Pricing and modelling under the Italian Direct Compensation Card Scheme

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

Solvency II premium risk modelling under 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


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


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


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.



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.



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.



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.



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


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



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



2007-2009 forfeit by province

Giorgio Spedicato Figure: 2007-2009 forfeit territorial structure Unicatt


2010-2011 forfeit by province

Giorgio Spedicato Figure: 2010-2011 forfeit territorial structure Unicatt


Overview of CARD components of claims
Within the CARD system up to three different ”components of claims”
may arise even together from a single loss occurrence:
1 No Card claims: severe bodily injuries or losses not caused 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).



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.


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.




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 [?, ?].



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.


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.



Example of CIDG claim cost distribution

Figure: CigG claim cost



Example of CIDG forfeit claim cost distribution

Figure: CigGF claim cost



Solvency II challenges for P&C Insurer

In addition to CARD scheme introduction, Italian insurers face 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.



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.



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



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.



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.



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.



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.



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.



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.



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



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.




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.




Figure: NoCard Frequency model for Cars




Figure: CIDG severity model for Cars



EVT employment to assess shock losses effect

EVT was used to model large losses over a defined threshold. 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.



Inﬂation rate by percentiles component of claims

Figure: Inﬂation rate by percentiles



GPD assessment for NoCard losses

Figure: NoCard GPD ﬁt



The purpose of the model

The purpose is to assess the premium risk capital charge on a 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.



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, . . . ).



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.



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.



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.




Figure: NoCard four wheels frequency GAMLSS residuals analysis



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.



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



Internal models capital charge distributions

Figure: Total loss distribution by internal model



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
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.



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.



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.


Bibliography Thanks

Outline

5 Bibliography

6 Thanks


Bibliography Thanks

Bibliography


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.


Pricing and modelling under the Italian Direct Compensation Card Scheme

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