ASTIN Colloquium 2011

Madrid, 19-22 June 2011

Stochastic Claim Reserving based on CRMfor Solvency II purposes

Nino Nino SavelliSavelli

CatholicCatholic UniversityUniversity, Milan, Milannino.savelli@unicatt.itnino.savelli@unicatt.it

Gian Paolo ClementeGian Paolo Clemente

CatholicCatholic UniversityUniversity, Milan, Milangianpaolo.clemente@unicatt.itgianpaolo.clemente@unicatt.it

AGENDA

� Introduction

� A Collective Risk Model for a claim reserve distrib ution

2Clemente-Savelli: “Stochastic Claim Reserving based on a CRM for SII purposes” ©

� Motor Third-Party Liabilities (MTPL): two case stud ies

� General Third-Party Liabilities (GTPL): two case st udies

� Motor Other Damages (MOD): two case studies

� Final Remarks

IntroductionIntroduction

3

� In the context of Solvency II project a new valuation of claims reserve has been introduced.

� Non-Life Insurers need to quantify claims reserve as the amount for which they could betransferred, or settled, between knowledgeable willing parties in an arm's length transaction

� This amount should be valued, for non-hedgeable liabilities (as claims reserve), separatelybetween best estimate and risk margin .

Loss Reservein Solvency II framework

between best estimate and risk margin .

• “...the best estimate shall correspond to the probability-weighted average of futurecash-flows, taking account of the time value of money (expected present value of futurecash-flows), using the relevant risk-free interest rate term structure” (art. 77,Solvency II-Directive)

• “… the risk margin shall be calculated by determining the cost of providing an amountof eligible own funds equal to the Solvency Capital Requirement necessary to supportthe insurance and reinsurance obligations over the lifetime thereof” (art. 76, SolvencyII Directive)

4

Best Estimate

• Best estimate must be obtained on discounted basis .

• QIS5 Technical Specifications provide the term structure useful for BE valuation.

• Discount factors are determined by the term structure of risk-free interest rates as at31/12/2009 including a 50% illiquidity premium (discount rates move roughly from1.5% to 4.3%)1.5% to 4.3%)

4.8 5 5.2 5.4 5.6 5.8 6

x 107

0

100

200

300

400

500

600

5

Risk Margin

• The risk margin (QIS5 TS) is obtained by:1. projecting the SCR until the complete run-off of the

overall liabilities (diversification assumed)The SCR captures only some risks:

- underwriting risk (only existing business)- default risk with respect to reinsurance contracts- operational risk

SCR

- unavoidable market risk (not material for non-life liabilities)

2. quantifying the cost of capital of each year (6%* SCRt)

3. discounting using the risk-free rates without illiquidity premium

Cost of Holding future SCR

RM

6

∑=

⋅+⋅=T

ttSCRtvCoCRM

0

)1,0(

QIS5 results show (for Solo Insurers) a ratio RM/BE equ al to:

Italy EIOPAMTPL 4%-5% 5.62%

GTPL 4%-5% 6.83%

MOD 6%-7% 7.85%

• The SCR should be determined as the economic capital to be held by insurance andreinsurance undertakings in order to ensure that ruin occurs no more often than once in every200 cases or, alternatively, that those undertakings will still be in a position, with a probability ofat least 99,5%, to meet their obligations to policyholders and beneficiaries over theforthcoming 12 months”. (art. 37 Solvency II-Directive)

Solvency Capital RequirementOne-Year Approach

• The One-Year approach is based on the assumption

7

• The One-Year approach is based on the assumptionthat, once estimated the initial claim reserve at time T=0(RT=0), two variability sources will be in force duringthe year , described by the next random variables:- the payments occurred in year 1, i.e. the elements

located on the next diagonal of the triangle (XT=1)- the new claim reserve estimated at time T=1

conditionally to additional informations availableduring the year (RT=1).

In this way a capital requirement can be computed and the CV of the random variable XT=1+RT=1 as well.

Figure reported in AISAM – ACME (2007), “Study on non-life long tail liabilities”

SCR for Reserve Risk(multiple choice)

•The capital requirement for reserve risk in QIS5 could be obtained by:

– a standard formula based on a market-wide approach – MW.The capital charge is derived through fixed volatility factors provided in QIS5 TS(equal to 9.5% for MTPL, 11% for GTPL and 10% for MOD)

– a standard formula based on an undertaking-specific approach – USP.The capital charge is obtained by a credibility weight between an undertaking specific

8

The capital charge is obtained by a credibility weight between an undertaking specificestimate of the variability (e.g. by using CV determined with the Merz-Wüthrichformula as prescribed in QIS5 TS) and the MW volatility factor.The credibility factor to be applied should be chosen according to the length of timeseries (equal to the dimension of the triangle used for Merz-Wüthrich approach).Considering a 12x12 triangle, the credibility factor is 87% for MTPL and GTPL and100% for MOD.(in case of 8x8 triangle, credibility = 81%)

–an internal model (IM) using a stochastic approach

A Collective Risk Model (CRM)A Collective Risk Model (CRM)for a claim reserve distribution

9

• To develop and implement a stochastic model for the loss reserve evaluation innon-life insurance in order to quantify estimate uncertainty and, consequently, thecapital requirement for reserve risk, based on the approach “frequency-severity” of the Collective Risk Model (following our previous paper presentedat ICA 2010).

• We strongly regard this as an additional stochastic model to reduce model riskand possibly having also a larger comparability with Premium Ri skevaluation when CRM are used )

The aim of this paper

• To calibrate the model’s parameters as to expected values of claim counts andaverage cost of future payments throughout the deterministic model “Fisher-Lange” (i.e. Incremental average cost method), based on the estimate of separatenumber of claims to be paid and future average cost (at this regard also otherfreq/sev deterministic models may be used)

• To get a reliable comparison , as much as possible, with other stochasticmodels developed in actuarial literature and already in use in practice as Chain-Ladder Bootstrapping

• To evaluate the claims reserve and the capital requirement for reserve riskunder the assumptions of Solvency II-QIS5 framework and, furthermore, tocompare results obtained by either Internal Model and Standard Formula (bothMW and USP approaches)

10

The Fisher-Lange MethodA brief description

• This method, quite widespread in Italy among actuarial practitioners and supervisor, isused to evaluate the booked loss reserve, estimating separately the future number ofclaims to be paid (according their duration) and the relative average cost (see Fisher-Lange (1973), Ottaviani (1980), Ferrara (1996), G.Olivieri (2000) and Savelli-Clemente(2010)).

• The incremental amounts of payment to be estimated in the lower part of thetriangle are obtainable through the next relation:

ˆˆˆ ∑ ∑+

=N N

XR ˆˆ)1(ˆˆˆ,,,

++≤+<+⋅= NNjiNwithMCpNX jijiji ∑ ∑= +−=

=N

i

N

iNhhiXR

1 2,

ˆˆ

Ri

0

01

,1

,1

==

>+=

+

+

+

+

N

N

XifNN

XifNNi: accident year (AY)j: development year (DY)i+j-1: calendar year

i / j 1 2 3 N N+

12

3 Xi,j

N

jiX ,ˆ

11

AY

DY

• The stochastic model is based on the well known approach of the CRM,usually used for modelling premium risk, and we assume that incrementalpaid amounts in each cell located at the bottom of the triangle can beexpressed as the next random variable:

A Collective Risk Model (CRM)

∑=jiK

hjiji ZX,

~

,,,

~~

where:– K i,j r.v. claim counts of the i-th AY and effectively paid in the future j-th

development year (Mixed Poisson distributed – Negative Binomial case);– Zi,j,h r.v. paid amount of h-th single claim of i-th AY and paid in the j-th

development year (Mixed Gamma distributed) .

∑=h

hjiji1

,,,

• Once either the independence assumption between claim count and single claimcosts and i.i.d. assumption for the single claim costs (in each cell of triangle’s portionto be estimated) are fulfilled, it is possible to obtain easily variance and skewness foreach cell of the triangle.

12

• The r.v. K i,j is firstly assumed distributed as a Poisson, with the uniqueparameter equal to the expected number of claims (Np) estimated by thedeterministic Fisher-Lange for the cell (i,j)

Claim count distributionfor each cell

)ˆ(~~

,, jiji pNPoissK

• In order to incorporate also the uncertainty linked to parameter estimate, it isintroduced a structure variable (in the next assumed as a Gammadistributed with equal parameters) with mean 1 and std σq:

• With the aim to consider the correlations between the claim counts in thetriangle, it’s considered only one r.v. that affects all the bottom par tof the triangle .

)~ˆ(~~

,, qpNPoissK jiji ⋅

q~

q~

13

• The single claim cost r.v. Zi,j, is assumed to be Gamma distributed (differentdistributions could be chosen obtaining different skewness of overall claims reserve, for asensitivity see Savelli & Clemente (2010)) with parameters obtained implicitly by themoments method assuming that:

• As made for claim count, we can introduce for the single claim cost the parameter

Single claim severity distributionfor each cell

jiji MCZE ,,ˆ)

~( =

jZ

ji

ji cZE

Z=

)~

(

)~

(

,

,σ

uncertainty by a multiplicative structure variable having an impact on the variabilitycoefficient czj

. In particular this uncertainty is described by the multiplicative randomvariable (here assumed, as well, Gamma distributed with the same parameters) havingmean 1 and std σr (equal for each development year), contributing in a limited way tothe overall variability :

• With the aim to consider the correlations between the claim costs in the samedevelopment year, it is considered one r.v. for each development year (i.e. eachrandom value occurred affects only the cells in the same column).It is to be pointed out as this assumption, connected with the hypothesis on q, producesthe dependency between the cells of the run-off triangle to be estimated.

jZZ rccjj

~~ ⋅=

r~

jr~

14

Motor Third-Party Liability (MTPL): (MTPL):

two case studies

15

MTPL: two Case StudiesThe stochastic model here proposed has been applied, by MonteCarlo simulations (50.000 simulations) to theMTPL triangles (12x12 + tail) of two Italian companies, having different dimensions. Clearly proportion of thereal data have been modified to save the confidentiality of the data.

In particular the Triangle SIFA is referred to a small-medium company whereas the triangle AMASES has as areference a company roughly 10 times larger.

As usual in actuarial literature, the tail (12+) has been added with the payments of the last development year (12)

Triangle SIFA (Incremental Paid Amounts Xij – thousands of Euro)

1 2 3 4 5 6 7 8 9 10 11 12 12+

1 28.446 29.251 12.464 5.144 2.727 2.359 1.334 1.238 941 860 282 727 1.068 2 31.963 36.106 13.441 5.868 2.882 2.422 918 1.076 734 458 456 3 37.775 40.125 12.951 6.034 3.010 1.264 1.250 1.135 904 559 4 40.418 44.499 15.370 5.594 2.616 1.984 2.137 1.184 873

DYAY

Triangle AMASES (Incremental Paid Amounts Xij – thousands of Euro)

4 40.418 44.499 15.370 5.594 2.616 1.984 2.137 1.184 873 5 44.116 45.490 15.339 5.478 2.541 2.906 1.294 1.124 6 50.294 48.040 17.843 7.035 3.934 2.726 2.267 7 49.620 49.991 19.570 10.047 5.750 3.313 8 46.410 49.694 20.881 8.202 4.714 9 48.295 49.354 18.304 8.833

10 52.590 50.606 18.604 11 58.599 53.743 12 60.361

1 2 3 4 5 6 7 8 9 10 11 12 12+

1 193.474 172.618 87.200 45.798 29.768 19.795 19.782 17.315 13.372 12.552 8.831 8.053 19.889 2 199.854 168.966 80.543 40.656 29.053 21.121 19.964 14.249 10.720 13.684 6.008 3 225.578 186.764 93.349 47.609 30.971 26.291 17.621 18.410 14.662 7.591 4 256.398 236.678 105.616 51.172 37.338 24.085 20.754 12.082 14.137 5 282.956 263.196 120.383 63.689 37.220 29.239 23.120 15.509 6 292.428 284.401 141.400 56.390 40.195 27.955 29.987 7 312.350 285.506 131.687 75.252 46.549 38.731 8 327.673 307.992 161.516 77.965 52.696 9 339.899 326.280 185.911 101.273

10 371.275 385.847 193.006 11 388.025 390.737 12 398.686 16

BECHL= 230 mill

BECHL= 2.566 mill

AYDY

Link ratios and Average paid costs

17

AY:

AY: AY:

AY:AY:

Parameter Calibrations

The CRM (Mixed Compound Poisson) has been applied to both companies assuming that:

� the number of claims (in each single cell of the bottom triangle) is distributed as a NegativeBinomial, obtained from the mixture of a Poisson with mean Npi,j (estimated by the Fisher-Langeassuming as settlement speed the average computed on the last three diagonals) and with onlyone structure variable described by the r.v. q distributed according to a Gamma having mean 1and std equal to 8% for S IFA and 3% for A MASES. Standard deviations have been estimated by theexact moments assuming the same correlations (roughly 0.1 and equal to that implicitly obtainedby Mack prediction error formula) between claim counts for the two insurers.by Mack prediction error formula) between claim counts for the two insurers.

� the costs of the single claims are distributed as a Gamma with mean equal to the average costof the cell CMi,j (estimated by Fisher-Lange depending on the average of all average costsavailable in the upper part of the triangle expressed at current values and projected by the futureclaim inflation structure) and using the next variability coefficients czj (obtainable from theClaim DataBase of the company) different for each development year and depending on boththe analysed line of business and the company’s portfolio characteristics:

� Furthermore czjare affected by a structure variable r j (one for each development year) distributed

according to a Gamma having mean 1 and std equal to 3% for both companies .

18

CRM-FL vs Bootstrapping-CHLTotal Run-Off

• Next tables show the comparison between the two companies (SIFA and AMASES) of CV and skewnessusing not only our model (CRM-FL) but also the well known Bootstrapping model (under a LogNormalassumption).

• Firstly it is to be noted a difference between averages (on discounted basis) , clearly due to differentassumptions implicitly included in the two deterministic methods (Fisher-Lange and Chain-Ladder Paid),both applied without any “professional judgement” (present instead in the real valuations).

TOTAL RUN-OFF

CRM

(FL)

Bootstrapping

(CHL)

Mack

(CHL)

CRM

(FL)

Bootstrapping

(CHL)

SIFA 8.7% 4.6% 3.4% 0.15 0.08

AMASES 3.3% 3.7% 3.0% 0.06 0.08

SkewnessCV

19

• Secondly, it is to be emphasized how the differences of the companies are affecting on the differentvariability provided by the stochastic methods and how there is no prevalence of one method on the otherone.

• Finally, for these case studies the skewness is not affected so much when the Bootstrap method is used,for both different dimensions.

One-Year Approach

• The application of the One-Year approach through both models emphasizes how the ratiobetween CVOY concerning the variability of year 1 is located between 70% and 80% of theCVTot related to total Run-Off variability (roughly 79-80% for SIFA and 72-74% for AMASES).

• This result is close to that obtained applying the closed formula proposed by Merz andWüthrich, while it is pointed out how different patterns are in force according to accident years(see Figures below).

• CRM shows a major increase in the skewness of the “One-Year ” reserve distribution.

N. Savelli – G.P.Clemente : "A Collective Risk Model for Claims Reserve Distribution ” © 20

SCR for Reserve Risk(CRM-FL vs Bootstrapping-CHL)

• Next tables show the Ratios between SCR for reserve risk and the BestEstimate of Loss Reserve, obtained by Bootstrap and CRM for both Insurersaccording to three different confidence levels and regarding the 99.5%, adoptedby Solvency II Directive, as our benchmark.

• As expected, the smaller insurer has a higher SCR ratio due to the morevariable and skewed claim reserve distribution.

21

variable and skewed claim reserve distribution.

• Differences between models are more significant for SIFA Insurer.

SCR ratio (SCR/BE)

SCR for Reserve Risk(Internal Model vs Standard Formula)

• Figure compares SCR ratios obtained by the Internal Models (Bootstrapping andCRM) to the Capital Requirement ratios derived under QIS5 Standard Formula.

• Both approaches market-wide (MW based on a volatility factor equal to 9.5%provided by QIS5 TS) and undertaking-specific (USP based on Merz andWüthrich formula (2008) ) have been determined.

22

• Furthermore SCR ratios have beenobtained applying the ρ(x)transformation to the variabilitycoefficients derived by CRM andBootstrapping in order to test thecorrectness of QIS5 LogNormalassumption (in practice skewLogNapproximately equal to 3*CV)

Some sensitivities for c Z

• The CRM model appears rather affected by the variation of parameters. In particular it is tobe emphasized how a different calibration of the variability coefficients of Z (cz) canlead to different results in terms of variability and capital requirement.

• Assuming the three following vectors of variability coefficient of claim size (c z) :

23

and re-obtaining the following standard deviations of structure variable in order to preserve a similar correlation:

q~

The effect on CV Claim Reserveof different c Z

24

General Third-Party Liability (GTPL): (GTPL):

two case studies

25

CV Behaviour

As expected GTPL ratios between One-Year and Total Run-Off CVs show more variable patterns than MTPL:

26

Accident Year Accident Year

SCR ratios(SCR Reserve Risk / Initial BE)

• GTPL shows higher SCR ratio than MTPL because of more significant CV.

• It is noteworthy that standard formula leads to lower capital requirement than Internal Model (either Bootstrapping or CRM) for small Insurer (SF-MW volatility = 11%).

• AMASES observes a greater reduction using methodologies based on Chain-Ladder assumptions.

• Finally, it is confirmed a lower capital for small insurers by the LogNormal assumption.

27

Motor Other Damages(MOD): (MOD):

two case studies

28

CoV Behaviour

29

Accident Year Accident Year

• Stochastic models lead to very different capital requirements for Motor Damages.

• Models based on Chain-Ladder assumptions (as Bootstrapping and Merz-Wüthrich Formula) provides higher SCR ratios for AMASES (SF-MWvolatility = 10%).

SCR ratios(SCR Reserve Risk / Initial BE)

• On the contrary, CRM reflects the dimensional effect returning a negligible capital for AMASES and a ratio higher than MW approach for SIFA.

30

Final RemarksFinal Remarks

31

Final Remarks

• According to us the CRM model seems to represent a valid stochastic model to beadded to models suggested at the moment by the variegate actuarial literature (in orderto reduce the Model Risk ).

• Parameters of the models may be easily estimated using claims DataBase and thedeterministic model “Fisher-Lange”, based on the estimate of separate number of claimsto be paid and future average costs.

• The same model may be also used by deterministic methods other than Fisher -Lange ,

32

• The same model may be also used by deterministic methods other than Fisher -Lange ,providing separate estimates for either number of future payments and average claimcosts (i.e. using other incremental average cost methods).

• The sensitivity analyses here reported put in evidence the relevance of a reliableparameter calibration and the strict link between parameter uncertainty and the variabilityof the overall claim reserve.

• In particular it needs to pay attention to the estimation of the standard deviation of thestructure variable q in order to properly capture the correlation inside the triangle.Variability coefficient of total run-off claims reserve distribution is significantly affected bythat value.

• We regard estimation of variability coefficients czj(obtainable from the Claim DataBase

of the company and different for each development year) as a key issue. As well knownthese data are available mainly as an internal/appointed actuary, while it is rather difficultto get them for an external actuary (e.g. involved in a take-over evaluation). This mightproduce minor diffusion of the approach compared to other methods based on aggregaterun-off triangles mainly.

• The One-Year approach is representing a key approach to be fully investigated in order

33

• The One-Year approach is representing a key approach to be fully investigated in orderto perform evaluations consistent with Solvency II.

• Nevertheless, according to us, this approach used for reserve risk might guarantee amajor consistency with the premium risk evaluation, when applicable.

• The need of an international Data Base on run-off triangles (for EU market first of all)

• The results given here for 3 LoBs must be taken with great care and numerousanalyses should be carried on in practice for consistent estimation of main parameters

• Further research improvements : stochastic inflation, new Direct Reimbursementregime for MTPL (CARD), analysis of tails

Main References

� Bühlmann H., De Felice M., Gisler A., Moriconi F., Wüthrich M .V. (2009): “Recursive Credibility Formula forChain-Ladder Factors and The Claims Development Results”, ASTIN Bulletin, Volume 39 n.1

� Diers D. (2009): “Stochastic Re-Reserving in multi-year internal models”, Astin Colloquium, Helsinki

� De Felice M., Moriconi F. (2003): “Risk Based Capital in P&C loss reserving or stressing the triangle”, WorkingPaper

� England P., Verrall R. (2006): Predictive Distributions of Outstanding Liabilities in General Insurance, Annals ofActuarial Science, Volume 1, Number 2, 221-270

� Fisher W.H., Lange J. T. (1973): “Loss Reserve Testing: a report year approach”, PCAS, LX, pp. 189-207� Fisher W.H., Lange J. T. (1973): “Loss Reserve Testing: a report year approach”, PCAS, LX, pp. 189-207� Hayne (1989): “Application of collective risk theory to estimate variability in loss reserves”, PCAS

� IAA (2004): “A Global Framework for Insurer Solvency Assessment, May 2004, Research Report of The InsurerSolvency Assessment Working Party

� Merz M., Wüthrich M.V. (2008): Modelling the Claims Development Results for Solvency Purposes, CAS E-Forum Fall, pag. 542-658

� Meyers G . (2008): “Stochastic Loss Reserving with the Collective Risk Model”, CAS Forum

� Ottaviani R. (1980): “Il metodo Fisher-Lange per la riserva sinistri”, Annals of Italian Statistics Society, Rome

� Savelli N., Clemente G.P. (2010): “A Collective Risk Model for Claims Reserve Distribution”, Presented at ICA(2010) and Published on Proceedings of Workshop in Risk Theory, Campobasso

� Savelli N., Clemente G.P. (2011): “Stochastic Claim Reserving based on CRM for Solvency Purposes”,Submitted

34

Thank You Thank You for your attention

35

Back UpBack Up

36

MTPL: MTPL: Discounting and Risk Margin

37

Claims Reserves(Solvency II vs Solvency I)

Figure compares the technical provisions of both companies estimated as follows:• under the current regulatory environment (Solvency I): claims reserve is equal to the

ultimate cost estimated to be paid (not discounted).• under Solvency II – QIS5 framework: claims reserve is equal to best estimate (on

discounted basis) plus risk margin:- Discount factors (useful for BE valuation) are

determined by the term structure of risk-free interestrates, given in QIS5 Technical Specifications (equal

38

rates, given in QIS5 Technical Specifications (equalfor non-life obligations to the risk-free rate at31/12/2009 obtained by swap rates and including a50% illiquidity premium).

- Risk margin is derived by the QIS5 methodologiesincluding SCR for reserve risk (by IM) and foroperational risk (by Standard Formula on TP). Nodefault risk is here considered (no reinsurance isindeed assumed) and no unavoidable market risk(equal to zero for Non-Life business). Finally weassume no diversification effects with other LoBs, weuse the proportional method on BE proposed, assimplification by QIS5, to obtain future SCRs and therisk-free rate without illiquidity premium fordiscounting (provided by QIS5)

Sl= SlI=

Risk Margin

• The lower technical provisions under the Solvency II are due to the discounting effect (more relevant for AMASES because of the lower settlement speed).

• Risk Margin appears very low (roughly 2% ) because of a minor SCR. Only the higher Capital Requirement, derived by CRM, leads to a significant Risk Margin (for SIFA only).

• However all results are lower than the prudential proxy provided in QIS5 TS (8% for MTPL)

39

MTPL)

t t

General Third-Party Liability (GTPL): case studies

40

Link Ratios and Average Paid Costs

41N. Savelli – G.P.Clemente : "A Collective Risk Model for Claims Reserve Distribution ” ©

AY:

AY: AY:

AY:AY:

Calibration and first results

• Assuming:– the next variability coefficients for claim size payments (equal to 150% of MTPL czj

):

– an only one r.v. q with σq equal to 19% and 8.5% for SIFA and AMASES respectively (calibrated assuming a correlation equal to roughly 0.08, that is implicitly obtained by Mack prediction error formula) Mack prediction error formula)

– a r.v. rj for each development year with σr equal to 3% for both Insurers.

• the distribution of One-Year GTPL claims reserve has the next characteristics:

• It has to be emphasized that Bootstrapping CVs are very far from that derived by Merz-Wüthrich formula (both approaches are based on Chain-Ladder assumptions).

42

One-Year Approach

As expected GTPL ratios between One-Year and Total Run-Off CVs show more variable patterns than MTPL:

43

Accident Year Accident Year

SCR ratios(SCR Reserve Risk / Initial BE)

• GTPL shows higher SCR ratio than MTPL because of more significant CV.

• It is noteworthy that standard formula leads to lower capital requirement than Internal Model (either Bootstrapping or CRM) for small Insurer (SF-MW volatility = 11%).

• AMASES observes a greater reduction using methodologies based on Chain-Ladder assumptions.

• Finally, it is confirmed a lower capital for small insurers by the LogNormal assumption.

44

Motor Other Damages (MOD): Motor Other Damages (MOD): case studies

45

Link Ratios and Average Paid Costs

46

DY: DY:

AY: AY:

Calibration and first results

• Assuming:– the next variability coefficients for claim size payments (equal to 50% of MTPL czj

):

– an only one r.v. q with σq equal to 7.5% and 1.5% for SIFA and AMASES respectively (calibrated assuming a correlation equal to roughly 0.04, that is implicitly obtained by Mack prediction error formula);Mack prediction error formula);

– a r.v. rj for each development year with σr equal to 3% for both Insurers.

• the distribution of MOD claims reserve, determined by one-year approach, has the following characteristics:

where for AMASES we get a CV larger than SIFA under the CHL technique (due also to link ratios variability for DY4) whereas the CRM provides a very low CV (1%), not completely explained by the different size (in MOD the size factor is roughly only 2 to 1)

47

CoV Behaviour

48

Accident Year Accident Year

• Stochastic models lead to very different capital requirements for Motor Damages.

• Models based on Chain-Ladder assumptions (as Bootstrapping and Merz-Wüthrich Formula) provides higher SCR ratios for AMASES (SF-MWvolatility = 10%).

SCR ratios(SCR Reserve Risk / Initial BE)

• On the contrary, CRM reflects the dimensional effect returning a negligible capital for AMASES and a ratio higher than MW approach for SIFA.

49

A comparison among the LoBsA comparison among the LoBsand the results after aggregation

50

One-Year Approach

SCR ratios

52

Technical Provisions

Savelli&Clemente: "A Collettive Risk Model for Clai ms Reserve Distribution

Aggregated SCR

Aggregated SCR has been obtained for each companyby using the linear aggregation formula provided byQIS5 (see correlation matrix).

It has to be emphasized how the big Insurer has asignificant reduction of capital requirement with respectto MW approach of Standard Formula, whereas SIFAhas a comparable reduction for SF-USP approach only.has a comparable reduction for SF-USP approach only.

Savelli&Clemente: "A Collettive Risk Model for Claims R eserve Distribution