A hierarchical model for micro-level stochastic loss reserving A hierarchical model for micro-level E.A. Valdez stochastic loss reserving joint work with K. Antonio 1 and E.W. Frees 2 Motivation Data 44th Actuarial Research Conference Literature Madison, Wisconsin 30 Jul - 1 Aug 2009 Data structure Statistical approach Time to events Payment type Payments Prediction Conclusion E.A. Valdez University of Connecticut Storrs, Connecticut 1 U. of Amsterdam 2 U. of Wisconsin – Madison page 1
A hierarchical model Dynamics of claims reserving for micro-level stochastic loss reserving E.A. Valdez Development occurrence present declaration Motivation payment Data settlement Literature Data structure RBNS Statistical approach Time to events Payment type Payments Prediction Conclusion IBNR Calendar Time page 2
A hierarchical model Synopsis for micro-level stochastic loss reserving E.A. Valdez Put focus on RBNS claims: Reported But Not Settled . Motivation Data Use micro-level data to predict future development of Literature open claims. Data structure Statistical approach Time to events Payment type Develop a hierarchical model. Payments Prediction Conclusion “A hierarchical model for micro–level stochastic loss reserving.” page 3
A hierarchical model The data for micro-level stochastic loss reserving Data are from the General Insurance Association of E.A. Valdez Singapore . Observations are from one company over 10-year period: Jan 1993 – Jul 2002. Motivation ⇒ “present moment” in this case–study is 25 Jul 2002. Data Literature Policy file : characteristics of policyholder and vehicle Data structure insured Statistical approach Time to events Payment type ⇒ age, gender, vehicle type, vehicle age, . . . Payments Prediction Claims file : keeps track of each accident claim filed with Conclusion the insurer ⇒ linked to policy file, contains accident date. Payments file : reports each payment made during observation period. ⇒ linked to claims file, with payment date, size and type. page 4
A hierarchical model The data for micro-level stochastic loss reserving E.A. Valdez A claim will have multiple payments during its run–off. Payment types may be: Motivation Data own damage ( O ) (including injury, property, fire, theft); Literature Data structure injury ( I ) to a party other than the insured; Statistical approach Time to events property damage ( P ). Payment type Payments Prediction Combinations of these types may also occur. Conclusion Frees and Valdez (2008, JASA) summarized the many payments per claim into one single claim amount. page 5
A hierarchical model The data for micro-level stochastic loss reserving Development of claim 7 Development of claim 9942 E.A. Valdez 8000 own closed 6000 injury open property 4000 Motivation 2000 Data Literature 0 0 Data structure 0 2 4 6 8 10 12 0 2 4 6 Statistical approach Time to events Acc. Date 12/14/1999 Acc. Date 08/18/2001 Payment type Payments Development of claim 21443 Development of claim 24076 Prediction 12000 Conclusion 6000 8000 4000 2000 0 0 0 20 40 60 80 −1 0 1 2 3 4 Acc. Date 04/25/1995 Acc. Date 01/04/1996 page 6
A hierarchical model The data for micro-level stochastic loss reserving E.A. Valdez Arrival Year 1993 Arrival Year 1998 own injury property own injury property Motivation Data Literature Data structure Statistical approach Time to events 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Payment type Months since occurrence Months since occurrence Payments Arrival Year 2000 Prediction Conclusion own injury property 0 20 40 60 80 100 120 Months since occurrence page 7
A hierarchical model A traditional actuarial display for micro-level stochastic loss reserving Run–off triangle : aggregate claims per arrival year (AY) E.A. Valdez and development year (DY) combination. Run–off triangle for property ( P ) payments: (in ’000s, non–cumulative) Motivation Data Literature Arrival Development Year Year 1 2 3 4 5 6 7 8 9 10 Data structure 1993 205.3 847.6 226.3 77.9 47.9 40.6 10.2 1.8 0.0 0.6 1994 1,081.3 1,750.4 534.7 153.8 73.0 51.1 16.2 37.3 5.8 Statistical approach 1995 900.9 1,822.7 578.5 202.0 54.1 48.2 9.5 1.3 Time to events 1996 1,272.8 1,816.9 583.7 255.2 44.2 24.1 11.4 Payment type 1997 1,188.7 2,257.9 695.2 166.8 92.1 12.9 Payments 1998 1,235.4 3,250.0 649.9 211.2 74.1 1999 2,209.8 3,718.7 818.4 266.3 Prediction 2000 2,662.5 3,487.0 762.7 2001 2,457.3 3,650.3 Conclusion 2002 673.7 Common statistical techniques: chain–ladder, distributional, Bayesian, GLMs, . . . Modeling individual claims run-off is less developed in the literature. page 8
A hierarchical model Micro–level data: literature for micro-level stochastic loss reserving E.A. Valdez Suggestions from actuarial literature: England and Verrall (2002), Taylor and Campbell (2002), Taylor, McGuire, and Motivation Sullivan (2006). Data Some actuarial papers: Literature Data structure Arjas (1989, ASTIN), Norberg Statistical approach (1993, ASTIN), Norberg (1999, Time to events ASTIN); Payment type Payments Haastrup and Arjas (1996, Prediction ASTIN); Conclusion Larsen (2007, ASTIN); Zhao, Zhou, and Wang (2009, IME). Statistical resource: Cook and Lawless (2007), Statistical analysis of recurrent events . page 9
A hierarchical model Observable data structure for micro-level stochastic loss reserving E.A. Valdez 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 Motivation ( T i 0 = 0 is accident date and T iN i is settlement date); Data Literature C i time of censoring ; Data structure Statistical approach E ij type of event j . We distinguish: Time to events Payment type - event type 1: direct settlement without any payments; Payments Prediction - event type 2: payment with settlement; Conclusion - 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
A hierarchical model Timing of events, per event type for micro-level stochastic loss reserving E.A. Valdez Event 1: direct settlement Event 2: payment with settlement 35 4000 30 25 3000 Frequency Frequency Motivation 20 2000 15 Data Literature 10 1000 Data structure 5 0 0 Statistical approach Time to events 0 20 40 60 80 100 0 20 40 60 80 100 Payment type Months since occurrence Months since occurrence Min=0; Max=87.56 Min=0; Max=103 Payments Event 3: payment −− no settlement Prediction 15000 Conclusion 10000 Frequency 5000 0 0 20 40 60 80 100 Months since occurrence Min=0; Max=111 page 11
A hierarchical model Time of settlement, number of payments, times between for micro-level stochastic loss payments reserving E.A. Valdez Time of settlement Number of payments 2500 20000 2000 15000 Motivation 1500 Frequency Frequency Data 10000 Literature 1000 Data structure 5000 500 Statistical approach Time to events 0 0 Payment type 0 20 40 60 80 100 1 2 3 4 5 6 7 8 Payments Months since occurrence Prediction Min=0; Max=103 Min=1; Max=8 Conclusion Time between payments (in months) 25000 20000 15000 Frequency 10000 5000 0 page 12 0 20 40 60 80 100
A hierarchical model Payment types for micro-level stochastic loss reserving E.A. Valdez Motivation Number of payments per type: Data Literature Data structure Claim Type (I) (O) (P) Statistical approach Time to events Number 1,417 (1.95%) 45,950 (63.3%) 21,775 (30%) Payment type (I,O) (I,P) (O,P) (O,I,P) Payments Number 107 (0.147%) 319 (0.439%) 3017 (4.16%) 9 (0.012%) Prediction Conclusion page 13
A hierarchical model Distribution of payments for micro-level stochastic loss reserving Pay_vI (<0) Ln_Pay_vI E.A. Valdez Frequency Frequency 80 3 0 0 −4000 −3000 −2000 −1000 0 2 4 6 8 10 12 Pay_vINeg log(Pay_vIPos) Motivation Pay_vP (<0) Ln_Pay_vP Data Frequency Frequency 3000 400 Literature 0 0 Data structure −30000 −20000 −10000 0 −5 0 5 10 Statistical approach Time to events Pay_vPNeg log(Pay_vPPos) Payment type Payments Pay_vO (<0) Ln_Pay_vO (Claim amount) Prediction Frequency Frequency 6000 300 Conclusion 0 0 −150000 −100000 −50000 0 −10 −5 0 5 10 Pay_vONeg log(Pay_vOPos) Ln_Pay_vO (Loss amount) Frequency 6000 0 −5 0 5 10 log(Pay_vONoExPos) page 14
A hierarchical model Model formulation for micro-level stochastic loss reserving E.A. Valdez A claim i ( i = 1 , . . . , n c ) is a combination of accident date (‘ AD i ’); set of covariates C i ; Motivation Data development process X i : Literature X i = ( { E i ( v ) , M i ( v ) , P i ( v ) } ) v ∈ [ 0 , T iNi ] ; Data structure Statistical approach Development process X i is a jump process. 3 building Time to events Payment type blocks are used: Payments Prediction E i ( t ij ) := E ij is the type of the j th event in the development of Conclusion 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
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