index of ranking for bonus-malus system
TRANSCRIPT
1
INDEX OF RANKING
FOR BONUS-MALUS SYSTEM
Dr. D. José A. Álvarez Jareño
Dr. D. Prudencio Muñiz Rodríguez
Valencia, 2011
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Company Max Min PEE RSAL
A 250 40 42,18 1,04%
B 300 50 50,65 0,26%
C 185 50 50,52 0,39%
D 300 45 59,59 5,72%
E 200 46 47,11 0,72%
F 300 45 45,79 0,31%
G 200 60 60,65 0,47%
H 400 45 46,18 0,33%
I 400 50 52,30 0,66%
Belgium 200 54 58,66 3,19%
Germany (New) 200 30 39,77 5,74%
Germany (Old) 200 40 45,04 3,15%
Switzerland 270 45 57,24 5,44%
United Kingdom 100 33 37,72 7,04%
Relative Stationary Average Level (RSAL)
3
Prima Media
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
0 5 10 15 20 25 30
Mea
n P
rem
ium
Insurance Annuities
A B C D E
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
0 5 10 15 20 25 30
Insurance Annuities
F G H I
4
Company Coefficient of variation %
A 0,10085654 1,58%
B 0,03952437 0,62%
C 0,03170115 0,50%
D 0,50795957 7,94%
E 0,08034399 1,26%
F 0,057869 0,90%
G 0,03301014 0,52%
H 0,082077 1,28%
I 0,10244722 1,60%
Belgium 0,18991748 2,97%
Germany (New) 0,25587601 4,00%
Germany (Old) 0,22383281 3,50%
Switzerland 0,41172235 6,43%
United Kingdom 0,24132563 3,77%
Coefficient of Variation of Premiums
5
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 5 10 15 20 25 30
Co
effi
cien
to
f V
ari
ati
on
Insurance Annuities
A B C D E
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 0,2 0,4 0,6 0,8 1 1,2
Co
effi
cien
to
f V
ari
ati
on
Claim Frequency
A B C D E
Coefficient of Variation of Premiums
6
Company Convergence Ratio Years
A 0,8989 + 30
B 0,5895 17
C 0,6054 24
D 0,9418 + 30
E 0,8110 + 30
F 0,5093 12
G 0,5994 16
H 0,7550 28
I 0,7493 24
Belgium 0,9231 + 30
Germany (New) 0,8417 + 30
Germany (Old) 0,8028 + 30
Switzerland 0,9193 + 30
United Kingdom 0,5319 13
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,2 0,4 0,6 0,8 1 1,2
Co
nv
erg
ence
Ra
tio
Claim Frequency
A B C D E
Convergence Ratio
7
Loimaranta Elasticity
Elasticidad de un sistema bonus-malus
Elasticidad De Pril Elasticidad de Loimaranta
8
0
0,5
1
1,5
2
2,5
3
0 0,2 0,4 0,6 0,8 1 1,2
Frecuencia Siniestral
A B C D E
0
0,5
1
1,5
2
2,5
3
3,5
0 0,2 0,4 0,6 0,8 1 1,2
Frecuencia Siniestral
F G H I
Loimaranta Elasticity
9
Compañía Loimaranta Global [0; 0,225]
A 0,0917 0,5689 0,0833
B 0,0165 0,3679 0,0098
C 0,0133 0,4587 0,0047
D 0,6897 0,3986 0,1832
E 0,0533 0,6385 0,0332
F 0,0230 1,1080 0,0090
G 0,0138 0,6657 0,0047
H 0,0471 1,0826 0,0256
I 0,0657 0,9648 0,0280
Belgium 0,2149 0,3537 0,1016
Germany (New) 0,2832 0,6842 0,0646
Germany (Old) 0,8485 2,6284 0,2073
Switzerland 0,4795 0,3896 0,1661
United Kingdom 0,1429 0,4104 0,0376
Eficiencia de Loimaranta
Eficiencia Global De Pril
Eficiencia Global Loimaranta
Loimaranta Elasticity
10
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0 0,2 0,4 0,6 0,8 1 1,2
Ela
stic
ida
d L
oim
ara
nta
Frecuencia siniestral
Bélgica
Alemania N
Alemania V
Suiza
UK
Loimaranta Elasticity
11
Compañía Discriminación (d2) Varianza mínima (c)
A 0,8705 0,0041
B 0,1751 0,0478
C 0,1751 0,0383
D 2,4171 0,1949
E 0,4772 0,0548
F 0,1989 0,0594
G 0,1751 0,0478
H 0,4352 0,0575
I 0,4340 0,0813
Belgium 1,5892 0,0884
Germany (New) 1,3286 0,0883
Germany (Old) 1,0735 0,0973
Switzerland 1,5166 0,1914
United Kingdom 0,3903 0,1457
Discriminatory Power
12
Discriminatory Power
0
5
10
15
20
25
30
0 0,2 0,4 0,6 0,8 1 1,2
Dis
crim
ina
tory
Po
wer
Claim Frequency
A B C D E
0
5
10
15
20
25
30
0 0,2 0,4 0,6 0,8 1 1,2
Claim Frequency
F G H I
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Measurements of the analysed bonus-malus systems
Company RSALCoefficient
of variation
Convergency
Ratio
Loimaranta
Elasticity
Discrimination
(d2)
A 1,04% 0,1009 0,8989 0,0917 0,8705
B 0,26% 0,0395 0,5895 0,0165 0,1751
C 0,39% 0,0317 0,6054 0,0133 0,1751
D 5.72% 0,5080 0,9418 0,6897 2.4171
E 0,72% 0,0803 0,8110 0,0533 0,4772
F 0,31% 0,0579 0,5093 0,0230 0,1989
G 0,47% 0,0330 0,5994 0,0138 0,1751
H 0,33% 0,0821 0,7550 0,0471 0,4352
I 0,66% 0,1024 0,7493 0,0657 0,4340
Belgium 3.19% 0,1899 0,9231 0,2149 1,5892
Germany (New) 5.74% 0,2559 0,8417 0,2832 1,3286
Germany (Old) 3.15% 0,2238 0,8028 0,8485 1,0735
Switzerland 5.44% 0,4117 0,9193 0,4795 1,5166
United Kingdom 7.04% 0,2413 0,5319 0,1429 0,3903
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Correlations between the five measurements & Explained Variance
RSALCoef.
Variation
Convergence
Ratio
Loimaranta
Elasticity
Discriminatio
n (d2)
RSAL 1 0,8639 0,3124 0,6092 0,6538
Coef. Variation 1 0,5357 0,7848 0,8518
Convergence Ratio 1 0,5531 0,8264
Loimaranta Elasticity 1 0,7597
Discrimination Power 1
Factor Explained Variance % Explained Variance % Accumulated Variance
1 3.7319 74.64% 74.64%
2 0,7905 15.81% 90,45%
3 0,3327 6.65% 97.10%
4 0,098 1,96% 99.06%
5 0,0469 0,94% 100,00%
5 100,00%
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Scree Diagram & Pattern of Factors
0
0,5
1
1,5
2
2,5
3
3,5
4
1 2 3 4 5
Eig
en
va
lues
Factors
Factor 1 Factor 2
RSAL 0,8031 -0,0844
Coefficient of Variation 0,9435 -0,3117
Convergence Ratio 0,7393 -0,9650
Loimaranta Elasticity 0,8641 -0,3871
Discrimination Power 0,9503 -0,7051
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Pattern of factors- ortogonal rotation applied & Representation of Factors
Factor 1 Factor 2
RSAL 0,9392 -0,4643
Coefficient of Variation 0,9677 -0,7176
Convergence Ratio 0,4496 -0,9534
Loimaranta Elasticity 0,8141 -0,7487
Discrimination (d2) 0,8031 -0,9402
-1
-0,5
0
0,5
1
-1 -0,5 0 0,5 1
Seco
nd
Fa
cto
r
First Factor
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Index of Ranking of bonus-malus systems
Bonus-malus system Index of Ranking
D 1,8952
Switzerland 1,2431
Germany (Old) 0,7905
Germany (New) 0,7291
Belgium 0,5667
U.K. 0,0254
A -0,2931
E -0,4075
I -0,4556
H -0,5178
G -0,8737
C -0,8752
B -0,8888
F -0,9381
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Conclusions
1. We have created an Index of Ranking of BMS in order to summarise their
main characteristics
2. To allow prospective policy-holders to carry out an objective evaluation
3. Difference between the systems applied in Spain (with the exception of
company D) and in other European
4. The BMS applied in other European countries and company D obtain a
positive Index of Ranking, so that we may say that they "personalize risk
better", "they have a greater differentiation of prices in function of claims
frequency" or they are "systems with less solidarity”
5. The systems used by the Spanish companies which we analysed, on the
other hand, obtain negative results for this Index, which could be explained
by saying that these systems "apply common rates", "little differentiation of
risk" or "mutualized systems or systems with more solidarity”