Estimating the Predictive Distribution for Loss Reserve Models

Glenn Meyers ISO Innovative Analytics CAS Annual Meeting November 14, 2007

S&P Report, November 2003

Insurance Actuaries – A Crisis in Credibility “Actuaries are signing off on reserves that turn out to be wildly inaccurate.”

Background to Methodology - 1

• Zehnwirth/Mack – Loss reserve estimates via regression –

y

= a

∙x

+ e • GLM –

E

[

Y

] =

f

( a

∙x

) – Allows choice of

f

and the distribution of – Choices restricted to speed calculations

Y

• Clark – Direct maximum likelihood – Assumes

Y

distribution has an Overdispersed Poisson

Background to Methodology - 2

• Heckman/Meyers – Used Fourier transforms to calculate aggregate loss distributions in terms of frequency and severity distributions.

• Hayne – Applied Heckman/Meyers to calculate distributions of ultimate outcomes, given estimate of mean losses

High Level View of Paper

• Combine 1-2 above – Use aggregate loss distributions defined in terms of Fourier transforms to (1) estimate losses and (2) get distributions of ultimate outcomes.

• Uses “other information” from data of ISO and from other insurers.

– Implemented with Bayes theorem

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 • Test the predictions of the methodology on several insurers 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

Do Incremental Development Factors Differ by Size of Insurer?

• Form loss triangles as the sum of the loss triangles for all insurers in each of the four industry segments defined above.

• Plot the payment paths

Segment 1 Segment 3 Segment 2 Segment 4 There is no consistent pattern in aggregate loss payment factors for the four industry segments.

Expected Loss Model

  Premium

AY



Lag

• Paid Loss is the

incremental

paid loss in the

AY

and

Lag

• •

ELR

is the Expected Loss Ratio

ELR

and

Dev Lag

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 Count has a negative binomial distribution with l = E[Paid Loss]/E[Claim Severity] and

c

= .01

• See Meyers - 2007 “The Common Shock Model for Correlated Insurance Losses” for background on this model.

Claim Severity Distributions

Lags 5-10 Lag 4 Lag 3 Lag 2 Lag 1

q

Calculating     1

X

 1 l

Z Lag

 1    1/

c

 Where

Z Lag



p

Lag

 claim severity random variable

Lag

vector  Z Lag  Fast Fourier Transform (FFT) of

p

Lag

l   1

X

   Inverse FFT 

Lag



h

 1 

th

component of

q

Likelihood Function for a Given Insurer’s Losses –



x

 Likelihood    

x

   10 11   

AY AY

 1

Lag

 1  where Premium

AY



Lag

Maximum Likelihood Estimates

• Estimate

ELR

and

Dev Lag

maximum likelihood simultaneously by • Constraints on

Dev Lag

–

Dev 1 ≤ Dev 2

–

Dev i ≥ Dev i+1

for i = 2,3,…,7 –

Dev

8 =

Dev

9 =

Dev

10 • Use R’s

optim

function to maximize likelihood – Read appendix of paper before you try this

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.

Testing the Compound Negative Binomial (

CNB

) Assumption

• Calculate the percentiles of each observation given E[Paid Loss].

– 55 observations for each insurer • If

CNB

is right, the calculated percentiles should be uniformly distributed.

• Test with PP Plot – Sort calculated percentiles in increasing order – Vector (1:

n

)/(

n

+1) where

n

is the number of percentiles – The plot of the above two vectors against each other should be on the diagonal line.

Interpreting PP Plots

Take 1000 lognormally distributed random variables with m = 0 and s = 2 as “data” If a whole bunch of predicted percentiles are at the ends, the predicted tail is too light.

If a whole bunch of predicted percentiles are in the middle, the predicted tail is too heavy.

If in general the predicted percentiles are low, the predicted mean is too high

Testing the

CNB

Assumptions

Insurer Ranks 1-40 (Large Insurers) This sample has 55 ×40 or 2200 observations. According to the Kolomogorov-Smirnov test, D statistic for a sample of 2200 uniform random numbers should be within ± 0.026 of the 45º line 95% of the time.

Actual D statistic = 0.042. As the plot shows, the predicted percentiles are slightly outside the 95% band. We are close.

Testing the

CNB

Assumptions

Insurer Ranks 1-40 (Large Insurers) Breaking down the prior plot by settlement lag shows that there could be some improvement by settlement lag.

But in general, not bad!

pp plots by settlement lag

Testing the

CNB

Assumptions

Insurer Ranks 41-250 (Smaller Insurers)

This is bad!

pp plots by settlement lag

Using Bayes’ Theorem

• Let W = {

ELR

,

Dev Lag

,

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.

 

Using Bayes’ Theorem

• Then using Bayes’ Theorem, calculate the posterior probability of each parameter set given the data.

    

Selecting Prior Probabilities

• For

Lag

, select the payment paths from the maximum likelihood estimates of the 40 largest insurers, each with equal probability.

• For

ELR

, first look at the distribution of maximum likelihood estimates of the

ELR

from the 40 largest insurers and visually “smooth out” the distribution. See the slide on

ELR

prior below.

• Note that

Lag

independent.

and

ELR

are assumed to be

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

– Expected value of future loss payments – Second moment 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 ( Lag or accident year.

AY,Lag

) cells, of any combination of cells such as a given

Predicting Future Loss Payments Using Bayes’ Theorem for Sums over Sets of {

AY

,

Lag

}

• If we assume losses are independent by AY and Lag

q

 1   

AY Lag

 1 l

Z Lag

 1    1/

c

   • Actually use the negative multinomial distribution – Assumes correlation of frequency between lags in the

q

same accident year  1  

AY E

   

Lag e

 l  

Z Lag



p Lag

  1     

Predicting Future Loss Payments Using Bayes’ Theorem

• Calculate the Statistic

S

for each model.

• Then the posterior estimate of

S

model estimate of

S

is the weighted by the posterior probability of each model Posterior Estimate of

S



i n

  1 

S

| model

i

  

i



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 1997-2001 • Evaluate predictions with pp plots.

PP Plots on Validation Data

KS 95% critical values = ±13.03%

Feedback

• If you have paid data, you must also have the posted reserves. How do your predictions match up with reported reserves? – In other words, is S&P right?

• 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?

Response – Expand the Original Scope of the Paper

• Could persuade more people to look at the technical details.

• Warning – Do not over-generalize the results beyond commercial auto in 1995 2001 timeframe.

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.

Predictive Percentiles of Reported Reserves • Conservatism is not evenly spread out.

• Conservatism appears to be independent of insurer size • Except for the evidence of conservatism, the reserves are spread out in a way similar to losses.

• Were the reserves equal to ultimate losses?

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

Number of Insurers Total Predictive Mean 1995 Reserve @ 1995 1995 Reserve @ 2001 Reported Reserve @ 1995 < Predictive Mean (000) > Predictive Mean (000) 66 926,134 43 872,660 803,175 856,393 1,173,124 985,711 • • • – – 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 Discussion??

Main Points of Paper

• How do we evaluate stochastic loss reserve formula? – Test predictions of future loss payments – Test on several insurers – Main Focus • 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 others to raise.

Subsequent Developments

• Paper completed in April 2006 • Additional critique • Describe recent developments • Describe ongoing research

PP Plots on Validation Data Clive Keatinge’s Observation

• Does the leveling of plots at the end indicate that the predicted tails are too light?

• The plot is still within the KS bounds and thus is not statistically significant.

• The leveling looks rather systematic.

A

2

Alternative to the KS Anderson-Darling Test

n j k

  0   1  

n j k

  0 

F y n n

   2    2   ln 

F

* 

F

*     ln 

F

*   

F

* • AD is more sensitive to tails.

• Critical values are 1.933, 2.492, and 3.857 for 10, 5 and 1% levels respectively.

• Value for validation sample is 2.966

• Not outrageously bad, but Clive has a point.

• Explanation – Did not reflect all sources of uncertainty??

  

Is Bayesian Methodology Necessary?

• “Thinking Outside the Triangle” – Paper in June 2007 ASTIN Colloquium • Works with simulated data on a similar model • Compares Bayesian with maximum likelihood predictive distributions

Maximum Likelihood Fitting Methodology PP Plots for Combined Fits • PP plot reveals the S-shape that characterizes overfitting.

• The tails are too light 0.0

0.2

0.4

0.6

Uniform Probability 0.8

1.0

Bayesian Fitting Methodology PP Plots for Combined Fits

Nailed the Tails

0.0

0.2

0.4

0.6

Uniform Probability 0.8

1.0

IN THIS EXAMPLE

• Maximum Likelihood method understates the true variability • I call this “overfitting” i.e. the model fits the data rather than the population • Nine parameters fit to 55 points • SPECULATION – Overfitting will occur in all maximum likelihood methods and in moment based methods – i.e. GLM and Mack

Expository Paper in Preparation

• Focus on the Bayesian method described in this paper • Uses Gibbs sampler to simulate posterior distribution of the results • Complete algorithm coded in R • Hope to increase population of actuaries who: – Understand what the method means – Can actually use the method