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MTPL as a challenge to actuaries

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MTPL as a challenge to actuaries

HOT TOPICS of MTPL from the perspective of a Czech actuary

- 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

Problems:

- demonopolisation
- new players on the market
- not optimal claims handling (training of loss adjusters, upgrading SW)
development factors are unstable

- not optimal claims handling (training of loss adjusters, upgrading SW)

- new players on the market
- 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

- unknown exposition + GF=new(unknown) entity within the system

- settlement of claims caused by
- REQUIRE: incorporation of stochasticity and dynamism into methods

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

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)

- special cases:

= extended link ratio family of regression models described by

G.Barnett & B. Zehnwirth (1999)

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

- gives the answer for “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))

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

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

- 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

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

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

- 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

- 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%.

- 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%)

- Classification of disabled people
criteria:

- seriousness
- partial disability
- complete disability

- seriousness
- main cause
- illness
- injury =traffic accidents, industrial accidents,...

- 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

- No problem:
- Pain and suffering
- Loss of social status

- Problem
- Home assistance (nurse, housmaid, gardner, ...)
depends upon:

- mortality
- future development of disability

- Home assistance (nurse, housmaid, gardner, ...)
- Loss of income
depends upon:

- mortality
- future development of disability
- structure of future income prediction of long term inflation and valorization

Loss of income in Czech Republic

= “valorized income before accident”

- “actual pension”

- “actual income (partially disabled)”
Needs:

- both depend upon economic and political factors

- depends upon economic factors

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

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?

Examle 2 (“realistic”):

Examle 3 (“a blessing in disguise”) – degressive pension system

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

- the difference in premium level applied by different insurers >10% holds for a large set of policyholders