Modeling dependence in the Internal Capital Assessment Framework

Universiteit van Amsterdam

Faculteit der Economische Wetenschappen en Econometrie

Modeling Dependence in the

Internal Capital Assessment

Framework

Y.V. Gangaram Panday

Actuarial Science

13 August 2008

Department: Quantitative Economics

Supervisors: Dr. A. Chen

University of Amsterdam

Prof. Dr. A.A.J. Pelsser

University of Amsterdam

Drs. S. Tolk AAG

Delta Lloyd Groep

Page of contents

1. Introduction 3

2. Economic capital and correlation matrix 6

2.1 Economic capital 6

2.1.1 Calculation of economic capital 7

2.2 The correlation matrix and GI risk components 7

3. Methods to quantify dependence 10

3.1 Linear correlation 10

3.1.1 Shortcomings of linear correlation 11

3.2 Understanding relationships using copulas 12

3.2.1 Properties of copulas 12

3.2.2 Sklar’s theorem (1959) 12

3.3 Dependence concepts 13

3.3.1 Comonotonicity 13

3.3.2 Concordance and discordance 14

3.3.3 Rank correlation 14 3.3.3.1 Kendall’s tau 15 3.3.3.2 Spearman’s rho 15 3.3.4 Tail dependence 16 3.4 Elliptical copulas 17 3.4.1 Gaussian copulas 17 3.4.2 t-copulas 17 3.5 Archimedean copulas 18

3.5.1 Properties of Archimedean copulas 19

3.5.2 Gumbel family 19

3.5.3 Clayton family 20

3.5.4 Frank family 20

3.5.5 Kendall’s tau for Archimedean copulas 21

3.6 The “chosen” method to quantify dependence 22

3.6.1 Estimation of parameters 24

3.7 Dichotomous copula model 26

4. Illustration: “Quantification of dependence” 28

4.1 Description of data 28

4.1.1 Empirical cumulative distribution function for complete data 29 4.1.2 Empirical cumulative distribution function for separated data interval 31 4.2 Research: “Quantification of dependence between risks” 32 4.2.1 Results for single and mixture copulas 32 4.2.2 Results for dichotomous copula model 34

4.3 Comparative analyses 35

4.4 Aggregation of economic capital 36

1. Introduction

Economic capital and the allocation of capital to lines of businesses are an important part of the financial and risk management of an insurance company, particularly when the insurer will be under the regulation of Solvency II. Economic capital is the solvency rule where all individual risks are measured and transformed into individual capital charges. The measurement of these risks is done on a market-consistent base using the best information of the portfolio. Fair value techniques are used and for risks with a long history a time series approach is used. If all the individual risks are transformed into capital charges, they are put together to create the final amount of economic capital for the company. Measurement and management of economic capital are taken into account by insurers across a wide range of risks and business products. Thus there is a need for

aggregation of the losses from various risk drivers. The importance of economic capital is mainly based on four reasons. Firstly it is important for the insurance company to have a safe level of capital. Secondly it should increase the efficiency of capital (the insurer can create diversification benefits). Thirdly economic capital is used to prevent under- or over-capitalized firms, and last it should improve the company’s risk management.

Aggregation of economic capital is driven by risk dependencies among losses from different risk factors and lines of businesses. Therefore, it is extremely important for insurers to have a complete and optimal understanding of the dependence structure among risk drivers. Tang et al (2006) describe the importance of aggregation taking into account the dependence structure. The current industry mostly uses linear correlation as a measure of dependence, but it does not describe the dependence structure among risks well. However, the increasing complexity of insurance products has led to an increasing interest of modeling dependence among risks in actuarial science (Wang (1997)). Due to these complex products and risks, linear correlation is insufficient to understand the dependence structure. A more sophisticated technique, namely copulas, has been introduced to the world of actuarial science to model dependence. Nelson (1999) gives a good introduction to the underlying theory of copulas.

At this stage Delta Lloyd Group is using two internal models to determine economic capital. These two are the Internal Capital Assessment Framework, where stress and scenario tests are used, and the Risk Based Capital Framework, where economic capital is determined with Monte Carlo

simulations. This research focuses on the Internal Capital Assessment Framework, where the results of the individual stress tests are aggregated to an overall level of economic capital, taking into consideration correlations among risks. The dependencies among the risks are currently described through linear correlation in a correlation matrix provided by Aviva (Parent company of Delta Lloyd Group). Due to the fact that Dutch data are obviously different from UK data, it is essential to determine the dependence structure based on Dutch data. This shall lead to an improvement of Delta Lloyd Group’s risk management. Therefore, knowledge, specification and estimation of the dependence structure are significant for Delta Lloyd Group.

This research aims to determine the dependence structure between the General Insurance reserve test and the General Insurance claim ratio volatility test. These two risks are the most important sources of insurance risk to which the General Insurance business of Delta Lloyd Group is exposed. For these risks stress and or scenario tests are performed to determine the level of economic capital. The research is mainly designed to answer two questions: How are these risks related to each other and what level of dependence should be taken into account when it comes to

aggregation of economic capital?

Frees et al (1997) used copulas to describe the dependence structure between risks with single copula models. Mixtures of copulas were used to determine the dependence structure between risks by Canela et al (2005). In this research these types of mixture copula models will be used. The dependence will also be quantified with a constructed dichotomous copula model. With the mixture copulas a more extensive dependence structure can be determined, because the single Archimedean copula models, namely Gumbel, Calyton and frank, are not capable to determine dependence in both tails. These mixtures will be constructed with single Archimedean copulas. If two single copulas are combined dependence can be determined in both tails, as well as Kendall’s tau. The mixtures can be constructed with a fixed weight for each single copula, or with the weight as an extra unknown parameter. In this thesis both cases will be studied.

In the dichotomous copula model the data is split into separated data intervals to study the

dependence and thereupon an overall dependence measure is determined. This way of splitting the data in two seperated intervals is inspired by the Dichotomous Asset Pricing Model (DAPM) by Zou (2005). In this model the market returns are split into an upper and a lower market. For each data interval an analysis is done with the DAPM model. In this thesis the data is split into two parts, where the probability for a stochastic random variable to be in each interval is 50%. After

assessing the dependence measure in each interval, it is aggregated to a measure of dependence for the complete dataset.

This study led to some key findings about the dependence structure between the reserves and the claim ratio’s. With the method of linear correlation, the correlation coeficient is equal to 57.02%. Determination of the dependence structure with copulas led to different results. The dependence measure obtained with copulas is 39.13% (best estimate). The corresponding 99.5% confidence inteval is (34.52%, 43.12%). Taking a certain level of prudence into account it is wise to use the upper bound, namely 43.12%, when it comes to aggregation of economic capital to an overall level. One could observe that linear correlation over-estimates the dependence between the

reserves and claim ratio’s. This overestimation also leads to an amount of economic capital which is 2.500.000 euro higher than in the case where economic capital is aggregated with 43.12%. With more knowledge about the dependence structure Delta Lloyd can increase it’s efficiency of their capital. Furthermore there is evidence of assymetric tail dependence between these two risks, that allows us to reject the elliptical copulas. With an analysis of the goodness of fit statistics there is evidence that the copula mixtures and the dichotomous copula model are superior to the single

Archimedean copulas. An extension can be made for the dichotomous copula model by selection of an optimal threshold, like in the case of the mixture copulas with unkown weights.

The setup of the thesis is as follows. Chapter 2 gives a brief overview of economic capital and describes the correlation matrix with its risk components. Chapter 3 introduces several methods to quantify dependence. Different dependence concepts are discussed including their advantages and disadvantages. The chosen method to give a reflection of dependence structure between two risk components is discussed along with the dichotomous copula model. Chapter 4 describes the dataset used during the research with the statistical characteristics of the risk components. Thereupon the dependence among the risk components is quantified according to the optimal method discussed and constructed in chapter three. The several models with their results are compared and a statement can be made about the “best” model. Finally the thesis concludes with an answer of the central question and some possible recommendations for Delta Lloyd Group.

2. Economic capital and correlation matrix

Solvency positions of insurance companies are very important. Economic capital has always been an important concept for bank regulations. With the introduction of Solvency II which advocates risk based solvency requirements for insurance companies, more attention has been attracted to economic capital in the insurance regulation. In this chapter the concept of economic capital is discussed along with some reasons why it is such an important concept. The importance of the dependence structure among risks is discussed as well. Furthermore this chapter gives a brief overview of the risks a General Insurer is exposed to. The dependence between these risks will be quantified during the research.

2.1 Economic capital

Economic capital is an important concept especially for banks and insurers. The concept of economic capital is very well used in the management of a company’s resources. The word “economic”, is mostly interpreted as either “realistic” or “market consistent valuation”, and capital refers to the discounted expected value of future cash flows. Finkelstein et al (2006) describe two ways to interpret this concept, namely the amount of economic capital a business believes it needs, which is known as Required Economic Capital, and the amount of economic capital the business actually has, also known as Available Economic Capital. The concept of economic capital is the capital required to be held to protect customer liabilities from risk events (market risk, insurance risk, operational risk etc) at a given tolerance level.

Economic capital allows insurers to better take into account the cost of risk when planning and reassessing their future investment strategies. It helps them to make better operational decisions when it comes to pricing and capital allocation. It also allows them to measure their own level of risk aversion and to ensure that they have enough capital to cover their outstanding liabilities and possible disasters in the future. According to Scott, the use of economic capital is based on four main reasons, namely

• To ensure a safe level of capital to meet regulatory requirements and possible disaster scenarios in the future.

• To ensure that risk management is appropriate and to assess whether insurance policies, reinsurance programs or risk controls are cost-efficient.

• To ensure that the firm is not under- or over-capitalized.

• To ensure that capital is used as efficiently as possible to generate optimal returns, to assess strategy and to support decision making.

These fundamental reasons behind the use of economic capital are supported if as much as knowledge is available about the dependence structure among several risk drivers to which

insurers are exposed . More knowledge about the dependence structure will lead to an optimization of these four main reasons. Dependence will be studied during this research in order to improve

2.1.1 Calculations of economic capital

As mentioned before economic capital gives better understanding of the risks an insurance company is involved in. It is important to quantify the risks the company is facing over a time period that is relevant for management, e.g. by analyzing what the potential losses could be, as well as the probability and the size of a particular loss. The risk aversion of the company should also be considered, to decide whether they are prepared to take on a certain level of risk in exchange for a certain level of return. After this stage the company can calculate the economic capital for their business. The level of economic capital is mainly based on some key contributors, i.e. credit risk, market risk, operational risk, insurance risk and liquidity risk. Economic capital should provide a quantified measure to sustain potential unexpected losses up to a specified confidence level. Focus and urgency have been given to economic capital by regulatory mandates of Basel II and Solvency II.

Delta Lloyd Group is using a framework called Internal Capital Assessment, which is a method to quantify risks with the use of stress and scenario testing. At this stage the Internal Capital

Assessment model does not satisfy the Solvency II regulations completely. The approach is based on a Value-at-Risk idea where the current Available Economic Capital (AEC) is compared to the Required Economic Capital (REC) within a one-year time horizon to remain in a solvency position at a confidence level of 99.5%

(P[AEC

>

REC]

=

99.5%).

This can be interpreted as a 1 in 200 year’s probability of insolvency. The Internal Capital Assessment is an internal framework for identifying the risks which the insurer is exposed to and which have their specific impacts on the economic capital. Stress and scenario tests are performed to determine capital regulatory associated with all the risks except operational risk. Economic capital is then quantified by combining the results of stress and scenario tests with an operational risk model. Delta Lloyd Group uses Internal Capital Assessment as their internal model for capital where all risks are taken into account on adequate basis. This model is quite similar to the internal model of “het Financieel Toetsingskader”.

2.2 The correlation matrix and GI risk components

In the Internal Capital Assessment model, Delta Lloyd Group has implemented the correlation matrix provided by its parent company, Aviva. The dependence measures in this matrix are obtained with linear correlation. These correlations are based on assumptions made by Aviva and are determined using United Kingdom data. The correlations matrix might fit UK Insurance but may not be suitable for Delta Lloyd as the characteristics of the Dutch portfolio are obviously different from the UK portfolios. It is therefore high time to study dependence among risks for Delta Lloyd Group based on their own portfolio, which shall lead to an improvement of the internal model when it comes to aggregation of economic capital for risks.

The following matrix is now implemented and this thesis will be based on the research among the most important risk factors in matrix 2.1. For Delta Lloyd Group, insurance risk is the most important risk factor, especially for general insurance. This insurance risk for general insurance companies is calculated by a catastrophe test; an expense test; a reserve test and a claim ratio volatility test. The dependence between the last two tests is measured in this thesis. Business Units should compute and implement the correlation coefficient between these two variables themselves. For this correlation a minimum is set of 40% by Aviva. In chapter three different method are discussed to quantify the correlations for Delta Lloyd Group.

Matrix 2.1

GI Catastrophes GI Reserves GI Claim ratio vol Expenses

GI Catastrophes 100% 0% 0% 0%

GI Reserves 100% 40% (minimum) 0%

GI Claim ratio vol 100% 0%

Expenses 100%

As the quantification of the correlations among important risks in the correlation matrix is the backbone of this research, in the following a brief description is given of these risk components for General Insurers.

GI Catastrophes

Insurers are exposed to possible disaster events in the future, thus they are involved with

catastrophe risk. Non-life insurers should assess the impact on their financial position under several catastrophic scenarios, for example the impact of a flood, a windstorm or an earthquake. The estimated losses due to catastrophic events are calculated by insurance companies through their own internal models. The use of such models can increase the level of understanding and

management of the risk in their portfolio. A drama in the past was the World Trade Centre attacks. These attacks led to huge claims at Swiss Re.

GI Reserves

Reserving risk is a part of insurance risk. Insurance risk is the inherit uncertainties as to the occurrence, amount and timing of insurance liabilities. Archer-Lock et al (2004) describe reserving risk as the risk which is associated with the uncertainty of the adequacy of claim reserves and provisions for unearned premiums and unexpected risks. This category of risk brought several insurance companies in difficulties in the 80’s when old claims on asbestos were suddenly reported. An increase of longevity also led to large losses in specialized insurance companies (Movir).

GI Claim ratio volatility

Claim behavior can be different from expected over time and the insurer should have extensive knowledge about the underlying volatility in claims of their different products. With Solvency II in mind, insurance companies will face more stringent capital requirements for products with a high volatility in their claims. Here the unexpected claims as a result of a disaster event should not be included because the category for catastrophe risk already takes this into consideration.

Expenses

There is a possibility that the level of expenses is higher than expected. The type of risk related to such an event should also be considered by the insurer.

The correlation between expenses and catastrophes, expenses and reserves and expenses and claim ratio volatility is set to 0% because Aviva assumes no reason to suppose any significant correlation. Aviva also assumes that eventual correlation between expenses and these risks can be diversified away. The catastrophe test is not related to the other stress tests, because unexpected catastrophic claims are not included in the reserves and claim ratio volatilities. These large claims are taken into account separately by the catastrophe test and therefore a correlation of 0% is considered. The research will focus on the dependence between the General Insurance Reserves and General Insurance Claim ratio volatility.

When it comes to aggregation of economic capital to one level, the diversification effect can be applied. With an increasment of knowledge about the dependence structure the insurer can improve it’s deversification benefits. It is only possible to optimally apply diversification if the dependence structure among risks is well understood. This concept of diversification is relatively hard because the future is not identical to the past. Dependence among risks is in normal times completely different from crash scenarios. Dependence in the tails should be studied and understood as much as possible to avoid huge problems in eventual extreme scenarios in the future. This is another motivation to study the dependence structure among risks for Delta Lloyd Group. Determination of a dependence structure should lead to an improvement of Delta Lloyd Group’s risk management.

In the following chapter several methods will be discussed to obtain dependence among risks in the bivariate case. The model which describes the dependence structure in an extensive way will be chosen to model dependence.

3. Methods to quantify dependence

This chapter gives an overview of some different methods to describe dependence between several risk factors. Linear correlation is one of the most popular dependence measures. Another way to quantify dependence is using copulas. There are several copulas with their respective

characteristics to determine the dependence structure. In the following a review is given of the diverse methods, including the innovated dichotomous copula model. After evaluating their own advantages and disadvantages, the “best” method will be chosen to quantify the dependence between the General Insurance risks in the bivariate case.

3.1 Linear correlation

One of the most frequently used measures of dependence in practice is linear correlation, also known as Pearson’s correlation. If

X

and

Y

are two real-valued random variables with non-zero and finite variances, then the correlation coefficient between X and Y is given as:

] [ * ] [ ] , [ ) , ( Y Var X Var Y X Cov Y X =

r

.

This linear correlation is a measure of linear dependence. If

X

and

Y

are completely independent then

r

( Y

X

,

)

=

0

, since

Cov

[ Y

X

,

]

=

0

. In the case where

X

and

Y

are perfectly dependent one

has

r

( Y

X

,

)

=

1

/

−

1

. An example for perfect dependence is the case whereY = aX+b, where

}

{

\ 0

R

a ∈

and

b

∈

R

. In the case of imperfect linear dependence, one has

r

(

X

,

Y

)

∈

(

−

1

,

1

)

.

Linear correlation also has the following property:

r

(

α

X

+

β

,

g

Y

+

δ

)

=

sign

(

αg

)

∗

r

(

X

,

Y

)

, If

β

,

δ

are constant.

The use of linear correlation in practice is explained in the following way. First of all, linear

correlation is very straightforward to calculate. It is just a case of calculating the second moments, namely covariance and variance, in order to determine the correlation coefficient. Another reason of its popularity is the ease to manipulate correlation and covariances under linear transformations, (See Embrechts et al (1999) for an explanation on the transformations):

E.g.

b

Bx

x

R

R

B

a

Ax

x

R

R

A

m n m n

+

→

→

+

→

→

,

:

,

:

, with m n m

R

b

a

R

B

A

∈

∈

×

,

,

_.

The third reason of its popularity is the fact that it is very natural as a measure of dependence in multivariate normal distributions. It is very simple to explain and understand the concept of linear correlation as a measure of dependence.

3.1.1 Shortcomings of linear correlation

Apart from its popularity, linear correlation also has its shortcomings. Embrechts et al (1999) describe some disadvantages of linear correlation. To sum up the disadvantages of linear

correlation,

X

and

Y

are again considered as real-valued random variables. The shortcomings can be summed up in the following way:

• To calculate linear correlation the variance of

X

and

Y

should be non-zero and finite, otherwise the linear correlation coefficient is undefined. For dependence measures this is not ideal and if heavy tailed distributions are involved, this leads to problems. For example, if an insurer deals with components of bivariate tv-distribution random factors, the

covariance and correlation are not defined for

v

≤

2

.

• Independence of two random variables implies that they are uncorrelated with each other and thus linear correlation will be equal to zero, but a correlation coefficient of zero does not imply independence in general. Here one can think of the following situation. If

) , ( ~N 01

X and

_{Y =}

_X

2_{, the covariance disappears despite strong dependence between}

X

and

Y

. The reason behind this is that the third moment of the standard normal distribution is zero. It is acceptable to interpret uncorrelatedness when it comes to

independence only in the multivariate normal case. Such an implication does not hold when only the marginal distributions are normal and the joint distribution is not normal.

• Another shortcoming of linear correlation is that it is not invariant under non-linear strictly increasing transformations

T

:

R

→

R

. In general one gets that

)

,

(

))

(

),

(

(

T

X

T

Y

r

X

Y

r

≠

for two real-valued random variables. In the case of bivariate standard normal distributions with ρ as its correlation, Joag-dev (1984) shows that













=

2

6

r

π

r

(

T

(

X

),

T

(

Y

))

arcsin

with transformation

T

(

x

)

=

Φ

(

x

)

(the standard normal distribution function). Furthermore Kendall and Stuart (1979) show that

|

)

,

(

|

|

))

(

),

(

(

|

r

T

X

T

Y

≤

r

X

Y

when it comes to bivariate normally distributed vectors and arbitrary transformations

T

:

R

→

R

.

Due to all these disadvantages of linear correlation, the use of linear correlation as a measure of dependence does not lead towards optimal understanding of dependence among several risk components. Another approach to quantify the dependence structure between riks is the copula approach. This concept is introduced in the next paragraph to better describe the underlying relations among risks.

3.2 Understanding relationships using copulas

In this paragraph some basic knowledge of the copula theory is reviewed. The word copula is a Latin noun that means “bond”. Sklar was the first to coin the term copula, to describe the function that “joins” univariate distribution functions to construct multivariate ones. The basic idea of the copula approach is that a joint distribution function of random variables can be expressed as a function of the marginal distributions. Let

X ,...,

1

X

_n be real-valued random variables. The

dependence among these variables can be described completely by their joint distribution function which is defined as

F

(

X

1

,...,

X

n

)

=

P

[

X

1

≤

x

1

,...,

X

n

≤

x

n

]

. The idea of splitting

F

into two

parts leads to the concept of a copula. The first part explains the dependence structure and the second part describes the marginal behavior. The first assumption of a copula is that all X ,...,1 X_n

have their continuous marginal distribution

F ,...,

1

F

_n.

3.2.1 Properties of copulas

The distribution function of a random vector in

R

n with uniform [0, 1] marginal is a copula and any copula

C

:

[

0

,

1

]

n

→

[

0

,

1

]

has the following properties (McNeil (2008)):

•

C

[

x

₁

,...,

x

_n

]

is increasing in each component of xi For all

i

∈

{

1

,...,

n

}

and

x

_i

∈

[ 1

0

,

]

. •

C

[

1

,...,

1

,

x

i

,

1

...,

1

]

= xi.

• For all

(

a

1

,...,

a

_n

),

(

b

1

,...,

b

_n

)

∈

[

0

,

1

]

n with

a ≤

i

b

i it holds:

0

1

1 1 1 1 2 1 2 1

≥

−

∑ ∑

= = + +

_[

_,...,

_]

)

(

...

... n n n ni i i i i i

_C

_x

_x

_{, where} 1 j j

x

a =

and

b =

j

x

j2 for all

}

,...,

{

n

j

∈

1

.

The first property is clear and the second one follows from the fact that the marginals are uniformly-(0, 1) distributed. The last property holds because the sum can be interpreted as:

]

,...,

[

a

X

b

a

n

X

n

b

n

P

₁

≤

₁

≤

₁

≤

≤

, which is non-negative. A detailed explanation of these properties can be found in Embrechts et al (1999).

3.2.2 Sklar’s theorem (1959)

Given a joint cumulative distribution function F(X1,...,Xn)for real-valued random variables

n

X

X ,...,

1 , with their respective marginal distribution functions

F

1

(

X

1

),...,

F

n

(

X

n

)

,

F

can be

written as a function of its marginal.

)]

(

)

(

),...,

(

)

(

[

]

,...,

[

)

,...,

(

X

X

n

P

X

x

X

n

x

n

P

F

X

F

x

F

X

n

F

x

n

F

₁

=

₁

≤

₁

≤

=

₁

≤

₁

≤

=

C

[

F

1

(

x

1

),...,

F

_n

(

x

_n

)]

The copula C is unique when all the cumulative distribution functions (marginals)

F ,...,

1

F

_n are

continuous. When the cumulative distribution functions are discrete, the copula C is uniquely determined on

Ran

[

F

1

]

*

...

*

Ran

[

F

n

]

, where

Ran

[

F

i

]

denotes the range of Fi. Sklar’s theorem

is very general, thus any joint distribution can be written in copula form. Furthermore if

F

_i and

C

are continuous and differentiable, the joint density function

f

(

x

₁

,...,

x

_n

)

can be written down as:

)]

(

),...,

(

[

*

)

(

*

...

*

)

(

)

,...,

(

x

x

n

f

x

f

n

x

n

c

F

x

F

n

x

n

f

1

=

1 1 1 1

Where

f

_i

(

x

_i

)

is the density function of the random variable

X

_iand

n n n n n n

x

C

F

x

_F

_F

F

x

F

x

F

c

∂

∂

∂

=

...

)]

(

),...,

(

[

)]

(

),...,

(

[

1 1 1 1

1 is the copula density.

This is an essential result which states that under appropriate conditions the joint density function can be written as a product of the marginal densities and the copula density. If the

X

_i

'

s

are completely independent, then

c

=

1

and

f

(

x

1

,...,

x

_n

)

=

f

1

(

x

1

)

*

...

*

f

_n

(

x

_n

)

. Here one can observe

that the copula density carries essential information about the dependence among

X

_i

'

s

and this is the reason why the copula density is also called dependence function sometimes (Clemen et al (1999)).

If

F

(

X

1

,...,

X

n

)

is an n-dimensional distribution function with

F ,...,

1

F

nas its continuous margins

and

C

the related copula, then for any

u

∈

[ 1

0

,

]

nthe copula can be constructed in the following way: C[u ,...,un] F(F (u1),...,Fn1(un)) 1 1 1 − − = . 3.3 Dependence concepts

This subparagraph aims to introduce some copula-related dependence measures, such as comonotonicity, concordance and rank correlation. A detailed discussion of these dependence concepts can be found in Embrechts et al (2001) and Thorsten (2006).

3.3.1 Comonotonicity

Before coming to the comonotonic concept, first the Frechet-Hoeffding bounds for joint distribution are introduced. These bounds are involved when there exists perfect dependence. Perfectly

negative dependence leads to the lower bound and the upper bound results when there is a perfectly positive dependence between a pair of random variables.

Lower bound:

C

_l

[

x

₁

,...,

x

_n

]

=

max{

x

₁

+

...

+

x

_n

+

1

−

n

,

0

}

Upper bound:

C

u

[

x

1

,...,

x

n

]

=

min{

x

1

,...,

x

n

}

When comonotonicity is involved, the Frechet-Hoeffding inequality which is written below holds for any copula

max{

x

1

+

...

+

x

n

+

1

−

n

,

0

}

≤

C

[

x

1

,...,

x

n

]

≤

min{

x

1

,...,

x

n

}

.

If

n

=

2

, the upper- and lower bound are also copulas. If

U

~ U

[

0

,

1

]

then

].

,

[

]

,

[

]

,

[

]

,

[

2 1 2 1 2 1 2 1

1

x

U

x

U

P

x

x

C

x

U

x

U

P

x

x

C

u l

≤

≤

=

≤

−

≤

=

Here

C

_u and

C

_l are bivariate distribution functions of the vectors _(U,1−U)T_{and (U ),U}T_{. (U ),U} T _is

the transposed

( U

U

,

)

vector. Yaari (1987) state that

X

and

Y

are comonotonic if

(

X )

,

Y

T has the copula

C

_u and counter monotonic if they have the copula

C

_l. If

F

1 and

F

2are continuous the

stronger version of the result can be stated:

)

(

)

(

1 1 2

1

F

F

T

X

T

Y

C

C

=

_l

⇔

=

⇒

=

−

_

−

_{: decreasing} 1 1 2

F

F

T

X

T

Y

C

C

=

_u

⇔

=

(

)

⇒

=

−



: increasing

(See Embrechts et al (2001). Other comonotonicity characteristics can be found in Denneberg (1994)).

3.3.2 Concordance and Discordance

A pair

X

and

Y

, both real-valued random variables, is called concordant if “large” values of one variable are associated with “large” values of the other, and analogously for “small” values. Let

)

,

( Y

X

be a vector of two random variables and

(

x

1

,

y

1

),

(

x

2

,

y

2

)

are two samples from

)

,

( Y

X

. The pair

(

x

1

,

y

1

)

and

(

x

2

,

y

2

)

are called concordant if

(

x

1

−

x

2

)(

y

1

−

y

2

)

>

0

and discordant if(x1 −x2)(y1 −y2)<0 holds.

3.3.3 Rank correlation

The use of rank correlation has some main advantage over ordinary correlation. Rank correlation is invariant under monotonic transformations. Two important measures of dependence which fall in this category are Kendall’s tau and Spearman’s rho. Linear correlation as a dependence measure does not describe non-elliptical distributions well. For such distributions, linear correlation is inappropriate and often quite misleading, while Kendall’s tau and Spearman’s rho have the property of symmetry, co- and countermonotonicity, and assume the value of zero in case of independence and thus Kendall’s tau and Spearman’s rho are better measures of dependence. A disadvantage of these dependence measures is that although they have the property of invariance under monotonic transformations and can capture perfect correlation, they are not simple functions of moments and therefore, more computation is needed. However, for some classes of parametric copulas, Kendall’s tau and Spearman’s rho can be calculated from the dependence parameter

3.3.3.1 Kendall’s tau

Kendall’s tau measures the strength of relation between variables. Like other correlation measures Kendall’s tau also takes values between -1 and +1 to describe the strength of dependence. If

)

,

(

X

1

Y

1 and

(

X

2

,

Y

2

)

are an independent pairs of random variables, Kendall’s tau is defined as:

]

)

)(

[(

]

)

)(

[(

₁

−

_{2 1}

−

₂

>

0

−

₁

−

_{2 1}

−

₂

<

0

=

P

X

X

Y

Y

P

X

X

Y

Y

τ

r

. It is defined as the difference

between the probability of concordance and the probability of discordance. The main advantages of Kendall’s tau are:

• The distribution of this statistic has better statistical properties. (Even non-elliptical distributions are handled well).

• Kendall’s tau gives a direct interpretation in terms of probabilities of observing concordant and discordant pairs. (Dependence between large losses or profits).

This measure of dependence can be expressed by the underlying copulas of the two random variables

X

and

Y

. If both

X

and

Y

, are real-valued continuous random values with copula C, then Kendall’s tau can be derived in the following way:

]

)

)(

[(

]

)

)(

[(

₁

−

_{2 1}

−

₂

>

0

−

₁

−

_{2 1}

−

₂

<

0

=

_P

_X

_X

_Y

_Y

_P

_X

_X

_Y

_Y

τ

r

.

1

)]

,

(

[

*

4

1

)

,

(

)

,

(

4

1

)

,

(

)

,

(

4

1

)

,

(

)

,

(

1

*

1

2

*

2

1 0 1 0 1 1 1 1 1 1 2 2 } { } {1 2 1 2

−

=

−

=

−

=

−

=

∫∫

∫ ∫

∫ ∫ ∫ ∫

∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − > >

V

U

C

E

v

u

dC

v

u

C

y

x

dF

y

x

F

y

x

dF

y

x

dF

y y x x

As seen above, Kendall’s tau can be derived with respect to the underlying copula of the two random variables. If the copula is constructed with

U,

V

~

Uniformly[

0,1]

distributed, the measure of dependence can be determined. Here u =F(x)and v =F(y). See Embrechts et al (1999) for more information on Kendall’s tau and its derivation.

3.3.3.2 Spearman’s Rho

Another copula related measure of dependence is Spearman’s Rho. If

(

X

₁

,

Y

₁

)

,

(

X

₂

,

Y

₂

)

are vectors of random and continuous variables with copula

C

, then Spearman’s rho can be given by the following formula: (see Embrechts et al (1999))

]}

)

)(

[(

]

)

)(

[(

{

*

0

0

3

₁

−

_{2 1}

−

₂

>

−

₁

−

_{2 1}

−

₂

<

=

P

X

X

Y

Y

P

X

X

Y

Y

s

r

3

12

3

12

3

12

1 0 1 0 1 0 1 0

−

=

−

=

−

=

∫ ∫

∫ ∫

)]

,

[(

*

)

,

(

)

,

(

V

U

E

dudv

v

u

C

v

u

dC

uv

Spearman’s rho can also be computed with the underlying copulas. In practice it is in almost all situations that the outcomes of Kendall’s tau as well as Spearman’s rho are very close to each other and lead to the same conclusions. As Kendall’s tau and Spearman’s rho are copula based measures of dependence, they have some common properties. If both

X

and

Y

are random variables and have distributions

F

1and

F

2, joint distribution

F

and copula

C

, then the following

properties hold for both Kendall’s tau and Spearman’s rho: •

)

,

(

)

,

(

)

,

(

)

,

(

X

Y

Y

X

X

Y

Y

X

s s τ τ

r

r

r

r

=

=

• If X and Y are completely independent, then

r

_s

(

X

,

Y

)

=

r

_τ

(

X

,

Y

)

=

0

•

−

1

≤

r

_s

(

X

,

Y

),

r

_τ

(

X

,

Y

)

≤

+

1

•

=

∫ ∫

−

1 0 1 0

1

4

C

(

u

,

v

)

dC

(

u

,

v

)

τ

r

• =

_{∫ ∫}

− 1 0 1 0 3 12 C u v dudv s ( , )

r

•

r

_s

(

X

,

Y

)

=

r

τ

(

X

,

Y

)

=

1

⇔

C

=

C

_u

⇔

Y

=

T

(

X

)

→

T

increasing transformation. •

r

s

(

X

,

Y

)

=

r

τ

(

X

,

Y

)

=

−

1

⇔

C

=

C

l

⇔

Y

=

T

(

X

)

→

T

decreasing transformation. Embrechts et al (1999) give an interpretation on these properties.

3.3.4 Tail dependence

The concept of tail dependence deals with the amount of dependence in the upper- and lower tail of a bivariate distribution. This concept is relevant to study for dependence of random variables in extreme situations. Tail dependence is also a copula-related dependence concept. The

mathematical expressions for computing tail dependence will be given in the next sections for each family of copulas. Both the Elliptical and Archimedean class where possible.

After a brief overview of some dependence measures, Elliptical and Archimedean copulas can be introduced. These two classes have their own characteristics and with one of these copula classes the dependence structure will be determined during this research.

3.4 Elliptical Copulas

Elliptical copulas are often used as a benchmark model. They are simply copulas of elliptical distributions. The Gaussian copula and the t-copula belong to this category. A brief description of both copulas is given in the following two sections.

3.4.1 Gaussian Copulas

The Gaussian copula is a copula of an n-variate normal distribution. This copula can be defined in the following way:

C

_RGa

(

u

)

_Rn

(

(

u

),...,

1

(

u

_n

))

1 1 − −

_Φ

Φ

Φ

=

Where:

Φ

n_R: is the joint distribution function of an n-variate normal distribution. R: is the linear correlation matrix for the n variables.

_Φ

−1 _{: is the inverse of the distribution function of a univariate distribution function.}

In the bivariate case the copula can be written as:

.

)

(

exp

)

(

)

,

(

( ) ( )

dsdt

R

t

st

R

s

R

v

u

C

Ga u v R

_















−

+

−

−

−

=

∫

∫

−

−

Φ ∞ − Φ ∞ − 122 2 12 2 2 1 2 12

1

2

2

1

2

1

1

1

π

R12 is the linear correlation of the corresponding bivariate normal distribution of the first and second stochastic variables. These Gaussian copulas do not take into account upper or lower tail dependence.

3.4.2 t-Copulas

Just like the Gaussian copula, the t-copula is also an elliptical copula. The difference lies in its dependence structure. This copula can be written as:

C

_vt_,T

(

u

,

v

)

t

_vn_,R

(

t

_v

(

u

),....,

t

_v1

(

u

_n

))

1

1 −

−

=

.

In the bivariate case the expression of the t-copula becomes:

.

)

(

)

(

)

,

(

) ( ( ) ,

dsdt

R

v

t

st

R

s

R

v

u

C

v u t t v t R v

v

v

2 2 2 12 2 12 2 2 1 2 12

1

2

1

1

2

1

1

1

+ − ∞ − −∞

















−

+

−

+

−

=

∫

∫

−

−

π

R12 is again the linear correlation of the corresponding bivariate normal distribution in this case. The t-copula takes into account upper- as well as lower tail dependence. Tail dependence between two continuous random variables is a copula property and it is invariant under strictly increasing transformations of

X

and

Y

.

Because of symmetry the upper and lower tail dependence are equal. L U

λ

λ

=

















+

−

+

=

=

>

=

+ ∞ → 12 12 1 1 2

1

1

1

2

)

|

(

lim

2

R

R

v

t

x

X

x

X

P

v x

Both the Gaussian and the t-copula are fast and easy to implement. They are parameterized by linear correlation matrices. The main difference between these two copulas is that Gaussian copula does not take tail dependence into consideration

3.5 Archimedean Copulas

Another class of copulas is the class of Archimedean copulas. In this class a generator function is used to express the copula function. The generator (

ϕ

) should have the following properties.

• The generator

ϕ

should be continuous.

• It must be strictly decreasing on [0, 1] to [0,

∞

)

.

t

∈

[ 1

0

,

]

and

ϕ

(

t

)

∈ 0

[

,

∞

)

. •

ϕ

(

1

)

=

0

.

•

ϕ

[ 1−]

:

the pseudo inverse function of

ϕ

.

:

] [ 1−

ϕ

[0, 1]  [0,

∞

) is given as:







∞

<

<

⇔

≤

≤

⇔

=

− −

t

t

t

t

)

(

)

(

)

(

)

(

] [

0

0

0

0

1 1

ϕ

ϕ

ϕ

ϕ

The following properties hold for

ϕ

[−1]

:

• Continuous • Decreasing on [0,

∞

)

• Strictly decreasing on [0,

ϕ

(

0

)

] •

ϕ

[−1]

(

ϕ

(

u

))

=

u

on [0, 1], and







∞

<

<

⇔

≤

≤

⇔

=

−

t

t

t

t

)

(

)

(

)

(

))

(

(

[ ]

0

0

0

0

1

ϕ

ϕ

ϕ

ϕ

ϕ

If

ϕ

(0

)

=

∞

, then

ϕ

[−1]

=

ϕ

−1 .

Given that the generator function is convex, the copula function can be defined as:

)).

(

)

(

(

)

,

(

_u

_v

[ ]

_u

_v

C

=

ϕ

−1

ϕ

+

ϕ

Here

C

is a copula if and only if

ϕ

is convex. If

ϕ

[−1]

₌

ϕ

−1 _{and C(u,v)}

_ϕ

₍

_ϕ

_(u)

_ϕ

_(v))

+

= −1 _{then C}

is a strict Archimedean copula. Within the class of Archimedean copulas there are several families. All these families, namely the Gumbel family, the Clayton family and the Frank family have their own generator and thus their own expression of their copula.

3.5.1 Properties of Archimedean Copulas

Archimedean copulas have the following properties: • C is symmetric.

• C is associative.

C

ϑ

(

C

(

u

,

v

),

w

)

=

Cϑ

(

u

,

C

(

v

,

w

))

Proof for associative:

C

ϑ

(

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

ϑ

(

u

,

C

(

v

,

w

))

.

3.5.2 Gumbel family

The Gumbel family is different from the other families in the sense that it can only model independence or positive dependence. This copula model illustrates upper tail dependence. An example for upper tail dependence is the fact that there is a stronger relation between big losses than small losses with another variable. One can think of a stock market crash with big losses. Let

ϕ

(

t

)

=

(

−

ln

t

)

ϑ, with

ϑ

≥

1

.

Clearly for t 0,

ϕ

(t

)

→

∞

and

ϕ

(

1

)

=

0

.

t

t

t

₌

₋

ϑ

₋

ϑ−1

1

ϕ

'

(

)

(

ln

)

, strictly decreasing on [0, 1] to [0,

∞

)

.

0

≥

)

(

'' t

ϕ

, convex.

With these facts the generator is a strict generator and thus the Gumbel family of copulas is a strict Archimedean copula. The copula function for this family can be written as:

=

)

,

( v

u

C

ϑ

C

(

u

,

v

)

=

ϕ

[−1]

(

ϕ

(

u

)

+

ϕ

(

v

))

exp{

[(

ln

)

ϑ

(

ln

)

ϑ

]

ϑ

}

1

v

u

+

−

−

−

=

.

Derivation of this copula function can be done in the same way as done for the Clayton family in the next section.

3.5.3 Clayton family Let

ϑ

ϕ

(

t)

=

t

−ϑ

−

1

,

ϑ

∈

[

−

1

,

∞

)

\

{

0

}

Proof of convexity of

ϕ

(t

)

: ) (

)

(

'

₌

₋

−ϑ+1

ϕ

t

t

. ) (

)

(

)

(

''

2

1

− +

+

=

_ϑ

ϑ

ϕ

t

t

=

ϑ

_(ϑ

+

₊₂

1

₎

t

. If

ϑ

∈

[

−

1

,

∞

)

\

{

0

}

, then

ϕ

'' t

(

)

≥

0

∀

t

.

The copula function is:

C

_ϑ

( v

u

,

)

=

max{[

1

]

,

0

}

1 ϑ ϑ ϑ − − −

₊

_v

₋

u

.

If there is a restriction

ϑ

≥

0

, the copula of this family is strict Archimedean and the expression is:

=

)

,

( v

u

C

ϑ ϑ ϑ ϑ 1

1

− − −

₊

₋

_]

[

u

v

.

Assuming

ϑ

≥

0

Clayton’s copula can be derived as follows:

ϑ

ϑ

ϕ

1 1

₁

− −]

₍

₎

₌

₍

₊

₎

[

_t

_t

))

(

)

(

(

)

,

(

_u

_v

[ ]

_u

_v

C

=

ϕ

−1

ϕ

+

ϕ

₌













+

−

=













−

₊

−

− − − − − −

ϑ

ϕ

ϑ

ϑ

ϕ

[1]

u

ϑ

1

v

ϑ

1

[1]

u

ϑ

v

ϑ

2

= ϑ

(

ϑ ϑ

)

ϑ ϑ ϑ

ϑ

ϑ

1 1

1

1

2

− _{− −} − − −

−

+

=













+

−

+

v

_u

_v

u

_*

_.

The Clayton family measures lower tail dependence. 3.5.4 Frank family

Another strict Archimedean copula is the Frank family of copulas. This family is the only one which satisfies symmetry and does not take tail dependence into consideration.

Here

1

1

−

−

−

=

−₋ϑ_ϑ

ϕ

e

e

t

)

ln

t

(

, where

ϑ

∈

[

−∞

,

∞〉

\

{

0

}

The copula function is:

Cϑ

( v

u

,

)

=

_











−

−

−

+

−

− ₋ −

1

1

1

1

1

ϑ ϑ ϑ

ϑ

e

e

e

u

₎₍

v

₎

(

ln

3.5.5 Kendall’s tau for Archimedean Copulas

If C is a copula, then Kendall’s tau can be expressed as a double integral of C. When it comes to the class of Archimedean copulas Kendall’s tau is much easier to calculate. This correlation coefficient is a one-dimensional integral. If X and Y are two random variables with Archimedean copula

C

_ϑ

( v

u

,

)

, then Kendall’s tau of X and Y is given by:

=

+

∫

1 0

4

1

dt

t

t

)

(

'

)

(

ϕ

ϕ

r

_τϑ

For any family in the Archimedean class the expression of Kendall’s tau is a simple expression of

ϑ

. The expression of Spearman’s rho in some cases is too complicated in this class of copulas and that is why they are left behind from now on. For some families there is no closed form expression for Spearman’s rho. To be consistent only Kendall’s tau will be determined as dependence

measure. Table 3.1 provides an overview of the closed-form expressions of Kendall’s tau. Table 3.1

Gumbel family Clayton family Frank Family Kendall’s tau

ϑ

ϑ

1

1

4

1

1 0

−

=

+

∫

t ln

t

dt

2

4

1

1 0 1

+

=

−

+

∫

−

ϑ

ϑ

ϑ

ϑ

dt

t

t

ϑ

ϑ

))

(

(

1

1

4

1

−

−

D

Upper tail dependence ϑ 1

2

2

−

No upper tail dependence No tail dependence Lower tail dependence No lower tail dependence ϑ 1

2

− No tail dependence Where

∫

−

=

x t k k k

dt

e

t

x

k

x

D

0

1

)

3.6 The “chosen” method to quantify dependence

In the previous paragraphs of this chapter several ways are discussed to quantify dependence. Linear correlation is very easy to explain and use, but it has its disadvantages. It does not provide a good understanding of how stochastic random variables are related to each other. Especially, understanding of dependence in the tails cannot be modelled by linear correlation. A more appropriate method than linear correlation is making use of copulas to understand the underlying relation. Within copulas there are two classes discussed, namely the Elliptical and the Archimedean class. The Elliptical copulas are distribution functions of component wise transformed elliptically distributed random vectors. They are easy to use, but still there are several shortcomings:

• They do not have closed-form expressions which make it a bit difficult to be understood, especially for risk management.

• They are restricted to radial symmetry.

Embrechts et al (2001) state that it is reasonable that in most cases there is a stronger dependence between big losses than big gains in finance and insurance applications. These asymmetric situations cannot be modelled with Gaussian or

t

-copulas. Thus it is worth studying the dependence structure with Archimedean copulas. This class is an important category for a number of reasons which are summed up below.

• A lot of interesting parametric families are Archimedean copulas.

• Archimedean copulas allow a greater variety of various dependence structures. These Archimedean copulas can have asymmetric tail dependence.

• These copulas are easily constructed and have closed form expressions, which makes the calculation more tractable.

So in this research the dependence will be quantified with this Archimedean class of copulas. For risk management it is also much easier to understand Kendall’s tau as it can be easily expressed trough the dependence parameter.

If dependence is quantified with the three given families of Archimedean copulas, the correlation coefficient can be calculated with the dependence parameter. By comparing the three different results one can get a clear understanding of the underlying dependence structure. Upper and lower tail dependence can be analyzed separately. As a single Archimedean copula is not able to combine both upper- and lower tail dependence, a mix of two copulas can be constructed in order to take into account the dependence in both tails. For instance, Gumbel can model upper tail dependence while Clayton takes into account dependence in the lower tail. If these two copulas are combined, dependence in both tails can be taken into account. The lower tail dependence is that of the Clayton component, while the Gumbel component provides upper tail dependence. Both copulas will be a weighted component of the copula mix. The mixture of these two copulas can be described as follows:

C

_mix

w

C

1

(

u

,

v

)

(

w

)

C

2

(

u

,

v

)

2

1

1

θ θ

+

−

∗

∗

=

,

w

∈

[ 1

0

,

]

.