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A Method for Projecting Individual Large Claims. Casualty Loss Reserving Seminar 11-12 September 2006 Atlanta. Karl Murphy and Andrew McLennan. Overview. Rationale for considering individual claims Outline of methodology Examples Data Requirements Assumptions Whole account variability

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A Method for Projecting Individual Large Claims

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A Method for Projecting Individual Large Claims

Casualty Loss Reserving Seminar

11-12 September 2006


Karl Murphy and Andrew McLennan


  • Rationale for considering individual claims

  • Outline of methodology

  • Examples

  • Data Requirements

  • Assumptions

  • Whole account variability

  • Case Study

  • Conclusion

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

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

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

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

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

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

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

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

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

Example Data

Claim D

  • 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

Claim D: Simulations

Claim E

  • Closed status as at development period 2

  • Similar to claim A, with no development

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

Claim F Simulations to Year 2

Claim F Simulations to Year 3

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


  • Data below shows the claim number triangle, and frequency of claims


  • Result for one simulation

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


  • 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

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

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


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

  • Expected ultimate of £300k

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


  • Occasionally the claim can grow very large, however


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

  • Still significant variability in ultimate cost

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

Distribution of Capped Reserve

Comparison with Mack Method

2003 Distribution

  • Higher proportion of large claims

  • One claim of £6m

  • Greater uncertainty than implied by aggregate projection

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

Netting Down

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

Example: Aggregate Deductible

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


  • 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

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