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TRANSCRIPT
APRA Risk Margin Analysis
prepared by Scott Collings and Graham White
Trowbridge Consulting
for
the Institute of Actuaries of Australia
XIIIth General Insurance Seminar
25-28 November 2001
TABLE OF CONTENTS
Main Report
1. Introduction
2. Variability Analysis
3. Correlation Analysis
Appendices
A. Variability Analysis Results
B. Correlation Analysis Results
C. Description of the Mack Method
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1. Introduction and Scope
APRA have recently released new standards for the determination of liability valuationand solvency for Australian general insurers. The regulations require that the centralestimate plus risk margin of the insurance liabilities (ie the sum of the outstanding claimliabilities and premium liabilities) secures the 75th percentile of the underlyingdistribution (subject to the risk margin being greater than or equal to half of thecoefficient of variation). Under the new standards the Approved Actuary of an insurer isresponsible for determining these risk margins.
The risk margin assessment is required at the company aggregate level, but in addition,the risk margin for the outstanding claim liabilities and premium liabilities for each classof the portfolio also have to be reported. The total of the risk margins for the individualclasses will not equal the overall risk margin, because there is not perfect correlationbetween the classes. The difference between the overall risk margin and the sum of theindividual risk margins is referred to as a diversification benefit and has to be allocatedback to the individual classes.
This research paper has been prepared to assist actuaries in the application of the newAPRA standards, in particular, in the area of determination of “risk margins”. It is basedon internal research that has been accumulated over a number of years and more recentlyadapted to specifically address the APRA requirements. It describes the research we haveundertaken and documents the conclusions we have reached. In particular it includes:
• a description of the alternate methodologies we employed to estimate variability
• the individual results of our analysis of 60 direct insurer portfolios
• sensitivity analysis of our results
• suggested benchmarks for risk margins by type and size of portfolio
• our assessment of the correlation between classes of business (for the purpose ofquantifying the diversification adjustment).
We do not intend for our results to be taken as definitive, far from it. By describing ouranalytical approach and disclosing our results in detail we are hoping to give readers anunderstanding of how we derived our results as well as giving them scope to draw theirown conclusions.
We anticipate fully that further research will be required. Despite our efforts here it isstill inevitably true that “there is uncertainty about the uncertainty” (thanks Bob).
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2. Variability Analysis
This section sets out:
• our methodology for analysing the variability in outstanding claims portfolios,including commentary on the alternative approaches that were considered;
• the results of our research into the variability of outstanding claims portfoliosincluding our recommended risk margin benchmarks for each major class ofbusiness; and
• details of our analysis of premium liability variability and a recommendedapproach to the selection of premium liability risk margins.
Methodology
The goal of this research is two-fold:
1. to provide guidance on what should generally be considered reasonable riskmargins for different types and sizes of portfolio; and
2. to provide an appropriate methodology to apply in developing estimates of therequired risk margin for individual portfolios.
The three methodologies we employed in the course of our analysis of outstanding claimsvariability were the Mack method, Bootstrapping and the Stochastic Chain Ladder. Abrief description of each of these methods is set out below.
Mack
This is a well-known stochastic reserving method and was developed by Thomas Mack ofMunich Re in the early 1990s1. It is based on the chain ladder method and derives anestimate of the standard error of the estimated liability using a distribution free approachand requires no simulation.
It employs a strictly mechanical approach without allowing for or requiring any userintervention (the mean and standard deviation of chain ladder factors are generateddirectly from the data using the weighted average of the historical experience). It may beapplied to either cumulative paid or incurred development data. The standard errors forthe outstanding claims liability for each accident year and overall are derived using
1 Mack, T (1993) Distribution free calculation of the standard error of chain ladder reserve
estimates, ASTIN Bulletin 23, 2, pp213-225
Mack, T (1994) Measuring the Variability of Chain Ladder Reserve Estimates, CAS 1994
Spring Forum, pp101-182
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formulae that are automatically parameterised from the triangular data. It is worth notingtwo important assumptions which are stated as underlying this method:
(i) no accident year will develop beyond the current stage of development of theoldest accident year; and
(ii) actual outstanding claim payments are assumed to be uncorrelated betweenaccident years.
To avoid confusion it needs to be noted that the formula Mack provides for the overallvariability includes an allowance for (process) correlation between the reserve estimatesof each accident year. This is purely a function of the chain ladder projection processitself, ie. the same development factor being applied across all accident years at the samestage of development during the reserve estimation process. The correlation referred to in(ii) above is not addressed by this adjustment.
Further details of the Mack method are set out in Appendix C.
Bootstrapping
This is a technique derived from classical simulation techniques of the 1970s2. Thebootstrapping methodology proceeds as follows:
1. The triangle of paid data is projected to ultimate, using the standard chain laddermethod. The user is free to select average chain ladder factors incorporating priorknowledge and other experience as appropriate.
2. Another triangle of data is derived by backwards-engineering the chain ladderfactors from the ultimates.
3. This is then compared to the original data and the differences, called residuals, arenoted.
4. It is assumed that this is a typical representation of the way in which the real dataand the model could differ. Thus, if this is thought of as being a realisation of arandom process, another set of data may be generated which differs from the modelat any point by any one of the residuals.
5. Pseudo-data triangles are then generated by randomising the set of residuals (ie thesame residual amounts are used by arranging them randomly in the residualtriangle) and adding these back to the model data triangle (at 2).
2 Lowe, J (1994) A practical guide to measuring reserve variability using: Bootstrapping,
Operational Time and a distribution free approach, Proceedings of the 1994 General Insurance
Convention, Institute of Actuaries and Faculty of Actuaries
England, PD & Verral, RJ (1999) “Analytic and Bootstrap Estimates of Prediction Errors in
Claims Reserving” Insurance: Mathematics and Economics 25
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6. These pseudo-data triangles are then projected to ultimate, using a mechanicalchain ladder approach. From each of these ultimates, the future reserve is derivedand the simulations produce a distribution of reserves.
The figure below shows the bootstrapping method diagrammatically.
Figure 2.1 – Bootstrapping Method
(Taken from the 1994 UK General Insurance Convention Papers)
Stochastic Chain Ladder Method
This method, as its name suggests, employs a chain ladder projection of outstandingclaims liabilities. Chain ladder factors are defined by distributions rather than pointestimates and simulation is used to generate the distribution of the outstanding claimsliability. It can be applied to either payment or incurred cost development data andrequires no particular form of distributional assumption.
Our particular adaptation of this methodology3 was originally developed to assist inderiving variability assumptions for use in dynamic financial analysis modelling. Theuser needs to make three sets of parameter selections from the data and associatedanalysis:
• average incurred chain ladder development factors;
• standard deviation of each chain ladder factor; and
3 Renshaw, AE and Verral, RJ (1994) A stochastic model underlying the chain ladder
technique. Proceedings XXV ASTIN Colloquium, Cannes
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• paid to date (as a percentage of incurred) at each development period.
Assumptions are required for the number of development years it is expected to take toultimately pay out the claims liabilities. Where the data is not sufficient or reliableenough to base parameter selections on, assumptions are still required in order not tounderestimate the variability. In such cases the approach would include somecombination of the use of fitted decay factors, hand smoothing and reference to similarportfolios with better data. Particular care should be taken in assuming the comparabilityof different portfolios (even where the class of business is the same). At a minimum,operational time adjustments must be considered, as should portfolio size.
The simulation process proceeds as follows:
• the model for each chain ladder factor uses the mean and standard deviationassumptions selected above and assumes a LogNormal distribution;
• for any given simulation a unique chain ladder factor is simulated for eachcombination of accident year and future development year; and
• a correlation matrix is used to create correlation between accident years for thefactors sampled in a given projection year (ie. diagonal).
Using these assumptions a deterministic projection can be made of the outstanding claimpayments (as a proportion of the projected ultimate claims cost) for an accident year atany stage of development. The process is then made stochastic by randomly simulatingchain ladder factors from the parameterised LogNormal distribution. This produces adistribution for the outstanding claims liabilities by accident year and in aggregate.
Within this process negative incurred claims cost development is allowed, however, aconstraint is imposed on the distribution of the chain ladder factors. The factor drawn canin the event of “good” experience, at most, reduce incurred costs by 50% of the currentcase estimates for any given accident year in any one development period. No constraintis placed on positive claims cost development.
As mentioned, explicit allowance is made for chain ladder factor correlation betweenaccident years in any given projection year (ie. along the diagonal). This was consideredimportant for long-tail classes where claims inflation has a tendency to operate in thisway. After some sensitivity analysis we decided on a simulation basis whereby eachaccident year is:
• 50% correlated with the two accident years one year either side;
• 25% correlated with the two accident years two years either side; and there is
• no further correlation.
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In developing risk margin benchmarks we decided that they should be based on “mature”portfolios. The model we developed allows for explicit weighting of the variability byaccident year (as well as the abovementioned correlation) in arriving at the aggregatevariability for a portfolio so that explicit adjustments could be made to analyse the impactof different degrees of maturity (high growth, run-off etc). The approach we have taken,therefore, is to use the stochastic chain ladder basis derived from the actual data for eachportfolio we analysed and to then derive aggregate CV results for benchmarking purposesusing accident year weights which assume a mature portfolio. A mature portfolio here isdefined as one that has been growing at a steady 10% per annum (consistent with industryexperience over the last decade).
Choosing a Methodology
We chose the SCL method as our main modelling approach. In summary, the reasons forthis choice were:
• chain ladder assumptions (mean and standard error) are made by the user ratherthan mechanically. This facilitates the use of smoothing techniques to avoid over-emphasising unusual features in the data;
• the method develops a model of both payment and incurred claims developmentsimultaneously rather than separately, thereby using more of the available data;
• explicit assumptions can be made about accident year correlation;
• it is straightforward to re-weight the accident year results to approximate the resultfrom a similar portfolio at different stages of maturity;
• the length of development can be chosen by the user to reflect their expectationsrather than being limited to the maximum development currently present in thedata; and
• it is a more open methodology which facilitates the development of greaterunderstanding of the sensitivities inherent in the modelling process.
Risk Margins for Outstanding Claims
Our analysis was applied to 60 different portfolios belonging to direct insurers. This datawas predominantly net of reinsurance recoveries. In each case we used the SCL methodas our primary modelling tool. The Mack method was also used for each portfolio as aform of reasonableness check (bearing in mind its limitations). Details of the results foreach portfolio are set out in Appendix A.
CTP
We had a large number of portfolios representing the privately underwritten states andsome public sector schemes. There was also a good range of large and small portfolios.
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The results demonstrate the expected link between decreasing size and increasingvariability. The SCL results are generally more consistent than those of the Mackmethod, the latter producing two fairly extreme outliers.
Figure 2.2 – CTP Variability Results
0%
10%
20%
30%
40%
50%
A B C D E F G H I J
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
93%
Fire
We had only five portfolios but with a good spread by size. There appears to be a stronginverse relationship between size and variability. The SCL and Mack results are broadlysimilar although Mack is consistently higher.
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Figure 2.3 – Fire Variability Results
0%
10%
20%
30%
40%
50%
60%
A B C D E
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
House
We had a good number and spread of Householders portfolios. The results appear almosttotally independent of portfolio size with only one portfolio (a very small one at that)standing out with the rest mostly hovering in the 5% to 8% CV range. The Mack resultsgenerally agree with the SCL results although there is one extreme outlier (due to theimpact of unusual development associated with Newcastle earthquake claims).
Figure 2.4 – House Variability Results
0%
5%
10%
15%
20%
25%
30%
A B C D E F G H
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
61%
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Liability
We had a half-dozen portfolios mostly in the medium size category. As expected, thereappears to be an inverse relationship between size and variability. The largest portfolio,however, was an obvious outlier (particularly for the Mack results) due, it seems, to anunstable development history.
Figure 2.5 – Liability Variability Results
0%
5%
10%
15%
20%
25%
30%
35%
40%
A B C D E F
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
161%
Motor Commercial
We had only a few portfolios here and the SCL and Mack results generally differed quitesignificantly.
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Figure 2.6 – Motor Commercial Variability Results
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
A B C
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
Motor Personal
We had a good number of predominantly small to medium sized portfolios. The assumedinverse relationship between size and variability is not as clear-cut as in other classes.The results fell into an incredibly narrow range, despite the small size of a number of theportfolios. The Mack results displayed a wider range but were not systematically higheror lower then the SCL results.
Figure 2.7 – Motor Personal Variability Results
0%
2%
4%
6%
8%
10%
12%
14%
A B C D E F G H I
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
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Professional Indemnity
Only three portfolios were analysed. The results indicated generally high variability anda clear inverse relationship between size and variability. The Mack results were close tothe SCL results with one (higher) exception.
Figure 2.8 – Professional Indemnity Variability Results
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
A B C
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
Workers’ Compensation
We had a good number of portfolios to analyse, all of which were in the small to mediumcategory. Each portfolio represents their total underwritten workers’ compensationbusiness and excludes the run-off states. The size/variability relationship is evidentdespite the odd exception. The Mack results were generally close to but lower than theSCL results and there was one extreme outlier (for the smallest portfolio).
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Figure 2.9 – Workers’ Compensation Variability Results
0%
5%
10%
15%
20%
25%
30%
A B C D E F G H
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
183%
Other
The contents of these portfolios are hard to characterise but typically they includeanything not covered by the rest of the classes analysed nor Marine which was keptseparate. In particular, they do not include any long-tail business, although a number ofthem do include bits of property insurance that might well be called house, fire or motor.
We had eight portfolios with a good range between small and large. The link betweensize and variability appears to be reasonably consistent despite one of the medium-sizedportfolios bucking the expected trend due to a quite unstable development history. TheMack results are generally similar to the SCL results with a tendency to be higher for thesmaller portfolios (including one extreme outlier).
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Figure 2.10 – Other Variability Results
0%
5%
10%
15%
20%
25%
30%
35%
40%
A B C D E F G H
Portfolios Ranked from Largest to Smallest (left to right)
CV
MACK SCL
118%
Based on the preceding analysis we propose the CV and risk margin benchmarks set outin the table below. We have grouped our benchmarks according to portfolio size usingnational market share of gross premium to divide the portfolios into large ( > 6%), small( < 2%) and medium (in between).
Table 2.1 – Summary of CV and Risk Margin Results
CV (%s) APRA Risk Margins Class
Large Medium Small Large Medium Small
Motor (Per) 5-6 6-8 8-11 3-4 4-5 5-6
House 5-7 7-9 9-12 3-4 4-5 5-7
Fire 6-10 10-20 20-30 4-6 6-11 11-15
Other 7-10 10-15 15-20 4-6 6-9 9-11
Motor (Comm) 7-11 11-15 15-20 4-7 7-9 9-11
Liability 8-12 12-16 16-22 5-8 8-9 9-12
Workers’ Comp 8-13 13-18 18-25 5-8 8-10 10-13
PI 8-13 13-20 20-30 5-8 8-11 11-15
CTP 8-13 13-20 20-30 5-8 8-11 11-15
The table below indicates the equivalent gross premium amount for the market sharesused to separate our definitions of large, medium and small portfolios.
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Table 2.2 – Portfolio Size Definitions ($m)
Market Share of Premium
100% 6% 2%
Motor (Per) 3,685 221 74
House 2,494 150 50
Fire 1,407 84 28
Other 1,503 90 30
Motor (Comm) 1,054 63 21
Liability 898 54 18
Workers’ Comp 859 52 17
PI 641 38 13
CTP 2,038 122 41
Use of Benchmarks
These benchmarks are only intended as a guide to what “normal” might be in relation toCVs and risk margins. We would always recommend that each individual portfolio bemodelled for an estimate of its own variability provided sufficient data exists.
Once a model has been developed for a portfolio it should be used to estimate the CV onan equivalent basis to that of the benchmarks (ie. using accident year weightings thatapproximate a mature state). Estimates of the portfolio size based on premium marketshare will enable the estimated “mature” CV to be compared with the appropriatebenchmark.
For every such analysis it will be important to consider the appropriateness of the adoptedmodel (whether or not on the first attempt your results exactly match the appropriatebenchmark). Important issues to consider mainly relate to data reliability. Are there anybreaks in the development patterns that are unrepresentative of future variability? Thiscould be due to:
• changes in case estimation practices;
• changes in underlying exposure (due to legislative changes or, for example, the mixof property/casualty losses);
• the presence of major event losses will usually distort loss development patterns forproperty portfolios, especially when they occur late in the accident year; and
• bouts of claims inflation. To some extent these must be allowed for in futurevariability (especially for long-term portfolios) but the issue is whether it isunder/over-represented in the historical data.
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These are all judgemental to some extent but not beyond the normal range of issuesfacing an actuary preparing outstanding claims estimates.
Another issue is the appropriateness of a chain ladder projection method for the liabilitiesin question. This is not a method generally used for non-proportional reinsuranceportfolios. This is a particular area where further research is currently underway.
Sensitivity Analysis
A range of sensitivity tests were carried out to assess the robustness of the results. Abrief summary of these tests is set out below.
Reconciliation to Mack
For a representative range of portfolios we modified our SCL model to mimic the Mackapproach in order to check that our results converged. The adjustments we made were to:
• remove assumed accident year correlations
• remove Standard Deviations from the tail beyond and including the last actual datapoint (ie assuming that the oldest accident year was fully developed and allowingno variability after this)
• replacing our manually selected chain ladder factors with the mechanically derivedfactors that Mack uses
• using the actual maturity profile of the portfolio rather than that assumed for ourbenchmarking exercise.
Uniformly our modified SCL result converged towards the Mack results generally leavingno more than 30% of the original difference unexplained. The residual difference is dueto the remaining basic structural differences between SCL and Mack.
Chain Ladder Basis
The sensitivity of our CV results to the selected chain ladder basis was tested byalternately scaling the mean and standard deviation of the first three chain ladder factors(CLF) by 1.2 and 0.8. The results are summarised below averaged across long and shorttail classes of business.
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Table 2.3 – Chain Ladder Basis Sensitivity
It can be seen that the short-tail CV results are much more sensitive to the selected meanCLF. This is because a 20% increase in projected ultimate cost produces a largerpercentage increase in outstanding claims for a short-tail portfolio than for a long-tailportfolio. With the CLF standard deviation unchanged, the CV of the liabilities thereforefalls further for a short-tail portfolio.
Similarly, changes in the CLF standard deviation have a larger impact on the CV of short-tail portfolios. Note that this effect is reduced in the second and subsequent developmentyears. Again, this is because the bulk of short-tail liabilities are expected to have alreadybeen paid out so there is less scope to change the CV of the liabilities through latevariability.
Choice of Distribution
We investigated the impact of using a Normal distribution instead of the LogNormal tomodel the chain ladder factors in the SCL method. In general the LogNormal produces aslightly higher CV but, due to its skewness, often produces a lower 75th percentile thanthe Normal distribution.
Portfolio Maturity
As described earlier our approach has been to develop risk margin benchmarks for whatwe call a “mature” portfolio. We have defined mature as a state whereby the portfoliohas been growing in terms of new claims costs at a steady 10% per annum and has beendoing so for more years than the length of the payment pattern of that portfolio.
Long-tail Short-tail Long-tail Short-tail
Scale by 1.2CLF1 0.98 0.65 1.02 1.14CLF2 0.96 0.54 1.06 1.04CLF3 0.96 0.47 1.05 1.02
Scale by 0.8CLF1 1.04 1.11 0.99 0.87CLF2 1.07 0.89 0.96 0.96CLF3 1.07 0.99 0.97 0.99
Impact on Overall CV due to scaling ofMean CLF Std Dev CLF
Ratio of Scenario CV to Benchmark CV
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Clearly not all insurer portfolios have reached this state. The SCL model we havedeveloped, however, allows us to apply the exact maturity profile of whatever portfolio isbeing analysed (and to test how its CV might change under certain growth assumptions).
To understand better the likely impact of the state of maturity on the CV and risk marginof a portfolio we tested a representative sample of portfolios under the followingscenarios:
• high growth: 30% growth per annum
• contraction: 10% decline per annum.
The accident year weightings of variability were adjusted to reflect each of thesescenarios. The results are set out below.
Table 2.4 – Portfolio Maturity Sensitivity
It is clear that the impact of higher growth is to increase the portfolio CV. For the long-tailed and more variable portfolios this effect is magnified.
Accident Year Correlations
Before deciding on our final basis we tested the impact of a range of accident yearcorrelation assumptions. The table below summarises the results of these tests anddemonstrates to what extent the CV (and hence risk margin) is affected by the assumptionof accident year correlation.
Table 2.5 – Accident Year Correlation Sensitivity
Portfolio Type High Growth Contraction
CTP 1.20 0.75Liability 1.14 0.86House 1.04 0.92
Ratio of Scenario CV to Benchmark CV
Growth Scenario
AdoptedClass Basis 25% 50% 100%
CTP 1.33 2.30Fire 1.18 1.55House 1.18 1.68Liability 1.38 1.46 1.82 2.43Motor (Com) 1.17 1.44Motor (Per) 1.11 1.10 1.18 1.35Other 1.22 1.89Workers' Comp 1.31 2.25
Uniform Correlation all Acc Yrs
Ratio of Scenario CV to Zero Correlation CV
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Our adopted basis increases the CVs by a factor of 1.1 to 1.2 for short-tail portfolios andby 1.3 to 1.4 for long-tail portfolios. For the two classes tested our adopted basis appearsclose to the result obtained by assuming a uniform 20% correlation between all accidentyears.
Risk Margins for Premium Liabilities
It is generally recognised that the volatility of the premium liabilities of a class will begreater than that for outstanding claims. This is because the exposure period for theseliabilities has not yet occurred and events such as future catastrophes need to be allowedfor. This general reasoning approach leads to the expectation that short tail classes willshow a larger difference between premium liabilities and outstanding claims volatilitiesthan long tail classes.
Premium liabilities should contain a slightly greater degree of variability to that of themost recent accident year. We extracted these values from the results of our SCL modelfor each of the portfolios analysed for outstanding claims variability. The table belowsets out a summary of the ratio of the CV for the latest accident year to the CV of the totaloutstanding claims. Again note that these figures are all based on our “mature” portfoliobasis.
Table 2.5 – Current Accident Year CV Ratios to Overall Class Ratio RangeLong-tail PI 2.2 - 3.0
CTP 2.0 - 2.2WC 1.6 - 1.8Liab 1.7 - 1.9
Short-tail Motor C 1.1 - 1.3Other 1.1 - 1.3Fire 1.0 - 1.2House 1.0 - 1.1Motor P 1.0 - 1.1
The observed ratios are significantly higher for long-tail than for short-tail. The short-tailresults are not surprising given that the vast majority of the outstanding claims liability(and associated variability) resides with the most recent accident year. This is much lessthe case for a mature long-tail portfolio.
The short-tail ratios shown above will understate the variability for premium liabilities.Separately we have conducted a detailed study of the variability of the annual loss ratio ofeach class of short-tail business based on both individual company portfolios and industryaggregate statistics. While based on accounting year “incurred” measures the variability
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of these loss ratios is a good proxy for accident year loss ratio variability. Allowing foran “average” level of catastrophe reinsurance protection we have used this analysis toadjust the ratios in the table above for the short-tail classes. The House and Fire classesdeserve the largest adjustment due to event exposure, followed by Other then Motor.
The long-tail ratios, while they exclude the impact of claim frequency uncertainty, morethan likely compensate for this with additional “modelling” error. The first chain ladderfactor in the projection of a long-tail portfolio has by far the highest standard deviation.We have supplemented our analysis here with separate research into long-tail accidentyear loss ratio variability. This research suggests that the ratios above are slightly higherthan required for long-tail premium liabilities.
The table below summarises our conclusions for “multipliers” that when applied to theCV of the outstanding claims liability will provide a reasonable estimate of the premiumliability CV.
Table 2.6 – Selected Premium Liability CV Multipliers Long-tail PI 2.0
CTP 1.8WC 1.6Liab 1.6
Short-tail Motor C 1.3Other 1.4Fire 1.6House 1.5Motor P 1.2
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3. Correlation Analysis
So far we have considered only the process of deriving estimates of the uncertainty (andhence risk margins) for the outstanding claims and premium liabilities of a singleportfolio. In order to derive the overall risk margin required for a company it is necessaryto incorporate an allowance for the degree of correlation between the liabilities of eachportfolio. The CV of a company’s aggregate liability for outstanding claims andunexpired risk (premium liability) can then be determined by combining these correlationcoefficients (in matrix form) with the CVs of the outstanding claims and premiumliabilities of each portfolio.
The correlations required were considered in the following categories:
• outstanding claims liabilities: between classes
• premium liabilities: between classes
• within each class: between outstanding claims and premium liabilities.
The derivation of such correlation coefficients is, not surprisingly, difficult. Appropriatehistorical data upon which to measure these correlations is almost impossible to come by.Where sufficient data exists to make a rough estimate the usual limitation about whetherthe past can be considered representative of the future can be a major obstacle. Past“blips” in claim development experience can either create or destroy measured correlationbetween classes depending on whether they coincide – but regardless the questionremains whether something fundamental was at work or whether it was mere coincidence.
Against this background we have taken a pragmatic route:
• measure correlation as best we can
• but, lean towards general reasoning and intuition.
The correlation matrix we have attempted to calibrate includes both outstanding claimsand premium liabilities. For any two classes there are 6 correlations that need to beconsidered. Ten classes were chosen, leading to a 20 by 20 matrix and 190 individualcorrelations to consider.
Our assessment of each category of correlation is set out below.
Between Class Correlations for Outstanding Claims
The correlations we require are measures of the tendency for the estimated outstandingclaims liabilities of different classes of business to together turn out to be too high or toolow relative to the amounts that are ultimately paid.
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The factors that might influence this correlation include:
Company-specific
• the reserving practices of the insurer: are case and actuarial estimates preparedcentrally and consistently or in a decentralised manner without regard forconsistency? What is the likelihood of reserves being moved up or down acrossthe company other than by pure coincidence?
• the quality of the data available: poor data generally leaves a lot of scope forjudgement to be exercised (in such situations the reasonable range of judgementsmay be quite wide). The tendency to be relatively pessimistic or optimistic in suchjudgements may well be consistently applied across the company
Systemic
• the drivers of claims cost inflation: What is the weight of influence of:
◗ CPI and AWE inflation
◗ social, medical and judicial inflation
◗ currency changes
◗ unemployment rates, interest rates or the economy in general
Non-systemic
• the complexity of the business: does it lend itself to more or less accurate reserveestimates?
• claim reporting delays: for classes where unreported claims represent a significantproportion of the liability estimate this can introduce considerable additionaluncertainty to the liability estimate. One needs to consider whether this source ofuncertainty is likely to be a force for or against correlation between classes.
In general we would view the company-specific and systemic factors as acting to createpositive correlation. The non-systemic factors are more likely to act randomly and have atendency to destroy correlation.
Our assessment of correlation included both qualitative and quantitative components.
Our qualitative assessment consisted of a survey of all of the experienced generalinsurance actuaries in our firm. Each was asked to identify the degree of correlationbetween outstanding claims liabilities for ten different classes of business as either High,Medium or Low (and positive or negative). Once the survey results were compiledseveral meetings were held during which the survey results were discussed for eachpairing of classes. Participants put forward their views as to the causes and extent ofcorrelation and a consensus view was formed. These results are shown in the tablebelow.
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Table 3.1 – Outstanding Claims Correlation: Initial Qualitative Assessment
It was generally agreed that these correlations represent systemic correlation only. Non-systemic correlations would have to be considered on a company-specific level.
There was also discussion of what level of correlation each of High, Medium and Lowrepresented. Note that negative correlations were not considered likely. Our initial viewswere as follows:
Low: 0 – 20%Medium: 20 – 40%High: 40 – 60%
It was generally agreed that degrees of correlation in excess of 60% were extremelyunlikely and in any event would warrant a separate category of Very High.
Some quantitative analysis has also been undertaken although there is still further work tobe done. In brief, we have estimated the correlation between matching cells (by accidentand development year) within the chain ladder triangles of pairs of different classes ofbusiness (for the same insurer). The main variable measured for correlation was theresidual left after deducting the “average” chain ladder factor (for that development year)from the actual chain ladder factor. We have analysed about 10 pairs of portfolios in thisway. So far the results do not appear stable enough to support a change to our qualitativeassessments.
Our final selected outstanding claims correlation matrix is shown below.
Home Marine Motor CTP WC Liability PI OtherFire M L L L L L L LHome L L L L L L LMarine L L L L L LMotor L L L L MCTP M M L LWC M L LLiability M LPI L
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Table 3.2 – Outstanding Claims Correlation: Final Selections
Between Class Correlations for Premium Liabilities
In addition to the sources of correlation already identified for outstanding claimsliabilities we would expect premium liabilities correlation to be further influenced mainlyby claim frequency. The ultimate cost of claims arising from the unexpired risk will beclosely linked to the number of claims incurred. Potential sources of correlation betweenclasses will arise where the claim frequency drivers are similar, eg. catastrophe and otherweather-related events will influence both house and fire classes and to varying extentsmotor, marine and other.
Once again our assessment of correlations included both qualitative and quantitativecomponents. Our qualitative assessment followed the same process as employed foroutstanding claims. The results of this survey process are set out in the table below.
Table 3.3 – Premiums Liabilities Correlations: Initial Qualitative Assessment
Our quantitative assessment involved analysis of the correlation between industryaggregate loss ratios. This was able to be taken directly from research already conductedby the DFA practice to support their efforts in DFA model calibration. The table belowshows the cross-class correlations derived from this loss ratio analysis.
Home Marine Motor CTP WC Liability PI OtherFire 40% 10% 20% 0% 0% 0% 0% 20%Home 10% 20% 0% 0% 0% 0% 20%Marine 10% 0% 0% 0% 0% 10%Motor 0% 0% 0% 0% 30%CTP 25% 25% 10% 0%WC 25% 15% 0%Liability 30% 0%PI 0%
Home Marine Motor CTP WC Liability PI OtherFire H L M L L L L MHome L M L L L L MMarine L L L L L LMotor M L L L MCTP M M L LWC M L LLiability M LPI L
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Table 3.4 – Industry Analysis of Historical Loss Ratio Correlations
Although some judgemental allowance was made for price correlations within theselections above, for the purpose of the premium liabilities all price correlations need tobe removed. Further technical analysis of price correlations was carried out byexamining premium rate movements over the last seven years. Unfortunately more datahistory was not available. The results of this analysis are shown below.
Table 3.5 – Assessment of Pure Price Correlation in Loss Ratio Analysis
Using this extra information led to a slight revision of the matrix derived from thehistorical loss ratios. The changes are highlighted by shading.
Table 3.6 – Restated Industry Loss Ratio Correlations
Home Marine Motor CTP WC Liability PI OtherFire 50% 40% 30% 20% 20% 30% n/a 10%Home 20% 30% 10% 10% 20% n/a 20%Marine 20% 10% 10% 40% n/a 40%Motor 0% 0% 0% n/a 20%CTP 40% 20% n/a 10%WC 30% n/a 0%Liability n/a 10%PI n/a
Home Marine Motor CTP WC Liability PI OtherFire H n/a H L L H M n/aHome n/a H nil nil M M n/aMarine n/a n/a n/a n/a n/a n/aMotor L L M M n/aCTP M M M n/aWC M M n/aLiability H n/aPI n/a
Key: H = High Corr, M = Medium Corr, L = Low Corr, nil = No Corr, n/a = No Info Available
Home Marine Motor CTP WC Liability PI OtherFire 45% 40% 20% 10% 0% 10% n/a 10%Home 20% 20% 0% 0% 10% n/a 20%Marine 20% 0% 0% 20% n/a 40%Motor 0% 0% 0% n/a 20%CTP 30% 10% n/a 10%WC 20% n/a 0%Liability n/a 10%PI n/a
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Using this matrix and incorporating the qualitatively assessed matrix, we derived thefollowing final results for our premium liability correlation matrix.
Table 3.7 – Premiums Liabilities Correlations: Final Selections
Overall Matrix
The two matrices derived in the previous sections need to be combined to produce anoverall correlation matrix. In addition to these two matrices, the following extrainformation is needed:
• The outstanding claim to premium liability correlations within the same class (intraclass).
• The outstanding claim to premium liability correlations between classes (interclass).
Our starting position was that within a given class the correlation between premiumliabilities and outstanding claims liabilities should be quite high. Premium liabilities afterall represent the next accident year of claims costs (weighted more towards the start of theyear). Working against the correlation is the fact that uncertainty in premium liabilitiesincludes the additional influences of claim frequency and pricing.
Considering the long-tail classes we would expect that the bulk of the ultimate claimscost uncertainty relates to movements in the cost of claims which may be several yearsaway and which will (when it happens) exert a strong influence on both today’s premiumliabilities and outstanding claims liabilities. Uncertainty due to claim frequency is arelatively less significant component for long-tail classes and unlikely to materiallydisturb the correlation between premium liabilities and outstanding claims liabilities inmost cases.
For short-tail classes the settlement conditions applying to outstanding claims are only amatter of months in advance of these applying for today’s premium liabilities. Averageclaims cost correlation should be relatively high. On the other hand, short-tail classeshave a significant exposure to catastrophes residing in their premiums liabilities. This has
Home Marine Motor CTP WC Liability PI OtherFire 50% 20% 30% 0% 0% 0% 0% 30%Home 20% 30% 0% 0% 0% 0% 30%Marine 10% 0% 0% 0% 0% 20%Motor 20% 0% 0% 0% 20%CTP 30% 20% 10% 0%WC 20% 10% 0%Liability 30% 0%PI 0%
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the very real potential to destroy correlation between premium liabilities and outstandingclaims liabilities.
As a generalisation we have assumed that short tail premium and outstanding claimliabilities are less positively correlated than long tail. We have selected 70% for long tail,50% for short tail and 60% for those classes in-between.
For the inter class correlations, we would expect a slightly lower correlation between thepremium liabilities of one class with the outstanding claim liabilities of another classcompared with the correlation assumed between the premium liabilities or between theoutstanding claims liabilities of the two classes.
We have assumed that the premium liability versus outstanding claim liability correlationof two classes will be roughly 10% lower than the minimum of the correspondingpremium liability versus premium liability or outstanding claims versus outstandingclaims correlations.
With these additional assumptions, the overall correlation matrix was constructed. This isshown in Appendix B.
Positive Definite Tests
Testing a matrix for positive definiteness involves looking at the eigenvalues of thatmatrix. This essentially tests the internal consistency of the matrix (eg if A is highlycorrelated with B and B is highly correlated with C, then A should be fairly stronglycorrelated with C). The detailed methodology can be found in most basic statistical texts.The tests performed showed that the overall correlation matrix selected is definitely,positively, positive definite.
Diversification Adjustment
APRA requires companies to report the risk margin based on the aggregate of technicalliabilities as well as at a portfolio level. Lack of perfect correlation between portfolioswill result in this aggregate risk margin being less than the sum of the portfolio riskmargins. This discount is referred to as the diversification adjustment.
To understand the size of the likely diversification adjustment we applied our selectedmatrix of correlations to a portfolio representing the current industry mix of unexpiredrisk and outstanding claims. Our CVs were based on our selections for large portfoliosand therefore relate predominantly to systemic risk rather than independent risk.
We found that such a portfolio would require an aggregate risk margin which is 52%lower than the sum of margins on the individual portfolio components (outstandingclaims and unexpired risk). By contrast assuming no correlations whatsoever resulted in
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a diversification adjustment of 62%. Even with reasonable looking correlationassumptions it therefore appears that a mature company with an industry mix of businessmight expect a diversification adjustment of the order of 50%.
App 1
Appendix A – Variability Analysis Results
App 2
Appendix B – Correlation Analysis Results
App 3
APPENDIX C - The Mack Method
Introduction
The Mack method derives a standard error for outstanding claim reserves based on thechain ladder method and without assuming any specific claim amount distribution.
The method may be applied to paids or incurreds. The method involves derivingultimates, so when applied to incurreds one gets a measure of IBNR/IBNER rather thanthe outstanding claim reserve. This is what happens in the numerical example providedin the paper.
Chain Ladder Factors (and Terminology)
The chosen chain ladder factors are mechanically derived, with no allowance forjudgement. The method is based on the weighted average of the individual factors. Thisis the “usual” chain ladder factor and is denoted by:
fk1 = (Σj Cj,k+1 )/ (Σj Cjk )
where:
k refers to a particular development period1 refers to the weighting used (it is a bit of Mack terminology and see below)Cjk are the cumulative incurred figures (or payments) in accident year j and developmentyear k.
The method also considers the unweighted average (denoted by fk2) and the averageweighted by the square of the individual Cjks (denoted by fk0), but these are only used ifthe “usual” method is inappropriate. These f’s are defined below:
fk0 = [Σj (Cjk Cj,k+1 )] / (Σj Cjk2 ) {terminology as above}
fk2 = [Σj (Cj,k+1/ Cjk )] * (n-k)-1 {terminology as above and n = total number ofdevelopment periods in triangle}
Assumptions Required
Some basic assumptions need to hold before this method may be applied. These are:
1. accident years are independent
2. the expected value of a development factor for a particular period is independent ofprior development
App 4
3. For a given accident year, the variance of the value in period n+1 is proportional tothe value in period n, irrespective of prior development.
If the third assumption above does not hold, then two alternatives are given to replace thechain ladder factors. These are an unweighted average of the individual factors (denotedby fk2 and defined above) and an average weighted by the square of the individualpayments (denoted by fk0 and defined above).
Tests are given to check the assumptions:
1. For a particular development period, identify whether or not individual chainladder factors are “small” or “large”. The method then uses some derived statisticsto test significance. This method tests for calendar year effects.
2. For a particular development period, rank the individual chain ladder factors. Aftersome manipulation, the method uses Spearman Rank test to test significance.
3. For each development period k, plot the values Cj,k+1 (for each accident year j)against the value Cjk, to see if there is an approximate linear relationship. Thisrelationship should be a line through the origin with a slope equal to fk1 (ie the“usual” chain ladder factor for period k). Plot the weighted residuals(ie [Cj,k+1 – (Cjk * fk1 )] / Cjk
1/2 ) against Cjk to check randomness. If these residualsare not random, a couple of other residual plots are suggested. If Cj,k+1 – (Cjk * fk0 )against Cjk is random then should consider using fk0. If [Cj,k+1 – (Cjk * fk2 )] / Cjk
against Cjk is random then should consider using fk2.
Mack also assumes that there is no further development past the furthest developmentwithin the data triangle. This may lead to an underestimate of the potential volatility,especially with regard to payment triangles for long tail classes.
Methodology
Once the three assumptions underlying the model have been verified, the variables αk2 are
calculated. These are the proportional constants between the variance of the values inperiod k+1 to the value in period k.
The value of αk2 is:
(n-k-1)-1 * [Σj {Cjk *(Cj,k+1/ Cjk - fk )2}] {terminology as above}
The above formula works for all k, except for k = n-1 (where n is the total number ofdevelopment periods in triangle). The value for αn-1
2 is defined as:
Min[(αn-24 / αn-3
2), min(αn-32 , αn-2
2)]
App 5
The αk2 ’s are used to derive the standard errors for the reserves, or IBNR allowance, for
each accident year. These are denoted by se(Ri)and [se(Ri)]2 has formula:
Cin * {Σk>=n+1-i [(αk2 / fk
2) * {1/ Cik + 1/ (Σj Cjk)}]} {terminology as above}
The final calculation relates to the standard error for the overall reserve/IBNR allowance(se(R)) , with [se(R)]2 having the formula:
Σk>=n+1-i { [se(Ri)]2 + Cin * (Σj>=I+1 Cjn) * (Σk>=n+1-i [(2αk
2 / fk2) / (Σj Cjk)]) }
{terminology as above}
Excel Model
An Excel model was created that replicated the numerical results presented in the Mackpaper. The model allows the Mack method to be applied to every portfolio analysed.