γ(α, v) is the quantile of order α of the distribution function C̃(u, v) that is to say
Γ(α, v) = inf {u ∈ [0, 1] | Γ(u, v) = α} and F−1

0 the generalized inverse of F0. In general,
neither the copula C̃, nor the marginal distribution functions F0 and F are known. They
must therefore be estimated. Using this method of estimating the conditional quantile
we will estimate the intermediate quantile. Concretely, we will use the Bernstein copula
estimator to estimate the copula C̃ and the empirical estimate for the marginal distribution
functions.

3. Mains Results

3.1. Intermediate conditional quantile estimating by copulas

Let αn be the order of the quantile such that αn → 1 when n → ∞. The conditional
quantile is said to be intermediate if n(1−αn) → ∞ and extreme if n(1−αn) → K where
K is a constant. In this work, we estimate an extreme VaR under the effects of covariates
and therefore an extreme conditional quantile. Extreme value theory provides an ideal
framework for these types of estimates.

One considers a sequence (βn) defined such that βn → 1 and n(1−βn) → ∞ when n →
∞, the conditional quantile of order βn is an intermediate quantile noted QY (βn | X = x).
We will thus estimate this intermediate conditional quantile by using the relation estab-
lished between the copula functions, the marginal distribution functions and the condi-
tional quantile functions.

According to the equation (14),

QY (βn | X = x) = F−1
0 (Γ(βn, F (x))

with Γ is the quantile of order βn of the distribution function C̃ , and F−1
0 is the generalized

inverse of F0. We will first estimate the copula C̃ which corresponds to the conditional



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2080

distribution and then we will estimate its quantile and finish by estimating the generalized
inverse of the partial distribution F0.

We first estimate the copula of the conditional distribution. By definition, we have:

FY (y | x) = C̃(F0(y), F (x)) =
∂C(F0(y), F (x))

∂F (x)
.

Like the copula C and the marginal distribution functions are generally not known, we will
estimate them by using respectively Bernstein’s copulas for the copula C and empirical
distribution functions for the marginal distribution functions. We call F0n and Fn the
empirical distribution functions corresponding to the marginal distributions F0 and F

The estimator of C using Bernstein’s copulas is given by

Cp,n(F0n(y), Fn(x)) =

p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p, k, F0n(y))B(p, l, Fn(x)) (15)

where Cn is an empirical estimator of the copula C and B(p, k, F0n(y)) (resp) B(p, l, Fn(x)
the Bernstein polynomials of order p in F0n(y) (resp) F0n(x)(x). We have B(p, l, Fn(x)) =
C l
p(Fn(x))

l(1− Fn(x))
p−l .

By partially derivating Cp,n with respect to Fn(x), we obtain the copula C̃p,n given by
the following result.

Proposition 1. The estimator of the copule of conditional distribution is given by

C̃p,n(F0n(y), Fn(x)) = A× Cp,n(F0n(y), Fn(x)). (16)

with A defined by A = −p2Fn(x) + 5pFn(x)− 2p

2Fn(x)(1− Fn(x))
.

If p = p(n) → ∞ and n
p log logn → k ≥ 0 then

∥ C̃p,n −AC ∥=| A | ×O(n− 1
2 (log log n)

1
2 ) a.s n → ∞ (17)

avec ∥ . ∥ the supremum norm

The following proof is for the proposition (1)

Proof. ∥ C̃p,n − AC ∥=| A |∥ Cp,n − C ∥. According to the theorem 1 of [19] ∥
Cp,n − C ∥= O(n− 1

2 (log log n)
1
2 . We then deduce the result

3.2. Estimation of the partial inverse of the copula C̃

This consists in finding the partial inverse of the copula C̃(F0(y), F (x)). It is defined
by

Γ(β(n), F (x)) = inf{F0(y) | C̃(F0(y), F (x)) = βn}.
The intermediate conditional quantile is given by

QY (βn | X = x) = F−1
0 (Γ(β(n), F (x))); (18)

where F−1
0 is the generalized inverse of F0.



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2081

Proposition 2. Let βn be the parameter of the intermediate conditional quantile. We fix
βn = 1− k

n such that k = k(n) → ∞ when n → ∞. The non-parametric estimator of the
intermediate conditional quantile of order βn is defined by

Q̂Y (βn | X = x) = F−1
0n

{
C−1
p,n

(
−βnA

−1, Fn(x)
)}

. (19)

Proof. Recall that for a Bernstein polynomial B(m, k, z) it comes that:

d

dz
B(m, k, z) = m [B(m− 1, k − 1, z)−B(m− 1, kz)] (20)

and that

Cp,n(F0(y), F (x)) = F0(y)F (x)

p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p−1, k−1, F0(y))B(p−1, l−1, F (x)).

(21)
By definition we have

C̃p,n(F0(y), F (x)) =

p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p, k, F0(y))

d

dF (x)
B(p, l, F (x)). (22)

Moreover from (20) we have equation (22) wich becomes:

C̃p,n(F0(y), F (x)) =

p∑
k=0

p∑
l=0

C

(
k

p
,
l

p

)
B(p, k, F0(y))p [B(p− 1, l − 1, F (x))−B(p− 1, l, F (x))]

(23)
Expanding and simplifying (23), we find

C̃p,n(F0(y), F (x)) = B −D; (24)

where

B = p

(
1− F (x)

1− F (x)

) p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p, k, F0(y))B(p−1, l−1, F (x))−B(p−1, l, F (x));

and

D =
F (x)

1− F (x)

p∑
k=0

p∑
l=0

lCn

(
k

p
,
l

p

)
B(p, k, F0(y))B(p− 1, l − 1, F (x)).

So, since we have

p∑
k=0

p∑
l=0

Cn

(
k

p
,
l

p

)
B(p, k, F0(y))B(p− 1, l − 1, F (x)) =

1

F (x)
Cp,n(F0(y), F (x));

Then, it comes that

B = p

(
1− F (x)

1− F (x)

)
1

F (x)
Cp,n(F0(y), F (x))



K. Ouédraogo, D. Barro / Eur. J. Pure Appl. Math, 15 (4) (2022), 2074-2085 2082

and

D =
p(p+ 1)

2(1− F (x))
Cp,n(F0(y), F (x)).

So, we obtain
C̃p,n(F0(y), F (x)) = B +D = ACp,n(F0(y), F (x)). (25)

Hence the value of A obtained in the previous definition. Using the formula

Γ(β(n), F (x)) = inf{F0(y) | C̃(F0(y), F (x)) = βn}

we obtain
F0(y) = C−1

p,n(βnA
−1, F (x)). (26)

Moreoer by inversion of F0 we obtain:

y = F−1
0 (C−1

p,n(βnA
−1, F (x))) (27)

Hence the value of the conditional quantile intermediate

Let us make the following assumption.

Hypothesis 1. We assume that the index of the extreme values γ of the conditional
quantile is a constant and consequently does not depend on the covariate X

Proposition 3. Consider a sample of n random variables for the response variable (Y1, X1), (Y2, X2), ..., (Yn, Xn).
Let αn be defined such that αn → 1 and n(1−αn) → K where K is a constant. The non-
parametric estimator of the extreme conditional quantile of order αn is defined by

Q̂Y (αn | X = x) =

(
1− αn

1− βn

)γF0

F−1
0

{
C−1
p,n

(
−βnA

−1, Fn(x)
)}

(28)

where γF0 is Hill estimator define by γF0 = 1
k

∑n
k=0 log(Yn−i,n)− log(Yn−k,n).

3.3. Asymptotic results

In this section, we establish the limiting behaviour of our different estimators.

Proposition 4. Assuming the conditions of Theorem 2 in [19] are satisfied

• If
√
n
p → 0 then for all (u, v) ∈ [0; 1]2

√
nA(C̃p,n(u, v)− C(u, v)) −→D N(0, A2σ2(u, v)).

• If
√
n
p → d avec 0 < d < ∞ alors pour tout (u, v) ∈ [0; 1]2

√
nA(C̃p,n(u, v)− C(u, v)) −→D N(db(u, v), A2σ2(u, v));



REFERENCES 2083

with

σ2(u, v) = C(u, v)(1− C(u, v)) + u(1− u)C2
u(u, v) + v(1− v)C2

v (u, v)(29)

-2(1− u)C(u, v)Cu(u, v) −2(1− v)C(u, v)Cv(u, v)

+2Cu(u, v)Cv(u, v) [C(u, v)− uv]

and

b(u, v) =
1

2
[u(1− u)Cuu(u, v) + v(1− v)Cvv(u, v)]

where

Cuu =
∂2

∂u2
C(u, v) and Cu =

∂

∂u
C(u, v)

Proof. According to the theorem 2 of [19], we have

• If
√
n
p → 0 then for all (u, v) ∈ [0; 1]2

√
n(Cp,n(u, v)− C(u, v)) −→D N(0, σ2(u, v))

we deduce the precedent result

• If
√
n
p → d avec 0 < d < ∞ alors pour tout (u, v) ∈ [0; 1]2

√
n(Cp,n(u, v)− C(u, v)) −→D N(db(u, v), σ2(u, v)),

we also deduce the second result

4. Conclusion

This paper allowed us to propose a new approach while estimating the extreme value at
risk for heavy-tailed distribution using the associated copula which by taking into account
the effects of certain covariates on the variable of interest. So, the extreme conditional
quantiles have been estimated for this subfamily of distributions. The model of Bernstein
copulas made it possible to estimate the intermediate conditional quantile. Then, the ex-
treme conditional quantile have been estimated and we provided the asymptotic properties
of this estimator.

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