Degree Project 

Marie Manyi Taku 2010-10-19 Subject: Mathematics Level: Master Course code: 5MA11E

Modelling Dependence of Insurance Risks

Modelling Dependence of Insurance

Risks

Marie Manyi Taku.

September 27, 2010

1

Abstract

Modelling one-dimensional data can be performed by different well-known ways. Modelling two-dimensional data is a more open question.There is no unique way to describe dependency of two dimensionaldata. In this thesis dependency is modelled by copulas.

Insurance data from two different regions (Göinge and Kronoberg)in Southern Sweden is investigated. It is found that a suitable modelis that marginal data are Normal Inverse Gaussian distributed andcopula is a better dependence measure than the usual linear correlationtogether with Gaussian marginals.

2

Contents

1 Introduction 4

2 Marginal distributions 5

2.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 52.2 NIG Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Edgeworth Expansion . . . . . . . . . . . . . . . . . . . . . . . 72.4 Empirical Comparisons . . . . . . . . . . . . . . . . . . . . . . 9

3 Model Dependence 11

3.1 Linear Correlation . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Rank Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Kendall’s tau . . . . . . . . . . . . . . . . . . . . . . . 143.3 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Tail Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Elliptical Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Gaussian Copulas . . . . . . . . . . . . . . . . . . . . . 213.5.2 t-copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . 223.6.1 Clayton Copula . . . . . . . . . . . . . . . . . . . . . . 253.6.2 Gumbel Copula . . . . . . . . . . . . . . . . . . . . . . 253.6.3 Properties of Archimedean Copula . . . . . . . . . . . 27

4 Data Characterisation and Analysis 28

4.1 Nature of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Results and Explanations . . . . . . . . . . . . . . . . . . . . 29

5 Conclusion 36

A Appendix:Matlab codes 38

A.1 Random variate generation, an example . . . . . . . . . . . . . 39

References 40

3

1 Introduction

The Concise Oxford English Dictionary defines risk as ”hazard, a chance ofbad consequences, loss or exposure to mischance”. In [8] A.J. McNeil, R.Frey and P. Embrechts define Financial risk as ”any event or action thatmay adversely affect an organisation’s ability to achieve its objectives andexecute its strategies” or alternatively, ”the quantifiable likelihood of loss orless-than-expected returns”. Types of risks include Market risks, Credit risksand Operational risks. These pertain mostly to banking. An additional riskcategory entering through insurance is underwriting risk, the risk inherentin insurance policies sold. Examples of risks factors that play a role here arechanging patterns of natural catastrophes or changing customer behaviour(such as payment patterns).

In risk theory, a field which has motivated a lot of interest during the lastdecade is the dependence between risks. It is sometimes amazing how depen-dence may entail large loss amounts. For example, hurricanes or earthquakesmay cause many types of claims such as damages to buildings, car crashes,accidental deaths, etc. In this case risks cannot be regarded as independent.So it is important to have appropriate models for modelling dependence.

The main aim of this thesis is to describe appropriate distributions tomodel risks and to collect and clarify the useful ideas of dependence; linearcorrelation, rank correlation and copulas which should be known by anyonewho wishes to model dependence phenomenae. A number of properties,advantages and drawbacks of these dependence measures are highlighted.What is of more interest is the concept of the copula to model dependencebetween risks due to the fact that it (the copula) maneuvers around thepitfalls of correlation. As an illustration, daily reported damages from aninsurance company is investigated for dependence.

Section 2 takes a look at the marginal distributions used in modelling thedamages. These include; Normal distribution, the Normal Inverse Guassian(NIG) distribution and Edgeworth expansions. We give properties and es-timate parameters. Section 3 examines some dependence measures; linearcorrelation, rank correlation and coefficient of tail dependence bringing outthe limitations of linear correlation as a dependence measure particularlywhen we leave the normal and elliptical distributions. The last part of sec-tion 3 looks at the issue of modelling dependence of risks using the conceptof copulas where we describe elliptical (Gaussian and t) and archimedean(Clayton and Gumbel) copulas with illustration in section 4.

4

2 Marginal distributions

In this section we will study the Normal distribution, the Normal InverseGuassian (NIG) distribution and Edgeworth expansions. The Normal distri-bution is known to be one of the most important distributions and is foundin many areas of study. The Normal Inverse Gaussian (NIG) distribution onits part is suitable for modelling stochastic payments(in our work paymentsreflect damages) in insurance as investigated by [2]. The NIG distributionis generally very interesting for applications in finance due to their specificcharacteristics- they are a continuous four parameter distribution family thatcan produce fat tails and skewness, the class is convolution stable under cer-tain conditions and the cumulative distribution function, density and inversedistribution functions can be computed sufficiently fast. It should be notedthat the normal distribution is not an approriate distribution for modellingstochastic payments due to the absence of skewness and excess kurtosis i.e.it underestimates both the thickness of the tails of the marginals and theirdependence structure.

2.1 Normal Distribution

The Normal distribution with parameters µ ∈ R and σ2 > 0 has a densityfunction given by

fNorm(x, µ, σ2) =

1√2πσ2

exp

(−(x− µ)

2

2σ2

). (1)

The parameters µ and σ2 are the mean and variance. The distributionwith µ = 0 and σ2 = 1 is the Standard Normal Distribution.

Moment generating function The moment generating function is givenby

MX(t) = E[etX ] = eut+

1

2σ2t2 . (2)

The Cumulant generating function (the logarithm of the moment gener-ating function) is given by

Φ(t) = log(MX(t)) = ut+1

2σ2t2.

Properties

The Normal distribution is symmetric around its mean, and always hasa kurtosis equal to 3. Skewness is 0. The first and second moments are

κ1 = E[X] = Φ′(0) = µ

κ2 = V ar[X] = Φ′′(0) = σ2.

5

All higher moments are zeros.

2.2 NIG Distribution

The Normal Inverse Gaussian (NIG) distribution with parameters α, β, δ, µhas density function given by

fNIG(x, α, β, δ, µ) =αδ

π· K1(α

√δ2 + (x− µ)2)√

δ2 + (x− µ)2· e{δ

√α2−β2+β(x−µ)} (3)

x, β, µ ∈ R, α, δ ∈ R+, |β| < α,where K1(ω) :=

12

∫∞0

exp(−12ω(t+ t−1))dt is the modified Bessel function of

the third kind with index 1. The four parameters can be interpreted as fol-lows: α determines how "steep" the density is (tail heavyness), β determineshow skewed it is, δ determines the scale, and µ the location.

Moment generating function

The moment generating function of the NIG distribution has a simpleform though the density function is quite complicated.

The moment generating function of a random variable X∼NIG(α,β,δ,µ)is

MX(t) = e

(

δ√

α2−β2−δ√

α2−(β+t)2)

+tµ, (4)

α > |β + t|. The Cumulant generating function is given by

Φ(t) = log(MX(t)) = δ{√α2 − β2 −

√α2 − (β + t)2}+ µt,

α > |β + t|.Properties and Parameter estimation of the NIG distribution

The main properties of the NIG distribution are the scaling property

X ∼ NIG(α, β, δ, µ) ⇒ cX ∼ NIG(α

c,β

c, cδ, cµ

),

and the closure under convolution for the independent random variables Xand Y,

X ∼ NIG (α, β, δ1, µ1) , Y ∼ NIG (α, β, δ2, µ2) ⇒ X+Y ∼ NIG (α, β, δ1 + δ2, µ1 + µ2) .

The moments obtained by evaluating the differential of the cumulative gen-erating function at 0 are as follows:

6

κ1 = E[X] = Φ′(0) =

δβ√α2 − β2

+ µ

κ2 = V ar[X] = Φ′′(0) =

δα2

(√α2 − β2)3

κ3 = E[(X − E[X])3] = Φ′′′(0) =3δα2β

(√α2 − β2)5

κ4 = E[(X − E[X])4]− 3(V ar[X])2 = Φ(4)(0) =3δα2(α2 + 4β2)

(√α2 − β2)7

Let γ1 =κ3√(κ2)3

and γ2 =κ4

(κ2)2. We then get:

α̂ =3√

3γ2 − 4γ21(3γ2 − 5γ21)

√κ2

β̂ =3γ1

(3γ2 − 5γ21)√κ2

δ̂ =3√κ2√3γ2 − 5γ21

3γ2 − 4γ21µ̂ = κ1 −

3γ1√κ2

3γ2 − 4γ21The NIG distributions have semi-heavy tails. The right tail behaves as

follows:fNIG (α, β, δ, µ) ∼ |x|−3/2e(−α+β)x as x → +∞ The tails of the distribution

are heavier than those of the Normal distribution.

2.3 Edgeworth Expansion

Edgeworth expansion is an expansion which relates the probability densityfunction, f , of a random variable, X, having expectation 0 and variance 1,to the probability density function of a Standard Normal distribution, usingthe Chebyshev-Hermite polynomials.

With the first four moments

κ1 = E[X] = µ

κ2 = V ar[X] = σ2 = s2

κ3 = E[(X − E[X])3]κ4 = E[(X − E[X])4]− 3(V ar[X])2,

7

and their respective estimates:

µ̂1 = x̄

σ̂2 =1

n− 1

n∑

1

(xi − x̄)2

κ̂3 =1

n− 1

n∑

1

(xi − x̄)3

κ̂4 =1

n− 1

n∑

1

(xi − x̄)4 − 3s2,

and Chebyshev-Hermite polynomials

H3 = x3 − 3

H4 = x4 − 6x2 + 3,

the Edgeworth expansion is given by

f(x) =1√2πσ

exp

[−(x− µ)

2

2σ2

] [1 +

κ33!σ3

H3

(x− µσ

)+

κ44!σ4

H4

(x− µσ

)+ ...

].

(5)See [12]. The approximation which stops at the term with H3 is called theEdgeworth expansion of the third order. Note that in many cases higher orderexpansions are not better approximations than the third order expansion seefor example [6].

8

2.4 Empirical Comparisons

Here real data X1, for Kronoberg and X2 for Göinge are modelled alongsidethe Normal and Normal Inverse Gaussian distributions, and the Edgeworthexpansion of the third order. The parameters are as estimated above. X1is the logarithm of home damages in Kronoberg with costs bigger than zeroand X2 is the logarithm of home damages in Göinge with costs bigger thanzero.

5 6 7 8 9 10 11 12 13 14 150

0.2

0.4

0.6

0.8

1

1.2

1.4

real datanormal cdf3 order Edgew cdfNIG cdf

Figure 1: Cumulative density function of the NIG distribution, Normal dis-tribution and the third order Edgeworth expansion for the region Kronoberg.

In both figures, the NIG seems to fit best. The second best fit is the thirdorder Edgeworth expansion and the Normal has the worst fit. The differencesof the real data and the models can be quantified by Kolmogorov-Smirnovtest. This test will be performed in Chapter 4.

9

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

real datanormal cdfEdgew cdfNIG cdf

Figure 2: Cumulative density function of the NIG distribution, Normal dis-tribution and the third order Edgeworth expansion for the region Göinge.

10

3 Model Dependence

This section focuses on modelling dependence between risks. It starts bylooking at dependence measures: linear correlation, and rank correlation.These dependence measures yield scalar measurements for any pair of ran-dom variables (X1, X2), although the nature and properties of the measuresmight be different in each case. Also described in this section is the conceptof copulas (an alternative measure of dependence) and tail dependence(animportant concept since it addresses the phenomenon of joint extreme valuesin several risk factors).

3.1 Linear Correlation

The purpose of a linear correlation analysis is to determine whether there isa relationship between two sets of variables. An example will be the rela-tionship between X1, Home damage costs in Göinge and X2, Home damagecosts in Kronoberg. This correlation could be because of seasonal or weatherreasons.

Definition 3.1. The correlation ρ (X1, X2) between random variables X1and X2 is defined as

ρ1,2 =Cov (X1, X2)√V (X1)V (X2)

, (6)

whereCov (X1, X2) = E [(X1 − E [X1]) (X2 − E [X2])]

andV (X1) = Cov (X1, X1) .

It is a measure of linear dependence and takes values in [−1, 1]. If X1and X2 are independent then

ρ (X1, X2) = 0,

but it is well known that the converse is false: the uncorrelatedness of X1 andX2 does not in general imply independence except for the case of normality.

Example 3.1: Consider the variate pair (X1, X2) which takes on values{(0, 1) , (0,−1) , (1, 0) (−1, 0)} with equal probability 1

4. The linear correla-

tion of the co-dependent variates X1 and X2 for this discrete distribution isidentically zero, but in fact, these variates are strongly dependent. If X1 = 0then X2 = 1 or X2 = −1. However, for X1 6= 0, y is fully determined: it hasto be zero.

11

If |ρ (X1, X2) | = 1, then this is equivalent to saying that X1 and X2 areperfectly linearly dependent, that is X2 = α + βX1 almost surely for someα ∈ R and β 6= 0, with β > 0 for positive linear dependence and β < 0 fornegative linear dependence.

12

Limitations of Linear Correlation as a Dependence Measure

• In strongly non-Normal distributions like in Example 3.1 above andnon-elliptical distributions, linear correlation can actually conceal thestrong co-dependence information contained in the full joint distribu-tion.

• The marginal distributions and pairwise correlations of a random vectordo not determine its joint distribution. Indeed, for a given pair ofmarginal distribution densities FX1 (X1) and FX2 (X2), there may noteven be a joint distribution density F (X1, X2) for every possible ρ ∈[−1, 1]

• The correlation is not invariant under nonlinear strictly increasingtransformations. That is, for two real-valued random variables wehave, in general ρ (T (X1) , T (X2))) 6= ρ (X1, X2) where T : R → Ris a nonlinear strictly increasing transformation. For example, log(X)and log(Y ) generally do not have the same correlation as X and Y .

• Correlation is only defined when the variances of X1 and X2 are finite.This restriction to finite-variance models is not ideal for a dependencemeasure and can cause problems when we work with heavy-tailed dis-tributions. For example, actuaries who model losses in different busi-ness lines with infinite-variance distributions may not describe the de-pendence of their risks using correlation. Indeed, the covariance andcorrelation between the two components of a bivariate t2-distributionrandom variable are not defined because of infinite second moments.

Advantages of linear Correlation

• Correlation is invariant under strictly linear transformations. That is,for β1, β2 > 0, ρ (α1 + β1X1, α2 + β2X2) = ρ (X1, X2).

• Correlation does not depend on scientific measurement units used.

• Has an easy algorithm and easy to interprete.

13

3.2 Rank Correlation

There are two types of rank correlations; Kendall’s tau and Spearman’s rho.They are better alternatives to linear correlation coefficient as a measure ofdependence for nonelliptical distributions, for which the linear correlationcoefficient is inappropriate and often misleading.

3.2.1 Kendall’s tau

Kendall’s tau is a co-dependence measure that focuses on the idea of con-cordance and discordance. Two points in R2, (X1, X2) and (Y1, Y2) aresaid to be concordant if (X1 − Y1)(X2 − Y2) > 0 and to be discordant if(X1 − Y1)(X2 − Y2) < 0.

Definition 3.2. Kendall’s tau for the random vector (X1, X2) is defined as

τk = τk(X1, X2) = P[(X1 − Y1)(X2 − Y2) > 0]− P[(X1 − Y1)(X2 − Y2) < 0]

where (Y1, Y2) is an independent copy of (X1, X2).

Hence Kendall’s tau for (X1, X2) is simply the probability of concordanceminus the probability of discordance.

Theorem 3.1. Given two variables X1 ∈ R and X2 ∈ R, their marginal dis-tribution densities fX1(X1) and fX2(X2), their respective cumulative marginaldistributions

FX1(X1) =

∫ X1

−∞f(x1)dx1, (7)

FX2(X2) =

∫ X2

−∞f(x2)dx2, (8)

and their joint distribution density function F (X1, X2) with

F (X1, X2) =

∫ X1

−∞

∫ X2

−∞f(x1, x2)dx2dx1. (9)

Then the Kendall’s tau for the continuous distribution densities is:

τk = 4

∫ ∫F (x1, x2)f(x1, x2)dx2dx1 − 1. (10)

14

Properties

1. Kendall’s tau is symmetric. That is τk(X1, X2) = τk(X2, X1).

2. τk(X1, X2) ∈ [−1, 1].

3. Kendall’s tau gives the value zero for independent random variables,although a rank correlation of 0 does not necessarily imply indepen-dence.

4. τk(X1, X2) = 1 when X1 and X2 are comonotonic (common monotonic-

ity). By monotonicity, we mean that (X1, X2)d=(F−1X1 (U), F

−1X2

(U)),

where ’d=’ stands for equality in distribution and U is a uniformly dis-

tributed random variable on the unit interval. (See [5]). Furthermore,τk(X1, X2) = −1 when X1 and X2 are countermonotonic.

The second limitation of linear correlation remains a pitfall under rank cor-relation: The marginal distributions and pairwise correlations of a randomvector do not determine its joint distribution. Indeed, for a given pair ofmarginal distribution densities FX1 (X1) and FX2 (X2), there may not evenbe a joint distribution density F (X1, X2) for every possible ρ ∈ [−1, 1]!

However, for any choice of continuous marginal distributions it is possibleto specify a bivariate distribution that has any desired rank correlation valuein [−1, 1].

15

3.3 Copulas

In [13], copulas are considered as ’functions that join or couple multivariatedistribution functions to their one-dimensional marginal distribution func-tions’ and as ’distribution functions whose one-dimensional margins are uni-form.’ But neither of these is a definition. [13] therefore goes ahead to give agood definition of copulas and some examples which we find in this section.

We are motivated to study copulas because they help in the understandingof dependence at a deeper level. They let us see the potential pitfalls ofapproaches to dependence that focuses only on correlation and show us howto define a number of useful alternative dependence measures. See [8]

The notion of an increasing function is used here. An increasing functionis a function f such that x ≤ y implies f(x) ≤ f(y). Also, DomH and RanH,denote the domain and range of the function H respectively.

Definition 3.3. Let H be a function such that DomH = S1×S2 is a subsetof R

2and RanH = is a subset of R. H is said to be 2-increasing if:

VH(B) := H(x2, y2)−H(x2, y1)−H(x1, y2) +H(x1, y1) ≥ 0,

for all rectangle B = [x1, x2]× [y1, y2] whose vertices lie in DomH

Definition 3.4. A function H from DomH = S1 × S2 into R is said to begrounded if H(x, a2) = 0 = H(a1, y) for all (x, y) in S1 × S2, where a1 anda2 are the least elements of S1 and S2 respectively.

Definition 3.5. A 2-dimensional copula is a function C from [0, 1]2 to [0, 1]with the following properties:

1. For every u, v in [0, 1],

C(u, 0) = 0 = C(0, v)

andC(u, 1) = u

C(1, v) = v;

2. For every u1, u2, v1, v2 in [0, 1] such that u1 ≤ u2 and v1 ≤ v2,

C(u2, v2)− C(u2, v1)− C(u1, v2) + C(u1, v1) ≥ 0.

A function C from [0, 1]2 to [0, 1] which is 2-increasing, grounded and saties-fies 1 and 2 above is also considered a copula. For n-dimensional copula, see[8], [13] and [10].

16

The following theorems are important results regarding copulas and itsapplications.

Theorem 3.2 (Sklar’s theorem). Let H be a joint distribution function withcumulative marginals FX1(x1) and GX2(x2). Then, there exists a copula Csuch that for all x1, x2 in R,

H(x1, x2) = C (FX1(x1), GX2(x2)) . (11)

If FX1(x1) and GX2(x2) are continuous, then C is unique; otherwise, C isuniquely determined on RanF × RanG. Conversely, C is a copula and Fand G are distribution functions, then the function H defined by (11) is ajoint distribution function with cumulative marginals F and G.

Proof. See [13]

From the above theorem, we see that for continuous multivariate distri-bution functions the univariate marginals and the multivariate dependencestructure can be seperated, and the dependence structure can be representedby a copula.

Theorem 3.3. Let H be a 2-dimensional distribution function with contin-uous marginals F and G and copula C (where C satifies (11)). Then, forany (u, v) in [0, 1]2 ,

C(u, v) = H(F−1X1 (u), G

−1X2(v)).

Theorem 3.4. For every copula C(u, v), we have the bounds

max (u+ v − 1, 0) ≤ C(u, v) ≤ min (u, v).Here is a general algorithm for random variate generation from copulas.

The properties of the specific copula family is often essential for the efficiencyof the corresponding algorithm. To generate observations (x, y) of a pair ofrandom variables (X, Y ) with a joint distribution function H, first generatea pair (u, v) of observations of uniform (0, 1) random variables (U, V ) whosejoint distribution function is C, the copula of X and Y (this is by virtue ofSklar’s theorem), and then transform those uniform variates using the inversetransform method. The conditional distribution method is used in this caseto generate the pair (u, v). The conditional distribution of V given U is givenby:

Cu(v) = P [V ≤ v |U = u] = lim∂u→0

C(u+ ∂u, v)− C(u, v)∂u

=∂C(u, v)

∂u. (12)

The following algorithm generates a random variate (u, v) from C. As usual,let U(0, 1) denote the uniform distribution on [0, 1].

17

Algorithm

1. Generate two independent U(0, 1) variates u and v.

2. Set v = C−1u (t) where C−1u denotes a quasi-inverse of Cu.

3. The desired pair is (u, v).

For other algorithms, see [7] and [9]. See [10] for a generalization.Example [13] Let X and Y be random variables whose joint distribution

function H (DomH = R2) is

H(x, y) =

(x+1)(ey−1)x+2ey−1 , (x, y) ∈ [−1, 1]× [0,∞]

1− ey, (x, y) ∈ (1,∞]× [0,∞]0, elsewhere

(13)

The copula C of X and Y is

C(u, v) =uv

u+ v − uvand so the conditional distribution function Cu and its inverse C

−1u are given

by

Cu(v) =∂C(u, v)

∂u=

(v

u+ v − uv

)2

and

C−1u =u√t

1− (1− u)√t.

Thus an algorithm to generate random variates (x, y) is:

1. Generate two independent U(0, 1) variates u and t;

2. Set v = u√t

1−(1−u)√t,

3. Set x = 2u− 1 and y = − ln(1− v)Note: x and y are the inverses of the marginals F and G given by

F (x) =

0, x < −1x+12, x ∈ [−1, 1]

1, x > 1(14)

and

G(y) =

{0, y < 01− e−y, y ≥ 0 (15)

18

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

Figure 3: Random variates generated from a sample of 500.

4. The desired pair is (x, y).

See [10] for another example.We now take a look at tail dependence and then study some copulas in

detail.

3.4 Tail Dependence

Tail dependence measures the dependence between the variables in the upper-right-quadrant tail or lower-left quadrant tail of [0, 1]2. It is a concept thatis relevant for the study of dependence between extreme values. Tail de-pendence between two continuous random variables X and Y is a copulaproperty and hence the amount of tail dependence is invariant under strictlyincreasing transformations of X and Y .

Definition 3.6. Let X and Y be continuous random variables with marginaldistribution functions F and G respectively. The coefficient of upper taildependence of X and Y is:

λU = limuր1

P[Y > G−1(u)|X > F−1(u)

],

provided that the limit λU ∈ [0, 1] exists. If λU ∈ (0, 1], X and Y are said tobe asymptotically dependent in the upper tail; if λU = 0, X and Y are saidto be asymptotically independent in the upper tail.

19

Similarly, coefficient of the lower tail dependence of X and Y is:

λL = limuց1

P[Y ≤ G−1(u)|X ≤ F−1(u)

],

provided that the limit λL ∈ [0, 1] exists.Since P [Y > G−1(u)|X > F−1(u)] can be written as1− P [X ≤ F−1(u)]− P [Y ≤ G−1(u)] + P [X ≤ F−1(u), Y ≤ G−1(u)]

1− P [X ≤ F−1(u)] ,

(See [15] [p.245]) an alternative and equivalent definition (for continuousrandom variables), from which it is seen that the concept of tail dependenceis indeed a copula property, is the following found in [4];

Definition 3.7. If a bivariate copula C is such that

limuր1

1− 2u+ C(u, u)1− u = λU

exists, then C has upper tail dependence if λU ∈ (0, 1], and upper tail inde-pendence if λU = 0.

In the last part of this section, we take a look at the Gaussian Copula,the t-copula, the Gumbel copula and the Clayton copula. The first twocopulas are elliptical while the Gumbel and Clayton copulas are Archimedeancopulas.

3.5 Elliptical Copulas

Elliptical copulas are simply the copulas of elliptical distributions. Since tosimulate from elliptical distributions is easy, it is easy to simulate from ellip-tical copulas making use of Sklar’s theorem. They are however, not withoutdrawbacks: They do not have closed form expressions and are restricted to

have radial symmetry(C = Ĉ

). In many finance and insurance applications

it seems reasonable that there is stronger dependence between big losses (e.ga stock market crash) than between big gains. Such asymmetries cannot bemodelled with elliptical copulas.

A general definition of elliptical distributions is the following:

Definition 3.8. [Elliptical Distributions] If X is an n-dimensional randomvector and, for some vector µ ∈ R2, some n × n nonnegative definite sym-metric matrix

∑and some function φ : [0,∞) → R, the characteristic func-

tion ϕX−µ of X − µ is of the form ϕX−µ(t) = φ(tT∑

t), we say that Xhas an elliptical distribution with parameter µ,

∑, and φ, and we write

X ∼ En(µ,∑

, φ).

20

When n = 1, the class of elliptical distributions coincides with the classof one-dimensional symmetric distributions. A function φ as in Definition3.8 is called a characteristic generator.

An important result to take note of is the following theorem which pro-vides a relation between the Kendall’s tau and the rank correlation matrix R(with Rij =

∑

ij√∑

ii

∑

jj

) for nondegenerate elliptical distributions. R is easily

estimated from data.

Theorem 3.5. Let X ∼ En(µ,∑

, φ) with P[Xi = µi] < 1 and P[Xj = µj] <1. Then

τ(Xi, Xj) =(1−

∑(P[Xi = µ])

2) 2πarcsin(Rij), (16)

where the sum extends over all atoms of the distribution of Xi. If in additionrank(

∑) ≥ 2, then equation 13 simplifies to

τ(Xi, Xj) =(1− (P[Xi = µ])2

) 2πarcsin(Rij). (17)

Proof. see [3]

Note: For 2-dimensional normal distribution with linear correlation coef-ficient ρ, τ = 2

πarcsin ρ.

3.5.1 Gaussian Copulas

The Gaussian copula of the n-variate normal distribution with linear corre-lation matrix R is

CGaR (u) = Φ−1R

(Φ−1(u1), ...,Φ

−1(un)),

where ΦnR denotes the joint distribution function of the n-variate standardnormal distribution function with linear correlation matrix R, and Φ−1 de-notes the inverse of the distribution function of the univariate standard nor-mal distribution. For the bivariate case, which we are more concerned with,the copula expression can be written as

CGaR (u, v) =

∫ Φ−1(u)

−∞

∫ Φ−1(v)

−∞

1

2π(1− ρ2) 12exp

{−s

2 − 2ρst+ t22(1− ρ2)

}dsdt.

Note: ρ = ρ(X1, X2).Tail dependence: The Gaussian copula do not have upper tail depen-

dence. See [10, Ex.3.4]. Since elliptical distributions are radially symmetric,the coefficient of the upper and lower tail dependence are equal. Hence Gaus-sian copulas do not have lower tail dependence.

21

3.5.2 t-copulas

The t-copula is conceptually very similar to the Gaussian Copula. It isgiven by the cumulative distribution function of the marginals of correlatedt-variates. If X has the stochastic representation.

X =d µ+

√υ√S

Z (18)

where µ ∈ Rn, S ∼ χ2ν and Z ∼ N(0,∑

) are independent, then X has an n-variate tν-distribution with mean µ (for ν > 1) and covariance matrix

νν−2∑

(for ν > 2). If ν ≤ 2 then Cov(X) is not defined. In this case we interpret∑as being the shape parameter of the distribution of X.The n-dimensional t-copula of X given by (15) is

Ctν,R(u) = tnν,R

(t−1ν (u1), ..., t

−1ν (un)

),

where R is a correlation matrix, tnν,R is the joint distribution function ofthe X with marginals tν . In the bivariate case the t-copula is expressed as

Ctρ(u, v) =

∫ t−1ν (u)

−∞

∫ t−1ν (v)

−∞

1

2π(1− ρ2) 12exp

{1 +

s2 − 2ρst+ t2ν(1− ρ2)

ν+22

}dsdt

where ρ is the linear correlation of the corresponding bivariate tν distributionif ν > 2.

Tail dependence: The t-copula has upper tail (and because of radialsymmetry) equal lower tail dependence.

3.6 Archimedean Copulas

Archimedean Copulas is a class of copulas worth studying for the followingreasons:

1. The ease with which they can be constructed.

2. The great variety of families of copula which belong to this class.

3. The many nice properties possessed by the members of this class; Theyhave closed form expressions as opposed to elliptical coplulas.

It should however, be noted that these copulas are not derived from multi-variate distribution functions using Sklar’s theorem. This acts as a drawbackwhen we try to extend Archimedean 2-copulas to n-copulas.

22

Definition 3.9. Let ϕ be a continuous, strictly decreasing function from[0, 1] to [0,∞] such that ϕ(1) = 0. The pseudo-inverse of ϕ is the functionϕ[−1] with Domϕ[−1] = [0,∞] and Ranϕ[−1] = [0, 1] given by

ϕ[−1](t) =

{ϕ−1(t), 0 ≤ t ≤ ϕ(0)0, ϕ(0) ≤ t ≤ ∞ (19)

ϕ[−1] so defined is continuous and nonincreasing on [0,∞], and strictlydecreasing on [o, ϕ(0)]. Furthermore, ϕ[−1](ϕ(u)) = u on [0, 1] and

ϕ(ϕ[−1](t)

)=

{t, 0 ≤ t ≤ ϕ(0)ϕ(0), ϕ(0) ≤ t ≤ ∞

= min(t, ϕ(0)).

Finally, if ϕ(0) = ∞, then ϕ[−1] = ϕ−1.

Lemma 3.1. Let ϕ be a continuous, strictly decreasing function from [0, 1]to [0,∞] such that ϕ(1) = 0 and let ϕ[−1] be the pseudo-inverse of ϕ definedby (19). Let C be the function from [0, 1]2 to [0, 1] given by

C(u, v) = ϕ[−1] (ϕ(u) + ϕ(v)) . (20)

Then C is 2-increasing if and only if for all v ∈ [0, 1], whenever u1 ≤ u2,

C(u2, v)− C(u1, v) ≤ u2 − u1. (21)

Proof. [13] Because (21) is equivalent to

C(u2, 1)−C(u2, v1)−C(u1, 1)+C(u1, v1) = u2−u1+C(u, v1)−C(u2, v1) ≥ 0,

it holds whenever C is 2-increasing. Hence assume that C satisfies (21).Choose v1, v2 in [0, 1] such that v1 ≤ v2, and note that C(0, v2) = 0 ≤ v1 ≤v2 = C(1, v2). But C is continuous (because ϕ and ϕ

[−1] are), and thus thereis a t in [0, 1] such that C(t, v2) = v1, or ϕ(v2) + ϕ(t) = ϕ(v1). Hence

C(u2, v1)− C(u1, v1) = ϕ[−1] (ϕ(u2) + ϕ(v1))− ϕ[−1] (ϕ(u1) + ϕ(v1)) ,= ϕ[−1] (ϕ(u2) + ϕ(v1) + ϕ(t))− ϕ[−1] (ϕ(u1) + ϕ(v1) + ϕ(t)) ,= C (C(u2, v2), t)− C (C(u1, v2), t) ,≤ C(u2, v2)− C(u1, v2),

so that C is 2-increasing.

23

Theorem 3.6. Let ϕ be a continuous, strictly decreasing function from [0, 1]to [0,∞] such that ϕ(1) = 0 and let ϕ[−1] be the pseudo-inverse of ϕ definedby (19). Let C be the function from [0, 1]2 to [0, 1] given by (20). Then C isa copula if and only if ϕ is convex.

Proof. [13]

1.C(u, 0) = ϕ[−1](ϕ(u) + ϕ(0)) = 0;

C(u, 1) = ϕ[−1](ϕ(u) + ϕ(1))

= ϕ[−1](ϕ(u)) = u.

By symmetry, C(u, v) = 0 and C(1, v) = v

2. To complete the proof, we show that (21) holds (by the last lemma) ifand only if ϕ is convex. We note tht ϕ is convex if and only if ϕ[−1] isconvex. Observe that (21) is equivalent to

u1 + ϕ[−1](ϕ(u2) + ϕ(v)) ≤ u2 + ϕ[−1](ϕ(u1) + ϕ(v))

for u1 ≤ u2, so that if we set a = ϕ(u1), b = ϕ(u2), and c = ϕ(v), then(21) is equivalent to

ϕ[−1](a) + ϕ[−1](b+ c) ≤ ϕ[−1](b) + ϕ[−1](a+ c), (22)

where a ≥ b and c ≤ 0. Now suppose (21) holds, i.e., suppose that ϕ[−1]satisfies (22). Choose any s, t in [0,∞] such that 0 ≤ s < t. If we seta = (s+ t)/2, b = s, and c = (t− s)/2 in (22), we have

ϕ[−1](s+ t

2

)≤ ϕ

[−1](s) + ϕ[−1](t)

2. (23)

Thus ϕ[−1] is midconvex, and because ϕ[−1] is continuous it follows thatϕ[−1] is convex.

In the other direction, assume ϕ[−1] is convex. Fix a, b, and c in [0, 1]such that a ≥ b and c ≤ 0; and let γ = (a− b)/(a− b+ c), and hence

ϕ[−1](a) ≤ (1− γ)ϕ[−1](b) + γϕ[−1](a+ c)

andϕ[−1](b+ c) ≤ γϕ[−1](b) + (1− γ)ϕ[−1](a+ c).

Adding these inequalities yields (22), which completes the proof.

24

Copulas of the form (20) are called Archimedean copulas. The functionϕ is called a generator of the copula. If ϕ(0) = ∞, we say that ϕ is a strictgenerator. In this case, ϕ[−1] = ϕ−1 and C(u, v) = ϕ−1(ϕ(u) + ϕ(v)) is saidto be a strict Archimedean copula.

For some examples of Archimedean Copulas, see and [13].

3.6.1 Clayton Copula

The Clayton Copula is generated by

ϕ(t) = (t−1 − 1)/θ,

where θ ∈ [−1,∞] 0 and defined as

Cθ(u, v) = max([

u−θ + v−θ − 1]−1/θ

, 0).

For θ > 0 the copulas are strict and the copula expression simplifies to

Cθ(u, v) =(u−θ + v−θ − 1

)−1/θ.

The Clayton family has lower tail dependence for θ > 0, and C−1 =max(u+ v, 0), limθ→0Cθ = uv and limθ→∞Cθ = min(u, v).

Kendall’s tau for this family is given by tθ =θ

θ+2.

3.6.2 Gumbel Copula

The Gumbel Copula (sometimes also refered to as the Gumbel-HougaardCopula) is controlled by a single parameter θ ≥ 1. It is generated by

ϕ(t) = (− ln t)θ,

and defined as

Cθ(u, v) = exp

(−[(− ln u)θ + (− ln v)θ

]1/θ).

Indeed:ϕ(t) is continuous and ϕ(0) = 0

ϕ′

(t) = −θ (− ln t)θ−1 1t, so ϕ is a strictly decreasing function from [0, 1] to

[0,∞].

25

ϕ′′

(t) ≥ 0 on [0, 1], so ϕ is convex. Moreover, ϕ(0) = ∞, so ϕ is a strictgenerator. Thus ϕ[−1](t) = ϕ−1(t) = exp(−t). From (17) we get

Cθ(u, v) = ϕ[−1](ϕ(u) + ϕ(v)) = exp

(−[(− ln u)θ + (−lnv)θ

]1/θ).

If θ = 1 then C1(u, v) = uv and limθ→∞ Cθ = min(u, v).Tail Dependence: Consider the bivariate Gumbel family of copulas

given above. Then by 3.7

1− 2u+ C(u, u)1− u =

1− 2u+ exp(21/θ ln u)1− u

=1− 2u+ u21/θ

1− u ,

and hence

limuր1

(1− 2u+ C(u, u)

1− u

)= lim

uր1

(−2 + 21/θu21/θ−1

−1

)

= 2− limuր1

(21/θu2

1/θ−1)

= 2− 21/θ.

Thus for θ > 1, Cθ has upper tail dependence.Kendall’s tau: With the generator for the Gumbel family given above,

ϕ(t)

ϕ′(t)=

t ln t

θ.

Using theorem 21 in [1], we can calculate Kendall’s tau for the Gumbel family.

tθ = 1 + 4

∫ 1

0

t ln t

θdt

= 1 +4

θ

([t2

2ln t

]1

0

− ∈ t10t

2dt

)

= 1 +4

θ(0− 1

4)

= 1− 1θ.

1tθ(u, v) = 4∫ 10

ϕ(t)

ϕ′ (t)

dt+ 1

26

3.6.3 Properties of Archimedean Copula

Stated below are two theorems concerning some algebraic properties of ArchimedeanCopulas.

Theorem 3.7. Let C be an Archimedean copula with generator ϕ. Then

1. C is symmetric i.e. C(u, v) = C(v, u) for all u, v in [0, 1].

2. C is associative i.e. C(C(u, v), w) = C(u, C(v, w)) for all u, v, w in[0, 1].

Proof. 1.

C(u, v) = ϕ[−1](ϕ(u) + ϕ(v))

= ϕ[−1](ϕ(v) + ϕ(u))

= C(v, u).

2.

C(C(u, v), w) = ϕ[−1](ϕ(ϕ[−1](ϕ(u) + ϕ(v))) + ϕ(w))

= ϕ[−1](ϕ(u) + ϕ(v) + ϕ(w))

= ϕ[−1](ϕ(u) + ϕ(ϕ[−1](ϕ(v) + ϕ(w))))

= C(u, C(v, w)).

Note: The associative property of Archimedean copulas is not shared bycopulas in general. See [10] for an example.

Theorem 3.8. Let C be an associative copula such that the diagonal sectionδc satisfies δc(u) < u for all u in (0, 1). Then C is Archimedean.

It that it is possible to extend Archimedean copulas to higher dimensions.However, these extensions suffer from lack of free parameter choice in thesense that some of the entries in the resulting rank correlation matrix areforced to be equal. For a possible multivariate extension see [10] as well as[13].

27

4 Data Characterisation and Analysis

4.1 Nature of Data

Table 1: Data for Kronoberg

No. Region Damage yr. ObjNo. Objtype Damagedate payment Damage cost1 Kronoberg 1998 1 Home 19980101 0 02 Kronoberg 1998 4 Holiday 19980102 192734 1927343 Kronoberg 1998 3 Villa 19980102 25679 256794 Kronoberg 1998 6 farms 19980102 10640 106405 Kronoberg 1998 4 Holiday 19980101 5315 53156 Kronoberg 1998 3 Villa 19980105 1000 10007 Kronoberg 1998 3 Villa 19980105 7715 77158 Kronoberg 1998 3 Villa 19980105 246339 2463399 Kronoberg 1998 3 Villa 19980105 16159 1615910 Kronoberg 1998 3 Villa 19980105 24621 2462111 Kronoberg 1998 3 Villa 19980105 14416 1441612 Kronoberg 1998 4 Holiday 19980105 8000 800013 Kronoberg 1998 6 Farms 19980107 15814 15814.. ......... .... .. .... .......... .... ......

28

Table 2: Data for Göinge

No. Region Damage yr. ObjNo. Objtype Damagedate payment Damage cost1 Göinge 1998 3 Villa 19980102 0 02 Göinge 1998 3 Villa 19980101 0 03 Göinge 1998 3 Villa 19980101 0 04 Göinge 1998 3 Villa 19980102 0 05 Göinge 1998 3 Villa 19980101 0 06 Göinge 1998 3 Villa 19980105 1588 15887 Göinge 1998 3 Villa 19980101 4000 40008 Göinge 1998 6 Farms 19980105 0 09 Göinge 1998 11 Business 19980102 3705 370510 Göinge 1998 6 Farms 19980102 0 011 Göinge 1998 4 Holiday 19980105 1200 120012 Göinge 1998 12 koff 19980102 17405 1740513 Göinge 1998 5 property 19980102 0 0.. ......... .... .. .... .......... .... ......

The data above is data of daily reported damages obtained from Läns-försäkringar Kronoberg. It is collected over a five year period, 1998-2002.The damage costs are from two regions in Sweden (Göinge and Kronoberg)and are of different types home/villa, farms, business and accidents. It shouldnot be surprising to see holiday as an entry with object number 4 since if anindividual is involved in an accident while on vacation outside Sweden, he iscovered by his home insurance.

The following table describes the column 4 of the data.

Table 3: Data Description

Insurance Type Object number

Home and Villa 01,02,03,04,20Farms 06Property,Company 05,07,08,09,11,12,13,14Accidents 30

4.2 Results and Explanations

It is sufficient to model dependence between damage costs of homes in Göingeand those of Kronoberg. The same models can be used for the other data

29

types. Copulas are the prefered dependence measure and the Normal andNIG are the Marginal distributions used.

4 6 8 10 12 14 164

6

8

10

12

14

16

datasim marg Norm Gauss cop

Figure 4: Simulated points from the Gaussian Copula using the Normalmarginals

4 6 8 10 12 14 162

4

6

8

10

12

14

16

18

datasim marg NIG Gauss cop

Figure 5: Simulated points from the Gaussian Copula using the NormalInverse Gussian marginals

Figures 4, 5 , 7 and 8 show samples from bivariate distributions withGaussian and t-coplulas. The Gaussian copula does not get the extreme jointtail observations while the t-copula seems to be able to do just that (thoughnot very clear from the figures). t-copulas yield dependence structures withtail dependence.

The dependence observed between Homes in Kronoberg and Homes inGöinge can be due to seasons or weather.

30

12 14 16 18 20 22 24 26 28 30 320

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

real dataN marg Gauss copNIG marg Gauss cop

20 22 24 26 28 30

0.65

0.7

0.75

0.8

0.85

0.9

0.95

real dataN marg Gauss copNIG marg Gauss cop

Figure 6: Cumulative density function of X1 +X2 (cdf zoom is below)

31

4 6 8 10 12 14 164

6

8

10

12

14

16

dataN marg t cop

Figure 7: Simulated points from the t Copula with Normal marginals

4 6 8 10 12 14 164

6

8

10

12

14

16

dataNIG marg t cop

Figure 8: Simulated points from the t Copula using the Normal InverseGaussian marginals

32

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

16

18

dataNIG marg Clayton cop

Figure 9: Simulated points from the Clayton copula using the NIG marginals

To summerize it all, consider the total home damage cost eX1 + eX2 . Thefigure 10 below illustrates each of the distributions with their correspondingcopulas perform. We should note that a good dependence is reflected by agood fit of the total cost.

One can quantify the error by a version of the two-sample Kolmogorov-Smirnov Test.

Table 4: Two-sample Kolmogorov-Smirnov test

Model p-value k-value

NIG marginals, Gaussian copula 0.9366 0.0178NIG marginals, Clayton copula 0.4600 0.0283Normal marginals, Gaussian copula 0.1389 0.0383NIG marginals, Gumbel copula 0.1180 0.0399Edgeworth marginals, Gaussian copula 0.0856 0.0417Normal marginals, Independence 0.0157 0.0517

Here k = maxx

∣∣∣F̂model(x)− F̂empirical(x)∣∣∣ , where the hats denote esti-

mates. The Gaussian copula with NIG marginals has a p-value of 0.9366meaning that the difference between this distribution and the real data isnot significant at the .05 level. The difference is k = 0.0178 which is theleast distance compared with the k values of the other models. Therefore,the Gaussian copula with NIG marginals is the best model. Take note thatwhen our variables modelled with the Normal marginas are considered in-dependent, it retains a p-value of 0.0157 and the largest distance k = .0517

33

102

103

104

105

106

107

108

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

real dataN marg Gauss copNIG marg Gauss copNIG marg Clayton copulaNormal marg indepEdg marg Gauss cop

Figure 10: Cumulative distribution function for eX1 + eX2 .

34

which means that real data deviate significantly from the model with inde-pendent Normal margins. This is not surprising.

35

5 Conclusion

The main objective of this thesis is to model dependency of two-dimensionaldata using the concept of copulas. The investigated data reflects damagecosts from two different regions.

The copula is realized to perform best where linear correlation fails. Thecopula joins together marginal distributions to form a joint distribution butpairwise correlation fails to do that. Therefore dependency is better definedby the copula. Nevertheless, the Gaussian Copula with Normal marginsperforms like linearly correlated random variables modelled with the Normalmarginals.

Furthermore, with the same copula different margins seem to performdifferently. The distribution of the data are approximated using the Normalmarginal, the Normal Inverse Gaussian marginal and Edgeworth marginalsamong some different types of copula. It is observed that the Normal In-verse Gaussian marginals performs best with the Gaussian copula to modeldependence.

36

Acknowledgement

I would like to thank my supervisor, R. Pettersson for his patience and enor-mous contribution to this work. R. Pettersson expresses gratitude to Profes-sor Sören Asmussen for suggesting to consider copulas for insurance data.

37

A Appendix:Matlab codes

Data in file damagedata.mat

load damagedata

X1=log(hem(hem>0&ghem>0));%Kronoberg

X2=log(ghem(hem>0&ghem>0));%Göinge

Estimated parameters:

[alpha beta delta mu]=cumnigest(X1);%1.5429 0.0261 3.7127 9.7679

[galpha gbeta gdelta gmu]=cumnigest(X2);%1.1875 0.3429 2.3686 8.5467

[r p]=corr(X1’,X2’)%0.1345 6.2524e-05

[rKendall pKendall]=corr(X1’,X2’,’type’,’Kendall’)%0.0869 1.1487e-04

tau=rKendall;

rho=sin(tau*pi/2);

n=length(X1)

m1=mean(X1);

sigma1=std(X1);

m2=mean(X2);

sigma2=std(X2);

cumulative density functions of approximated distribu-

tions:

[F1 x1]=ecdf(X1);

plot(x1,F1,x1,normcdf(x1,m1,sigma1),’--’,x1,

FEdg1_3(x1,m1,sigma1),’:’,x1,nigcdf(x1,alpha,beta,delta,mu),’-.’),

legend(’real data’,’normal cdf’,’3 order Edgew cdf’,’NIG cdf’)

print -depsc densX1Edge.eps

[F2 x2]=ecdf(X2);

plot(x2,F2,x2,normcdf(x2,m2,sigma2),’--’,x2,FEdg2_3(x2,m2,sigma2),’:’,x2,

nigcdf(x2,galpha,gbeta,gdelta,gmu),’-.’),

legend(’real data’,’normal cdf’,’Edgew cdf’,’NIG cdf’)

print -depsc densX2Edge.eps

38

A.1 Random variate generation, an example

n=500;

u=rand(n,1);

t=rand(n,1);

v=u.*sqrt(t)./(1-(1-u)).*sqrt(t);

x=2*u-1; y=-log(1-v);

plot(x,y,’.’)

print -depsc copex.eps

Normal marginals Gaussian copula:

%Xnorm=[norminv(U(:,1),m1,sigma1) norminv(U(:,2),m2,sigma2)]

XNmargGcop=[m1+sigma1*norminv(U(:,1)) m2+sigma2*norminv(U(:,2))]

%figure(1);

plot(X1,X2,’.’,XNmargGcop(:,1),XNmargGcop(:,2),’+’)

legend(’data’,’norm marg Gauss cop’)

%print -depsc NmargGausscop.eps

NIG marginals Gaussian copula:

XNIGmargGcop=[niginv(U(:,1),alpha,beta,delta,mu) niginv(U(:,2),

galpha,gbeta,gdelta,gmu)]

%subplot(211),plot(X1,X2,’.’)

%subplot(212),plot(X(:,1),X(:,2),’.’)

%figure(2);

plot(X1,X2,’.’,XNIGmargGcop(:,1),XNIGmargGcop(:,2),’+’)

legend(’data’,’NIG marg Gauss cop’)

%print -depsc NIGmargGausscop.eps

Models together:

semilogx(xe,Fe,xNmargGcop,FNmargGcop,’--’,xNIGmargGcop,FNIGmargGcop,’:’,

xNIGmargCcop,FNIGmargCcop,’-.’,xNmargind,FNmargind,’-’,

xEdgmarGcop,FEdgmargGcop,’--’)

legend(’real data’,’N marg Gauss cop’,’NIG marg Gauss cop’,

’NIG marg Clayton copula’,’Normal marg indep’,’Edg marg Gauss cop’)

print -depsc sumGcopcdf.eps

More codes can be obtained from author upon request.

39

References

[1] C. Genest and J. Mackay The Joy of Copulas: Bivariate Distributionswith Uniform Marginals, The American Statistician, 280-283, Vol. 40,1986.

[2] E. Novikova, Modellering av försäkringsdata med normal invers gaussisk(NIG)-fördelning. Magister Thesis, Växjö Universitet,2006.

[3] F. Lindskog, A. McNeil, and U. Schmock Kendall’s Tau for EllipticalDistributions,2001.

[4] H. Joe Multivariate Models and Dependence Concepts. Chapman andHall, London, 1997.

[5] J. Dhaene, S. Vanduffel, M. Goovaerts Comotonicity. 2007.

[6] J. Ekström Uppskattning av svansscannolikhet från försäkringsdata medEdeworthutveckling. Magister Thesis, Växjö Universitet, 2004.

[7] Johnson M. Multivariate Statistical Simulation. Wiley, New York, 1987.

[8] J. McNeil, R. Frey and P. Embrechts Quantitative Risk Manage-ment.Concepts, Techniques and Tools. Princeton University Press, 2005.

[9] L. Devroye Non-Uniform Random Variate Generation. Springer, NewYork, 1986.

[10] P. Embrechts, F. Lindskog and A. McNeil, Modelling Dependence withCopulas and Applications to Risk Management. 2001.

[11] P. Jäckel Monte Carlo Methods in Finance. Wiley, 2002.

[12] P. Hall The Bootstrap and Edgeworth Expansion. Springer, New York,1992.

[13] R. B. Nelsen An Introduction to Copulas. Springer, New York, 2006.

[14] S. Cambanis, S. Huang and G. Simons On the Theory of Elliptical Con-toured Distribution , Journal of Multivariate Analysis,11,368-385, 1981.

[15] S. Ross A first Course in Probability. Prentice Hall, California, 1998.

[16] Wim Schoutens Levy Processes in Finance. Pricing Financial Deriva-tives, John Wiley & Sons, Ltd,2003.

40

SE-351 95 Växjö / SE-391 82 Kalmar Tel +46-772-28 80 00 dfm@lnu.se Lnu.se

Degree Project

Modelling Dependence of Insurance Risks

SE-351 95 Växjö / SE-391 82 Kalmar

Tel +46-772-28 80 00

dfm@lnu.se

Lnu.se

Marie Manyi Taku

2010-10-19

Subject: Mathematics

Level: Master

Course code: 5MA11E