joint work with K. Antonio 1 and E.W. Frees 2 44th Actuarial Research Conference Madison, Wisconsin 30 Jul - 1 Aug 2009

Transcription

1 joint work with K. Antonio 1 and E.W. Frees 2 44th Actuarial Research Conference Madison, Wisconsin 30 Jul - 1 Aug 2009 University of Connecticut Storrs, Connecticut 1 U. of Amsterdam 2 U. of Wisconsin Madison page 1

2 Dynamics of claims Development present occurrence declaration payment settlement RBNS IBNR Calendar Time page 2

3 Synopsis Put focus on RBNS claims: Reported But Not Settled. Use micro-level data to predict future development of open claims. Develop a hierarchical model. for micro level. page 3

4 The data are from the General Insurance Association of Singapore. Observations are from one company over 10-year period: Jan 1993 Jul present moment in this case study is 25 Jul Policy file: characteristics of policyholder and vehicle insured age, gender, vehicle type, vehicle age,... Claims file: keeps track of each accident claim filed with the insurer linked to policy file, contains accident date. file: reports each payment made during observation period. linked to claims file, with payment date, size and type. page 4

5 The data A claim will have multiple payments during its run off. s may be: own damage (O) (including injury, property, fire, theft); injury (I) to a party other than the insured; property damage (P). Combinations of these types may also occur. Frees and Valdez (2008, JASA) summarized the many payments per claim into one single claim amount. page 5

6 The data Development of claim 7 Development of claim own injury property closed open Acc. Date 12/14/1999 Acc. Date 08/18/2001 Development of claim Development of claim Acc. Date 04/25/1995 Acc. Date 01/04/1996 page 6

7 The data Arrival Year 1993 Arrival Year 1998 own injury property own injury property Months since occurrence Arrival Year Months since occurrence own injury property Months since occurrence page 7

8 A traditional actuarial display Run off triangle: aggregate claims per arrival year (AY) and development year (DY) combination. Run off triangle for property (P) payments: (in 000s, non cumulative) Arrival Development Year Year , , , , , , , , , , , , , , , Common statistical techniques: chain ladder, distributional, Bayesian, GLMs,... Modeling individual claims run-off is less developed in the literature. page 8

9 Micro level data: literature Suggestions from actuarial literature: England and Verrall (2002), Taylor and Campbell (2002), Taylor, McGuire, and Sullivan (2006). Some actuarial papers: Arjas (1989, ASTIN), Norberg (1993, ASTIN), Norberg (1999, ASTIN); Haastrup and Arjas (1996, ASTIN); Larsen (2007, ASTIN); Zhao, Zhou, and Wang (2009, IME). Statistical resource: Cook and Lawless (2007), Statistical analysis of recurrent events. page 9

10 Observable data structure total number of claims in the data set is n = 43, 729; N i, number of events in development period of claim i; T ij, time of event j, in months since the accident date (T i0 = 0 is accident date and T ini is settlement date); C i time of censoring; E ij type of event j. We distinguish: - event type 1: direct settlement without any payments; - event type 2: payment with settlement; - event type 3: payment without settlement. M ij type of payment for event j of claim i. P ijk size of payment of type k (k being own damage (O), injury (I) or property (P)) for event j of claim i. page 10

11 Timing of events, per event type Event 1: direct settlement Event 2: payment with settlement Months since occurrence Min=0; Max=87.56 Months since occurrence Min=0; Max=103 Event 3: payment no settlement Months since occurrence Min=0; Max=111 page 11

12 Time of settlement, number of payments, times between payments Time of settlement Number of payments Months since occurrence Min=0; Max=103 Min=1; Max=8 Time between payments (in months) page 12

13 s Number of payments per type: Claim Type (I) (O) (P) Number 1,417 (1.95%) 45,950 (63.3%) 21,775 (30%) (I,O) (I,P) (O,P) (O,I,P) Number 107 (0.147%) 319 (0.439%) 3017 (4.16%) 9 (0.012%) page 13

14 Distribution of payments 0 3 Pay_vI (<0) 0 80 Ln_Pay_vI Pay_vINeg log(pay_vipos) Pay_vP (<0) Ln_Pay_vP Pay_vPNeg log(pay_vppos) Pay_vO (<0) Ln_Pay_vO (Claim amount) Pay_vONeg log(pay_vopos) Ln_Pay_vO (Loss amount) log(pay_vonoexpos) page 14

15 Model formulation A claim i (i = 1,..., n c ) is a combination of accident date ( AD i ); set of covariates C i ; development process X i : X i = ({E i (v), M i (v), P i (v)}) v [0,TiNi ]; Development process X i is a jump process. 3 building blocks are used: E i (t ij ) := E ij is the type of the jth event in the development of claim i, occurring at time t ij ; If this event includes a payment, its payment is given by M i (t ij ) := M ij ; Corresponding payment vector is P ij. page 15

16 Intensity modeling with single type of events at times t ij : N i ( τi ) L i = λ i (t ij ) exp λ i (u)du. 0 j=1 [0, τ i ] is the period of observation of subject i with τ i = min (T ini, C i ). λ i (t) is the event intensity (or hazard rate) at time t for subject i. For multitype events: each subject is at risk of m different types of recurrent events. Specify intensity function for each type of event (k = 1,..., m) with λ ik (t). page 16

17 How to specify the intensity functions λ 1 (t) (for event 1), λ 2 (t) (for event 2) and λ 3 (t) (for event 3)? Techniques from survival analysis: (k = 1, 2, 3) exponential: λ k (t) := λ k ; Weibull: λ k (t) := α k γ k t α k 1 e γ k t αk ; Cox model: λ k (t) := λ 0k (t) exp (z kβ k ); piecewise constant: λ k (t) = λ k1 for 0 t < t k1 λ k2 for t k1 t < t k2. λ kd for t kd 1 t < t kd. page 17

18 Hazard rates per event type Hazard Rate Type 1 Hazard Rate Type 2 h.gridw const. Weibull piec. const. (12m) piec. const. (3m) h.gridw const. Weibull piec. const. (12m) piec. const. (3m) t.grid t.grid Hazard Rate Type 3 h.gridw const. Weibull piec. const. (12m) piec. const. (3m) t.grid page 18

19 M ij represents the combination of payments observed at t ij. 7 combinations are possible: I, O, P, (I, O), (I, P), (O, P) and (O, I, P). Claim type is modeled with multinomial logit model: Pr(M ij = m ij ) = with V ij,m = x ij β M,m. exp V ij,m 7 s=1 exp (V ij,s), Covariate information used in multinomial model: Type of vehicle, vehicle age, age of driver; Arrival Year, Development Year. page 19

20 Given M ij for the event at time t ij, P ij gives corresponding severities. For the sign of a payment, use: { 1 if P ijk > 0 I ijk = 0 if P ijk < 0, and s ijk = Pr(I ijk = 1). Use logistic regression to model the sign of P ijk : logit(s ijk ) = x ijβ S,k. Covariate information used in logistic models: Development year; Number of previous injury/own damage/property payments. page 20

21 Negative part of payments Burr regression: f P (p) = λβλ τp τ 1 (β + p τ ) λ+1, with τ ijk = exp (x ijk β P,k) with k for payment type. used for Property and Own Damage payments GB2 regression: f P (p) = α p αγ 1 1 β αγ 2 B(γ 1, γ 2 )(β α + p α ) γ 1+γ 2, with α 0, β, γ 1, γ 2 > 0, B(α 1, α 2 ) the usual beta function and β ijk = exp (x ij β P,k). used for Injury payments page 21

22 Positive part of payments Inspired by the histograms of the positive payments, we used a mixture of lognormal regression models: log (P) w 1 N 1 (µ 1, σ 2 1) + w 2 N 2 (µ 2, σ 2 2) + w 3 N 3 (µ 3, σ 2 3), where w 1, w 2 and w 3 are weights, specified as w 1 = w 2 = w 3 = exp (a) exp (a) + exp (b) + exp (c), exp (b) exp (a) + exp (b) + exp (c), exp (c) exp (a) + exp (b) + exp (c), and N i (µ i, σ 2 i ) is a normal distribution with mean µ i and variance σ 2 i. Covariate information is incorporated in the weights and parameters µ i and σ 2 i (i = 1, 2, 3). page 22

23 QQ plots on the negative payments empirical quantile empirical quantile Own damage theoretical quantile Injury theoretical quantile empirical quantile Property theoretical quantile page 23

24 y y y Histograms of the positive payments - own damage Positive Own Damage (log scale) Positive Own Damage (log scale) x Sample: DY=1 x Sample: DY=2 Positive Own Damage (log scale) x Sample: DY>2 page 24

25 of RBNS claim reserves Step 1: simulate the next event s time interval Step 2: simulate the exact time of the next event Step 3: simulate the event type Step 4: simulate payment type Step 5: simulate payments Step 6: stop or continue, if necessary - depending on whether settled or not page 25

26 Resulting predictive distributions of reserves - by type Reserve Own Damage Reserve Injury e e e e+06 4e+06 6e+06 8e+06 1e+07 Reserve Property e e e e e+07 page 26

27 Concluding remarks Main idea: claims using statistics for recurrent events. The hope is to improve the prediction of reserves using detailed micro-level recorded information. the cost is the additional complexity in the modeling involved. Additional work to be done: comparing the results with traditional methods. Similar methodology to other areas of actuarial statistics e.g. recurrent episodes in workers compensation. page 27

28 Thank you! page 28