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Compartmental Reserving
a process-based Bayesian reserving approach
Jake Morris Rob Murray
22 October 2015 1
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• Background– Motivations
• Methodology
• Case studies– Semi-Bayesian– Fully Bayesian
• Conclusions
Agenda
2
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• Intuitive parameters including case reserve robustness measure
• Provides coherent measure of reserve uncertainty
• Supports negative development
• Can capture calendar effects
• Independent of DFM / BF
• Incorporates judgement
OverviewFeatures
3
Models the claims generation process
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BackgroundMotivations
4
1. Parsimony– extract signal from noise– description of individual cohort vs. average
2. Quantification of reserve uncertainty– incorporate multiple information sources– isolate drivers of uncertainty
3. Interpretability & Extensibility– meaningful parameters– option to capture specific process features Model
P3P2 P1
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Mixed-effects / hierarchical modelling
*Also known as a mixture of random effects and fixed effects
Cohorts
Parameters a mixture of those varying across cohort
and those not*
Only estimate mean and s.d. of the variable parameters
Cohort P1 P2 P3 P4
1 P1,1
P2
P3,1
P4
2 P1,2 P3,2
… … …
N P1,N P3,N
Background1. Parsimony
5
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Background2. Reserve uncertainty
6
)();()|( θθθ pyLyp ∝Posterior ∝ Likelihood x Prior
)();()|( ULRpincurredULRLincurredULRp ∝
Objective:
Bayes’ theorem:
“Given any value (estimate of future payments) andour current state of knowledge, what is the probability that
final payments will be no larger than the given value?”
- Casualty Actuarial Society (2004)Working Party on Quantifying Variability in Reserve Estimates
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• These concepts have been applied to loss reserving:
BackgroundExisting research
7Key idea: fit a nonlinear parametric curve to cumulative paid triangles in a mixed-effects modelling framework
A Bayesian non-linear model for forecasting insurance loss payments
Yanwei Zhang Vanja Dukic James Guszcza
………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………………………………………
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Background3. Interpretability/Extensibility
Meaningful parameters and extensibility
Reformulate to model claims generation process8
• Models provide control over complexity (vs. methods)• Drug developers use modelling & simulation to predict exposure/response:
“Compartmental” Pharmacokinetic models
Blood TissuesDrug dose
Con
cent
ratio
n
Time
Pharmacokinetics
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*ODEs: a collection of simultaneous Ordinary Differential Equations
MethodologyCompartmental loss reserving model
Structural model
• Cash flows between compartments governed by ODEs*
• Fit to paid and outstanding triangles– Simultaneously– Explicitly estimating tails
Premiums Written
9
Supports negative development OS
& P
aid
Cla
ims
Development period
OS Cumulative Paid
Exposed to risk
Claims reported
Claimspaid
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MethodologyParameters
Parameters have natural interpretations
Reported loss ratio (“RLR”)
Rate of earning + reporting (“ker”)
Reserve robustness factor (“RRF”)
Rate of payment (“kp”)
RLR RRF
ker kp
ULR = RLR*RRF
Estimate parameters in a mixed-effects framework
Base model parameters for a single cohort
Premiums Written
10
Exposed to risk
Claims reported
Claimspaid
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MethodologyRates → Patterns
11
EtR to Reported Pattern Reported to Paid Pattern
t
ker
t
kp
RLR RRF
ker kp
Premiums Written
t = development time
Pattern % = 1 – e-rate*t
Exposed to risk
Claims reported
Claimspaid
t t
% %
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MethodologyRates → Patterns
12
EtR to Reported Pattern Reported to Paid Pattern
t
ker
t
kp
RLR RRF
ker kp
Premiums Written
t = development time
Pattern % = 1 – e-rate(t)*t
Exposed to risk
Claims reported
Claimspaid
t t
% %
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Case Study 1LSM
• Class X – Underwriting cohorts (1 – 5)– Ultimate premiums & writing patterns– Paid and incurred claims development
• Objectives– Fit semi-Bayesian* compartmental model– Extrapolate fits to ultimate– Compare ULRs to LSM
13*Specify parameter starting values but not distributions
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Case Study 1Selected model
14
t
ker
Exposed to risk
Claims reported
Claimspaid
RLR RRF
ker kp
U/W cohort model:
t
kp
Premiums Written
Variable rates Full random effects structureCohort RLR ker RRF kp
1 RLR1 ker1 RRF1 kp1
2 RLR2 ker2 RRF2 kp2
… … … … …
5 RLR5 ker5 RRF5 kp5
Fit model and explore diagnostics…
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Case Study 1Model diagnostics
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Case Study 1O/S fits
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Case Study 1Paid fits
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Case Study 1Incurred fits
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Case Study 1Incurred fits
19
Signal or Noise?
BCL assumes signal
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Case Study 1Results Summary
20
-1%
0%
1%
2%
3%
4%
1 2 3 4 5 1 - 5
ULR
Diff
eren
ce (%
)
UW Cohort
Comp Res ULR - LSM ULR
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Case Study 2Why do Bayesian data analysis?
21
“Modern Bayesian methods provide richer information, with greater flexibility and broader applicability than 20th
century methods.
Bayesian methods are intellectually coherent and intuitive …[and] readily computed...”
- John K. Kruschke Open Letter extolling the benefits of the Bayesian approach
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Case Study 2Why do Bayesian loss reserving?
22
1) Estimate full probability distributions of quantities of interest:
2) Incorporate judgement:
3) Model structure flexibility*:
“Given our current state of knowledge…”
Alternative distributions
Calendar effects
Autocorrelation
External information
ULRBF = f(incurred, exposure, IELR)
*Provided by Gibbs sampling
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Case Study 2Data & Objectives
• Workers’ Comp Schedule P data– Accident year cohorts (1988 – 1997)– Earned premiums– Paid and incurred claims development
• Objectives– Fit Bayesian compartmental model– Extrapolate fits & posterior predictive intervals
(“PPIs”) to time 10*– Compare fits & PPIs to lower triangles
0102030405060708090
0 1 2 3 4 5 6 7 8 9 10
Incu
rred
Cla
ims
($00
0)
Development Year
Incurred Claims 1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
23*PPIs account for Parameter and Process uncertainty
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Case Study 2Incurred data visualisation
24
Calendar effect?
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Case Study 2Model 1
25
Exposed to risk
Claims reported
Claimspaid
RLR RRF
ker kp
Base model
AY RLR ker RRF kp
1988 RLR1
ker
RRF1
kp
1989 RLR2 RRF2
… … …
1997 RLR10 RRF10
t
ker
t
kp
(extended):
ker
Constant rates 2 random effects
Fit model and explore diagnostics…
Premiums Written
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Case Study 2Model 1 Diagnostics
26
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Case Study 2Model 1 O/S fits
27
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Case Study 2Model 1 paid fits
28
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Case Study 2Model 1 incurred fits
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Case Study 2Model 1 O/S vs hold out sample
30
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Case Study 2Model 1 paid vs hold out sample
31
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Case Study 2Model 1 incurred vs. hold out sample
Under-reserving?Late reporting?
32
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Case Study 2Model 1 Summary (1)
-5
0
5
10
15
20
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997
Diff
eren
ce fr
om T
ime
10 In
curr
ed ($
000)
Accident Year
Actual – Modelled Time 10 Incurred
Model 1 Incurred Chain Ladder
BCL projects over-reserving into the future*
33*BF method used in practice
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0%
20%
40%
60%
80%
100%
120%
140%
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997
Perc
enta
ge
Accident Year
Estimated Reported Loss Ratios and Reserve Robustness Factors
RLRRRF
Case Study 2Model 1 Summary (2)
Model estimates less over-reserving over time…
34
(Case) Reserving Cycle
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0%
20%
40%
60%
80%
100%
120%
140%
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997
Perc
enta
ge
Accident Year
Estimated Reported Loss Ratios and Reserve Robustness Factors
RLRRRF
Case Study 2Model 1 Summary (2)
Model estimates less over-reserving over time……but note under-reserving in 1997!*
*In practice: discuss with case handlers and sensitivity test prediction to changes in RRF 35
(Case) Reserving Cycle
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Case Study 2Model 2
Estimate case reserve % increases/decreases
36
RLR RRF
ker kp
Explicitly model calendar effect:
Case reserve review
C
Full random effects structure
Exposed to risk
Claims reported
Claimspaid
Premiums Written
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Case Study 2Diagnostics: Model 1 vs Model 2
37
21
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Case Study 2Diagnostics: Model 1 vs Model 2
38
Over fitted21
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Case Study 2Model 2 incurred fits
39
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Case Study 2Model 2 incurred vs. hold out sample
40
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Conclusions (1)A modeller’s notes*
*Mike K Smith – Pharmacometrician, Pfizer
• “Fitting nonlinear mixed-effects models can be a tricky (and frustrating) business”– Is the model appropriate?– Convergence ⇏ correctness– Different fitting methods ⇒ different results
• “Model diagnostics are (even more) important for these models”– Does the model describe all cohorts reasonably well?
“There are some general rules for fitting these models……but experience is the best guide”
41
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Conclusions (2)Bayesian compartmental reserving
42
Exposed to risk
Claims reported
Claimspaid
RLR RRF
ker kp
Premiums Written
)();()|( ULRpincurredULRLincurredULRp ∝
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Conclusions (2)Bayesian compartmental reserving
Try it out for yourself!
• Strengths of compartmental reserving:– Independent stochastic method– Meaningful parameters– Parsimonious yet extensible
• Weaknesses of compartmental reserving:– Model shape constraints with volatile data– Sensitivity to starting values / priors– Learning curve
[email protected]@lcp.uk.com
43
supports negative incurred developmentincluding measure of reserve robustness
can capture calendar effects
(strength!)
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Acknowledgements
for sponsoring intermediary development• Independent
Including former colleagues:
• Robert Ruiz– Leading semi-Bayesian mathematical documentation– Advocating the method
• Rob Murray, Charl Cronje, Matthew Pearlman, Richard Holloway, Matt Locke and Charlie Stone– Support at various stages– Sounding boards
44
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Compartmental Reserving
a process-based Bayesian reserving approach
Jake Morris Rob Murray
22 October 2015 45
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Appendix
46
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Mixed-effects / hierarchical modelling
*Also known as a mixture of random effects and fixed effects
Cohorts
Parameters a mixture of those varying across cohort
and those not*
Only estimate mean and s.d. of the variable parameters
Cohort P1 P2 P3 P4
1 P1,1
P2
P3,1
P4
2 P1,2 P3,2
… … …
N P1,N P3,N
Background1. Parsimony
47
“BorrowingStrength”
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ImplementationThe modelling cycle
48
(re)Define model
Specify starting values*
Run model
Examine diagnostics
*/prior distributions
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ImplementationA departure from Excel
• Nonlinear mixed-effects models require complex solver algorithms:
• “f” is derived by solving ODEs:
Response y {OS,PD}=
Non-linear function f of (Parameter vector ϕ and time t)
+ Noise w
dEX/dt = -ker · EXdOS/dt = ker · RLR · EX - kp · OS
dPD/dt = kp · RRF · OS
Exposedto risk
Claims reported
Claimspaid
Semi-Bayesian:
∏∏∫∈ ∈
ℜ∈⋅⋅
=ℑ
Ii Ccb
iiiciSizePi dbbpdfbypdf
L
)(
0
)()()(0
),(
)(0
),|(),,|)((
)|,,(
σησβω
σηβ ω
EX OS PD
We don’t have to worry about this!
49
Fully Bayesian:
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Case Study 2Bayesian loss reserving in practice
50
• Autoregressive sub-models– for consecutive under/over fits
• Log-normal distributions– for claims process parameters
• Prior distributions – for all other uncertain parameters
ULRi
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ImplementationSemi-Bayesian
Cohort RLR ker RRF kp
1 RLR1
ker
RRF1
kp
2 RLR2 RRF2
… … …
N RLRN RRFN
51
t*
ker
Judgementally select parameter starting values
Exposed to risk
Claims reported
Claimspaid
RLR RRF
ker kp
Base model:
t
kp
*Development time
Semi-Bayesian
Constant rates 2 random effects
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AY RLR ker RRF kp
1988 RLR1
ker
RRF1
kp
1989 RLR2 RRF2
… … …
1997 RLR10 RRF10
Case Study 2Model 1.5
Fit new model and explore diagnostics
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Exposed to risk
Claims reported
Claimspaid
RLR RRF
ker kp
Base model
AY RLR ker RRF kp
1988 RLR1 ker1 RRF1 kp1
1989 RLR2 ker2 RRF2 kp2
… … … … …
1997 RLR10 ker10 RRF10 kp10
t
ker
t
kp
(extended):
4 random effects2 random effects
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Case Study 2Model 1.5 Diagnostics
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Reject model?
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Case Study 2Model 1.5 O/S vs hold out sample
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Case Study 2Model 1.5 paid vs hold out sample
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Case Study 2Model 1.5 incurred vs. hold out sample
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Case Study 2Model 2 O/S vs hold out sample
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Case Study 2Model 2 paid vs hold out sample
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