APPLIED COPULA IN FINANCIAL RISK MEASUREMENT ( khóa luận tốt nghiệp)

25 2.3 Value at Risk method using Conditional Copula.. To study applications of copula theory, we use some methods to estimate andcompare VaR of portfolio contains two stocks namely FPT

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Nguyen Vu Linh

APPLIED COPULA

IN FINANCIAL RISK MEASUREMENT

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Hanoi - 2012

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Nguyen Vu Linh

APPLIED COPULA

IN FINANCIAL RISK MEASUREMENT

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Thesis Advisor: Dr Tran Trong Nguyen

Hanoi - 2012

AcknowledgementsFirstly, I want to express my sincere gratitude to my thesis advisor Dr TranTrong Nguyen, who has introduced me to the eld of Financial Mathematics

I am especially grateful for your continual availability for discussions, your tience and ability of making Financial Mathematics so easy to be applied You aresuch a wonderful and great supervisor

pa-Great deals appreciated go to the contribution of my faculty- Faculty of ematics, Mechanics and Informatics

Math-Lastly to my family, especially to my parents for their support and ment

encourage-Ha Noi, October, 2012Nguyen Vu Linh

1.1 Basics concepts of copula 5

1.1.1 Denition and properties 5

1.1.2 Sklar's Theorem 11

1.1.3 The copula and Transformations of Random Variables 13

1.1.4 Dependence concepts 15

1.2 Conditional copula 20

2 Applied copula in nancial risk measurement 23 2.1 About VaR model 23

2.2 Unbias estimate method and Riskmetrics method 25

2.3 Value at Risk method using Conditional Copula 28

2.3.1 Estimation of the marginal distributions 28

2.3.2 Estimation of the copula and Monte Carlo simulations 31

3 Some results for portfolio of FPT and STB stocks 34 3.1 About FPT and STB stocks 34

3.2 Applied Copula in VaR measurement 37

3.2.1 Results for unbias estimate and Riskmetrics method 38

3.2.2 Results for conditional copula method 40

3.3 Comparision of the value at risk estimates 43

Copula methods are used to study nancial risk measurement problems Copulas havebeen rst applied to credit risk modeling, and have been later applied to the multi-dimensional non-normality problems in nancial investment In this thesis, we studyapplications of copula in measuring a very important measure in nancial namely Value

at risk (VaR) In order to estimate VaR of nancial portfolios, traditional VaR modelsassume that nancial return series are independent normally distributed Then wecan easily nd the joint distribution of returns series and VaR of portfolio However,these assumptions are usually violated in real portfolios Copula methods provide veryuseful tools to solve this problem By using ARMA-GARCH models and some families

of copula, we can estimate VaR under weak conditions

The aim in writing this thesis is to present the fundamentals of copula theory in theclearest possible way and give some simple application of copula theory in VaR mea-surement of a nancial portfolio In the theory part, we recall some basics concepts ofcopula that is necessary in order to understand the meaning and the applications ofcopula To study applications of copula theory, we use some methods to estimate andcompare VaR of portfolio contains two stocks namely FPT and STB (We use data onhttp://www.cophieu68.com) The structure of this thesis is as follows:

Chapter 1 provides a short review on copula theory with denitions and some erties of copula We also give denitions and simple properties of some importantmeasures which is usually used in applications of copula theory This chapter also con-tains basics concepts for conditional copula which have many applications in nancialrisk measurement

prop-Chapter 2 provides denition of VaR which is one of the most important nancial surements, and some models to estimate VaR In order to show advantages of copulamethod, we introduce three methods to estimate VaR: unbias estimate, Riskmetricsand conditional copula method We also explain how to use these three methods toestimate Value at risk of portfolio contains two stocks

mea-Chapter 3 is main part of this thesis There are some papers which said that tional copula method give very good results in case of unnormal series So, I repaircodes on http://www.mathworks.com and write some functions to write a program

condi-in matlab to process conditional copula method condi-in case of two series I also suggestprograms to process unbias estimate and riskmetrics method Using Jarqua-Bera test,

we see that rate of returns series of FPT and STB stocks are unnormal Then, weuse portfolio contains FPT and STB stocks with the same weights to check whetherconditional copula is good or not Codes and results can be found in appendices parts

at the end of thesis

By comparing three methods, we see that conditional copula is the best method forportfolio contains FPT and STB stocks We can use these programs to nd best methodwith any portfolio contains two assets But nancial portfolios usually contain morethan two assets So, we look forward to some programs which can process arbitraryportfolios

Chapter 1

Introduction to Copula

The basic idea of a copula is to separate the dependence and the marginal distributions

in a multivariate distribution In 1940s, Hoeding studied properties of multivariatedistributions And the notion copula appears for the rst time in a paper of Sklar in

1959 Since 2004, some insurance companies and nancial institutions have started touse copulas as a risk management tool

1.1 Basics concepts of copula

In this section, we recall some basics concepts that is necessary in order to understandthe meaning and the use of copula All of this material can be found in [9] and [10]

1.1.1 Denition and properties

Denition 1.1 Let S1, S2, Sn be non-empty subsets of R, where R denotes the theextended real line [−∞, +∞] Let H be a real function of n variables such that DomH =

S1×S2×S3× ×Snand for a ≤ b (ak ≤ bkfor all k = 1, 2, , n) where a = (a1, a2, , an),

b = (b1, b2, , bn) and a, b ∈ DomH Let B = [a, b] ([a1, b1] × [a2, b2] × × [an, bn]) be

an n-box whose vertices are in DomH The H-volume of B is given by

Where the sum is taken over all vertices c = (c1, c2, , c3) (ci = {ai, bi}) of B andsign(c)is given by

sign(c) =

(

1 if ck= ak for even number of k0s,

−1 if ck= ak for odd number of k0s (1.2)Equivalently, the H-volume of an n-box B = [a, b] is the n-th order dierence of H onB

for x in Sk.Higher-dimensional margins are dened in an obvious way One-dimensionalmargins are just called margins.

Example 1.2 Let H be the function with domain [−1, 1] × [0, ∞] × [0,π

2] given byH(x, y, z) = (x + 1)(e

y− 1) sin z

x + 2ey − 1 .Then H is grounded because H(x, y, 0) = H(x, 0, z) = H(0, y, z) = 0; H has one-dimensional margins H1(x), H2(y), H3(z)given by:

Lemma 1.3 Let S1, S2, , Sn be non-empty subsets of R, and let H be a grounded,

n-increasing function with domain S1 × S2 × × Sn Then H is increasing in eachargument, i.e if (t1, t2, tk−1, x, tk+1, , tn) and (t1, t2, tk−1, y, tk+1, , tn) are in DomHand x ≤ y Then

H(t1, t2, tk−1, x, tk+1, , tn) ≤ H(t1, t2, tk−1, y, tk+1, , tn) (1.3)Lemma 1.4 Let S1, S2, , Sn be non empty subsets of R, and let H be a grounded,

n-increasing function with margins and domain S1 × S2 × × Sn Then, if x =(x1, x2, , xn) and y = (y1, y2, , yn) are any point in S1× S2× × Sn

|H(x) − H(y)| ≤ Σnk=1|Hk(xk) − Hk(yk)| (1.4)

For the proof, see [9].

Denition 1.5 An n-dimensional distribution function is a function H with domain

Rn such that H is grounded, n-increasing and H(∞, ∞, , ∞, 1, ∞, , ∞) = 1

It follows from Lemma 1.3 that the margins of an n-dimensional distributionfunction are distribution functions, which denotes F1, F2, , Fn

Denition 1.6 An n-dimensional sub-copula (or n-subcopula) is a function C0

withthe following properties:

Note that for every u in DomC0

, 0 ≤ C0

≤ maxi{ui} ≤ 1 (i = 1, 2, , n) , sothat RanC0

prop-ˆ For every u in In, C(u) = 0if at least one coordinate of u is 0 and if all coordinates

of u are 1 except uk, then C(u) = uk

ˆ For every a and b in In such that a ≤ b, VC([a, b]) ≥ 0

Note that for any n-copula C, n ≥ 3, each k-dimensional margin of C is a k-copula

We show this claim by using induction as follows:

First, we consider (n − 1)-margins of n-copula

C1,2, ,k−1,k+1, ,n(x1, x2, , xk−1, xk+1, , xn) = C(x1, x2, , xk−1, 1, xk+1, , xn)

It is easy to see that C1,2, ,k−1,k+1, ,n is grounded and has margin Cj(uj) = uj, j ∈{1, 2, , k − 1, k + 1, , n}.

For any (n − 1) boxes

B = [a1, b1] × [a2, b2] × × [ak−1, bk−1] × [0, 1] × [ak+1, bk+1] × × [an, bn]

Then C1,2, ,k−1,k+1, ,n is (n − 1)-increasing

It implies that C1,2, ,k−1,k+1, ,n is (n − 1)-copula By induction, we can show that if

n ≥ 3, then for any k, 2 ≤ k ≤ n, all Ck

n k-margins of C are k-copulas

Example 1.3 Let C(u, v, w) = w min(u, v) It is easy to see that C satises conditions

of Denition 1.7, and the H-volume of the 3-boxes and B = [a1, b1] × [a2, b2] × [a3, b3]is

Since b1 ≥ min(b1, a2), and a1 ≥ min(a1, b2).Then C(u, v, w) is a 3-copula

The 1-dimensional margins of C are the 1-copulas

C1(u) = C(u, 1, 1) = u, C2(v) = C(1, v, 1) = v, and C3(w) = C(1, 1.w) = w And the

2-margins of C are 2-copulas

C1,2(u, v) = C(u, v, 1) = min(u, v), C2,3(v, w) = C(1, v, w) = vw,

and

C1,3(u, w) = C(u, 1, w) = uw

The following theorem can be proved by using result of Lemma1.4

Theorem 1.8 Let C be an n-copula Then for every u = (u1, u2, , un) and v =(v1, v2, , vn) in [0, 1]n

|C(u) − C(v)| ≤ Σn

Hence C is uniformly continuous on [0, 1]n

The Frechet-Hoeding Theorem states that for any Copula C : [0, 1]n → [0, 1]and any (u1, u2, , un) ∈ [0, 1]n the following bounds hold:

W (u1, u2, , un) ≤ C(u1, u2, , un) ≤ M (u1, u2, , un)where function W is called lower Frechet-Hoeding bound and is dened as

C such that C(u) = W (u) However, W is a copula only in two dimensions

In two dimensions, i.e the bivariate case, the Frechet-Hoeding Theorem states:

max{u + v − 1, 0} ≤ C(u, v) ≤ min{u, v} (1.6)

de-F1, F2, , Fn are distribution functions, then the function H dened by (1.7) is an

n-distribution function with margins

In order to prove Sklar's theorem, we need 2 following lemmas The proofs ofthese lemmas can be found in Appendix 1

Lemma 1.10 Let H be a joint distribution function with margins F1, F2, , Fn.Thenthere exists a unique sub-copula C0

such that

ˆ DomC0

= RanF1× RanF2× × RanFn,

ˆ For all (x1, x2, , xn)in Rnwe have H(x1, x2, , xn) = C0(F1(x1), F2(x2), , Fn(xn))

Lemma 1.11 Let C0

be an n-subcopula Then there exists a copula C such that

C0(u1, u2, , un) = C(u1, u2, , un), (1.8)for all (u1, u2, , un) in DomC0

, i.e any n-subcopula can be extended to a copula Theextension is generally non-unique

To prove Sklar's theorem, we use Lemma1.10to show that there is a sub-copula

C0 exists and satised (1.10) for all (x1, x2, , xn)in Rn

.If F1, F2, , Fnare continuous.Then RanF1 = RanF2 = = RanFn = I, so that the unique sub-copula in Lemma1.11 is a copula Otherwise, we use Lemma 1.11 to show that there exists copula Cwhich is uniquely determined on on RanF1× RanF2× × RanFn

The converse can be process as follows:

(1.7) implies that H(x1, x2, , xn) is grounded and n-increasing and since Fi(∞) = 1then H(∞, ∞, , ∞) = C(1, 1, , 1) = 1 It is complete the proof of Sklar's theorem.Equation (1.7) gives an expression for joint distribution functions in term of a copulaand n univariate distribution functions But (1.7) can be inverted to express copulas

in term of a joint distribution function and the " inverses " of the n margins However,

if a margin is not strictly increasing, then it does not possess an in verses in the usualsense Thus we need to dened " quasi-inverses " of distribution function

Denition 1.12 Let F be a distribution function Then a quasi-inverse of F is anyfunction F(−1) with domain I such that:

ˆ if t is in RanF, then F(−1)(t)is any number x in R such that F−1(x) = t, i.e, forall t in RanF,

We consider the bivariate copula to illustrate how to nd the quasi-inverse

Example 1.4 Let H be the joint distribution function given by

C given by

C(u, v) = uv

u + v ư uv.Example 1.6 (Gumbel's bivariate exponential distribution)

Let Hθ be the joint distribution function given by

exponen-in I Hence, the correspondexponen-ing copula is

Cθ(u, v) = u + v ư 1 + (1 ư u)(1 ư v)e(ưθ ln(1ưu) ln(1ưv))

1.1.3 The copula and Transformations of Random Variables

From now we focus on bivariate case for simplicity, but it should be noted that thetheory of copulas is applicable to the more general multivariate case We also assumethat the marginal distribution functions F and G are continuous (It follows that thecopula in (1.7) exists uniquely) We denote the distribution (or c.d.f ) of a randomvariable using an upper case letter, and the corresponding density (or p.d.f ) using thelower case letter Also note that we will denote the extend real line as R = R ∪ {±∞}.Let U = F (X) and V = G(Y ) We then say that U and V are the probability integral

transforms of X and Y The distribution of the probability integral transform is given

in the following theorem

Theorem 1.14 If F and G are continuous distribution function, then U ≡ F (X) ∼

U nif (0, 1) and V ≡ G(Y ) ∼ Unif(0, 1)

Since, F and G are strictly increasing and continuous, we have that X = F−1(U )and Y = G−1(V ), and ∂X

∂U = (∂X∂U)−1 = (∂F (X)∂X )−1 = f (X)−1 and ∂Y

∂V = (∂V∂Y)−1 =(∂G(Y )∂Y )−1 = g(Y )−1 Note that ∂X

∂V = ∂Y∂U = 0 Then,

c(u, v) = h(X(u), Y (v))

we can obtain a rst result on the properties of copulas: If X and Y are independent,then the copula density takes the value 1 every where, since in that case the jointdensity is equal to the product of the marginal densities Since we know that themarginal densities of U and V are uniform, by Theorem 1.14, we thus have that if Xand Y are independent the joint distribution of U and V is the bivariate Uniform(0, 1)distribution

We can also use the equation (1.12) to derive an expression for h as a function of xand y instead:

h(F−1(u), G−1(v)) = f (F−1(u)).g(G−1(v)).c(u, v) (1.13)

h(x, y) = f (x).g(y).c(F (x), G(y))

Equation (1.13) is the density version of Sklar's (1959) theorem: the joint density, h,can be decomposed into product of the marginal densities, f and g, and the copuladensity, c Sklar's theorem holds under more general conditions than the ones weimposed for this illustration, and we give the general proof in the below subsection

1.1.4 Dependence concepts

Copulas provide a natural way to study and measure dependence between randomvariables In this subsection, we recall some copula based measures of dependencewhich will be used later in this thesis

Linear correlation is a measure of linear dependence In the case of perfect lineardependence, i.e., Y = aX + b almost surely for a ∈ R {0}, b ∈ R, we have |ρ(X, Y )| =

1 More important is that the converse also holds Otherwise, −1 < ρ(X, Y ) < 1.Furthermore, linear correlation have property that

The following theorem can be found in Nelson (1999) Many of the results in thissubsection are direct consequences in this theorem

Theorem 1.16 Let (X, Y )T and (X, Y )T be independent vectors of continuous randomvariables with joint distribution function H and H, respectively, with common margins

F (of X and X) and G (of Y and Y ) Let C and C denote the copulas of (X, Y )T

and (X, Y )T, respectively, so that H(x, y) = C(F (x), G(y)) and H = C(F (x), F (y)).Let Q denotes the dierence between the probability of concordance and discordance of

(X, Y )T and (X, Y )T, i.e let

Proof Since the random variables are all continuous, P {(X − X)(Y − Y ) < 0} =

1 − P {(X − X)(Y − Y ) > 0} and hence Q = 2P {(X − X)(Y − Y ) > 0}-1 But

P {(X − X)(Y − Y ) > 0} = P {X > X, Y > Y } + P {X < X, Y < Y }, and theseprobabilities can be evaluated by integrating over the distribution of one of the vectors(X, Y )T or (X, Y )T Hence

Corollary 1.17 Let C, C and Q be given in Theorem 1.16 Then

1, Q is symmetric in it arguments: Q(C, C) = Q(C, C)

2, Q is non-decreasing in each arguments: C ≺ C0

, then Q(C, C) ≤ Q(C0

, C).The following denition can be found in Scarnisi (1984)

Denition 1.18 A real valued measure κ of dependence between two continuousrandom variables X and Y whose copula is C is a measure of concordance if it satisesthe following properties:

1, κ is dened for every pair X, Y of continuous random variables

2, −1 ≤ κX,Y ≤ 1, κX,X = 1 and κX,−X = −1

3, κX,Y = κY,X

4, If X and Y are linear independent, then κX,Y = κΠ= 0

5, κX,−Y = κ−X,Y = −κX,Y

6, If C and C are copulas such that C ≺ C, then κC ≺ κC

7, If {(Xn, Yn)}is a sequence of continuous random variables with copulas Cn, and

if {Cn} converges pointwise to C then limn→∞κCn = κC

Let κ be a measure of concordance for continuous random variables X and Y.As a consequence of Denition 1.18, if Y is almost surely an increasing function of

X ,then κX,Y = κM = 1 and if Y is almost surely a decreasing function of X ,then

κX,Y = κW = −1 Moreover, if α and β are almost surely strictly increasing functions

on RanX and RanY respectively, then κα(X),β(Y ) = κX,Y

Kendall's tau and Spearman's rho

In this subsection we discuss two important measure of dependence (concordance) know

as Kendall's tau and Spearman's rho

Denition 1.19 Kendall's tau for the random vector (X, Y )T is dened as

τ (X, Y ) = P {(X − X)(Y − Y ) > 0} − P {(X − X)(Y − Y ) < 0},

where (X, Y )T is an independent copy of (X, Y )T

Hence Kendall's tau for (X, Y )T is simply the probability of concordance minusthe probability of discordance.

Theorem 1.20 Let (X, Y )T be a vector of continuous random variables with copula

C Then Kendall's tau for (X, Y )T is given by

Denition 1.21 Spearman's rho for the random vector (X, Y )T is dened as

ρS(X, Y ) = 3(P {(X − X)(Y − Y0) > 0} − P {(X − X)(Y − Y0) < 0}),

where (X, Y )T, (X, Y )T and (X0

, Y0)are independent copies

Note that X and Y are independent Then, we can show that C(X, Y ) =Π(X, Y ) = F (X)G(Y ) Using Theorem 1.16 and the rst part of Corollary 1.17 weobtain the following result

Theorem 1.22 Let (X, Y )T be a vector of continuous random variables with copula

C Then Spearman's rho for (X, Y )T is given by

= ρ(F (X), G(Y ))

In next theorem we will see that Kendall's tau and Spearman's rho are concordancemeasures according to Denition1.18

Theorem 1.23 If X and Y are continuous random variables whose copula is C ,thenKendall's tau and Spearman's rho satisfy the properties in Denition1.18for a measure

of concordance

For a proof, see Nelsen (1999)

Example 1.7 Kendall's tau and Spearman's rho for the random vector (X, Y )T areinvariant under strictly increasing componentwise transformations This property doesnot hold for linear correlation It is not dicult to construct examples, the followingconstruction is instructive in its own right Let X and Y be standard exponential ran-dom variables with copula C ,where C is a member of the Farlie-Gumbel-Morgensternfamily, i.e C is given by

C(u, v) = uv + θuv(1 − u)(1 − v),for some θ ∈ [1, 1] The joint distribution function H of X and Y is given by

H(x, y) = C(1 − e−x, 1 − e−y)

Let ρ denote the linear correlation coecient Then

ρ(X, Y ) = E(XY ) − E(X)E(Y )

pVar(X)Var(Y) = E(XY ) − 1,where

E(XY ) =

Z ∞ 0

Z ∞ 0

xydH(x, y)

=

Z ∞ 0

Z ∞ 0

xy((1 + θ)e−x−y− 2θe−2x−y− 2θe−x−2y+ 4θe−2x−2y)dxdy

= 1 + θ

4.Hence ρ(X, Y ) = θ/4 But

C = M ⇒ τC = ρC = 1,

C = W ⇒ τC = ρC = −1

The following theorem states that the converse is also true

Theorem 1.24 Let X and Y be continuous random variables with copula C ,and let

κ denote Kendall's tau or Spearman's rho Then the following are true:

κ(X, Y ) = 1 ⇐⇒ C = M,κ(X, Y ) = −1 ⇐⇒ C = W

For a proof, see Embrechts, McNeil, and Straumann (1999) Kendall's tau andSpearman's rho are measures of dependence between two random variables Howeverthe extension to higher dimensions is obvious, we simply write pairwise correlations in

an n × n matrix in the same way as is done for linear correlation

1.2 Conditional copula

The extension to conditional copula consists in expressing the Sklar's theorem forconditional c.d.f, i.e, conditional to a sigma algebra = generated by all past information.Denition 1.25 A conditional bivariate distribution function is a right continuousfunction H : R2 → [0, 1] with the properties:

1 H(x, −∞ | =) = H(−∞, y | =) = 0 and H(∞, ∞ | =),

distribu-in the region [x1, x2] × [y1, y2] is non negative The conditional marginal distributions

of X and Y are dened as F (x | =) = H(x, ∞ | =), and G(y | =) = H(∞, y | =) Wenow dened the focus of this section; the conditional copula

Denition 1.26 A two-dimensional conditional copula is a function C : [0, 1]×[0, 1] →[0, 1] with the following properties:

1 C(u, 0 | =) = C(0, v | =) = 0, and C(u, 1 | =) = u and C(1, v | =) = v, for every

u, v in [0, 1],

2 VC([u1, u2] × [v1, v2] | =) ≡ C(u2, v2 | =) − C(u1, v2 | =) − C(u2, v1 | =) + C(u1, v1 |

=) ≥ 0 for all u1, u2, v1, v2 ∈ [0, 1], such that u1 ≤ u2 and v1 ≤ v2 Where = issome conditioning set

The rst condition of Denition 1.26 provides the lower bound on the bution function ensures that the marginal distributions, C(u, 1 | =) and C(1, v | =),are uniform The condition that VC is non-negative has the same interpretation as thesecond condition of Denition1.25: it simply ensures that the probability of observing

distri-a point in the region [0, 1] × [0, 1] is non-negdistri-ative

We now move on to an extension of the key result in the theory of copulas: Sklar's(1959) theorem for conditional distributions:

Theorem 1.27 Let H be a conditional bivariate distribution function with continuousmargins F and G, and let = be some conditional set Then there exists a uniqueconditional copula C : [0, 1] × [0, 1] → [0, 1] such that

H(x, y | =) = C(F (x | =), G(y | =) | =), ∀x, y ∈ R (1.14)

Conversely, if C is a conditional copula and F and G are conditional distributionfunctions of two random variables X and Y , then the function H dened by equation(1.14) is a bivariate conditional distributional distribution function with margins F and

G

The density function equivalent of (1.14) is useful for maximum likelihood ysis, and is obtained quite easily, provided that F and G are dierentiable, and H and

anal-G are twice dierentiable

where u ≡ F (x | =), and v ≡ G(y | =)

We can also obtain a corollary to Theorem 1.27, analogous to that of Nelson's (1999)corollary to Sklar's Theorem, which enables us to extract the conditional copula fromany conditional bivariate distribution function, but rst we need the denition of thequasi-inverse of a conditional distribution function

Denition 1.28 The quasi-inverse, F−1, of a conditional distribution function F isdened as:

F−1(u | =) = inf {x : F (x | =) ≥ u}, ∀u ∈ [0, 1] (1.16)

If F is strictly increasing then the above denition returns the usual functionalinverse of F , but more importantly it allows us to consider inverse of non-strictly in-creasing functions

Corollary 1.29 Let H be any conditional bivariate distribution with continuous marginaldistributions, F and G, and let F−1 and G−1 denote the (quasi-) inverses of themarginal distributions Finally, let = be some conditioning set Then there exists aunique conditional copula C : [0, 1] × [0, 1] → [0, 1] such that

C(u, v | =) = H(F−1(u | =), G−1(v | =)), ∀u, v ∈ [0, 1] (1.17)

This corollary completes the idea that a bivariate distribution function may bedecompose into three parts Given any two marginal distributions and any copula wehave a joint distribution, and from any given joint distribution we can extract the im-plied marginal distributions and copula

cop-in this area In this thesis, we use results of [5] to study the application of conditionalcopula in estimating the Value at risk (VaR) of portfolio with two assets.

2.1 About VaR model

In order to study what is VaR, we consider the following problem:

We want to invest Vt USD in the portfolio with two assets at day t when we knownthe history data from day t − m to day t of two assets We assume that the amount ofmoney invest in two assets are the same, Vi,t = 1/2Vt, i = 1, 2 for simplicity And wewant to predict the value of this portfolio Vt+1at day t+1 By using some mathematicstechnique we can estimate the value MVaR such that 100(1 − α)% the value of portfolio

at the at day t + 1 will be large than MVaR or in mathematics way P (Mt≤ MVaR) = α

In fact, the distribution function of value of assets are dicult to study then we usuallystudy the rate of return of assets We denoted the value of two asset at day t by S1,t

and S2,t, respectively We also denote number of stocks by n1, n2, respectively Then

S1,j+1− S1,j

S1,j

+12

represents the past information from day t − m to day t Then the VaR of the portfolio

at time t, with condent level 1 − α, where α ∈ (0, 1) is dened as

VaRt(α) = inf{s : Ft(s | =) ≥ α} (2.4)The following gure illustrates VaR and α

Figure 2.1: The value at risk VaR and level α

2.2 Unbias estimate method and Riskmetrics method

In this section, we introduce some traditional methods which require normality of jointdistributions of two rate of return series We also study their required conditions inorder to show the advantage of VaR method using conditional copula

Condition for Unbias estimate method and Riskmetrics method

Unbias estimators method require the rate of returns series have normality and arity properties And Riskmetrics method require rate of returns series have normalityproperty Stationary series are ones whose statistical properties such as mean, vari-ance, autocorrelation, etc are all constant over time And they are easy to predict:

Station-we simply predict that its statistical properties will be the same in the future as theyhave been in the past But this assumption is routinely violated in nancial markets.Normality is also a very strict assumption that mean times series in nancial marketsrare follow normal distribution

There are some way to test the normality and stationarity of series In this thesis, weuse Jarqua-Bera(JB) test and Unit root test for normality and stationarility, respec-tively

Testing for normality assumption

To know rate of return series ri have either normal distribution or not one can use χ2

distribution test The most popular is Jarqua-Bera(JB):

JB = n[S

2

6 +(K − 3)2

where S is non-symmetric rate and K is kustoris For n large enough JB distributionapproximates χ2(2) (Chi-squared distribution with 2 degrees of freedom).

Consider the hypothesis testing

H0 : ri have normal distributions , (2.6)

H1 : ri do not have normal distributions

We refuse H0 if JB > χ2

α(2), where α is given signication level Otherwise do notrefuse H0 (when JB < χ2

α(2) )

Testing for stationarity assumption

To study the stationarity we use Unit root test Consider the model

Yt= ρYt−1+ ut, ut is white noise (2.7)

If ρ = 1, Yt is random walk, and Yt is unstationary series Otherwise, Yt is stationaryseries (when ρ < 1)

There are some test for stationarity assumption In this thesis we consider the testgiven by Dickey-Fuller which is a very useful test

Using unbias estimate method to estimate VaR

For this method, we need assumptions that ri,t, i = 1, 2, is stationary and have normaldistribution i.e ri,t ∼ N (µi,t, σi,t | =) or r i,t −µ i,t

2Cov(r1, r2 | =),Cov(r1, r2 | =) = 1

expec-in real problems

Using Riskmetrics method to estimate VaR

Under normality and unstationarity assumptions we can estimates VaR by using Metrics model

Risk-We consider the assumptions for this model: Rate of returns ri,t, i = 1, 2 under givenconditional set = is normal distribution: (ri,t | =) ∼ N (0, σi,t)

ri,t = i,t,

σ2i,t = λσ2i,t−1+ (1 − λ)2i,t−1

Since, we work with daily data Then we can xed λ = 0.94 (or λ = 0.97 for monthlydata) It is easy to see that σ2

2Cov(r1, r2 | =),

To process this class of problems one can use non-parametric models.

The most popular non-parametric method is Historical simulation model The aim ofthis method is to generate a future scenatios based on historical data The data usuallyconsist of daily return for all possible assets, reaching back over a certain period Thesimulation of daily return for a day in the future them simply is done by choosing bychance (uniform distribution) one of the historical returns

This method is very simply to implement Its advantages are that it naturally porates any correlation between assets and any non-normal distribution of asset prices.The main disadvantage is that it require a lot of historical data that may correspond

incor-to completely dierent economic circumstances than those that currently apply

In this thesis, we pay concentrate to method that use conditional copula to estimate

ri,t, i = 1, 2, and rt Then we use Monte Carlo process to estimate VaR of portfolio

2.3 Value at Risk method using Conditional Copula

The most advantage of this model is that ones do not need normality assumption fordistributions function of assets By using conditional copulas, we can also omit thestationarity assumption Furthermore, we also can generalize this model to portfolio

of more than two assets by using multiple copula

2.3.1 Estimation of the marginal distributions

[5] and [10] provide models to estimate VaR using several kind of copulas In thisthesis, we use Gaussian and Student copula which is the most popular copula in order

to estimate VaR of portfolio

Gaussian and Student copula

We only consider the bivariate case for simplicity Ones can read [10] for multiple cases

Let ρ ∈ [−1, 1] be the linear correlation between x and y and Φ2

ρthe bivariate Gaussiandistribution correspond to Φ:

2 − 2ρ2st + t2

2(1 − ρ2) )dsdt. (2.10)Denition 2.2 The bivariate Gaussian copula is the function:

2+ t2− 2ρstν(1 − ρ2) )

− ν+1

Denition 2.3 The bivariate Student-t copula (or briey t copula ) is the function

2− 2ρst + t2

ν(1 − ρ2) }−(ν+2)/2dsdt,where t−1

ν is the inverse of the univariate t distribution with ν degree of freedom, and

ρ is linear correlation between x = t−1

ν (u)and y = t−1

ν (v).The model

In specifying the bivariate model we must specify the two models for the marginalvariables and the model for the conditional copula The models for the univariatevariables must take into account the characteristics of the variables Return serieshave been successfully modeled by ARMA-GARCH models For instance, for an AR(1)-GARCH(1, 1) the models for the margins are given by:

ri,t = µi+ φiri,t−1+ i,t; (2.15)

i,t = σi,tηi,t;

σ2i,t = αi+ βi2i,t−1+ γiσ2i,t−1;

where i = 1, 2, {η1,t} and {η2,t} are white noise processes ( or {η1,t} and {η2,t} areuncorrelated, have zero means and nite variances), αi, βi, γi satises the condition

of GARCH model: βi + γi < 1, for i = 1, 2 The conditional distribution of thestandardized innovations

ηi,t = i,t

σi,t|=i,t−1, i = 1, 2,was modeled by white noses and denoted by Fi,t in general case (the marginal distribu-tions) In this thesis we consider case ηi,t, i = 1, 2 are standard normal distributions and

tdistributions with the same degree of freedom We denote these models, repectively byGARCH-N, GARCH-t Then the joint distribution of innovation vector ηt= (η1,t, η2,t)

is model by copula

Let uj = F1,t(η1,j|=), vj = F2,t(η2,j|=), j = t − m, t − 1, F1,t and F2,t are marginaldistributions conditioned to =, the information available up to time t−1 If the modelswere correctly specied then both series uj, and vj will be standard uniform series

2.3.2 Estimation of the copula and Monte Carlo simulations

Estimation of the copula

In order to apply the copula models we need to specify the conditional marginal bution of the residuals (estimates of the standardized innovations) According to theIFM method the selected conditional copula Ct(F1,t(η1,t|=), F2,t(η2,t|=) | =) functionswill be tted to these residuals series ηi,j, j = t − 1, t − 2, (estimates of the standard-ized innovations) Denote the conditional set by = = {η1,t−1, η1,t−2, , η2,t−1, η2,t−2 }The estimation of the models can be done as follows:

distri-Assume that (η1,t; η2,t | = ) has multivariate distribution function Ht(η1,t; η2,t | =) andcontinuous univariate marginal distribution functions F1,t(η1,t | =) and F2,t(η2,t | =)

In order to investigate the residual dependence, we t copula-based models of the type

Ht(η1,t, η2,t; θ1,t, θ2,t | =) = Ct(F1,t(η1,t | =), F2,t(η2,t | =); θ1,t, θ2,t | =), (2.16)where θ1,t is the margins' parameters and θ2,t is copula's parameters of copula function

Ct Since, the margins distributions are continuous, Ct exists uniquely by Sklar'sTheorem (Sklar (1959 )) The corresponding model density is the product of theconditional copula density ct and the marginal densities f1,t and f2,t

ht(η1,t, η2,t | =; θ1,t, θ2,t) = ct(F1,t(η1,t | =), F2,t(η2,t | =); θ1,t, θ2,t)Π2i=1fi,t(ηi,t | =),where ct is the conditional copula density of model (2.16) and is given by

ct(ut, vt| =; θ1,t, θ2,1) = ∂

2Ct(ut, vt| =; θ1,t, θ2,t)

∂ut∂vt ,where ut= F1,t(η1,t | =), and vt= F2,t(η2,t | =)

We estimates bθ1,t, bθ2,t by using IFM ( inference for the margins) method:

1 At rst step, we estimate the margin's parameters bθ1,t by performing the tion of the univariate marginal distributions:

2 As a second step, given bθ1,t, we perform the estimation of the copula parameterb

If the marginal distributions F1,t and F2,t are two standard normal distributions and Ct

is a bivariate Gaussian copula with parameter θ2,t = ρt, then the bivariate distributionfunction Ht dene by Ht(η1,t, η2,t | =) = Ct(F1,t(η1,t | =), F2,t(η2,t | =) | =) is thebivariate normal distribution with linear correlation coecient ρt More precisely, thenormal density function is given by:

Using equation (2.18), (2.20) and information set =, we can estimate ρt In this case we

do not need to use (2.17) and (2.19) because we assume that the marginal distributionsare standard normal distributions

If the marginal distributions F1,t and F2,t are two Student-t distributions with same

θ1,t = νt degrees of freedom and Ct is a Student t-copula with parameter θ2,t = ρt,then the bivariate distribution function Ht dene by Ht(η1,t, η2,t | =) = Ct(F1,t(η1,t |

=), F2,t(η2,t | =) | =) is the bivariate t distribution with linear correlation coecient ρt

and νt degree of freedom More precisely, the Student t density function is given by:

fνt(ηi,t) = Γ((ν√ t+ 1)/2)

πνtΓ(νt/2)(1 +

η2 i,t

h(x, y) = f (x).g(y).c(F (x), G(y))

Equation (1 .13) is the density version of Sklar''s (1 959) theorem: the joint density, h,can... X) and G (of Y and Y ) Let C and C denote the copulas of (X, Y )T

and (X, Y )T, respectively, so that H(x, y) = C(F (x), G(y)) and H = C(F (x), F (y)).Let...

H(x, y) = C(1 − e−x, − e−y)

Let ρ denote the linear correlation coecient Then

ρ(X, Y ) = E(XY ) − E(X)E(Y )

pVar(X)Var(Y) = E(XY