Modeling. joint work with Jed Frees, U of Wisconsin - Madison. Travelers PASG (Predictive Analytics Study Group) Seminar Tuesday, 12 April 2016

Transcription

1 joint work with Jed Frees, U of Wisconsin - Madison Travelers PASG (Predictive Analytics Study Group) Seminar Tuesday, 12 April 2016 claim Department of Mathematics University of Connecticut Storrs, Connecticut page 1

2 A collection of work Frees and Valdez (2008),, Journal of the American Statistical Association, Vol. 103, No. 484, pp Frees, Shi and Valdez (2009), Actuarial Applications of a Model, ASTIN Bulletin, Vol. 39, No. 1, pp Young, Valdez and Kohn (2009), Multivariate Probit Models for Conditional Claim Types, Insurance: Mathematics and Economics, Vol. 44, No. 2, pp Antonio, Frees and Valdez (2010), A Multilevel Analysis of Intercompany Claim Counts, ASTIN Bulletin, Vol. 40, No. 1, pp claim page 2

3 Location of Singapore claim page 3

4 Car ownership in Singapore Owning and driving a car in Singapore is quite expensive. Government has put up measures to manage car ownership: certificate of entitlement (COE) vehicle quota system (VQS) registration fees, road taxes (annual/semi-annual) electronic road pricing (per use, driving in certain areas) Every motor vehicle must have a valid insurance policy: minimum required is coverage for personal injury to other parties. three major types available in the marketplace: third party, third part + fire & theft, and comprehensive. claim page 4

5 Basic data set-up Policyholder i is followed over time t = 1,..., 9 years Unit of analysis it a registered vehicle insured i over time t (year) Have available: exposure e it and covariates (explanatory variables) x it covariates often include age, gender, vehicle type, driving history and so forth Goal: understand how time t and covariates impact claims y it. Statistical methods viewpoint basic regression set-up - almost every analyst is familiar with: part of the basic actuarial education curriculum claim incorporating cross-sectional and time patterns is the subject of longitudinal data analysis - a widely available statistical methodology page 5

6 More complex data set-up Some variations that might be encountered when examining insurance company records For each it, could have multiple claims, j = 0, 1,..., 5 For each claim y itj, possible to have one or a combination of three (3) types of losses: 1 losses for injury to a party other than the insured y itj,1 - injury ; 2 losses for damages to the insured, including injury, property damage, fire and theft y itj,2 - own damage ; and 3 losses for property damage to a party other than the insured y itj,3 - third party property. Distribution for each claim is typically medium to long-tail The full multivariate claim may not be observed: Distribution of Claims, by Claim Type Observed Value of M Claim by Combination (y 1) (y 2) (y 3) (y 1, y 2) (y 1, y 3) (y 2, y 3) (y 1, y 2, y 3) Number ,216 2, , Percentage claim page 6

7 The hierarchical insurance claims model Traditional to predict or estimate insurance claims distributions: Cost of Claims = Frequency Severity Joint density of the aggregate loss can be decomposed as: f (N, M, y) = f (N) f (M N) f (y N, M) joint = frequency conditional claim-type conditional severity. This natural decomposition allows us to investigate and model each component separately. claim page 7

8 Allows for risk rating factors to be used as explanatory variables that predict both the frequency and the multivariate severity components. Helps capture the long-tail nature of the claims distribution through the GB2 distribution model. Provides for a two-part distribution of losses - when a claim occurs, not necessary that all possible types of losses are realized. Allows to capture possible dependencies of claims among the various types through a t-copula specification. claim page 8

9 Literature on claims frequency/severity There is large literature on modeling claims frequency and severity Klugman, Panjer and Willmot (2004) - basics without covariates Kahane and Levy (JRI, 1975) - first to model joint frequency/severity with covariates. Coutts (1984) postulates that the frequency component is more important to get right. Many recent papers on frequency, e.g., Boucher and Denuit (2006) Applications to motor insurance: Brockman and Wright (1992) - good early overview. Renshaw (1994) - uses GLM for both frequency and severity with policyholder data. Pinquet (1997, 1998) - uses the longitudinal nature of the data, examining policyholders over time. claim considered 2 lines of business: claims at fault and not at fault; allowed correlation using a bivariate Poisson for frequency; s used were lognormal and gamma. Most other papers use grouped data, unlike our work. page 9

10 Observable data Model is calibrated with detailed, micro-level automobile insurance records over eight years [1993 to 2000] of a randomly selected Singapore insurer. Year 2001 data use for out-of-sample prediction Information was extracted from the policy and claims files. Unit of analysis - a registered vehicle insured i over time t (year). The observable data consist of number of claims within a year: N it, for t = 1,..., T i, i = 1,..., n type of claim: M itj for claim j = 1,..., N it the loss amount: y itjk for type k = 1, 2, 3. exposure: e it vehicle characteristics: described by the vector x it The data available therefore consist of claim {e it, x it, N it, M itj, y itjk }. page 10

11 Risk factor rating system Insurers adopt risk factor rating system in establishing premiums for motor insurance. Some risk factors considered: vehicle characteristics: make/brand/model, engine capacity, year of make (or age of vehicle), price/value driver characteristics: age, sex, occupation, driving experience, claim history other characteristics: what to be used for (private, corporate, commercial, hire), type of coverage The no claims discount (NCD) system: rewards for safe driving discount upon renewal of policy ranging from 0 to 50%, depending on the number of years of zero claims. claim These risk factors/characteristics help explain the heterogeneity among the individual policyholders. page 11

12 Year: the calendar year ; treated as continuous variable. Vehicle Type: automotive (A) or others (O). Vehicle Age: in years, grouped into 6 categories - 0, 1-2, 3-5, 6-10, 11-15, <=16. Vehicle Capacity: in cubic capacity. Gender: male (M) or female (F). Age: in years, grouped into 7 categories - ages 21, 22-25, 26-35, 36-45, 46-55, 56-65, 66. The NCD applicable for the calendar year - 0%, 10%, 20%, 30%, 40%, and 50%. claim page 12

13 Random effects negative binomial count model ( ) Let λ it = e it exp x λ,it β λ be the conditional mean parameter for the {it} observational unit, where x λ,it is a subset of x it representing the variables needed for frequency modeling. Negative binomial distribution model with parameters p and r: ( ) k + r 1 Pr(N = k r, p) = p r (1 p) k. r 1 Here, σ = 1 is the dispersion parameter and r p = p it is related to the mean through 1 p it p it = λ it σ = e it exp(x λ,itβ λ )σ. claim page 13

14 Multinomial claim type Certain characteristics help describe the claims type. To explain this feature, we use the multinomial logit of the form exp(v m ) Pr(M = m) = 7 s=1 exp(v s), where V m = V it,m = x M,it β M,m. For our purposes, the covariates in x M,it do not depend on the accident number j nor on the claim type m, but we do allow the parameters to depend on type m. Such has been proposed in Terza and Wilson (1990). An alternative model to claim type, multivariate probit, was considered in: Young, Valdez and Kohn (2009) claim page 14

15 Severity We are particularly interested in accommodating the long-tail nature of claims. We use the generalized beta of the second kind (GB2) for each claim type with density f (y) = where z = (ln y µ)/σ. exp (α 1 z) y σ B(α 1, α 2 ) [1 + exp(z)] α 1+α 2, µ is a location, σ is a scale and α 1 and α 2 are shape parameters. With four parameters, distribution has great flexibility for fitting heavy tailed data. Introduced by McDonald (1984), used in insurance loss modeling by Cummins et al. (1990). Many distributions useful for fitting long-tailed distributions can be written as special or limiting cases of the GB2 distribution; see, for example, McDonald and Xu (1995). claim page 15

16 GB2 Distribution claim Source: Klugman, Panjer and Willmot (2004), p. 72 page 16

17 Heavy-tailed regression models Loss - Actuaries have a wealth of knowledge on fitting claims distributions. (Klugman, Panjer, Willmot, 2004) (Wiley) are often heavy-tailed (long-tailed, fat-tailed) Extreme values are likely to occur Extreme values are the most interesting - do not wish to downplay their importance via transformation Studies of financial asset returns is another good example Rachev et al. (2005) Fat-Tailed and Skewed Asset Return Distributions (Wiley) Healthcare expenditures - Typically skewed and fat-tailed due to a few yet high-cost patients (Manning et al., 2005, J. of Health Economics) claim page 17

18 GB2 regression We allow scale and shape parameters to vary by type and thus consider α 1k, α 2k and σ k for k = 1, 2, 3. Despite its prominence, there are relatively few applications that use the GB2 in a regression context: McDonald and Butler (1990) used the GB2 with regression covariates to examine the duration of welfare spells. Beirlant et al. (1998) demonstrated the usefulness of the Burr XII distribution, a special case of the GB2 with α 1 = 1, in regression applications. Sun et al. (2008) used the GB2 in a longitudinal data context to forecast nursing home utilization. We parameterize the location parameter as µ ik = x ik β k: Thus, β k,j = ln E (Y x) / x j Interpret the regression coefficients as proportional changes. claim page 18

19 Claim losses by type of claim Table 2.3. Summary Statistics of Claim Losses, by Type of Claim Statistic Third Party Own Damage (y 2 ) Third Party Injury (y 1 ) non-censored all Property (y 3 ) Number ,974 20,503 6,136 Mean 12, , , , Standard Deviation 39, , , , Median 1,700 1, , , Minimum Maximum 336, , ,183 56, claim page 19

20 Dependencies among claim types We use a parametric copula (in particular, the t copula). Suppressing the {i} subscript, we can express the joint distribution of claims (y 1, y 2, y 3 ) as F(y 1, y 2, y 3 ) = H (F 1 (y 1 ), F 2 (y 2 ), F 3 (y 3 )). Here, the marginal distribution of y k is given by F k ( ) and H( ) is the copula. the joint distribution of the simultaneous occurrence of the claim types, when an accident occurs, provides the unique feature of our work. Some references are: Frees and Valdez (1998), Nelsen (1999). claim page 20

21 The calendar year effect Table 3.2. Number and Percentages of Claims, by Count and Year Percentage by Year Percent Count Number of Total , , , Number 4,976 5,969 5,320 8,562 19,344 19,749 28,473 44,821 62, , by Year claim page 21

22 The effect of vehicle type and age Table 3.3. Number and Percentages of Claims, by Vehicle Type and Age Percentage by Count Count Count Count Count Count Count Percent =0 =1 =2 =3 =4 =5 Number of Total Vehicle Type Other , Automobile , Vehicle Age (in years) , , , to , to , to , and older Number 178,080 19,224 1, , by Count claim page 22

23 The effect of gender, age, NCD Table 3.4. Number and Percentages of Claims, by Gender, Age and NCD for Automobile Policies Percentage by Count Count Count Count Count Count Count Percent =0 =1 =2 =3 =4 =5 Number of Total Gender Female , Male , Person Age (in years) 21 and younger , , , , , and over , No Claims Discount (NCD) , , , , , , Number 139,183 14,774 1, , by Count claim page 23

24 Comparing the alternative conditional claim s Table 3.7. Comparison of Fit of Alternative Claim Type Models Model Variables Number of -2 Log Parameters Likelihood AIC BIC Intercept Only 6 25, , ,538.5 Automobile (A) 12 24, , ,042.2 A and Gender 24 24, , ,159.2 Year 12 25, , ,462.0 Year , , ,406.3 A and Year , , ,950.3 VehAge2 (Old vs New) 12 25, , ,542.9 VehAge2 and A 18 24, , ,984.2 A, VehAge2 and Year , , ,939.5 claim page 24

25 The fitted copula model Table 3.8. Fitted Copula Model Type of Copula Parameter Independence Normal copula t-copula Third Party Injury σ (0.124) (0.138) (0.120) α (1.482) (1.671) (1.447) α ( ) ( ) ( ) βc,1,1 (intercept) (2.139) (4.684) (4.713) Own Damage σ (0.031) (0.022) (0.029) α (1.123) (0.783) (0.992) α (42.021) (22.404) (59.648) βc,2,1 (intercept) (1.009) (0.576) (1.315) βc,2,2 (VehAge2) (0.025) (0.025) (0.025) βc,2,3 (Year1996) (0.035) (0.035) (0.035) βc,2,4 (Age2) (0.032) (0.032) (0.032) βc,2,5 (Age3) (0.064) (0.064) (0.064) Third Party Property σ (0.032) (0.031) (0.031) α (0.111) (0.101) (0.101) α (0.372) (0.402) (0.401) βc,3,1 (intercept) (0.136) (0.140) (0.139) βc,3,2 (VehAge2) (0.043) (0.042) (0.042) βc,3,3 (Year1) (0.011) (0.011) (0.011) Copula ρ (0.115) (0.115) ρ (0.112) (0.111) ρ (0.024) (0.024) r ( ) Model Fit Statistics log-likelihood -31, , , number of parms AIC 62, , , Note: Standard errors are in parenthesis. claim page 25

26 Concluding remarks Allows for covariates for the frequency, type and severity components Captures the long-tail nature of severity through the GB2. Provides for a two-part distribution of losses - when a claim occurs, not necessary that all possible types of losses are realized. Allows for possible dependencies among claims through a copula Allows for heterogeneity from the longitudinal nature of policyholders (not claims) Other applications Could look at financial information from companies Could examine health care expenditure Compare companies performance using multilevel, intercompany experience claim page 26

27 - Thank you - claim page 27