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Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerationsTh...
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerationsTh...
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Pricing and modelling under the Italian Direct Compensation Card Scheme

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Transcript of "Pricing and modelling under the Italian Direct Compensation Card Scheme"

  1. 1. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Solvency II premium risk modelling under the direct compensation CARD system Seminario di Statistica Assicurativa Giorgio Spedicato, Ph.D Universit´ Cattolica a Milan, Italy August 27, 2011Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  2. 2. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsTable of contents 1 Overview 2 Recent challenges in insurance business The DR system in Italy Overview of the CARD scheme Pricing MTPL policies within the CARD scheme Solvency II 3 An internal model for MTPL UW premium risk The general framework Theoretical models The empirical application Data set description Model output 4 Final considerations Model discussion Extension ad developmentsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  3. 3. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOutline 1 Overview 2 Recent challenges in insurance business The DR system in Italy Overview of the CARD scheme Pricing MTPL policies within the CARD scheme Solvency II 3 An internal model for MTPL UW premium risk The general framework Theoretical models The empirical application Data set description Model output 4 Final considerations Model discussion Extension ad developmentsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  4. 4. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsIntroduction This Ph.D thesis presents a possible model to assess UW premium risk 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 [?]. Multivariate techniques are currently used to price MTPL contracts. Italian MTPL insurance regulation has known frequent changes since MTPL compulsory requirement in 1969. MTPL pricing has been liberalized in 1994. More recently, in 2007 the regulation has been strongly revised by the introduction of the direct reimbursement scheme with the so called CARD agreement. In addition, a deep revision of the Bonus Malus transition rules has been put in force.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  5. 5. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsIntroduction The forthcoming introduction of Solvency II pulls insurers to better assess risks within portfolio with the ultimate objective to determine the distribution of the capital at risk (CaR) arising from the insurance operation. Underwriting CaR estimation requires not only the expected value but also the volatility of policyholders’ portfolio to be properly assessed.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  6. 6. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsIntroduction Literature and practical applications exist regarding both Solvency II underwriting risk component of the CaR and regarding multivariate classification techniques used to price MTPL tariffs. Nevertheless the introduction of DR scheme in the Italian MTPL business practice brought relevant complications in the process of pure premium estimation and total loss distribution assessment. This Ph.D thesis deepens the impact of CARD DR scheme on pricing and capital modelling showing one possible approach to model premium risk under a DR scheme.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  7. 7. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOutline 1 Overview 2 Recent challenges in insurance business The DR system in Italy Overview of the CARD scheme Pricing MTPL policies within the CARD scheme Solvency II 3 An internal model for MTPL UW premium risk The general framework Theoretical models The empirical application Data set description Model output 4 Final considerations Model discussion Extension ad developmentsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  8. 8. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe CARD system Regulation reforms have affected relevantly MTPL underwriting and actuarial practices since 2007. They are the second Bersani law and the introduction of DR scheme. Bersani laws have halted the structure of Italian experience rating allowing policyholders with few if any driving experience to inherit the best BM class within their household. A second provision of Bersani laws allowed tied agents to use their allocated discount budget more freely. Nevertheless, this Ph.D thesis will not take into account the effect of such provisions. [?] carries a comprehensive introduction on these topics.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  9. 9. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe 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 responsible part indemnifies directly its own policyholder by the full claim amounts it has suffered for most non responsible claims. The responsible part insurer then 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 usually different from the amount effectively paid to the non - responsible part. CARD actuarial challenges arise from: Negative claim amounts are possible as received forfeit is generally different 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 forfeit usually does not offset suffered claim severity by the same amount. The shortness of historical experience period affects credibility of pricing and reserving estimates. Regulatory environment frequently changes, especially in forfeitGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  10. 10. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe CARD system Since 2010 received forfeit due to damages to the non-responsible driver depends by class of vehicle and territory. Figures 1 and 2 show forfeit structure by territory since 2007. Forfeit due damages to other passengers (and CID bodily iis set according to a fair complicated rule containing deductible and coinsurance clauses. Formula 1 show CTT forfeit rule for 2007-2009. Effective amount for CID and CTT forfeit are reported in tables 1 and 2 respectively. ˜ X ≤ 500 → F = 0 F = ˜ ˜ ˜ (1) X > 500 → F = 3250 + max 0; X − 5000 −max 500; min 0.1 ∗ X ; 20000Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  11. 11. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsCID and CTT forfeit tables AY cluster 1 cluster 2 cluster 3 split 2007 2300 2000 1800 none 2008 1670 1373 1175 BI and PD 2009 1658 1419 1162 BI and PD 2010 (4077) 2152 (3789) 1871 (3410) 1589 (two wheels) all other 2011 (4040) 2183 (3741) 1883 (3367) 1627 (two wheels) all other Table: Synoptic CID forfeit structure by AY and territorial cluster AY two wheels all other 2007 - 2009 3250 3250 2010 4011 3150 2011 3959 3143 Table: Synoptic CTT forfeit structure by AY and class of vehicleGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  12. 12. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations2007-2009 forfeit by provinceGiorgio Spedicato Figure: 2007-2009 forfeit territorial structure UnicattSolvency II premium risk modelling under the direct compensation CARD system
  13. 13. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations2010-2011 forfeit by provinceGiorgio Spedicato Figure: 2010-2011 forfeit territorial structure UnicattSolvency II premium risk modelling under the direct compensation CARD system
  14. 14. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOverview 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 by collisions between two vehicles. 2 CID claims: slight bodily injuries and property damages arising from collision between two vehicles. They can be split into CID caused (CIDD) and CID suffered (CIDG) components of claims. CIDG amounts are paid in full by the handling company, that receive a compensating forfeit amount (CIDGF). 3 CTT claims: losses regarding property damage an bodily injuries suffered by passengers. They can be also spit into CTT caused (CTTD) and CTT suffered (CTTG) components of claims. CTTG amounts are paid in full by the handling company, that receive a compensating forfeit amount (CTTGF).Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  15. 15. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOverview of CARD system No Card components of claim can be modelled by the classical actuarial approach, as only one type of frequency and claim cost need to be assessed. On the other hand, the assessment of the CARD components of claim requires: 1 an evaluation of the frequency of both caused and suffered component of claims. 2 an evaluation of the corresponding forfeit. 3 the handling of negative claim cost when received forfeit is greater than 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 UnicattSolvency II premium risk modelling under the direct compensation CARD system
  16. 16. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsPricing MTPL coverage under the CARD scheme Traditional MTPL rate making requires an overall rate adequacy analysis and a risk classification step (e.g. using GLMs). [?, ?] provide an adequate literature about this topic. Moreover MTPL tariffs 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 negative claim amounts lead to a revision of classical risk classification analysis in the CARD system. Moreover even if it is possible to adjust classification rate-making in order to provide a coherent estimate of burning cost, the assessment of the risk premium distribution requires greater care.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  17. 17. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsPricing MTPL coverage under the CARD scheme Generalized linear models (GLMs) are the standard method used in P&C actuarial practice to set tariff relativities. Log-linear overdispersed Poisson and Gamma regressions represents the most used models for frequency and severity assessment. It is worth to stress that the severity is modelled instead of the cost of a single claim as the dependent variable is the severity weighted for the number of claim occurred in the cell defined by the rate-making factors used 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 [?, ?].Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  18. 18. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsPricing MTPL coverage under the CARD scheme The following steps may provide a coherent estimate of MTPL coverage relativities under a CARD scheme. 1 Build classical frequency - severity models on handled components of 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 modelled using a very simple model that uses only forfeit zone as predictors. Gamma log-linear link or even normal identity link GLMs may be used. 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 GLM and appropriate offsets on specific variables a priori set.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  19. 19. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsFailure of standard classification ratemaking in assessingthe total loss distribution The mathematical scope of GLMs is to model the expected value of the stochastic risk indicator (frequency, severity or burning cost) conditional to the risk characteristics. The dispersion of the dependent variable is not an important issue when building the risk premium models. Another bias arises from two typical adjustment are used when the claim cost is modelled within standard classification analysis [?]: 1 The standard loss cost distribution is modified by capping losses to a specified threshold (e.g. 99th percentile) before being modelled though GLMs. A offset multiplier based on the ratio of excess claim on capped claim is obtained and applied on the capped claim cost amount. This trick is used to avoid the distortion 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 a single policy (weighted by the number of incurred claims). The purpose of these adjustment is to change the shape of the dependent variable distribution to obtain estimate of tariff relativities more robust to outliers.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  20. 20. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsExample of CIDG claim cost distribution Figure: CigG claim costGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  21. 21. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsExample of CIDG forfeit claim cost distribution Figure: CigGF claim costGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  22. 22. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsSolvency II challenges for P&C Insurer In addition to CARD scheme introduction, Italian insurers face the upcoming enforcement of Solvency II directives. Solvency II directives require a risk based calculation of solvency capital. The calculation considers the joint contribution of all risk sources (underwriting, market, credit and operational) that the insurer bears. Underwriting risk represents the risk arising from the insurance core business. It is further divided into three sub - modules: premium, reserving and catastrophe risk.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  23. 23. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsSolvency II challenges for P&C Insurer Solvency II standard formulas is expected to increase the Solvency Required Capital relevantly with respect to current regulatory environment (see [?] for details). Standard formula usually leads to a conservative estimation of solvency required capital. Besides standard formula, insurers may assess their capital requirement by an internal model approved by the regulator. Internal models development is encouraged as they drive entities toward better 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.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  24. 24. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsSolvency II NL premium risk standard formula Formula 2 represents NL premium risk standard formula according to QIS4 framework. Volume measure is defined as Vj lob = max P t,written j,lob ; P t,earned j,lob ; 1.05P t−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 credibility weight. The ¡credibility weight depends by number of years used into the experience period, that is σprem,lob = clob σU,prem/lob 2 + (1 − clob ) σM,prem/lob 2 . NLpr = ρ (σ) V exp z0.995 ln σ 2 + 1 (2) ρ (σ) = −1 σ2 + 1 Plob y (LRlob y − µlob )2 y σprem,lob = (3) (nlob − 1) Vprem,lobGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  25. 25. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsCurrent Solvency II premium risk literature Underwriting risk internal models usually build a framework for the total cost of claim distribution of each LOB within the company. LOB results are therefore aggregated allowing to obtain and underwriting risk capital charge. [?] exemplifies the use of collective risk theory in modelling UW risk 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 taken into account. In fact the frequency and cost of claim distributions are assumed with the same parameter on all risks within a LOB. On the other hand MTPL portfolios present a risk heterogeneity so relevant that standard distributions cannot model properly.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  26. 26. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOutline 1 Overview 2 Recent challenges in insurance business The DR system in Italy Overview of the CARD scheme Pricing MTPL policies within the CARD scheme Solvency II 3 An internal model for MTPL UW premium risk The general framework Theoretical models The empirical application Data set description Model output 4 Final considerations Model discussion Extension ad developmentsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  27. 27. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe modeling idea The modelling framework we have implemented takes into account key characteristics of DR MTPL portfolios: the risk heterogeneity and the structural presence of negative claim cost. Four internal models will be presented. These models lies within the presented framework and differs whether: Component of claims are separately modelled or the total net loss payment is taken into account. Large claims are considered separately from attritional claims.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  28. 28. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsAccounting for risk heterogeneity Risk heterogeneity is handled dividing the portfolio under assessment into more homogeneous clusters of policyholders. GAMLSS predictive models have been estimated, yielding to models of expected value and volatility of the frequency and cost of claims that return different values according to the insured’s cluster. We have chosen to define clusters according to rate-making factors levels instead of use standard clustering algorithm to keep consistency with standard pricing practice.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  29. 29. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsNegative claim amounts handling Loss amounts modelling requires a particular care when being modelled as structurally negative claim amounts might arise. No analytical distribution of the claim payment exists under the CARD system, even if insured are clustered as much finest as possible. Re-sampling from an empirical sample has been found as the only suitable mathematical solution to approximate the distribution of compensating CidG and CttG forfeits and caused CidD and CttD amounts. The sampling data set was stratified by class of vehicle, CARD territorial zone and type of component of claims.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  30. 30. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsNegative claim amounts handling Analytical solutions to CARD cost of claims net payments would have been found both within a particular parametrization of the Tweedie distribution or in the skew normal family. Implementation issues have led not to follow these ways as: [?] quotes that when the p parameter lies p < 0, the domain lies on the whole real line (but, interestingly, µ > 0). Estimating a marginal distribution for CARD losses would be the first application known for such distribution. Nevertheless no software estimates Tweedie distribution when p < 0. Skew normal distribution [?] would approximated CARD losses as parameters may be found to be positively skew and having domain in the negative part of the real line also. Nevertheless skew normal regression in the GAMLSS [?] packages is experimental and failed convergence when tested.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  31. 31. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsAttritional vs large claims Split the analysis between attritional and large claims is a common choice in P&C actuarial practice, especially in capital modeling. Large losses distort estimated relativities when used in the standard predictive models, as detailed in [?]. Moreover a more precise assessment of the tail of the loss distribution is coherent with the final purpose of VaR type capital allocation implemented in Solvency II. GPD distribution has been used to model large claims distribution. Large claim modeling is a not easy task as it requires: a choice of the threshold and the parameters estimation. Parameter risk is an issue in GPD modeling. Relevant literature is [?]. The algorithm chosen to estimate GPD parameters was the minimum Anderson - Darling statistic (in order to maximize the fit on the tail) provided in the POT package [?]. The sensitivity of premium risk capital charge implied by the use of separate modelling for attritional and large claims has been assessed reporting distinct capital charge estimates.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  32. 32. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsModel total payments vs components of claim Another modelling choice consisted whether modelling component of claims separately of modelling the total net payments. Separate modelling of the component of claims requires the assumption of independence between the components of claims within any cluster of policyholders. This assumption may be very strong, even if a part of the dependency have been already considered by dividing the portfolio into homogeneous clusters. Modelling component of claim residual dependency within cluster is howerer a very difficult task. The strongest advantage of such approach lies in a insight of the effect of policy rate making variable on the specific component of claims. Modelling the net payment requires use of re-sampling for the claim cost distribution as no available regression modelling exists. On the other hand the is no more need to consider the dependency between component of claims.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  33. 33. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe GAMLSS regression models GAMLSS have been introduced and discussed in [?]. Few actuarial applications of GAMLSS model exist in actuarial literature: [?] and [?]. Nevertheless, no applications with focus on ERM exist until now. The rationale under GAMLSS is to extend GLMs by estimating regression equations to predict up to four parameter of a distribution, that are mean, location, shape and scale.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  34. 34. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe GAMLSS regression models GAMLSS models allow to extend GLMs methodology to a wide set of no - exponential family distribution (e.g. log-normal, logistic, Weibull, etc...). The GAMLSS framework allows also be non parametric elements (e.g. splines) to be included in the regression equations. Moveover also mixed models can be estimated within GAMLSS family. Step-wise approach can be implemented to select parsimonious dependency relationships using minimum AIC criterion can be used to compare competing models. Model assumptions can be assessed by analysis of normalized quantile residual, as described in [?]. Figure 5 and 6 show µ regressions equation parameters estimated for the NoCard frequency and the CidG severity of four wheels vehicles.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  35. 35. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe GAMLSS regression models Figure: NoCard Frequency model for CarsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  36. 36. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe GAMLSS regression models Figure: CIDG severity model for CarsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  37. 37. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsEVT employment to assess shock losses effect EVT was used to model large losses over a defined threshold. It has been applied both when separated component of claims were modelled and when the net payments were modelled. 2008 and 2009 losses were pooled together after having put on 2009 level 2008 losses by a proper inflation correction factor. As ratio of percentiles in usually a more consistent estimator of inflation than mean and as inflation rate usually differs by size of loss the ratio of percentiles of 2008 and 2009 component of claims losses has been analysed, as in figure 7. Extreme losses distributions by component of claims have been fitted assuming a GPD. Goodness of fit analysis of estimated models lead to acceptable results, as exemplified in figure 8 for NoCard losses.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  38. 38. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsInflation rate by percentiles component of claims Figure: Inflation rate by percentilesGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  39. 39. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsGPD assessment for NoCard losses Figure: NoCard GPD fitGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  40. 40. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe purpose of the model The purpose is to assess the premium risk capital charge on a real MTPL portfolio handled with MTPL. UW premium risk capital ˜ charge has been defined according to formula 4, where S represent the total loss amount of the underlying portfolio. ˜ ˜ NLprRisk = S99.5% − E S (4) A real MTPL portfolio was provided by a major insurer. Provided data bases contain data from exposures, rate-making variables and claims transactions for last three calendar / accident years (2007 - 2009). Moreover MTPL LOB written premium, earned premium and losses last seven years time series were provided in order to estimated standard formula premium risk capital charge.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  41. 41. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsExposures classification variables In addition to exposures and losses aggregate by Calendar Year / Accident Year, following classification variables were selected by vehicle category: four wheels: age crossed by sex, horsepower, feed, territory class, 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 have found significant in at least one regression. Calendar year has been inserted into all models even if found insignificant in order to absorb any specific CY effect (claim cost inflation, legal environment changes, . . . ).Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  42. 42. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsStructure of data and assumption Following hypotheses have been assumed when calculating the 2010 underwriting premium risk capital charge: 1 No handled cost inflation. 2 No change in the CARD forfeit structure with respect to 2009. 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 been assumed for ultimate cost applied to claim cost to account for IBNR and IBNER. 5 A supplementary charge of +3.5% has been added to account for unallocated loss adjustment expenses (ULAE), as claim cost data accounts for loss and ALAE only.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  43. 43. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsThe 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. Attritional claims were modelled by standard predictive modelling, whilst large losses modelling was based GPD distribution. 2 the cost of single loss was modelled directly. Compensating and caused forfeit distribution are difficult to be modelled by a standard distribution. Therefore re-sampling on the empirical 2009 forfeit 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.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  44. 44. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsGAMLSS models fit issues When GAMLSS predictive models have been estimated on each component of claim, good fit has been obtained on frequency model, e.g. see 9. On the other hand bad fit has been found on the models for the cost of claims, e.g. see 10. A not good fit on the cost of claim was expected due to the low maturity of claims (maximum maturity 12 month) and tabulated value of case reserves. With respect to frequency and cost of claim modelling of each component of claims we have assumed negative binomial and gamma distributions and only one variable was used to model dispersion parameter following [?] and to not develop unreasonably complicated models.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  45. 45. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsGAMLSS models fit issues Figure: NoCard four wheels frequency GAMLSS residuals analysisGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  46. 46. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsGAMLSS models fit issuesGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  47. 47. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsInternal model results overview Table 3 and figure 11 show total loss distributions and capital charges by applied model. Following conclusion may be drawn: Internal models premium risk capital charges are comparable with undertaking specific standard model. When components of claim have been separately modelled, resulting capital charges are generally lower than capital charges resulting when the net payment were used. Higher capital charge may be due to a significant positive residual dependency between component of claims not considered in the model. When GPD has been used to model large losses, resulting capital charges are lower. Thin tailed nature of MTPL losses may be drawn.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  48. 48. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsCapital 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 expected losses) and CVGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  49. 49. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsInternal models capital charge distributions Figure: Total loss distribution by internal modelGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  50. 50. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsOutline 1 Overview 2 Recent challenges in insurance business The DR system in Italy Overview of the CARD scheme Pricing MTPL policies within the CARD scheme Solvency II 3 An internal model for MTPL UW premium risk The general framework Theoretical models The empirical application Data set description Model output 4 Final considerations Model discussion Extension ad developmentsGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  51. 51. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsDrawbacks 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 the underwriting risk volatility have been not takend into account. Also lack of stable historical data does not allow credible IBNR analysis eventually split by component of claims. Reserve are considered at un-discounted basis. It is difficult to update model on time to account for forfeit rules revision as forfeit changes are decided close to year end. The claim cost distributions by component of claims are difficult to be approximated by a suitable loss distribution even clustering the portfolio to account for risk heterogeneity. Nevertheless this issue is common in personal line empirical data. The probability distributions for risk components have been chosen negative binomial for frequency and gamma for severity and a log-linear link have been assumed. The link function and the regression function might be build more precisely.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  52. 52. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsAdvantages Most relevant advantages of the model are: it provides a premium risk capital charge coherent with the CARD system. it allows to model heterogeneous portfolios. it allows to model non - static portfolio even with change of portfolio mix.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  53. 53. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerationsExtension and developments Valuable applications of the presented framework are: Capital allocation across sub - lines (Cars, Truck and four wheels). See [?] for an overview. Risk based pricing allowing to determine a profit loading depending by profile variability. Suggested research directions with respect to the discussed issues are: Claim reserve analysis under the CARD system. Pricing methodology deepening.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  54. 54. Bibliography ThanksOutline 5 Bibliography 6 ThanksGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  55. 55. Bibliography ThanksBibliographyGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  56. 56. Bibliography ThanksOutline 5 Bibliography 6 ThanksGiorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
  57. 57. Bibliography ThanksAcknowledgments I whis to thank my PhD supervisor, Prof. Nino Savelli for the patience and suggestions that have lead to these results. Moreover, I’m grateful with my actuarial supervisors, Gloria Leonardi and Garnier Stella. Finally I wish to thank my employer, AXA Assicurazioni, for having provided me a sample dataset on which calibrate the model. Nevetheless any considerations appearing in this paper are responsibility of myself alone. In publishing these contents AXA Assicurazioni takes no position on the opinion expressed by myself and disclaims all responsibility for any opinion, incorrect information or legal error found therein.Giorgio Spedicato UnicattSolvency II premium risk modelling under the direct compensation CARD system
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