A Method for Projecting Individual Large Claims

Casualty Loss Reserving Seminar11-12 September 2006Atlanta

Karl Murphy and Andrew McLennan

2

Overview

• Rationale for considering individual claims

• Outline of methodology• Examples• Data Requirements• Assumptions• Whole account variability• Case Study• Conclusion

3

Rationale for Considering Individual Claims

• Last few years has seen a significant change in requirements from actuaries in terms of understanding variability around results

• Partially driven by a greater understanding by board members that things can go wrong, and partly by the increased use of DFA models

• Much work done based on aggregate triangles, but very little on stochastic individual claims development

• Weaknesses in methods for deriving consistent gross and net results

4

Traditional Netting Down Methods• How do you net down gross reserves?• Could assume reinsurance ultimate reserves = reinsurance

current reserves– Prudent if deficiencies in reserves– Optimistic if redundancies

• Analyse net data, and calculate net results from this• Disadvantages:

– Retentions may change• look at data on consistent retention• lots of triangles! Ensuring consistency between gross and various nets

difficult– Indexation of retention

• need assumption of payment pattern– Aggregate deductibles

• need assumption of ultimate position of individual claims

• Another option – model excess claims above a threshold, and calculate average deficiency of excess claims – i.e. IBNER on those above threshold. Apply average IBNER loading to open claims to get ultimate

5

Deterministic Netting Down Methods Tend to Understand Effect of Reinsurance• Example: excess IBNER of £0.5m, two

claims of incurred of £250k, and retention of £500k

• Deterministic development factor of 2, so gross-up claims to ultimate of £500k each

• Calculate reinsurance recoveries: 500k-500k = 0 – no reinsurance recoveries

• Net reserves = gross reserves

6

Deterministic Netting Down Methods Tend to Understand Effect of Reinsurance• Because of the one-sided nature of

reinsurance, this will understate the reinsurance recoveries:

• Above example:– one claim settles for 250k, one for 750k

• same gross result• Net reserves = gross reserves – 250k

• Need method that allows for distribution of ultimate individual claims to allow for reinsurance correctly

7

Traditional Variability Methods• Traditional Methods:

– Methods based on log(incremental data), i.e. lognormal models– Mack’s model – based on cumulative data– Provide mean and variance of outcomes only

• Bootstrapping– Provides a full predictive distribution – not just first two

moments– Bootstrap any well specified underlying model

• Over-dispersed Poisson (England & Verrall)• Mack’s model

– Characteristics• Usually applied to aggregate triangles• Works well with stable triangles• However, large claims can influence volatility unduly

• Bayesian Methods:– Like Bootstrapping, provides a full predictive distribution– Ability to incorporate expert judgement with informative priors

8

Traditional Variability Methods

• No allowance made for the number of large claims in an origin period, and no allowance made for the status (i.e. open/closed)

• No linkage between variability of gross and net of reinsurance reserves

• No information about the distribution of individual claims – will have same problems of netting down gross results as deterministic methods

9

Outline of Methodology

• Our methodology simulates large claims individually

• Separately simulate known claims (for IBNER) and IBNR claims

• Consider dependencies between IBNER and IBNR claims

• For non-large claims, use an aggregate “capped” triangle– when a individual claim reaches the capping level,

ignore any development in excess of the capping– index the capping threshold at an appropriate level– use a “traditional” stochastic method– consider dependency between the run-off of capped

and excess claims

10

Outline of Methodology: IBNER• Take latest incurred position and status of claim• Simulate next incurred position and status of claim

based on movement of a similar historic claim• Allows for re-openings, to the extent they are in

the historic data• Projects individual claims from the point they

become “large”• Claims are considered “similar” by:

– Status of claim (open / closed)– Number of years since a claim became large

(development period)– Size of claim – e.g. a claim with incurred of £10m will

behave differently to a claim with incurred of £1m – claims are banded into layers

11

Outline of Methodology: IBNR

• IBNR large claims can be either genuine IBNR, or claims previously not reported as large

• Apply “standard” stochastic methods to numbers triangles

• Alternatively, simulate based on an assumed frequency per unit of exposure

• For severity, can sample from the (simulated) known large claims, or simulate from an appropriately parameterised distribution

12

Example Data

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

Incurred Amounts

Claim

Development YearClaim

Year 1 to Year 2

Year 2 to Year 3

A 2 1B 3.2 0.53C 1 1.5D 2.5E 0.67F

Development Factors

1 2 3

A Open Closed ClosedB Open Open ClosedC Open Open ClosedD Open OpenE Open ClosedF Open

Claim Status

ClaimDevelopment Year

13

Claim D

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

Incurred Amounts

ClaimDevelopment Year

• Need to simulate into development period 3

• Open status as at development period 2

• Similar to claims B and C, with development factors of 0.53 and 1.5

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

ClaimDevelopment Year

Incurred Amounts

14

Claim D: Simulations

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000 265,625E 300,000 200,000F 150,000

Incurred Amounts

ClaimDevelopment Year

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000 750,000E 300,000 200,000F 150,000

Incurred Amounts

ClaimDevelopment Year

15

Claim E

• Closed status as at development period 2

• Similar to claim A, with no development

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

ClaimDevelopment Year

Incurred Amounts

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

Incurred Amounts

ClaimDevelopment Year

16

Claim F

• Open status as at development period 1

• For development into year 2, can consider any of A to E

• Consider also the status

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

Incurred Amounts

ClaimDevelopment Year

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000

ClaimDevelopment Year

Incurred Amounts

17

Claim F Simulations to Year 2

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000 300,000

Incurred Amounts

ClaimDevelopment Year

1 2 3

A Open Closed ClosedB Open Open ClosedC Open Open ClosedD Open OpenE Open ClosedF Open Closed

Claim Status

ClaimDevelopment Year

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000 150,000

ClaimDevelopment Year

Incurred Amounts

1 2 3

A Open Closed ClosedB Open Open ClosedC Open Open ClosedD Open OpenE Open ClosedF Open Open

ClaimDevelopment Year

Claim Status

18

Claim F Simulations to Year 3

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000 300,000 800,000

Incurred Amounts

ClaimDevelopment Year

1 2 3

A 400,000 800,000 800,000B 500,000 1,600,000 850,000C 1,000,000 1,000,000 1,500,000D 200,000 500,000E 300,000 200,000F 150,000 150,000 79,688

Incurred Amounts

ClaimDevelopment Year

1 2 3

A Open Closed ClosedB Open Open ClosedC Open Open ClosedD Open OpenE Open ClosedF Open Open Closed

Claim Status

ClaimDevelopment Year

1 2 3

A Open Closed ClosedB Open Open ClosedC Open Open ClosedD Open OpenE Open ClosedF Open Closed Closed

Claim Status

ClaimDevelopment Year

19

IBNR Claims

• Two sources of IBNR claims:– True IBNR claims– Known claims which are not yet large

• Triangle of claims that ever become large• Calculate frequency of large claims in

development period• Simulate number of large claims going

forward• Simulate IBNR claim costs from historic

claims that became large in that period

20

IBNR• Data below shows the claim number

triangle, and frequency of claimsDevelopment Year

Accident Year 1 2 3 4 5 6 7 8 9 101994 92 94 63 61 25 15 1 4 1 11995 38 33 31 24 8 4 7 1 01996 56 55 29 24 7 7 5 11997 57 40 35 14 3 5 31998 10 12 6 7 4 21999 28 29 19 22 82000 59 55 52 262001 102 111 452002 35 262003 89

Claims Per Unit of ExposureDevelopment Year

Accident Year 1 2 3 4 5 6 7 8 9 101994 0.0463% 0.0471% 0.0316% 0.0309% 0.0126% 0.0077% 0.0007% 0.0021% 0.0007% 0.0007%1995 0.0127% 0.0110% 0.0103% 0.0082% 0.0028% 0.0013% 0.0024% 0.0004% 0.0000%1996 0.0126% 0.0124% 0.0064% 0.0054% 0.0016% 0.0016% 0.0011% 0.0003%1997 0.0092% 0.0064% 0.0056% 0.0022% 0.0006% 0.0008% 0.0006%1998 0.0012% 0.0015% 0.0008% 0.0009% 0.0005% 0.0002%1999 0.0035% 0.0037% 0.0024% 0.0028% 0.0011%2000 0.0102% 0.0094% 0.0089% 0.0044%2001 0.0213% 0.0232% 0.0095%2002 0.0068% 0.0050%2003 0.0145%

Mean 0.0106% 0.0096% 0.0066% 0.0048% 0.0018% 0.0014% 0.0011% 0.0007% 0.0003% 0.0007%

21

IBNR

• Result for one simulationDevelopment Year

Accident Year 1 2 3 4 5 6 7 8 9 101994 92 94 63 61 25 15 1 4 2 21995 38 33 31 24 8 4 7 1 0 31996 56 55 29 24 7 7 5 1 0 31997 57 40 35 14 3 5 3 4 2 51998 10 12 6 7 4 2 6 3 3 41999 28 29 19 22 8 16 5 4 0 62000 59 55 52 26 7 4 10 4 0 72001 102 111 45 30 8 5 7 1 3 12002 35 26 32 29 11 9 5 1 0 62003 89 49 33 37 10 7 6 5 2 4

22

Data Requirements

• Individual large claim information:– Full incurred and payment history– Historic open status of claims– Claims that were ever large, not just currently

large

• Accident year exposure• Definition of “large” depends on:

– Historic retentions– Number of claims above threshold– Consider having two thresholds – e.g. all claims

above $100k, but then calculate excess above $200k – allows for claims developing just below the layer

23

Assumptions

• Historic claims provide the full distribution of possible chain ladder factors for claims

• Development by year is independent• No significant changes to case estimation

procedures– Can allow for this by standardising the historic chain

ladder factors, as is done in aggregate modelling• Historic reopening and settlement experience is

representative of future• Method cannot be applied blindly – it is not a

replacement of gross aggregate best estimate modelling, rather a tool to analyse variability around the aggregate modelling, and netting down of results

24

Variability of Whole Account

• Simulate variability of small claims via “capped” triangle, using existing methods

• Capped triangles preferred to triangles which totally exclude large claims– if claims are taken out once they become large,

we see negative development– if history of claim is taken out, then triangles

change from analysis to analysis– becomes difficult to allow for IBNR large claims

• Add gross excess claims from individual simulations for total gross results, with appropriate dependency structure

• Add net excess claims for total net results

25

Case Study

• UK auto account• 16 years of data• Individual claims > £100k• 2 layers used to simulate IBNER

claims, 80% in lower layer, 20% in upper layer

26

IBNER

• Distribution of one individual claim, current incurred £125k

• Expected ultimate of £300k

• 90% of the time, ultimate cost of claim doesn’t exceed £700k

27

IBNER

• Occasionally the claim can grow very large, however

28

IBNER

• Progression of one claim that has been large for 4 years, and is still open

• Still significant variability in ultimate cost

29

Ultimate Loss Development Factors

• Graph shows ultimate LDF (ultimate / latest incurred) for “big” and “little” claim from same point in development

• Probability of observe an large LDF (>4) 60% higher for small claim than large claim

• Average LDF for small claim 1.1, for big claim 0.87

Percentile: Cumulative Chain Ladder Factors

0

10

20

30

40

50

60

70

80

90

100

0 4 8 12 16 20 24 28 32 36 40Value

Pe

rce

nti

le

Cumulative Chain LadderFactors[big claim]

Cumulative Chain LadderFactors[little claim]

30

Distribution of Capped Reserve

31

Comparison with Mack Method

Accident YearMean

Reserve C.o.V.75th

percentile95th

percentile C.o.V.75th

percentile95th

percentile

1998 (and Prior) 499,653 60.28% 120.51% 212.55% 15.60% 110.21% 126.61%1999 2,836,912 16.52% 107.26% 135.45% 8.65% 105.73% 114.29%2000 4,525,560 21.08% 109.64% 137.95% 25.06% 116.66% 141.34%2001 6,582,895 24.46% 112.33% 144.18% 23.08% 115.10% 138.96%2002 7,073,142 28.99% 114.39% 153.25% 27.41% 117.91% 146.37%

2003 12,608,970 32.81% 113.21% 161.58% 18.46% 112.19% 130.81%

2004 12,265,893 25.46% 113.75% 147.09% 29.15% 118.97% 149.76%

2005 15,134,996 30.66% 114.44% 154.19% 30.97% 120.17% 153.40%

Total 61,528,020 12.53% 107.29% 121.70% 13.56% 109.04% 123.04%

Individual Claim Projection Method Mack Bootstrap

32

2003 Distribution

• Higher proportion of large claims

• One claim of £6m• Greater

uncertainty than implied by aggregate projection

33

2004 and 2005 Distributions

• Distributions from individual claims distributions slightly heavier tailed than aggregate method

• Caused by increase in large claims proportions over time, not adequately allowed for in aggregate methods

34

Netting Down

35

Reinsurance Structures

• Even simple portfolios can have reinsurance structures that are difficult to model– Aggregate Deductibles– Loss Occurring During vs Risk Attaching

coverages– Partial Placements– Indexation Clauses

• By having individual claims, can explicitly allow for any structure

36

Example: Aggregate Deductible

• Graph shows percentile chart of the usage of a £2.25m aggregate deductible attaching to layer £400k XS £600k

37

Conclusion

• Existing stochastic methods work well for homogenous data, but some lines of business are dominated by small number of large claims

• Treating these claims separately allows existing methods to be used on the attritional claims, with our individual claims simulation technique allowing for variability in these large claims explicitly

• This allows net and gross results to be calculated on a consistent basis, allowing explicitly for any reinsurance structures in place