ClaimsAgency metrics

1
Claims/Agency metrics

• Greg Taylor
• Taylor Fry Consulting Actuaries
• University of Melbourne
• University of New South Wales
• Casualty Actuarial Society
• Special Interest Seminar on Predictive Modeling
• Boston, October 4-5 2006

2
Overview

• Individual claim models
• Paids models
• Incurreds models
• Numerical results
• Adaptive models

3
Why individual claim models?
4
Example problem

• Classical workers compensation cost centre
  allocation problem
• Claim numbers at the leaves of this tree may be
  small

Total claim cost
. . .
Cost centre 1
Cost centre 2
Cost centre m



5
Measuring claims performance

• Consider measuring claims performance in a
  segment of a long tail portfolio
• Likely that adopted metric will require an
  estimate of the amount of losses incurred but as
  yet unpaid (loss reserve)
• e.g. metric is expected ultimate losses per
  policy for a specific underwriting period
• Paid to date unpaid losses
• Number of policy-years of exposure
• average PTD per policy-year average unpaid
  per policy-year

6
Measuring claims performance in large portfolio
segments

• Let there be n policy-years of exposure and
• ui i-th amount unpaid
• Consider the ui to be random drawings from some
  distribution
• Average amount unpaid is
• ui S ui /n S Eui ui - Eui/n
• Eui S ui - Eui/n
• ? Eui as n?8
• by the large of large numbers

d
7
Measuring claims performance in large portfolio
segments (contd)

• ui ? Eui as n?8
• Eui expected size of a randomly drawn claim
• This will be the result produced by most
  conventional actuarial methods, e.g.
• Paid chain ladder
• Even incurred chain ladder at early development
• While Eui may be a good approximation to ui for
  large sample sizes, it may be very poor for small
  ones
• Leading to a highly distorted cost allocation

d
8
Measuring claims performance in small portfolio
segments

• Effective estimation of small sample average
  claim cost must somehow take account of the
  properties of the individual claims

9
There is a need to change from this

Data
Fitted
Model
Forecast
Forecast

• Conventional actuarial analysis of loss
  experience
• Call such models aggregate models

10
to this

Forecast
Model
Special case of individual claim reserving
statistical case estimation
11
Individual claim models
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Form of such a model

Forecast g( )
Model Yf(ß)e
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Form of such a model

Forecast g( )
Model Yf(ß)e
Yi f(Xi ß) ei Yi size of i-th completed
claim Xi vector of attributes (covariates) of
i-th claim ß vector of parameters that apply to
all claims ei vector of centred stochastic
error terms
14
Form of individual claim model

• Yi f(Xi ß) ei
• Convenient practical form is
• Yi h-1(XiT ß) ei GLM form

h link function
Error distribution from exponential dispersion
family
Linear predictor linear function of the
parameter vector
15
Form of individual claim model more specifically

• How might one create an individual claim model of
  the paids type?
• Aggregate paids model usually takes the form
• Yjk f(j,k ß) ejk
• for
• j accident period
• k development period
• Compare with
• Yi f(Xi ß) ei

Not always formulated
16
Form of paids individual claim model

• Possible to mimic aggregate model by defining
  individual model as just
• Yi h-1(ji,ki ß) ei

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Form of paids individual claim model

• Possible to mimic aggregate model by defining
  individual model as just
• Yi h-1(ji,ki ß) ei
• But often possible to improve on this, e.g.
• Replace development period j with operational
  time ti (proportion of accident periods incurred
  claims completed) at completion of i-th claim
• Example
• Yi exp ß0ß1tiß2max(0,ti-0.8) ei

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Example of paids individual claim model

• Yi exp ß0ß1tiß2max(0,ti-0.8) ei
• EYi exp ß0ß1tiß2max(0,ti-0.8)

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Example of paids individual claim model (contd)

• Yi exp ß0ß1tiß2max(0,ti-0.8) ei
• Include superimposed inflation
• Let qjkcalendar period of claim completion
• Extend model
• Yi exp ß0ß1tiß2max(0,ti-0.8)ß3qi ei
• Superimposed inflation at rate expß3 per period

20
Example of paids individual claim model (contd)

• Yi exp ß0ß1tiß2max(0,ti-0.8)ß3qi ei
• We might wish to model superimposed inflation as
  beginning at period qq0
• Yi exp ß0ß1tiß2max(0,ti-0.8)ß3max(0,qi-q0)
  ei

21
Example of paids individual claim model (contd)

• Yi exp ß0ß1tiß2max(0,ti-0.8)ß3qi ei
• We might wish to model superimposed inflation as
  beginning at period qq0
• Yi exp ß0ß1tiß2max(0,ti-0.8)ß3max(0,qi-q0)
  ei
• and we might wish to model superimposed
  inflation with a rate that decreases with
  increasing operational time
• Yi exp ß0ß1tiß2max(0,ti-0.8)(ß3-ß4ti)
  max(0,qi-q0) ei
• etc etc

22
Example of paids individual claim model (contd)

• Paids estimate of loss reserve scaled to
  baseline 1,000M
• Prediction CoV 5.3
• Mack (incurreds) estimate is 887M with CoV
  10.5
• Mack estimate produces negative reserves for the
  old years of origin
• Paids chain ladder fails completely

23
Example of paids individual claim model (contd)

• This model is very economical
• Contains only 9 parameters to represent many
  thousands of claims

24
Further extension of paids individual claim
model

• Yi exp ß0ß1tiß2max(0,ti-0.8)(ß3-ß4ti)
  max(0,qi-q0) ei
• May include claim characteristics other than
  time-related, e.g.
• Nature of injury
• Claim severity (MAIS scale)
• Pre-injury earnings
• Yi exp ß0ß1tiß2max(0,ti-0.8)(ß3-ß4ti)
  max(0,qi-q0) more terms ei

25
Example of incurreds individual claim model

• Similar to paids model
• Basic set-up is still
• Yi h-1(ji,ki,qi,ti,otherß) ei
• Example
• Yi exp(Ci,ji,ki,qi,ti,otherß) ei
• where Ci current manual estimate of
  incurred cost of i-th claim

26
Example of incurreds individual claim model
(contd)

• In fact, the model requires more structure than
  this because of claims and estimates for nil cost
• Let (for an individual claim)
• U ultimate incurred (may 0)
• C current estimate (may 0)
• X other claim characteristics

Model of ProbU0C,X
ProbU0
ProbUgt0
Model of UUgt0,C0,X
Model of U/CUgt0,Cgt0,X
If C0
If Cgt0
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Example of incurreds individual claim model
(contd)

• Paids estimate of loss reserve 1,000M
• Prediction CoV 5.3
• Incurreds estimate of loss reserve 1,040M
• Prediction CoV 5.3

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Adaptive reserving
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Static and dynamic models

• Return for a while to models based on aggregate
  (not individual claim) data
• Model form is still Yf(ß)e
• Example
• j accident quarter
• k development quarter
• EYjk a kb exp(-ck) exp aßln k - ?k
• (Hoerl curve for each accident period)

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Static and dynamic models (contd)

• Example
• EYjk a kb exp(-ck) exp aßln k - ?k
• Parameters are fixed
• This is a static model
• But parameters a, ß, ? may vary (evolve) over
  time, e.g. with accident period
• Then
• EYjk exp a(j)ß(j) ln k - ?(j) k
• This is a dynamic model, or adaptive model

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Illustrative example of evolving parameters

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Formal statement of dynamic model

• Suppose parameter evolution takes place over
  accident periods
• Y(j)f(ß(j)) e(j) observation equation
• ß(j) u(ß(j-1)) ?(j) system equation
• Let (js) denote an estimate of ß(j) based on
  only information up to time s

Some function
Centred stochastic perturbation
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Adaptive reserving

q-th diagonal
(1q)
(2q)
Forecast at valuation date q
(qq)
34
Adaptive reserving (contd)

• Reserving by means of an adaptive model is
  adaptive reserving
• Parameter estimates evolve over time
• Fitted model evolves over time
• The objective here is robotic reserving in
  which the fitted model changes to match changes
  in the data
• This would replace the famous actuarial
  judgmental selection of model

35
Special case of dynamic model DGLM

• Y(j)f(ß(j)) e(j) observation equation
• ß(j) u(ß(j-1)) ?(j) system equation
• Special case
• f(ß(j)) h-1(X(j) ß(j)) for matrix X(j)
• e(j) has a distribution from the exponential
  dispersion family
• Each observation equation denotes a GLM
• Link function h
• Design matrix X(j)
• Whole system called a Dynamic Generalised Linear
  Model (DGLM)

36
Adaptive form of individual claim models

• Individual claim models can also be converted to
  adaptive form
• Just subject parameters to evolutionary model
• We have experimented with this type of model and
  adaptive reserving
• Moderately successful

37
Conclusions

• Effective forecast of costs of small samples of
  claims requires individual claim models
• Such models condition the forecasts on much more
  information than aggregate models
• Even for large samples, individual claim models
  may yield considerably more efficient forecasts
• Lower coefficient of variation
• This may save real money
• Lower uncertainty implies lower capitalisation
• Adaptive forms of individual claim models may
  further improve the tracking of claims experience
  over time