DEPENDENCE MATTERS! BY

DOREEN STRAßBURGER Institut für Mathematik

Carl von Ossietzky Universität, Oldenburg

AND

DIETMAR PFEIFER Institut für Mathematik

Carl von Ossietzky Universität, Oldenburg

and AON Jauch&Hübener, Hamburg, Germany

"The new system should provide supervisors with the appropriate tools to assess the “overall solvency” of an insurance undertaking. This means that the system should not only consist of a number of quantitative ratios and indicators, but also cover qualitative aspects that influence the risk-standing of an undertaking (management, internal risk control, competitive situation etc.). … The solvency system should encourage and give an incentive to insurance undertakings to measure and manage their risks. In this regard, there is a clear need for developing common EU principles on risk management and supervisory review. Furthermore the quantitative solvency requirements should cover the most significant risks to which an insurance undertaking is exposed. This risk-oriented approach would lead to the recognition of internal models (either partial or full) provided these improve the undertaking’s risk management and better reflect its true risk profile than a standard formula."

European Commission, Internal Market DG, March 2003

important topic here:

recognizing and modeling of dependencies between risks

1

Definition. A copula is a function of d variables on the unit d-cube with the following properties:

C [0,1]d

[0,1]( )u [0,1]d

k= uC ≤a [0,1]d

Cba C 1 1[ , ] [ ,d da b a b= × ×a b

(1 ) 0.d da bε+ − ≥

2

1. the range of C is the unit interval ; 2. is zero for all in for which at least one coordinate equals

zero; ( )C u

C u

3. u if all coordinates of are 1 except the k-th one; 4. is d-increasing in the sense that for every b in the measure

assigned by to the d-box ] is nonnegative, i.e. ∆ [ , ]

( )( ) { }

1

1

1 1 1 1, , 0,1

: ( 1) (1 ) , ,

d

ii

dn

d dC C a bε

ε ε

ε ε ε=

∈

∑∆ = − + −∑b

a

In other words:

A copula C is a multivariate distribution function of a random vector that has uniform margins.

Sklar’s Theorem. Let H denote a n-dimensional distribution function with margins .F Then there exists a n-copula C such that for all real ( , )1, , nF 1 , nx x ,

( )11 1( , , ) ( ,) ( .,n n nF FC )H x x x=

1 , , nF F−

, nu u

), ( )n nnFu uC FuHu − −=

x If all the margins are continuous, then the copula is unique, and is determined uniquely on the ranges of the marginal distribution functions otherwise. Moreover, the converse of the above statement is also true. If we denote by

1− the generalized inverses of the marginal distribution functions, then for every ) in the unit n-cube,

1

1( ,

( )1 111 1( , , ) ( .

3

Fréchet-Hoeffding bounds:

1 1

, ,1 1( , )

max( 1,0) ( , ,d d

d du u

u u d C u u 1

( , )

) min( , , )d

u u

u= =

+ + − + ≤ ≤W M

du uW 3d ≥

1 2( ,u uM

1 2( ,u uW

u

,1( , ) is generally not a copula for , but inequality is sharp.

Two random variables with copula ) are called co-monotonic (highest degree of positive dependence), two random variables with copula ) are called counter-monotonic (highest degree of negative dependence). Wide-spread fallacy: Co-monotonic risks increase total risk, counter-monotonic risks decrease total risk (→ diversification effect)

This is wrong in general!

4

Definition. Let and define intervals , ∈d n1 , , 1

1( :)

d

dj

i i j

i iI n

n n=

⎛ ⎤−⎜ ⎥⎜⎝

= ⎜ ⎥⎜ ⎦×

{ }d ni i N∈ =

, j for all

possible choices .n If n are non-negative real numbers with the property

1, , : 1, ,1 , , ( )

di ia

( )1 , ,

1( )di ia n

n

,11 , , ( )i dd ii Iia n

1 1, , | ( ,( )di i ia n

1( , , ) kdi i J i∈

=∑

{1, , and 1,kk d i∈ ∈ ( ) ( ){ }1: , , | ,dk n n k kJ i j j N j i= ∈ =

, ( )( , , )

:d n

di inn

N

c∈

= ∑, ) d

d ni N∈

A

for all }n with

then the function 1 is the density of a d-dimensional

copula, called grid-type copula with parameters . Here

denotes the indicator random variable of the event A, as usual.

{ } , ,

1

{ }1

5

It is easy to see that in case of an absolutely continuous d-dimensional copula C, with continuous density

( ) ( ) ( )1 11

, , , , , ( )1, , 0,1d

dd d d

d

u u u u u uu u

c C∂= ∈∂ ∂

,

c can be approximated arbitrarily close by a density of a grid-type copula. The classical multivariate mean-value-theorem of calculus tells us here that we only have to choose

( )1

1

1 , , ( )di ia n 1 1 1

1 1

: , , , ,

d

n

n

d

i in n

d d d nii

nn

u u du du C i i Nc− −

= =∆ ∈∫ ∫ βα ,

with 1, , 1, .k knk nk

i i k dn n

α β−= = =

6

,

Lemma. Let U be independent standard uniformly distributed random variables and let

1, ,U d

df and denote the density and cumulative distribution

function of ,iU resp., for Then

dF

1

d

i

S=

=∑ .d ∈:d

( )0

0

1( ) ( 1) ( ) 1 sgn(2( 1)!

1( ) ( 1) ( ) ( ) )2 !

dk d

dk

dk d

dk

df x x k

kd

dF x k x k k

kd

−

=

=

⎛ ⎞⎟⎜= − ⎟ −⎜ ⎟⎜ ⎟⎜− ⎝ ⎠⎛ ⎞⎟⎜= − ⎟ − + − −⎜ ⎟⎜ ⎟⎜⎝ ⎠

∑

∑0 .x d≤ ≤

7

)

sgn(d

x k

x

− for

This follows e.g. from USPENSKY (1937), Example 3, p.277, who attributes this result already to Laplace.

Theorem. Let ( )1, , dX X

()

,(

, )i

dd ni i

nnN

c∈

df n ( ; )dF n i1

:d

di

S=

=∑

1( , , )

( , , )

( ; )

( ; )

dd n

dd n

d

d ii

di i N

di

d

d iN ii

f n x n

F n

be a random vector whose joint cumulative distribution function is given by a grid-type copula with density

.= ∑ 1 Then the density and cumulative distribution

function ) and , resp., for the sum iX is given by

, ,11 , , ( ):id di i Ia n

1

1

, ,

, ,

)(

( )

d

d

i i

i i

1

( ; i

1

1

1

f nx d j

nxx d j

=

=

∈

∈

⎛ ⎞⎟⎜ + − ⎟⎜ ⎟⎜ ⎟=⎝ ⎠⎛ ⎞⎟⎜ + − ⎟⎜ ⎟⎜ ⎟⎝ ⎠

=

∑

∑∑

∑0 ,

F

a n

a n

⋅

⋅x d≤ ≤ for with

( )0

0

1( ) ( 1) ( ) 1 sgn(2( 1)!

1( ) ( 1) ( ) ( ) )2 !

dk d

dk

dk d

dk

df x x k

kd

dF x k x k k

kd

−

=

=

⎛ ⎞⎟⎜= − ⎟ −⎜ ⎟⎜ ⎟⎜− ⎝ ⎠⎛ ⎞⎟⎜= − ⎟ − + − −⎜ ⎟⎜ ⎟⎜⎝ ⎠

∑

∑0 .x d≤ ≤

)

sgn(d

x k

x

− for

8

Example. 1/ 3

2 / 3 4 22 / 3 3

a ba b ca b c

( ) ( ) 1 4 2 21/ 3 2 / 3 4 2

ij

a bA n a n c a b c

a c a b c

⎡ ⎤− −⎢ ⎥⎡ ⎤ ⎢ ⎥+ + += = − − − −⎢ ⎥⎣ ⎦ ⎢ ⎥

⎢ ⎥− − −− − − + + +⎣ ⎦

[0,1/ 3a b c ∈

1,

with suitable real numbers ]. It follows that the covariance of the corresponding random variables 2

, ,X X is given by

( ) ( ) ( ) ( )3 3

1 2 1 21 1

( )2 1 (2 1)

36iji j

i jE X X E X E X a n

= =

− −− = =∑∑

1,

0

i.e. the risks 2X X are uncorrelated but dependent.

9

The density and cumulative distribution function of the aggregated risk 2S X X= + are thus, by the above Theorem, given by 2 1:

{ } { }

{ } { }

{ } { }

{ } { }

{ } { }

2

9 ,

3(2 ) 9( ) ,

9(4 3 ) 10 3(5 18 12

(3; ; ) 32 9(16 7 ) 9(14 6 3

28 3(52 19 ) 3(6 33 12

6(2 9 3 ) 3( 2 9 3 )

0,

ax

a b c a b c x

a b c a b c

f x a b c a b c

a b c a b

a b c a b c

γ

− + + − + +

+ + − + − − +

= − + + + + + − ≤

− + + + + − −

− − + + − + + +

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

103

1 23 32) , 13

4) , 13

4 5) ,3 35, 23oth

x

x

x x

x x

c x x

x x

≤ ≤

≤ ≤

≤ ≤

≤

+ ≤ ≤

≤ ≤

erwise;

10

{ } { } { }

{ } { }

{ }

{ } { }

{ }

2

2

2

2

2

0,9 1, 02 3

9 1( ) 3(2 ) ( 22 25 (3 18 12 ) ( 10 36 27 )2

1 (20 666

9 ( 3 14 6 ) (32 144 63 )(3; ; ) 2

1 ( 106 237 26

9 (2 222

a x x

a b c x a b c x a b c

a b c x a b c x

a b

a b c x a b c xF x

a b

γ

− + + + − + + − + +

− − + + − + + + +

+ − −

− + + + + − − + +=

+ − + +

− { } { }

0

1 2),3 3

2 1357 ),

41313 ),

x

x

xc

xc

≤

≤ ≤

≤ ≤

≤ ≤+

≤ ≤+

{ }

{ } { }

{ }

2

2

4 ) ( 28 156 57 )

1 (134 726 267 ),6

3 ( 6 9 3 ) 3(4 6 18 )2

( 11 541,

a b c x a b c x

a b c

a b c x a b c x

a b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ − + + − + − + +⎪⎪⎪⎪⎪⎪ + − − +⎪⎪⎪⎪⎪⎪ − + + + + − − + +⎪⎪⎪⎪⎪ + − + +⎪⎪⎪⎪⎪⎩

4 53 3

5 2318 ),

2.

x

xc

x

≤ ≤

≤ ≤+

≥

{ } { } { }

{ } { }

{ }

{ } { }

{ }

2

2

2

2

2

0,9 1, 02 3

9 1( ) 3(2 ) ( 22 25 (3 18 12 ) ( 10 36 27 )2

1 (20 666

9 ( 3 14 6 ) (32 144 63 )(3; ; ) 2

1 ( 106 237 26

9 (2 222

a x x

a b c x a b c x a b c

a b c x a b c x

a b

a b c x a b c xF x

a b

γ

− + + + − + + − + +

− − + + − + + + +

+ − −

− + + + + − − + +=

+ − + +

− { } { }

0

1 2),3 3

2 1357 ),

41313 ),

x

x

xc

xc

≤

≤ ≤

≤ ≤

≤ ≤+

≤ ≤+

{ }

{ } { }

{ }

2

2

4 ) ( 28 156 57 )

1 (134 726 267 ),6

3 ( 6 9 3 ) 3(4 6 18 )2

( 11 541,

a b c x a b c x

a b c

a b c x a b c x

a b

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ − + + − + − + +⎪⎪⎪⎪⎪⎪ + − − +⎪⎪⎪⎪⎪⎪ − + + + + − − + +⎪⎪⎪⎪⎪ + − + +⎪⎪⎪⎪⎪⎩

4 53 3

5 2318 ),

2.

x

xc

x

≤ ≤

≤ ≤+

≥

11

The following graphs show five different densities and cumulative distribution functions for the sum 2 for various choices of ( , , ).a b cγ =

2 (3; ; )f γ i

2 (3; ; )γ i 2 1 ,X= +F S X

( )1 10, ,

4 4

( )2 3 1, ,

18 18 18γ =

( )1 1, , 0

6 6γ =

γ =

( )2, 0, 0

9γ =

( )2 20, ,

9 9γ =

Plots of for various choices of 2 (3; ; )f γ i γ

12

( )2 3 1, ,

18 18 18γ =

( )2 20, ,

9 9γ =

( )1 10, ,

4 4=

( )2, 0, 0

9γ =

γ

( )1 1, , 0

6 6γ =

Plots of for various choices of 2 (3; ; )F γ i γ

13

For finding the “worst” VaR scenario in this setup we have to minimize the cumulative distribution function

{ } { }2

2

3(3; ; ) ( 6 9 3 ) 3(4 6 18 ) (

2{ }11 54 18 )F x a b c x a b c xγ = − + + + + − − + + a b c− + + +

5 ,3

=

2 2b+ +

at the point x which is the solution of the following linear programming

problem: min! 6a c under the conditions

24 2324 23

, , 0.

a b c

a b c

a b c

+ + ≥

+ + ≥

≥

14

131323

a b

a

a b

+

+ +3

c

c

≤

+ ≤

≤

The solution of this problem is given by all ) fulfilling the condition (0, ,b cγ =49

. In particular, b c+ = 29

b c= = is a feasible solution, which is among the

cases shown above (blue line). In a similar way, we can determine the “best” VaR scenario here, which is

uniquely determined by the parameter 2 ,0,09

γ ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

(3; ; )Q γ i

(3; ; )γ i

15

.⎛ ⎞

This case is also shown

above (pink line). Naturally, it is also possible to express the quantile function for all choices of γ in an explicit way, by solving the appropriate quadratic equations in the representation of above. 2F

For the three cases “worst” VaR scenario, independence and “best” VaR scenario, we obtain

4 1 7 89 7,3 3 9 9

82 1 , 197 1(3; ; ) 2 2(1 ), 19 9

5 1 72(1 ), 13 2 9

u u

u u

Q u u u

u u

γ γ

⎧⎧⎪⎪⎪⎪ + − ≤ ≤⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪ − − ≤ ≤⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪= − − ≤ ≤⎨⎪⎪⎪⎪⎪⎪ − − ≤ ≤⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2 20, ,9 9,

1 1, , ,9 9

2, ,0,0 .9

γ

γ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

16

Likewise, for the Expected Shortfall1, we obtain

3

1

38 36 2 9 7 7 827 81 ,1 91 82 1 1 ,3 9

1 1 7ES(3; ; ) (3; ; ) 2 2(1 ),1 3 9

5 1 72(1 ),3 3 9

u

u u

u

u u

u Q v dv uu

u u

γ γ

⎧⎧ −⎪⎪⎪⎪ − −⎪⎪⎪⎪⎪⎪⎪⎪ −⎨⎪⎪ ⎛ ⎞⎪ ⎟⎜ − −⎪ ⎟⎜⎪ ⎟⎜⎝ ⎠⎪⎩

= = − − ≤⎨−

− −

∫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2 29 0, ,9 9

1,

1 1 11, , ,9 9 921, ,0,0 .9

u

u

γ

γ

γ

≤ ≤ ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠≤ ≤

⎛ ⎞⎟⎜≤ = ⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜≤ ≤ = ⎟⎜ ⎟⎜⎝ ⎠

|d du ES n u E Sγ == ( ; ;Q n uγ 0 1.u< <

1 .uε = −

17

1 Here we denote u with ) for In the literature, one often finds the complementary notation with

( )ES ( ; ; ) S VaR≥ uVaR =

VaR and Expected Shortfall for “worst” case [1],

independence case [2], and “best” case [3]

VaR [1]

VaR [2]

VaR [3]

18

ES [3]

ES [1]

ES [2]

For instance, we obtain

worst case independence best case

VaR0,9 1,6838 1,5528 1,4430

ES0,9 1,7892 1,7019 1,5269

VaR0,99 1,9000 1,8586 1,5960

ES0,99 1,9333 1,9057 1,6225

19

Example. Suppose that the risks 1X 2X and are each uniformly distributed and their joint cumulative distribution function is a copula of Clayton or Gumbel type, resp., i.e.

( ) ( )( ) { }( )

1/

1

1/

2

; , 1 (

; , exp ( ln ) ( ln ) (

C u v u v

C u v u v

θθ θ

θθ θ

θ

θ

−− −= + −

= − − + −

Clayton) or

Gumbel) for ( )2( , ) 0,1u v ∈

1.

2 1:S X X= +

20

,

with θ≥

We consider the distribution of the aggregated risk 2. The following graph shows the calculated densities for under these copulas, for a grid-type copula approximation, with 10000 subsquares of the unit interval of equal area each.

2S

Clayton copula, θ 1=Clayton copula, 2θ=Gumbel copula, 2θ=

el copula, 3θ=Gumb

approximation of densities for two aggregate dependent risks

21

Example (extension to joint distribution with compact support). We consider five dependent risks , ,1 5X X with different marginal distributions and joint density given by

20 20 20 20 20

1 51 1 1 1 1

( , , ) ( , , , , ) ( , , , , ) 1 5( , ,I i j k l mi j k l m

)f x x i j k l m xα= = = = =

= ⋅∑∑∑∑∑ 1

] (( , , , , ) : 1, 1, 1, 1, 1,

x

with ]( ] ( ] ( ] (I i j k l m i i j j k k l l m m= − × − × − × − × − and

2 3

1 3( , , , , )i j

i j k l mK i j k l

α+ +

= ⋅+ + +

{ }, , , , 1, , 20i j k l m∈

4 sin( )

4

k

m

+

+ − for ,

with the normalizing constant

20 20 20 20 20

2 31 1 1 1 1

4 sin(3

i j k l m

i jK

i j k l= = = = =

+ +=

+ + + +∑∑∑∑∑)

12198.4

k

m

+≈

−

5 3200000= 5

22

(support of the joint distribution: disjoint hypercubes in ) 20

plot of marginal densities

2g

5g

1 3g g=

23

4g

copulandenceindepe

density of five aggregated risks, dependent vs. independent case

24

independencecopula

25cumulative distribution function, dependent vs. independent case

Lemma. Suppose that the risks 1X 2X and follow a Pareto distribution with density

(1 )( ) (1 ) ,f x x xββ − += + ≥ ( 0)β>

2β =

0

each. Then the density g and cumulative distribution function G of the aggregated risk 2:S X X= + can be explicitly calculated in the following cases: 2 1

1/ :

Case 1: 1X and 2X are independent:

32

1 1( ) , ( ) 1 2 ,2(2 ) 1 1

z zg z G z zzz z z

+= ≈ = − ≥++ + +

26

0

Case 2: 1X and 2X are co-monotonic:

3 31 1( ) , ( ) 21 ,

24 1 / 2 2 1g z G z z

zz z= ≈ = − ≥

++ +0

Case 3: 1X and 2X are counter-monotonic:

33

1 ,1

, 6.

zz

zz

≈+

≥

27

2

4 2 3( )6 4 3 3 2

8 12 (8 4 ) 3( )

2

z zg zz z z

z z zG z

z

+ − +=+ − + + +

+ + − + +=

+

VaR’s for the three cases

Case 1: independence

Case 2:(bes

co-monotonicityt case)

-monotonicity

28

Case 3: counter(worst case)

1:β =

Case 1: 1X and 2X are independent:

2

3 2 2

ln(1 ) 2 2 2( ) 4 , ( )(2 ) (1 )(2 ) (1 )

z z zg z G zz z z z+ += + ≈ =+ + + + 2

2ln(1 ) , 0(2 )z z z

z− + ≥+

Case 2: 1X and 2X are co-monotonic:

2 2 2

1 2 2( ) , ( ) 21 ,2(1 / 2) (2 ) (1 ) 2

g z G z zz z z z

= = ≈ = − ≥+ + + +

0

Case 3: 1X and 2X are counter-monotonic:

23/ 2

2 2( ) , ( ) 2 , 2(1 ) 22(2 )

zg z G z zz zz z

−= ≈ = ≥+ +− +

29

.

Case 1: independence

VaR’s for the three cases

Case 2: co-monotonicity

Case 3: counter-monotonicity

30

2 :β =

Case 1: 1X and 2X are independent:

( )22 3

4 ,) (1 )

ln(1 ), 0)

zz

z z

≈+

+ ≥

5 4

3 2

3 5

4 10 1048ln(1 )( )(2 ) (2 ) (1

7 16 6 12( )(2 ) (1 ) (2

z zzg zz z z

z z zG z zz z z

+ ++= ++ + ++ + += −+ + +

Case 2: 1X and 2X are co-monotonic:

3 3

8 8( ) , ( ) 1 2

4 ,(2 ) (1 ) (2 )

g z G z zz z z

= ≈ = − ≥+ + +

0

Case 3: 1X and 2X are counter-monotonic:

( )22 2 2 2

4 2 22

4(2 )( )4 2 2 4 5 4 5 1

1( ) (2 ) 4(2 ) 8 8 (2 ) 1,

3

4 ,(1 )4 5

.(2 )

zg zzz z z z z z z z

G z z z z zz

+= ≈++ + − + + + + − + +

= + − + − − + ++ 31

2≥

Case 1: independence

VaR’s for the three cases

Case 2: co-monotonicity

Case 3: counter-monotonicity

32

empirical copula for windstorm vs. flooding, from real data

33

The data can be well fitted to a 4 4× grid-type copula represented by the following weight matrix:

13 12136 136

8 15136 136

8 7 7

8 1136 136

7 4136 136

1136 136 136 136

5 12 170136 136 136

A2

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

: :V= +

34

From this , we obtain the following density f and cumulative distribution function F for the aggregated risk S U

26 1, 017 414 3 1 1,17 17 4 222 1 1 3,17 17 2 432 22 3, 117 17 4

( ) ( )28 18 5, 117 17 426 27 5 3,17 17 4 233 14 3 7,17 17 2 4

74 2 , 24

0, otherwise

x x

x x

x x

x x

f x F xx x

x x

x x

x x

⎧⎪⎪ ≤ ≤⎪⎪⎪⎪⎪⎪ + ≤ ≤⎪⎪⎪⎪⎪⎪ − ≤ ≤⎪⎪⎪⎪⎪⎪⎪ − ≤ ≤⎪⎪⎪⎪=⎨ − ≤ ≤⎪⎪⎪⎪⎪⎪⎪ − ≤ ≤⎪⎪⎪⎪⎪⎪ − ≤ ≤⎪⎪⎪⎪⎪⎪ − ≤ ≤⎪⎪⎪⎪⎪⎪⎩

2

2

2

2

2

2

2

2

0,13 , 0177 3

17 1711 117 17

11 3217 179 28

17 1713 2717 17

7 3317 17

4 3

1,

x x

x x

x x

x x

x x

x x

x x

x x

⎧⎪⎪⎪⎪⎪⎪⎪⎪

+ − ≤

− +

− +=⎨

− +

−

− +

− + − ≤

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

014

3 1 1,136 4 2

5 1 3,136 2 4

47 3, 168 439 5, 168 4

197 5 3,136 4 2

163 3 7,136 2 4

7, 24

2.

x

x

x

x

x

x

x

x

x

<

≤ ≤

≤

≤ ≤

− ≤ ≤

− ≤ ≤

+ ≤ ≤

− ≤ ≤

≤

>

35

VaR, dependecase

nt

ndependent case

R, independent case

ES, dependent caseES, i

Va

Dependence matters!

VaR and ES, dependence vs. independence

36