this presentation has been prepared for the 2016 general … · 2016-11-13 · this presentation...

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!