Stochastic Loss Reserving with Bayesian MCMC Models...w w w . I C A 2 0 1 4 . o r g Stochastic Loss...
Transcript of Stochastic Loss Reserving with Bayesian MCMC Models...w w w . I C A 2 0 1 4 . o r g Stochastic Loss...
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w w w . I C A 2 0 1 4 . o r g
Stochastic Loss Reserving with Bayesian MCMC Models
Revised March 31 Glenn Meyers – FCAS, MAAA, CERA, Ph.D.
April 2, 2014
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The CAS Loss Reserve Database Created by Meyers and Shi With Permission of American NAIC
• Schedule P (Data from Parts 1-‐4) for several US Insurers – Private Passenger Auto – Commercial Auto – Workers’ CompensaEon – General Liability – Product Liability – Medical MalpracEce (Claims Made)
• Available on CAS Website hKp://www.casact.org/research/index.cfm?fa=loss_reserves_data
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Illustrative Insurer – Incurred Losses
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IllustraEve Insurer – Paid Losses
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Criteria for a “Good” Stochastic Loss Reserve Model • Using the upper triangle “training” data,
predict the distribution of the outcomes in the lower triangle – Can be observations from individual (AY, Lag)
cells or sums of observations in different (AY,Lag) cells.
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Criteria for a “Good” Stochastic Loss Reserve Model
• Using the predictive distributions, find the percentiles of the outcome data.
• The percentiles should be uniformly distributed. – Histograms – PP Plots and Kolmogorov Smirnov Tests
• Plot Expected vs Predicted Percentiles • KS 95% critical values = 19.2 for n = 50 and 9.6 for n = 200
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IllustraEve Tests of Uniformity
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• List of insurers available from me. • 50 Insurers from four lines of business
– Commercial Auto – Personal Auto – Workers’ CompensaEon – Other Liability
• Criteria for SelecEon – All 10 years of data available – Stability of earned premium and net to direct premium raEo
• Both paid and incurred losses
Data Used in Study
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Test of Mack Model on Incurred Data
Conclusion – The Mack model predicts tails that are too light.
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Test of Mack Model on Paid Data
Conclusion – The Mack model is biased upward.
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Test of Bootstrap ODP on Paid Data
Conclusion – The Bootstrap ODP model is biased upward.
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• The “Black Swans” got us again! – We do the best we can in building our models, but the real world keeps throwing curve balls at us.
– Every few years, the world gives us a unique “black swan” event.
• Build a beKer model. – Use a model, or data, that sees the “black swans.”
Possible Responses to the Model Failures
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Proposed New Models are Bayesian MCMC
• Bayesian MCMC models generate arbitrarily large samples from a posterior distribution.
• See the limited attendance seminar tomorrow at 1pm.
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• w = Accident Year w = 1,…,10 • d = Development Year d = 1,…,10 • Cw,d = CumulaEve (either incurred or paid) loss • Iw,d = Incremental paid loss = Cw,d – Cw-‐1,d
NotaEon
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• Use R and JAGS (Just Another Gibbs Sampler) packages
• Get a sample of 10,000 parameter sets from the posterior distribuEon of the model
• Use the parameter sets to get 10,000, , simulated outcomes
• Calculate summary staEsEcs of the simulated outcomes – Mean – Standard DeviaEon – PercenEle of Actual Outcome
Bayesian MCMC Models
!!Cwd
w=1
10
∑
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The Correlated Chain Ladder (CCL) Model
• logelr ~ uniform(-5,0) • αw ~ normal(log(Premiumw)+logelr, ) • β10 = 0, βd ~ uniform(-5,5), for d=1,…,9 • ai ~ uniform(0,1)
• Forces σd to decrease as d increases
• µ1,d = α1 + βd • C1,d ~ lognormal(µ1,d, σd) • ρd ~ uniform(-1,1) • µw,d = αw + βd + ρd·(log(Cw-1,d) – µw-1,d) for w = 2,…,10 • Cw,d ~ lognormal(µw,d, σd)
σd = ai
i=d
10
∑
! 10
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Predicting the Distribution of Outcomes
• Use JAGS software to produce a sample of 10,000 {αw}, {βd},{σd} and {ρ} from the posterior distribution.
• For each member of the sample – µ1,10 = α1 + β10 – For w = 2 to 10
• Cw,10 ~ lognormal (αw + β10 + ρd·(log(Cw-1,10) – µw-1,10)),σ10)
– Calculate
• Calculate summary statistics, e.g.
• Calculate the percentile of the actual outcome by counting how many of the simulated outcomes are below the actual outcome.
=∑10
,101
ww
C
!!E Cw ,10
w=1
10
∑"
#$
%
&'!and!Var Cw ,10
w=1
10
∑"
#$
%
&'
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Done in JAGS
The First 5 of 10,000 Samples on Illustrative Insurer with ρd = ρ
Done in R
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The Correlated Chain Ladder Model Predicts Distributions with Thicker Tails
• Mack uses point estimations of parameters. • CCL uses Bayesian estimation to get a
posterior distribution of parameters. • Chain ladder applies factors to last fixed
observation. • CCL uses uncertain “level” parameters for
each accident year. • Mack assumes independence between
accident years. • CCL allows for correlation between
accident years, – Corr[log(Cw-1,d),log(Cw,d)] = ρd
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Examine Three Behaviors of ρd
1. ρd = 0 - Leveled Chain Ladder (LCL)
2. ρd = ρ ~ uniform (-1,1) (CCL)
3. ρd = r0�exp(r1�(d-1)) (CCL Variable ρ) – r0 ~ uniform (0,1) – r1 ~ uniform (-log(10) –r0, –log(r0)/9) – This makes ρd monotonic (0,1) ∈
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Case 2 - Posterior Distribution of ρ for Illustrative Insurer
ρ
Frequency
−1.0 −0.5 0.0 0.5 1.0
01000
ρ is highly uncertain, but in general positive.
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Generally Positive Posterior Means of ρ for all Insurers
Commercial Auto
Mean ρ
Frequen
cy
−1.0 −0.5 0.0 0.5 1.0
04
812
Personal Auto
Mean ρ
Frequen
cy
−1.0 −0.5 0.0 0.5 1.0
04
812
Workers' Compensation
Mean ρ
Frequen
cy
−1.0 −0.5 0.0 0.5 1.0
05
10
Other Liability
Mean ρ
Frequen
cy
−1.0 −0.5 0.0 0.5 1.0
05
1015
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Case 3 - Posterior Distributions of r0 and r1 - ρd = r0�exp(r1�(d-1))
ρd > 0
ρd is generally monotonic decreasing Illustrative
Insurer
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Generally Monotonic Decreasing ρd for all Insurers
Commercial Auto
Mean r1
Freq
uenc
y
−5.0 −4.5 −4.0 −3.5 −3.0
08
Personal Auto
Mean r1
Freq
uenc
y
−5.0 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0
015
Workers' Compensation
Mean r1
Freq
uenc
y
−5.0 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5
015
Other Liability
Mean r1
Freq
uenc
y
−5.0 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5
015
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Results for the Illustrative Insured With Incurred Data
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Results for the Illustrative Insured With Incurred Data
Rank of Std Errors Mack < LCL < CCL-VR ≈ CCL-CR
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Compare SDs for All 200 Triangles
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Test of Mack Model on Incurred Data
Conclusion – The Mack model predicts tails that are too light.
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Test of CCL (LCL) Model on Incurred Data ρd = 0
Conclusion – Predicted tails are too light
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Test of CCL Model on Incurred Data ρd = ρ
Conclusion – Plot is within KS Boundaries
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Test of CCL Model on Incurred Data Variable ρd
Conclusion – Plot is within KS Boundaries
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• Accomplished by “pumping up” the variance of Mack model.
What About Paid Data? • Start by looking at CCL model on
cumulative paid data.
Improvement with Incurred Data
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Test of Bootstrap ODP on Paid Data
Conclusion – The Bootstrap ODP model is biased upward.
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Test of CCL on Paid Data
Conclusion Roughly the same performance as bootstrapping
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• Look at models with payment year trend. – Ben Zehnwirth has been championing these
for years. • Payment year trend does not make sense
with cumulative data! – Settled claims are unaffected by trend.
• Recurring problem with incremental data – Negatives! – We need a skewed distribution that has
support over the entire real line.
How Do We Correct the Bias?
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X ~ Normal(Z,δ), Z ~ Lognormal(µ,σ)
The Lognormal-Normal (ln-n) Mixture
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• µw,d = αw + βd + τ·(w + d – 1) • Zw,d ~ lognormal(µw,d, σd) subject to σ1<σ2< …<σ10 • I1,d ~ normal(Z1,d, δ) • Iw,d ~ normal(Zw,d + ρ·(Iw-1,d – Zw-1,d)·eτ, δ)
• Estimate the distribution of • “Sensible” priors
– Needed to control σd – Interaction between τ , αw and βd.
The Correlated Incremental Trend (CIT) Model
=∑10
,101
ww
C
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CIT Model for Illustrative Insurer
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Posterior Mean τ for All Insurers Commercial Auto
Mean τ
Freque
ncy
−0.06 −0.04 −0.02 0.00 0.02
05
15
Personal Auto
Mean τ
Freque
ncy
−0.06 −0.04 −0.02 0.00 0.02
010
20
Workers' Compensation
Mean τ
Freque
ncy
−0.06 −0.04 −0.02 0.00 0.02
05
10
Other Liability
Mean τ
Freque
ncy
−0.06 −0.04 −0.02 0.00 0.02
04
812
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Test of Bootstrap ODP on Paid Data
Conclusion – The Bootstrap ODP model is biased upward.
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Test of CIT on Paid Data
Better than when ρ = 0 – Comparable to Bootstrap ODP - Still Biased
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Why Don’t Negative τs Fix the Bias Problem?
6.0 7.0 8.0−0.15
−0.05
0.05α6+β1
τ5.0 6.0 7.0−0
.15
−0.05
0.05
α6+β3
τ
3.0 4.0 5.0 6.0−0.15
−0.05
0.05
α6+β5
τ
2 3 4 5−0.15
−0.05
0.05
α6+β7
τ
Low τ offset by Higher α + β
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The Changing Settlement Rate (CSR) Model
• logelr ~ uniform(-5,0) • αw ~ normal(log(Premiumw)+logelr, ) • β10 = 0, βd ~ uniform(-5,5), for d = 1,…,9 • ai ~ uniform(0,1)
• Forces σd to decrease as d increases
• µw,d = αw + βd·(1 – γ)(w - 1) γ ~ Normal(0,0.025)
• Cw,d ~ lognormal(µw,d, σd)
σd = ai
i=d
10
∑
! 10
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The Effect of γ
• µw,d = αw + βd·(1 – γ)(w - 1)
• βs are almost always negative! (β10 = 0) • Positive γ – Speeds up settlement • Negative γ – Slows down settlement • Model assumes speed up/slow down
occurs at a constant rate.
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Distribution of Mean γs
Predominantly Positive γs
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CSR Model for Illustrative Insurer
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Test of CSR on Paid Data
Conclusion - Much better than CIT Varying speedup rate???
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Test of CIT on Paid Data
Better than when ρ = 0 – Comparable to Bootstrap ODP - Still Biased
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Calendar Year Risk Calendar Year Incurred Loss
= Losses Paid In Calendar Year
+ Change in Outstanding Loss
in Calendar Year
Important in One Year Time Horizon Risk
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• 0 – Prior Year, 1 – Current Year • IP1 = Loss paid in current year • CPt = Cumulative loss paid through year t • Rt = Total unpaid loss estimated at time t • Ut = Ultimate loss estimated at time t
Ut = CPt + Rt
Calendar Year Incurred Loss =
IP1 + R1 – R0 = CP1 + R1 – CP0 – R0 = U1 – U0 =
Ultimate at time 1 – Ultimate at time 0
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Estimating the Distribution of The Calendar Year Risk
• Given the current triangle and estimate U0
• Simulate the next calendar year losses – 10,000 times
• For each simulation, j, estimate U1j
• CCL takes about a minute to run.
10,000 minutes????
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A Faster Approximation • For each simulation, j
– Calculate U0j from {α, β, ρ and σ}j parameters – Simulate the next calendar year losses CY1j
• Let T = original triangle • Then for each i
– Calculate the likelihood Lij = L(T,CY1i|{α, β, ρ and σ}j)
– Set !!pij = Lij Lij
j∑ !and!U1i = pij ⋅U0 j
j∑
Bayes Theorem
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A Faster Approximation
• {U1i - U0} is a random sample of calendar year outcomes.
• Calculate various summary statistics – Mean and Standard Deviation – Percentile of Outcome (From CCL)
_
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Illustrative Insurer Constant ρ Model
Figure 1 − 'Ultimate' Incurred Losses at t=0
Mean = 39162 Standard Deviation = 1906
Freq
uenc
y
40000 50000 60000 70000 80000
020
0040
00
Figure 2 − Next Calendar Year Incurred Losses at t=0
Mean = −56 Standard Deviation = 1248
Freq
uenc
y
0 10000 20000 30000 40000
030
0060
00
Outcome = -54.62 Percentile = 53.14
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Illustrative Insurer Variable ρ Model
Figure 1 − 'Ultimate' Incurred Losses at t=0
Mean = 39091 Standard Deviation = 1906
Freq
uenc
y
40000 50000 60000 70000 80000
020
00
Figure 2 − Next Calendar Year Incurred Losses at t=0
Mean = −90 Standard Deviation = 1295
Freq
uenc
y
0 10000 20000 30000 40000
030
0060
00
Outcome = -71.01 Percentile = 51.29
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Test of CCL Constant ρ Model Calendar Year Risk
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Test of CCL Variable ρ Model Calendar Year Risk
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Short Term Conclusions Incurred Loss Models
• Mack model prediction of variability is too low on our test data.
• CCL model correctly predicts variability at the 95% significance level.
• The feature of the CCL model that pushed it over the top was between accident year correlations.
• CCL models indicate that the between accident year correlation decreases with the development year, but models that allow for this decrease do not yield better predictions of variability.
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Short Term Conclusions Paid Loss Models
• Mack and Bootstrap ODP models are biased upward on our test data.
• Attempts to correct for this bias with Bayesian MCMC models that include a calendar year trend failed.
• Models that allow for changes in claim settlement rates work much better.
• Claims adjusters have important information!
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Short Term Conclusions on Quantifying Calendar Year Risk • Models with explicit predictive distributions
provide a faster approximate way to predict the distribution of calendar year outcomes.
• Even though the original models accurately predicted variability, the variability predicted by the calendar year model was just outside the 95% significance level.
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Long Term Recommendations New Models Come and Go
• Transparency - Data and software released • Large scale retrospective testing on real data
– While individual loss reserving situations are unique, knowing how a model performs retrospectively should influence ones choice of models.
• Bayesian MCMC models hold great promise to advance Actuarial Science. – Illustrated by the above stochastic loss reserve
models. – Allows for judgmental selection of priors.