Measuring tail dependence for collateral losses using bivariate L - PowerPoint PPT Presentation

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 coe ffi cient of (upper) tail dependence using Fast Fourier transform.

Bivariate aggregate losses • Insurance companies are experiencing dependent losses from one spe- ci fi c event such as fl ood, windstorm, hail, earthquake and terrorist attack. So for bivariate risk case, we can model N t X L (1) = X i , t i =1 N t X L (2) = Y i , t i =1

where L (1) is the total losses arising from risk type 1, L (2) is the total t t losses arising from risk type 2 and N t is the total number of collateral losses up to time t . X i and Y i , 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, N t follows a Pois- son process with loss frequency rate µ . It is also assumed that is independent of X i and Y i . • We employ the Farlie-Gumbel-Morgenstern family copula, that is given by C ( u, v ) = uv + θ uv (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 de fi ne a copula more formally, consider u = ( u 1 , · · · , u n ) 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 de fi nition, we see that ¢ = 0 ¡ u 1 , · · · , u k − 1 , 0 , u k +1 , · · · , u n C and C (1 , · · · , 1 , u k , 1 , · · · , 1) = u k 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, F 1 , F 2 . Then there exists a two-dimensional copula C such that for all 2 − x ∈ R , F ( x 1 , x 2 ) = C ( F 1 ( x 1 ) , F 2 ( x 2 )) . (1) • If F 1 and F 2 are continuous then C is unique, i.e. C ( u 1 , u 2 ) = F ( F − 1 ( u 1 ) , F − 1 ( u 2 )) , 1 2

where F − 1 , F − 1 denote the quantile functions of the univariate margins 1 2 F 1 , F 2 . Otherwise C is uniquely determined on Ran F 1 × Ran F 2 . • Conversely, if C is a copula and F 1 and F 2 are distribution functions, then the function F de fi ned by (1) is a two-dimensional distribution function with margins F 1 and F 2 .

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 − α x ( α > 0 , x > 0) and H ( y ) = 1 − e − β y ( β > 0, y > 0), then the joint distribution function F ( x, y ) is given by F ( x, y ) = C (1 − e − α x , 1 − e − β y ) = 1 − e − β y − e − α x + e − α x − β y + θ e − α x − β y − θ e − α x − 2 β y − θ e − 2 α x − β y + θ e − 2 α x − 2 β y .

• And its derivative is given by dF ( x, y ) = dC (1 − e − α x , 1 − e − β y ) = (1 + θ ) αβ e − α x − β y − 2 θαβ e − α x − 2 β y − 2 θαβ e − 2 α x − β y +4 θαβ e − 2 α x − 2 β y . (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 coe ffi cient of upper tail dependence, λ U , used by Em- brechts, Lindskog and McNeil (2003), ( ) L (2) ( u ) | L (1) > G − 1 > G − 1 u % 1 P lim ( u ) = λ U t t L (2) L (1) t t provided that the limit λ U ∈ [0 , 1] exists, where G L (1) and G L (2) are t t marginal distribution functions for L (1) and L (2) . t t

µ ¶ L (1) , L (2) The generator of the process , t t t µ ¶ L (1) , L (2) • The generator of the process acting on a function , t t t ³ ´ l (1) , l (2) , t f belonging to its domain is given by ³ ´ l (1) , l (2) , t A f ∂ f = ∂ t ⎡ ⎤ ∞ ∞ Z Z ³ ´ ³ ´ ⎢ ⎥ l (1) + x, l (2) + y, t l (1) , l (2) , t + µ f dC ( H ( x ) , H ( y )) − f ⎦ . ⎣ 0 0

A suitable martingale • Considering constants ν ≥ 0 and ξ ≥ 0, ⎡ ⎤ t µ ¶ µ ¶ Z ⎢ ⎥ − ν L (1) − ξ L (2) exp exp exp ⎣ µ { 1 − ˆ c ( ν , ξ ) } ds ⎦ t t 0 ∞ ∞ R R e − ν x e − ξ y dC ( H ( x ) , H ( y )). is a martingale where ˆ c ( ν , ξ ) = 0 0

The joint Laplace transform of the distribution of L (1) and L (2) t t • Using the martingale obtained above, the joint Laplace transform of the distribution of L (1) and L (2) at time t is given by t t ½ ¾ µ ¶ µ ¶ e − ν L (1) t e − ξ L (2) t | L (1) 0 , L (2) − ν L (1) − ξ L (2) = exp exp E 0 0 0 ⎡ ⎤ t Z ⎢ ⎥ × exp ⎣ − µ { 1 − ˆ c ( ν , ξ ) } ds ⎦ . 0

• For simplicity, we assume that L (1) = 0 and L (2) = 0, then it is given 0 0 by ⎡ ⎤ t ½ ¾ Z e − ν L (1) t e − ξ L (2) ⎢ ⎥ E = exp ⎣ − µ { 1 − ˆ c ( ν , ξ ) } ds ⎦ , t 0 ∞ ∞ R R e − ν x e − ξ y dC ( H ( x ) , H ( y )) . where ˆ c ( ν , ξ ) = 0 0

• In order to obtain the explicit expression of the joint Laplace transform of the distribution of L (1) and L (2) at time t, let us use the joint density t t function f ( x, y ) driven by (2), then it is given by ½ ¾ e − ν L (1) t e − ξ L (2) E t " ( ) # ( αξ + βν + νξ ) (2 α + ν ) (2 β + ξ ) − θαβ νξ = exp − µ t . ( α + ν ) ( β + ξ ) (2 α + ν ) (2 β + ξ ) (3)

• If we set ξ = 0 , then the Laplace transform of the distribution of L (1) t is given by ½ ¾ ½ µ ¶ ¾ e − ν L (1) ν = exp (4) E − µ t t α + ν and if we set ν = 0, then the Laplace transform of the distribution of L (2) t is given by ( Ã ! ) ½ ¾ ξ e − ξ L (2) E = exp − µ t , (5) t β + ξ

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 ½ ¾ ½ ¾ ½ ¾ e − ν L (1) t e − ξ L (2) e − ν L (1) e − ξ L (2) E 6 = E E . t t t

When θ = 0,.i.e. no dependence in loss sizes • If θ = 0 , then we have " ( ) # ½ ¾ ( αξ + βν + νξ ) e − ν L (1) t e − ξ L (2) E = exp − µ t , (6) t ( α + ν ) ( β + ξ ) 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

other, we can easily derive the explicit expression of the joint Laplace transform of the distribution of L (1) and L (2) at time t , i.e. t t ½ ¾ ½ ¾ ½ ¾ e − ν L (1) t e − ξ L (2) e − ν L (1) e − ξ L (2) E = E E t t t ( Ã ! ) ½ µ ¶ ¾ ν ξ − µ ( x ) − µ ( y ) = exp t exp t . α + ν β + ξ (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 ½ ¾ ½ ¾ ½ ¾ e − ν L (1) t e − ξ L (2) e − ν L (1) e − ξ L (2) = E E E t t t ( Ã ! ) ½ µ ¶ ¾ ν ξ = exp − µ t exp − µ t α + ν β + ξ " ( ) # ( αξ + βν + 2 νξ ) = exp (8) − µ t . ( α + ν ) ( β + ξ ) 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 of the distribution of L (1) and L (2) at time t expressed by (8) are the case t t that they are occurring independently, not collaterally like (6).

Covariance and linear correlation of collateral losses • Di ff erentiating (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) and L (2) at time t , i.e. t t ½ ¾ ³ ´ = µ 2 L (1) L (2) αβ t 2 + µ 1 + θ E t. t t αβ 4 • Also from (4) and (5) we can easily derive the expectation of L (1) and t L (2) at time t , i.e. t ½ ¾ ½ ¾ L (1) L (2) = µ = µ and β t . E α t E t t