Transcript Reserving methods for MTPL

MTPL as a challenge to actuaries
HOT TOPICS of MTPL from the perspective
of a Czech actuary
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 system
unstable development factors
significant trend in incurred claims
REQUIRE: incorporation of stochasticity and
dynamism into methods
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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:




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expected value
standard error
higher moments
distribution
Reserving methods for MTPL
Dynamism:
 regression methods - a natural extension of Chain-ladder
2
Y(i,j)=b*Y(i,j-1)+e(i), Var(e)= Y(i,j-1)
 special cases:
=1 (chain-ladder) 

 Y (i, j)
b
 Y (i, j  1)
i
i
=2 
b
1
Y (i, j )

n i Y (i, j  1)
=0 (ordinary least sq. regression) 
 Y (i, j)  Y (i, j  1)
b
 Y (i, j  1)
i
2
i
2
 extension: Y(i,j)=a0+a1*i+b*Y(i,j-1)+e(i), Var(e)= Y(i,j-1)
=
extended link ratio family of regression models described by
G.Barnett & B. Zehnwirth (1999)
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
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:
j
i j
k 1
t 2
Y (i, j )   i    k    t   i , j
development years j=0,…,s-1; accident years i=1,…,s; payment years t=1,…,s
= probabilistic trend family (G.Barnett & B. Zehnwirth (1999))
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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
j
i j
k 1
t 2
Y (i, j )   i    k   t   i , j , where  i , i  1; 16 ,  j , j  1; 15 ,  t , t  2; 16 
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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
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Reserving methods for MTPL - example
i  const  model with intercept

usually possible to assume

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-W ilk W =,98418, p=,15747
30
25
No. of obs.
20
15
10
5
0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
X <= Category Boundary
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1.5
2.0
2.5
3.0
3.5
Reserving methods for MTPL - example
Predicted vs. Residual Scores
Dependent variable: lnY
2,0
1,5
1,0
Residuals
0,5
0,0
-0,5
-1,0
-1,5
-2,0
-2,5
-1
0
1
2
3
Predicted Values
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4
5
6
95% confidence
7
Reserving methods for MTPL - example
Predicted vs. Observed Values
Dependent variable: lnY
7
6
Observed Values
5
4
3
2
1
0
-1
-1
0
1
2
3
Predicted Values
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4
5
6
95% confidence
7
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
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 nonexistence of exposure measure
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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
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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
14%

13%
12%
12%
11%
Share from
total amount
of all claims
10%
8%
Share from
total number

of all claims
6%
4%
0.12%
2%
0%
2000
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0.10%
2001
0.09%
2002
2003
3%
0.04%
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%.
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%)

40%
36%
35%
30%
25%
Share from
total amount
of all claims
Share from
total number
of all claims
20%
20%
15%
10%
9%
10%
5%
0.18%
0.29%
0.07%
0.05%
0%
2000
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2001
2002
2003
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
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Reserving large BI claims - mortality
 Comparison of mortality of regular and disabled people
Thousands CZK
20 years old man
900
250%
average person
800
partially disabled
700
200%
completely disabled
600
150%
difference "regulardisabled" in % (right axis)
500
400
100%
300
200
50%
100
0
0%
0
5
10
15
20
25
30
35
40
45
50
55
It’s reasonable to assume that „illness“ disability implies higher
mortality than “accident” disability  proper reserve is probably
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60
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
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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
• estimate of future valorization of pensions
... vI(t)
... vP(t)
 both depend upon economic and political factors
• estimate of future inflation of incomes
 depends upon economic factors
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... ii(t)
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) 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
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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?
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Reserving large BI claims – loss of income
Income before accident (I):
Initial disability pension (P):
P / I:
Initial payment of ins. company:
Average valorization of P:
Average valorization of I:
Expected interest income:
10 000
6 709
67.1%
3 291
2.0%
3.0%
4.0%
Implied average interest rate on reserve
2.0%
1.5%
1.0%
0.5%
0.0%
1
3
5
7
9
11
13
15
17
19
21
23
25
Thousands
Structure of future income of injured person
25
100%
20
80%
15
60%
10
40%
5
20%
4.5%
0
0%
4.0%
1
4
7 10 13 16 19 22 25
6.0%
5.5%
5.0%
1 4 7 10 13 16 19 22 25
Government
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Average valorization of
undiscounted payments of
insurer
1
Insurance company
4
7 10 13 16 19 22 25
Reserving large BI claims – loss of income
Examle 2 (“realistic”):
Income before accident (I):
Initial disability pension (P):
P / I:
Initial payment of ins. company:
Average valorization of P:
Average valorization of I:
Expected interest income:
15 000
7 699
51.3%
7 301
6.0%
7.0%
3.0%
Implied average interest rate on reserve
6.0%
5.5%
5.0%
4.5%
4.0%
1
3
5
7
9
11
13
15
17
19
21
23
25
Thousands
Structure of future income of injured person
100
100%
80
80%
60
60%
40
40%
20
20%
7.5%
0
0%
7.0%
1
4
7 10 13 16 19 22 25
9.0%
8.5%
8.0%
1 4 7 10 13 16 19 22 25
Government
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Average valorization of
undiscounted payments of
insurer
1
Insurance company
4
7 10 13 16 19 22 25
Reserving large BI claims – loss of income
Examle 3 (“a blessing in disguise”) – degressive pension system
Income before accident (I):
Initial disability pension (P):
P / I:
Initial payment of ins. company:
Average valorization of P:
Average valorization of I:
Expected interest income:
7 000
5 930
84.7%
1 070
6.0%
7.0%
3.0%
Implied average interest rate on reserve
10.0%
9.5%
9.0%
8.5%
8.0%
7.5%
7.0%
6.5%
6.0%
1
3
5
7
9
11
13
15 17
19 21
23
25
Thousands
Structure of future income of injured person
40
100%
30
80%
14.0%
13.5%
13.0%
12.5%
12.0%
11.5%
11.0%
10.5%
10.0%
9.5%
9.0%
60%
20
40%
10
20%
0
0%
1
4
7 10 13 16 19 22 25
1 4 7 10 13 16 19 22 25
Government
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Average valorization of
undiscounted payments of
insurer
1 4 7 10 13 16 19 22 25
Insurance company
Segmentation – problem of asymmetric information
Split of risks using rating
factors of company A
Set of all risks
Men
100 000 vehicles
Risk premium for
individual segments
12
8
5
12
9
7
50.000
Number of
risks in
segments
16 667 16 667
16 667 16 667
16 667 16 667
Fair/real market risk premium:
883 333
Written risk premium:
Total written risk premium:
138 889
138 889
0
433 333
Total written market risk premium:
775 000
Loss/profit:
Total loss/profit:
-61 111
5 556
0
-100 000
Total market loss/profit:
-108 333
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Women
50.000
Risk premium
8.3
Split of risks using rating
factors of company B
Cities above
500.000
Towns 50.000500.000
33.000
Others
33.000
33.000
Risk premium
12.0
8.5
6.0
9.3
883 333
155 556
0
0
0
0
100 000
341 667
0
141 667
100 000
-44 444
0
0
0
0
16 667
-8 333
0
-8 333
-16 667
-14.0%
relatively to written market risk premium
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
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