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Pricing motor quota share treaties
Technical publishing
Motorfahrz_Tarifierung_en_dc
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Pricing motor quota share treaties
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Contents
Foreword
4
1
Introduction
5
2
Expected loss ratio
7
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Motor third party liability (TPL)
Estimating the future claims frequency
Proportion of bodily injury claims
The average property damage claim
The average bodily injury claim
Discounting
The average premium
Fluctuation loading
Expense loading
8
8
9
10
12
14
16
18
19
4
4.1
4.2
4.3
4.4
Motor own damage (MOD)
Covered loss events
Claims frequency, average claim, discounting and average premium
Allowing for natural hazards
The loadings for fluctuation and expenses
20
20
20
21
22
5
The fixed commission
23
Appendix 1: Mathematical deductions
A
Minimum number of claims required for a frequency estimate
B
Minimum number of bodily injury claims for the estimate of the
proportion of bodily injury claims
C
Minimum number of claims for the estimate of the expected claim
D
Variance loading
E
Variance loading for natural catastrophe claims
24
24
24
24
25
25
Appendix 2: References
26
Appendix 3: Questions for cedents
A
Questionnaire for motor third party liability (TPL)
B
Questionnaire for motor own damage (MOD)
26
26
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Foreword
The quota share is an older and seemingly simpler form of reinsurance than the
excess of loss treaty. While rating of excess of loss treaties has always been a
matter for specialists, scant attention has been paid to the pricing of quota
share treaties for some time. In regulated markets with universally binding premium rates, this was not really necessary anyway, because the order of the
expected loss ratio was known, as was the level of commissions that would make it
possible to operate the reinsurance business at a profit over a sufficiently long
period. In deregulated markets, however, and in new markets where the basis for
the calculation of primary insurance premiums is still uncertain, there is no reliable
empirical foundation for pricing quota share treaties.
This publication will deal with how to determine the proper commission for
motor quota share treaties in such situations. The procedure described was developed by Hans Schmitter and Pamela Hall in the year 2000 from their analysis of
data provided by a Swiss Re cedent. This analysis identifies which of the data
available to a primary insurer are important to reinsurance pricing at all; these are
compiled in the two questionnaires in Appendix 3. The methods derived in this
publication for predicting a loss ratio or an investment income are not limited to
reinsurance only, but can also be used in primary insurance. The publication is
thus intended for reinsurers who have to price reinsurance covers in practice and
for primary insurers who require such prognoses for their planning calculations.
All calculations described in this publication and needed for pricing can be performed with the aid of the Excel file motorproppricing.xls developed by Pamela
Hall, which is programmed in Visual Basic. Copies are available free of charge from
the e-mail address [email protected]
Thomas Hiltmann
Head of Group Product Management Casualty
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1 Introduction
Motor quota share treaties normally cover motor third party liability (TPL) and
motor own damage (MOD) business. Passenger accident insurance may also be
included, but it is insignificant in terms of volume and will not be discussed in this
publication.
All numerical examples used to illustrate the pricing procedure presented here are
based on figures provided by a cedent of Swiss Re. Data from the years 1988 to
1999 were available in the year 2000 for pricing treaties covering the year 2001.
These years naturally lose their relevance in the course of time and will eventually
become out of date. The pricing procedure is, of course, independent of the time
period used, ie from 1988 to 2000, in this example. Nevertheless, to avert any
suspicion that the method itself may be obsolescent just because it is explained
on the basis of out-of-date figures, the years have been renumbered in the text:
the year with the oldest data is referred to as year 1, followed by year 2, etc. Thus,
data are available for the twelve years 1, 2, …12; the calculations are performed in
year 13 and serve as a prognosis for the next year, ie year 14.
Unless otherwise specified, all monetary units are in Swiss francs (CHF). Only in
pointed instances, as in Table 3, for example, does CHF 1 000 serve as the monetary unit.
In the chapter on motor third party liability insurance, the computations are performed in all detail to make them easy to follow. To avoid repetition, the chapter
on motor own damage (MOD) insurance does not go into such detail, except
where the calculation procedures differ significantly from those used for TPL.
In reinsurance, a distinction is sometimes made between the technical premium
and the commercial premium. The technical premium is the premium the reinsurer
needs to charge in order to carry on its business. The commercial premium is the
premium that is actually charged for a specific treaty in a given instance. Depending on market conditions, this may differ considerably from the technical premium.
The publication in hand deals only with the technical premium.
Every technical reinsurance premium is the sum of the risk premium, a fluctuation
loading and an expense loading. The risk premium is the average claims burden
expected over the treaty period. In lines of business in which there may be several
years between the time at which the premiums are collected and the time at
which the claims payments are made – as in motor TPL – a distinction is made
between the discounted and non-discounted risk premium.
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The discounted risk premium is that amount which, when invested in interestbearing assets, is on average, together with its investment yield, just enough to pay
for the claims. The fluctuation loading is that component of the premium which
the reinsurer charges for absorbing the fluctuations in the claims burden. In the
long term, it remains with the reinsurer as a profit. The expense loading, finally, is
the amount from which the expenses associated with the operation of the reinsurance business are paid, that is to say wages, office rentals, taxes, etc. The reinsurance premium for a quota share treaty is never expressed as a straight technical
premium, but always indirectly via a commission. The relationship between the
technical premium and the commission is as follows:
earned premium – technical premium
= fixed commission
earned premium
This publication will not deal with sliding-scale – as opposed to fixed – commissions.
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2 Expected loss ratio
The ratio between the claims burden and the earned premium is known as the
loss ratio. The average of all possible loss ratios of the year for which pricing is to
be performed is referred to as the expected loss ratio. This is the ratio between
the expected average claims burden – ie the risk premium – and the earned
premium.
The expected loss ratio is thus:
expected loss ratio =
risk premium
earned premium
In this equation, the risk premium can be replaced by the product of the average
number of claims and the average claim size. This gives:
expected loss ratio =
average number of claims average claim size
earned premium
If the numerator and the denominator of this fraction are divided by the same
number, eg the number of policies, its value remains unchanged. Thus, the
expected loss ratio can also be expressed as:
expected loss ratio =
average number of claims
average claim size
number of policies
earned premium
number of policies
The ratio between the average number of claims and the number of policies is
referred to as the claims frequency or frequency for short; the ratio in the denominator – the earned premium divided by the number of policies – is the average
premium per policy. Using these two terms, the expected loss ratio can be calculated as:
expected loss ratio =
claims frequency average claim size
average premium
In this form, the expected loss ratio can be calculated independent of the premium
volume and premium growth.
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3 Motor third party liability (TPL)
3.1
Estimating the future claims frequency
The future claims frequency is estimated on a statistical basis. Table 1 shows an
example compiled from observations from the six latest years available, ie the
years 7 to 12. These enable a prognosis of the year for which pricing is to be performed (the “pricing year”).
The year in which a loss occurs is called its occurrence year. This is given in the
first column in Table 1. The second column, under the heading “Annual risks”, in
principle gives the number of insured vehicles. “In principle” because both vehicles
leaving the insured portfolio and new vehicles entering it before the end of the
year will not be counted in full but only in proportion to the number of days they
were actually insured. Normally, not all claims are reported to the insurer before
the end of their occurrence year – claims that occur on 31 December, for example.
About 8% of all claims are not reported until the year following the occurrence
year, which is the first development year.
In the second development year, only a few further claims from the original occurrence year are reported. For example, the number of claims reported from occurrence year 9 increases only from 18 345 to 18 383 in the course of its second
development year. As of the fourth development year, the insurer knows practically
all the claims. The occurrence year itself is sometimes referred to as development
year zero.
Table1
Occurrence Annual
year
risks
7
8
9
10
11
12
252 266
255 894
273 711
275 728
272 217
272 668
Reported claims
Development year
0
1
17 566
19 006
17 694
19 327
16 763
18 345
17 206
18 641
16 758
18 058
17 346
Frequency
2
19 048
19 369
18 383
18 680
3
19 055
19 375
18 388
4
19 057
19 378
5
19 057
0.0753
0.0755
0.0670
0.0676
0.0663
Not every claim reported ends up being indemnified by the insurance company.
Some claims are settled without payment. These are the so-called nil cost claims
and may be the case if the policyholder was not liable at all, for example. Or if the
policyholder pays a small claim him/herself, so as not to forfeit part of the no-claim
bonus.
It would be possible to leave the nil cost claims out of compilations of the kind
shown in Table 1, but it is easier to keep them in. This leads to an elevated estimate of the claims frequency, but that does not necessarily entail an error in the
pricing of the premiums: because if the nil cost claims are also included in the calculation of the average claim size, they result in a lower average. The overstatement of the claims frequency and the understatement of the average claim size
cancel each other out.
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Practically all claims have been reported at the end of the first development year
so that the claims frequencies can be calculated with sufficient accuracy. Figure 1
shows the claims frequencies for the occurrence years 7 to 11, calculated from
Table 1 on the basis of the reported claims at the end of the first development year.
10
Figure 1
9
Claims frequency in %
8
7
6
5
4
3
2
1
0
7
Estimate used in pricing
Frequency 6.63%
8
9
10
Occurrence year
11
The changes from occurrence year to occurrence year are surprisingly small.
A slight decrease in the claims frequency in the course of time can be observed
– other markets may occasionally show an increase. If an explanation is found for
such a trend and indicates that, for example, a further decrease is to be expected,
the estimate for the future frequency will be lower than the last observed value.
Normally, however, the last known value will be the most reliable estimate for the
future. In our case, this would be 6.63%, the frequency in year 11.
Statistics of the kind shown in Table 1 will provide a reliable basis for predicting
claims frequencies only if the number of claims observed is large enough. As a
rule of thumb, 1 500 claims are sufficient for a reasonable estimate. Claims frequencies estimated on the basis of only a few hundred claims are unreliable. The mathematical reasons for this are explained in Appendix 1.
If experience from the past is used to estimate the future claims frequency, this
implicitly assumes that the risk composition of the portfolio is going to remain
essentially unchanged over time. However, if, for example, a small insurer takes on
a large fleet of rental cars from one year to the next, or if it launches a special
campaign to establish a profile as a taxi insurer, this assumption becomes questionable. The effects of such changes in the risk composition of the portfolio must
always be assessed separately.
3.2 Proportion of bodily injury claims
Just as the change in the claims frequency is only slight from occurrence year to
occurrence year, so, too, is the change in the proportion of bodily injury claims over
longer periods. A bodily injury claim is defined as any motor TPL claim in which at
least one person is killed or injured. Thus, the bodily injury claim consists not only
of the payments and reserves for fatalities or injured persons, but also the payments
and reserves for property damage due to the same accident, which the motor TPL
insurance has to indemnify.
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Table 2 shows the number of bodily injury claims for the occurrence years 7 to 11,
as reported from development year to development year. After the first development
year, the number of bodily injury claims increases just as insignificantly as the
overall number of reported claims (Table 1).
Table 2
Occurrence
year
7
8
9
10
11
Reported bodily injury claims
Development year
0
1
1 713
1 880
1 733
1 899
1 572
1 738
1 607
1 793
1 676
1 856
2
1 889
1 911
1 749
1 805
3
1 892
1 913
1 750
4
1 893
1 915
5
1 893
Figure 2 shows the proportion of the bodily injury claims relative to the total of all
reported claims from Table 1 at the end of the first development year.
12
Proportion of bodily
injury claims in %
Figure 2
10
8
6
4
2
0
7
8
9
10
Occurrence year
11
Figure 2 shows no clear trend. Nor is there any real reason why the proportion
of bodily injury claims should systematically increase or decrease from year to year.
For this reason, it is expedient to estimate the future figure on the basis of the
average of the last few years observed. The average of the years 7 to 11 is 9.8%.
Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Like the estimate of the claims frequency, the estimate of the proportion of bodily
injury claims is again unreliable, if it is based on too small a number of observations. As a rule of thumb: 1 400 bodily injury claims are sufficient for a reasonable
estimate (the mathematical derivation is given in Appendix 1).
3.3 The average property damage claim
Table 3 shows the sum of the paid and reserved property damage claims from
the occurrence years 7 to 12, the unit being CHF 1 000. The ultimate claims burden in the last column is estimated with the aid of incurred but not reported
(IBNR) techniques, in the present example by means of the chain ladder method.
Details of the most commonly used IBNR techniques are given in the references
provided in Appendix 2, (Boulter and Grubbs).
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Occurence Claims burden
year
Development year
0
1
7
49 557
44 229
8
49 315
44 211
9
43 262
41 018
10
42 964
39 955
11
41 178
39 941
12
43 143
2
42 515
40 873
39 493
39 048
3
41 154
40 370
38 283
4
41 216
39 986
5
41 344
Estimated
ultimate
claims burden
41 344
40 110
38 250
38 039
37 190
37 167
Property damage claims are settled relatively quickly. By the end of the second
development year, the difference between the known claims burdens and the
ultimate claims burdens is already down to only a few percent, and after the fourth
development year practically all the property damage claims have been paid.
Representations such as Tables 1, 2 and 3 are called run-off triangles: they show
how a quantity (eg the number of reported claims or the claims burden) originating
in a given occurrence year changes from development year to development year.
Anybody who is familiar with such triangles from pricing excess of loss treaties
normally expects the claims burden to increase from development year to development year. In Table 3, by contrast, it declines, which is nothing unusual in primary
insurance, however.
The average property damage claim is equal to the estimated ultimate claims
burden divided by the number of property damage claims. The number of property
damage claims is the difference between the total number of reported claims
(Table 1) and the number of bodily injury claims (Table 2). Since the number of
reported claims only increases slightly after the first development year, it can be
calculated with sufficient accuracy from the data of the first development year.
Thus, for occurrence year 10, for example, the number of property damage claims
obtained is 16 848 (18 641 – 1 793 = 16 848) and, accordingly, the average
property damage claim is 2 258 (38 039 000 / 16 848 = 2 258). Figure 3 shows
the average property damage claims for the occurrence years 7 to 11.
Average property
damage claim
2 500
Figure 3
2 000
1 500
1 000
500
0
7
Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
8
9
10
Occurrence year
11
The average property damage claim of the pricing year, year 14, must now be
estimated on the basis of the five latest averages observed. The average in occurrence year 7, which is the oldest and thus the least representative for the future
estimate, is somewhat higher than the others. The averages in the later occurrence years, by contrast, lie practically on a horizontal line. A statistical analysis
likewise shows no significant trend. Under these circumstances, the most recent
average (that in year 11), yields the best prognosis for the future. This figure is
2 295.
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As for the prediction of a claims frequency or of the proportion of bodily injury
claims, there is also a rule of thumb for forecasting the average property damage
claim: an average based on at least 8 000 property damage claims is reliable
enough (the mathematical reasoning is given in Appendix 1).
If possible, the average property damage claim of the individual insurer should be
used for pricing, rather than a market average. This is because the amount of the
property damage claims and thus their average depends in part on claims settlement practice and therefore differs from one insurer to the next. It must be based
on a portfolio of around 150 000 annual risks: the expected number of claims is
then 9 000 to 10 500 at a frequency of 6% to 7%. If the property damage claims
account for around 90% of all claims, 8 100 to 9 450 property damage claims
can be expected.
3.4 The average bodily injury claim
By analogy with Table 3, Table 4 shows the bodily injury claims burden from the
occurrence years 1 to 12. The monetary unit is again CHF 1 000. In this case, too,
the ultimate claims burdens in the last column are estimated with the aid of the
chain ladder method.
Table 4
Occurrence Claims burden
year
Development year
0
1
1
30 995
39 325
2
37 713
46 778
3
39 214
47 350
4
40 880
51 485
5
44 025
54 152
6
41 741
51 666
7
40 841
51 836
8
48 770
57 404
9
50 687
58 297
10
54 184
64 170
11
58 829
70 328
12
60 587
2
41 933
47 860
50 974
55 328
57 151
57 737
55 032
62 267
58 190
65 467
3
42 208
49 939
53 669
57 270
61 659
60 880
56 139
64 767
58 595
4
44 498
51 897
55 342
61 742
61 489
61 908
56 084
65 237
5
44 808
52 108
60 270
62 456
61 876
62 921
56 492
6
45 928
54 219
60 551
64 928
63 014
62 647
7
46 271
55 464
59 961
64 913
63 387
8
47 546
56 029
61 258
66 130
9
47 569
56 045
62 298
10
47 866
57 336
Estimated
ultimate
11
claims burden
47 827 47 827
57 289
63 201
67 528
65 971
65 507
60 245
70 982
65 421
75 897
86 132
90 158
A comparison with Table 3 shows two fundamental differences between property
and bodily injury claims:
bodily injury claims take longer to run off than property damage claims. A runoff triangle showing not more than five development years (such as Table 3),
would not be sufficient for predicting the average bodily injury claim. This normally requires ten or more development years;
the bodily injury claims burden normally increases from development year to
development year, whereas the property damage claims burden tends to be
overestimated in the early years (though, depending on claims reserving practice, it is also possible for the property damage claims burden to increase as
run-off progresses).
Dividing the estimated ultimate claims burden by the number of bodily injury
claims gives the average bodily injury claim. The number of bodily injury claims is
shown in Table 5. Figure 4 shows the average bodily injury claim of occurrence
years 1 to 11.
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Occurrence
year
Bodily injury claims
Development year
0
1 953
1 959
1 939
1 854
1 769
1 658
1 713
1 733
1 572
1 607
1 676
1
2
3
4
5
6
7
8
9
10
11
1
2 120
2 146
2 153
2 051
1 912
1 797
1 880
1 899
1 738
1 793
1 856
50 000
Figure 4
Average bodily injury claim
45 000
40 000
35 000
30 000
25 000
20 000
15 000
10 000
5 000
0
1
3
5
7
Occurrence year
9
11
Unlike the average property damage claim, the average bodily injury claim shows
a distinct upward trend. On the basis of Figure 4, the average bodily injury claim
of the pricing year, year 14, would be forecast at around 50 000. The trend line in
Figure 4 is calculated by simple linear regression of the average bodily injury claims
on the occurrence year, as explained in any statistics textbook. The trend value for
the pricing year, year 14, would give a good prognosis for the future, if the number
of claims on which the computation of the trend is based is large enough.
As the rule of thumb elaborated in Appendix 1 shows, the computation base
should be around 21 000 bodily injury claims per occurrence year. In our example,
with only some 2 000 observations per occurrence year, the trend value obtained
for the pricing year is not a reliable prognosis. In this case, it would be better to
calculate the average bodily injury claim per occurrence year from the data of several companies together. In Switzerland, for example, where roughly 22 000 bodily
injury claims are reported every year, the average of the entire market would have
to be used. For further calculations, the trend value of the market as a whole for
occurrence year 14 is assumed to be 50 000.
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Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
Average bodily injury claim 50 454
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The question remains as to how representative the average bodily injury claims of
years 1 to 11 are for the future. Most markets have not experienced any catastrophic
claims on the scale of the Gotthard, Mont Blanc or Tauern tunnel losses in any of
the old occurrence years. Nevertheless, such catastrophes can occur and pricing
must allow for them. If we assume, for example, that a loss event triggering claims
totalling CHF 100 million occurs once every ten years in Switzerland, the estimate of
the average bodily injury claim increases from 50 000 to 50 454 as the following
calculation shows:
50 000 22 000 10 + 100 000 000
= 50 454
22 000 10 + 1
This figure is used as the basis for the following calculations.
3.5 Discounting
To calculate the average motor TPL claim from the average property damage claim
and the average bodily injury claim, the two types of claim have to be weighted
and added. In line with Section 3.2, the weights in our example are 90.2% for the
property damage claims and 9.8% for the bodily injury claims. The estimate of the
average claim of the pricing year is thus:
0.902 2 295 + 0.098 50 454 = 7 015
However, the amount of 7 015 per claim need not be available in the pricing year;
it is sufficient to have a smaller amount which later, in the course of run off, is
incremented by the return on its investment to attain 7 015 at the time of payment. This smaller amount is called the discounted average claim. In the following,
we use the example of the average bodily injury claim to show how the discounted value is calculated.
Table 6 shows the paid bodily injury claims burdens for occurrence years 1 to 12.
As in Table 4, the unit is CHF 1 000.
Table 6
Occurrence Paid bodily injury claims
year
Development year
0
1
1
5 839
12 289
2
6 721
15 461
3
7 067
15 449
4
7 673
17 099
5
7 006
15 019
6
7 002
16 253
7
7 135
14 837
8
6 985
15 076
9
6 625
14 370
10
6 635
15 242
11
7 506
15 673
12
7 421
2
16 343
20 071
20 300
22 673
20 674
21 886
19 176
20 734
18 812
20 263
3
19 625
24 408
23 864
27 484
25 019
26 197
23 712
24 855
22 504
14
4
22 616
28 027
27 674
31 377
29 424
30 425
27 571
29 371
5
24 891
31 321
30 677
35 654
33 857
35 691
31 858
6
27 482
34 920
37 419
38 565
37 984
40 063
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7
30 138
38 515
41 497
42 784
42 950
8
33 775
41 202
44 058
45 861
9
34 902
43 373
47 227
10
36 986
45 781
11
38 457
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Table 7
Occurrence
year
1
2
3
4
5
6
7
8
9
10
11
12
Average:
Part of paid claims in ultimate claims burden
Development year
0
1
2
3
12.2%
25.7%
34.2%
41.0%
11.7%
27.0%
35.0%
42.6%
11.2%
24.4%
32.1%
37.8%
11.4%
25.3%
33.6%
40.7%
10.6%
22.8%
31.3%
37.9%
10.7%
24.8%
33.4%
40.0%
11.8%
24.6%
31.8%
39.4%
9.8%
21.2%
29.2%
35.0%
10.1%
22.0%
28.8%
34.4%
8.7%
20.1%
26.7%
8.7%
18.2%
8.2%
10.4%
23.3%
31.6%
38.8%
4
47.3%
48.9%
43.8%
46.5%
44.6%
46.4%
45.8%
41.4%
5
52.0%
54.7%
48.5%
52.8%
51.3%
54.5%
52.9%
6
57.5%
61.0%
59.2%
57.1%
57.6%
61.2%
7
63.0%
67.2%
65.7%
63.4%
65.1%
8
70.6%
71.9%
69.7%
67.9%
9
73.0%
75.7%
74.7%
10
77.3%
79.9%
11
80.4%
45.6%
52.4%
58.9%
64.9%
70.0%
74.5%
78.6%
80.4%
Table 7 is derived by dividing the paid claims by the associated estimated ultimate
claims burdens (in the last column of Table 4).
On the assumption that all payments are made mid-year and the reserve of the 11th
development year is paid in the middle of the 12th development year, the average
run-off period in years prior to payment is obtained as follows:
10.4% 0.5 + (23.3% – 10.4%) 1.5 + … + (100% – 80.4%) 12.5 = 6.2
The run-off period thus calculated could be used to discount the average bodily
injury claim, if the reinsurer received the premiums on 1 January of the pricing
year, which, of course, is never the case. For this reason, the average investment
period is as follows:
average investment period = average run-off period – average delay in receipt of
premiums.
If premiums are received with one year’s delay, the average investment period
in our example is 5.2 years. A 4% interest rate, for example, yields a discounting
factor of:
1 / 1.045.2 = 0.816 oder 81.6%
It is advisable to discount with the same interest rate as is used in pricing excess
of loss treaties, with the five-year government bond rate, for example. The discounting factors for the average property damage claim are calculated in exactly
the same way. However, the average run-off period of the property damage claims
is much shorter than that of the bodily injury claims. As a result, the average
investment period will also be shorter, and the discounting factor higher. The following calculations are based on the assumption that the average run-off period
for the property damage claims is 0.9 years. In this case, the investment period is
0.9 – 1 = –0.1 years. This means that the reinsurer is advancing money to the
primary insurer for the duration of 0.1 years to settle the property damage claims.
The discounting factor is then 1.040.1 or 100.4%.
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Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
Average bodily injury claim 50 454
Average discounted claim 6 113
10:18 Uhr
Seite 16
Applying the two discounting factors identified above, the average discounted
motor TPL claim is obtained as follows:
0.902 2 295 1.004 + 0.098 50 454 0.816 = 6 113
Only the average discounted claim is of importance to pricing; however, the nondiscounted average (7 015 in our example), is also of interest for planning purposes.
3.6 The average premium
The denominator in the equation for the expected loss ratio in Section 2, the
average premium, must also be predicted for the pricing year, as it changes from
year to year for three reasons:
1 the composition of the portfolio changes, and since not all risks pay the same
tariff premium, the average tariff premium also changes;
2 increases and reductions in the tariff affect the average premium;
3 every risk changes its bonus/malus (no-claim bonus) class every year and thus
pays a different bonus/malus level, that is to say a different percentage of the
tariff premium or 100% premium every year. The only exceptions are those
risks in the category with the lowest premium rate that have experienced
another claim-free year and those in the category with the highest premium rate
that have caused yet another accident. As a result, the average bonus/malus
level of the overall insured portfolio changes from occurrence year to occurrence
year: all markets that operate bonus/malus systems exhibit a gradual decline in
the average bonus/malus level, as more and more risks move towards the lower
levels.
Table 8 shows the premium development in years 1 to 12. The average premium
is the ratio between the earned premiums and the number of annual risks. If the
average premium is then divided by the average bonus/malus level, the average
100% premium is obtained.
Figure 5 shows the development of the average bonus/malus level. Up to year 8,
the year-on-year decline is in the order of half a percent. The blip in year 8 can be
traced to the liberalisation of the market: in the data on which this publication is
based, year 8 represents the calendar year 1995, the last year prior to liberalisation
in Switzerland. To attract the vast numbers of accident-free drivers, new premium
discounts were introduced, throwing the system slightly off balance. Since then,
the average bonus/malus level has been dropping more rapidly, which makes it
necessary – sooner or later – to revise the tariffs upward.
Table 8
Occurrence
year
Annual
risks
Earned
premium
Average
premium
1
2
3
4
5
6
7
8
9
10
11
12
225 551
233 099
240 598
246 004
247 890
248 637
252 266
255 894
273 711
275 728
272 217
272 668
102 852 692
118 085 646
129 129 440
132 631 564
140 247 736
142 626 124
144 283 632
146 001 541
144 213 834
138 609 053
132 005 901
129 622 873
456
507
537
539
566
574
572
571
527
503
485
475
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Average
bonus/malus
level in %
60.7
60.2
60.1
59.6
59.0
58.3
57.7
57.3
52.7
50.6
49.2
47.7
Average
100%
premium
751
842
893
905
959
984
991
996
1 000
993
986
997
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Figure 5
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Average bonus/malus level in %
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Seite 17
70
60
50
40
30
20
10
0
2
4
6
Year
8
10
12
Figure 6 shows the average premiums and the average 100% premiums for
years 1 to 12.
1 000
Figure 6
Average 100%-premium
Premium
800
600
Average premium
400
200
0
2
4
6
Year
8
10
12
To calculate the premium estimate for the pricing year, year 14, we need:
the last known value,
an estimation of the effects of changes in the tariff (if planned) and
an estimation of the average bonus/malus level in the pricing year.
The last known value is 475, the average premium for year 12. We assume that
changes in the tariff will lead to an average increase of 8% in premiums between
the years 12 and 14.
If the average bonus/malus level changes on the same scale as from year 11 to
year 12, it will decline by 1.5% per year, reaching 44.7% in the year 14.
Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
Average bodily injury claim 50 454
Average discounted claim 6 113
Average premium 481
Discounted loss ratio 84.26%
Hence, we obtain:
475 1.08 44.7
= 481
47.7
The expected discounted loss ratio is then:
6.63% 6 113
= 84.26%
481
The expected non-discounted loss ratio would be:
6.63% 7 015
= 96.69%
481
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3.7 Fluctuation loading
The various approaches commonly used for calculating the fluctuation loading in
pricing excess of loss treaties can also be used in proportional pricing. The loading
which Swiss Re has been successfully applying in non-proportional motor business
for many years is a variance loading. For further information on the practical application of the variance loading, in particular on how to determine the loading factor,
refer to Appendix 2 (Schmitter and Bütikofer), in which the product of the loading
factor and 1 000 000 is defined as the fluctuation factor. For proportional business, the loading takes the following form:
Fluctuation loading in % of premium = loading factor 1 000 000
discounted expected loss ratio
average discounted claim in CHF million
quota share in %
65
Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
Average bodily injury claim 50 454
Average discounted claim 6 113
Average premium 481
Discounted loss ratio 84.26%
Fluctuation loading 1.26%
For a loading factor of, for example, 0.15 10–6 and a quota share of 25%, the
fluctuation loading will be 1.26% in our case:
0.15 0.8426 0.006113 25% 65 = 1.26%
The derivation is explained in Appendix 1. It is important to use the same loading
factor for both excess of loss and quota share treaties. This makes the two types of
treaty directly comparable: an excess of loss with a fluctuation loading of likewise
1.26% raises the reinsurer’s fluctuation by the same margin as the quota share in
the present numerical example. The reinsurer then has no reason to give one of the
treaties preference over the other. However, the discounted risk premium of the XL
treaty will be about ten times lower than the 84.26% of the quota share treaty,
since excess of loss treaties typically absorb high fluctuations even at low risk premiums – in return for high fluctuation loadings charged.
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3.8 Expense loading
Expense loading represents the expenditure (wages, office rentals, taxes, etc)
incurred by the reinsurer in connection with the individual treaty. Its amount is
independent of the size of the treaty, even if in practice it is often (rather simplistically) expressed as a percentage of the risk premium. It is calculated as follows:
expense loading =
amount of expenses
expected premiums quote share
Assuming that the expenses amount to CHF 100 000, the quota share is 25% and
the cedent’s expected premium income for the pricing year is 125 000 000, the
expense loading is obtained as follows:
100 000
= 0.32%
125 000 000 0.25
Estimates used in pricing
Frequency 6.63%
Proportion of bodily injury claims 9.8%
Average property damage claim 2 295
Average bodily injury claim 50 454
Average discounted claim 6 113
Average premium 481
Discounted loss ratio 84.26%
Fluctuation loading 1.26%
Cedent’s earned premiums CHF 125 million
Expense loading 0.32%
The maximal possible commission is the difference between 100% and the technical premium. The technical premium as the sum of the discounted risk premium,
fluctuation and expense loadings in our example is:
84.26% + 1.26% + 0.32% = 85.84%
Thus, the maximal possible fixed commission on motor TPL alone is:
100% – 85.84% = 14.16%
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4 Motor own damage (MOD)
Pricing for a motor own damage cover is essentially the same as pricing for the
liability cover: The expected loss ratio is calculated from the claims frequency,
average claim and average premium; then fluctuation loading and expense loading
are added. In the following, we highlight only the major differences from liability
pricing, without going into all the details of the calculations and estimations.
4.1
Covered loss events
Unlike compulsory third party liability insurance, to which statutory regulations
apply, motor own damage insurance is less uniform. It covers damage to the policyholder’s vehicle for which no one else is liable: losses and damage due to collision,
theft, hail, storms, fire, etc. In Switzerland, Germany and Austria, there are two
standard products which will be called Inclusive of collision (IC) and Exclusive of
collision (EC) for the purposes of this publication. The IC product covers the types
of damage mentioned inclusive of collision, whereas the EC product covers the
same damages exclusive of collision. In other markets, other combinations are
common, collision, glass and theft, for example. Before pricing these products, it is
important to check back with the primary insurer about what the various products
actually cover. The claims frequency, average claim and average premium must
then be determined separately for each product.
4.2 Claims frequency, average claim, discounting and average premium
Apart from natural hazards such as hail, storm or flooding, which must be treated
separately, the claims frequencies for the various products are determined in the
same way as for liability: at the end of the first development year, the number of
claims is known and is divided by the number of annual risks. As in TPL, the last
known value is normally the most reliable estimate for the future. As a rule of
thumb, and as in TPL, 1 500 claims are sufficient for a reliable prognosis.
The average claims of the various products depend on the covered perils, the
insured vehicles and the amount of any policyholder deductibles. Run-off of claims
is similar to the run-off of property damage claims in TPL: by the end of the first
development year, practically all claims have been paid. Again, trend computations
are recommended for predicting the average claim size. The minimum number
of claims per occurrence year needed for a reliable prognosis depends on the
product. In EC in Switzerland, the reliability threshold is taken to be roughly 6 000,
in IC 9 500 (the reasoning is given in Appendix 1).
Because of the rapid claims settlement, as in the case of property damage claims
under TPL, discounting is of no practical importance.
The average premium of the pricing year is calculated in exactly the same way as
for liability. In particular, in the case of products with a bonus/malus system, the
development of the average bonus/malus level must also be taken into account.
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4.3 Allowing for natural hazards
Natural hazards such as hail, storm or flooding are responsible for the greatest
fluctuations in the motor own damage claims burden. The assumptions compiled
in Table 9 concerning the MOD portfolio illustrate the further computations with
concrete numerical examples.
Table 9
Estimate for the pricing year
IC
EC
Frequency without natural hazards
Average discounted claim
Average premium
Earned premiums
Number of annual risks
27.4%
2 780
1 015
66 000 000
65 025
12.5%
1 645
316
35 000 000
110 759
The average annual claims burden due to natural hazards depends on the return
period of the covered natural events, the average number of risks affected and the
average claim per affected risk. The primary insurer’s risk portfolio influences each
of these three variables:
an insurer operating only locally will be less frequently affected by a hailstorm
than one whose portfolio is spread over a whole country;
a natural event, a hailstorm, for example, affects a larger proportion of a small
local insurer’s risks than that of a large insurer whose risks are spread over the
whole country;
the average claim per risk depends on any policyholder deductibles that may be
applicable and on the individual insurer’s claims settlement practice.
The following assumptions apply to the further calculations:
the return period of natural events is 4 years;
the proportion of risks affected per natural event is 5%;
the average claim per risk is 1 645 (the average EC or similar claim provides a
reasonable estimate, if more precise data are not available).
This yields the average annual claims burden due to natural hazards:
proportion of risks affected number of annual risks average claim size
return period
or per risk:
proportion of risks affected average claim size
return period
In our numerical example we obtain:
0.05 1 645
= 21
4
Thus, all the variables required for predicting the expected discounted loss ratio are
known, and insertion of the values from Table 9 yields:
27.4% 2 780 + 21
= 77.12% for IC
1 015
12.5% 1 645 + 21
= 71.72% for EC
316
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4.4 The loadings for fluctuation and expenses
From the various possibilities available for calculating fluctuation loading, we
again choose the variance loading, as in the case of TPL. No loading is necessary
for the fluctuations in the claims burden without natural events, because these
fluctuations are negligible. However, the fluctuations as a result of natural events
are significant. The loading in % of the premiums is calculated as follows (the derivation is given in Appendix 1):
Fluctuation loading in % of premium = loading factor 1 000 000
average catastrophe loss in CHF million
maximum catastrophe loss in CHF million
/ return period
/ earned premiums in CHF million
quota share in %.
On the basis of the above assumptions, the average catastrophe loss is:
0.05 (65 025 + 110 759) 1 645 = 14 458 234
or, in CHF million, 14. 458 234
Estimates used in pricing
Natural hazards:
Return period 4 years
Proportion of risks affected per
natural event 5%
Average claim per risk 1 645
IC:
Frequency without natural hazards 27.4%
Average discounted claim 2 780
Average premium 1 015
Number of annual risks 65 025
Discounted loss ratio 77.12%
EC:
Frequency without natural hazards 12.5%
Average discounted claim 1 645
Average premium 316
Number of annual risks 110 759
Discounted loss ratio 71.72%
Fluctuation loading 3.36%
Expense loading 0.24%
It should be noted that the loadings must be calculated collectively for all products affected by the same natural catastrophe, and not for each product separately. In our example, IC and EC are calculated together. The maximum catastrophe
loss is likewise the overall loss of all products. If the maximum loss is taken as
25 000 000, our numerical example yields the following result:
Fluctuation loading in % of premium = 0.15
14. 458 234
25
/ 4
/ (66 + 35)
25%
= 3.36%
The expense loading is calculated in exactly the same way as for TPL. Assuming that
the expenses amount to CHF 60 000, the loading for IC and EC together is 0.24%:
60 000
= 0.24%
(66 000 000 + 35 000 000) 0.25
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5 The fixed commission
Table 10 compiles all the data necessary for calculating the maximal possible
fixed commission for a motor quota share treaty covering liability, IC and EC. The
discounted expected loss ratio, the fluctuation loading and the expense loading in
the bottom line are weighted averages from the corresponding figures for liability,
IC and EC insurance, the estimated premiums serving as the weights.
Table 10
Class
Estimated
premiums in
millions
Discounted
expected loss
ratio in %
Fluctuation
loading in %
Expense loading
in %
TPL
IC
EC
Total
125
66
35
226
84.26
77.12
71.72
80.23
1.26
3.36
3.36
2.20
0.32
0.24
0.24
0.28
The maximal possible commission which the reinsurer can pay on this motor
treaty is:
100% – 80.23% – 2.20% – 0.28% = 17.29%
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Appendix
Appendix 1:
Mathematical deductions
A
Minimum number of claims required for a frequency estimate
Let the number N of claims from a portfolio of s risks follow a Poisson distribution.
Let the frequency f = E[N] / s be estimated by N / s. N has to be sufficiently high so
that N / s will not deviate more than c = 5% from f with a probability of 95%. For
the Poisson distribution, Var[N] = E[N], and the distribution of (N – E[N]) / √E[N]
is asymptotically normal with mean 0 and standard deviation 1.
From | N / s – f | ≤ c · f
follows | N – f · s | / √(f · s) ≤ c · √(f · s).
For c = 0.05 we obtain from the normal distribution 0.05 · √(f · s) = 1.96,
or, resolved for f · s, f · s = 1 537.
B
Minimum number of bodily injury claims for the estimate of the
proportion of bodily injury claims
Let the probability of a claim being a bodily injury claim be r. Let the number K
of bodily injury claims out of n claims follow a binomial distribution with mean n · r
and variance n · r · (1 – r). K has to be sufficiently high so that K / n will not deviate
more than c = 5% from r with a probability of 95%. (K – n · r) / √[n · r · (1 – r)] is
asymptotically normal with mean 0 and standard deviation 1.
From | K / n – r | ≤ c · r
follows | K – n · r | /√[n · r · (1 – r)] ≤ c · n · r/√[n · r · (1 – r)].
For c = 0.05 we obtain from the normal distribution 0.05 · √(n · r) = 1.96 · √(1 – r),
or, resolved for n · r, n · r = 1 537 · (1 – r). For r ≈ 0.1, n · r ≈ 1 383.
C
Minimum number of claims for the estimate of the expected
total claim
Let the n claims X1, ..., Xn be independent and identically distributed with expected
value E and coefficient of variation v. n has to be sufficiently high so that ∑ Xi / n
will not deviate more than c = 5% from E with a probability of 95%. According to
the central limit theorem, ∑ Xi / n follows an asymptotic normal distribution with
mean E and variance E2 · v2 / n.
From | ∑ Xi / n – E| ≤ c · E
follows | ∑ Xi / n – E| · √n / (E · v) ≤ c · √n / v.
For c = 0.05 we obtain from the normal distribution 0.05 · √n / v = 1.96,
or, resolved for n, n = (1.96 · v / 0.05)2.
In Western Europe, the coefficients of variation in motor TPL are about 2.3 for
property damage claims and 3.7 for bodily injury claims. This yields n = 8 129 for
property damage claims and n = 21 037 for bodily injury claims. In Switzerland,
the coefficient of variation for EC claims is about 2, for IC claims, about 2.5. This
yields n = 6 147 and n = 9 604, respectively.
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Seite 25
D
Variance loading
The loading recommended in this publication is proportional to the variance
assumed by the reinsurer. Define as:
f the loading factor, if the Swiss franc is used as the monetary unit
E the discounted expected individual claim
V the variance of the discounted individual claim
v the coefficient of variation
the expected number of Poisson-distributed claims
r the expected discounted loss ratio
P the earned premium
q the quota share in %.
Then the fluctuation loading in % of the premiums is
loading =
f · · (E2 + V) · q2
P·q
= f · (1 + v2) · r · E · q
In Western Europe, the coefficient of variation in motor TPL is around v = 8. Thus,
1 + v2 = 65. The loading factor f depends on the reinsurer’s capital cost and profit
target and can change from year to year. It lies in the order of 10–7 CHF–1. In
Appendix 2 (Schmitter and Bütikofer), the product f · 106 is referred to as the fluctuation factor.
1.5
Variance loading for natural catastrophe claims
The fluctuation loading for natural catastrophe claims is based on the following
simple estimate of E2 + V: let the catastrophe loss X be limited by the maximum
claim size M, X ≤ M. Then, E2 + V = E[X2] ≤ E[X · M] = E · M. If this upper bound
is used as the estimate for E2 + V, the fluctuation loading in percent of the premiums is obtained as follows:
loading =
25
f··E·M·q
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Appendix 2:
References
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Seite 26
Boulter, Anthony and Grubbs, Dawson (2000): Late claims reserves in
reinsurance. Swiss Re.
Schmitter, Hans and Bütikofer, Peter (1997): Estimating property excess
of loss risk premiums by means of the Pareto model. Swiss Re.
Appendix 3:
Questions for cedents
A
Questionnaire for motor third party liability (TPL)
Claims
number of annual risks for the last 3 years
number of claims (including nil cost claims) for the last 3 years
prediction of claims frequency. If this differs from the most recent observed
frequency, please state reasons.
number of bodily injury claims for the last 3 years
run-off triangles for the property damage claims burden over the last 7 years,
separately for paid and reserved amounts
run-off triangles for the bodily injury claims burden over the last 12 years,
separately for paid and reserved amounts
Premiums
last year’s earned premiums
are any changes to the tariff planned? How many policies will be affected
by these changes? How will they affect the premium level?
spread of risks over the bonus/malus classes
average bonus/malus level
prediction of the average bonus/malus level for the pricing year
B
Questionnaire for motor own damage (MOD)
Claims
Perils covered: for each combination of covered perils:
number of annual risks for the last 3 years
number of claims for the last 3 years
prediction of claims frequency. If this differs from the most recent observed
frequency, please state reasons.
claims burdens (splitting into paid and reserved amounts not necessary)
for the last 3 years
has there been a natural catastrophe in the last 3 years? Return period
of natural catastrophes? Percentage of policies affected?
Premiums
Same questions as for motor TPL
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Seite 27
Other publications in the “Technical publishing”
category include:
Late claims reserves in reinsurance
“Late claims reserves in reinsurance” is a
second and thoroughly revised edition of the
popular 1989 Swiss Re publication. Targeted
at general management, claims managers,
underwriters and accountants, the publication
gives insight into why reinsurers need to hold
additional claims reserves over and above the
claims estimates that the ceding insurer supplies.
Order no.: 207_8955_en/de
Estimating property excess of loss risk
premiums by means of the Pareto model
This is the revised edition of the highly popular
title “Estimating property excess of loss premiums by means of the Pareto model” (1978).
The Pareto model is often used to estimate risk
premiums in property excess of loss with great
loss priority, as in such treaties, loss experience
is often inconsistant and can be confusing.
Order no.: 207_9798_en/de
Setting optimal reinsurance retentions
The primary goal of reinsurance is to maintain,
at an acceptable level, the random fluctuations in the results of primary insurers. The
Swiss Re publication “Setting optimal reinsurance retentions” reveals how this can be effectively achieved. The work is aimed equally at
primary insurers, reinsurers and reinsurance
brokers who advise their clients on arranging
reinsurance programmes. The concepts
described are applicable to every class of insurance and are not limited to the casualty or
property damage classes.
Order no.: 207_01288_en/de
The economics of insurance
How insurers create value for shareholders
The greatest shareholder value is generated
by those insurers who identify and capitalise
on the best business opportunities and have
optimum operating efficiency. Although this
publication is founded on a detailed economic
study, it caters specifically to practitioners.
Examples that clearly demonstrate how to
apply the structure in practice are provided,
making it invaluable for insurance management
seeking to improve long-term profitability.
Order no.: 207_01310_en/de
An introduction to reinsurance
This teaching aid explains the reinsurance system to prespective underwriters and introduces
working methods involved using examples and
graphics. The publication does not pretend to
be self-explanatory but is intended as a support
for both trainer and trainees.
Order no.: 207_9682_en/de/es/pt
The insurance cycle as an
entreprenuerial challenge
The fact that insurance buyers, cedents, reinsurers and industrial lines insurers have to live
with fluctuating market prices does not mean
that the individual company has to sit out the
cycle compliantly. There are supply-side strategies which ensure better results over the cycle
than by passively standing by and allowing the
cycle to take its course.
Order no.: 1492633_02_en/de
The brochure entitled Swiss Re Publications
includes a complete overview of all available
Swiss Re publications.
Order no.: 1492220_02_en
Non-proportional reinsurance accounting
This publication offers a general and easy-tounderstand overview of the essentials and
processes of non-proportional treaty business.
The first part of the two-part publication presents the business process using a fictitious
portfolio and the reinsurance programme developed for it. The teaching material is supplemented by a workbook containing case studies which can also be used for reviewing and
consolidating the information.
Order no.: 207_00218_en/de
Publications in the “Technical
publishing/Casualty” series:
Introduction to rating casualty business
Excess-of-loss rating in longtail branches such
as motor or general liability is no easy matter
since estimates must be made of claims
incurred which will not in fact be determined
by court ruling until 10 or 20 years later. Every
round of renewals presents the new challenge
of calculating estimates of future claims burdens based on previous data or current risk
profiles. This publication shows the considerations on which this procedure is based and
serves as an introduction to the essential steps
involved in this demanding field of work.
Order no.: 207_99211_en/fr/de
Liability and liability insurance:
Yesterday – today – tomorrow
Increasingly, the momentous changes brought
about by technological progress are contributing to a marked rise in compensation claims,
notably when technological progress is seen
to have failed. To illustrate the changes that
have taken place in liability insurance, this publication covers the net of liable parties and the
types of liability loss for which claims are made.
The judicial interpretation of policy wordings,
the types of triggers for coverage and the definition of insured liability loss are each examined.
Order no.: 205_01292_en/de
How to order
To order a copy of a publication, please send
an email to [email protected].
You can also place your order via our portal at
www.swissre.com. Please include the title of
the publication, the order number and the language abbreviation. Language versions are
indicated at the end of the order number as
follows:
English _en
German _de
French _fr
Italian _it
Portuguese _pt
Spanish _es
Swiss Re’s P&C publications are edited and
produced by Technical Communications,
Chief Underwriting Office.
Motorfahrz_Tarifierung_en_dc
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Seite 28
Hans Schmitter
Chief Underwriting Office
Dr Hans Schmitter is a specialist in casualty business
at Swiss Re’s Chief Underwriting Office. He studied
mathematics and lingusitics at the University of Bern.
Following two years as a Fortran programmer at the
Swiss Federal Office of Statistics, Hans Schmitter
joined Swiss Re in 1973 to work in the area of pricing
concepts and tool development. In 1991, he moved
to the Europe Department of Winterthur insurance
company, where he was active in primary insurance
pricing. Returning to Swiss Re, he has worked since
1999 in the Product Management Casualty unit. In
addition to the publication in hand, Hans Schmitter
has also authored the following: “Estimating property
excess of loss risk premiums by means of the Pareto
model”, and “Setting optimal reinsurance retentions”.
© 2003
Swiss Reinsurance Company
Title:
Pricing motor quota share treaties
Author:
Hans Schmitter
Chief Underwriting Office
Translation by:
Swiss Re Group Language Services
Editing and realisation:
Technical Communications
Chief Underwriting Office
Graphic design:
Galizinski Gestaltung, Zurich
Photograph:
Alan Schein, Blue Planet/The Stock Market
The material and conclusions contained in this publication are for information purposes only, and the
author(s) offers no guarantee for the accuracy and
completeness of its contents. All liability for the
integrity, confidentiality or timeliness of this publication
or for any damages resulting from the use of information herein is expressly excluded. Under no circumstances shall Swiss Re Group or its entities be liable
for any financial or consequential loss relating to this
product.
Order number: 1493126_02_en
Property & Casualty, 03/03, 3000 en
Motorfahrz_Tarifierung_en_dc
Swiss Reinsurance Company
Mythenquai 50/60
P.O. Box
CH-8022 Zurich
Switzerland
Telephone +41 43 285 2121
Fax +41 43 285 5493
[email protected]
Swiss Re publications can
also be downloaded from
www.swissre.com
25.02.2003
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Seite 30