1998 CASUALTY LOSS RESERVE SEMINAR
Intermediate Track II - Techniques
SEPTEMBER 28, 1998
RESERVE MODELS
Average Hindsight Reserve Method
(Slides 2 - 12)
Average Incremental Paid Method
(Slides 13 - 19)
Bornhuetter-Ferguson Method
(Slides 20 - 30)
Slide 1
AVERAGE HINDSIGHT RESERVE METHOD
Goal:
 Estimate The Average Future Settlement Value Per Claim For
Recent Accident Years, Both For Claims Already Reported
and Future Claim Reports
.
.
Based On:
 Projected Ultimate Losses and Hindsight
Average Values For More Mature Accident Years
.
 Hindsight (Current Ultimate-Historical Payments) is the
Implied Unpaid Amount
Slide 2
.
AVERAGE HINDSIGHT RESERVE METHOD
Data Needed

Cumulative Paid Loss Triangle

Cumulative Closed (Paid) Claim Count Triangle

Estimated Ultimate Number of Claims for All Accident Years

Estimated Ultimate Losses For Several Mature Accident Years
Slide 3
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Cumulative Paid Losses ($000)
Accident
Year
12
1991 $50.0
1992
60.2
1993
75.5
1994
91.9
1995 115.0
1996 146.5
1997 181.1
*To be estimated
Slide 4
24
$80.0
97.0
120.1
147.1
184.1
233.4
Months Of Development
36
48
60
72
$98.2 $107.8 $113.2 $117.2
118.5
130.7
136.6 141.0
147.0
162.4
171.0
180.2
197.0
226.4
.
84
Ultimate
$119.7 $119.7
143.8
178.7
220.1
*
*
*
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Cumulative Number Of Closed Claims
Accident
Year
1991
1992
1993
1994
1995
1996
1997
12
50
55
63
70
80
93
105
24
75
83
94
105
120
139
Months Of Development
36
48
60
88
94
97
97
104
107
110
118
122
123
131
141
*Estimated using claim count development factors.
Slide 5
.
72
99
109
84
100
Ultimate*
100
110
125
140
160
185
210
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Calculation Of Average Outstanding Losses At 36 Months
Purpose: Project Future Settlement Dollars For 1995
Estimated
Paid
Accident Ultimate Losses at
Year
Losses 36 Months
.
(1)
(2)
1991 $119,700
1992 143,800
1993 178,700
1994 220,100
(3)
(4)
(5)
(6)
$98,200
118,500
147,000
180,200
$21,500
25,300
31,700
39,900
100
110
125
140
88
97
110
123
*Includes IBNR Claims
Slide 6
Number
Estimated Estimated Number of to Settle Average
Future
Ultimate
Closed
Beyond
Future
Payments Number
Claims at 36 Mos* Payment
= (2)-(3) of Claims 36 Months =(5)-(6) =(4)/(7)
(7)
12
13
15
17
Fitted forecasted value for AY 1995
(8)
$1,792
1,946
2,113
2,347
= $2,549
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Estimated Ultimate Losses: Accident Year 1995
(1) Forecasted Average Future Payment [Slide 6]
(2) Number of Future Claims to Settle [Slide 5]
(Ultimate - Closed Claims) = 160 - 141
(3) Estimated Future Loss Payments [(1) x (2)]
(4) Paid Losses to Date [Slide 4]
(5) Estimated Ultimate Losses [(3) + (4)]
Slide 7
=
$2,549
=
19
= $ 48,431
= $226,400
= $274,831
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Calculation Of Average Outstanding Losses At 24 Months
Purpose: Project Future Settlement Dollars For 1996
Estimated
Paid
Accident Ultimate Losses at
Year
Losses 24 Months
.
(1)
(2)
(3)
1992 $143,800 $97,000
1993 178,700 120,100
1994 220,100 147,100
1995 274,831 184,100
*Includes IBNR Claims
Slide 8
Number
Estimated Estimated Number of to Settle Average
Future
Ultimate
Closed
Beyond
Future
Payments Number
Claims at 24 Mos* Payment
= (2)-(3) of Claims 24 Months =(5)-(6) =(4)/(7)
(4)
$46,800
58,600
73,000
90,731
(5)
(6)
(7)
(8)
110
125
140
160
83
94
105
120
27
31
35
40
$1,733
1,890
2,086
2,268
Fitted forecasted value for AY 1996
= $2,484
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Estimated Ultimate Losses: Accident Year 1996
(1) Forecasted Average Future Payment [Slide 8]
(2) Number of Future Claims to Settle [Slide 5]
(Ultimate - Closed Claims) = 185 - 139
(3) Estimated Future Loss Payments [(1) x (2)]
(4) Paid Losses to Date [Slide 4]
(5) Estimated Ultimate Losses [(3) + (4)]
Slide 9
=
$2,484
=
46
= $ 114,264
= $233,400
= $347,664
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Calculation Of Average Outstanding Losses At 12 Months
Purpose: Project Future Settlement Dollars For 1997
Estimated
Paid
Accident Ultimate Losses at
Year
Losses 12 Months
.
(1)
(2)
(3)
1993 $178,700 $75,500
1994 220,100 91,900
1995 274,831 115,000
1996 347,664 146,500
*Includes IBNR Claims
Slide 10
Number
Estimated Estimated Number of to Settle Average
Future
Ultimate
Closed
Beyond
Future
Payments Number
Claims at 12 Mos* Payment
= (2)-(3) of Claims 12 Months =(5)-(6) =(4)/(7)
(4)
$103,200
128,200
159,831
201,164
(5)
(6)
(7)
125
140
160
185
63
70
80
93
62
70
80
92
Fitted forecasted value for AY 1997
(8)
$1,665
1,831
1,998
2,187
=
$2,397
AVERAGE HINDSIGHT RESERVE METHOD
XYZ Auto Insurance Company
Estimated Ultimate Losses: Accident Year 1997
(1) Forecasted Average Future Payment [Slide 10]
(2) Number of Future Claims to Settle [Slide 5]
(Ultimate - Closed Claims) = 210 - 105
(3) Estimated Future Loss Payments [(1) x (2)]
(4) Paid Losses to Date [Slide 4]
(5) Estimated Ultimate Losses [(3) + (4)]
Slide 11
=
$2,397
=
105
= $ 251,385
= $181,100
= $432,785
AVERAGE HINDSIGHT RESERVE METHOD
ADVANTAGES



Relatively Unaffected by Changes in Case Reserving Practices
Can Easily Adjust Trend Assumption
Allows Separate Analysis of Frequency and Severity
DISADVANTAGES
 Sensitive to Payment Pattern Shifts
 Averages Highly Variable When Only a Few Claims
 May be Insufficient if Business has Significantly
Changed (Example: Retentions Dramatically Increase)
 Too “Formula” Driven
Slide 12
.
.
AVERAGE INCREMENTAL PAID METHOD
Incremental Paid Losses per Ultimate Claim (Actual)
Accident
Year
1991
1992
1993
1994
1995
1996
1997
0-12
500
547
604
656
719
792
862
12-24
300
335*
357
394
432
470
* 335 = (97,000-60,200) / 110
Slide 13
Months Of Development
24-36
36-48
48-60
182
195
215
236
264
96
111
123
120
54
54
69
.
60-72
72-84
40
40
25
AVERAGE INCREMENTAL PAID METHOD
Trend Factors
Accident
Year
1992/91
1993/92
1994/93
1995/94
1996/95
1997/96
Select
0-12
12-24
9.4%
10.4%
8.6%
9.6%
10.2%
8.8%
9.0%
11.7%
7.1%
6.6%* 10.3%
10.4%
9.8%
9.6% 11.9%
8.8%
* 6.6% = 357 / 335 - 1
Slide 14
Months Of Development
24-36
36-48
48-60
9.0%
9.0%
.
60-72
15.6%
10.8%
- 2.4%
0.0%
27.8%
0.0%
9.0%
9.0%
9.0%
72-84
9.0%
AVERAGE INCREMENTAL PAID METHOD
Incremental Paid Losses per Ultimate Claim (On-level)
Accident
Year
0-12
12-24
1991
1992
1993
1994
1995
1996
1997
Select
914
917
929
926
931
941
940
N/A
503
515*
504
510
513
512
280
275
278
280
288
136
144
146
131
510
280
140
* 515 = 335 x (1.09 ^ 5)
Slide 15
Months Of Development
24-36
36-48
48-60
.
60-72
72-84
70
64
75
48
44
27
70
46
27
AVERAGE INCREMENTAL PAID METHOD
Incremental Paid Losses per Ultimate Claim (Projected)
Accident
Year
1991
1992
1993
1994
1995
1996
1997
0-12
12-24
510
* 83 = 70 x (1.09 ^ 2)
Slide 16
Months Of Development
24-36 36-48 48-60 60-72
280
305
140
152
166
70
76
83*
91
46
50
55
60
65
.
72-84
27
29
32
35
38
42
Total
0
27
75
152
306
613
1,179
AVERAGE INCREMENTAL PAID METHOD
Projected Ultimate Losses
Accident
Year
(1)
Paid
Severity
as of
12/31/97
1991
1992
1993
1994
1995
1996
1997
$1,197
1,282
1,368
1,407
1,415
1,262
862
.
Slide 17
(2)
Projected
Future
Severity
(Slide 16)
$0
27
75
152
306
613
1,179
(3)
Projected
Ultimate
Severity
(1) x (2)
(4)
Projected
Ultimate
Claims
(Slide 5)
(5)
Projected
Ultimate
Losses ($000)
(3) x (4)
$1,197
1,309
1,443
1,559
1,721
1,875
2,041
100
110
125
140
160
185
210
$119.7
144.0
180.4
218.3
275.4
346.9
428.6
AVERAGE INCREMENTAL PAID METHOD
Summary
Step 1: Project Ultimate Claims
Step 2: Calculate Historical Severities
Step 3: Select Trend Factor
Step 4: Calculate On-level Severities
Step 5: Select On-level Severities
Step 6: Project Future Severities
Step 7: Multiply Total Severities by Ultimate Claims
Slide 18
AVERAGE INCREMENTAL PAID METHOD
ADVANTAGES

Allows separate analysis of frequency and severity trends.

Can be modified to account for changes affecting accident year severity (e.g.
deductibles, benefit changes).
.

.
Model can accommodate different trends by accident year, calendar year, or
development age.
DISADVANTAGES

Very dependent upon estimate of future inflation rates.

Less accurate for low frequency lines of business.

Could be distorted if payout patterns change.
Slide 19
BORNHEUTTER-FERGUSON METHOD
Data Needed

Earned Premium or Exposure By Year

A priori Expected Loss Ratio or Pure Premium For Each Year

An Estimate of the Percent of Dollars Unreported, Usually
Based on Loss Development Factors (LDFs)
.
Slide 20
BORNHUETTER-FERGUSON METHOD
“IBNR” RESERVES
.
As of the evaluation date, there are four categories of future claims activity that
may not be reflected in either claim payments or case reserves.
1. Losses Not Yet Reported To The Company
2. Claims In Transit (Reported But Not Recorded)
3. Future Development On Known Open Claims
4. Reopenings on Claims Currently Closed
The Bornhuetter-Ferguson method and most accident year methods estimate
“broad” IBNR which includes all four categories.
NOTE: The sum of (1) and (2) is termed “True” or “Pure” IBNR.
Slide 21
BORNHUETTER-FERGUSON METHOD
Basic Formulas
Expected Losses = Loss Ratio x Earned Premium
= Pure Premium x Exposure
= Frequency x Severity
IBNR Reserve = IBNR Factor x Expected Losses
Slide 22
BORNHUETTER-FERGUSON METHOD
XYZ Auto Insurance Company
Accident Year
1995
1996
1997
.
(1) Earned Premium
$1,250
$1,600
$2,000
65%
70%
75%
$813
$1,120
$1,500
1.350
1.650
2.000
[1 - [1 / (4)]
26%
39%
50%
(3) x (5)
$211
$437
$750
(7) Reported Losses
$600
$700
$1,000
(8) Estimated Ultimate Losses (6) + (7)
$811
$1,137
$1,750
(2) Expected Loss Ratio
(3) Initial Expected Losses
(1) x (2)
(4) Development Factor
(5) IBNR Factor
(6) IBNR Reserve
Slide 23
BORNHUETTER-FERGUSON METHOD
IBNR Factor Derivation
IBNR Factor = IBNR / Ultimate Losses
= Ultimate - Incurred to Date
Ultimate
= 1.000 - (Incurred to Date/Ultimate)
= 1.000 - [1.000/(LDF-to-Ultimate)]
= 1.000 - Percent Unreported
= Percent Unreported
Slide 24
BORNHUETTER-FERGUSON METHOD
XYZ Auto Insurance Company
1995
Accident Year
1996
1997
.
(1) Ultimate Losses
a. Expected (Slide 23, Line 3)
b. Estimated (Slide 23, Line 8)
813
811
1,120
1,137
1,500
1,750
(2) Reported Losses at 12/31/97
a. Expected (Slide 23, Line 3 - Line 6)
b. Actual
602
600
683
700
750
1,000
(3) Loss Ratio
a. Expected (Slide 23, Line 2)
b. Estimated (Slide 23, Line 8/Line 1)
65.0%
64.9%
70.0%
71.1%
Slide 25
75.0%
87.5%
BORNHEUTTER-FERGUSON METHOD
Considerations

Premium Adequacy and Expected Loss Ratios
- Sources: pricing assumptions, historical data such as
Schedule P, industry data
 Changes in Operations:
- Reinsurance
- Longer-Tailed Lines (LDF selection more critical)
- Underlying Limits, Deductibles
- Claims Made versus Occurrence
- Claims Handling
 Changes in Mix of Business That May Impact Either Loss
Ratios, and/or Development Patterns
Slide 26
BORNHUETTER-FERGUSON METHOD
Advantages








Easy to Use
Compromises Between Loss Development and Expected Loss
Ratio Methods
Avoids Overreaction: Doesn’t apply Development Factors to
an Unusual Claim Occurrence
Suitable for New or Volatile Lines of Business
Can be Used with No Internal Loss History
Can also be Used with Paid Data
Disadvantages
Depends on Expected Loss Ratio or A Priori Pure Premium
Requires Development Factors
Slide 27
BORNHUETTER-FERGUSON METHOD
XYZ Auto Insurance Company
Illustration Of Tempering Effect
Expected
EXPECTED LOSS RATIO METHOD
(1) Earned Premium
$2,000
(2) Expected Loss Ratio
75%
(3) Projected Ultimate Losses (1) x (2)
$1,500
LOSS DEVELOPMENT METHOD
(4) Actual Reported to Date
$750
(5) Development Factor
2.00
(6) Projected Ultimate Losses (4) x (5)
$1,500
Slide 28
One Extra
Large Claim
of $150
$2,000
75%
$1,500
$900
2.00
$1,800
BORNHUETTER-FERGUSON METHOD
XYZ Auto Insurance Company
Illustration Of Tempering Effect
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Expected
BORNHUETTER-FERGUSON METHOD
Earned Premium
$2,000
Expected Loss Ratio
75%
Expected Ultimate Losses (1) x (2)
$1,500
Actual Reported to Date
$750
Development Factor
2.00
IBNR Factor 1 - [1 / (5)]
50%
Projected Ultimate Losses
$1,500
(4) + [(3) x (6)]
Slide 29
One Extra
Large Claim
of $150
$2,000
75%
$1,500
$900
2.00
50%
$1,650
BORNHUETTER-FERGUSON METHOD
XYZ Auto Insurance Company
Illustration Of “Tempering” Effect
Expected
One Extra
Large Claim
of $150
(1) Expected Loss Ratio Method
$1,500
$1,500
(2) Loss Development
$1,500
$1,800
(3) Bornhuetter-Ferguson
$1,500
$1,650
Method
Slide 30