MTPL as a challenge to actuaries

HOT TOPICS of MTPL from the perspective of a Czech actuary

MTPL as a challenge to actuaries

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

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 systemunstable development factorssignificant trend in incurred claims

REQUIRE: incorporation of stochasticity and dynamism into methods

Jakub Strnad


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

Jakub Strnad


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)

i

i

jiY

jiYb

)1,(

),(

i jiYjiY

nb

)1,(),(1

i

i

jiY

jiYjiYb 2)1,(

)1,(),(

Jakub Strnad


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))

ji

ji

tt

j

kkijiY ,

21

),(

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

16 ;2,,15 ;1,,16 ;1, where,),( ,21

tjijiY tjiji

ji

tt

j

kki

Jakub Strnad


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

Jakub Strnad

Reserving methods for MTPL - exampleconsti 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

K-S d=,07700, p> .20; Lilliefors p<,10

Shapiro-Wilk W=,98418, p=,15747

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

X <= Category Boundary

0

5

10

15

20

25

30

No.

of

obs.

Jakub Strnad

Reserving methods for MTPL - examplePredicted vs. Residual Scores

Dependent variable: ln Y

-1 0 1 2 3 4 5 6 7

Predicted Values

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

Res

idua

ls

95% confidence

Jakub Strnad


Predicted vs. Observed Values

Dependent variable: ln Y

-1 0 1 2 3 4 5 6 7

Predicted Values

-1

0

1

2

3

4

5

6

7

Ob

serv

ed

Va

lue

s

95% confidence

Jakub Strnad


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

Jakub Strnad


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

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

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%.

13%

0.12%

12%

0.10%

11%

0.09%

3%

0.04%

0%

2%

4%

6%

8%

10%

12%

14%

2000 2001 2002 2003

Share fromtotal amountof all claims

Share fromtotal numberof all claims

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%)

36%

0.29%

20%

0.18%

9%

0.07%

10%

0.05%

0%

5%

10%

15%

20%

25%

30%

35%

40%

2000 2001 2002 2003

Share fromtotal amountof all claims

Share fromtotal numberof all claims

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

Jakub Strnad

Reserving large BI claims - mortality Comparison of mortality of regular and disabled people

It’s reasonable to assume that „illness“ disability implies highermortality than “accident” disability proper reserve is probably

20 years old man

0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25 30 35 40 45 50 55 60

Th

ou

san

ds

CZ

K

0%

50%

100%

150%

200%

250%

average person

partially disabled

completely disabled

difference "regular-disabled" in % (right axis)

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 incomedepends upon:

mortality future development of disability

structure of future income prediction of long term inflation and valorization

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 ... 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

Jakub Strnad


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) and i 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

Jakub Strnad


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?

Jakub Strnad

Reserving large BI claims – loss of incomeIncome before accident (I): 10 000Initial disability pension (P): 6 709

P / I: 67.1%Initial payment of ins. company: 3 291

Average valorization of P: 2.0%Average valorization of I: 3.0%

Expected interest income: 4.0%

Government Insurance company

Structure of future income of injured person

0

5

10

15

20

25

1 4 7 10 13 16 19 22 25

Th

ou

san

ds

0%

20%

40%

60%

80%

100%

1 4 7 10 13 16 19 22 25

Implied average interest rate on reserve

0.0%

0.5%

1.0%

1.5%

2.0%

1 3 5 7 9 11 13 15 17 19 21 23 25

Average valorization of undiscounted payments of

insurer

4.0%

4.5%

5.0%

5.5%

6.0%

1 4 7 10 13 16 19 22 25

Jakub Strnad

Reserving large BI claims – loss of incomeExamle 2 (“realistic”):

Income before accident (I): 15 000Initial disability pension (P): 7 699






0

20

40

60

80

100

1 4 7 10 13 16 19 22 25

Th

ou

san

ds

0%

20%

40%

60%

80%

100%

1 4 7 10 13 16 19 22 25


4.0%

4.5%

5.0%

5.5%

6.0%

1 3 5 7 9 11 13 15 17 19 21 23 25


insurer

7.0%

7.5%

8.0%

8.5%

9.0%

1 4 7 10 13 16 19 22 25

Jakub Strnad

Reserving large BI claims – loss of incomeExamle 3 (“a blessing in disguise”) – degressive pension system

Income before accident (I): 7 000Initial disability pension (P): 5 930






0

10

20

30

40

1 4 7 10 13 16 19 22 25

Th

ou

san

ds

0%

20%

40%

60%

80%

100%

1 4 7 10 13 16 19 22 25


6.0%6.5%7.0%7.5%8.0%8.5%9.0%9.5%

10.0%

1 3 5 7 9 11 13 15 17 19 21 23 25


insurer

9.0%9.5%

10.0%10.5%11.0%11.5%12.0%12.5%13.0%13.5%14.0%

1 4 7 10 13 16 19 22 25

Jakub Strnad

Segmentation – problem of asymmetric information

Men WomenCities above

500.000 Towns 50.000-

500.000

Others

12 12 16 667 16 6678 9 16 667 16 6675 7 16 667 16 667

Fair/real market risk premium: 883 333 883 333

Written risk premium: 138 889 155 556 0 0138 889 0 0 141 667

0 0 100 000 100 000Total written risk premium: 433 333 341 667

Total written market risk premium: 775 000

Loss/profit: -61 111 -44 444 0 05 556 0 0 -8 333

0 0 16 667 -16 667Total loss/profit: -100 000 -8 333

Total market loss/profit: -108 333 -14.0% relatively to written market risk premium

Set of all risksSplit of risks using rating

factors of company ASplit of risks using rating

factors of company B

100 000 vehicles 50.000 50.000

33.000

33.000

33.000

Risk premium for individual segments

Number of risks in

segmentsRisk premium Risk premium

8.3 9.312.08.56.0

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

Documents

MTPL as a challenge to actuaries