Estimating the Predictive Distribution
for
Loss Reserve Models
Glenn Meyers
Casualty Loss Reserve Seminar
September 12, 2006
Objectives of Paper
• Develop a methodology for predicting the
distribution of outcomes for a loss reserve model.
• The methodology will draw on the combined
experience of other “similar” insurers.
– Use Bayes’ Theorem to identify “similar” insurers.
• Illustrate the methodology on Schedule P data for
several insurers.
• Test the predictions of the methodology with data
from later Schedule P reports.
• Compare results with reported reserves.
A Quick Description of the Methodology
• Expected loss is predicted by chain
ladder/Cape Cod type formula
• The distribution of the actual loss around
the expected loss is given by a collective
risk (i.e. frequency/severity) model.
A Quick Description of the Methodology
• The first step in the methodology is to get the
maximum likelihood estimates of the model
parameters for several large insurers.
• For an insurer’s data
– Find the likelihood (probability of the data) given
the parameters of each model in the first step.
– Use Bayes’ Theorem to find the posterior
probability of each model in the first step given
the insurer’s data.
A Quick Description of the Methodology
• The predictive loss model is a mixture of
each of the models from the first step,
weighted by its posterior probability.
• From the predictive loss model, one can
calculate ranges or statistics of interest
such as the standard deviation or various
percentiles of the predicted outcomes.
The Data
• Commercial Auto Paid Losses from 1995
Schedule P (from AM Best)
– Long enough tail to be interesting, yet we
expect minimal development after 10 years.
• Selected 250 Insurance Groups
– Exposure in all 10 years
– Believable payment patterns
– Set negative incremental losses equal to zero.
16 insurer groups
account for one half of
the premium volume
Look at Incremental Development Factors
• Accident year 1986
• Proportion of loss paid in the “Lag”
development year
• Divided the 250 Insurers into four industry
segments, each accounting for about 1/4
of the total premium.
• Plot the payment paths
Incremental Development Factors - 1986
Incremental development
factors appear to be
relatively stable for the 40
insurers that represent
about 3/4 of the premium.
They are highly unstable
for the 210 insurers that
represent about 1/4 of the
premium.
The variability appears to
increase as size
decreases
Expected Loss Model
E Paid LossAY ,Lag  Premium AY  ELR  Dev Lag
• Paid Loss is the incremental paid loss in the AY and Lag
• ELR is the Expected Loss Ratio
• ELR and DevLag are unknown parameters
– Can be estimated by maximum likelihood
– Can be assigned posterior probabilities for Bayesian analysis
• Similar to “Cape Cod” method in that the expected loss
ratio is estimated rather than determined externally.
Distribution of Actual Loss
around the Expected Loss
• Compound Negative Binomial Distribution (CNB)
– Conditional on Expected Loss – CNB(x | E[Paid Loss])
– Claim count is negative binomial
– Claim severity distribution determined externally
• The claim severity distributions were derived from
data reported to ISO. Policy Limit = $1,000,000
– Vary by settlement lag. Later lags are more severe.
Claim Severity Distributions
Lags 5-10
Lag 4
Lag 3
Lag 2
Lag 1
Likelihood Function for a Given
Insurer’s Losses – x AY ,Lag 


Likelihood  x AY ,Lag 


10 11 AY
  CNB  x
AY 1 Lag 1
AY ,Lag
| E Paid Loss AY ,Lag 

where
E Paid LossAY ,Lag   PremiumAY  ELR  Dev Lag
Maximum Likelihood Estimates of
Incremental Development Factors
Loss development factors
reflect the constraints on
the MLE’s described in
prior slide
Contrast this with the
observed 1986 loss
development factors on
the next slide
Incremental Development Factors - 1986
(Repeat of Earlier Slide)
Loss payment factors
appear to be relatively
stable for the 40 insurers
that represent about 3/4
of the premium.
They are highly unstable
for the 210 insurers that
represent about 1/4 of the
premium.
The variability appears to
increase as size
decreases
Maximum Likelihood Estimates of
Expected Loss Ratios
Estimates of the ELRs are
more volatile for the
smaller insurers.
Using Bayes’ Theorem
• Let W = {ELR, DevLag, Lag = 1,2,…,10} be
a set of models for the data.
– A model may consist of different “models” or
of different parameters for the same “model.”
• For each model in W, calculate the
likelihood of the data being analyzed.
 Pr data | model
Using Bayes’ Theorem
• Then using Bayes’ Theorem, calculate the
posterior probability of each parameter set
given the data.
Posterior model | data 

Pr data | model  Prior model
Prior Distribution of
Loss Payment Paths
Prior loss payment paths
come from the loss
development paths of the
insurers ranked 1-40, with
equal probability
Posterior loss payment
path is a mixture of prior
loss development paths.
Prior Distribution of
Expected Loss Ratios
The prior distribution
of expected loss ratios
was chosen by visual
inspection.
Predicting Future Loss Payments
Using Bayes’ Theorem
• For each model, estimate the statistic of choice,
S, for future loss payments.
• Examples of S
– Moments of future loss payments.
– The probability density of a future loss payment of x,
– The cumulative probability, or percentile, of a future
loss payment of x.
• These examples can apply to single (AY,Lag)
cells, of any combination of cells such as a given
Lag or accident year.
– Use FFT’s to calculate distribution of sum of cells
Predicting Future Loss Payments
Using Bayes’ Theorem
• Calculate the Statistic S for each model.
• Then the posterior estimate of S is the
model estimate of S weighted by the
posterior probability of each model
Posterior Estimate of S

n
 S | model   Posterior model
i 1
i
i
| data 
Sample Calculations
for Selected Insurers
• Coefficient of Variation of predictive
distribution of unpaid losses.
• Plot the probability density of the predictive
distribution of unpaid losses.
Predictive Distribution
Insurer Rank 7
Predictive Mean = $401,951 K
CV of Total Reserve
= 6.9%
Predictive Distribution
Insurer Rank 97
Predictive Mean = $40,277 K
CV of Total Reserve
= 12.6%
CV of Unpaid Losses
Validating the Model on Fresh Data
• Examined data from 2001 Annual Statements
– Both 1995 and 2001 statements contained losses
paid for accident years 1992-1995.
– Often statements did not agree in overlapping years
because of changes in corporate structure. We got
agreement in earned premium for 109 of the 250
insurers.
• Calculated the predicted percentiles for the
amount paid from 1996 to 2001
• If model works, the predicted percentiles should
be uniformly distributed
PP Plots on Validation Data
Plot sorted predicted
percentiles against
uniform distribution.
Significant differences
given by KolomogorovSmirnov test.
Critical values @ 95%
= ±13.03%
Feedback
• If you have paid data, you must also have the
posted reserves. How do your predictions
match up with reported reserves?
• Your results are conditional on the data
reported in Schedule P. Shouldn’t an actuary
with access to detailed company data (e.g.
case reserves) be able to get more accurate
estimates?
Predictive and Reported Reserves
• For the validation sample, the predictive mean (in
aggregate) is closer to the 2001 retrospective reserve.
• Possible conservatism in reserves. OK?
• “%” means % reported over the predictive mean.
• Retrospective = reported less paid prior to end of 1995.
Reported Reserves More Accurate?
•
Divide the validation sample in to two groups and
look at subsequent development.
1. Reported Reserve < Predictive Mean
2. Reported Reserve > Predictive Mean
•
Expected result if Reported Reserve is accurate.
–
•
Reported Reserve = Retrospective Reserve for each
group
Expected result if Predictive Mean is accurate?
–
–
Predictive Mean  Retrospective Reserve for each
group
There are still some outstanding losses in the
retrospective reserve.
Subsequent Reserve Changes
Group 1
Group 2
• Group 1
• 50-50 up/down
• Ups are bigger
• Group 2
• More downs than
ups
• Results are
independent of
insurer size
Subsequent Reserve Changes
Reported Reserve @ 1995
< Predictive Mean (000)
> Predictive Mean (000)
66
43
Total Predictive Mean
926,134
872,660
1995 Reserve @ 1995
803,175
1,173,124
1995 Reserve @ 2001
856,393
985,711
Number of Insurers
•
The CNB formula identified two groups where:
–
–
•
Group 1 tends to under-reserve
Group 2 tends to over-reserve
Incomplete agreement at Group level
–
Some in each group get it right
Main Points of Paper
• How do we evaluate stochastic loss reserve
formula?
– Test predictions of future loss payments
– Test on several insurers
– Main focus is the testing
• Are there any formulas that can pass these tests?
– Bayesian CNB does pretty good on CA Schedule P data.
– Uses information from many insurers
– Are there other formulas? This paper sets a bar for
additional research to raise.