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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
1
Overview
• Individual claim models
• “Paids” models
• “Incurreds” models
• Numerical results
• Adaptive models
2
Why individual claim models?
3
Example problem
• Classical workers compensation cost centre
allocation problem
Total claim
cost
Cost centre
1
…
...
Cost centre
2
Cost centre
m
…
• Claim numbers at the leaves of this tree may be small
…
4
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
5
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
ūi = Σ ui /n = Σ {E[ui] + ui - E[ui]}/n
= E[ui] + Σ {ui - E[ui]}/n
d
 E[ui] as n∞
by the large of large numbers
6
Measuring claims performance in
large portfolio segments (cont’d)
d
ūi  E[ui] as n∞
• E[ui] = 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 E[ui] may be a good approximation to ūi for
large sample sizes, it may be very poor for small
ones
• Leading to a highly distorted cost allocation
7
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
8
There is a need to change from
this…
Data
Model
Fitted
Forecast
Forecast
Conventional actuarial analysis of loss experience
• Call such models “aggregate models”
9
…to this
Claim 1
Claim 1
Claim 2
Claim 2
Claim 3
Claim 1
Forecast
Claim 2
Claim 3
Claim 3
:
:
:
:
:
:
:
:
:
Claim n
Claim n
Claim n
Model
Special case of individual claim reserving – statistical
case estimation
10
Individual claim models
11
Form of such a model
Claim 1
Claim 1
Claim 2
Claim 2
Claim 3
Model
:
:
Claim 3
:
Y=f(β)+ε
Forecast
g(̂ )
Claim 1
Claim 2
Claim 3
:
:
:
:
:
:
Claim n
Claim n
Claim n
12
Form of such a model
Claim 1
Claim 1
Claim 2
Claim 2
Claim 3
Model
Claim 3
:
:
:
Y=f(β)+ε
Forecast
g(̂ )
Claim 1
Claim 2
Claim 3
:
:
:
:
:
:
Claim n
Claim n
Claim n
Yi = f(Xi; β) + εi
Yi = size of i-th completed claim
Xi = vector of attributes (covariates) of i-th claim
β = vector of parameters that apply to all claims
εi = vector of centred stochastic error terms
13
Form of individual claim model
Yi = f(Xi; β) + εi
• Convenient practical form is
Yi = h-1(XiT β) + εi
h = link function
Linear predictor =
linear function of the
parameter vector
[GLM form]
Error distribution
from exponential
dispersion family
14
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; β) + εjk
for
j = accident period
k= development period
Not always
formulated
• Compare with
Yi = f(Xi; β) + εi
15
Form of “paids” individual claim
model
• Possible to mimic aggregate model by
defining individual model as just
Yi = h-1(ji,ki; β) + εi
16
Form of “paids” individual claim
model
• Possible to mimic aggregate model by
defining individual model as just
Yi = h-1(ji,ki; β) + εi
• But often possible to improve on this, e.g.
• Replace development period j with operational
time ti (proportion of accident period’s incurred
claims completed) at completion of i-th claim
• Example
Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi
17
Example of “paids” individual claim
model
Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi
E[Yi] = exp [β0+β1ti+β2max(0,ti-0.8)]
Expected claim size as a function of operational time
Linear predictor of expected claim size as a function of operational
time
$70,000
12
$60,000
10
$50,000
8
Operational time
Operational time
18
1
0.96
0.92
0.88
0.8
0.84
0.76
0.72
0.68
0.6
0.64
0.56
0.52
0.48
0.4
0.44
0.36
0.32
0.28
0.2
0.24
0.16
0.12
0.08
0
1
0.96
0.92
0.88
0.8
0.84
0.76
0.72
0.68
0.6
0.64
0.56
0.52
0.48
0.4
0.44
0.36
0.32
0.28
$0
0.24
0
0.2
$10,000
0.16
2
0.12
$20,000
0.08
4
0
$30,000
0.04
6
0.04
$40,000
Example of “paids” individual claim
model (cont’d)
Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi
• Include superimposed inflation
• Let q=j+k=calendar period of claim
completion
• Extend model
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi
• Superimposed inflation at rate exp[β3] per
period
19
Example of “paids” individual claim
model (cont’d)
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi
• We might wish to model superimposed inflation as
beginning at period q=q0
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi
20
Example of “paids” individual claim
model (cont’d)
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi
• We might wish to model superimposed inflation as
beginning at period q=q0
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi
• …and we might wish to model superimposed
inflation with a rate that decreases with increasing
operational time
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi
etc etc
21
Example of “paids” individual claim
model (cont’d)
• “Paids” estimate of
250
Mack
Outstanding Liability ($M)
200
Mack -1 s.d
Mack +1 s.d
Time Paids
150
•
Time Paids -1 s.d
Time Paids +1 s.d
100
•
50
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Accident Year
•
•
loss reserve scaled to
baseline $1,000M
Prediction CoV =
5.3%
Mack (incurreds)
estimate is $887M
with CoV = 10.5%
Mack estimate
produces negative
reserves for the old
years of origin
“Paids” chain ladder
fails completely 22
Example of “paids” individual claim
model (cont’d)
• This model is very
economical
• Contains only 9
parameters to
represent many
thousands of claims
Name
Mean
l_Qtrly optime(^1)
Value
Standard Error Standard Error (%)
9.755
0.18697
1.9
0.0086
0.00167
19.5
-0.0158
0.00049
3.1
l_Finalisation year<1981(^1)
0.1917
0.02029
10.6
l_Finalisation year1981-87(^1)
0.0795
0.00844
10.6
l_Finalisation year1987-90(^1)
0.0244
0.01281
52.4
l_Finalisation year>1990(^1)
0.0386
0.00379
9.8
l-Qtrly optime<70(^1)
0.0034
0.00204
60.3
l_Finalisation year>1990(^1).l-Qtrly optime<70(^1)
0.0008
0.0001
12.7
l_Qtrly optime<12(^2)
23
Further extension of “paids”
individual claim model
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi
• May include claim characteristics other than timerelated, e.g.
• Nature of injury
• Claim severity (MAIS scale)
• Pre-injury earnings
Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0) +
more terms] + εi
24
Example of “incurreds” individual
claim model
• Similar to “paids” model
• Basic set-up is still
Yi = h-1(ji,ki,qi,ti,other;β) + εi
• Example
Yi = exp(Ci,ji,ki,qi,ti,other;β) + εi
where Ci = current manual estimate of
incurred cost of i-th claim
25
Example of “incurreds” individual
claim model (cont’d)
• In fact, the model requires more structure than this because
of claims and estimates for nil cost
• Let (for an individual claim)
• U = ultimate incurred (may = 0)
• C = current estimate (may = 0)
• X = other claim characteristics
Model of
Prob[U=0|C,X]
Prob[U=0]
Prob[U>0]
Model of
U|U>0,C=0,X
If
C=0
If
C>0
Model of
U/C|U>0,C>0,X
26
Example of “incurreds” individual
claim model (cont’d)
• “Paids” estimate of
1,000
Outstanding Liability ($M)
100
10
1
Time Paids
Time Paids -1 s.d
0
Time Paids +1 s.d
Incurreds
Incurreds -1 s.d
0
Incurreds +1 s.d
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Accident Year
loss reserve =
$1,000M
• Prediction CoV =
5.3%
• “Incurreds” estimate
of loss reserve =
$1,040M
• Prediction CoV =
5.3%
27
Adaptive reserving
28
Static and dynamic models
• Return for a while to models based on
aggregate (not individual claim) data
• Model form is still Y=f(β)+ε
• Example
• j = accident quarter
• k = development quarter
• E[Yjk] = a kb exp(-ck) = exp [α+βln k - γk]
• (Hoerl curve for each accident period)
29
Static and dynamic models (cont’d)
• Example
E[Yjk] = a kb exp(-ck) = exp [α+βln k - γk]
• Parameters are fixed
• This is a static model
But parameters α, β, γ may vary (evolve) over
time, e.g. with accident period
Then
• E[Yjk] = exp [α(j)+β(j) ln k - γ(j) k]
• This is a dynamic model, or adaptive model
30
Illustrative example of evolving
parameters
Separate curves represent different accident periods
70
60
50
40
30
20
10
0
1
3
5 7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59
Development period
31
Formal statement of dynamic model
• Suppose parameter evolution takes place over
accident periods
• Y(j)=f(β(j)) +ε(j)
• β(j) = u(β(j-1)) + ξ(j)
Some function
[observation equation]
[system equation]
Centred stochastic perturbation
• Let ̂(j|s) denote an estimate of β(j) based on only
information up to time s
32
Adaptive reserving
q-th diagonal
̂(1|q)
̂ (2|q)
̂(q|q)
̂
Forecast at
valuation date
q
33
Adaptive reserving (cont’d)
• 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
34
Special case of dynamic model:
DGLM
• Y(j)=f(β(j)) +ε(j)
• β(j) = u(β(j-1)) + ξ(j)
• Special case:
[observation equation]
[system equation]
• f(β(j)) = h-1(X(j) β(j)) for matrix X(j)
• ε(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)
35
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
36
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
37