EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 194-207
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Value-at-Risk Modeling with Conditional Copulas in
Euclidean Space Framework

Vini Yves Bernadin Loyara1, Diakarya Barro2,∗

1 ESPK, BP 174 Kaya, Burkina Faso
2 UFR-SEG, U.O2, 12 BP:417 Ouaga 12, Burkina Faso

Abstract. This paper aims to establish an analytic relation between a time-varying conditional
copula and the value at risk modeled by the underlying. Specically, under the assumption that the
space is euclidean we use scalar product to clarify a link between the conditional copula varying
with time and norms. It is then established a new expression on the geometric yield.

2010 Mathematics Subject Classifications: 51A50, 91C15, 60G57, 60H30, 62H00

Key Words and Phrases: Copulas, Euclidean space, Scalar product, VaR

1. Introduction

Modeling the risk of portfolio is to highlight the different methods or protocols to
minimize the loss of values of a portfolio. The use of multivariate copulas in this modeling,
is a contemporary approach, to develop indicators for the evaluation of the dependence
between the different assets of this portfolio.

In multivariate theory of probability a pioneer theorem (Sklar,1959). Abe SKlar showed
that the copula function enables to capture and to piece together the univariate models

(Sklar’s Theorem). Therefore, every n-dimensional continuous distribution H can be
canonically parameterized by its univariate marginal H1; ...;Hn using [0, 1]n a copula C

defined on the unit cube [0; 1] , such as

H (x1, ..., xn) = C [H (x1) , ...,Hn (xn)] ; with (x1, ..., xn) ∈ R̄n = [−∞,+∞]n . (1)

Under additional assumptions, differentiating the formula (1) shows that the density
function of the copula is equal to the ratio of the joint density h of H to the product of n

marginal densities hi such as, for all (u1, ..., un) ∈ [0, 1]n,

c (u1, ..., un) =
∂nC (u1, ..., un)

∂u1 × ...× ∂un
=

h
[
H−1
n (u1) , ...,H−1

n (un)
]

h1

[
H−1

1 (u1)× ...×H−1
n (un)

] . (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3347

Email addresses: loyarayves@outlook.com (V. Y. B. Loyara), dbarro2@gmail.com (D. Barro)

http://www.ejpam.com 194 c© 2019 EJPAM All rights reserved.



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 195

where H−1
i is the quantile function of Hi; that is, H−1

i (u) = inf {x ∈ R, H (xi) ≥ u}.
In stochastic financial analysis, from the definition in the univariate case we know that
the quantile function provides a point that accumulates a probability for the left tail and
for the right tail. The univariate quantile function QX (α) is used in the risk theory to
define the univariate risk measure : the value at risk is defined.

More generally in multivariate study, for a random vector X satisfying the regularity
conditions, we define the multidimensional VaR at probability level α by :

V aRα (X) = E [X | X ∈ ∂L (α)] (3)

where ∂L (α) is the boundary of the α − level set of F , the univariate component of
the vector : V aRα (X) are such as, for all portfolio Xi;V aRα (X) = {z/FXi (z) ≥ α} =
F−1
Xi

(α), F−1
Xi

being the right continuous inverse of FX .
In this paper, it is matter of the notion of multivariate risk coupled with that of

conditional copula varying with time. We have established, subject of being in a Euclidean
space, a relation between the conditional copula varying with time and the scalar product
or norm. We are inspired by Patton’s work on conditional copulas varying with time.
Indeed, from proposition and Sklar’s theorem adapted to the conditional copula (which
are all from Patton) we use the relation (5) to obtain our different results. Finally, a new
expression on the geometric yield is established.

2. Preliminaries

In this section we have grouped together the different notion definitions, propositions
and theorems which will be useful thereafter. We need Sklar’s theorem and its adaptation
in the conditional case proposed by Patton (2006) to elaborate the different results we
found.

2.1. A survey of Conditional Copulas

In copulas theory Joe (1997) or Nelsen (2007) provide detailed and readable intro-
ductions to copulas and their statistical and mathematical foundations while Bouy et al.
(2000) or Cherubini et al. (2004) deal with applications of copulas to different levels of
financial issues and derivatives pricing.

A n− dimensional copula is a multivariate distribution function C : [0, 1]n −→ [0, 1]
satisfying the following properties

i) Grounded : C (u1, ..., ui−1, 0, ui+1, ..., un) = 0 for all i and all (u1, ..., ui−1, ui+1, ..., un) ∈
[0, 1]n−1 .

ii) copula marginal : C (u1, ..., ui−1, 1, ui+1, ..., un) is an (n− 1) copula for all i ∈
{1, ..., n}.

iii) n− increasingness : the volume VB of any rectangle B = [a, b] ⊆ [0, 1]n is positive,

VB =

∫
B
dC (u1, ..., un) =

[
i1=1]2

∑
...

[
in=1]2

∑
(−1)i1+...+in C (ui1 , ..., uin) ≥ 0. (4)



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 196

Using the above relation (4) (positiveness of the volume of any hyper-rectangle of R̄n
) Barro et all.(2012) provide the following result by extending a proposition of Patton
(2002) both to space-varying case and to higher dimensional framework.

Proposition 1. Let Ft,wt denote the joint distribution of
(
X̃t,n−1,Wt

)
, t ∈ T with

X̃t,n−1 = (Xt,1, ..., Xt,n−1) ,then the conditional time-varying distribution of
(
X̃t,n−1,Wt

)
is given, for all ỹt ∈

(
R̄
)n−1

by

Ht,wt (xt/wt) = f−1
w (wt)

∂Ht,wt (xt,1, ..., xt,n−1, wt)

∂wt
(5)

where fw is the spacial density of the law of Wt. Moreover, the following properties are
satisfied

(i) Ht,wt (xt,1, ...,−∞, ..., xt,n−1, wt) = 0 for all ỹt ∈
(
R̄
)n−1

.

(ii) H (∞, ...,∞/wt) = 1 for all ỹt ∈
(
R̄
)n−1

.

(iii) For all x̃
(1)
t =

(
x

(1)
t,1 , ..., x

(1)
t,n−1

)
∈
(
R̄
)n−1

and ỹ
(1)
t =

(
x

(2)
t,1 , ..., x

(2)
t,n−1

)
∈
(
R̄
)n−1

such

as x
(1)
t,j ≤ x

(2)
t,j then

∑
(i1,...,in)∈{1,2}n

(−1)
[

j=1]n
∑
jj Hn−1,wt

(
x

(i1)
t,1 , ..., x

(in−1)
t,n−1 , wt

)
≥ 0. (6)

Let’s consider a linear portfolio of consisting of n different financial instruments (risks,
actions) X = (X1, ..., Xn). Further, let p0 = (p0,1, ..., p0,n) the initial value of the portfolio

is given by V0 =
[

i= 1]n
∑
xip0,i for a realization x = (x1, ..., xn) of X. At the next date t

the uncertain Profit and Loss function of the portfolio is given by

Ft (x1, ..., xn) =
[

i=1]n
∑

xi (p0,i − pt,i) =
[

i=1]n
∑

xipt,i (ezt,i − 1) . (7)

where Zt = (zt,i, ..., zt,n) is the Log Price vector such as zt,i = logpt,i . Particularly, from
the integral probability transforms we can associate to Ft, a parametric copula Ct such
as, for all (ut,1, ..., ut,n) ∈ [0, 1]n ,

Ct (ut,1, ..., ut,n) = P (Ft,1 (X1) ≤ ut,1, ..., Ft,n (Xn) ≤ ut,n) (8)

2.2. Scalar product and copulas application on VaR

According to Karl Friedrich Siburg et al. (1975) , the restrictions of 〈, 〉, ‖‖ and d to Cn
are called the sobolev scalar product, the Sobolev norm and the Sobolev distance function
on Cn, respectively. But we suppose a new norm in a space a Euclidean vector space is a
prehilbert space of finite dimension. It is complete.



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 197

Definition 1. Consider E = Rn a vector space with the scalar product 〈, 〉. In the following
we consider ourselves in finite dimension and (E, 〈, 〉) is Euclidean space. The application

x 7−→ ‖x‖ =
√
〈x, x〉

defines on E a norm, called euclidean norm and noted ‖‖. ∀ (x, y) ∈ E2 we have :

• Cauchy-Schwartz inequality :

|〈x, y〉| ≤ ‖x‖ ‖y‖ (9)

• Cauchy-Schwartz equality (in the case where (x, y) do not form a free family):

|〈x, y〉| = ‖x‖ ‖y‖ (10)

• Polarization identity :

〈x, y〉 =
1

4

(
‖x+ y‖2 + ‖x− y‖2

)
(11)

In Euclidean space there is an orthogonal basis and the Gram-Schimdt process allows to
build it. In a Euclidean space, any orthogonal family can be completed in an orthogonal
basis. Let an orthogonal base of a euclidean vector space (E, 〈, 〉) and u an endomorphism
from E:

3. Main Results

Let’s consider a linear portfolio of consisting of n different financial instruments (risks,
actions) X = (X1, ..., Xn) and let pt = (p1,t, p2,t, ..., pn,t) at a given date measured at
given time t. Further, let p0 = (p0,1, ..., p0,n) the initial value of the portfolio is given by

V0 =
[

i= 1]n
∑
xip0,i for a realization x = (x1, ..., xn) of X.

pt = (p1,t, ..., pn,t) =
[

i=1]n
∑
〈pi,t, ei〉 ei, and ‖pt‖ =

√
[

i=1]n
∑

p2
i,t =

√
[

i=1]n
∑
〈pt, ei〉2.

Consider

Pt = (p1,te
zt,1 , ..., pn,te

zn,1) =
[

i=1]n
∑
〈pi,tezt,i , ei〉 ei,

and

‖Pt‖ =

√
[

i=1]n
∑

p2
i,te

2zt,i =

√
[

i=1]n
∑
〈Pt, ei〉2.

The concept of scalar product that allowed us to highlight a link between the condi-
tional copula and the notion of norms in the metric spaces. Let assume that we are in an
euclidean space and that all necessary conditions are fulfilled. We use the characteristic
elements of metric spaces (euclidean space), to establish with the notion of scalar product
or norm a relation between the VaR and the conditional time-dependent copula.



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 198

3.1. Scalar product and copulas applications on the VaR

The following sub-section proposal was inspired by the Gram-Schmidt process. We
think it is necessary to depend the proposition 2 afterwards.

Proposition 2. Let E be a Euclidean space and (e1, ..., en) be a base of E in which. Then,
there is a only base ξ = (ξ1, ..., ξn) such that; if

V aRu (X) = (V aRu1 (X1) , ..., V aRun (Xn)) =
[

i=1]n
∑
〈V aRui , ei〉 ei

then

〈V aRu (X) , ξ〉−
[

i=2]n
∑V aRuiϑi

‖ϑi‖
= V aRu1

e1

‖e1‖
(12)

with ξ1 = e1
‖e1‖ and ∀i ∈ {1, ..., n− 1} , and

ξi+1 =
ϑi+1

‖ϑi+1‖
with ϑi+1 = ei+1−

[
k=1]i

∑
〈ei+1, ξk〉 ξk.

Proof. By assumption E is a Euclidean space and let (e1, e2, ..., en) be a base of E such
that

V aRu (X) = (V aRu1 (X1) , ..., V aRun (Xn)) =
[

i=1]n
∑
〈V aRui , ei〉 ei.

So, it follows that,

‖V aRu (X)‖ =

√
[

i=1]n
∑

V aR2
ui =

√
[

i=1]n
∑
〈V aRu, ei〉2

The orthogonalization process of Gram-Schimdt (1875) allows us to say that there is
only one base (ξ1, ..., ξn) as

ξ1 =
e1

‖e1‖
and ∀i ∈ {1, ..., n− 1} , ξi+1 =

ϑi+1

‖ϑi+1‖
withϑi+1 = ei+1−

[
k=1]i

∑
〈ei+1, ξk〉 ξk.

(13)
Futhermore, it comes that :

〈V aRu (X) , ξ〉 =
[

i=1]n
∑

V aRuiξi

〈V aRu (X) , ξ〉−
[

i=2]n
∑V aRuiϑi

‖ϑi‖
= V aRu1

e1

‖e1‖

Let’s consider a linear portfolio of consisting of n different financial instruments (risks,
actions) X = (X1, ..., Xn) and let pt = (p1,t, p2,t, ..., pn,t) at a given date measured at
given time t. Further, let p0 = (p0,1, ..., p0,n) the initial value of the portfolio is given by

V0 =
[

i= 1]n
∑
xip0,i.



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 199

Theorem 1. For a realization x = (x1, ..., xn) of X, at the next date t the uncertain Profit
and Loss function of the portfolio is given by

Ft (x1, ..., xn) =
[

i=1]n
∑

xi (p0,i − pt,i) =
[

i=1]n
∑

xipt,i (ezt,i − 1) ;

then
Ct (u1, ..., un) = ‖V aRu (X)‖ ‖Pt − pt‖ . (14)

Furthermore,

Ct (u1, ..., un) =
1

4

(
‖V aRu (X) + Pt − pt‖2 + ‖V aRu (X)− (Pt − pt)‖2

)
(15)

and where Pt = (p1,te
zt,1 , ..., pn,te

zn,1),

V aRu (X) = (V aRu1 (X1) , ..., V aRun (Xn)) .

is a Value at risk of the X and ‖·‖ euclidean norm and noted.

Proof. Consider the following relation

Ct (u1, ..., un) = FWt

(
F

(−1)
1,Wt

(u1) , ..., F
(−1)
n,Wt

(un)
)
.

Ff we consider relation (7) we obtain,

Ct (u1, ..., un) =
[

i=1]n
∑

V aRui (Xi) (pt,ie
zt,i − pt,i) .

Consider Pt = (p1,te
zt,1 , ..., pn,te

zn,1) and V aRu (X) = (V aRu1 (X1) , ..., V aRun (X)). Then,

Ct (u1, ..., un) = 〈V aRu (X) , Pt − pt〉

Value at risk is intrinsically linked to the portfolio and therefore to the initial amount
and the amount at a given time t. We will suppose linked vector V aRu (X) and vector
pt − Pt. The relation 10 we give

|〈V aRu (X) , Pt − pt〉| = ‖V aRu (X)‖ ‖Pt − pt‖ .

Then,
Ct (u1, ..., un) = ‖V aRu (X)‖ ‖Pt − pt‖

and equality (11) given

Ct (u1, ..., un) =
1

4

(
‖V aRu (X) + Pt − pt‖2 + ‖V aRu (X)− (Pt − pt)‖2

)
The following result allows us to obtain;



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 200

Proposition 3. Let pt = (p1,t, p2,t, ..., pn,t) at a given date measured at given time t. And
suppose these risks represent potential losses in dependent lines of business for example
an insurance company. Then

C (u1, ..., un/wt) = f−1
w (wt)

〈
∂

∂wt
(V aRu (X/wt)) , (Pt − pt)

〉
(16)

and where Pt = (p1,te
zt,1 , ..., pn,te

zn,1) , V aRu (X/wt) is a Value at risk of the X such that
wt and ‖‖ euclidean norm and noted.

Proof. Let FWt be any conditional time-varying conditional distribution with marginal
{Fi,Wt ; 1 ≤ i ≤ n}. Then there exists a only copula C : [0, 1]n −→ [0, 1] such as

C (u1, ..., un/wt) = FWt

(
F

(−1)
1,Wt

(u1/w) , ..., F
(−1)
n,Wt

(un/w)
)

(17)

where F
(−1)
i,Wt

(ui/w) = inf {x : Fi,Wt (x/w) ≥ ui} for each ui and wt ∈ Wt. Then, it
follows that :

C (u1, ..., un/wt) =
∂FWt

(
F

(−1)
1,Wt

(u1/wt) , ..., F
(−1)
n−1,Wt

(un−/wt) , wt

)
∂wt

.

By considering the equality (7) int the following relation

C (u1, ..., un/wt) = FWt

(
F

(−1)
1,Wt

(u1/wt) , ..., F
(−1)
n,Wt

(un/wt) /wt

)
are obtains :

C (u1, ..., un/wt) = = f−1
w (wt)×

[
[

i=1]n
∑∂(V aRui (Xi/wt))

∂wt
pi,te

zt,i

−
[

i=1]n
∑∂(V aRui (Xi/wt))

∂wt
pi,t

]
Consider Pt = (p1,te

zt,1 , ..., pn,te
zn,1), we have :

C (u1, ..., un/wt) = f−1
w (wt)×

〈
∂

∂wt
V aRu (X/wt) , Pt

〉
−f−1

w (wt)×
〈

∂

∂wt
V aRu (X/wt) , pt

〉

C (u1, ..., un/wt) =

〈
∂

∂wt
V aRu (X/wt) , f

−1
w (wt) (Pt − pt)

〉
if f−1

w ∈ C1 we can write, then

C (u1, ..., un/wt) = f−1
w (wt)

〈
∂

∂wt
(V aRu (X/wt)) , (Pt − pt)

〉



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 201

Proposition 4. Let these risks represent potential losses in dependent lines of business
for an insurance company for example. In the following we consider ourselves in finite
dimension and (E, 〈, 〉) is Euclidean space. Further, the conditional copula of is given by

C (u1, ..., un/wt) =
∣∣f−1
w (wt)

∣∣ ∥∥∥∥ ∂

∂wt
(V aRu (X/wt))

∥∥∥∥ ‖Pt − pt‖ (18)

Furthermore

C (u1, ..., un/wt) = 1
4f
−1
w (wt)

(∥∥∥ ∂
∂wt

(V aRu (X/wt)) + (Pt − pt)
∥∥∥2

+
∥∥∥ ∂
∂wt

(V aRu (X/wt))− (Pt − pt)
∥∥∥2
) . (19)

Proof. For this proof, consider the following relation :

C (u1, ..., un/wt) = f−1
w (wt)

〈
∂

∂wt
(V aRu (X/wt)) , Pt − pt

〉
;

Moreover considering equality (10) it came that;∣∣∣∣f−1
w (wt)

〈
∂

∂wt
(V aRu (X/wt)) , Pt − pt

〉∣∣∣∣ =
∣∣f−1
w (wt)

∣∣ ∥∥∥∥ ∂

∂wt
(V aRu (X/wt))

∥∥∥∥ ‖Pt − pt‖
so

C (u1, ..., un/wt) =
∣∣f−1
w (wt)

∣∣ ∥∥∥∥ ∂

∂wt
(V aRu (X/wt))

∥∥∥∥ ‖Pt − pt‖
and if we take it the relation (11)

〈V aRu (X/wt) , Pt − pt〉 = 1
4

(∥∥∥ ∂
∂wt

(V aRu (X/wt)) + (Pt − pt)
∥∥∥2

+
∥∥∥ ∂
∂wt

(V aRu (X/wt))− (Pt − pt)
∥∥∥2
) .

Then, it follows that :

f−1
w (wt)

〈
∂
∂wt

(V aRu (X/wt)) , Pt − pt
〉

= 1
4

∂
∂wt

(
f−1
w (wt)

)(∥∥∥ ∂
∂wt

(V aRu (X/wt)) + (Pt − pt)
∥∥∥2

+
∥∥∥ ∂
∂wt

(V aRu (X/wt))− (pt − Pt)
∥∥∥2
)

hence the result :



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 202

C (u1, ..., un/wt) = 1
4f
−1
w (wt)

(∥∥∥ ∂
∂wt

(V aRu (X/wt)) + (Pt − pt)
∥∥∥2

+
∥∥∥ ∂
∂wt

(V aRu (X/wt))− (Pt − pt)
∥∥∥2
) .

3.2. The CVaR in an Euclidean space

The Tail-VaR (TVaR) is derivative coherent risk measure of the VaR. For a given
confidence level α ∈ ]0, 1[, it follows that

TV aRα (X) =
1

1− α

∫ 1

α
V aRξ (X) dξ =

1

1− α

[
E [X]−

∫ α

0
V aRξ (X) dξ

]
(20)

The XTVaR is the average amount of ruins beyond the VaR;

XTV aRα (X) = TV aRα (X)− V aRα (X) (21)

the relation 21 we get the following proposition.

Proposition 5. The Tail-VaR (TVaR) is derivative coherent risk measure of the VaR.
For a given confidence level α ∈ ]0, 1[, it follows that

C (u1, ..., un/wt) = f−1
w (wt)

〈
∂
∂wt

(TV aRu (X/wt)) , (Pt − pt)
〉

− f−1
w (wt)

〈
∂
∂wt

(XTV aRu (X/wt)) , (Pt − pt)
〉 (22)

Furthermore

C (u1, ..., un/wt) = f−1
w (wt)

1

2

(∥∥∥∥ ∂

∂wt
(TV aRu (X/wt))

∥∥∥∥2

−
∥∥∥∥ ∂

∂wt
(XTV aRu (X/wt))

∥∥∥∥2
)

(23)

Proof. For relation (21) and the Proposition (3) we obtain the following equality :

C (u1, ..., un/wt) = f−1
w (wt)

[〈
∂
∂wt

(TV aRu (X/wt)) , (Pt − pt)
〉

−
〈

∂
∂wt

(XTV aRu (X/wt)) , (Pt − pt)
〉]



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 203

Then, it follows that :

C (u1, ..., un/wt) = f−1
w (wt)

1
4

[∥∥∥ ∂
∂wt

(TV aRu (X/wt)) + (Pt − pt)
∥∥∥2

−
∥∥∥ ∂
∂wt

(XTV aRu (X/wt)) + (Pt − pt)
∥∥∥2
]

= f−1
w (wt)

1
4

[
2

(∥∥∥ ∂
∂wt

(TV aRu (X/wt))
∥∥∥2

+ ‖Pt − pt‖2
)

− 2
(
‖XTV aRu (X/wt)‖2 + ‖(pt − Pt)‖2

)]

it comes that :

C (u1, ..., un/wt) = f−1
w (wt)

1

2

(∥∥∥∥ ∂

∂wt
(TV aRu (X/wt))

∥∥∥∥2

−
∥∥∥∥ ∂

∂wt
(XTV aRu (X/wt))

∥∥∥∥2
)
.

3.3. Performance Measures and the Distribution of L&P of a Portfolio

Proposition 6. Let w = (w1, ..., wn)T ∈ Rn a portfolio consisting of n capital (the alloca-
tion of capital) and St = (S1,t, ..., Sn,t)

T the non-negative random vector representing the
capital at the moment t. Then geometric yield

Rt = log

(
([(σ(St))

2+(E(St))
2])

1/2

([(σ(St−1))2+(E(St−1))2])
1/2

+ n×∆t

([(σ(w))2+(E(w))2])
1/2×([(σ(St−1))2+(E(St−1))2])

1/2

) (24)

where σ (·) is a standard deviation and E (·) is a mean and with ∆t all the interim payments
obtained between the dates t− 1 and t.

The distribution of (Pt+τ − Pt) is called profit distribution loss that expresses the
change in the value of the portfolio.

Proof. The Pt value of the portfolio is given by :

Pt =
n∑
j=1

wjSj,t. (25)



V. Y. B. Loyara, D. Barro / Eur. J. Pure Appl. Math, 12 (1) (2019), 194-207 204

The profits and losses associated with holding the asset are then defined by the difference
:

P&L = Pt + ∆t − Pt−1 =
n∑
j=1

wj (Sj,t − Sj,t−1) + ∆t. (26)

The equality 26 becomes using the scalar product: :

P&L = 〈w, St − St−1〉 = ‖w‖ ‖St − St−1‖+ ∆t (27)

Indeed on the same types of considerations of the section 3.1 (w and St are linked). Then
:

P&L =

 n∑
j=1

w2
j

1/2

×

 n∑
j=1

(Sj,t − Sj,t−1)2

1/2

+ ∆t (28)

These losses and profits are expressed in the form of a geometric return noted Rt :

Rt = log

(
Pt + ∆t

Pt−1

)
= log


(

n∑
j=1

w2
j

)1/2

×

(
n∑
j=1

S2
j,t

)1/2

+ ∆t(
n∑
j=1

w2
j

)1/2

×

(
n∑
j=1

S2
j,t−1

)1/2

 (29)

if we use the following relation;

E
(
S2
t

)
=

[
1

n
(σ (St))

2 + (E (St))
2

]1/2

.

It comes that:

Rt = log

(
([(σ(St))

2+(E(St))
2])

1/2

([(σ(St−1))2+(E(St−1))2])
1/2

+ n×∆t

([(σ(w))2+(E(w))2])
1/2×([(σ(St−1))2+(E(St−1))2])

1/2

) (30)

It is assumed that the return on the date t, i.e. Rt , is a real random variable.

Corollary 1. Let w = (w1, ..., wn)T ∈ Rn a portfolio consisting of n capital (the allocation
of capital) and St = (S1,t, ..., Sn,t)

T the non-negative random vector representing the capital
at the moment t. Then geometric yield

Rt '
n×∆t([

(σ (w))2 + (E (w))2
])1/2

×
([

(σ (St))
2 + (E (St))

2
])1/2

(31)

where σ (·) is a standard deviation and E (·) is a mean and with ∆t all the interim payments
obtained between the dates t− 1 and t.



REFERENCES 205

Proof. By taking the relationship (24) and as St and St−1 are of the same nature we
have :

σ (St) ' σ (St−1) and E (St) ' E (St−1) .

and view the limited developmental formula in the neighborhood of zero log(1 + x) ' x it
comes that

Rt = log

1 +
n×∆t([

(σ (w))2 + (E (w))2
])1/2

×
([

(σ (St))
2 + (E (St))

2
])1/2


Hence the result.

3.4. Conclusion and Discussion

We thought to introduce the notion of product in the stochastic modeling of the copula
and the value at risk. Because it allows us to combine precisely these two notions (Copula,
VaR). One of the characteristics of the scalar product is the fact that it makes it possible
to move the diferential from one component to another. The different relations that we
obtained in this document allow to establish a close link between copula and the value at
risk through an analytical expression with the norms. It becomes quite easy to calculate
the values of VaR when we know the value of the copula and the type of copula.

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