Individual claims reservingWhat’s new? What’s not?

Chairs Days, 11 June 2018

Agenda

2

1 Introduction to individual reserving

3 Medical malpractice case study

2 Some results

4 Perspectives

IntroductionFrom aggregate… (1/2)

3

The current reserving practice consists, in most cases, of using methods based on claimsdevelopment triangles for point estimate projections as well as for capital requirementcalculations.

The triangles are organized by origin (e.g., accident, underwriting) and development period.

In the context of an increasing need within the reserving practice for more accurate models,taking advantage of the information embedded in individual claims data is a promisingalternative compared with the traditional aggregate triangles.

Traditional reserving methods (Chain-Ladder, Mack, Wüthrich,…) work well in relatively stablecontexts and for standard business lines

Today, however, the awareness of the insurance market about some possible limitations of traditionalaggregate models to provide robust and realistic estimates in more variable contexts has reached a levelwhich should be noted

Several potential limits of aggregate models based on triangles have indeed already been highlighted both from a practical and a theoretical point of view:

Huge estimation error for the latest development periods due to the lack of observed aggregate amounts,

The difficulty of these models in identifying and capturing the various sub-risks,

Over/under-estimation of the distribution when back-testing realized amounts with forecasts.

Past

Future

IntroductionFrom aggregate… (2/2)

Recall that the Mean Square Error of Prediction (MSEP) from Mack (1993) measures both process andestimation errors all along the projection period (here displayed for a given origin year 𝑖) as

The replicative bootstrap method is depicted below, which provides a full distribution of futurepayments:

𝑀𝑆𝐸𝑃𝑀𝐴𝐶𝐾 = መ𝐶𝑖,𝑁2 ( ො𝜎𝑁−𝑖+1

𝐼 )2/ መ𝑓𝑁−𝑖+1𝐼 2

𝐶𝑖,𝑁−𝑖+1+( ො𝜎𝑁−𝑖+1

𝐼 )2/ መ𝑓𝑁−𝑖+1𝐼 2

σ𝑘=1𝑖−1 𝐶𝑘,𝑁−𝑖+1

+

𝑗=𝑁−𝑖+2

𝑁−1( ො𝜎𝑗

𝐼)2/ መ𝑓𝑗𝐼 2

መ𝐶𝑖,𝑗+

ො𝜎𝑗𝐼 2/ መ𝑓𝑗

𝐼 2

σ𝑘=1𝑁−𝑗

𝐶𝑘,𝑗

Original

triangle

Bootstrap iteration n°1

Bootstrap iteration n°b

Bootstrap iteration n°K

ki

kb

kik

b

kiC

rff,

2)(

,

)(

,

ˆˆ

Step 2 : Re-sampling the upper

triangle by bootstrapping the residuals

of individual development factors

Step 3 :

calculation of

Chain Ladder

factors

Step 4 : calculation

of the expected

future payments

Step 5 : simulation

of future payments

of the lower-

diagonal

...)1( 2)1(

1 II PP

Step 6 : recursive

simulation of future

payments of the lower-

triangle

Step 1 : estimation of

Mack parameters )ˆ,ˆ( kkf

o

)1(

,

)1(

1, )( iNiNiiNi fCCE

)1(

kf

2

,

)1(

1,ˆ)( iNiNiiNi CCVar

)1(

1Nf)1(

1f

)1()1(

,

)1(

1, )( kkiki fCCE 2)1(

,

)1(

1,ˆ)( kkiki CCVar

fij (1)

……..

……..

first year process error first year estimation error next years process error

Output: full distribution of future payments → allows to select the percentile related to the Risk Adjustment confidence level 𝜶

next years estimation error

Note that this term does not appear in the

Wüthrich-Merz one-yearversion

pure estimation error assessment additional process error assessment

Reference: A. Boumezoued, Y. Angoua,

L. Devineau, J.-P. Boisseau, 2011. One-

year reserve risk including a tail factor:

closed formula and bootstrap approaches.

Note that the variance of the distribution matches the Mack closed-formula

4

Introduction…to individual-based modelling

5

As noted in the report on worldwide non-life reserving practices from the ASTIN Working Partyon Non-Life Reserving (June, 2016), there is ‘an increase in the need to move towardsindividual claims reserving and big data, to better link the reserving process with thepricing process and to be able to better value non-proportional reinsurance.’

The first research papers on individual models are concomitant with the development ofstochastic methods on aggregated triangles:

Norberg (1983, 1993, 1999), Jewell (1987), Arjas (1989) et Hesselager (1994) are among the earliestpapers which introduced a proper probabilistic setting for individual claims reserving –current references are: Larsen (2007), Antonio & Plat (2014), Boumezoued & Devineau (2017)

To be compared with the stochastic models for triangles in Mack (1993) and following contributions

In a context of increasing need for more reliable quantification and management of reserverisk, it appears promising to take advantage of the information contained in detailedclaims data; this alternative requires models adapted to reveal the information inherent inthese data.

Time

Occurrence Reporting Payments Closing

Past Future

Why using individual reserving methods?A better estimate of reserves and their components

6

The primary objective of using individual models is to improve the Best Estimate of reserves, compared to the triangles-based methods.

The predictive capacity of these models relies on the fact that the whole life of each claim is taken into account (occurrence, reporting, payments, closing, …) as well as its covariates

Beyond the quantification of reserves, these models allow a separate estimation of reserve amounts for IBNR (Incurred But Not Reported) and RBNS (Reported But Not Settled)

Note that this distinction is, by construction, more natural in the context of individual models, whereas it is non-trivial (and less easy to justify) for methods based on triangles

Adequate valuation of non-proportional reinsurance: due to the non-linearity of treaty valuation formulas, a stochastic individual approach is needed to produce an unbiased pricing

The stochastic simulation of claims paths beyond a basic frequency x severity model makes it possible to solve adequately this problem.

The claims pathClaim path and associated sub-risks – specification details

7

The individual claims paths are modelled with continuous time stochastic processes

Payments and settlement events are modelled using three types of events (**): (1) settlement without payment at settlement(2) settlement with payment at settlement(3) payment without settlementEach type of event (1, 2, or 3) occurs according to its specific intensity parameter ℎ1(𝑣), ℎ2(𝑣) or ℎ3 𝑣 .→ If an event 𝑖 ∈ {2,3} occurs 𝑣 time units after reporting, then random payments 𝑌𝑖(𝑣) are generated

1

2

3

1

2

3

1

2

3

(*) The corresponding interpretation is that over all event types in the claims pool, the proportion of event of type 𝑖 ∈

{1,2,3} is given by ℎ𝑖

ℎ1+ℎ2+ℎ3. In addition, this gives information on the timing of these events, as for example, the time

to wait between two intermediary payments (3) is 1/ℎ3 in average, and the time to wait between two events (whatever their type) is 1/(ℎ1 + ℎ2 + ℎ3)

Based on occurrence and reporting parameters, stochastic IBNyR occurrences and reporting delays can be simulated.

The payments and settlement model parameters allow simulation of the stochastic future payments paths to closing for both IBNyR and RBNS claims.

Time

Occurrence Reporting Payments Settlement

Claims occur (*) at times 𝑇𝑛according to some Poisson process with intensity 𝜆(𝑡)

Claims are reported with a delay with distribution 𝑝𝑈∣𝑡(𝑑𝑢)

Occurrence and reporting distributions have to be estimated jointly as observation is biased due to hidden Incurred But Not yet Reported claims (IBNyR)

𝑡𝑢 𝑣

(**) Representation by Antonio & Plat (2014)

Agenda

8

1 Introduction to individual reserving

3 Medical malpractice case study

2 Some results

4 Perspectives

Prediction error decompositionForecasting: closed-formulas and simulation algorithm

9

Process error Estimation error

Simulations Algorithm called thinning for the simulation

of all types of events involved in claim path

Simulation based on the variance-covariance

matrix of Maximum Likelihood Estimators

(asymptotic normality result)

Closed-formulasClosed-form formulas established in

Boumezoued & Devineau (2017)

Asymptotic normality result and the use of the

Delta method for obtaining closed-forms

The total prediction error for the future claim trajectories includes the following sub-components, as classical for reserve risk:

Process error: pure stochasticity due to the intrinsic randomness of future paths,

Estimation error: linked to the uncertainty on the value of the parameters estimated

Both types of error can be quantifies by simulation or alternatively by using closed-formulas,see Boumezoued & Devineau (2017):

Simulation algorithmForecasting IBNyR occurrences – thinning algorithm (1/2)

10

Forecasting the IBNyR claims

Having estimated the occurrence frequency 𝜆(𝑡) and the delay distribution 𝑝𝑈∣𝑡(𝑑𝑢), it is proved in the scientific paper that IBNyR occurrences can be simulated on the observation period 0, 𝜏 according to a Poisson process with time-dependent intensity

𝜆𝐼𝐵𝑁𝑦𝑅 ∶ 𝑡 ↦ 𝜆 𝑡 𝑝𝑈∣𝑡 [𝜏 − 𝑡,∞)

The simulation can be done by means of the thinning algorithm: (which can be seen as a generalization of the acceptance/rejection method)

1. First consider a bound on the intensity – take ҧ𝜆 = sup 𝜆𝐼𝐵𝑁𝑦𝑅(𝑡)

2. Then draw the sequence (𝑆𝑘) on [0, 𝜏] as a Poisson process with intensity ҧ𝜆 - for example by drawing (𝑆𝑘+1 − 𝑆𝑘) as iid increments that are exponentially distributed with parameter ҧ𝜆

3. For each time 𝑆𝑘, draw a Bernoulli random variable 𝑃𝑘 which is equal to one with probability 𝜆𝐼𝐵𝑁𝑦𝑅 𝑆𝑘

ഥ𝜆, and zero otherwise

→ if 𝑃𝑘 = 1, this corresponds to the acceptance of the time 𝑆𝑘 as being a ‘true’ IBNyR claim occurrence. In other words, we simulate the fact that an IBNyR claim occurred at time 𝑆𝑘 .

→ if 𝑃𝑘 = 0, then the ‘virtual’ occurrence time 𝑆𝑘 is omitted (deleted)

Simulation algorithmForecasting IBNyR occurrences – thinning algorithm (2/2)

11

Below are examples of the simulation for two non-homogeneous Poisson processes

ҧ𝜆 Poisson process withintensity ҧҧ𝜆

Poisson process withintensity 𝜆(𝑡)𝜆(𝑡)

Poisson process with

intensity ҧҧ𝜆

Poisson process withintensity 𝜆(𝑡)

ҧ𝜆

𝜆(𝑡)

The intensity 𝜆(𝑡) is given, one has to simulate a Poisson process with such

intensity

Thinning procedure: assume that this

intensity is bounded, that is 𝜆 𝑡 ≤ ҧ𝜆

-> One is able to easily simulate a Poisson

process with intensity ҧ𝜆 as a sequence of ത𝑇𝑛 𝑛≥1 such that the ത𝑇𝑛 − ത𝑇𝑛−1 are iid

exponentially distributed with parameter ҧ𝜆

-> Then, select each occurrence ത𝑇𝑛 with probability 𝜆 ത𝑇𝑛 / ҧ𝜆

Closed form formulasClosed-form formulas for the future payments

12

Proposition: the expected payments and related process error can be written as

𝜇 𝑣 = 𝔼 𝑋 𝑣,∞ ∣ 𝑆 𝑣 = 3 = න𝑣

∞

𝑦2 𝑠 ℎ2(𝑠) + 𝑦3 𝑠 ℎ3 𝑠 exp −න𝑣

𝑠

ℎ1 + ℎ2 d𝑠

𝛾 𝑣 = 𝑉𝑎𝑟 𝑋 𝑣,∞ ∣ 𝑆 𝑣 = 3 = න𝑣

∞

𝐻(𝑣) exp −න𝑣

𝑠

ℎ1 + ℎ2 d𝑣

𝐻 ≡ ℎ1𝜇2 + ℎ2 𝜎2

2 + 𝑦2 − 𝜇2 + ℎ2 𝜎3

2 + 𝑦32

𝔼 𝑋𝜏𝐼𝐵𝑁𝑦𝑅

= න0

𝜏

න𝜏−𝑡

∞

𝜇(𝑠) 0 𝜆 𝑠 d𝑠 p𝑈∣𝑠 d𝑢

𝑉𝑎𝑟 𝑋𝜏𝐼𝐵𝑁𝑦𝑅

= න0

𝜏

න𝜏−𝑡

∞

𝛾 𝑠 (0) + 𝜇(𝑠) 0 2 𝜆 𝑠 d𝑠 p𝑈∣𝑠 d𝑢

Similar results hold for RBNS claims, see Section 3.4 of the scientific paper

Closed form formulasClosed-form formulas for the future payments – Example 1

13

In the homogeneous Markov setting, under which the payment frequencies areconstant, that is ℎ𝑖 𝑣 ≡ ℎ𝑖 The mean of future payments on a single claim path is written as:

𝜇 =𝑦2ℎ2 + 𝑦3ℎ3ℎ1 + ℎ2

The variance of future payments on a single claim path is written as:

𝛾 =ℎ1𝜇

2 + ℎ2 𝜎22 + 𝑦2 − 𝜇

2 + ℎ2 𝜎32 + 𝑦3

2

ℎ1 + ℎ2

14

Assuming ℎ2 = 0 (which is the case in the case study), and log-normal payments (with parameters 𝜇 and 𝜎), we obtain in the standard setting the following closed form for the expected payments for IBNyR:

𝔼 𝑃𝐼𝐵𝑁𝑦𝑅 =𝜆

𝜃1 − exp −𝜃𝜏

ℎ3ℎ1exp 𝜇 +

𝜎2

2

The variance (process error) can be written as:

𝑉𝑎𝑟 𝑃𝐼𝐵𝑁𝑦𝑅 =𝜆

𝜃1 − exp −𝜃𝜏

ℎ3ℎ1exp 2𝜇 + 𝜎2 2

ℎ3ℎ1

+ exp 𝜎2

Occurrence and reportingClaims pathPayments volume

Closed form formulasClosed-form formulas for the future payments – Example 2

Assumptions: 𝜆 is constant and the delays are exponential 𝑝𝑈∣t 𝑑𝑢 = 𝜃𝑒

−𝜃𝑢𝑑𝑢

Agenda

15

1 Introduction to individual reserving

3 Medical malpractice case study

2 Some results

4 Perspectives

Medical malpractice case studyDatabase description

16

The database used is made of Medical malpractice claims of several origins (classified into 8 groups).

There are 19870 observed claims in the database, including 6121 settled claims and 13749 reported but not yet settled (RBNS).

For each claim, we potentially have the following information:

Occurrence date

Declaration date

Payment dates and associated amounts

Closing date

Observed occurrence dates Observed reporting delays

IBNyR-related observation biasLine of business with particularly long

reporting delays

Medical malpractice case studyOccurrence/reporting model calibration

17

Calibration of occurrence frequencies and reporting delays are given below for each of the 8 groups considered:

The reporting delays calibrated by the model (in

red) capture the specific reporting dynamics for

each of the 8 groups

A 'naive' estimate (in blue) would lead to an

underestimation of the reporting delay (claims with a

low reporting delay are over-represented in the

sample).

Average reporting delayOccurrence frequency

The model captures the disparities in occurrence and reporting dynamics among the different claim groups.

The frequency of claims occurrence is captured (in

red) by the model, restoring the bias related to

non-observation of IBNyR.

A calculation directly based on observation (in blue)

would underestimate the frequency of claims

occurrence.

Medical malpractice case studyIBNR simulation, backtesting and comparison with Mack model

18

In order to have a comparative dataset, the individual model and the Mack Chain Ladder model (1993) are calibrated on a common history of 5 years.

Predicted values are then compared to real observed values of IBNyR (backtesting)

Higher predictive power of the individual model in this context

The Mack Chain Ladder approach seems to overestimate the number of IBNyR claims

The Mack Chain Ladder approach involves relatively huge uncertainty around the prediction

Prediction of IBNyR number with Mack Chain Ladder model

Prediction of IBNyR number with individual model

Results per group Total results

Medical malpractice case studyCalibration of payments and settlement frequencies (1/2)

19

Computation of the frequency parameters related to payments and settlement.

1

2

3

1

2

3

1

2

3

ℎ1

Settlement without payment Settlement with payment

ℎ2 ℎ3

Payment without settlement

• Both settlement frequencies (1) and payments without settlement (3) are of the same

order of magnitude (note that this database contains a negligible number of settlements

with payment (2)).

• One can identify that group C is characterized by both a shorter claims history (1) and a

low payment frequency (3). This contrasts with group B, with a larger time to settlement,

and more frequent payments during the claims paths.

1 2 3

20

Refinement of the calibration to allow for time-varying frequency parameters: the frequencies are allowed to vary from one claims development period to the next.

This makes it possible to have a more realistic model and to anticipate the future dynamics of claims according to their development time.

1

2

3

1

2

3

1

2

3

Closing frequencies (1) are maximum for the majority of groups for claims of age around 3 years; these

estimates characterize claims with a high potential for long-term development: these are rather 8-9

years old and belong to groups A to D (low frequencies)

Payments (without settlement) frequency (3) show a very different pattern: here they indicate that an

IBNR (zero age) claim has potentially more future payments in the short term than claims that have been

developed for 2 to 3 years

Settlement without payment Payment without settlement

Group → Group →

Fre

qu

en

cy

Fre

qu

en

cy

Medical malpractice case studyCalibration of payments and settlement frequencies (2/2)

1 3

Medical malpractice case studyComputation of future payments for each type of claim

21

The closed formulas that we have developed make it possible to instantly compute the expected payments and the associated variance (prediction error) for each type of claim (group) and according to its duration of development.

Expected future payments

Number of development years

Depending on the duration of development, the expected future payments decrease aver the first 3 years,

due to the combination of a decrease in payment frequency and a growth in the settlement frequency (see

previous slides).

Expected payments then increase for more developed claims, due to a reduction in the settlement frequency

after the 3-year barrier.

This example illustrates the benefit of an analysis of development times, and in particular the bias due to the

use of standard frequency x severity models (notably for the evaluation of reinsurance).

Group:

Variance of future payments

Number of development years

Group:

22

We assess separately the IBNyR and RBNS reserves based on the individual model; this allows for computing the relative importance of the IBNyR both in number and amount:

The comparison between methods shows

that Chain-Ladder provides higher reserves

compared to the individual model

The differentiated estimation makes the

individual model particularly adapted to the

lines of business with complex and/or

atypical development.

We compare the total reserve provided by the individual model with the Chain-Ladder prediction:

Medical malpractice case studyAnalysis of Best Estimate reserves and comparison with Chain-Ladder

Although the expected number of IBNyR represents

around 20% of the claims number to be paid (both RBNS

and IBNyR), in terms of amounts this represents here

around 50%.

The use of the individual model makes it possible to

quantify the IBNR reserves in a coherent and appropriate

way, in particular for the lines of business with high

reporting delays.

GroupGroup

23

The graph below represents the coefficient of variation (standard deviation divided by expectation) for the estimation error and the process error. It also makes the comparison between the individual model the Mack model.

Medical malpractice case studyPrediction error and comparison with Mack model

The estimation error is clearly reduced with the individual model, which is a fundamental property : the model

takes advantage from detailed claims information (‘line-by-line’) to make calibration more reliable, and relies

on a more ‘natural’ specification of the model into several sub-blocks (occurrence, reporting, payment, flow,

…) which avoids errors of parametrization at the aggregated level.

The process error is also reduced in the individual model, thanks to its specification: Poisson modeling is

intrinsically less dispersed compared to Mack’s Markov chain model.

Current modeling can easily be extended to more dispersed stochastic frameworks, depending on the

calculation objectives and the risk indicators to be considered.

Group

Agenda

24

1 Introduction to individual reserving

3 Medical malpractice case study

2 Some results

4 Perspectives

25

PerspectivesWhat use of Machine Learning techniques?

PRÉDICTION DES COÛTS ULTIMES RBNS

Observedpayments

Machine Learning for Prediction Predicting

RBNS reserve

MODELLING IBNR CLAIMS

Unsupervised clustering

Stochastic IBNR model

Observedclaims

Class n°1

Class n°K

PredictingIBNR occurrence and reporting

Stochastic IBNR model

¨PREDICTING IBNR RESERVE

Some references on Machine Learning applied to Individual claims reserving: Olivier Lopez, Xavier Milhaud, Pierre-Emmanuel Therond. (2015) Tree-based censored regression with

applications to insurance. Baudry, M., & Robert, C. Y. (2017). Non parametric individual claim reserving in insurance. Wüthrich, M. V. (2018). Machine learning in individual claims reserving. Scandinavian Actuarial Journal, 1-16.

Towards mixing “classical” and Machine Learning for Individual Claims Reserving:

26

References

Antonio, Katrien, Richard Plat. 2014. Micro-level stochastic loss reserving for general

insurance. Scandinavian Actuarial Journal 2014(7)

Arjas, Elja. 1989. The claims reserving problem in non-life insurance: Some structural ideas.

Astin Bulletin 19(2)

Boumezoued, A. & L. Devineau. 2017. Individual claims reserving: a survey. Submitted.

Hesselager, Ole. 1994. A Markov model for loss reserving. Astin Bulletin 24(02)

Jewell, William S. 1989. Predicting IBNYR events and delays: I. continuous time. Astin Bulletin

19(01)

Mack, Thomas. 1993. Distribution-free calculation of the standard error of chain ladder reserve

estimates. Astin bulletin 23(02) 213225.

Norberg, Ragnar. 1993. Prediction of outstanding liabilities in non-life insurance. Astin Bulletin

23(01) 95115.

Norberg, Ragnar. 1999. Prediction of outstanding liabilities II. model variations and extensions.

Astin Bulletin 29(01) 525

Pigeon, Mathieu, Katrien Antonio, Michel Denuit. 2013. Individual loss reserving with the

multivariate skew normal framework. ASTIN Bulletin: The Journal of the IAA 43(3)

27

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