RESERVING TECHNIQUES
By Lorie Darrow
Select Actuarial
1
Topics
Fundamentals
Situational Reserving
• Internal/External Forces
• Business Context/Pricing
The opinions expressed in this document are solely those of the presenter.
2
Fundamentals of Loss Reserving
• Definitions/Terminology
• Basic Reserving Techniques
– Paid Loss Development Method (PLDM)
– Incurred Loss Development Method (ILDM)
– Expected Loss Ratio (ELR)
– Bornhuetter-Ferguson (BF)
3
Pure
IBNR
IBNR
Development
of the Known
Outstanding
Incurred
Paid
Ultimate
Definitions
4
Basic Reserving Techniques
Definitions
Loss Development
The financial activity on claims from the time they occur to
the time they are eventually settled and paid
Triangles
Compiled to measure the changes in cumulative claim
activity over time in order to estimate patterns of future
activity
Loss Development Factor
The ratio of losses at successive evaluations for a defined
group of claims (e.g. accident year)
5
Examples of Reserving Issues
•Immature Years
•Claim Department Changes
•Environmental Changes
•Unique Exposures
6
Basic Reserving Techniques
Compilation of Paid Loss Triangle
The losses are sorted by the year in which the accident
occurred
The losses are summed at the end of each year
Losses paid to date are shown on the most recent diagonal
The data is organized in this way to highlight historical
patterns
7
Basic Reserving Techniques
Calendar Year Paid Losses by Accident Year
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
Total
12
3,780
3,780
($000 Omitted)
Development Stages in Months
24
36
48
60
2,891
1,485
1,049
785
4,212
3,329
1,810
1,288
4,901
3,963
2,123
5,708
4,560
6,093
7,103
9,715
12,530
14,849
72
518
897
1,471
2,431
5,079
6,962
17,358
8
Basic Reserving Techniques
Compilation of Paid Loss Triangle
The goal is to estimate the total amount that will
ultimately be paid
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12
3,780
4,212
4,901
5,708
6,093
6,962
Cumulative Paid Losses ($000 Omitted)
Development Stage in Months
24
36
48
60
6,671
8,156
9,205
9,990
7,541
9,351
10,639
11,536
8,864
10,987
12,458
10,268
12,699
11,172
Ultimate
Total
72
Cost
10,508
???
???
???
???
???
???
9
Basic Reserving Techniques:
Paid Loss Development Factors
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12-24
1.765
1.790
1.809
1.799
1.834
Evaluation Interval in Months
24-36
36-48
48-60
60-72
1.223
1.129
1.085
1.052
1.240
1.138
1.084
1.240
1.134
1.237
72 to Ultimate
Sample Calculation for Accident Year 2:
12-to-24 Months
1.240
9,351 / 7,541
Loss Development Factors (LDFs) are also known as:
Age-to-Age factors
Link Ratios
10
Basic Reserving Techniques:
Paid Loss Development Factors
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12-24
1.765
1.790
1.809
1.799
1.834
24-36
1.223
1.240
1.240
1.237
Evaluation Interval in Months
36-48
48-60
1.129
1.085
1.138
1.084
1.134
60-72
1.052
72 to Ultimate
Simple Average - All Years
1.800
1.235
1.134
1.085
1.052
1.134
xxx
xxx
1.239
1.134
xxx
xxx
1.235
1.134
1.085
1.052
1.134
1.085
1.052
Simple Average - Latest 3 Years
1.814
1.239
Simple Average - Excluding High & Low
1.800
Weighted Average - All Years
1.803
Selected Loss Development Factors
1.800
1.235
1.070
11
Basic Reserving Techniques:
Paid Loss Development Factors
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12-24
1.765
1.790
1.809
1.799
1.834
1.800
24-36
1.223
1.240
1.240
1.237
1.235
1.235
Evaluation Interval in Months
36-48
48-60
1.129
1.085
1.138
1.084
1.134
1.085
1.134
1.085
1.134
1.085
1.134
1.085
60-72
1.052
1.052
1.052
1.052
1.052
1.052
72 to Ultimate
1.070
1.070
1.070
1.070
1.070
1.070
Simple Average - All Years
1.800
1.235
1.134
1.085
1.052
1.134
xxx
xxx
1.239
1.134
xxx
xxx
1.235
1.134
1.085
1.052
1.134
1.085
1.052
Simple Average - Latest 3 Years
1.814
1.239
Simple Average - Excluding High & Low
1.800
Weighted Average - All Years
1.803
Selected Loss Development Factors
1.800
1.235
1.070
12
Basic Reserving Techniques:
Application of Paid LDM
Evaluation Interval in Months
12-24
LDFs
1.800
24-36
36-48
1.235
1.134
48-60
1.085
Cumulative Paid Losses ($000 Omitted)
Development Stage in Months
24
36
48
6,671
8,156
9,205
60-72
1.052
72 to
Ultimate
1.070
60
9,990
72
10,508
Ultimate
Total
Cost
11,244
10,639
11,536
12,136
12,985
10,987
12,458
13,517
14,220
15,215
10,268
12,699
14,401
15,625
16,437
17,588
11,172
13,797
15,646
16,976
17,859
19,109
12,532
15,477
17,550
19,042
20,032
21,435
Accident
Year
Yr 1
12
3,780
Yr 2
4,212
7,541
9,351
Yr 3
4,901
8,864
Yr 4
5,708
Yr 5
6,093
Yr 6
6,962
Sample Calculations for Accident Year 2003:
At 24 Months:
At 36 Months:
or
12 to Ult
3.079
24 to Ult
12,532 = 6,962 x 1.800
15,477 = 12,532 x 1.235
15,477 = 6,962 x 1.800 x 1.235
Cumulative Development Factors
36 to Ult
48 to Ult
1.710
1.385
1.221
60 to Ult
1.126
72 to Ult
1.070
13
Basic Reserving Techniques:
Paid LDM Projections & Reserves
 Loss
Reserve Estimate @ Yr 6 = $32.241 million
Actual
Accident
Year
(1)
Yr 1
Paid
Losses
Selected
Cumulative
Development
Factors to
@ Yr 6
(2)
LDFs
(3)
Ultimate
(4)
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
10,508
11,536
12,458
12,699
11,172
6,962
Total
65,335
1.070
1.052
1.085
1.134
1.235
1.800
1.070
1.126
1.221
1.385
1.710
3.079
Estimated
Estimated
Ultimate
Losses
Loss
Reserves
[(2) x (4)]
(5)
[(5) - (2)]
(6)
11,244
12,985
15,215
17,588
19,109
21,435
736
1,449
2,757
4,889
7,937
14,473
97,576
32,241
14
Selection of Tail Factors
How much difference does the tail factor selection make?
Effect on Estimates Given a 2% Increase in Paid Losses Tail Factor
Paid
Accident
Year
Losses
@ Yr 6
Estimated
Selected LDF's
LDF
Age to Ult.
Unpaid
Ultimate
Earned
Losses
Premium
Losses
@ Yr 6
Yr 1
10,508
1.070
1.090
11,454
18,168
Yr 2
11,536
1.052
1.147
13,232
21,995
1,696
Yr 3
12,458
1.085
1.244
15,498
24,173
3,040
Yr 4
12,699
1.134
1.411
17,918
25,534
5,219
Yr 5
11,172
1.235
1.743
19,473
31,341
8,301
Yr 6
6,962
1.800
3.137
21,840
38,469
14,878
99,415
159,680
34,080
Total
65,335
Estimated Unpaid Losses Based on Original PLDM
946
32,241
(Without the 2% Tail Factor Increase)
Increase in Estimated Unpaid Losses Due to Increased Tail Factor
6%
15
Basic Reserving Techniques:
Compilation of Incurred Loss Triangle
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12
5,657
6,428
7,074
7,635
8,376
9,599
12
9,437
10,640
11,975
13,343
14,469
16,561
Case Reserves ($000 Omitted)
Development Stage in Months
24
36
48
60
4,176
2,936
1,987
1,145
4,664
3,300
2,051
1,189
4,968
3,251
1,955
5,174
3,367
5,604
Cumulative Reported Losses* ($000 Omitted)
Development Stage in Months
24
36
48
60
10,847
11,092
11,192
11,135
12,205
12,651
12,690
12,725
13,832
14,238
14,413
15,442
16,066
16,776
72
742
72
11,250
Ultimate
Total
Cost
???
???
???
???
???
???
* = paid losses + case reserves
16
Basic Reserving Techniques:
Selected Incurred LDFs
Accident
Year
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
Yr 6
12-24
1.149
1.147
1.155
1.157
1.159
24-36
1.023
1.037
1.029
1.040
Evaluation Interval in Months
36-48
48-60
1.009
0.995
1.003
1.003
1.012
60-72
1.010
72 to Ultimate
Simple Average - All Years
1.153
1.032
1.008
0.999
1.010
1.008
xxx
xxx
1.033
1.009
xxx
xxx
1.033
1.008
0.999
1.010
1.008
1.000
1.000
1.000
1.000
1.000
Simple Average - Latest 3 Years
1.157
1.035
Simple Average - Excluding High & Low
1.154
Weighted Average - All Years
1.154
Selected Loss Development Factors
1.154
1.035
Selected Cumulative Development Factors to Ultimate
1.204
1.043
1.008
1.000
17
Basic Reserving Techniques:
Incurred LDM Projections & Reserves
Actual
Accident
Year
(1)
Estimated
Actual
Estimated
Reported
Development
Ultimate
Paid
Loss
Losses
Factors to
Losses
Losses
Reserves
Ultimate
[(2) x (3)]
(3)
(4)
(5)
@ Yr 6
(2)
@ Yr 6
[(4) - (5)]
(6)
Yr 1
11,250
1.000
11,250
10,508
742
Yr 2
12,725
1.000
12,725
11,536
1,189
Yr 3
14,413
1.000
14,413
12,458
1,955
Yr 4
16,066
1.008
16,195
12,699
3,496
Yr 5
16,776
1.043
17,502
11,172
6,330
Yr 6
16,561
1.204
19,939
6,962
12,977
Total
87,791
92,023
65,335
26,688
18
Basic Reserving Techniques:
Issues to Consider for LDM
Issues to Consider
Examples
Have there been any changes
which might make the older
years irrelevant?
There are more motorcycle
losses in the oldest years; Typical
P&C no longer insures
motorcycles.
Are the more recent years
better predictors of the
future?
Typical P&C has begun writing
more business in state X.
Are there outlier points that
need to be ignored or
adjusted?
In one year, there were bad ice
storms at the end of December.
late reporting caused unusually
high development in the next
year
19
Comparison of Loss Development Methods
Underlying Assumptions
•PLDM: No changes in the payment pattern
•ILDM: No changes in case reserve adequacy
Advantages
Disadvantages
PLDM: “Hard” data; no
estimates involved
PLDM: May generate large,
volatile loss development factors &
take longer to develop to ultimate
ILDM: Uses all available
information
ILDM: Uses case reserves, which
are estimates, to develop
estimates of ultimate losses
20
Expected Loss Ratio Method
Expected Loss Ratio (ELR)
The anticipated ratio of projected ultimate losses to
earned premiums
Sources:
•Pricing assumptions
•Historical data such as Schedule P
•Industry data
21
Expected Loss Ratio Method
Use when you have no history such as:
New product lines
Radical changes in product lines
Immature accident years for long tailed lines
Can generate negative reserves or negative IBNR if
Ultimate Losses<Paid Losses-MOST LIKELY
ILLOGICAL!!!
Ultimate Losses<Incurred Losses
22
Expected Loss Ratio
Earned Expected
Accident Premium Loss Ratio
Year
(1)
(2)
Yr 1
18,000
60.0%
Yr 2
20,000
62.0%
Yr 3
22,000
64.0%
Yr 4
24,000
66.0%
Yr 5
26,000
68.0%
Yr 6
28,000
70.0%
Total
138,000
Ultimate
[(1)x(2)]
(4)
10,800
12,400
14,080
15,840
17,680
19,600
90,400
23
Expected Loss Ratio
Advantages
Disadvantages
Straight forward
Not pure
Relies on pricing
Relies on pricing
24
Bornhuetter-Ferguson Method
Reserves Base on ELR and Actual Loss
(EP x ELR) x (IBNR Factor)
= (IBNR Reserves)
Where IBNR Factor
= (1.000-1.000/CDF)
Actual + IBNR Reserve
= Ultimate Losses
Case Reserves + IBNR Reserve
= Total Reserve
The IBNR Factor is the percent of the expected losses
unreported.
25
Paid Bornhuetter-Ferguson
Earned
Accident Premium
Year
(1)
Yr 1
18,000
Yr 2
20,000
Yr 3
22,000
Yr 4
24,000
Yr 5
26,000
Yr 6
28,000
Total
138,000
ELR
(2)
60.0%
62.0%
64.0%
66.0%
68.0%
70.0%
Ultimate
[(1) x (2)] = (3)
10,800
12,400
14,080
15,840
17,680
19,600
90,400
1/LDF
(4)
93%
89%
82%
72%
58%
32%
Expected
1-(1/LDF)
[(3) x (4)] = (5)
(6)
10,093
6.5%
11,016
11.2%
11,529
18.1%
11,437
27.8%
10,336
41.5%
6,366
67.5%
60,778
Reserve
[(6) x (3)] = (7)
707
1,384
2,551
4,403
7,344
13,234
29,622
Ultimate
Actual
Total Cost
(8)
[(8) + (7)] = (10)
10,508
11,215
11,536
12,920
12,458
15,009
12,699
17,102
11,172
18,516
6,962
20,196
65,335
94,957
26
Incurred Bornhuetter-Ferguson
Earned
Accident Premium
Year
(1)
Yr 1
18,000
Yr 2
20,000
Yr 3
22,000
Yr 4
24,000
Yr 5
26,000
Yr 6
28,000
Total
138,000
ELR
(2)
60.0%
62.0%
64.0%
66.0%
68.0%
70.0%
Ultimate
[(1) x (2)] = (3)
10,800
12,400
14,080
15,840
17,680
19,600
90,400
LDF
(4)
100.0%
100.0%
100.0%
99.2%
95.9%
83.1%
Expected
[(3) x (4)] = (5)
10,800
12,400
14,080
15,714
16,947
16,280
86,221
1-LDF
(6)
0.0%
0.0%
0.0%
0.8%
4.1%
16.9%
Reserve
[(6) x (3)] = (7)
126
733
3,320
4,179
Ultimate
Actual
Total Cost
(8)
[(8) + (7)] = (10)
11,250
11,250
12,725
12,725
14,413
14,413
16,066
16,192
16,776
17,509
16,561
19,881
87,791
91,970
27
Bornhuetter-Ferguson Method
Advantages
Disadvantages
Compromise between loss
development and expected loss
ratio methods
Assumes that case
development is unrelated to
reported losses
Avoids overreaction to unexpected
incurred losses to date
Relies on accuracy of expected
loss ratio
Suitable for new or volatile line of
business
Less responsive to losses incurred
to date
Can be used with no internal loss
history
Relies on accuracy of earned premium
Easy to use
28
Comparison of all Methods
Accident Earned
Year
Premium
Yr 1
18,000
Yr 2
20,000
Yr 3
22,000
Yr 4
24,000
Yr 5
26,000
Yr 6
28,000
Total
138,000
Paid
LDF
11,244
12,985
15,215
17,588
19,109
21,435
97,576
Incurred
LDF
11,250
12,725
14,413
16,195
17,502
19,939
92,023
ELR
10,800
12,400
14,080
15,840
17,680
19,600
90,400
B-F
Paid
11,215
12,920
15,009
17,102
18,516
20,196
94,957
B-F
Incurred
11,250
12,725
14,413
16,192
17,509
19,881
91,970
Min
10,800
12,400
14,080
15,840
17,502
19,600
90,222
Max
Selection
11,250
11,247
12,985
12,855
15,215
14,711
17,588
16,647
19,109
17,680
21,435
19,600
97,582
92,740
29
Comparison of all Methods
Comparison of Methods
22,000
20,000
Ultimate
18,000
Paid LDF
Incurred LDF
16,000
ELR
Paid B-F
14,000
Incurred B-F
12,000
10,000
Yr 1
Yr 2
Yr 3
Yr 4
Accident Year
Yr 5
Yr 6
30
Situation
Claim Department Changes
•Paid to Incurred relationship changes – Munich Chain
•Reserve strengthening or weakening - Berquist - Sherman
31
Situation
Exposure Change
•Frequency and severity
•Rate per unit of exposure
•Mergers - mix change (state, line, segment)
•Contractors - seasonality (summer vs. winter)
32
Situation
Environment
•Small claims go away - type of claim (mix shift)
•Frequency
Severity
will change
the development pattern
33
Unique Exposures : Methodology in estimating
Unique liabilities
•Asbestos – Science/no claims
-Ground up analysis
•Credit Crunch – D&O and E&O
•Construction Defect
34
35