Transcript basic methods – baseball example

METODOLOGÍAS Y PRÁCTICAS
EN RESERVAS TÉCNICAS
PARA SEGUROS DE SALUD
Y SEGUROS GENERALES
■ Basic Methods for Evaluating Reserves ■
LIMA – 31 DE MAYO, 2007
APESEG
Presentado por: APESEG & Milliman, Inc.
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LOSS RESERVING METHODS
 In practice, many methods estimate “Ultimate
Losses”
 Loss Reserves are Obtained By Subtraction
Estimated
Ultimate
Losses
Total
Reserve
-
Paid to
date
Losses
=
Total
Reserve
-
Case
Reserve
=
IBNR
Reserve
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LOSS RESERVING METHODS
 Expected Loss (Loss Ratio)
 Loss Development (Chain Ladder)
─Tail Factors
─Inflation Adjustments
 Bornhuetter-Ferguson
 Generalized Cape Cod
 Many, many others available
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BASIC METHODS – BASEBALL EXAMPLE
 Given the following information, how many homeruns will
Sammy Sosa finish with in the 2007 season?
─ 2004 total homeruns: 40
─ 2005 total homeruns: 35
─ 2006 total homeruns: 45
─ 2007: 20 homeruns through 40 games of the 160
game season
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
Basic Methods Comparison
Baseball Example
Loss Reserving
(1)
Expected Value
Expected Loss (Loss Ratio)
(2) Development Projection Loss Development (Chain Ladder)
(3) Bornhuetter-Ferguson
Bornhuetter-Ferguson
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
 Based on the prior 3 years, the Expected Value would be
40 homeruns.
Expected Value Method
(1)
(2)
Year
Homeruns
2004
2005
2006
40
35
45
Average
40
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
Development Projection Method
(1) 40 games played out of 160 is 1/4 of the season.
(2) To project season total from current statistics,
need to multiply current statistics by a
Development Projection Factor.
(3) Development Projection Factor = 4.000
(4) He has already hit 20 homeruns through 40 games.
(5) 20 * 4.000 = 80 Homeruns
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
Basic Methods Comparison
(1)
(2)
(3)
Method
Season Homeruns Estimate
Expected Value
Development Projection
Bornhuetter-Ferguson
40
80
???
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
 What is the Bornhuetter-Ferguson (B-F) Method?
─ Named after paper written by Ron Bornhuetter and
Ron Ferguson published in the Casualty Actuarial
Society Proceedings 1972.
─ Compromise between the Expected Loss and Loss
Development Methods.
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the
2007 season?
 What data is needed for the B-F Method?
─ Expected Ultimate Value (40)
─ Factor to Project Actual Data to Ultimate (4.000)
─ Actual Data to Date (20)
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the 2007
season?
B-F Projection: Ultimate Value =
(Expected Value*IBNR Factor)+(Inc. to Date)
 IBNR Factor = 1.000 - (1.000/LDF) = 1.000 - (1.000/4.000) = .75
(In Other Words, 75% of the season is left to be played)
 Ultimate Value = (40 * .75) + 20 = 50
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the 2007
season?
Expected Value Method
Games 0-40
Games 41-80
Games 81-120
Games 121-160
10 Home Runs
10 Home Runs
10 Home Runs
10 Home Runs
Development Projection Method
Games 0-40
Games 41-80
Games 81-120
Games 121-160
20 Home Runs
20 Home Runs
20 Home Runs
20 Home Runs
Bornhuetter-Ferguson Method
Games 0-40
Games 41-80
Games 81-120
Games 121-160
20 Home Runs
10 Home Runs
10 Home Runs
10 Home Runs
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BASIC METHODS – BASEBALL EXAMPLE
 How many homeruns will Sammy Sosa finish with in the 2007
season?
Basic Methods Comparison
(1)
(2)
(3)
Method
Season Homeruns Estimate
Expected Value
Development Projection
Bornhuetter-Ferguson
40
80
50
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BASIC METHODS – BASEBALL EXAMPLE
 Comparison of Methods
90
Home Runs IBNR
80
Home Runs Incurred
70
60
50
40
30
20
10
0
ELR
B-F
LDF
If Actual 10
After 40
Games
ELR
B-F
LDF
If Actual 20
After 40
Games
ELR
B-F
LDF
If Actual 5
After 40
Games
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LOSS RESERVING METHODS
 Each Method is Based on Idealized Model of Reality
 Models have Implicit Assumptions
 Such Assumptions are Rarely Satisfied
─ No single method or “Formula” will work
►Use multiple methods
►Use judgment in selecting values
─ Actuary to test Assumptions
►Review Diagnostics
►Possibly Use Additional Methods to “Adjust” Data
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LOSS RATIO METHODS
Earned
Premium
x
Expected
Loss Ratio
E(LR)
=
Ultimate
Losses
►Problem. I know the earned premium, but how do I
estimate the Expected Loss Ratio?
─ Use Pricing Assumptions
─ Use Available Industry Information
►Useful when Company does not have relevant
information to analyze
─ New Business
─ Small Volume
─ Late Reporting of Losses (e.g., high excess of
loss).
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EXPECTED LOSS RATIO METHOD
Commissions
Taxes
General Expenses
Profit
Total
Percent of
Premium
20.0%
5.0%
12.0%
-2.0%
35.0%
Expected Loss Ratio
65.0%
(Available for Loss and Loss Adjustment Expense)
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EXPECTED LOSS METHOD EXAMPLE
Expected Loss Ratio Method
(1)
(2)
(3)
Accident
Year
Earned
Premium
Loss
Ratio
2,104,744
2,095,230
2,869,303
2,897,637
3,251,632
0.65
0.65
0.65
0.65
0.65
2002
2003
2004
2005
2006
Totals
13,218,545
(4)
Estimated
Ultimate
Loss
(2)X(3)
1,368,084
1,361,899
1,865,047
1,883,464
2,113,561
8,592,054
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DEVELOPMENT METHOD
 Also Called the Chain Ladder Method and the
Completion Factor Method
 Data Triangles
─ Paid Loss
─ Incurred Loss
─ Claim Counts
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TRIANGLE EXAMPLE
20
TRIANGLE EXAMPLE
Cumulative Paid Losses
Accident
Year
2002
2003
2004
2005
2006
12
235,200
253,024
262,202
288,075
361,188
Months from Beginning of Year
24
36
48
60
825,442 1,095,966 1,203,738 1,259,943
869,487 1,282,677 1,462,804
897,625 1,332,810
1,006,209
Diagonals represent a common
valuation date. Latest Diagonal is
as of 12/31/2006.
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LOSS DEVELOPMENT METHOD EXAMPLE
Cumulative Paid Losses
Year
2002
2003
2004
2005
2006
Year
2002
2003
2004
2005
2006
Average:
Column Average:
Selected LDF:
12
24
36
48
60
235,200 825,442 1,095,966 1,203,738 1,259,943
253,024 869,487 1,282,677 1,462,804
262,202 897,625 1,332,810
288,075 1,006,209
1,332,810
1.485 
361,188
897, 625
12-24
3.510
3.436
3.423
3.493
24-36
1.328
1.475
1.485
36-48
1.098
1.140
48-60 60-Ultimate
1.047
3.466
3.465
3.465
1.429
1.432
1.430
1.119
1.121
1.120
1.047
1.047
1.047
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“TAIL” FACTORS
 Previous Example Showed Development to 60
Months
 Suppose Process is Not Complete
─ Curve Fits
─ Industry Information
─ Paid to Incurred Ratios
─ Judgment
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LOSS DEVELOPMENT METHOD EXAMPLE
Cumulative Paid Losses
Accident
Year
2002
2003
2004
2005
2006
Accident
Year
2002
2003
2004
2005
2006
12
24
36
48
235,200 825,442 1,095,966 1,203,738
253,024 869,487 1,282,677 1,462,804
262,202 897,625 1,332,810
288,075 1,006,209
361,188
60
1,259,943
12-24
3.510
3.436
3.423
3.493
24-36
1.328
1.475
1.485
36-48
1.098
1.140
48-60 60-Ultimate
1.047
Average:
Column Average:
Selected LDF:
Cumulative LDF:
3.466
3.465
3.465
6.262
1.429
1.432
1.430
1.807
1.119
1.121
1.120
1.263
1.047
1.047
1.047
1.128
1.077
Percent of Ultimate:
16.0%
55.3%
79.2%
88.7%
92.8%
Tail Factor
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LOSS DEVELOPMENT METHOD EXAMPLE
Paid Loss Development Method
(1)
Accident
Year
2002
2003
2004
2005
2006
Totals
(2)
Cumulative
Paid
Losses
1,259,943
1,462,804
1,332,810
1,006,209
361,188
5,422,953
(3)
Cumulative
Paid
LDF
1.077
1.128
1.263
1.807
6.262
(4)
Estimated
Ultimate
Loss
(2)X(3)
(5)
Reserve
(4)-(2)
1,357,354
97,411
1,649,481
186,677
1,683,559
350,750
1,818,073
811,864
2,261,597 1,900,410
8,770,065 3,347,112
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IMPORTANCE OF “TAIL” FACTOR
No Tail Estimate
(1)
(2)
(3)
(4)
Cumulative Cumulative Estimated
Accident
Paid
Paid
Ultimate
Year
Losses
LDF
Loss
(2)X(3)
2002
2003
2004
2005
2006
Totals
1,259,943
1,462,804
1,332,810
1,006,209
361,188
5,422,953
1.000
1.047
1.173
1.678
5.814
(5)
Reserve
(4)-(2)
1,260,310
367
1,531,551
68,748
1,563,193 230,384
1,688,090 681,881
2,099,905 1,738,717
8,143,050 2,720,097
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IMPORTANCE OF “TAIL” FACTOR
No Tail Reserve: 2,720,097
Reserve with Tail: 3,347,112
Error:
Error %:
627,015
23%
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PAYMENT/REPORTING PATTERNS
Payment and Reporting Patterns
120%
100%
80%
60%
Paid Loss
Incurred Loss
40%
20%
0%
12
24
36
48
60
Age in Months
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EXAMPLE PAYMENT PATTERNS
Auto Physical Damage
Auto Liability
90%
90%
70%
70%
50%
50%
30%
30%
10%
10%
-10%
1
5
10
15
-10%
1
Workers Compensation
90%
70%
70%
50%
50%
30%
30%
10%
10%
1
5
10
10
15
10
15
General Liability
90%
-10%
5
15
-10%
1
5
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EMERGENCE AND SETTLEMENT PATTERNS
Products
Liability
Workers
Compensation
Automobile
Bodily Injury
Reporting Delay
Settlement Delay
Auto Physical
Damage
0
2
4
6
8
10
12
14
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BRINGING RESULTS TOGETHER
 Make Tentative Selections
─ Loss Development Factors
─ Tail Factors
─ Expected Loss Ratios
─ Generalized Cape Cod Trend and Decay
─ Weights for Various Methods
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BRINGING RESULTS TOGETHER
Selected Ultimate Loss
(1)
(2)
Expected
Accident
Loss
Year
Method
2002
2003
2004
2005
2006
Totals
1,368,084
1,361,899
1,865,047
1,883,464
2,113,561
(3)
(4)
(5)
Paid Loss Methods
Dev.
BF
Cape Cod
Method
Method
Method
1,357,354
1,649,481
1,683,559
1,818,073
2,261,597
1,358,124
1,616,934
1,721,370
1,847,274
2,137,203
1,354,967
1,618,822
1,731,302
1,899,398
2,337,749
(6)
(7)
(8)
(9)
Incurred Loss Methods
Selected
Dev.
BF
Cape Cod Ultimate
Method
Method
Method
Loss
1,449,483
1,752,102
1,628,666
1,857,183
2,366,689
1,368,084
1,361,899
1,865,047
1,883,464
2,113,561
1,449,483
1,739,344
1,664,483
1,861,305
2,230,682
1,389,582
1,623,097
1,715,738
1,861,116
2,241,247
8,592,054 8,770,065 8,680,905 8,942,237 9,054,123 8,592,054 8,945,297 8,830,780
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BRINGING RESULTS TOGETHER
Indicated Reserve
(1)
(2)
(3)
(4)
Cumulative Cumulative Selected
Accident
Paid
Incurred
Ultimate
Year
Losses
Losses
Losses
2002
2003
2004
2005
2006
Totals
1,259,943
1,462,804
1,332,810
1,006,209
361,188
1,401,236
1,669,478
1,448,951
1,520,611
1,166,209
1,389,582
1,623,097
1,715,738
1,861,116
2,241,247
(5)
(6)
(7)
Case
Reserve
(3)-(2)
IBNR
Reserve
(4)-(3)
Total
Reserve
(5)+(6)
141,293
-11,654 129,639
206,674
-46,381 160,293
116,142 266,787 382,928
514,402 340,505 854,907
805,021 1,075,038 1,880,059
5,422,953 7,206,485 8,830,780 1,783,531 1,624,296 3,407,827
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BRINGING RESULTS TOGETHER
 Check Ultimate Losses for Reasonableness by Reviewing
additional Diagnostics
─ Ultimate Loss Ratios
─ Frequency and Severity
─ Percent of Total Loss by Year
►Paid, Case Reserve, IBNR
─ Ratios of Ceded Reserves to Total Reserves
─ Average Hindsight Outstanding Cost Per Open Claim
─ Many Others
 Sensitivity Test
─ Reasonable Ranges for Key Selections
34
BRINGING RESULTS TOGETHER
Ultimate Severity
2,900
2,813
2,700
2,634
2,500
2,300
2,441
2,143
2,122
2,100
1,900
1,700
1,500
2002
2003
2004
2005
2006
35
BRINGING RESULTS TOGETHER
Frequency per 1,000 Exposure Units
300
251
250
208
200
172
150
181
185
100
50
0
2002
2003
2004
2005
2006
36
BRINGING RESULTS TOGETHER
Ratio of Reserves to Premium
0.700
0.600
IBNR
0.500
Total
Reserve
0.400
0.300
0.295
0.331
0.200
0.100
0.000
-0.100
-0.006
2002
0.093
0.118
-0.022
2003
2004
2005
2006
37
BRINGING RESULTS TOGETHER
Losses By Type
2,500,000
IBNR Reserve
Case Reserve
2,000,000
Paid
1,500,000
1,000,000
500,000
0
2002
2003
2004
2005
2006
38
BASIC METHODS REVIEW
 Basic Methods
─ Expected Loss
─ Loss Development
─ Bornhuetter-Ferguson
─ Cape Cod
 Reasonability and Sensitivity of Estimates
─ Tail Factor Selection
─ Review of resulting Ultimate Loss Ratios, Claim
Severities and Frequencies, and other Diagnostics
─ Sensitivity of Results to Key Assumptions
 Loss Adjustment Expenses
 Documentation
39