Pricing Large Insureds
Christopher Claus
American Re-Insurance
2001 Seminar on Ratemaking
Reasons Data for Large Insureds is
Not Fully Credible (not an exhaustive list)
 Lack of Volume/Size of Insured
 Changes in Operations/Mergers &
Acquis.
 Significant Growth/Discontinued
Operations
 Changes in Limits Profile over Time for
Group of Insureds
 Long Tailed Lines Lacking Sufficient
Historical Experience
Reasons Data for Large Insureds is
Not Fully Credible (Continued)
 Changes in Technology/Exposures
 Lack of Complete Data
 Changes in Geographic Spread
 Expansion into New Markets/Products
 Changes in Claims Handling
Areas Where Data
May Be Blended
 LDF’s
•
•
•
•
•
•
•
•
Unlimited vs. Limited, Excess vs Ground-up
Changes in Reserving and Payment Patterns
Changes in Underlying Exposures
Selecting Tail Factors
Adjustments Using Claim Count Information
Report Year vs. Accident Year Triangles
Review ALAE Separately
Closed-to-Reported Ratios
Adjust Industry LDFs based on
Trended Closed to Reported Ratios
Reported Loss+ALAE
Accident
Year
Loss+ALAE Ratios
Reported Claim Count
Claim Count Ratios
150,000
250,000
150,000
250,000
150,000
250,000
150,000
250,000
xs
xs
xs
xs
xs
xs
xs
xs
100,000
250,000
100,000
250,000
100,000
250,000
100,000
250,000
96%
98%
98%
87%
83%
68%
100%
79%
41%
100%
0%
88%
100%
100%
100%
100%
81%
100%
100%
100%
15%
0%
0%
90%
94%
95%
97%
81%
86%
78%
100%
77%
64%
100%
0%
87%
100%
100%
100%
100%
71%
100%
100%
100%
33%
0%
0%
90%
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
2,595,582 1,196,660
Total
12,566,049 4,843,488
1,812,281
839,886
1,777,596
506,768
1,671,172
611,854
1,916,049
990,352
634,921
88,711
452,105
41,192
1,010,119
219,295
685,872
348,771
10,352
0
0
0
33
11
22
5
29
5
32
4
35
7
18
1
7
1
13
2
11
3
1
0
0
0
201
39
Areas Where Data
May Be Blended
 Trend
• ISO Sources
• WC - Medical Cost Inflation, Wage Inflation,
Benefit Levels
• Trend Based on Client Data
• Annual Trend Rate vs Actual Indication
• Remember to Review Frequency Trend
Areas where Data
May Be Blended Data
 Severity Distributions
• ISO Data
• Fit to Client Data
• Blending Client Fit with Industry
Example of Blending
Severity Distributions

Situation: Pricing High Excess Layer
• Can Price Lower Layers
• But Need Severity Distribution to Price High Layers

Problem: How to Adjust Severity Distribution
to Reflect Clients Data
One Method of Blending
Severity Distributions
(1) Parameters fitted to individual client data
Pareto Parameters
Alpha
1.599
Lambda
306,605
(2) Alternative:
Selecting factors based on industry & LEV given a tail parameter (alpha)
(1)
Low -End
Limit
[T]
(A)
(B)
1,000,000
2,000,000
(2)
High-End
Limit
(L)
2,000,000
3,000,000
Pareto
Parameters
Alpha
Lambda
(3)
Company
Empirical
E[X|X>T,L]
(4)
HK
Pareto
Fitted
E[X|X>T,L]*
1,700,000
2,800,000
1,700,000
2,800,000
(A)
Selected
Modified
Industry
1.500
(B)
Selected
Modified
Industry
1.500
682,273
926,650
(5)
Abs [(4)-(5)]
Absolute
Difference
0
0
Graph of Empirical vs
Possible Pareto Distributions
PROBABILITY OF LOSS GIVEN GREATER
THAN $500,000
COMPARISON OF CLIENT EMPIRICAL TO FITTED PARETO DIST
18.0%
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.53
0.71
0.96
1.30
1.75
2.36
3.19
4.30
5.81
7.84
10.59
14.29
SIZE OF LOSS (IN $MM)
Pareto Fit to Client Individual Claims
Actual
Pareto Based on 1MM xs 1MM LEV Matching
Pareto Based on 1MM xs 2MM LEV Matching
Comparison of
Pareto Relativities
Pareto Parameters
(1)
Alpha
Lambda
Limit
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
1.599
306,605
(A)
1.500
682,273
(B)
1.500
926,650
LEV Based on Ground-up Claims
296,983
495,547
568,007
358,921
676,344
810,459
388,550
777,179
952,989
406,562
843,664
1,049,537
418,909
891,714
1,120,477
428,014
928,527
1,175,437
Implied XS Severities Based on Ground-up Claims
1MM
61,937
180,797
242,452
2MM
29,629
100,835
142,530
3MM
18,012
66,485
96,548
4MM
12,347
48,050
70,940
5MM
9,105
36,813
54,960
1MM xs
1MM xs
1MM xs
1MM xs
1MM xs
1MM
1MM
1MM
1MM
1MM
xs
xs
xs
xs
xs
Implied Excess Relativities
1MM
1.000
1.000
2MM
0.478
0.558
3MM
0.291
0.368
4MM
0.199
0.266
5MM
0.147
0.204
1.000
0.588
0.398
0.293
0.227
Comparison of
Excess Losses
Based On
Client Fit
(000)
LAYER
1,000 XS 1,000
1,000 XS 2,000
1,000 XS 3,000
1,000 XS 4,000
1,000 XS 5,000
BASE
1,000 XS 1,000
2,800,000
1,339,425
814,260
558,188
411,587
BASE
1,000 XS 2,000
1,250,000
760,983
520,397
384,414
SELECTED
EXPECTED
LOSSES
2,800,000
1,295,000
788,000
539,000
398,000
5,000 XS 1,000
Based On
1MM xs 1MM
LEV Matching
1,000 XS 1,000
1,000 XS 2,000
1,000 XS 3,000
1,000 XS 4,000
1,000 XS 5,000
5,820,000
2,800,000
1,561,634
1,029,656
744,153
570,116
1,250,000
824,182
595,653
456,346
5,000 XS 1,000
Weight
2,800,000
1,406,000
927,000
670,000
513,000
6,316,000
50%
50%
= Base layer from which higher layers were priced.
A Report Year Approach
 Report Year Approach to Price
Accident Year Coverage
Company A - Accident Year
Reported Loss+ALAE Development
Accident
Year
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
12:24
2.112
1.971
2.559
2.291
1.994
2.663
2.677
3.842
3.474
2.988
24:36
1.546
1.501
1.814
1.969
1.447
1.784
1.882
1.901
2.050
36:48
1.467
1.474
1.898
1.603
1.577
1.808
1.570
1.648
48:60
1.179
1.576
1.343
1.453
1.832
1.285
1.211
60:72
1.302
0.932
1.096
1.064
1.174
1.125
72:84
1.261
1.060
1.076
1.247
1.212
84:96
1.034
1.194
1.060
1.011
All Yr Avg
Avg Ex H & L
Last 2 Years
Last 3 Years
Last 5 Years
2 Yr Wghtd
5 Yr Wghtd
Weighted
2.657
2.595
3.231
3.435
3.129
3.183
3.120
2.781
1.766
1.771
1.975
1.944
1.813
1.973
1.865
1.826
1.631
1.613
1.609
1.675
1.641
1.613
1.642
1.636
1.411
1.374
1.248
1.443
1.425
1.243
1.367
1.362
1.115
1.115
1.150
1.121
1.078
1.146
1.097
1.121
1.171
1.178
1.230
1.178
1.171
1.228
1.186
1.186
1.075
1.047
1.035
1.088
1.058
1.057
1.027
1.058
1.004
1.031
1.033
0.985
1.056
1.068
0.985
1.031
Selected
Cumulative
3.200
23.628
1.950
7.384
1.620
3.787
1.250
2.337
1.160
1.870
1.160
1.612
1.080
1.390
1.050
1.287
1.040
1.225
1.040
1.178
96:108 108:120 120:132 132:144 144:Ult.
1.121
0.940
1.031
0.997
1.068
1.057
1.031
1.004
1.030
1.133
1.100
Company A - Report Year
Reported Loss+ALAE Development
Report
Year
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
12:24
1.353
1.368
1.755
1.576
1.633
1.280
1.767
2.013
1.360
1.387
24:36
1.393
1.100
1.171
1.708
1.053
1.162
1.335
1.246
1.309
36:48
1.336
1.423
1.331
1.194
1.249
1.356
1.188
1.304
48:60
0.842
1.154
1.063
0.963
1.203
1.054
0.935
60:72
1.076
1.014
1.139
1.188
0.925
1.073
72:84
1.125
1.031
1.028
1.065
0.977
84:96
1.016
1.166
0.932
0.980
All Yr Avg
Avg Ex H & L
Last 2 Years
Last 3 Years
Last 5 Years
2 Yr Wghtd
5 Yr Wghtd
Weighted
1.549
1.525
1.374
1.587
1.561
1.376
1.525
1.530
1.250
1.221
1.278
1.297
1.221
1.278
1.250
1.278
1.260
1.280
1.246
1.283
1.258
1.245
1.253
1.274
1.021
1.021
0.994
1.064
1.043
0.976
1.018
1.013
1.065
1.068
0.999
1.062
1.068
1.003
1.067
1.068
1.021
1.025
1.021
1.023
1.045
1.026
1.041
1.041
1.009
0.982
0.956
1.026
0.952
0.929
1.028
0.994
0.936
0.876
0.972
0.961
1.013
1.024
0.996
0.982
1.530
2.953
1.300
1.930
1.250
1.485
1.050
1.188
1.050
1.131
1.025
1.077
1.020
1.051
1.010
1.030
1.010
1.020
Selected
Cumulative
96:108 108:120 120:132 132:144
0.928
0.876
1.032
0.930
1.067
1.126
1.003
1.019
1.005
1.010
1.003
1.005
Company A
Report Year Loss+ALAE Severity
(1)
(2)
(3)
(4)
(5)
Loss
Trend
Factor
2.252
2.105
1.967
1.838
1.718
1.606
1.501
1.403
1.311
1.225
1.145
(3) x (4)
Trended
Ultimates
Loss+ALAE
(in 000's)
25,516
22,490
29,115
23,119
28,166
47,858
48,701
40,494
54,085
76,662
70,825
(1) x (2)
Report
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Reported
Loss+ALAE
(in 000's)
11,273
10,579
14,508
12,205
15,598
27,667
28,691
24,310
27,794
32,426
20,951
LDFs
1.005
1.010
1.020
1.030
1.051
1.077
1.131
1.188
1.485
1.930
2.953
Ultimates
Loss+ALAE
(in 000's)
11,329
10,685
14,801
12,575
16,393
29,804
32,451
28,872
41,261
62,579
61,862
(6)
(7)
(5) /(6)
Report
Year
Claim
Count
2,600
2,301
2,835
2,452
2,800
4,683
4,452
3,986
5,498
7,832
6,983
Trended
Average
Severity
9,814
9,774
10,270
9,429
10,059
10,220
10,939
10,160
9,837
9,788
10,143
Latest 5 Year Weighted Average
All Year Weighted Average
10,123
10,061
Selected Prospective Report Year Severity
10,100
Company A
Accident Year Claim Count
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Cumulative LDF
Ult. Counts
Accident Year
12
24
36
48
60
72
84
400
460
560
700
840
980
1,100
1,240
1,300
1,400
1,300
800
920
1,120
1,400
1,680
1,960
2,200
2,480
2,600
2,800
1,100
1,265
1,540
1,925
2,310
2,695
3,025
3,410
3,575
1,400
1,610
1,960
2,450
2,940
3,430
3,850
4,340
1,580
1,817
2,212
2,765
3,318
3,871
4,345
1,680
1,932
2,352
2,940
3,528
4,116
1,780
2,047
2,492
3,115
3,738
96
108
120
132
1,860 1,920 1,960 1,980
2,139 2,208 2,254
2,604 2,688
3,255
5.000 2.500 1.818 1.429 1.266 1.190 1.124 1.075 1.042 1.020 1.010
6,500 7,000 6,500 6,200 5,500 4,900 4,200 3,500 2,800 2,300 2,000
1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987
Company A
Estimated 1999 AY Loss+ALAE
Accident
Year
1,987
1,988
1,989
1,990
1,991
1,992
1,993
1,994
1,995
1,996
1,997
Ultimate
Counts
2,000
2,300
2,800
3,500
4,200
4,900
5,500
6,200
6,500
7,000
6,500
Exposure
Base
63,952
64,707
66,335
62,363
59,463
53,395
53,446
52,044
45,345
47,544
49,003
(1) Selected Frequency
(2) Prospective Exposure
(3) Prospective Claim Count (1) x (2)
(4) Estimated Report Year Severity
(5) Estimated Years Between Average
Acc. Date of Acc. Year & Report Year
(6) Trend Rate
(7) Estimated Loss+ALAE (3) x (4) x[1+(6)]^(5)
Frequency
Per Exposure
0.031
0.036
0.042
0.056
0.071
0.092
0.103
0.119
0.143
0.147
0.133
0.140
50,500
7,070
10,100
2.700
7%
85,718,979
Company A - Comparison of
Accident Year Development Estimate
(1)
Reported
Accident Loss+ALAE
Year
(in 000's)
1987
8,774
1988
12,631
1989
17,132
1990
18,757
1991
21,309
1992
25,208
1993
27,313
1994
21,858
1995
17,424
1996
10,025
1997
4,162
(2)
LDFs
1.13
1.18
1.23
1.29
1.39
1.61
1.87
2.34
3.79
7.38
23.63
(3)
(1) x(3)
Ultimate
Loss+ALAE
(in 000's)
9,940
14,884
20,995
24,135
29,612
40,636
51,074
51,092
54,338
58,678
63,597
(4)
Trend
Factor
2.252
2.105
1.967
1.838
1.718
1.606
1.501
1.403
1.311
1.225
1.145
(5)
(3) x (4)
(6)
Trended
Ultimate
Loss+ALAE
(in 000's) Exposures
22,388
63,952
31,328
64,707
41,300
66,335
44,372
62,363
50,879
59,463
65,253
53,395
76,648
53,446
71,659
52,044
71,226
45,345
71,883
47,544
72,812
49,003
Latest 6 Year Average
Latest 3 Year Average
(10) Selected 1999 Pure Premium
(11) 1999 Exposures
(12) Selected 1999 Loss+ALAE in 000's (10) x (11)
(7)
(8)
(9)
(5) /
1999
(8) / (4)
[(6)x1000] Expected Expected
Pure
Pure
Pure
Premium Premium
Premium
For BF
For BF
350
484
623
712
856
1,222
1,434
1,450
1,377
1,571
1,106
1,512
1,184
1,486
1,266
1,434
1,523
1,500
50,500
75,750
Review of Results
 Historical Loss Ratios
 Historical Pure Premiums