1998 CLRS
Intermediate Case Study
Discussion Material For
GL Insurance Company
Intermediate Case Study
Company Facts
•
•
•
•
Stock insurance company
Only general liability policies written
In business for over 20 years
Stable book of business (essentially the
same group of insureds)
• Well managed
• Healthy balance sheet
GL-1
Intermediate Case Study
Balance Sheet @ 12/31/97 ($000)
Assets
Bonds
Liabilities & Surplus
Stocks
$48,262
Loss & LAE Reserves
Case Reserves
IBNR Reserves
Cash
$11,028
Unearned Premium Reserve
$84,196
$522,678
Other Liabilities
$24,965
Agents' Balance
$19,799
Total Liabilities
$515,517
Other Assets
$35,662
Policyholders' Surplus
Total Assets
$578,139
Total Invested Assets
GL-2
$463,388
Total Liabilities & Surplus
$208,052
$198,304
$62,622
$578,139
Intermediate Case Study
Reserving Information
• GL Insurance Company has a small actuarial staff
headed by an actuarial student.
• The department calculated year-end reserves using
both the paid and incurred loss development
methods.
• The staff supplements this analysis with the use of
expected loss techniques, if needed.
GL-3
Intermediate Case Study
GLIC Actuarial Study
Slides GL-5 and GL-6 display the paid loss
development method.
 Slides GL-7 and GL-8 display the incurred
loss development method.
 Tail factor selections are based upon
reviews of industry data, as well as curve
fits of selected loss development factors
(Slides GL-6 and GL-8).

GL-4
GL INSURANCE COMPANY
Total General Liability
Paid Loss Development Method
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
12
$1,340
1,857
2,024
2,781
3,439
3,714
4,652
5,292
6,818
9,337
15,073
12-24
24
$3,188
4,297
4,891
6,655
8,272
9,039
11,236
12,974
16,984
23,263
24-36
36
48
$5,072
6,864
7,790
10,671
13,325
14,638
18,109
21,106
27,677
$6,973
9,438
10,773
14,738
18,551
20,326
25,239
29,611
36-48
48-60
Paid Losses ($000)
60
72
84
$8,677
11,820
13,792
18,022
23,386
26,117
31,250
$10,008
13,594
16,071
20,795
26,861
30,643
$11,802
14,783
17,695
23,179
29,409
Age-to-Age Development Factors
60-72
72-84
84-96
96
108
120
132
$12,606
15,710
18,886
24,597
$13,174
16,439
19,735
$13,596
16,972
$14,033
96-108
108-120
120-132
132+
2.379
2.314
2.417
2.393
2.405
2.434
2.415
2.452
2.491
2.491
1.591
1.597
1.593
1.603
1.611
1.619
1.612
1.627
1.630
1.375
1.375
1.383
1.381
1.392
1.389
1.394
1.403
1.244
1.252
1.280
1.223
1.261
1.285
1.238
1.153
1.150
1.165
1.154
1.149
1.173
1.179
1.087
1.101
1.115
1.095
1.068
1.063
1.067
1.061
1.045
1.046
1.045
1.032
1.032
1.032
3 Avg
3 Wtd Avg
5 Avg
Mid 3 of 5
Wtd Avg
2.478
2.482
2.457
2.459
2.443
1.623
1.624
1.620
1.619
1.615
1.395
1.396
1.392
1.391
1.390
1.261
1.259
1.257
1.260
1.255
1.159
1.160
1.158
1.156
1.159
1.104
1.103
1.115
1.104
1.109
1.064
1.064
1.065
1.065
1.064
1.045
1.045
1.045
1.045
1.045
1.032
1.032
1.032
1.032
1.032
1.032
1.032
1.032
Selected
2.491
1.623
1.395
1.261
1.152
1.104
1.065
1.045
1.032
1.032
GL-5
GL INSURANCE COMPANY
Total General Liability
Analysis of Development Patterns-- Paid Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
Y Variable
Age-to-Age
Factors
12
24
36
48
60
72
84
96
108
120
2.491
1.623
1.395
1.261
1.152
1.104
1.065
1.045
1.032
1.032
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.09)
(0.73)
(1.10)
(1.46)
(1.96)
(2.31)
(2.77)
(3.12)
(3.46)
(3.46)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
(0.67)
(0.04)
0.23
0.45
0.71
0.86
1.03
1.15
1.25
1.25
660
66
(20.45)
(2.05)
39.95
4.00
6.21
0.62
Parameter Estimates
N=
A=
B=
10
2.961
0.968
10
0.061
0.857
R^2=
0.984
0.996
Power Model
Fitted Values
X
Y
Fitted Tail Factor from 132 to Ultimate
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
2.085
1.644
1.400
1.256
1.167
1.110
1.073
1.049
1.033
1.022
1.015
1.010
1.007
1.005
1.003
1.002
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
1.047
1.072
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.135
Selected Tail Factor
1.075
GL-6
2.492
1.652
1.367
1.228
1.150
1.102
1.071
1.050
1.036
1.026
1.019
1.014
1.010
1.007
1.005
1.004
1.003
1.002
1.002
1.001
1.001
1.001
1.001
1.000
GL Insurance Company
Total General Liability
Incurred Loss Development Method
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
12
24
$5,662
6,975
8,345
10,652
13,647
15,549
18,260
22,029
28,730
39,637
55,297
$8,879
10,897
13,012
17,073
21,807
24,872
29,200
35,312
46,297
64,628
12-24
24-36
36
48
$11,006
13,556
16,304
21,391
27,086
31,261
36,605
44,500
58,061
$12,396
15,303
18,417
23,978
30,684
35,432
41,696
50,322
36-48
48-60
Incurred Losses ($000)
60
72
84
$13,067
16,271
19,507
25,469
32,600
37,460
44,488
$13,526
16,861
20,224
26,443
33,807
38,965
$13,838
17,252
20,677
27,073
34,584
Age-to-Age Development Factors
60-72
72-84
84-96
96
108
120
132
$14,075
17,565
21,077
27,550
$14,315
17,883
21,465
$14,573
18,208
$14,778
96-108
108-120
120-132
132+
1.568
1.562
1.559
1.603
1.598
1.600
1.599
1.603
1.611
1.630
1.240
1.244
1.253
1.253
1.242
1.257
1.254
1.260
1.254
1.126
1.129
1.130
1.121
1.133
1.133
1.139
1.131
1.054
1.063
1.059
1.062
1.062
1.057
1.067
1.035
1.036
1.037
1.038
1.037
1.040
1.023
1.023
1.022
1.024
1.023
1.017
1.018
1.019
1.018
1.017
1.018
1.018
1.018
1.018
1.014
3 Avg
3 Wtd
5 Avg
Mid 3 of 5
Wtd
1.615
1.618
1.609
1.605
1.605
1.256
1.256
1.253
1.255
1.253
1.134
1.134
1.131
1.132
1.131
1.062
1.062
1.062
1.061
1.062
1.038
1.039
1.038
1.037
1.038
1.023
1.023
1.023
1.023
1.023
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.014
1.014
1.014
1.018
1.014
Selected
1.621
1.256
1.134
1.062
1.038
1.023
1.018
1.018
1.018
1.014
GL-7
GL INSURANCE COMPANY
Total General Liability
Analysis of Development Patterns-- Incurred Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
Y Variable
Age-to-Age
Factors
12
24
36
48
60
72
84
96
108
120
1.621
1.256
1.134
1.062
1.038
1.023
1.018
1.018
1.018
1.014
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.73)
(1.48)
(2.07)
(2.81)
(3.29)
(3.78)
(4.03)
(4.03)
(4.03)
(4.28)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
(0.04)
0.46
0.76
1.04
1.20
1.33
1.40
1.40
1.40
1.45
660
66
(30.52)
(3.05)
39.95
4.00
10.40
1.04
Parameter Estimates
N=
A=
B=
10
1.484
0.968
10
0.193
0.671
R^2=
0.888
0.973
Power Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
Fitted Tail Factor from 132 to Ultimate
1.306
1.198
1.130
1.086
1.058
1.039
1.026
1.018
1.012
1.008
1.005
1.004
1.002
1.002
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
1.017
1.024
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.037
Selected Tail Factor
1.025
GL-8
1.562
1.244
1.134
1.081
1.052
1.034
1.024
1.016
1.012
1.008
1.006
1.004
1.003
1.002
1.002
1.001
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
Intermediate Case Study
Summary of Loss Development Projections
Accident Earned
Year
Premium
(1)
(2)
Actual Losses
@ 12/31/97
Paid
Incurred
(3)
(4)
Cumulative
Estimated
Estimated Ultimate
LDF
Ultimate Losses
Loss Ratio
Paid Incurred Paid LDM Inc. LDM Paid LDM Inc. LDM
(5)
(6)
(7)
(8)
(9)
(10)
1.075
1.109
1.145
1.196
1.274
1.407
1.621
2.043
2.851
4.627
11.525
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
22,122
26,474
30,286
37,741
45,691
50,562
60,349
75,972
97,616
131,861
168,391
14,033
16,972
19,735
24,597
29,409
30,643
31,250
29,611
27,677
23,263
15,073
14,788
18,208
21,465
27,550
34,584
38,965
44,488
50,322
58,061
64,628
55,297
Total
747,065
262,263
428,346
GL-9
1.025
1.039
1.058
1.077
1.096
1.122
1.164
1.237
1.402
1.761
2.855
15,085
18,829
22,595
29,428
37,473
43,106
50,641
60,509
78,898
107,629
173,715
15,147
18,924
22,711
29,674
37,921
43,707
51,799
62,224
81,414
113,822
157,866
0.682
0.711
0.746
0.780
0.820
0.853
0.839
0.796
0.808
0.816
1.032
0.685
0.715
0.750
0.786
0.830
0.864
0.858
0.819
0.834
0.863
0.937
637,907
635,212
0.854
0.850
Intermediate Case Study
Applications of Bornhuetter-Ferguson Technique
AY 1996
AY 1997
Paid
Paid
Incurred
Estimate Estimate
Estimate
1. Loss Development Factor to Ultimate
4.627
11.525
2.855
131,861
168,391
168.391
86%
90%
90%
113,400
151,552
151,522
5. Actual Losses @ 12/31/97
23,263
15,073
55,297
6. Estimated Losses @ 12/31/97
(i) Paid or Reported [(4)/(1)]
(ii) Unpaid or Unreported [(4) - (6i)]
24,508
88,892
13,150
138,402
53,085
98,467
112,155
153,475
153,764
2. Earned Premium
3. Expected Loss Ratio
4. Expected Losses [(2) x (3)]
7. Revised Ultimate Losses [(5) + (6ii)]
GL-10
Intermediate Case Study
Estimated IBNR Reserves
Accident
Year
(1)
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Total
Earned
Premium
(2)
Selected
Ultimate Losses
Paid
Incurred
(3)
(4)
22,122
15,085
26,474
18,829
30,286
22,595
37,741
29,428
45,691
37,473
50,562
43,106
60,349
50,641
75,972
60,509
97,616
78,898
131,861 112,153*
168,391 153,475*
747,065
622,192
Estimated Ultimate
Loss Ratio
Paid
Incurred
(5)
(6)
15,147
18,924
22,711
29,674
37,921
43,707
51,799
62,224
81,414
113,822
153,764*
0.682
0.711
0.746
0.780
0.820
0.853
0.839
0.796
0.808
0.851
0.911
0.685
0.715
0.750
0.786
0.830
0.864
0.858
0.819
0.834
0.863
0.913
631,109
0.833
0.845
Selected IBNR RESERVES @ 12/31/97 (AVG of columns 7 & 8)
* From Bornhuetter-Ferguson Method (Slide GL-10)
GL-11
Estimated
Required IBNR
Paid LDM Inc. LDM
(7)
(8)
307
621
1,130
1,878
2,889
4,141
6,153
10,187
20,837
47,525
98,178
193,846
396
716
1,246
2,124
3,337
4,742
7,311
11,902
23,353
49,194
98,467
202,763
198,304
Intermediate Case Study
GLIC’s Carried Reserves
 The
company’s carried reserves are based
upon the straight average of the paid and
incurred loss projection methods after
application of expected loss techniques.
GL-12
Intermediate Case Study
Management’s Observations
 In
previous years, the paid and incurred loss
projections were almost identical.
 Recently, differences between the two
estimates are emerging.
GL-13
Intermediate Case Study
Assignment
 GLIC
has employed you, a consulting
actuary, to complete their 1997 reserve
certification and critique the actuarial work
done by GLIC’s actuarial department.
GL-14
Intermediate Case Study
Initial Task
 You
begin the assignment by interviewing the
vice president of each of the following
insurance disciplines.
Claims
Marketing
Underwriting
GL-15
Intermediate Case Study
Vice President of Claims
 Staff
and procedures have remained the
same for as long as anyone can remember.
 Systems have not changed, and there have
been no accounting or other changes that
would have impacted year-end processing.
GL-16
Intermediate Case Study
Vice President of Marketing
 The
client base is extremely stable.
 Growth has come primarily from increase in
business from existing clients, as opposed to
new clients.
 GLIC’s clients represent almost all US
distributors of Widgets.
 These clients are expanding into other areas,
generating the growth in premium.
GL-17
Intermediate Case Study
Vice President of Marketing
 Given
the company’s understanding of the
product and their sensible approach to
pricing (small annual increases), they have
captured and retained their niche market
GL-18
Intermediate Case Study
Vice President of Underwriting
 The
VP is concerned about the deterioration
in the loss ratio (including ALAE) from
81% in 1994 to 91% in 1997 on an accident
year basis.
 They attribute at least part of the problem to
the heavier GL exposures being accepted
from their long-term clients.
GL-19
Intermediate Case Study
Distribution of Earned Premium
Accident
Year
GL-20
Total
Earned Premium ($000)
Heavy
Light
% Heavy
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
22,122
26,474
30,286
37,741
45,691
50,562
60,349
75,972
97,616
131,861
168,391
192
822
2,499
5,101
9,987
12,065
15,174
22,537
35,455
59,999
86,337
21,930
25,652
27,787
32,640
35,704
38,497
45,175
53,435
62,161
71,862
82,054
0.9%
3.1%
8.3%
13.5%
21.9%
23.9%
25.1%
29.7%
36.3%
45.5%
51.3%
Total
747,065
250,168
496,897
33.5%
Intermediate Case Study
Vice President of Underwriting
 The
underwriting department, with the help
of the actuarial staff, will be conducting
separate rate analyses for Heavy GL versus
Light GL later in the year.
 Reports by class of business have just been
provided via an ad hoc request to the data
processing department.
GL-21
Intermediate Case Study
Vice President of Underwriting
 Although
the analysis has not yet been
completed, the underwriting department
suspects that Heavy GL rates need to
increase by more than the traditional 5%
annual increase taken in previous years for
Total GL.
GL-22
GL INSURANCE COMPANY
Light General Liability
Paid Loss Development Method
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
12
24
36
48
$1,329
1,812
1,886
2,463
2,795
2,956
3,643
3,632
4,661
5,544
10,484
$3,159
4,177
4,517
5,810
6,565
7,012
8,561
9,331
11,154
13,128
$5,023
6,654
7,150
9,220
10,399
11,149
13,535
14,836
17,679
$6,902
9,129
9,838
12,631
14,288
15,263
18,584
20,444
12-24
24-36
36-48
48-60
60
Paid Losses ($000)
72
84
$8,586
11,420
12,543
15,359
17,831
19,384
22,765
$9,900
13,122
14,575
17,647
20,345
22,563
$11,682
14,250
16,018
19,606
22,176
Age-to-Age Development Factors
60-72
72-84
84-96
96
108
120
132
$12,476
15,134
17,075
20,763
$13,037
15,830
17,826
$13,454
16,337
$13,885
96-108
108-120
120-132
132+
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
2.377
2.305
2.395
2.359
2.349
2.372
2.350
2.373
2.393
2.368
1.590
1.593
1.583
1.587
1.584
1.590
1.581
1.590
1.585
1.374
1.372
1.376
1.370
1.374
1.369
1.373
1.378
1.244
1.251
1.275
1.216
1.248
1.270
1.225
1.153
1.149
1.162
1.149
1.141
1.164
1.180
1.086
1.099
1.111
1.090
1.068
1.062
1.066
1.059
1.045
1.046
1.044
1.032
1.032
1.032
3 Avg.
3 Wtd.
5 Avg.
Mid 3 of 5
Wtd.
2.378
2.378
2.371
2.371
2.367
1.585
1.585
1.586
1.586
1.587
1.373
1.374
1.373
1.372
1.373
1.248
1.246
1.247
1.248
1.245
1.151
1.152
1.153
1.153
1.153
1.100
1.100
1.113
1.100
1.108
1.062
1.062
1.064
1.064
1.063
1.045
1.045
1.045
1.045
1.045
1.032
1.032
1.032
1.032
1.032
1.032
1.032
1.032
Selected
2.378
1.601
1.358
1.248
1.153
1.100
1.064
1.045
1.032
1.032
GL-23
GL INSURANCE COMPANY
Total General Liability
Analysis of Development Patterns--Paid Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
Y Variable
Age-to-Age
Factors
12
24
36
48
60
72
84
96
108
120
2.378
1.601
1.358
1.248
1.153
1.100
1.064
1.045
1.032
1.032
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.14)
(0.75)
(1.18)
(1.15)
(1.95)
(2.35)
(2.78)
(3.12)
(3.46)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
(0.61)
(0.02)
0.29
0.48
0.70
0.87
1.03
1.15
1.25
1.25
660
66
(20.71)
(2.07)
39.95
4.00
6.39
0.64
Parameter Estimates
N=
A=
B=
10
2.769
0.969
10
0.069
0.830
R^2=
0.983
0.997
Power Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
Fitted Tail Factor from 132 to Ultimate
2.010
1.613
1.388
1.252
1.166
1.111
1.075
1.051
1.035
1.024
1.016
1.011
1.008
1.005
1.004
1.002
1.002
1.001
1.001
1.001
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
1.052
1.077
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.135
Selected Tail Factor
1.080
GL-24
2.388
1.616
1.350
1.219
1.145
1.100
1.070
1.050
1.036
1.026
1.019
1.014
1.011
1.008
1.006
1.004
1.003
1.002
1.001
1.001
1.001
1.001
1.001
1.000
GL INSURANCE COMPANY
Light General Liability
Incurred Loss Development Method
Accident
Year
12
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
5,612
6,752
7,642
9,187
10,611
11,775
13,600
15,388
18,143
21,383
29,195
12-24
24
8,794
10,520
11,837
14,561
16,569
18,475
21,311
24,082
28,394
33,614
24-36
36
10,896
13,066
14,773
18,128
20,491
22,964
26,404
29,934
35,038
36-48
48
12,269
14,738
16,649
20,249
23,073
25,857
29,863
33,586
48-60
Incurred Losses ($000)
60
72
12,932
15,666
17,615
21,464
24,434
27,253
31,744
13,385
16,230
18,249
22,258
25,289
28,289
84
96
108
120
132
13,693
16,603
18,650
22,770
25,845
13,926
16,902
19,004
23,157
14,163
17,206
19,346
14,418
17,516
14,620
Age-to-Age Development Factors
60-72
72-84
84-96
96-108
108-120
120-132
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1.567
1.558
1.549
1.585
1.570
1.569
1.567
1.565
1.565
1.572
1.239
1.242
1.248
1.245
1.230
1.243
1.239
1.243
1.234
1.126
1.128
1.127
1.117
1.126
1.126
1.131
1.122
1.054
1.063
1.058
1.060
1.059
1.054
1.063
1.035
1.036
1.036
1.037
1.035
1.038
1.023
1.023
1.022
1.023
1.022
1.017
1.018
1.019
1.017
1.017
1.018
1.018
1.018
1.018
1.014
3 Avg.
3 Wtd.
5 Avg.
Mid 3 of 5
Wtd.
1.567
1.568
1.568
1.567
1.568
1.239
1.238
1.238
1.239
1.240
1.126
1.126
1.124
1.125
1.125
1.059
1.059
1.059
1.059
1.059
1.037
1.037
1.036
1.036
1.036
1.022
1.022
1.023
1.023
1.023
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.018
1.014
1.014
1.014
1.018
1.014
Selected
1.567
1.239
1.126
1.059
1.037
1.022
1.018
1.018
1.018
1.014
GL-25
132+
GL INSURANCE COMPANY
Light General Liability
Analysis of Development Patterns--Incurred Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
Y Variable
Age-to-Age
Factors
12
24
36
48
60
72
84
96
108
120
1.567
1.239
1.126
1.059
1.037
1.022
1.018
1.018
1.018
1.014
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.80)
(1.54)
(2.13)
(2.86)
(3.32)
(3.83)
(4.03)
(4.03)
(4.03)
(4.28)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
660
66
(30.83)
(3.08)
39.95
4.00
Power Model
Fitted Values
X
Y
0.02
0.50
0.78
1.06
1.20
1.35
1.40
1.40
1.40
1.45
10.55
1.05
Parameter Estimates
N=
A=
B=
10
1.439
0.969
10
0.218
0.646
R^2=
0.884
0.972
Fitted Tail Factor from 132 to Ultimate
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
1.283
1.186
1.124
1.084
1.057
1.038
1.026
1.018
1.012
1.008
1.060
1.004
1.003
1.002
1.001
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
1.018
1.025
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.037
Selected Tail Factor
1.025
GL-26
1.510
1.224
1.124
1.075
1.049
1.033
1.023
1.016
1.011
1.008
1.006
1.005
1.003
1.003
1.002
1.001
1.001
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
Intermediate Case Study
Bornhuetter-Ferguson Technique - Light GL
Accident Year 1997
Paid
Incurred
Estimate Estimate
1. Loss Development Factor
2. Earned Premium
10.446
2.690
82,054
82,054
83.5%
83.5%
4. Expected Losses [(2) x (3)]
68,515
68,515
5. Actual Losses @ 12/31/97
10,484
29,195
6. Estimated Losses @ 12/31/97
(I) Paid or Reported [(4) / (1)]
(ii) Unpaid or Unreported [(4) - (6i)]
6,546
61,969
25,467
43,048
7. Revised Ultimate Losses [(5)+ (6ii)]
72,453
72,243
3. Expected Loss Ratio
GL-27
Intermediate Case Study
IBNR Reserves Estimate - Light GL
Estimated
Ultimate Losses
Paid
Incurred
(3)
(4)
Accident
Year
(1)
Earned
Premium
(2)
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
21,930
25,652
2,787
32,640
35,704
358,497
45,175
53,435
62,161
71,862
82,054
14,996
18,209
20,504
24,957
28,361
31,742
36,926
41,385
48,600
57,779
72,453*
496,897
395,912
342,981
265,680
Total
1987-95
14,986
18,205
20,504
24,942
28,339
31,701
36,889
41,332
48,552
57,711
72,243*
Estimated
Estimated
Loss Ratio
Required IBNR
Paid
Incurred Paid LDM Inc. LDM
(5)
(6)
(7)
(8)
0.684
0.710
0.738
0.765
0.794
0.825
0.817
0.774
0.782
0.804
0.883
0.683
0.710
0.737
0.764
0.794
0.823
0.817
0.774
0.781
0.803
0.880
376
693
1,158
1,800
2,516
3,453
5,182
7,799
13,562
24,165
43,258
366
689
1,123
1,785
2,494
3,412
5,145
7,746
13,514
24,097
43,048
395,369
0.797
0.796
103,962
103,419
265,415
0.775
0.774
36,539
36,274
* From Bornhuetter-Ferguson Method (Slide GL-27)
GL-28
GL INSURANCE COMPANY
Heavy General Liability
Paid Loss Development Method
Accident
Year
12
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
11
45
138
318
644
758
1,009
1,360
2,157
3,793
4,589
12-24
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
2.636
2.667
2.710
2.657
2.651
2.674
2.651
2.679
2.703
2.672
24
29
120
374
845
1,707
2,027
2,675
3,643
5,830
10,135
24-36
1.690
1.750
1.711
1.717
1.714
1.721
1.710
1.721
1.715
36
49
210
640
1,451
2,926
3,489
4,574
6,270
9,998
36-48
1.449
1.471
1.461
1.520
1.457
1.451
1.455
1.462
48
71
309
935
2,107
4,263
5,063
6,655
9,167
48-60
1.282
1.294
1.336
1.264
1.303
1.330
1.275
Paid Losses ($000)
60
72
91
400
1,249
2,663
5,555
6,733
8,485
108
472
1,496
3,148
6,516
8,080
84
96
108
120
132
120
533
1,677
3,573
7,233
130
576
1,811
3,834
137
609
1,909
142
635
148
Age-to-Age Development Factors
60-72
72-84
84-96
1.187
1.180
1.980
1.182
1.173
1.200
1.111
1.129
1.121
1.135
1.110
1.083
1.081
1.080
1.073
96-108
1.054
1.057
1.054
108-120
120-132
1.036
1.043
1.042
3 Avg.
3 Wtd. Avg.
5 Avg.
Mid 3 of 5
Wtd. Avg.
2.685
2.682
2.676
2.675
2.676
1.715
1.716
1.716
1.717
1.716
1.456
1.457
1.455
1.455
1.457
1.303
1.300
1.302
1.303
1.298
1.185
1.187
1.187
1.187
1.187
1.122
1.119
1.121
1.120
1.119
1.078
1.076
1.079
1.08
1.076
1.055
1.055
1.055
1.054
1.055
1.040
1.042
1.040
1.042
1.042
1.042
1.042
1.042
Selected
2.684
1.717
1.456
1.303
1.187
1.122
1.078
1.055
1.04
1.039
GL-29
132+
GL INSURANCE COMPANY
Heavy General Liability
Analysis of Development Patterns--Paid Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
12
24
36
48
60
72
84
96
108
120
Y Variable
Age-to-Age
Factors
2.684
1.717
1.456
1.303
1.187
1.122
1.078
1.055
1.040
1.039
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.01)
(0.62)
(0.98)
(1.33)
(1.76)
(2.16)
(2.59)
(2.93)
(3.24)
(3.26)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
(0.76)
(0.14)
0.15
0.38
0.61
0.80
0.97
1.08
1.18
1.19
660
66
(18.88)
(1.89)
39.95
4.00
5.46
0.55
Parameter Estimates
N=
A=
B=
10
3.246
0.969
10
0.052
0.875
R^2=
0.985
0.996
Power Model
Fitted Values
X
Y
Fitted Tail Factor from 132 to Ultimate
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
2.241
1.738
1.461
1.297
1.195
1.130
1.087
1.059
1.040
1.027
1.019
1.013
1.009
1.006
1.004
1.003
1.002
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
1.060
1.100
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.135
Selected Tail Factor
1.100
GL-30
2.724
1.761
1.433
1.273
1.182
1.126
1.088
1.063
1.046
1.033
1.025
1.018
1.014
1.010
1.008
1.006
1.004
1.003
1.002
1.002
1.001
1.001
1.001
1.001
1.000
GL INSURANCE COMPANY
Heavy General Liability
Incurred Loss Development Method
Accident
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Accident
Year
12
50
223
703
1,465
3,036
3,774
4,660
6,641
10,587
18,254
26,102
12-24
24
85
377
1,175
2,512
5,148
6,397
7,889
11,230
17,903
31,014
24-36
110
490
1,531
3,263
6,595
8,297
10,201
14,566
23,023
36-48
1.155
1.153
1.155
1.143
1.154
1.154
1.160
1.149
48
127
565
1,768
3,729
7,611
9,575
11,833
16,736
48-60
1.063
1.071
1.070
1.074
1.073
1.066
1.077
Incurred Losses ($000)
60
72
135
605
1,892
4,005
8,166
10,207
12,744
141
631
1,975
4,185
8,518
10,676
84
96
108
120
132
145
649
2,027
4,303
8,739
149
663
2,073
4,393
152
677
2,119
155
692
158
Age-to-Age Development Factors
60-72
72-84
84-96
1.044
1.043
1.044
1.045
1.043
1.046
1.028
1.029
1.026
1.028
1.026
1.028
1.022
1.023
1.021
96-108
1.020
1.021
1.022
108-120
120-132
1.020
1.022
1.019
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1.700
1.691
1.671
1.715
1.696
1.695
1.693
1.691
1.691
1.699
3 Avg.
3 Wtd.
5 Avg.
Mid 3 of 5
Wtd. Avg.
1.694
1.695
1.694
1.693
1.695
1.292
1.291
1.291
1.292
1.291
1.154
1.154
1.152
1.152
1.153
1.072
1.072
1.072
1.072
1.072
1.045
1.045
1.044
1.044
1.045
1.027
1.027
1.027
1.028
1.027
1.022
1.021
1.023
1.022
1.022
1.021
1.022
1.021
1.021
1.021
1.021
1.022
1.021
1.019
1.019
1.019
1.022
1.019
Selected
1.694
1.292
1.154
1.072
1.044
1.027
1.022
1.022
1.022
1.017
GL-31
1.294
1.300
1.303
1.299
1.281
1.297
1.293
1.297
1.286
36
132+
GL INSURANCE COMPANY
Heavy General Liability
Analysis of Development Patterns -- Incurred Losses
Using "The Method of Least Squares"
Actual Values
X Var.
Age
Y Variable
Age-to-Age
Factors
12
24
36
48
60
72
84
96
108
120
1.694
1.292
1.154
1.072
1.044
1.027
1.022
1.022
1.022
1.017
Total
Average
Power Model
Curve: Y =
A ^ (B ^ X)
Transformed Values
X'
Y'
X
ln[ln(Y)]
Weibull Model
Curve: Y =
1/ [1-EXP(-A * X ^ B)]
Transformed Values
X'
Y'
ln (X)
ln[ln(Y/(Y-1)]
12
24
36
48
60
72
84
96
108
120
(0.64)
(1.36)
(1.94)
(2.67)
(3.15)
(3.63)
(3.83)
(3.83)
(3.83)
(4.08)
2.48
3.18
3.58
3.87
4.09
4.28
4.43
4.56
4.68
4.79
(0.11)
0.40
0.70
0.99
1.15
1.29
1.35
1.35
1.35
1.41
660
66
(28.95)
(2.89)
39.95
4.00
9.87
0.99
Parameter Estimates
N=
A=
B=
10
1.54
0.969
10
0.175
0.683
R^2=
0.885
0.971
Power Model
Fitted Values
X
Y
Fitted Tail Factor from 132 to Ultimate
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
1.344
1.225
1.149
1.100
1.067
1.046
1.031
1.021
1.014
1.010
1.007
1.005
1.003
1.002
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Weibull Model
Fitted Values
X
Y
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
1.022
1.030
Broader Data Source - Bests Casualty Loss Reserve Development Series
1.037
Selected Tail Factor
1.030
GL-32
1.625
1.275
1.152
1.093
1.060
1.040
1.028
1.020
1.014
1.010
1.007
1.005
1.004
1.003
1.002
1.002
1.001
1.001
1.001
1.001
1.000
1.000
1.000
1.000
1.000
Intermediate Case Study
Bornhuetter-Ferguson Technique - Heavy GL
Accident Year 1997
Paid
Incurred
Estimate Estimate
1. Loss Development Factor
15.74
3.25
2. Earned Premium
86,337
86,337
3. Expected Loss Ratio
102.0%
102.0%
4. Expected Losses [(2) x (3)]
88,064
88,064
5. Actual Losses @ 12/31/97
4,589
26,102
6. Estimated Losses @ 12/31/97
(I) Paid or Reported [(4) / (1)]
(ii) Unpaid or Unreported [(4) - (6i)]
5,595
82,469
27,129
60,934
7. Revised Ultimate Losses [(5)+ (6ii)]
87,058
87,036
GL-33
Intermediate Case Study
IBNR Reserve Estimate - Heavy GL
Accident
Year
(1)
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Total
1987-95
Earned
Premium
(2)
Estimated
Ultimate Losses
Paid
Incurred
(3)
(4)
Estimated Ultimate
Estimated
Loss Ratio
Required IBNR
Paid
Incurred Paid LDM Inc. LDM
(5)
(6)
(7)
(8)
192
822
2,499
5,101
9,987
12,065
15,174
22,537
35,455
59,999
86,337
163
726
2,269
4,808
9,778
12,255
15,276
21,504
34,149
59,437
87,058*
163
725
2,269
4,806
9,772
12,260
15,279
21,509
34,146
59,430
87,036*
0.849
0.883
0.908
0.943
0.979
1.016
1.007
0.954
0.963
0.991
0.837
0.849
0.882
0.908
0.942
0.979
1.016
1.007
0.954
0.963
0.991
0.981
5
34
150
415
1,039
1,579
2,532
4,768
11,126
28,423
60,956
5
33
150
413
1,033
1,584
2,535
4,773
11,123
28,416
60,934
250168
160365
160359
0.954
0.954
111,027
110,999
0.991
0.991
21,648
21,649
103,832
100,928
100,929
* From Bornhutter-Ferguson Method (Slide GL-33)
GL-34
Intermediate Case Study
Summary of IBNR Estimates
Paid
Incurred
Estimates Estimates
Total General Liability
193,846
202,763
Sum of Components
Heavy GL
Light GL
Total GL
111,027
103,962
214,989
110,999
103,419
214,419
Selected by Consulting Actuary
214,704
GLIC's Carried IBNR Reserves (Slide GL-11)
198,304
Indicated Deficiency @ 12/31/97
(16,399)
GL-35
Intermediate Case Study
Balance Sheet @ 12/31/97 ($000)
Assets
Bonds
Liabilities & Surplus
Stocks
$48,262
Loss & LAE Reserves
Case Reserves
IBNR Reserves
Cash
$11,028
Unearned Premium Reserve
$84,196
$522,678
Other Liabilities
$24,965
Agents' Balance
$19,799
Total Liabilities
$531,917 *
Other Assets
$35,662
Policyholders' Surplus
Total Assets
$578,139
Total Invested Assets
*Changes from slide GL-2
GL-36
$463,388
Total Liabilities & Surplus
$208,052
$214,704 *
$46,223 *
$578,139