A copula-based model for risk aggregation

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1 Actuarial Research Conference 2014 & Christian Genest Department of Mathematics and Statistics, McGill University July 14, 2014

2 Objective Introduction To model adequately dependent insurance portfolios in order to evaluate the overall risk exposure and capital allocations. Risks: 1, 2: Québec Automobile insurance Subsidiary A, B 3, 4: Québec Home insurance Subsidiary A, B 5, 7: Ontario AB coverage for Subsidiary C, D 6, 8: Ontario TPL coverage for Subsidiary C, D

3 Current modelling options 1 Multivariate model for the 8 risks Archimedean copulas: Clayton, Frank, Gumbel, etc. Elliptical distributions: Normal, Student t, etc. 2 Hierarchical model Nested Archimedean copulas Vine copulas Because we are interested in the sum of the claims, we can simplify the model using partial sums!

4 Introduction Simple case with three risks Conditional independence assumption Tree structure Inference Components of the model: Tree structure Marginal distribution for each risk Bivariate copula for each aggregation step

5 Example with three risks Simple case with three risks Conditional independence assumption Tree structure Inference Tree structure Parametric marginal distributions F 1, F 2 and F 3 Bivariate copulas C {1,2} and C {1,2,3}

6 Simple case with three risks Conditional independence assumption Tree structure Inference Simple case X i has mean 0 and variance 1, for i = 1, 2, 3 C {1,2} and C {1,2,3} are copulas inducing correlations ρ 1 and ρ 2, respectively cov(x 2, X 3 ) = a

7 Simple case with three risks Conditional independence assumption Tree structure Inference Simple case the joint distribution is not unique Then var(x 1 + X 2 ) = 2(1 + ρ 1 ) and cov(x 1, X 3 ) = cov(x 1 +X 2, X 3 ) cov(x 2, X 3 ) = ρ 2 2(1 + ρ1 ) a. As cov(x 1, X 3 ) = corr(x 1, X 3 ), we have max{ρ 2 2(1 + ρ1 ) 1, 1} a min{ρ 2 2(1 + ρ1 ) + 1, 1}.

8 Simple case with three risks Conditional independence assumption Tree structure Inference Simple case the joint distribution is not unique 2.0 Difference rho rho Figure: Dierence between lower and upper bound for corr(x 1, X 3 )

9 Assumption required for uniqueness Simple case with three risks Conditional independence assumption Tree structure Inference All descendants of a given node are conditionally independent of non-descendants.

10 Simple case with three risks Conditional independence assumption Tree structure Inference Example with three risks: joint distribution Under the conditional independence assumption, the joint density of X 1, X 2, X 3 (assuming it exists) is f X1,X 2,X 3 (x 1, x 2, x 3 ) = c {1,2,3} {F {1,2} (x 1 +x 2 ), F 3 (x 3 )}c {1,2} {F 1 (x 1 ), F 2 (x 2 )} where F {1,2} (s) = Pr(X 1 + X 2 s) = s 0 3 f i (x i ), i=1 F X1,X 2 (x, s x)dx.

11 Simple case with three risks Conditional independence assumption Tree structure Inference Number of tree structures If X 1,..., X d are d 2 risks, the number M d of tree structures can be computed recursively by setting M 1 = 1 and, for k {2,..., d}, M k = 1 2 k 1 ( ) d M i M k i. i i=1 Table: Number of trees to aggregate d variables and partial sums therein, two at a time. d M d , ,135 2,027,025

12 Determine the structure Simple case with three risks Conditional independence assumption Tree structure Inference Idea (inspired from vine copulas): rst model the greatest dependencies, either positive or negative. Distance based on Kendall's tau: d(x, Y ) = 1 τ 2 (X, Y ). Alternatives: Spearman's rho or Pearson's correlation. Hierarchical clustering method: 1 Compute distances between all risks. 2 Sum the two risks that are the closest. 3 Repeat until only one risk is left.

13 Inference Introduction Simple case with three risks Conditional independence assumption Tree structure Inference Classical rank-based methods can be used! Data and partial sums are all observed. Choice of copulas, parameter estimation, goodness-of-t tests.

14 Results Monthly data: January 2004 to June 2012 Ultimate claims. Loss ratios are used to reect volume and ination. Assumption: consistent pricing in the period For aggregation, each loss ratio is rst multiplied by its earned premium (assumed to be known). Reform of legislation in Ontario eective September 1, 2010.

15 Results Overview of marginal distributions Risk Trend or Time Distribution Seasonality Series 1 Seasonality White noise Skewed t 5 2 Seasonality ARMA(1, 1) Skewed t 5 3 None White noise Skewed t 8 4 None White noise Skewed t 16 6 None White noise Gamma 8 None White noise Gamma

16 Results Overview of marginal distributions AB coverages Risk Period Distribution 5 before reform ln(lr 5,t ) N ( t, σ 1 ) after reform LN ( 0.782, σ 1 ) 7 before reform LN ( 0.309, σ 2 ) after reform LN ( 1.070, σ 2 )

17 Introduction Results Dependence structure before and after reform Cluster Dendrogram Height Cluster Dendrogram Height ResidualMatrixb clust (*, "") Figure: Before the reform ResidualMatrixa clust (*, "") Figure: After the reform

18 Copulas Introduction Results Risks Copula Kendall's τ (X 3, X 4 ) Galambos 0.68 (X 1, X 2 ) t (X 1 + X 2, X 3 + X 4 ) tev (X 7, X 8 ) before/after reform t / 0.44 (X 5, X 6 ) before/after reform Gaussian 0 / 0.49 (X X 4, X 7 + X 8 ) Product 0 (X X 4 + X 7 + X 8, X 5 + X 6 ) Product 0

19 Results Model adequacy: Global test on the 8-dimensional copula Compare simulations from the tted model to the data. Test of Rémillard & Scaillet (2009) H 0 : simulated and original samples have the same copula p-values are 51% and 69% for before and after reform.

20 Results Impact of the reform Results We are interested in two senarios: I. There is no reform for AB coverage (model before the reform). II. The reform happens (model after the reform). Results are obtained by simulation, with the algorithm described in Côté & Genest (2014). Table: Proportion of premium per $1000 in the portfolio i Q i

21 Results Results for September 2010 to June months $1000 premium. The observed result for that period is $11,694. Table: Risk measures for the overall portfolio Expectation Standard Deviation VaR 95% TVaR 95% I 14, ,545 15,903 II 12, ,586 13,946 Table: TVaR 95% for the overall portfolio after the reform Model Independence Comonotonicity II 13,946 13,504 14,857

22 TVaR-based capital allocations Results After the reform, Ontario risks represent a smaller proportion of total capital, but the allocation to risk 6 (TPL, C) is increased. Figure: TVaR-based capital allocations in percentage of TVaR 0.95 (S)

23 Introduction References The model is exible and easy to apply to insurance portfolios. The conditional independence assumption seems reasonable. Formal validation of this assumption is still to be developed. The simulation algorithm used is only partially validated by Arbenz et al. (2012).

24 References Introduction References Arbenz, P., Hummel, C. & Mainik, G. (2012). Copula-based hierarchical risk aggregation through sample reordering. Insurance: Mathematics and Economics, 51, Côté, M.-P. & Genest, C. (2014) Copula-based risk aggregation model. Submitted for publication. Genest, C. & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, Rémillard, B. & Scaillet, O. (2009). Testing for equality between two copulas. Journal of Multivariate Analysis, 100,

25 References Appendix Joint distribution of the risks X 1,..., X d with marginal densities f 1,..., f d A i,1 and A i,2 is the partition of A i such that S Ai,1 and S Ai,2 are modelled at node N i, with S A = j A X j c Ai is the density of the copula at node N i Given a tree structure characterized by the sets A 1,..., A d 1, then for all x 1,..., x d R d 1 f D (x 1,..., x d ) = c Ai F A i,1, F Ai,2 d f i (x i ). i=1 j A i,1 x j j A i,2 x j i=1

CONDITIONAL, PARTIAL AND RANK CORRELATION FOR THE ELLIPTICAL COPULA; DEPENDENCE MODELLING IN UNCERTAINTY ANALYSIS

CONDITIONAL, PARTIAL AND RANK CORRELATION FOR THE ELLIPTICAL COPULA; DEPENDENCE MODELLING IN UNCERTAINTY ANALYSIS D. Kurowicka, R.M. Cooke Delft University of Technology, Mekelweg 4, 68CD Delft, Netherlands