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Microlevel stochastic loss reserving Katrien Antonio & Richard Plat Dept. of Quantitative Economics University of Amsterdam The Netherlands August 29 2011 1 / 29 Context Consider the claims reserving problem for a branch of

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Micro–level stochastic loss reserving

Katrien Antonio & Richard Plat

  • Dept. of Quantitative Economics

University of Amsterdam The Netherlands

August 29 2011

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Context

◮ Consider the claims reserving problem for a branch of

insurance products known as non–life insurance (Europe), general insurance (UK) and property and casualty insurance (USA).

◮ Examples of LoBs: motor insurance, property (e.g. against

fire), liability insurance, . . .

◮ Insured receives financial coverage against the random

  • ccurrence of well–specified events, in return for paying a

premium to the insurance company.

◮ For consistent financial statements: all claims with accident

year ‘xx’ have to be matched to premium earned in ‘xx’.

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Dynamics of claims reserving

Run–off process of a non–life claim t1 t2 t3 t4 t5 t6 t7 t8 t9 Occurrence Notification Loss payments Closure Re–opening Payment Closure

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Dynamics of claims reserving

t1 t2 t3 t4 t5 Closed claim time Occurrence Notification Loss Payments Closure DATE OF ANALYSIS

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Dynamics of claims reserving

t1 t2 t3 t4 RBNS claim time Occurrence Notification Loss Payments DATE OF ANALYSIS Uncertainty

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Dynamics of claims reserving

t1 IBNR claim time Occurrence DATE OF ANALYSIS Uncertainty

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Claims reserving: aims

◮ Two types of incomplete claims:

  • IBNR: Incurred But Not Reported;
  • RBNS: Reported But Not Settled.

◮ Predict the unknown development of these claims. ◮ Not just a point estimate of outstanding amount, but real

interest is in predictive distribution.

◮ The measurement of future cash–flows and their

uncertainty becomes more and more important: see Solvency 2 (in 2012) and IFRS 4 Phase 2 (in 2013).

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Micro–level run–off data

◮ Non–life insurance companies have data bases with detailed

information:

  • exposure measure;
  • information about the claim event, the policy (holder) (eg

policy limit) and the reporting delay;

  • payments: date and severity, type;
  • explanatory variables (eg case estimates by experts).

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Micro–level run–off data: example

◮ European general liability insurance portfolio: bodily

injury claims and material damage claims.

◮ Observation period is Jan. 1997 – August 2009. ◮ File consists of 1,525,376 records corresponding with 474,634

claims.

◮ Structure of the data:

  • Policy file: exposure per month from January 2000 till August

2009.

  • Claims file: accident date + details, open/closed.
  • Payments file: each payment made during observation period.

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Micro–level run–off data: example

Development of 4 random material claims:

2 4 6 8 500 1500 time since origin of claim (in months)

Development of claim 327002

  • Acc. Date 15/03/2006

2 4 6 8 10 500 1500 time since origin of claim (in months)

Development of claim 331481

  • Acc. Date 24/04/2006

−1 1 2 3 4 500 1500 time since origin of claim (in months)

Development of claim 434833

  • Acc. Date 01/10/2008

2 4 6 500 1000 1500 time since origin of claim (in months)

Development of claim 34127

  • Acc. Date 02/04/1998

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Micro–level run–off data: example

Development of 4 random injury claims:

2 4 6 8 500 1000 time since origin of claim (in months)

Development of claim 12

  • Acc. Date 01/01/1997

2 4 6 8 500 1500 time since origin of claim (in months)

Development of claim 173876

  • Acc. Date 19/06/2002

5 10 15 200 600 time since origin of claim (in months)

Development of claim 216542

  • Acc. Date 05/02/2003

2 4 6 8 10 12 400 800 time since origin of claim (in months)

Development of claim 349176

  • Acc. Date 17/09/2006

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Traditional actuarial display

◮ Actuarial techniques for claims reserving are based on data

aggregated in run–off triangles.

Accident Development Year j Year i 1 2 3 4 . . . j . . . J 1 . . . . . . Observations Cij, Xij . . . (i + j ≤ I) . . . i . . . . . . I − 2 Predicted Cij, Xij (i + j > I) I − 1 I

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Traditional actuarial display

◮ Actuarial techniques for claims reserving are based on data

aggregated in run–off triangles.

◮ Drawbacks/Questions:

  • useful information at individual claim and policy level is

ignored;

  • limited amount of data is analyzed;
  • how to distinguish IBNR and RBNS claims?
  • how to distinguish small and large claims?
  • zero cells, negative cells, how to combine paid and incurred

data?

  • how should reinsurance companies approach the reserving

problem?

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Micro–level loss reserving model

◮ A claim i is a combination of

  • an accident date Ti;
  • a reporting delay Ui;
  • a set of covariates Ci;
  • a development process Xi: Xi = ({Ei(v), Pi(v)})v∈[0,ViNi ];

◮ In the development process we use:

  • Ei(vij) := Eij the type of the jth event in development of claim

i;

  • occurs at time vij, in months after notification date;
  • corresponding payment vector Pi(vij) := Pij.

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Micro–level loss reserving model

◮ Run–off process of a non–life claim on a time axis

Ti Ti + Ui Ti + Ui + Vi1 Vi1 Vi2 . . . ViNi

  • Occurrence

Notification Loss payments Closure

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Micro–level loss reserving model

◮ Say outstanding liabilities are to be predicted at calendar time

τ.

◮ Observed data: development up to time τ of claims reported

before τ. (T o

i , Uo i , X o i )i≥1. ◮ Development of claim i is censored τ − T o i − Uo i time units

after notification.

◮ Likelihood of the observed claim development process:

Λ(obs) ∝ {

  • i≥1 λ(T o

i )PU|t(τ−T o i )} exp(−

τ

0 w(t)λ(t)PU|t (τ−t)dt)

×

  • i≥1

PU|t (dUo i ) PU|t (τ−To i )

  • ×

i≥1 P τ−To i −Uo i X|t,u

(dX o

i ). 16 / 29

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Micro–level loss reserving model

◮ Building blocks in the model used in Antonio & Plat,

following Norberg (1993, 1999):

  • a distribution for the reporting delay;
  • a filtered Poisson process driving the occurrence of claims

(IBNR + RBNS);

  • the claims development process: recurrent events and payment

severities;

  • (recurrent) events?

⇒ settlement with payment, settlement without payment, intermediate payment.

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Micro–level loss reserving model

◮ Likelihood uses the following building blocks:

(1) the reporting delay:

i≥1 PU|t(dUo

i )

PU|t(τ−T o

i );

(2) the occurrence times (given the reporting delay distribution):   

  • i≥1

λ(T o

i )PU|t(τ − T o i )

   exp

  • −

τ w(t)λ(t)PU|t(τ − t)dt

  • ;

(3) the development process – event part:

  • i≥1

Ni

j=1

  • h

δij1 se

(Vij)

  • ×h

δij2 sep (Vij)×h δij3 p

(Vij)

  • exp(−

τi

0 (hse(u)+hsep(u)+hp(u))du);

(4) the development process – severity part:

  • i≥1
  • j

Pp(dVij).

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Calibration: reporting delay

◮ Reporting delay distribution. ◮ Combine a Weibull distribution with degenerate components

at 0 days delay, 1 day delay, . . . , 8 days delay:

8

  • k=0

pkIU=k + (1 −

  • k

pk)fU|U>8(u).

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Calibration: occurrence of claims

◮ Poisson process driving the occurrence of claims. ◮ A piecewise constant specification for the occurrence rate

λ(t).

◮ Material damage (left) and injury (right) claims:

20 40 60 80 100 120 0.04 0.05 0.06 0.07 time lambda(t) 20 40 60 80 0.0005 0.0010 0.0015 0.0020 time lambda(t)

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Calibration: development of claims

◮ Claims development: occurrence and type of events. ◮ Piecewise constant specification of the hazard rates.

20 60 100 0.00 0.04 0.08 0.12

Type 1 − Injury

t.grid h.grid first later 20 60 100 0.000 0.010 0.020 0.030

Type 2 − Injury

t.grid h.grid 20 60 100 0.05 0.10 0.15 0.20 0.25

Type 3 − Injury

t.grid h.grid 5 10 20 30 0.1 0.2 0.3 0.4 0.5

Type 1 − Mat

t.grid h.grid 5 10 20 30 0.0 0.1 0.2 0.3 0.4

Type 2 − Mat

t.grid h.grid 5 10 20 30 0.05 0.10 0.15 0.20

Type 3 − Mat

t.grid h.grid

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Calibration: severities

◮ Severities distribution. ◮ Lognormal distributions with µ and σ depending on:

  • the development period: 0-12 months after notification, 12-24

months . . . (for injury) and 0-4 months, 4-8 months . . . (for material);

  • the initial reserve (set by company experts): categorized.

◮ Policy limit of 2,500,000 euro is implemented.

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Calibration: severities

◮ Severities distribution: material damage (left) and injury

(right).

−10 −5 5 −4 −2 2 4

Normal QQplot Payments: Material

  • Emp. Quant.
  • Theor. Quant.

−4 −2 2 4 −4 −2 2 4

Normal QQplot Payments: Injury

  • Emp. Quant.
  • Theor. Quant.

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Forecasting

◮ Using these building blocks we can easily:

  • simulate the time to a next event, the corresponding type and

severity for an RBNS claim;

  • simulate the number of IBNR claims that will show up, their
  • ccurrence time and their development.

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Results

◮ Example of a back–test: fit model to data 1/1/1997 till

1/1/2004 and compare predictions with real outcomes.

◮ Results obtained with micro–model are compared with those

from traditional techniques (i.e. overdispersed Poisson and lognormal regression model with chain–ladder structure).

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Results

Injury claims, results for calendar year 2006.

IBNR claims Number Frequency 5 10 15 20 25 30 35 500 1000 1500 IBNR + RBNS Reserve Reserve Frequency 1000 2000 3000 4000 5000 6000 200 400 600 800 1000 Total events Number Frequency 700 800 900 1000 1100 200 400 600 800 1000 Type 1 events Number Frequency 100 120 140 160 180 200 220 500 1000 1500 Type 2 events Number of events Frequency 20 40 60 80 200 400 600 800 1000 1200 Type 3 events Number Frequency 500 550 600 650 700 750 800 850 200 400 600 800 1000

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Results

Material damage claims, total reserve: micro–level,

  • verdispersed Poisson (triangle), lognormal (triangle) model.

Micro−level: Total Reserve Frequency 2000 3000 4000 5000 6000 7000 100 200 300 400 500 600 700 Aggregate Model − ODP: Total Reserve Frequency 2000 3000 4000 5000 6000 7000 200 400 600 800 1000 1200 Aggregate Model − Lognormal: Total Reserve Frequency 5000 10000 15000 20000 25000 30000 500 1000 1500

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Results

Injury claims, total reserve: micro–level, overdispersed Poisson (triangle), lognormal (triangle) model.

Micro−level: Total Reserve Frequency 4000 6000 8000 10000 12000 14000 500 1000 1500 Aggregate Model − ODP: Total Reserve Frequency 6000 8000 10000 12000 14000 16000 500 1000 1500 Aggregate Model − Lognormal: Total Reserve Frequency 6000 8000 10000 12000 14000 16000 200 400 600 800 1000 1200

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Conclusion and outlook

◮ Development of a micro–model for claims reserving in non–life

insurance, including:

  • calibration to a realistic data base from practice;
  • forecasting;
  • back–testing, in comparison with results from traditional

techniques.

◮ On–going work:

  • aggregate data ≪ ≫ individual data with development

aggregated in cells of e.g. one year ≪ ≫ micro–level data in continuous time;

  • the reinsurance point of view;
  • combination with extreme value statistics.

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