MTPL as a challenge to actuaries

1. MTPL as a challenge to actuaries HOT TOPICS of MTPL from the perspective of a Czech actuary

2. Contents • Dynamism and stochasticity of loss reserving methods • Regression methods • Bootstrapping • Appropriate reserving of large bodily injury claims • Practical implications of segmentation • Simultaneous co-existence of different rating factors on one market • Price sensitivity of Czech MTPL policy holders

3. Reserving methods for MTPL Problems: • demonopolisation • new players on the market • not optimal claims handling (training of loss adjusters, upgrading SW)  development factors are unstable • guarantee fund (GF) • settlement of claims caused by • uninsured drivers • unknown drivers • unknown exposition + GF=new(unknown) entity within the system • unstable development factors • significant trend in incurred claims • REQUIRE: incorporation of stochasticity and dynamism into methods

4. Reserving methods for MTPL Stochasticity: • “easy” but reasonable way = bootstrap • fitting a preferred projection method to a data triangle • comparison of original data and projection  residuals • sampling residuals and generation of many data triangles • derivation of ultimates from these sampled triangles • statistical analysis of ultimates/IBNRs/RBNSes: • expected value • standard error • higher moments • distribution

5. Reserving methods for MTPL Dynamism: • regression methods - a natural extension of Chain-ladder Y(i,j)=b*Y(i,j-1)+e(i), Var(e)=2Y(i,j-1) • special cases: =1 (chain-ladder) =2  =0 (ordinary least sq. regression) • extension: Y(i,j)=a0+a1*i+b*Y(i,j-1)+e(i), Var(e)=2Y(i,j-1) = extended link ratio family of regression models described by G.Barnett & B. Zehnwirth (1999)

6. Reserving methods for MTPL Modelling trends in each “direction”: • accident year direction • in case of adjustment for exposure  probably little changes over time • in case of unavailability of exposure  very important • development year direction • payment year direction • gives the answer for “inflation” • if data is adjusted by inflation, this trend can extract implied social inflation • MODEL: development years j=0,…,s-1; accident years i=1,…,s; payment years t=1,…,s = probabilistic trend family (G.Barnett & B. Zehnwirth (1999))

7. Reserving methods for MTPL - example Construction of PTF model using STATISTICA (data analysis software system) • Data set • claim numbers caused by uninsured drivers in Czech Republic 2000-2003 • triangle with quarterly origin and development periods • Exposure – unknown • Full model: • applied on Ln(Y) • 46 parameters

8. Reserving methods for MTPL - example Complete design matrix • necessary to exclude intercept • too many parameters necessary to create submodel GOAL: description of trends within 3 directions and changes in these trends optimal submodels = submodels adding together columns (“columns-sum submodels (CSS)”) • How to create submodels: • manually • use forward stepwise method • it is necessary to transform final model into CSS submodel, this model will still have too many parameters (problem of multi-colinearity + bad predictive power) • necessity of subsequent reduction of parameters

9. Reserving methods for MTPL - example • usually possible to assume  model with intercept • final model for Czech guarantee fund: • 7 parameters • R2=91% • tests of normality of standardized residuals • autocorrelation of residuals rejected

10. Reserving methods for MTPL - example

11. Reserving methods for MTPL - example

12. Reserving methods for MTPL - example Statistics of total ultimate for 2000-3 • bootstrap method based upon assumptions of regression model • predict future values (i+j>16)  mean,quantiles  st. dev. • bootstrap future data (assumption of normality) • descriptive statistics based upon bootstrapped samples

13. Reserving methods for MTPL Conclusions: • we got a reasonable model using PTF model for describing and predicting incurred claims of guarantee fund • model reasonably describes observed trend in data and solves the problem of non-existence of exposure measure

14. Reserving large bodily injury claims • Importance of properly reserving large bodily injury (BI) claims • Mortality of disabled people • Sensitivity of reserve for large BI claim upon estimation of long term inflation/valorization processes

15. Reserving large BI claims - importance • More than 90% of large claims consists from large BI claims • Proportion of large BI claims on all MTPL claims measured relatively against: • number of all claims • amount of all claims • Decreasing trend is only due to: • long latency of reporting BI claims to insurer • not the best reserving practice. • It’s reasonable to assume that share of BI claims is aprox. 20%.

16. Reserving large BI claims - importance • Due to the extreme character of large BI claims the importance of appropriate reserving is inversely proportional to the size of portfolio  Example: proportion of large BI claims on all claims of Czech Insurers Bureau („market share“ approx. 3%)

17. Reserving large BI claims - mortality • Classification of disabled people  criteria: • seriousness • partial disability • complete disability • main cause • illness • injury =traffic accidents, industrial accidents,... • Availability of corresponding mortality tables in Czech Republic

18. Reserving large BI claims - mortality • Comparison of mortality of regular and disabled people It’s reasonable to assume that „illness“ disability implies higher mortality than “accident” disability  proper reserve is probably

19. Reserving large BI claims – types of damage • No problem: • Pain and suffering • Loss of social status • Problem • Home assistance (nurse, housmaid, gardner, ...) depends upon: • mortality • future development of disability • Loss of income depends upon: • mortality • future development of disability • structure of future income  prediction of long term inflation and valorization

20. Reserving large BI claims – loss of income Loss of income in Czech Republic = “valorized income before accident” - “actual pension” • “actual income (partially disabled)” Needs: • estimate of future valorization of incomes ... vI(t) • estimate of future valorization of pensions ... vP(t) • both depend upon economic and political factors • estimate of future inflation of incomes ... ii(t) • depends upon economic factors

21. Reserving large BI claims – loss of income Notation: • income before accident ... IB • pension ... P • income after accident ... IA • vI(t), vP(t), ii(t) • inflation ... i (used for discounting future payments) • Small differences among vI(t), vP(t), ii(t) andi can imply dramatic changes in needed reserve  Proportion of IB , P and IA is crucial Assumptions: • dependence upon mortality is not considered • complete disability  IA=0 • vI(t), vP(t) and ii(t) are constant over time

22. Reserving large BI claims – loss of income Examle 1: • income before accident ... IB = 10 000 CZK • pension ... P = 6 709 CZK • initial payment of ins. company = 3 291 CZK • vI(t)=3% • vP(t)=2% • i = 4% • expected interest rate realized on assets of company is higher than both valorizations Question: • Will the payments of ins. company increase faster or slower than interest rate?

23. Reserving large BI claims – loss of income

24. Reserving large BI claims – loss of income Examle 2 (“realistic”):

25. Reserving large BI claims – loss of income Examle 3 (“a blessing in disguise”) – degressive pension system

26. Segmentation–problem of asymmetric information

27. Segmentation – problem of asymmetric information During 2000-2003: • identical rating factors used by all insurers • partial regulation of premium • real spread of premium +/- 5% within given tariff category annual fluctuation of policyholders = more than 5% of all registered vehicles From the beginning of 2004: • beginning of segmentation • the difference in premium level applied by different insurers >10% holds for a large set of policyholders  probability of loss due to assymetric information grows