Generalized linear mixed models (GLMMs) for dependent compound risk models

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1 Generalized linear mixed models (GLMMs) for dependent compound risk models Emiliano A. Valdez, PhD, FSA joint work with H. Jeong, J. Ahn and S. Park University of Connecticut Seminar Talk at Yonsei University Seoul, Korea 15 May 2017 Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

2 Outline Introduction Data structure The frequency-severity model Exponential dispersion family Covariates Compound risk models Generalized linear mixed models Model specifications Model estimates Claim frequency Claim severity Tweedie models Model comparison Gini index Conclusion Appendix A - Singapore Insurance market Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

3 Introduction Data structure Data structure Policyholder i is followed over time t = 1,..., T i years, where T i is at most 9 years. Unit of analysis it a registered vehicle insured i over time t (year) For each it, could have several claims, k = 0, 1,..., n it Have available information on: number of claims n it, amount of claim y itk, exposure e it and covariates (explanatory variables) x it covariates often include age, gender, vehicle type, driving history and so forth We will model the pair (n it, c it ) where is the observed average claim size. 1 n it y itk, n it > 0 c it = n it k=1 0, n it = 0 Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

4 The frequency-severity model The frequency-severity model Traditional to predict/estimate insurance claims distributions: Cost of Claims = Frequency Severity The joint density of the number of claims and the average claim size can be decomposed as f(n, C x) = f(n x) f(c N, x) joint = frequency conditional severity. This natural decomposition allows us to investigate/model each component separately and it does not preclude us from assuming N and C are independent. For purposes of notation, we will use the notation C = NC to be the aggregate claims. Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

5 The frequency-severity model Exponential dispersion family Exponential dispersion family We say Y comes from an exponential dispersion family if its density has the form [ ] yθ ψ(θ) f(y) = exp + c(y; φ). φ where θ and ψ are location and scale parameters, respectively, and b(θ) and c(y; ψ) are known functions. The following well-known relations hold for these distributions: mean: µ = E[Y ] = ψ (θ) variance: V ar[y ] = φψ (θ) = φv (µ) Reproductive EDF: If Y 1,..., Y N are mutually independent belonging to the EDF(θ, φ), then its average Y also belongs to EDF(θ, φ/n). Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

6 The frequency-severity model Exponential dispersion family Examples of members of this family Normal N(µ, σ 2 ) with θ = µ and φ = σ 2, V (µ) = 1 Gamma(α, β) with θ = β/α and φ = 1/α, V (µ) = µ 2 Inverse Gaussian(α, β) with θ = 1 2 β2 /α 2 and φ = β/α 2, V (µ) = µ 3 Poisson(µ) with θ = log µ and φ = 1, V (µ) = 1, V (µ) = µ Binomial(m, p) with θ = log [p/(1 p)] and φ = 1, V (µ) = µ(1 µ/m) Negative Binomial(r, p) with θ = log(1 p) and φ = 1, V (µ) = µ(1 + µ/r) Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

7 The frequency-severity model Exponential dispersion family The link function and linear predictors The linear predictor is a function of the covariates, or predictor variables. The link function connects the mean of the response to this linear predictor in the form η = g(µ). g(µ) is called the link function with µ = E(Y ) = linear predictor. The transformed mean follows a linear model as η = x iβ = β 0 + β 1 x i1 + + β p x ip with p predictor variables (or covariates). You can actually invert µ = g 1 (x i β) = g 1 (η). Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

8 The frequency-severity model Covariates Covariates Vehicle Type: automotive (A) or others (O). Marital status: married (M), single (S), others (O) Gender: male (M) or female (F). Coverage type: Comprehensive (Comp) or Others Age: in years, grouped into 3 categories Young (Y), Middle Aged (M), Old (O) Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

9 Compound risk models Compound risk models See Garrido, et al. (2016) In the case of independence, we have: Mean: E(C) = E(N)E(C) Variance: V ar(c) = E(N)V ar(c) + V ar(n)[e(c] 2 In case we do not assume independence, we have Mean: E(C) = E[NE(C N))] E(N)E(C) Variance: V ar(c) = E[NV ar(c N))] + V ar[ne(c N)] Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

10 Compound risk models Generalized linear mixed models Generalized linear mixed models GLMMs extend GLMs by allowing for random, or subject-specific, effects in the linear predictor which reflects the idea that there is a natural heterogeneity across subjects. For insurance applications, our subject i is usually a policyholder observed for a period of T i periods. Given the vector b i with the random effects for each subject i, Y it belongs to the EDF: [ ] yit θ it ψ(θ it ) f(y it b i ) = exp + c(y it ; φ) φ with the following conditional relations: mean: µ it = E[Y it b i ] = ψ (θ it ) variance: V ar[y b i ] = φψ (θ + it) = φv (µ it ) Link function: g(µ it ) = x it β + z it b i Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

11 Model specifications Model specifications We calibrated models with the following specifications: For the count of claims (frequency): negative binomial for our baseline model more suitable than the traditional Poisson model because it can handle overdispersion and individual unobserved effects For the severity of claims: Gamma (due to its reproductive property) For the random effects for GLMMs: Normal with mean zero and unknown variances different variances for frequency and claim severity The Tweedie model is a special case of the GLM family and it has mass at zero (avoids modeling frequency and severity separately). for some case, its density has no explicit form, but the variance function has the form V (µ) = µ p e.g. compound Poisson Gamma with 1 < p < 2. Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

12 Model estimates Claim frequency Negative binomial model for claim frequency NB GLM NB GLMM (Intercept) (0.08) (0.12) VTypeCar (0.07) (0.12) MaritalM (0.35) (0.54) SexM (0.02) (0.04) Comp (0.03) (0.05) AgeYoung (0.08) (0.13) AgeMid (0.02) (0.03) AIC BIC Log Likelihood Deviance Num. obs Num. groups: ID Variance: ID.(Intercept) 0.28 Variance: Residual 2.27 p < 0.001, p < 0.01, p < 0.05 NB GLM NB GLMM MAE MSE Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

13 Model estimates Claim severity Gamma distributions models for claim frequency Gamma-indep GLM Gamma-dep GLM Gamma-dep GLMM (Intercept) (0.56) (0.31) (0.31) VTypeCar (0.53) (0.29) (0.28) MaritalM (2.29) (1.25) (1.24) SexM (0.15) (0.08) (0.08) Comp (0.23) (0.12) (0.12) AgeYoung (0.56) (0.31) (0.30) AgeMid (0.14) (0.07) (0.07) log(freq) (0.04) (0.04) AIC BIC Log Likelihood Deviance Num. obs Num. groups: Year 8 Variance: Year.(Intercept) 0.01 Variance: Residual p < 0.001, p < 0.01, p < 0.05 Gamma-indep GLM Gamma-dep GLM Gamma-dep GLMM MAPE MAE MSE Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

14 Model estimates Tweedie models Tweedie model estimates Tweedie GLM Tweedie GLMM (Intercept) VTypeCar MaritalM SexM Comp AgeYoung AgeMid AIC BIC Log Likelihood Deviance Num. obs Num. groups: ID Variance: ID.(Intercept) 2.33 Variance: Residual Tweedie GLM Tweedie GLMM MAE MSE Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

15 Model comparison Gini index Using the Gini index to compare models See Frees, et al. (2016). In computing the Gini index, we employ the following steps: 1 Sort observed claims Y i i = 1,, M according to risk score S i i = 1,, M (in our case, estimated value from each method) in ascending order. That is, calculate R i i = 1,, M, where each R i is the rank of S i between 1 and M so that R 1 = arg max (S i ). 2 Compute F score (m/m) = M i=1 1 (R i m)/m, F loss (m/m) = M i=1 Y i1 M (Ri m)/ i=1 Y i for each m = 1,, M. 3 Plot F score (m/m) on x-axis, and F loss (m/m) on y-axis. Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

16 Model comparison Gini index Comparing the Gini index for the two-part models Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

17 Model comparison Gini index Comparing the Gini index for the Tweedie models Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

18 Conclusion Concluding remarks We are still in the early stages of this work. In particular, this is an early exploration of the promise of using GLMMs to account for dependency between frequency and severity. This work hopes to extend the literature on this type of dependency: H. Jeong (2016) work on Simple Compound Risk Model with Dependent Structure Garrido, et al. (2016) Frees and Wang (2006), Frees, et al. (2011) Czado (2012) Shi, et al. (2015) Further work is needed to measure the financial impact of using GLMMs versus other models: risk classification a posteriori prediction and applications in bonus-malus systems modified insurance coverage and reinsurance applications Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

19 Appendix A - Singapore A bit about Singapore Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

20 Appendix A - Singapore A bit about Singapore 1 Singa Pura: Lion city. Location: km N of equator, between latitudes 103 deg 38 E and 104 deg 06 E. [islands between Malaysia and Indonesia] Size: very tiny [647.5 sq km, of which 10 sq km is water] Climate: very hot and humid [23-30 deg celsius] Population: nearly 5 mn. Age structure: 0-14 yrs: 16%, yrs: 76%, 65+ yrs 8% Birth rate: 9.34 births/1,000. Death rate: 4.28 deaths/1,000; Life expectancy: 81 yrs; male: 79 yrs; female: 83 yrs Ethnic groups: Chinese 74%, Malay 13%, Indian 9%; Languages: Chinese, Malay, Tamil, English 1 Updated: February 2010 Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

21 Appendix A - Singapore Insurance market Insurance market in Singapore As of : market consists of 45 general ins, 8 life ins, 7 both, 17 general reinsurers, 2 life reins, 7 both; also the largest captive domicile in Asia, with 59 registered captives. Monetary Authority of Singapore (MAS) is the supervisory/regulatory body; also assists to promote Singapore as an international financial center. Insurance industry performance in 2009: total premiums: 11.4 bn; total assets: bn [20% annual growth] life insurance: annual premium = mn; single premium = mn general insurance: gross premium = 1.9 bn (domestic = 0.9; offshore = 1.0) Further information: 2 Source: wikipedia Jeong/Ahn/Park/Valdez (U. of Connecticut) Seminar Talk - Yonsei University 15 May / 21

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