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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

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

page 1

A hierarchical model for micro-level stochastic loss reserving

joint work with K. Antonio1 and E.W. Frees2 44th Actuarial Research Conference Madison, Wisconsin 30 Jul - 1 Aug 2009 E.A. Valdez University of Connecticut Storrs, Connecticut

  • 1U. of Amsterdam
  • 2U. of Wisconsin – Madison
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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Dynamics of claims reserving

  • ccurrence

declaration payment settlement

present IBNR RBNS

Calendar Time Development

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Synopsis Put focus on RBNS claims: Reported But Not Settled. Use micro-level data to predict future development of

  • pen claims.

Develop a hierarchical model. “A hierarchical model for micro–level stochastic loss reserving.”

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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The data Data are from the General Insurance Association of Singapore. Observations are from one company over 10-year period: Jan 1993 – Jul 2002. ⇒ “present moment” in this case–study is 25 Jul 2002. 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. Payments file: reports each payment made during

  • bservation period.

⇒ linked to claims file, with payment date, size and type.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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The data A claim will have multiple payments during its run–off. Payment types may be:

  • wn 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.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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The data

2 4 6 8 10 12 2000 6000

Development of claim 7

  • Acc. Date 12/14/1999
  • wn

injury property 2 4 6 4000 8000

Development of claim 9942

  • Acc. Date 08/18/2001

closed

  • pen

20 40 60 80 4000 8000 12000

Development of claim 21443

  • Acc. Date 04/25/1995

−1 1 2 3 4 2000 6000

Development of claim 24076

  • Acc. Date 01/04/1996
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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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The data

20 40 60 80 100 120

Arrival Year 1993

Months since occurrence

  • wn

injury property 20 40 60 80 100 120

Arrival Year 1998

Months since occurrence

  • wn

injury property 20 40 60 80 100 120

Arrival Year 2000

Months since occurrence

  • wn

injury property

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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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 1 2 3 4 5 6 7 8 9 10 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 1995 900.9 1,822.7 578.5 202.0 54.1 48.2 9.5 1.3 1996 1,272.8 1,816.9 583.7 255.2 44.2 24.1 11.4 1997 1,188.7 2,257.9 695.2 166.8 92.1 12.9 1998 1,235.4 3,250.0 649.9 211.2 74.1 1999 2,209.8 3,718.7 818.4 266.3 2000 2,662.5 3,487.0 762.7 2001 2,457.3 3,650.3 2002 673.7

Common statistical techniques: chain–ladder, distributional, Bayesian, GLMs, . . . Modeling individual claims run-off is less developed in the literature.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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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.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Observable data structure total number of claims in the data set is n = 43, 729; Ni, number of “events” in development period of claim i; Tij, time of event j, in months since the accident date (Ti0 = 0 is accident date and TiNi is settlement date); Ci time of censoring; Eij 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.

Mij type of payment for event j of claim i. Pijk size of payment of type k (k being ‘own damage’ (O), ‘injury’ (I) or ‘property’ (P)) for event j of claim i.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Timing of events, per event type

Event 1: direct settlement

Min=0; Max=87.56 Months since occurrence Frequency 20 40 60 80 100 5 10 15 20 25 30 35

Event 2: payment with settlement

Min=0; Max=103 Months since occurrence Frequency 20 40 60 80 100 1000 2000 3000 4000

Event 3: payment −− no settlement

Min=0; Max=111 Months since occurrence Frequency 20 40 60 80 100 5000 10000 15000

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Time of settlement, number of payments, times between payments

Time of settlement

Min=0; Max=103 Months since occurrence Frequency 20 40 60 80 100 500 1000 1500 2000 2500

Number of payments

Min=1; Max=8 Frequency 1 2 3 4 5 6 7 8 5000 10000 15000 20000

Time between payments (in months)

Frequency 20 40 60 80 100 5000 10000 15000 20000 25000

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Payment types 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%)

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Distribution of payments

Pay_vI (<0)

Pay_vINeg Frequency −4000 −3000 −2000 −1000 3

Ln_Pay_vI

log(Pay_vIPos) Frequency 2 4 6 8 10 12 80

Pay_vP (<0)

Pay_vPNeg Frequency −30000 −20000 −10000 400

Ln_Pay_vP

log(Pay_vPPos) Frequency −5 5 10 3000

Pay_vO (<0)

Pay_vONeg Frequency −150000 −100000 −50000 300

Ln_Pay_vO (Claim amount)

log(Pay_vOPos) Frequency −10 −5 5 10 6000

Ln_Pay_vO (Loss amount)

log(Pay_vONoExPos) Frequency −5 5 10 6000

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Model formulation A claim i (i = 1, . . . , nc) is a combination of

accident date (‘ADi’); set of covariates Ci; development process Xi: Xi = ({Ei(v), Mi(v), Pi(v)})v∈[0,TiNi ];

Development process Xi is a jump process. 3 building blocks are used:

Ei(tij) := Eij is the type of the jth event in the development of claim i, occurring at time tij; If this event includes a payment, its payment is given by Mi(tij) := Mij; Corresponding payment vector is Pij.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Time to events Intensity modeling with single type of events at times tij: Li =  

Ni

  • j=1

λi(tij)   exp

  • −

τi λi(u)du

  • .

[0, τi] is the period of observation of subject i with τi = min (TiNi , Ci). λi(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 λik(t).

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Time to events How to specify the intensity functions λ1(t) (for event 1), λ2(t) (for event 2) and λ3(t) (for event 3)? Techniques from survival analysis: (k = 1, 2, 3)

exponential: λk(t) := λk; Weibull: λk(t) := αkγktαk −1e−γk tαk ; Cox model: λk(t) := λ0k(t) exp (z

′

kβk);

piecewise constant: λk(t) =            λk1 for 0 ≤ t < tk1 λk2 for tk1 ≤ t < tk2 . . . λkd for tkd−1 ≤ t < tkd.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Hazard rates per event type

20 40 60 80 100 120 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Hazard Rate −− Type 1

t.grid h.gridW const. Weibull

  • piec. const. (12m)
  • piec. const. (3m)

20 40 60 80 100 120 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Hazard Rate −− Type 2

t.grid h.gridW const. Weibull

  • piec. const. (12m)
  • piec. const. (3m)

20 40 60 80 100 120 0.00 0.05 0.10 0.15 0.20

Hazard Rate −− Type 3

t.grid h.gridW const. Weibull

  • piec. const. (12m)
  • piec. const. (3m)
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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Payment type Mij represents the combination of payments observed at tij. 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(Mij = mij) = exp Vij,m 7

s=1 exp (Vij,s)

, with Vij,m = x

′

ijβM,m.

Covariate information used in multinomial model:

Type of vehicle, vehicle age, age of driver; Arrival Year, Development Year.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Payments Given Mij for the event at time tij, Pij gives corresponding severities. For the sign of a payment, use: Iijk =

  • 1 if Pijk > 0

0 if Pijk < 0, and sijk = Pr(Iijk = 1). Use logistic regression to model the sign of Pijk: logit(sijk) = x

′

ijβS,k.

Covariate information used in logistic models:

Development year; Number of previous injury/own damage/property payments.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Negative part of payments Burr regression: fP(p) = λβλτpτ−1 (β + pτ)λ+1 , with τijk = exp (x

′

ijkβP,k) with k for payment type.

used for ’Property’ and ’Own Damage’ payments

GB2 regression: fP(p) = |α|pαγ1−1βαγ2 B(γ1, γ2)(βα + pα)γ1+γ2 , with α = 0, β, γ1, γ2 > 0, B(α1, α2) the usual beta function and βijk = exp (x

′

ijβP,k).

used for ’Injury’ payments

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Positive part of payments Inspired by the histograms of the positive payments, we used a mixture of lognormal regression models: log (P) ∼ w1N1(µ1, σ2

1) + w2N2(µ2, σ2 2) + w3N3(µ3, σ2 3),

where w1, w2 and w3 are weights, specified as w1 = exp (a) exp (a) + exp (b) + exp (c), w2 = exp (b) exp (a) + exp (b) + exp (c), w3 = exp (c) exp (a) + exp (b) + exp (c), and Ni(µi, σ2

i ) is a normal distribution with mean µi and

variance σ2

i .

Covariate information is incorporated in the weights and parameters µi and σ2

i (i = 1, 2, 3).

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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QQ plots on the negative payments

−6 −4 −2 2 4 6 8 −10 −5 5 10 Own damage − theoretical quantile empirical quantile −10 −8 −6 −4 −2 −10 −8 −6 −4 −2 Injury − theoretical quantile empirical quantile −8 −6 −4 −2 2 4 −8 −6 −4 −2 2 4 Property − theoretical quantile empirical quantile

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Histograms of the positive payments - own damage

2 4 6 8 10 12 14 0.0 0.1 0.2 0.3 0.4

Positive Own Damage Payments (log scale)

Sample: DY=1 x y 2 4 6 8 10 12 14 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Positive Own Damage Payments (log scale)

Sample: DY=2 x y 2 4 6 8 10 12 14 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Positive Own Damage Payments (log scale)

Sample: DY>2 x y

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Prediction 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

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Resulting predictive distributions of reserves - by type

Reserve Own Damage

Frequency −2.0e+07 −1.0e+07 0.0e+00 5.0e+06 100 200 300 400 500 600

Reserve Injury

Frequency 4e+06 6e+06 8e+06 1e+07 50 100 150

Reserve Property

Frequency 7.0e+06 8.0e+06 9.0e+06 1.0e+07 1.1e+07 50 100 150

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Concluding remarks Main idea: claims reserving 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 reserving methods.

Similar methodology to other areas of actuarial statistics e.g. recurrent episodes in workers’ compensation.

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A hierarchical model for micro-level stochastic loss reserving E.A. Valdez Motivation Data Literature Data structure Statistical approach

Time to events Payment type Payments

Prediction Conclusion

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Thank you!