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Measuring tail dependence for collateral losses using bivariate L´ evy process

Jiwook Jang Actuarial Studies Faculty of Commerce and Economics University of New South Wales Sydney, AUSTRALIA

Actuarial Studies Research Symposium 11 November 2005

Overview

• Collateral losses : In worldwide, once a storm or earthquake arrives, it

brings about damages in properties, motors and interruption of busi- nesses. It occurred a couple of losses simultaneously from the World Trade Centre (WTC) catastrophe.

• Bivariate L´

evy process with a copula, i.e. bivariate compound Pois- son process with a member of Farlie-Gumbel-Morgenstern copula for dependence between losses.

• Calculation of the coefficient of (upper) tail dependence using Fast

Fourier transform.

Bivariate aggregate losses

• Insurance companies are experiencing dependent losses from one spe-

cific event such as flood, windstorm, hail, earthquake and terrorist attack. So for bivariate risk case, we can model L(1)

t

=

Nt

X

i=1

Xi, L(2)

t

=

Nt

X

i=1

Yi,

where L(1)

t

is the total losses arising from risk type 1, L(2)

t

is the total losses arising from risk type 2 and Nt is the total number of collateral losses up to time t. Xi and Yi, i = 1, 2, · · · , are the loss amounts, which are to be dependent each other, where H(x) be the identically distribution function of X and H(y) be the identically distribution function of Y .

A point process and a copula

• We assume that the collateral loss arrival process, Nt follows a Pois-

son process with loss frequency rate µ. It is also assumed that is independent of Xi and Yi.

• We employ the Farlie-Gumbel-Morgenstern family copula, that is given

by C(u, v) = uv + θuv(1 − u)(1 − v), where u ∈ [0, 1], v ∈ [0, 1] and θ ∈ [−1, 1] , to capture the dependence of collateral losses of X and Y .

Copula

• A general approach to model dependence between random variables is

to specify the joint distribution of the variables using copulas.

• Dependence between random variables is usually completely described

by their multivariate distribution function, To define a copula more formally, consider

u = (u1, · · · , un)

belongs to the n-cube [0, 1]n. A copula, C(u), is a function,with support [0, 1]n and range [0, 1], that is multivariate cumulative distribution function whose univariate marginals are uniform U(0, 1).

• As a consequence of this definition, we see that

C

¡u1, · · · , uk−1, 0, uk+1, · · · , un ¢ = 0

and C (1, · · · , 1, uk, 1, · · · , 1) = uk for all k = 1, 2, · · · n. Any copula C is therefore the distribution of a multivariate uniform random vector.

Sklar theorem

• Let F be a two-dimensional distribution function with margins, F1,

F2. Then there exists a two-dimensional copula C such that for all x ∈

−

R

2

, F(x1, x2) = C(F1(x1), F2(x2)). (1)

• If F1 and F2 are continuous then C is unique, i.e.

C(u1, u2) = F(F −1

1

(u1), F −1

2

(u2)),

where F −1

1

, F −1

2

denote the quantile functions of the univariate margins F1, F2. Otherwise C is uniquely determined on Ran F1× Ran F2.

• Conversely, if C is a copula and F1 and F2 are distribution functions,

then the function F defined by (1) is a two-dimensional distribution function with margins F1 and F2.

Farlie-Gumbel-Morgenstern family copula with exponential margins

• In order to obtain the explicit expression of the function F(x, y), that is

a two-dimensional distribution function with margins H (x) and H(y), we let X and Y be exponential random variables, i.e. H (x) = 1−e−αx (α > 0, x > 0) and H(y) = 1 − e−βy (β > 0, y > 0), then the joint distribution function F(x, y) is given by F(x, y) = C(1 − e−αx, 1 − e−βy) = 1 − e−βy − e−αx + e−αx−βy + θe−αx−βy −θe−αx−2βy − θe−2αx−βy + θe−2αx−2βy.

• And its derivative is given by

dF(x, y) = dC(1 − e−αx, 1 − e−βy) = (1 + θ) αβe−αx−βy −2θαβe−αx−2βy − 2θαβe−2αx−βy +4θαβe−2αx−2βy. (2)

Upper tail dependence of collateral losses

• We examine upper tail dependence of collateral losses X and Y as

insurance companies’ concerns are on extreme losses in practice.

• we adopt the coefficient of upper tail dependence, λU, used by Em-

brechts, Lindskog and McNeil (2003), lim

u%1P

(

L(2)

t

> G−1

L(2)

t

(u) | L(1)

t

> G−1

L(1)

t

(u)

)

= λU provided that the limit λU ∈ [0, 1] exists, where GL(1)

t

and GL(2)

t

are marginal distribution functions for L(1)

t

and L(2)

t

.

The generator of the process

µ

L(1)

t

, L(2)

t

, t

¶

• The generator of the process

µ

L(1)

t

, L(2)

t

, t

¶

acting on a function f

³

l(1), l(2), t

´

belonging to its domain is given by A f

³

l(1), l(2), t

´

= ∂f ∂t +µ

⎡ ⎢ ⎣

∞

Z

∞

Z

f

³

l(1) + x, l(2) + y, t

´

dC(H (x) , H (y)) − f

³

l(1), l(2), t

´ ⎤ ⎥ ⎦ .

A suitable martingale

• Considering constants ν ≥ 0 and ξ ≥ 0,

exp

µ

−νL(1)

t

¶

exp

µ

−ξL(2)

t

¶

exp

⎡ ⎢ ⎣µ

t

Z

{1 − ˆ c (ν, ξ)} ds

⎤ ⎥ ⎦

is a martingale where ˆ c (ν, ξ) =

∞

R

∞

R

e−νxe−ξydC(H (x) , H (y)).

The joint Laplace transform of the distribution of L(1)

t

and L(2)

t

• Using the martingale obtained above, the joint Laplace transform of

the distribution of L(1)

t

and L(2)

t

at time t is given by E

½

e−νL(1)

t e−ξL(2) t |L(1)

0 , L(2)

¾

= exp

µ

−νL(1)

¶

exp

µ

−ξL(2)

¶

× exp

⎡ ⎢ ⎣−µ

t

Z

{1 − ˆ c (ν, ξ)} ds

⎤ ⎥ ⎦ .

• For simplicity, we assume that L(1)

= 0 and L(2) = 0, then it is given by E

½

e−νL(1)

t e−ξL(2) t

¾

= exp

⎡ ⎢ ⎣−µ

t

Z

{1 − ˆ c (ν, ξ)} ds

⎤ ⎥ ⎦ ,

where ˆ c (ν, ξ) =

∞

R

∞

R

e−νxe−ξydC(H (x) , H (y)).

• In order to obtain the explicit expression of the joint Laplace transform
• f the distribution of L(1)

t

and L(2)

t

at time t, let us use the joint density function f(x, y) driven by (2), then it is given by E

½

e−νL(1)

t e−ξL(2) t

¾

= exp

"

−µ

(

(αξ + βν + νξ) (2α + ν) (2β + ξ) − θαβ νξ (α + ν) (β + ξ) (2α + ν) (2β + ξ)

)

t

#

. (3)

• If we set ξ = 0, then the Laplace transform of the distribution of L(1)

t

is given by E

½

e−νL(1)

t

¾

= exp

½

−µ

µ

ν α + ν

¶

t

¾

(4) and if we set ν = 0, then the Laplace transform of the distribution of L(2)

t

is given by E

½

e−ξL(2)

t

¾

= exp

(

−µ

Ã

ξ β + ξ

!

t

)

, (5)

which are the Laplace transform of the distribution of the compound Pois- son process with exponential loss sizes. Due to the dependence of col- lateral losses of X and Y with sharing loss frequency rate µ, it is obvious that E

½

e−νL(1)

t e−ξL(2) t

¾

6= E

½

e−νL(1)

t

¾

E

½

e−ξL(2)

t

¾

.

When θ = 0,.i.e. no dependence in loss sizes

• If θ = 0, then we have

E

½

e−νL(1)

t e−ξL(2) t

¾

= exp

"

−µ

(

(αξ + βν + νξ) (α + ν) (β + ξ)

)

t

#

, (6) which is the case that two losses X and Y occur at the same time from a sharing loss frequency rate µ, but their sizes are independent each other.

• If loss X occurs with its frequency rate µ(x) and loss Y occurs with

its frequency rate µ(y) respectively and everything is independent each

• ther, we can easily derive the explicit expression of the joint Laplace

transform of the distribution of L(1)

t

and L(2)

t

at time t, i.e. E

½

e−νL(1)

t e−ξL(2) t

¾

= E

½

e−νL(1)

t

¾

E

½

e−ξL(2)

t

¾

= exp

½

−µ(x)

µ

ν α + ν

¶

t

¾

exp

(

−µ(y)

Ã

ξ β + ξ

!

t

)

. (7)

• If we set µ = µ(x) = µ(y), i.e. frequency rate for loss X and Y are

just the same, then (7) becomes E

½

e−νL(1)

t e−ξL(2) t

¾

= E

½

e−νL(1)

t

¾

E

½

e−ξL(2)

t

¾

= exp

½

−µ

µ

ν α + ν

¶

t

¾

exp

(

−µ

Ã

ξ β + ξ

!

t

)

= exp

"

−µ

(

(αξ + βν + 2νξ) (α + ν) (β + ξ)

)

t

#

. (8) Equation (8) looks similar to (6) as loss size X and Y are independent and their frequency rates are the same. However the joint Laplace transform

• f the distribution of L(1)

t

and L(2)

t

at time t expressed by (8) are the case that they are occurring independently, not collaterally like (6).

Covariance and linear correlation of collateral losses

• Differentiating (3) w.r.t. ν and ξ and set ν = 0 and ξ = 0, then we

can easily derive the joint expectation of L(1)

t

and L(2)

t

at time t, i.e. E

½

L(1)

t

L(2)

t

¾

= µ2

αβt2 + µ αβ

³

1 + θ

4

´

t.

• Also from (4) and (5) we can easily derive the expectation of L(1)

t

and L(2)

t

at time t, i.e. E

½

L(1)

t

¾

= µ

αt

and E

½

L(2)

t

¾

= µ

βt.

• The higher moments of L(1)

t

and L(2)

t

at time t can be obtained by differentiating it further, i.e. V ar

½

L(1)

t

¾

= 2µ α2t and V ar

½

L(2)

t

¾

= 2µ β2t.

• The covariance between L(1)

t

and L(2)

t

at time t is given by Cov(L(1)

t

, L(2)

t

) = E

½

L(1)

t

L(2)

t

¾

− E

½

L(1)

t

¾

E

½

L(2)

t

¾

= µ αβ

µ

1 + θ 4

¶

t. (9)

• As linear correlation (or Pearson’s correlation) has been most popularly

used in practice as a measure of dependence, we present the expression

• f the linear correlation coefficient for L(1)

t

and L(2)

t

at time t, denoted by ρ

µ

L(1)

t

, L(2)

t

¶

, ρ

µ

L(1)

t

, L(2)

t

¶

= Cov(L(1)

t

, L(2)

t

)

∙

V ar

½

L(1)

t

¾¸1

2 ∙

V ar

½

L(2)

t

¾¸1

2

=

µ αβ

³

1 + θ

4

´

t

³2µ

α2t

´1

2

µ

2µ β2t

¶1

2

= 4 + θ 8 . (10)

Example 1

• The parameter values used to calculate the covariance and linear cor-

relation using (9) and (10) are µ = 4, α = 1, β = 0.5, t = 1. From (9) and (10), the calculations of covariance and linear correlation between L(1)

t

and L(2)

t

at time t are shown in Table 1 and Table 2 respectively.

Table 1. θ Cov(L(1)

t

, L(2)

t

) −1 6 −0.5 7 8 0.5 9 1 10 Table 2. θ ρ

µ

L(1)

t

, L(2)

t

¶

−1 0.375 −0.5 0.4375 0.5 0.5 0.562 5 1 0.625

The coefficient of upper tail dependence, λU

• As it is not possible for us to obtain the joint distribution of L(1)

t

and L(2)

t

explicitly, we invert the joint Fast Fourier transform obtained from the joint Laplace transform of collateral losses to approximate the coefficient of (upper) tail dependence (Castleman 1996; Gonzalez and Woods 2002 and Gonzalez et al. 2004), i.e. F(u, v) =

1 MN M−1

P

x=0 N−1

P

y=0

f(x, y)e−j2π(ux

M +vy N ) and

f(x, y) =

M−1

P

u=0 N−1

P

v=0

F(u, v)ej2π(ux

M +vy N )

for x = 0, 1, 2, · · · , M − 1 and y = 0, 1, 2, · · · , N − 1.

• The below figures are the joint distribution of collateral losses and

their contours at each value of θ.

The joint distribution of collateral losses with θ = 1

The contour of the joint distribution of collateral losses with θ = 1

The joint distribution of collateral losses with θ = 0.5

The contour of the joint distribution of collateral losses with θ = 0.5

The joint distribution of collateral losses with θ = 0

The contour of the joint distribution of collateral losses with θ = 0

The joint distribution of collateral losses with θ = −0.5

The contour of the joint distribution of collateral losses with θ = −0.5

The joint distribution of collateral losses with θ = −1

The contour of the joint distribution of collateral losses with θ = −1

• Using Matlab, the calculations of the coefficients of (upper) tail de-

pendence for collateral losses are shown in Table 3, Table 4, Table 5 and Table 6 using the different VaR at 90%, 95%, 99% and 99.9%.

Table 3. θ

P

½

L(1)

t

> 7.84, L(2)

t

> 15.68

¾

P

½

L(1)

t

> 7.84 | L(2)

t

> 15.68

¾

= P

½

L(2)

t

> 15.68 | L(1)

t

> 7.84

¾

1 0.038425 0.38425 0.5 0.034221 0.34221 0.030239 0.30239 −0.5 0.026450 0.26450 −1 0.022831 0.22831 where P

½

L(1)

t

> 7.84

¾

= P

½

L(2)

t

> 15.68

¾

= 0.1

Table 4. θ

P

½

L(1)

t

> 9.37, L(2)

t

> 18.74

¾

P

½

L(1)

t

> 9.37 | L(2)

t

> 18.74

¾

= P

½

L(2)

t

> 18.74 | L(1)

t

> 9.37

¾

1 0.014593 0.29187 0.5 0.012565 0.25129 0.010681 0.21363 −0.5 0.0089309 0.17862 −1 0.0073023 0.14605 where P

½

L(1)

t

> 9.37

¾

= P

½

L(2)

t

> 18.74

¾

= 0.05

Table 5. θ

P

½

L(1)

t

> 12.61, L(2)

t

> 25.22

¾

P

½

L(1)

t

> 12.61 | L(2)

t

> 25.22

¾

= P

½

L(2)

t

> 25.22 | L(1)

t

> 12.61

¾

1 0.0015290 0.15290 0.5 0.0012168 0.12168 0.00094405 0.094405 −0.5 0.00070781 0.070781 −1 0.00050554 0.050554 where P

½

L(1)

t

> 12.61

¾

= P

½

L(2)

t

> 25.22

¾

= 0.01

Table 6. θ

P

½

L(1)

t

> 16.81, L(2)

t

> 33.62

¾

P

½

L(1)

t

> 16.81 | L(2)

t

> 33.62

¾

= P

½

L(2)

t

> 33.62 | L(1)

t

> 16.81

¾

1 0.000059365 0.059365 0.5 0.000042203 0.042203 0.000028667 0.028667 −0.5 0.000018274 0.018274 −1 0.000010590 0.010590 where P

½

L(1)

t

> 16.81

¾

= P

½

L(2)

t

> 33.62

¾

= 0.001

Further Research

• Dependence in interarival time of losses.
• In practice, we might need to employ one of the heavy-tailed distrib-

utions for jump sizes, H (x) and H(y) to deal with extreme losses.

• Employing other copulars.
• Nt can be the Cox process, rather than the Poisson process, that has

stochastic jump frequency rate µ(t).