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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, 2011




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Table 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 developments
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Outline
       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 developments
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Introduction

       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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Introduction




       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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Introduction



       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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Outline
       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 developments
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



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 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 forfeit
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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 ; 20000




Giorgio Spedicato                                                                                                  Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business     An internal model for MTPL UW premium risk      Final considerations



CID 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 vehicle




Giorgio Spedicato                                                                                                    Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



2007-2009 forfeit by province




Giorgio Spedicato           Figure: 2007-2009 forfeit territorial structure                                     Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



2010-2011 forfeit by province




Giorgio Spedicato           Figure: 2010-2011 forfeit territorial structure                                     Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



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 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Overview 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Pricing 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Pricing 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Pricing 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations


Failure of standard classification ratemaking in assessing
the 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Example of CIDG claim cost distribution




                                             Figure: CigG claim cost



Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Example of CIDG forfeit claim cost distribution




                                            Figure: CigGF claim cost



Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Solvency 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Solvency 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview      Recent challenges in insurance business          An internal model for MTPL UW premium risk               Final considerations



Solvency 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,lob




Giorgio Spedicato                                                                                                                       Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Current 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Outline
       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 developments
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Accounting 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Negative 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Negative 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Attritional 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Model 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The GAMLSS regression models




                               Figure: NoCard Frequency model for Cars

Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The GAMLSS regression models




                                  Figure: CIDG severity model for Cars

Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



EVT 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Inflation rate by percentiles component of claims




                                    Figure: Inflation rate by percentiles




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



GPD assessment for NoCard losses




                                            Figure: NoCard GPD fit


Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



The 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                                                                                                Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Exposures 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Structure 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview       Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



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. 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                                                                                                 Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



GAMLSS 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



GAMLSS models fit issues




             Figure: NoCard four wheels frequency GAMLSS residuals analysis




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



GAMLSS models fit issues




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Internal 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



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 expected
       losses) and CV




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Internal models capital charge distributions




                          Figure: Total loss distribution by internal model




Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Outline
       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 developments
Giorgio Spedicato                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



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 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Advantages



       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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Overview     Recent challenges in insurance business   An internal model for MTPL UW premium risk   Final considerations



Extension 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                                                                                               Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Bibliography                                                                   Thanks



Outline




       5       Bibliography



       6       Thanks




Giorgio Spedicato                                                              Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Bibliography                                                                   Thanks



Bibliography




Giorgio Spedicato                                                              Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Bibliography                                                                   Thanks



Outline




       5       Bibliography



       6       Thanks




Giorgio Spedicato                                                              Unicatt
Solvency II premium risk modelling under the direct compensation CARD system
Bibliography                                                                   Thanks



Acknowledgments


       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                                                              Unicatt
Solvency II premium risk modelling under the direct compensation CARD system

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Pricing and modelling under the Italian Direct Compensation Card Scheme

  • 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, 2011 Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 2. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Table 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 developments Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 3. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Outline 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 developments Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 4. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Introduction 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 5. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Introduction 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 6. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Introduction 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 7. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Outline 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 developments Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 8. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 9. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 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 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 forfeit Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 10. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 ; 20000 Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 11. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations CID 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 vehicle Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 12. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 2007-2009 forfeit by province Giorgio Spedicato Figure: 2007-2009 forfeit territorial structure Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 13. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 2010-2011 forfeit by province Giorgio Spedicato Figure: 2010-2011 forfeit territorial structure Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 14. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 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 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 15. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Overview 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 16. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Pricing 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 17. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Pricing 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 18. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Pricing 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 19. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Failure of standard classification ratemaking in assessing the 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 20. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Example of CIDG claim cost distribution Figure: CigG claim cost Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 21. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Example of CIDG forfeit claim cost distribution Figure: CigGF claim cost Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 22. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Solvency 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 23. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Solvency 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 24. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Solvency 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,lob Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 25. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Current 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 26. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Outline 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 developments Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 27. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 28. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Accounting 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 29. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Negative 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 30. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Negative 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 31. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Attritional 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 32. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Model 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 33. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 34. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 35. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The GAMLSS regression models Figure: NoCard Frequency model for Cars Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 36. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The GAMLSS regression models Figure: CIDG severity model for Cars Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 37. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations EVT 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 38. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Inflation rate by percentiles component of claims Figure: Inflation rate by percentiles Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 39. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations GPD assessment for NoCard losses Figure: NoCard GPD fit Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 40. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations The 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 41. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Exposures 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 42. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Structure 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 43. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 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. 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 44. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations GAMLSS 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 45. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations GAMLSS models fit issues Figure: NoCard four wheels frequency GAMLSS residuals analysis Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 46. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations GAMLSS models fit issues Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 47. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Internal 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 48. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 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 expected losses) and CV Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 49. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Internal models capital charge distributions Figure: Total loss distribution by internal model Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 50. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Outline 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 developments Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 51. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations 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 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 52. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Advantages 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 53. Overview Recent challenges in insurance business An internal model for MTPL UW premium risk Final considerations Extension 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 54. Bibliography Thanks Outline 5 Bibliography 6 Thanks Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 55. Bibliography Thanks Bibliography Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 56. Bibliography Thanks Outline 5 Bibliography 6 Thanks Giorgio Spedicato Unicatt Solvency II premium risk modelling under the direct compensation CARD system
  • 57. Bibliography Thanks Acknowledgments 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 Unicatt Solvency II premium risk modelling under the direct compensation CARD system