Analysis of a Bivariate Risk Model

Transcription

1 Jingyan Chen 1 Jiandong Ren 2 July 23, MSc candidate, Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada. 2 Associate Professor, Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada.

2 Outline Introduction

3 Outline Introduction Model

4 Outline Introduction Model Distribution and Method

5 Outline Introduction Model Distribution and Method Ruin Probability and Joint Ruin Probability

6 Outline Introduction Model Distribution and Method Ruin Probability and Joint Ruin Probability Numerical Examples

7 Outline Introduction Model Distribution and Method Ruin Probability and Joint Ruin Probability Numerical Examples Conclusion and Question

8 Introduction In this paper, we introduce a bivariate compound loss model describing two dependent lines of insurance businesses. The dependencies are both between claim frequencies and among claim sizes;

9 Introduction In this paper, we introduce a bivariate compound loss model describing two dependent lines of insurance businesses. The dependencies are both between claim frequencies and among claim sizes; a method for computing the distribution of the total loss random variable S;

10 Introduction In this paper, we introduce a bivariate compound loss model describing two dependent lines of insurance businesses. The dependencies are both between claim frequencies and among claim sizes; a method for computing the distribution of the total loss random variable S; a method to obtain the joint distribution for the loss of two lines of businesses S (1) and S (2) ;

11 Introduction In this paper, we introduce a bivariate compound loss model describing two dependent lines of insurance businesses. The dependencies are both between claim frequencies and among claim sizes; a method for computing the distribution of the total loss random variable S; a method to obtain the joint distribution for the loss of two lines of businesses S (1) and S (2) ; the impact of dependencies on the ruin probabilities related to the bivariate risk model.

12 Model Consider the case of a book of business divided into two dependent classes of business. It is assumed that the number of claims only in the jth (j = 1, 2) class of business is the sum of two random variables, that is, N (1) = N 1 + N 0, (1) N (2) = N 2 + N 0, (2) where N 0 is produced by a common shock effect. Let N (12) be the number of claims happened in both classes, then N (12) = N 12 + N 0. (3)

13 Model Consider the case of a book of business divided into two dependent classes of business. It is assumed that the number of claims only in the jth (j = 1, 2) class of business is the sum of two random variables, that is, N (1) = N 1 + N 0, (1) N (2) = N 2 + N 0, (2) where N 0 is produced by a common shock effect. Let N (12) be the number of claims happened in both classes, then N (12) = N 12 + N 0. (3) N i Poisson(λ i ). N i s are independent with each other.

14 Model Suppose the loss of two lines of businesses is S (1) and S (2). S (j) = X (j) N (12) k + Z j,k (j = 1, 2) N (j) k=1 k=1 where X (j) k s are independent with each other and (Z 1,k, Z 2,k ) for all k and j. Z 1,k and Z 2,k are correlated. Then the total loss is S = S (1) + S (2).

15 Distribution of S To obtain the mass probability of total claim amount S, 1 Characteristic Function(chf ) of S can be written as where λ = λ 1 + λ 2 + λ 12 + λ 0 and φ S (t) = e λ(φ X (t) 1), (4) φ X (t) = λ 1 λ φ X (1)(t) + λ 2 λ φ X (2)(t) + λ 12 λ φ Z 1 +Z 2 (t) + λ 0 λ φ X (1)(t)φ X (2)(t)φ Z 1 +Z 2 (t).

16 Distribution of S To obtain the mass probability of total claim amount S, 2 Take the discretization of F X (j) (j = 1, 2) and F Z1 +Z 2 and take the Fourier transform of them. 3 Get φ X (t) and insert it into equation (4) to obtain φ S(t). 4 Use FFT method to invert φ S(t) and get the mass probability of S.

17 Distribution of S F X (x)

18 Distribution of S F X (x) discretize mass prob of X

19 Distribution of S F X (x) discretize mass prob of X FFT φ X (t)

20 Distribution of S F X (x) discretize mass prob of X FFT φ X (t) φ S (t) = eλ(φ X (t) 1) φ S (t)

21 Distribution of S F X (x) discretize mass prob of X mass prob of S FFT inv FFT φ X (t) φ S (t) = eλ(φ X (t) 1) φ S (t)

22 Ruin Probability Denote the surplus of an insurer at time n (n = 0, 1, 2,...) as U n and U n = u + cn S n where u is the initial surplus, c is the premium received in each period and S n is the total claim amounts during period 1 to n. T = inf (n, U n < 0), which is the time of ruin. ψ(u, 1, n) is the finite-time ruin probability over the periods 1 to n, ψ(u, 1, n) = P(T n). ϕ(u, 1, n) = 1 ψ(u, 1, n) is the finite-time horizon non-ruin probabilities.

23 Ruin Probability Let u,c,j be integers. Then, u+c ϕ u,1,n = ϕ u+c j,1,n 1 f j (n = 2, 3,...), j=0 u+c ϕ u,1,1 = F u+c = f j. f j is the mass probabilities of the total claim amount in each period. j=0

24 Numerical Example N (1) Poisson(1) N (2) Poisson(1) N (12) Poisson(1) X (1) Lognormal(µ 1, σ 2 1 ) X (2) Lognormal(µ 2, σ 2 2 ) [ µ1 (Z 1, Z 2 ) BivariateLognormal( µ 2 ] [, σ1 2 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2 ] ). where µ 1 = 2, µ 2 = 1, σ 1 = 1, σ 2 = 2. We try to compare the ruin probability with different ρ N and ρ. ρ N is the correlation between N (j) (j = 1, 2, 12). ρ is the parameter of the Bivariate Lognormal distribution (Z 1, Z 2 ).

25 Numerical Example The ruin probabilities ψ(u, 1, 20) with dependent claim amounts (ρ = 0.3) are showing below, u (initial surplus) ψ(u, 1, 20, 0) ψ(u, 1, 20, 0.25) ψ(u, 1, 20, 0.75) ψ(u,1,20,0.25) ψ(u,1,20,0) 1(%) ψ(u,1,20,0.75) ψ(u,1,20,0) 1(%) Note: ψ(u, 1, n, ρ N ) here denotes the ruin probability over period 1 to n.

26 Numerical Example The ruin probabilities ψ(u, 1, 20) with dependent number of claims (ρ N = 0.25) are showing below, u ρ = ρ = ρ = ψ(u,ρ=0.3) ψ(u,ρ=0) 1(%) ψ(u,ρ=0.9) ψ(u,ρ=0) 1(%) Note: ρ N is the correlation between N (j) (j = 1, 2, 12). ρ is the parameter of the Bivariate Lognormal distribution (Z 1, Z 2 ).

27 Joint Ruin Probability T min = min(t 1, T 2 ) the first ruin time of the two classes of business S (1) and S (2) T max = max(t 1, T 2 ) the last ruin time of S (1) and S (2).

28 Joint Ruin Probability The cdf funtion of T min can be expressed as F Tmin (t) = 1 P(T min > t) = 1 ϕ (S (1),S (2) ),1,t (u 1, u 2 ), which is the joint non-ruin probability during period 1 to t for S (1) and S (2). Also, F Tmax (t) = P(T 1 < t, T 2 < t) = 1 P(T 2 > t) P(T 1 > t) + P(T 1 > t, T 2 > t).

29 Joint Ruin Probability Let u 1, u 2, c, j be integers. Then, the joint non-ruin probability from period 1 to n is ϕ u1,u 2,1,n = where u+c u+c ϕ u1 +c j 1,u 2 +c j 2,n 1f j1,j 2 (n = 2, 3,...), j 1 =0 j 2 =0 ϕ u1,u 2,1,1 = u+c u+c f j1,j 2. j 1 =0 j 2 =0 Here, f j1,j 2 is the joint probabilities of the total claim amount in each period for (S (1), S (2) ). (u 1, u 2 ) is the initial surplus for the two classes of business and c is the premium for each period.

30 Numerical Example Probabilities for T max 5 with dependent frequencies (ρ N = 0.25) and initial surplus u 1 = u 2 = u are showing below, u ρ = ρ = ρ = P(u,ρ=0.3) P(u,ρ=0) 1(%) P(u,ρ=0.9) P(u,ρ=0) 1(%) Note: ρ N is the correlation between N (j) (j = 1, 2, 12). ρ is the parameter of the Bivariate Lognormal distribution (Z 1, Z 2 ).

31 Numerical Example Probabilities for T max 5with dependent claim amounts (ρ = 0.3) and initial surplus u 1 = u 2 = u are showing below, u ρ N = ρ N = ρ N = ρ N =0.25 ρ N =0 1(%) ρ N =0.75 ρ N =0 1(%) Note: ρ N is the correlation between N (j) (j = 1, 2, 12). ρ is the parameter of the Bivariate Lognormal distribution (Z 1, Z 2 ).

32 Conclusion The ruin probability ψ(u, 1, 20) increases as ρ N increases from table 1. But the increase slows down when u is large. We can also notice the increase of the ruin probability as the parameter ρ increases, that is, the more dependencies among claim amounts corresponds to the higher ruin probability..

33 Conclusion The cdf of the last ruin time for S (1) and S (2) increases as ρ N increases. The cdf of the last ruin time for S (1) and S (2) increases as the parameter ρ increases, that is, the more dependencies among claim amounts corresponds to the higher ruin probability.

34 Question

35 Question Thank you!