this presentation has been prepared for the 2016 general … · 2016-11-13 · this presentation...
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This presentation has been prepared for the 2016 General Insurance Seminar. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily
those of the Institute and the Council is not responsible for those opinions.
This presentation has been prepared for the 2016 General Insurance Seminar. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily
those of the Institute and the Council is not responsible for those opinions.
Self-assembling insurance claim models
Greg Taylor School of Risk and Actuarial Studies,
UNSW Australia Sydney, Australia
Gráinne McGuire Hugh Miller
Taylor Fry Analytics & Actuarial Consulting, Sydney, Australia
Agenda 1. Motivation
2. Regularized regression for GLMs
3. Simulated data examples
4. Real data example
5. Conclusions & future work
Motivation Problem: Conventional loss reserving with triangles
e.g. paid losses, incurred losses, etc.
One option: chain ladder model
• Simple structure
• Sometimes inadequate in capturing effects – ad hoc changes
Another option: Generalized linear models (GLMs) McGuire, 2007; Taylor & McGuire, 2004, 2016
• Time-consuming and expensive
Is there a (better) third way?
Regularized regression for GLMs
GLM estimator is defined as:
𝒚𝒊 = 𝒉−𝟏 𝑿𝒊𝑻𝜷 + 𝜺𝒊
𝑦𝑖
ℎ−1
𝑋𝑖𝑇𝛽
𝜀𝑖
The regularized fit then minimises:
−ℓ 𝒚𝒊; 𝑿𝒊, 𝜷
𝒊
+ 𝝀 𝜷𝒋𝒑
𝒋
Response variable: Here the observed values in the upper triangle
Link function: Converts full number line to restricted domain (e.g. positive values)
Linear predictor: sum of input variables (e.g. Dev. year effects)
Stochastic error (exponential family)
Negative log-likelihood: As per standard GLM. Is squared loss for Normal distribution.
Penalty term: Introduces a bias against lots of large parameter values. Experiments today use 𝑝 = 1 (lasso)
Tuning constant: balances the two terms.
• Three simulated incremental quarterly paid claim triangles, 40x40
• Potential accident qtr (𝑘), development qtr (𝑗) and payment quarter
(𝑡 = 𝑘 + 𝑗 − 1) effects
• 𝜇𝑘𝑗 = 𝐸 𝑌𝑘𝑗 , 𝜎𝑘𝑗2 = 𝑉𝑎𝑟 𝑌𝑘𝑗 with:
– 𝑙𝑛 𝜇𝑘𝑗 = 𝛼𝑘 + 𝛽𝑗 + 𝛾𝑡
– 𝜎𝑘𝑗2 consistent with the Mack model for chain ladders
• Basis functions are a large set of:
Simulated data examples
Steps & Ramps
The general process is:
• Generate the data (upper and lower triangles)
• Define family of basis functions
• Define the rule for ‘best’ model – usually lowest error on 8-fold cross-
validation on the upper triangle
• Repeat lots of times (re-generate the triangle 100 times, say)
We compare the forecast payments (lower triangle) to true answer, as well
as standard chain ladder results
Simulated data examples (2)
• Observations simulated as 𝑌𝑘𝑗~𝑂𝐷𝑃 𝜇𝑘𝑗 , 𝜙 .
• Main effect setup with Hoerl curve for DQ, Linear
splines for AQ. No SI
• 2,400 basis functions (AQ and DQ ramps, plus step
interactions)
Dataset 1: Basic Recall 𝑙𝑛 𝜇𝑘𝑗 = 𝛼𝑘 + 𝛽𝑗 + 𝛾𝑡
0 5 10 15 20 25 30 35 40
AQ (log) effect αk
0 5 10 15 20 25 30 35 40
DQ (log) effect βk
Selected model has 152 nonzero parameters
Results 1
0
100
200
300
400
500
600
700
800
900
1000
0
1
10
100
1,000
10,000
100,000
1,000,000
1 41 81 121
nu
mb
er
of
pa
ram
ete
rs in
mo
de
l
Erro
r sc
ore
Model number
Error in train and test data sets for each model
trainerror testerror AIC CV error nparams
17
18
19
20
21
0 5 10 15 20 25 30 35 40
Ln(a
mo
un
t)
Accident quarter
AQ effect when DQ = 20
True mean
Actuals in sim
Lasso fit
ProjectionHistorical
12
14
16
18
20
22
0 5 10 15 20 25 30 35 40
Ln(a
mo
un
t)
Development quarter
DQ effect when AQ = 20
True mean
Actuals in sim
Lasso fit
ProjectionHistorical
Linear predictor tracking
Results 1
2
4
6
8
10
12
14
0 5 10 15 20 25 30 35 40
Ou
tsta
nd
ing
p
ay
me
nts
($
b)
Accident quarter
True outs
8qtr Chain ladder
Lasso
AQ reserve – single simulation
Distribution of reserve – 1,000 sims
• A per #1 but with a superimposed inflation effect
• About 3,200 basis functions (extra PQ ramps)
• An unpenalised constant SI term 𝛾𝑡 = 𝑡
Dataset 2: Extra SI
0 5 10 15 20 25 30 35 40
SI effect γk
• Tracking still looks good
• Lasso less biased and more stable,
albeit with a few outliers
• 84 non-zero terms
Results 2
• Setup as per #2 but with an interaction:
– LP ↑ by 0.3 if 𝑘 > 16 and 𝑗 > 20
• Big reform, but have to ‘pick’ it from 10
cells out of 820
• Same basis functions as #2
Dataset 3: A tricky reform
j>20
k>16
Results 3
16.0
17.0
18.0
19.0
20.0
21.0
22.0
0 10 20 30 40
True mean Sim actual lasso
DQ tracking at AQ 25 (linear predictor)
DQ tracking surprisingly accurate
AQ tracking at DQ 25
Though under-estimation of tail at higher AQs. Conflation of PQ-AQ effects
12.0
14.0
16.0
18.0
20.0
22.0
24.0
0 10 20 30 40
True mean Sim actual lasso
Results 3 Loss reserve by AQ, single sim
CL under-estimation even more severe
Both approaches are quite volatile
100 sims
0
5
10
15
20
25
30
35
40
0 10 20 30 40
Out
stan
ding
pay
men
ts ($
b)
Accident period
Actual 8 period CL Lasso
Results 3 Chain ladder
Stability of lasso even more pronounced at AQ basis (100 sims)
Lasso
Results 3
Unanticipated
Anticipated
0
5
10
15
20
25
30
35
40
0 10 20 30 40
Out
stan
ding
pay
men
ts ($
b)
Accident period
Actual 8 period CL Lasso
0
5
10
15
20
25
30
35
40
0 10 20 30 40
Out
stan
ding
pay
men
ts ($
b)
Accident period
Actual 8 period CL Lasso
Part of the issue is the penalisation of the (𝑗 > 20) 𝑘 > 16 interaction. If the reform is anticipated, could apply no penalty to this particular basis function
• Motor bodily injury. Average finalisation size in 37 x 37 triangle
• Main features:
– Major reform in quarter 35 (cost and development speed) results for 34x34 triangle shown here
– Material SI trends (negative followed by positive)
• Use operational time (OT) instead of DQ. Proportion of claims in AQ closed.
• Data used in CAS paper (Taylor and McGuire, 2004) – detailed GLM fit
Real data example
Early investigations found that
• Need to balance the OT and PQ effects via standardisation
• Need some PQ and OT interactions … but not too many
• Also need some AQ and OT interactions
• SI extrapolation still an issue requiring judgement
Real data example (2)
Real data - fit
Actual versus expected
Main effect Min 1se
PQ 14 1
OT 15 2
AQ*OT 12 0
PQ*OT 10 0
Total (incl intercept) 52 4
# Parameters
Real data – fit
1 6 11 16 21 26 31
Accident quarter
actual min 1se
1 6 11 16 21 26 31
Payment quarter
actual min 1se
1 6 11 16 21 26 31
Development quarter
actual min 1se
Marginal totals AvE
Real data – projection
Assumes 1% SI – choice of future SI still requires judgement.
Also performed some testing for the post-reform quarters. Able to accept many of the required shape changes.
1 6 11 16 21 26 31
Ou
tsta
nd
ing
rese
rve
($
m)
Accident quarter
lasso.min
CL 8
GLM
• The lasso appears promising as a platform for self-assembling models
– Time efficient (~1 hour for routine calibration, diagnostics and one or two ad hoc changes e.g. superimposed inflation, legislative change)
– Plausible extrapolation on the lower half of the triangle
– Good flexibility
– Good theoretical basis
• The lasso appears to track eccentric features of the data reasonably well
– Including in scenarios where the chain ladder has little hope of an accurate forecast
• Further experimentation required
Conclusions
• Further scenarios
• Different basis functions: Hoerl curve basis functions for DQ effects, or standardisation variants
• Bayesian lasso formulations
• Robustification and choice of optimal model
• Multi-line reserving (with dependencies)
• Adaptive reserving
Future work
• McGuire, G. 2007. “Individual Claim Modelling of CTP Data.” Institute of Actuaries of Australia XIth Accident Compensation Seminar, Melbourne, Australia. http://actuaries.asn.au/Library/6.a_ACS07_paper_McGuire_Individual%20claim%20modellingof%20CTP%20data.pdf
• Taylor, G., and G. McGuire. 2004. “Loss Reserving with GLMs: A Case Study.” Casualty Actuarial Society 2004 Discussion Paper Program, pp 327-392.
• Taylor, G., and G. McGuire. 2016. “Stochastic Loss Reserving Using Generalized Linear Models”. CAS Monograph Series, Number 3. Casualty Actuarial Society, Arlington VA.
• Detailed version of this presentation (CAS Loss reserving seminar, 2016)
http://cas2016clrsiframe.azurewebsites.net/SessionDetail.aspx?id=50820
References
This presentation has been prepared for the 2016 General Insurance Seminar. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily
those of the Institute and the Council is not responsible for those opinions.
Thanks for
listening!