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

2. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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)= 2 Y(i,j-1)   special cases: =1 (chain-ladder)  =2  =0 (ordinary least sq. regression)   extension: Y(i,j)=a 0 +a 1 *i+b*Y(i,j-1)+e(i), Var(e)= 2 Y(i,j-1)  = extended link ratio family of regression models described by G.Barnett & B. Zehnwirth (1999)

6. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad Reserving methods for MTPL - example  usually possible to assume  model with intercept  final model for Czech guarantee fund:  7 parameters  R 2 =91%  tests of normality of standardized residuals  autocorrelation of residuals rejected

10. Jakub Strnad Reserving methods for MTPL - example

11. Jakub Strnad Reserving methods for MTPL - example

12. Jakub Strnad Reserving methods for MTPL - example Statistics of total ultimate for 2000-3  bootstrap method based upon assumptions of regression model 1)predict future values (i+j>16)  mean,quantiles  st. dev. 2)bootstrap future data (assumption of normality) 3)descriptive statistics based upon bootstrapped samples

13. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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. Jakub Strnad 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... v I (t) estimate of future valorization of pensions... v P (t)  both depend upon economic and political factors estimate of future inflation of incomes... i i (t)  depends upon economic factors

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

22. Jakub Strnad Reserving large BI claims – loss of income Examle 1:  income before accident... I B = 10 000 CZK  pension... P = 6 709 CZK  initial payment of ins. company = 3 291 CZK  v I (t)=3%  v P (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. Jakub Strnad Reserving large BI claims – loss of income

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

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

26. Jakub Strnad Segmentation – problem of asymmetric information

27. Jakub Strnad 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