Approximation and Aggregation of Risks by Variants of Panjer s Recursion

Transcription

1 Approximation and Aggregation of Risks by Variants of Panjer s Recursion Uwe Schmock Vienna University of Technology, Austria Version of Slides: Feb. 8, 2013 Presentation based on joint work with Stefan Gerhold and Richard Warnung and work in progress with Cordelia Rudolph Financial support: OeNB, Bank Austria, OeKB, CDG, WWTF,... Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 1

2 Outline 1 Collective risk model and variants of Panjer s recursion 2 Generalized gamma convolutions Error bounds and approximation Examples 3 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 2

3 The collective model of risk theory Task: Calculate (fast and in a numerically stable way if possible) the distribution of the random sum where S = N n=1 {X n } n N is a sequence of N 0 -valued i.i.d. random variables, N is an N 0 -valued random variable independent of {X n } n N. Applications: X n N insurance claims with sizes X 1, X 2,... N credit losses, X n equals the loss given default minus recovery N operational losses with sizes X 1, X 2,... Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 3

4 The simple-minded solution Start with S 1 := X 1 and calculate the distribution of S k := X X k = S k 1 + X k for k 2 recursively by convolution (due to independence of S k 1 and X k ) P(S k = n) = n P(S k 1 = n j) P(X k = j), n N 0. j=0 For the distribution of S = N k=1 X k = S N, due to independence of N and {S k } k N, just sum up P(S = n) = P(N = k) P(S k = n), n N 0. k=0 This is numerically stable but very time consuming. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 4

5 More sophisticated approaches Approximations based on clever use of limit theorems (cf. textbooks on risk theory). Fast Fourier Transform (FFT): Can be problematic for heavy-tailed distributions (see later). FFT with exponential tilting: Critical choice of tilting parameter, numerical instabilities are possible (see later). I will concentrate on: Recursive methods, in particular variants and extensions involving Panjer s recursion. Requires restrictions on the distribution of claim number N. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 5

6 Panjer class distributions (for claim number N) Definition A probability distribution {q n } n N0 is in the Panjer(a, b, k) class with a, b R and k N 0 if q 0 = q 1 = = q k 1 = 0 and q n = ( a + b ) q n 1 for all n N with n k + 1. n Note: Same distribution on both sides, see later... Determination of all distributions: k = 0: Sundt and Jewell (1981) k = 1: Willmot (1988) General k N 0 : Hess, Liewald and Schmidt (2002) Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 6

7 Basic Panjer class distributions Bin(m, p) Panjer( p q, m+1 q p, 0) with m N and p [0, 1) Poisson(λ) Panjer(0, λ, 0) with λ 0 NegBin(α, p) Panjer(q, (α 1)q, 0) with α > 0 and p (0, 1) Log(q) Panjer(q, q, 1) with q (0, 1) and q n = qn for all n N n log(1 q) Extended logarithmic distribution: Given k N \ {1} and q (0, 1], define q 0 = = q k 1 = 0 and q n = ( n ) 1q n k l=k ( l ) 1q l k for n k. ExtLog(k, q) is in Panjer(q, kq, k), has heavy tails for q = 1. Closed-form expression for the series is available in our paper. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 7

8 Basic Panjer class distributions (cont.) Extended Negative Binomial Distribution: For k N, α ( k, k + 1) and p [0, 1) define q = 1 p, q 0 = = q k 1 = 0 and ( α+n 1 ) q n q n = n p α k 1 j=0 ( α+j 1 ) j q j for n k. ExtNegBin(α, k, p) is in Panjer(q, (α 1)q, k). It has heavy tails for q = 1, which is good for reinsurance companies. Theorem (Hess, Liewald and Schmidt, 2002) Let Q = {q n } n N0 be non-degenerate. Then are equivalent: Q is in Panjer(a, b, k). Q is the k-truncation of a basic Panjer(a, b, k ) distribution Q = {q n} n N0 with k k and c := n=k q n > 0, i.e., q n = 0 for n {0, 1,..., k 1} and q n = q n/c for all n k. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 8

9 (Extended) Panjer recursion Theorem (Panjer, 1981; Hess, Liewald and Schmidt, 2002) Assume that the probability distribution {q n } n N0 of N belongs to the Panjer(a, b, k) class and a P(X 1 = 0) 1. Then the distribution {p n } n N0 of S = X X N can be calculated by { q 0 if P(X 1 = 0) = 0, p 0 = ϕ N (P(X 1 = 0)) = E [ (P(X 1 = 0)) N] otherwise, where ϕ N (s) = n N 0 q n s n is the probability generating function of N, and the recursion formula p n = ( 1 P(S k = n)q k + 1 a P(X 1 = 0) n ( j=1 for all n N, where S k := X X k. a + bj n }{{} 0? ) ) P(X 1 = j)p n j }{{} same dist. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 9

10 Historical comment on Panjer s recursion For α R and a power series f (s) = k=0 a ks k with a 0 0, the coefficients {b n } n N0 of the power series f α (s) satisfy the recursion b n = 1 na 0 n ( ) (1 α)k n ak b n k, n N. k=1 Gould (1974) has traced this remarkable, often rediscovered recurrence back to Euler (1748). Using the probability generating functions of the binomial, negative binomial, and extended negative binomial claim number distributions and ϕ S = ϕ N ϕ X1, the above formula applied to f (s) = 1 qϕ X1 (s) gives the corresponding Panjer recursions. Panjer (1981) introduced the recursion to actuarial science. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 10

11 Numerical stability of Panjer s recursion Panjer s recursion is certainly numerically stable when a + bj n 0 for all j {1,..., n}. This is the case when a 0 and b a, hence for Poisson distribution, Negative binomial distribution, Logarithmic distribution, Trancations of the above. It is potentially unstable for Binomial distribution, Extended negative binomial distribution, Extended logarithmic distribution. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 11

12 Example for numerical instability due to cancellation Take N ExtNegBin(α, k, p) with k N, ε, p (0, 1) and α = k + ε. Consider the loss distribution P(X 1 = 1) = P(X 1 = l) = 1/2 with l 3. Then k(l 1) + εk p k+l = q k + l q ( qk k(l 1) εl k + l With ε = 1/10 000, k = 1, l = 5, p = 1/10: 2 k+1 + q ) k+l 1 k2 k+l q k 2 k+1. p 6 = = Panjer s recursion with five significant digits gives p 6 = Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 12

13 Theorem (Gerhold, S., Warnung, 2010) Fix l N, consider N {q n } n N0 and Ñi { q i,n } n N0 such that q i,0 = = q i,k+l i 1 = 0 for all i {1,..., l} and one k N 0. Assume that there exist a 1,..., a l, b 1,..., b l R such that q n = l ( i=1 a i + b i n ) q i,n i for n k + l. Define S = X X N {p n } n N0 and S (i) = X XÑi { p i,n } n N0 for i {1,..., l}. Then p 0 = ϕ N (P(X 1 = 0)) and, for n N, k+l 1 p n = P(S j = n)q j + j=1 l n ( i=1 j=0 a i + b ij in ) P(S i = j) p i,n j. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 13

14 Example: Combination of truncated distributions Lemma (Gerhold, S., Warnung, 2010) Fix k N 0, l N. For all i {1,..., l} assume that α i 0, β i iα i (at least one ) and that the N 0 -valued Ñ i satisfies P(Ñ i < k + l i) = 0. Consider q 0,..., q k+l 1 0 with q q k+l 1 1. Define q n = c c = l ( i=1 α i + β i n ( k+l 1 1 n=0 ) P(Ñ i = n i) for n k + l, q n )/ l i=1 ( [ α i + β i E 1 i + Ñ i ]). Then {q n } n N0 is a probability distribution satisfying the recursion condition of the theorem with a i = cα i and b i = cβ i and the calculation of {p n } n N0 is numerically stable. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 14

15 Weighted convolution for extended logarithmic dist. Corollary Let k N and q (0, 1). Let N ExtLog(k + 1, q) and Ñ ExtLog(k, q), where ExtLog(1, q) means Log(q). Define S = X X N and S = X XÑ. Then, with an explicit b 1 > 0, P(S = n) = b 1 n n j P(X 1 = j) P( S = n j), n N. j=1 Algorithm (for ExtLog(k, q), numerically stable, q 1) Panjer s recursion for N Log(q) k 1 weighted convolutions: Log(q) ExtLog(2, q) ExtLog(k 1, q) ExtLog(k, q) Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 15

16 Numerically stable algorithm for ExtLog(2, q) with q = 1 Lemma (Gerhold, S., Warnung, 2010) Let N ExtLog(2, 1). For S = X X N we have P(S = 0) = P(X 1 = 0) + P(X 1 1) log P(X 1 1) with 0 log 0 := 0 and, in the case P(X 1 1) > 0, P(S = n) = 1 n n j P(X 1 = j)r n j, n N, j=1 where r 0 = log P(X 1 1) and, recursively for n N, ( 1 r n = P(X 1 = n) + 1 n 1 j P(X 1 = n j)r j ). P(X 1 1) n j=1 Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 16

17 Weighted convolution for extended negative binomial dist. Corollary Let k N 0, α ( k, k + 1) and p (0, 1). Let N ExtNegBin(α 1, k + 1, p) and Ñ ExtNegBin(α, k, p), where ExtNegBin(α, 0, p) := NegBin(α, p). Let S = X X N and S = X XÑ. Then, with an explicit b 1 > 0, P(S = n) = b 1 n n j P(X 1 = j)p( S = n j), n N. j=1 Algorithm (for ExtNegBin(α, k, p), numerically stable, p 0) Panjer recursion for N NegBin(α + k, p) k weighted convolutions: NegBin(α + k, p) ExtNegBin(α + k 1, 1, p) ExtNegBin(α + 1, k 1, p) ExtNegBin(α, k, p) Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 17

18 Numerically stable algorithm for ExtNegBin(α 1, 1, 0) Lemma (Gerhold, S., Warnung, 2010) Let N ExtNegBin(α 1, 1, 0) with α (0, 1). For S = X X N we have P(S = 0) = 1 ( P(X 1 1) ) 1 α and in the non-trivial case P(X 1 1) > 0 P(S = n) = 1 α n n j P(X 1 = j)r n j, n N, j=1 where r 0 = ( P(X 1 1) ) α and, recursively for n N, r n = 1 P(X 1 1) n j=1 n j + αj P(X 1 = j)r n j. n Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 18

19 Application: Poisson tempered-α-stable mixtures Definition (τ-tempered α-stable distribution F α,σ,τ ) For index α (0, 1), scale σ > 0 and tempering τ 0 define F α,σ,τ (y) := E[e τy 1 {Y y} ]/E[e τy ], y R. where Y is α-stable on [0, ) with Laplace transform E[exp( sy )] = exp( γ α,σ s α ) for s 0, where γ α,σ = Theorem (Gerhold, S., Warnung, 2010) σ α cos(απ/2). Let Λ F α,σ,τ and L(N Λ) a.s. = Poisson(λΛ) with λ > 0. Then N d = N N M with independent M Poisson(γ α,σ ((λ + τ) α τ α )) and N m ExtNegBin ( ) τ α, 1, λ+τ for m N. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 19

20 Application: Poisson tempered α-stable mixtures (cont.) Let Λ F α,σ,τ and L(N Λ) a.s. = Poisson(λΛ) with λ > 0. Then the stochastic representation N d = N N M leads to S = N j=1 X j d = M N 1 + +N i i=1 j=n 1 + +N i 1 +1 X j N i d M = X i,j, i=1 j=1 where {X i,j } i,j N are i.i.d. with X i,j d = X1. Algorithm (numerically stable, τ 0) Panjer recursion for Ñ NegBin ( 1 α, ) τ λ+τ Weighted convolution: N 1 ExtNegBin ( α, 1, ) τ λ+τ Panjer recursion for M Poisson(γ α,σ ((λ + τ) α τ α )) If τ = 0, use the special algorithm for N 1 ExtNegBin( α, 1, 0). Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 20

21 Examples for τ-tempered 1 2-stable distributions Definition (Lévy distribution with scale parameter σ > 0) A density of F 1/2,σ,0 is f Lévy,σ (x) = ( σ ) 1/2 exp ( σ ) 2πx 3, x > 0. 2x Definition (inverse Gaussian distribution, parameters µ, σ > 0) Define σ = µ 2 / σ 2 and τ = 1/(2 σ 2 ). A density of F 1/2,σ,τ is f IG,µ, σ (x) = ( ) µ 2π σ 2 x exp (x µ)2 3 2 σ 2, x > 0. x Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 21

22 Additional examples of probability distributions for the Poisson mixture we can handle Generalized τ-tempered α-stable distributions (one additional parameter m N 0 ) Inverse gamma distribution (with half-integer shape parameter) Generalized inverse Gaussian distribution (with additional half-integer parameter m ) With an additional convolution: Reciprocal generalized inverse Gaussian distribution (with additional half-integer parameter m ) Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 22

23 Generalized gamma convolutions Error bounds and approximation Examples 1 Collective risk model and variants of Panjer s recursion 2 Generalized gamma convolutions Error bounds and approximation Examples 3 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 23

24 Generalized gamma convolutions Error bounds and approximation Examples Definition of a generalized gamma convolution These distributions arise as the weak limit of sums of independent gamma distributed random variables. Definition (cf. Bondesson 1992, Lecture Notes in Statistics) An (a, U)-generalized gamma convolution (g.g.c.) is a probability distribution F on R + = [0, ) with moment generating function ( ( ) ) t M(s) = e sx F (dx) = exp as + ln U(dt), t s 0 (0, ) for s 0, where a 0 and U is a locally finite non-negative measure on (0, ) satisfying 1 ln t U(dt) <, U(dt) <. t (0,1] (1, ) Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 24

25 Generalized gamma convolutions Error bounds and approximation Examples Examples of generalized gamma convolutions Consider a finite sum Y = a + n j=1 Y j with a 0 of independent random variables with Y j Gamma(α j, β j ), j {1,..., n}. Then E [ e sy ] = e as n j=1 ( ) αj βj = exp( as+ β j s n j=1 ( )) βj α j ln β j s for s < min{β 1,..., β n }, hence U = n j=1 α jδ βj. Pareto distribution, τ-tempered α-stable distribution, lognormal distribution, inverse Gaussian distribution, etc. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 25

26 A closure theorem Generalized gamma convolutions Error bounds and approximation Examples This theorem is useful for the construction of an approximation. Theorem (cf. Bondesson 1992, Thorin 1977) Let {F n } n N be a sequence of (a n, U n )-generalized gamma convolutions and F a probability distribution. Then {F n } n N converges weakly to F as n and F is an (a, U)-generalized gamma convolution if and only if 1 U n U vaguely on (0, ) as n, ( 2 a = lim A lim n a n + ) 1 (A, ) t U n(dt), 3 lim ε 0 lim sup n (0,ε) ln t U n(dt) = 0. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 26

27 Generalized gamma convolutions Error bounds and approximation Examples Approximation by finite gamma convolutions Proposition For every (a, U)-generalized gamma convolution F with Λ F there exists a weakly convergent sequence {Λ n } n N {F n } n N of (a, U n )-generalized gamma convolutions with U n = n i=1 α (n) i δ (n) β, i which converges vaguely to U as n. Then F n = δ a Gamma ( α (n) 1, ) ( β(n) (n) 1 Gamma α n, β n (n) ). Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 27

28 Generalized gamma convolutions Error bounds and approximation Examples Representation and convergence of Poisson mixtures Lemma (Rudolph & S.) Fix λ > 0. Let Λ and Λ n as in the Proposition. For each n N let N n be a random variable such that L(N n Λ n ) a.s. = Poisson(λΛ n ) and where P Poisson(a) and ( NegBin R (n) j M n := P + n j=1 α (n) j, 1 1+λ/β (n) j R (n) j, ), j {1,..., n}, are independent. Then M n d = Nn for all n N and {N n } n N converges weakly to some random variable N satisfying L(N Λ) a.s. = Poisson(λΛ). Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 28

29 Generalized gamma convolutions Error bounds and approximation Examples Total variation distance for Poisson g.g.c. mixtures (I) Assumption 1 λ > 0 and Λ denotes an (a, U)-g.g.c. and Ψ a (b, V )-g.g.c., U(dt) 2 (0,T ] t+λ V (dt) (0,T ] t+λ for all T > 0, 3 the random variables N and M satisfy L(N Λ) a.s. = Poisson(λΛ) and L(M Ψ) a.s. = Poisson(λΨ), 4 {X i } i N is a sequence of i.i.d. non-negative random variables independent of N and M, 5 S := N i=1 X i and T := M i=1 X i. Remark Assumption 2 is satisfied if U((0, T ]) V ((0, T ]) for all T > 0. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 29

30 Generalized gamma convolutions Error bounds and approximation Examples Total variation distance for Poisson g.g.c. mixtures (II) Theorem (Rudolph & S.) Let the assumption be satisfied. Then the total variation distance is where µ = d TV (L(S), L(T )) 3 2 µ ν + µ ν 2 (0, ) ( ) t + λ ln U(dt), ν = ln t (0, ) µ = µ + aλ and ν = ν + bλ. Remark If a = b = 0, then d TV (L(S), L(T )) 2(µ ν). + λ aν bµ 2ν, ( t + λ t ) V (dt), Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 30

31 Sketch of the proof Generalized gamma convolutions Error bounds and approximation Examples d TV (L(S), L(T )) d TV (L(M), L(N)), see Lemma 3.1 in Chaudhuri and Vellaisamy (1996). Representation as compound Poisson distributions: Let M CPoi(ν, H) and N CPoi(µ, F ). Applying Corollary 3.2 in Chaudhuri and Vellaisamy (1996) gives d TV (L(M), L(N)) min{ µ ν, µ ν } + min{µ, ν }d TV (F, H). Determine the distributions F and H using the moment generating function of Λ and Ψ, respectively. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 31

32 Generalized gamma convolutions Error bounds and approximation Examples Approximation quality by finite gamma convolutions Lemma (Rudolph & S.) For λ > 0, n N and U denotes the measure of a g.g.c. 0 = β 0 < β 1 < < β n with atoms U({β 1 }),..., U({β n }) α i = β i +λ (β i 1,β i ) t+λ U(dt) for i {1,..., n} V := n i=1( αi + U({β i }) ) δ βi, ν := 0 ln ( ) t+λ t V (dt), the previous theorem is applicable and n ( ( t + λ 0 µ ν = ln i=1 (β i 1,β i ) t ( t + λ + ln t (β n, ) ) U(dt) ln ) U(dt). ( ) βi + λ )α i β i Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 32

33 Generalized gamma convolutions Error bounds and approximation Examples Algorithm for computing the approximation (I) For ε > 0 choose β 1,..., β n such that µ ν ε and use the algorithm given in Gerhold, S. and Warnung (2010): M n = P + ( ) n i=1 R 1 i, where R i NegBin α i, 1+λ/β i R i = P i j=1 L i,j where P i Poisson(α i ln(1 + λ/β i )) and ) L i,j Log( λ/βi 1+λ/β i are Panjer class distributions. S n = M n i=1 X i d = P i=1 X i + n Ri i=1 Let S i,j = L i,1 + +L i,j k=l i,1 + +L i,j 1 +1 X i,k k=1 X d i,k, and X i = Xi,k. d = L i,1 k=1 X i,k. Compute L(S i,1 ) for i {1,..., n} by numerically stable Panjer recursions. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 33

34 Generalized gamma convolutions Error bounds and approximation Examples Algorithm for computing the approximation (II) The probability generating function of is S n = P X i + i=1 G Sn (z) = exp ( a(g X1 (z) 1) ) n i=1 n P i i=1 j=1 S i,j = exp ( (a + ν)(g(z) 1) ), z 1, ( ) exp α i ln(1 + λ/β i ) (G Si,1 (z) 1) with ν = n i=1 α i ln(1 + λ/β i ) and convex combination G(z) := a n a + ν G α i ln(1 + λ/β i ) X 1 (z) + G Si,1 (z). a + ν i=1 G Si,1 for i {1,..., n} were computed previously. Conduct another numerically stable Panjer recursion for CPoi(a + ν, F G ). Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 34

35 Generalized gamma convolutions Error bounds and approximation Examples Comparison of distributions obtained by FFT with exponential tilting and iterated Panjer recursion Probabilities Panjer FFT Tilted n Figure : Approximations of S = X X N, where X 1 Poisson(30) and L(N Λ) a.s. = Poisson(20Λ) with Λ Pareto(0.5, 2.5) and ε = Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 35

36 Generalized gamma convolutions Error bounds and approximation Examples Comparison of FFT and iterated Panjer recursion Probabilities 3 x Panjer exact Panjer approx FFT Number of Atoms Figure : Approximations of the distribution of S = X X N, where X 1 Poisson(10) and L(N Λ) a.s. = Poisson(30Λ) with Λ F 0.4,50,2 and ε = 0.1. Here F α,σ,τ denotes a τ-tempered α-stable distribution. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 36

37 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms 1 Collective risk model and variants of Panjer s recursion 2 Generalized gamma convolutions Error bounds and approximation Examples 3 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 37

38 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms New task: Calculate (in a fast and numerically stable way if possible) the distribution of the sum of random sums where N 1 N m S = X 1,j + + j=1 j=1 X m,j the losses {X i,j } j N with i {1,..., m} are independent sequences of N 0 -valued i.i.d. random variables, (N 1,..., N m ) is an N m 0 -valued random vector, independent of all individual losses but with possibly dependent components. Remark: Independence is lost, no convolutions of random sums. Question: Which multivariate distributions for the claim numbers (N 1,..., N m ) can we handle with recursive methods? Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 38

39 Random scenarios of linear dependence Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Let A 1,..., A k [0, ) m n be matrices with non-negative entries describing dependence scenarios. Let J be a random variable selecting one of the k scenarios. Let R 1,..., R n be independent and non-negative random variables (risk factors), independent of J. Define random Poisson intensities by (Λ 1,..., Λ m ) = A J (R 1,..., R n ). Let N = (N 1,..., N m ) be a random vector with conditionally independent components given Λ 1,..., Λ m such that L(N i Λ 1,..., Λ m ) a.s. = L(N i Λ i ) a.s. = Poisson(λ i Λ i ), where λ i 0 for each i {1,..., m}. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 39

40 Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Equivalent stochastic representation of claim numbers Theorem (Rudolph & S.) Let {E j h,l } h N for each scenario j {1,..., k} and risk l {1,..., n} be independent sequences of i.i.d. random vectors, independent of all other random variables, where E j 1,l Multinomial ( 1; λ i a j i,l /λ j,l, i = 1,..., m ) with λ j,l := m d=1 λ da j d,l > 0. Define the N m 0 -valued random vector M = M k n j,l 1 {J=j} E j h,l, j=1 l=1 h=1 where L ( ) a.s. M j,l J, R 1,..., R n = L ( ) a.s. M j,l R l = Poisson(λ j,l R l ) for j {1,..., k} and l {1,..., n} are conditionally independent given J, R 1,..., R n. Then M and N have the same distribution. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 40

41 Application to recursive algorithms Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms Rewrite the random sum using the equivalent representation to restore independence via scenarios and risk factors: S = m N i M k n j,l d X i,j = 1 {J=j} X h,j,l, i=1 j=1 j=1 l=1 h=1 where {X h,j,l } h N are independent sequences of i.i.d. random d variables such that X 1,j,l = m i=1 E j,i 1,l X i,1 For the random sums, if L ( ) M j,l permits, recursive methods are applicable, 1 see the previous sections. If every L ( M j,l ) is a compond Poisson distribution, then the n 1 convolutions might be replaced by a convex combination and one Panjer recursion. 1 FFT and FFT with tilting can also be applied. Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 41

42 References Claim numbers and random scenarios of linear dependence Equivalent stochastic representation of claim numbers Application to recursive algorithms L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statistics, vol. 76, Springer, P. Embrechts and M. Frei, Panjer recursion versus FFT for compound distributions, Math. Meth. Oper. Res., 69, ( ), S. Gerhold, U. Schmock, and R. Warnung, A generalization of Panjer s recursion and numerically stable risk aggregation, Finance Stoch., 14, (81 128), G. Giese, Dependent risk factors, in M. Gundlach, F. Lehrbass (eds.), CreditRisk + in the Banking Industry, ( ), Springer, O. Thorin, On the infinite divisibility of the Pareto distribution, Scand. Actuar. J., 1, (31 40), P. Vellaisamy, and B. Chaudhuri, Poisson and compound Poisson approximations for random sums of random variables, J. Appl. Probab., 33, ( ), Uwe Schmock (TU Vienna) Approximation and Aggregation of Risks by Variants of Panjer s Recursion 42