Pricing and modelling under the Italian Direct Compensation Card Scheme

Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations

Solvency II premium risk modelling under thedirect compensation CARD system

Seminario di Statistica Assicurativa

Giorgio Spedicato, Ph.D

Universita CattolicaMilan, Italy

August 27, 2011

Giorgio Spedicato Unicatt

Solvency II premium risk modelling under the direct compensation CARD system


Table of contents

1 Overview

2 Recent challenges in insurance businessThe DR system in Italy

Overview of the CARD schemePricing MTPL policies within the CARD scheme

Solvency II

3 An internal model for MTPL UW premium riskThe general frameworkTheoretical modelsThe empirical application

Data set descriptionModel output

4 Final considerationsModel discussionExtension ad developments




Outline

1 Overview



Solvency II







Introduction

This Ph.D thesis presents a possible model to assess UW premiumrisk capital charge on a motor third party liability (MTPL)portfolio handled under a direct reimbursement (DR) scheme.MTPL is the most relevant Italian P&C Market line of business(LOB), accounting for 57% of GWP in 2008 [?]. Multivariatetechniques are currently used to price MTPL contracts.Italian MTPL insurance regulation has known frequent changessince MTPL compulsory requirement in 1969. MTPL pricing hasbeen liberalized in 1994. More recently, in 2007 the regulation hasbeen strongly revised by the introduction of the directreimbursement scheme with the so called CARD agreement. Inaddition, a deep revision of the Bonus Malus transition rules hasbeen put in force.




Introduction

The forthcoming introduction of Solvency II pulls insurers to betterassess risks within portfolio with the ultimate objective todetermine the distribution of the capital at risk (CaR) arising fromthe insurance operation. Underwriting CaR estimation requires notonly the expected value but also the volatility of policyholders’portfolio to be properly assessed.




Introduction

Literature and practical applications exist regarding both SolvencyII underwriting risk component of the CaR and regardingmultivariate classification techniques used to price MTPL tariffs.Nevertheless the introduction of DR scheme in the Italian MTPLbusiness practice brought relevant complications in the process ofpure premium estimation and total loss distribution assessment.This Ph.D thesis deepens the impact of CARD DR scheme onpricing and capital modelling showing one possible approach tomodel premium risk under a DR scheme.




Outline

1 Overview



Solvency II







The CARD system

Regulation reforms have affected relevantly MTPL underwritingand actuarial practices since 2007. They are the second Bersanilaw and the introduction of DR scheme.Bersani laws have halted the structure of Italian experience ratingallowing policyholders with few if any driving experience to inheritthe best BM class within their household. A second provision ofBersani laws allowed tied agents to use their allocated discountbudget more freely. Nevertheless, this Ph.D thesis will not takeinto account the effect of such provisions. [?] carries acomprehensive introduction on these topics.




The CARD system

DR scheme has been put in force in Italy by the so called ”CARD”regulation. According to CARD rules, the insurer of the non responsiblepart indemnifies directly its own policyholder by the full claim amounts ithas suffered for most non responsible claims. The responsible part insurerthen indemnifies the not responsible part insurer by a forfeit amount.The reverse happens when the insured is responsible for the claim.The received forfeit is calculated by a known rule Therefore it is usuallydifferent from the amount effectively paid to the non - responsible part.CARD actuarial challenges arise from:

Negative claim amounts are possible as received forfeit is generallydifferent from actual suffered claim cost.

The frequency and the cost of suffered claims needs to be modelled,in addition to caused claim ones. In fact the average received forfeitusually does not offset suffered claim severity by the same amount.

The shortness of historical experience period affects credibility ofpricing and reserving estimates.

Regulatory environment frequently changes, especially in forfeitdetermination rules.

Actuarial literature on CARD system is currently scarce, even in the

fourth year since DR introduction. Relevant documents discussing CARD

topic are [?],[?],[?], [?],[?] and [?].




The CARD system

Since 2010 received forfeit due to damages to the non-responsible driverdepends by class of vehicle and territory. Figures 1 and 2 show forfeitstructure by territory since 2007. Forfeit due damages to otherpassengers (and CID bodily iis set according to a fair complicated rulecontaining deductible and coinsurance clauses. Formula 1 show CTTforfeit rule for 2007-2009. Effective amount for CID and CTT forfeit arereported in tables 1 and 2 respectively.

F =

{X ≤ 500→ F = 0

X > 500→ F = 3250 + max(

0; X − 5000)−max

(500; min

(0.1 ∗ X ; 20000

)) (1)




CID and CTT forfeit tables

AY cluster 1 cluster 2 cluster 3 split2007 2300 2000 1800 none2008 1670 1373 1175 BI and PD2009 1658 1419 1162 BI and PD2010 (4077) 2152 (3789) 1871 (3410) 1589 (two wheels) all other2011 (4040) 2183 (3741) 1883 (3367) 1627 (two wheels) all other

Table: Synoptic CID forfeit structure by AY and territorial cluster

AY two wheels all other2007 - 2009 3250 32502010 4011 31502011 3959 3143

Table: Synoptic CTT forfeit structure by AY and class of vehicle




2007-2009 forfeit by province

Figure: 2007-2009 forfeit territorial structureGiorgio Spedicato Unicatt



2010-2011 forfeit by province

Figure: 2010-2011 forfeit territorial structureGiorgio Spedicato Unicatt



Overview of CARD components of claims

Within the CARD system up to three different ”components of claims”may arise even together from a single loss occurrence:

1 No Card claims: severe bodily injuries or losses not caused bycollisions between two vehicles.

2 CID claims: slight bodily injuries and property damages arising fromcollision between two vehicles. They can be split into CID caused(CIDD) and CID suffered (CIDG) components of claims. CIDGamounts are paid in full by the handling company, that receive acompensating forfeit amount (CIDGF).

3 CTT claims: losses regarding property damage an bodily injuriessuffered by passengers. They can be also spit into CTT caused(CTTD) and CTT suffered (CTTG) components of claims. CTTGamounts are paid in full by the handling company, that receive acompensating forfeit amount (CTTGF).




Overview of CARD system

No Card components of claim can be modelled by the classical actuarialapproach, as only one type of frequency and claim cost need to beassessed.On the other hand, the assessment of the CARD components of claimrequires:

1 an evaluation of the frequency of both caused and sufferedcomponent of claims.

2 an evaluation of the corresponding forfeit.

3 the handling of negative claim cost when received forfeit is greaterthan suffered claim cost.

Whilst the cost of suffered component of claims is easily modelled ,

finding an analytical form for forfeit distributions is not possible, due to

the mixed discrete / continuous nature of the distribution as shown in

figure . Figure 3 and 4 show the 2009 distribution of CIDG gross amount

and compensating forfeit respectively.Giorgio Spedicato Unicatt



Pricing MTPL coverage under the CARD scheme

Traditional MTPL rate making requires an overall rate adequacyanalysis and a risk classification step (e.g. using GLMs). [?, ?]provide an adequate literature about this topic. Moreover MTPLtariffs tipically contain claim history sensitive variables (e.g. BM)that requires ad hoc analysis. In [?] a brief discussion is presented.The composite structure of losses and the presence of negativeclaim amounts lead to a revision of classical risk classificationanalysis in the CARD system.Moreover even if it is possible to adjust classification rate-makingin order to provide a coherent estimate of burning cost, theassessment of the risk premium distribution requires greater care.





Generalized linear models (GLMs) are the standard method used in P&Cactuarial practice to set tariff relativities. Log-linear overdispersedPoisson and Gamma regressions represents the most used models forfrequency and severity assessment.It is worth to stress that the severity is modelled instead of the cost of asingle claim as the dependent variable is the severity weighted for thenumber of claim occurred in the cell defined by the rate-making factorsused as independent variables.

The final relativities are obtained by fitting two separate models on the

frequency and the severity of claims respectively. Therefore an initial

estimate of burning cost per policyholder group is obtained. A final

model is estimated by a final log-linear gamma regression. In this last

model, a-priori restrictions on specific rate-making factors may be set

adding appropriate offsets in GLMs formula as described in [?, ?].





The following steps may provide a coherent estimate of MTPL coveragerelativities under a CARD scheme.

1 Build classical frequency - severity models on handled componentsof claim: NoCard, CidG and CttG.

2 Use a standard ODP model to model CidD and CttD frequencies.

3 Forfeits amounts (CidGF, CttGF, CidDF, CttDF) may be modelledusing a very simple model that uses only forfeit zone as predictors.Gamma log-linear link or even normal identity link GLMs may beused.

4 An initial pure premium for the i-th risk can be estimated as follows:ppi = frNoCard ∗ sevNoCard + frCidG ∗ (sevCidG − sevCidGF ) + frCttdG ∗(sevCttdG − sevCttdGF ) + frCidD ∗ sevCidDF + frCttD ∗ sevCttDF .

5 A final model on ppi can be finally estimate using a Gamma GLMand appropriate offsets on specific variables a priori set.




Failure of standard classification ratemaking in assessingthe total loss distribution

The mathematical scope of GLMs is to model the expected value of the stochastic riskindicator (frequency, severity or burning cost) conditional to the risk characteristics.The dispersion of the dependent variable is not an important issue when building therisk premium models.Another bias arises from two typical adjustment are used when the claim cost ismodelled within standard classification analysis [?]:

1 The standard loss cost distribution is modified by capping losses to a specifiedthreshold (e.g. 99th percentile) before being modelled though GLMs. A offsetmultiplier based on the ratio of excess claim on capped claim is obtained andapplied on the capped claim cost amount. This trick is used to avoid thedistortion on estimated relativities arising from shock or catastrophe losses.

2 The cost of single loss is usually not model directly, instead of the severity for asingle policy (weighted by the number of incurred claims).

The purpose of these adjustment is to change the shape of the dependent variabledistribution to obtain estimate of tariff relativities more robust to outliers.




Example of CIDG claim cost distribution

Figure: CigG claim cost




Example of CIDG forfeit claim cost distribution

Figure: CigGF claim cost




Solvency II challenges for P&C Insurer

In addition to CARD scheme introduction, Italian insurers face theupcoming enforcement of Solvency II directives. Solvency IIdirectives require a risk based calculation of solvency capital. Thecalculation considers the joint contribution of all risk sources(underwriting, market, credit and operational) that the insurerbears.Underwriting risk represents the risk arising from the insurancecore business. It is further divided into three sub - modules:premium, reserving and catastrophe risk.




Solvency II challenges for P&C Insurer

Solvency II standard formulas is expected to increase the SolvencyRequired Capital relevantly with respect to current regulatoryenvironment (see [?] for details).Standard formula usually leads to a conservative estimation of solvencyrequired capital. Besides standard formula, insurers may assess theircapital requirement by an internal model approved by the regulator.Internal models development is encouraged as they drive entities towardbetter assessing the risk sources they bear.

Our work will discuss premium risk under a CARD portfolio, that evaluate

the potential shortfall between actual losses and earned premium.

Reserve risk will be not assessed in this phase, even if reserve risk analysis

in a CARD environment would present interesting challenges.




Solvency II NL premium risk standard formula

Formula 2 represents NL premium risk standard formula according to QIS4 framework. Volume measure is defined

as Vjlob = max

(Pt,written

j,lob ; Pt,earnedj,lob ; 1.05Pt−1,written

j,lob

), while volatility estimate is determined

through a credibility weighted average of entity historical time series 3 and a tabulated complement of credibilityweight. The ¡credibility weight depends by number of years used into the experience period, that is

σprem,lob =√

clobσU,prem/lob2 + (1− clob)σM,prem/lob

2.

NLpr = ρ (σ) V

ρ (σ) =

exp

(z0.995

√ln(σ2 + 1

))√σ2 + 1

− 1

(2)

σprem,lob =

√√√√√∑y

Ploby (LRlob

y − µlob)2

(nlob − 1) Vprem,lob

(3)




Current Solvency II premium risk literature

Underwriting risk internal models usually build a framework for thetotal cost of claim distribution of each LOB within the company.LOB results are therefore aggregated allowing to obtain andunderwriting risk capital charge.[?] exemplifies the use of collective risk theory in modelling UWrisk capital charge for a multi - line insurer.The collective risk theory approach is standard in capital modelling.Nevertheless within LOB risk heterogeneity is rarely if never takeninto account. In fact the frequency and cost of claim distributionsare assumed with the same parameter on all risks within a LOB.On the other hand MTPL portfolios present a risk heterogeneity sorelevant that standard distributions cannot model properly.




Outline

1 Overview



Solvency II







The modeling idea

The modelling framework we have implemented takes into accountkey characteristics of DR MTPL portfolios: the risk heterogeneityand the structural presence of negative claim cost.Four internal models will be presented. These models lies withinthe presented framework and differs whether:

Component of claims are separately modelled or the total netloss payment is taken into account.

Large claims are considered separately from attritional claims.




Accounting for risk heterogeneity

Risk heterogeneity is handled dividing the portfolio underassessment into more homogeneous clusters of policyholders.GAMLSS predictive models have been estimated, yielding tomodels of expected value and volatility of the frequency and costof claims that return different values according to the insured’scluster.We have chosen to define clusters according to rate-making factorslevels instead of use standard clustering algorithm to keepconsistency with standard pricing practice.




Negative claim amounts handling

Loss amounts modelling requires a particular care when beingmodelled as structurally negative claim amounts might arise. Noanalytical distribution of the claim payment exists under the CARDsystem, even if insured are clustered as much finest as possible.Re-sampling from an empirical sample has been found as the onlysuitable mathematical solution to approximate the distribution ofcompensating CidG and CttG forfeits and caused CidD and CttDamounts. The sampling data set was stratified by class of vehicle,CARD territorial zone and type of component of claims.




Negative claim amounts handling

Analytical solutions to CARD cost of claims net payments would havebeen found both within a particular parametrization of the Tweediedistribution or in the skew normal family. Implementation issues have lednot to follow these ways as:

[?] quotes that when the p parameter lies p < 0, the domain lies onthe whole real line (but, interestingly, µ > 0). Estimating a marginaldistribution for CARD losses would be the first application knownfor such distribution. Nevertheless no software estimates Tweediedistribution when p < 0.

Skew normal distribution [?] would approximated CARD losses asparameters may be found to be positively skew and having domainin the negative part of the real line also. Nevertheless skew normalregression in the GAMLSS [?] packages is experimental and failedconvergence when tested.




Attritional vs large claims

Split the analysis between attritional and large claims is a common choicein P&C actuarial practice, especially in capital modeling. Large lossesdistort estimated relativities when used in the standard predictive models,as detailed in [?]. Moreover a more precise assessment of the tail of theloss distribution is coherent with the final purpose of VaR type capitalallocation implemented in Solvency II.GPD distribution has been used to model large claims distribution. Largeclaim modeling is a not easy task as it requires: a choice of the thresholdand the parameters estimation. Parameter risk is an issue in GPDmodeling. Relevant literature is [?]. The algorithm chosen to estimateGPD parameters was the minimum Anderson - Darling statistic (in orderto maximize the fit on the tail) provided in the POT package [?].The sensitivity of premium risk capital charge implied by the use ofseparate modelling for attritional and large claims has been assessedreporting distinct capital charge estimates.




Model total payments vs components of claim

Another modelling choice consisted whether modelling component ofclaims separately of modelling the total net payments.Separate modelling of the component of claims requires the assumptionof independence between the components of claims within any cluster ofpolicyholders. This assumption may be very strong, even if a part of thedependency have been already considered by dividing the portfolio intohomogeneous clusters. Modelling component of claim residualdependency within cluster is howerer a very difficult task. The strongestadvantage of such approach lies in a insight of the effect of policy ratemaking variable on the specific component of claims.Modelling the net payment requires use of re-sampling for the claim costdistribution as no available regression modelling exists. On the otherhand the is no more need to consider the dependency betweencomponent of claims.




The GAMLSS regression models

GAMLSS have been introduced and discussed in [?]. Few actuarialapplications of GAMLSS model exist in actuarial literature: [?] and[?]. Nevertheless, no applications with focus on ERM exist untilnow. The rationale under GAMLSS is to extend GLMs byestimating regression equations to predict up to four parameter ofa distribution, that are mean, location, shape and scale.





GAMLSS models allow to extend GLMs methodology to a wide setof no - exponential family distribution (e.g. log-normal, logistic,Weibull, etc...). The GAMLSS framework allows also be nonparametric elements (e.g. splines) to be included in the regressionequations. Moveover also mixed models can be estimated withinGAMLSS family.Step-wise approach can be implemented to select parsimoniousdependency relationships using minimum AIC criterion can be usedto compare competing models. Model assumptions can be assessedby analysis of normalized quantile residual, as described in [?].Figure 5 and 6 show µ regressions equation parameters estimatedfor the NoCard frequency and the CidG severity of four wheelsvehicles.





Figure: NoCard Frequency model for Cars





Figure: CIDG severity model for Cars




EVT employment to assess shock losses effect

EVT was used to model large losses over a defined threshold. Ithas been applied both when separated component of claims weremodelled and when the net payments were modelled.2008 and 2009 losses were pooled together after having put on2009 level 2008 losses by a proper inflation correction factor. Asratio of percentiles in usually a more consistent estimator ofinflation than mean and as inflation rate usually differs by size ofloss the ratio of percentiles of 2008 and 2009 component of claimslosses has been analysed, as in figure 7.Extreme losses distributions by component of claims have beenfitted assuming a GPD. Goodness of fit analysis of estimatedmodels lead to acceptable results, as exemplified in figure 8 forNoCard losses.




Inflation rate by percentiles component of claims

Figure: Inflation rate by percentiles




GPD assessment for NoCard losses

Figure: NoCard GPD fit




The purpose of the model

The purpose is to assess the premium risk capital charge on a realMTPL portfolio handled with MTPL. UW premium risk capitalcharge has been defined according to formula 4, where S representthe total loss amount of the underlying portfolio.

NLprRisk = S99.5% − E[S]

(4)

A real MTPL portfolio was provided by a major insurer. Provideddata bases contain data from exposures, rate-making variables andclaims transactions for last three calendar / accident years (2007 -2009).Moreover MTPL LOB written premium, earned premium andlosses last seven years time series were provided in order toestimated standard formula premium risk capital charge.




Exposures classification variables

In addition to exposures and losses aggregate by Calendar Year /Accident Year, following classification variables were selected byvehicle category:

four wheels: age crossed by sex, horsepower, feed, territoryclass, calendar year.

two wheels: age, engine volume, territory, calendar year.

trucks: weight crossed by use, age, territory, calendar year.

These variables affects significantly MTPL peril [?] and all havefound significant in at least one regression. Calendar year has beeninserted into all models even if found insignificant in order toabsorb any specific CY effect (claim cost inflation, legalenvironment changes, . . . ).




Structure of data and assumption

Following hypotheses have been assumed when calculating the2010 underwriting premium risk capital charge:

1 No handled cost inflation.

2 No change in the CARD forfeit structure with respect to2009. This assuptions did not hold in reality.

3 No change in the portfolio business mix with respect to 2009.

4 A cumulative development factor (CDF) of 1.2 have beenassumed for ultimate cost applied to claim cost to account forIBNR and IBNER.

5 A supplementary charge of +3.5% has been added to accountfor unallocated loss adjustment expenses (ULAE), as claimcost data accounts for loss and ALAE only.




The premium risk internal models variations

Two different approaches were tested in order to assess the claim cost:

1 split claims between attritional claims and large claims. Attritionalclaims were modelled by standard predictive modelling, whilst largelosses modelling was based GPD distribution.

2 the cost of single loss was modelled directly.

Compensating and caused forfeit distribution are difficult to be modelledby a standard distribution. Therefore re-sampling on the empirical 2009forfeit distribution was used.

The sample forfeit data set has been stratified by class of vehicle and

forfeit cluster. The frequencies of forfeit were modelled directly by the

corresponding frequency models for caused and suffered component of

claims.




GAMLSS models fit issues

When GAMLSS predictive models have been estimated on eachcomponent of claim, good fit has been obtained on frequencymodel, e.g. see 9. On the other hand bad fit has been found onthe models for the cost of claims, e.g. see 10. A not good fit onthe cost of claim was expected due to the low maturity of claims(maximum maturity 12 month) and tabulated value of casereserves.With respect to frequency and cost of claim modelling of eachcomponent of claims we have assumed negative binomial andgamma distributions and only one variable was used to modeldispersion parameter following [?] and to not develop unreasonablycomplicated models.





Figure: NoCard four wheels frequency GAMLSS residuals analysis





Figure: CidG two wheels cost of claims GAMLSS residuals analysis




Internal model results overview

Table 3 and figure 11 show total loss distributions and capitalcharges by applied model. Following conclusion may be drawn:

Internal models premium risk capital charges are comparablewith undertaking specific standard model.

When components of claim have been separately modelled,resulting capital charges are generally lower than capitalcharges resulting when the net payment were used. Highercapital charge may be due to a significant positive residualdependency between component of claims not considered inthe model.

When GPD has been used to model large losses, resultingcapital charges are lower. Thin tailed nature of MTPL lossesmay be drawn.




Capital charges by model

model on EP on EL CV

SF market wide 27.8% n.a. n.a.SF undert. spec 18.6% n.a. n.a.

component of claims, GPD 16.0% 21.2% 7.9%component of claims, no GPD 20.8% 26.8% 9.7%

net payments, GPD 20.5% 24.9% 8.9%net payments, no GPD 23.6% 30.5% 9.7%

Table: Premium risk capital charges ( on earned premiums and expectedlosses) and CV




Internal models capital charge distributions

Figure: Total loss distribution by internal model




Outline

1 Overview



Solvency II







Drawbacks

Relevant assumptions of the model are:

Deterministic CDFs for losses evaluated at 12th month of development.Therefore the claim emergence and settlement process contribution to theunderwriting risk volatility have been not takend into account. Also lack ofstable historical data does not allow credible IBNR analysis eventually split bycomponent of claims.

Reserve are considered at un-discounted basis.

It is difficult to update model on time to account for forfeit rules revision asforfeit changes are decided close to year end.

The claim cost distributions by component of claims are difficult to beapproximated by a suitable loss distribution even clustering the portfolio toaccount for risk heterogeneity. Nevertheless this issue is common in personalline empirical data.

The probability distributions for risk components have been chosen negativebinomial for frequency and gamma for severity and a log-linear link have beenassumed. The link function and the regression function might be build moreprecisely.




Advantages

Most relevant advantages of the model are:

it provides a premium risk capital charge coherent with theCARD system.

it allows to model heterogeneous portfolios.

it allows to model non - static portfolio even with change ofportfolio mix.




Extension and developments

Valuable applications of the presented framework are:

Capital allocation across sub - lines (Cars, Truck and fourwheels). See [?] for an overview.

Risk based pricing allowing to determine a profit loadingdepending by profile variability.

Suggested research directions with respect to the discussed issuesare:

Claim reserve analysis under the CARD system.

Pricing methodology deepening.



Bibliography Thanks

Outline

5 Bibliography

6 Thanks



Bibliography Thanks

Bibliography



Bibliography Thanks

Outline

5 Bibliography

6 Thanks



Bibliography Thanks

Acknowledgments

I whis to thank my PhD supervisor, Prof. Nino Savelli for thepatience and suggestions that have lead to these results. Moreover,I’m grateful with my actuarial supervisors, Gloria Leonardi andGarnier Stella.Finally I wish to thank my employer, AXA Assicurazioni, for havingprovided me a sample dataset on which calibrate the model.Nevetheless any considerations appearing in this paper areresponsibility of myself alone. In publishing these contents AXAAssicurazioni takes no position on the opinion expressed by myselfand disclaims all responsibility for any opinion, incorrectinformation or legal error found therein.



Economy & Finance

Pricing and modelling under the Italian Direct Compensation Card Scheme