Introduction to
Casualty Actuarial Science
November 2005
Ken Fikes, FCAS, MAAA
1
Casualty Actuarial Science
Two major areas are measuring
1. Written Premium Risk
 Pricing
2. Earned Premium Risk
 Reserving
Ken Fikes, FCAS, MAAA
2
Pure Premium Ratemaking
Pure Premium = P = C/E * L/C = L/E
Claims (C)
Car Years (E)
Losses (L)
1,000
10,000
$1,100,000
P = (1,000 / 10,000) * ($1.1M / 1,000) = $110
Ken Fikes, FCAS, MAAA
3
Pure Premium Ratemaking
PF
Rate  R 
1V  Q
Fixed expense per exposure (F)
$10
Variable expense factor (V)
17.5%
Profit and Contingencies factor (Q) 2.5%
R = ($110 + $10) / (1 – 0.175 – 0.025) = $150
Ken Fikes, FCAS, MAAA
4
Trends in the Pure Premium

Severity
» Inflation, Jury Awards, Medical Expenses

Frequency
» Court Decisions, Legal/Social Pressures

Exposure
» Payroll, Auto Values
Ken Fikes, FCAS, MAAA
5
Definitions

What is a Loss Reserve?
Amount necessary to settle unpaid claims

Why are Loss Reserves Important?
Accurate evaluation of financial condition &
underwriting income
Ken Fikes, FCAS, MAAA
6
Definitions

Accounting Aspects of Loss Reserves
Balance Sheet
Assets
Liabilities
Surplus
Ken Fikes, FCAS, MAAA
7
Definitions

Case Reserves
» Claim reported but not yet paid
» Assigned a value by a claims adjuster or by formula

Bulk + IBNR reserves include:
»
»
»
»
Reserves for claims not yet reported (pure IBNR)
Claims in transit
Development on known claims
Reserves for reopened claims
Ken Fikes, FCAS, MAAA
8
Life Cycle of a Claim Reserve
7/11/01
Accident reported
4/2/01
Accident occurs
Claims in Transit
8/1/01
Accident entered
into records as $1,000 Formula
Reserve
Pure IBNR
8/18/02
Settlement agreed
1/1/02
Estimate revised
$30,000 Case
Reserve
$25,000 Case Reserve
$10,000 Case Reserve
8/25/02
Payment sent
9/2/02
Claim draft clears
$30,000 Case Reserve
Closed
Ken Fikes, FCAS, MAAA
10/5/01
Individual reserve
established
9
Other Considerations

Factors Affecting Loss Reserves
» Internal or Operational
– Reinsurance programs
– Claims handling practices
– Business growth
– Case reserve adequacy
– Mix of business
– Underwriting
– Contract changes
– Structured settlements
– Portfolio characteristics
Ken Fikes, FCAS, MAAA
10
Other Considerations

Factors Affecting Loss Reserves
» External or Environmental
– Society
– Regulation
– Judiciary
– Seasonality
– Residual Market
– Inflation
– Economy
Ken Fikes, FCAS, MAAA
11
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).
Ken Fikes, FCAS, MAAA
12
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.
Ken Fikes, FCAS, MAAA
13
Basic Reserving Techniques:
Compilation of Paid Loss Triangle
 The goal is to estimate the total amount that will ultimately be paid
Accident
Year
12
1996
1997
1998
1999
2000
2001
3,780
4,212
4,901
5,708
6,093
6,962
Ken Fikes, FCAS, MAAA
Cumulative Paid Losses ($000 Omitted)
Development Stage in Months
24
36
48
60
6,671
7,541
8,864
10,268
11,172
8,156
9,351
10,987
12,699
9,205
10,639
12,458
9,990
11,536
72
10,508
Final
Total
Cost
???
???
???
???
???
???
14
Basic Reserving Techniques:
Paid Loss Development Factors
Evaluation Interval in Months
Accident
Year
1996
1997
1998
1999
2000
2001
12-24
1.765
1.790
1.809
1.799
1.834
24-36
1.223
1.240
1.240
1.237
36-48
1.129
1.138
1.134
48-60
1.085
1.084
60-72
1.052
72 to
Ultimate
???
Sample Calculation for Accident Year 1997:
12-to-24 Months
1.790
=
7,541 / 4,212
From the end of the accident year (at 12 months) to the end of the following year (at 24 months), paid
losses for 1997 grew 79%. During the next year (from 24 to 36 months), paid losses experienced an
additional 24% growth (or development) and so forth.
Loss Development Factors (LDFs) are also known as:
Age-to-Age factors
Link Ratios
Ken Fikes, FCAS, MAAA
15
Basic Reserving Techniques:
Paid Loss Development Factors
Evaluation Interval in Months
Accident
Year
12-24
24-36
36-48
48-60
60-72
1996
1.765
1.223
1.129
1.085
1.052
1997
1.790
1.240
1.138
1.084
1998
1.809
1.240
1.134
1999
1.799
1.237
2000
1.834
2001
Simple Average - All Years
1.799
1.235
1.134
1.085
1.052
Simple Average - Latest 3 Years
1.814
1.239
1.134
Simple Average - Excluding High & Low
1.799
1.239
1.134
Weighted Average - All Years
1.803
1.235
1.134
Selected Loss Development Factors
1.800
1.235
1.134
Ken Fikes, FCAS, MAAA
XXX
XXX
XXX
XXX
1.085
1.052
1.085
1.052
72 to
Ultimate
???
1.070
16
Basic Reserving Techniques:
Application of Paid LDM
Evaluation Interval in Months
12-24
LDFs
Accident
Year
1996
1997
1998
1999
2000
2001
24-36
1.800
1.235
36-48
1.134
48-60
60-72
1.085
1.052
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
13,517
10,268
12,699
14,401
15,625
11,172
13,797
15,646
16,976
12,532
15,477
17,550
19,042
12
3,780
4,212
4,901
5,708
6,093
6,962
72 to
Ultimate
1.070
72
10,508
12,136
14,220
16,437
17,859
20,032
Final
Total
Cost
11,244
12,985
15,215
17,588
19,109
21,435
Sample Calculations for Accident Year 2001:
At 24 Months:
At 36 Months:
12 to Ult
3.079
Ken Fikes, FCAS, MAAA
12,532 = 6,962 x 1.800
15,477 = 12,532 x 1.235
or 15,477 = 6,962 x 1.800 x 1.235
Cumulative Development Factors
24 to Ult 36 to Ult 48 to Ult 60 to Ult
1.710
1.385
1.221
1.126
72 to Ult
1.070
17
Basic Reserving Techniques:
Paid LDM Projections & Reserves
 Loss
Accident
Year
(1)
Reserve Estimate @ 12/31/01 = $32.241 million
Actual
Paid
Losses
@ 12/31/01
(2)
1996
1997
1998
1999
2000
2001
10,508
11,536
12,458
12,699
11,172
6,962
Total
65,335
Ken Fikes, FCAS, MAAA
Selected
LDFs
(3)
1.070
1.052
1.085
1.134
1.235
1.800
Cumulative
Development
Factors to
Ultimate
(4)
1.070
1.126
1.221
1.385
1.710
3.079
Estimated
Ultimate
Losses
[(2) x (4)]
(5)
Actual
Paid
Losses
@ 12/31/01
(6)
Estimated
Loss
Reserves
{(5) - (6)}
(7)
11,244
12,985
15,215
17,588
19,109
21,435
10,508
11,536
12,458
12,699
11,172
6,962
736
1,449
2,757
4,889
7,937
14,473
97,576
65,335
32,241
18
Basic Reserving Techniques:
Compilation of Incurred Loss Triangle
Accident
Year
1996
1997
1998
1999
2000
2001
Accident
Year
1996
1997
1998
1999
2000
2001
12
5,557
6,328
6,974
7,635
8,376
9,599
12
9,337
10,540
11,875
13,343
14,469
16,561
Case Reserves ($000 Omitted)
Development Stage in Months
24
36
48
60
4,176
2,936
1,987
1,245
4,664
3,200
2,051
1,189
4,968
3,251
1,955
5,274
3,367
5,604
72
742
Cumulative Reported Losses* ($000 Omitted)
Development Stage in Months
24
36
48
60
72
10,847
11,092
11,192
11,235
11,250
12,205
12,551
12,690
12,725
13,832
14,238
14,413
15,542
16,066
16,776
Final
Total
Cost
???
???
???
???
???
???
* = paid losses + case reserves
Ken Fikes, FCAS, MAAA
19
Basic Reserving Techniques:
Selected Incurred LDFs
Evaluation Interval in Months
Accident
Year
12-24
24-36
1996
1.162
1.023
1997
1.158
1.028
1998
1.165
1.029
1999
1.165
1.034
2000
1.159
2001
Simple Average - All Years
1.162
1.029
48-60
1.004
1.003
1.011
1.004
1.001
Simple Average - Latest 3 Years
1.163
1.030
1.011
XXX
XXX
Simple Average - Excluding High & Low
1.162 1.029
1.011
XXX
XXX
Weighted Average - All Years
1.162
1.029
1.011
1.003
1.001
Selected Loss Development Factors
1.162
1.030
1.011
1.003
1.001
Selected Cumulative Development factors to Ultimate
1.215
1.045
1.015
1.004
Ken Fikes, FCAS, MAAA
72 to
60-72 Ultimate
1.001
???
36-48
1.009
1.011
1.012
1.001
1.000
1.000
20
Basic Reserving Techniques:
Incurred LDM Projections & Reserves
 Loss
Reserve Estimate @ 12/31/01 = $27.090 million
Accident
Year
(1)
Ken Fikes, FCAS, MAAA
Actual
Estimated
Actual
Estimated
Reported Development Ultimate
Paid
Loss
Losses
Factors to
Losses
Losses
Reserves
@ 12/31/01
Ultimate
[(2) x (3)] @ 12/31/01 {(4) - (5)}
(2)
(3)
(4)
(5)
(6)
1996
1997
1998
1999
2000
2001
11,250
12,725
14,413
16,066
16,776
16,561
Total
87,791
1.000
1.001
1.004
1.015
1.045
1.215
11,250
12,738
14,471
16,308
17,539
20,119
10,508
11,536
12,458
12,699
11,172
6,962
742
1,202
2,013
3,609
6,367
13,157
92,425
65,335
27,090
21
Key Assumptions &
Potential Problems
Assumptions
Sample Problems
Claims settlement patterns unchanging
Increasing delays in claim closing rates
Case reserving practices & philosophies
unchanging
Conscious effort to improve case reserve adequacy;
Introduction of new case reserving procedures
No claim processing changes
Change in data processing;
Revised claim payment recording procedures
Policy limits have no impact on loss
development
Increasing frequency of full policy limits claims;
Changing policy limits
Ken Fikes, FCAS, MAAA
22
Key Assumptions &
Potential Problems
Assumptions
Sample Problems
Loss development unaffected by changing
loss cost trends
Surges in inflation;
Increased litigation;
Diminished policy defenses
No change in mix of business
Changes in reinsurance coverages;
Increased long-tail exposures;
Introduction of new or revised coverages
No cyclical loss development
Claims settlement or reserving impacted
by business underwriting cycles
No data anomalies
Catastrophic or unusual losses reflected in
loss experience;
Unusual claim settlement/reporting delays
Ken Fikes, FCAS, MAAA
23
Comparison of Estimated
Reserves
Accident
Year
1996
1997
1998
1999
2000
2001
Estimated Loss Reserves Based on:
Paid
Incurred
Average =
LDM
LDM
Selected
Paid Method
Incurred MethodAverage
736
742
739
1,449
1,202
1,326
2,757
2,013
2,385
4,889
3,609
4,249
7,937
6,367
7,152
14,473
13,157
13,815
Total
32,241
27,090
29,666
Components of Selected Reserve
@ 12/31/01
16,000
14,000
12,000
10,000
IBNR
Case
8,000
6,000
4,000
2,000
20
01
20
00
19
99
19
98
19
97
19
96
-
Accident Year
Ken Fikes, FCAS, MAAA
24
Example
You are given the following losses evaluated at 12/31/2004. Use the paid loss
development method to estimate the required reserves by accident year.
Assume all losses are fully developed at 60 months.
Accident
Year
2000
2001
2002
2003
2004
Ken Fikes, FCAS, MAAA
12
3,000
3,200
3,500
3,800
5,000
Cumulative Paid Losses ($000 Omitted)
Development Stage in Months
24
36
48
6,000
6,400
7,000
7,600
9,000
9,600
10,500
10,800
11,520
60
11,340
25
Solution
12-24
2.000
Age-to-Age Development Factors
24-36
36-48
48-60
1.500
1.200
1.050
60-Ult
1.000
12 to Ult
3.780
Cumulative Development Factors
24 to Ult
36 to Ult
48 to Ult
1.890
1.260
1.050
60 to Ult
1.000
Accident
Year
(1)
Paid
Losses
@ 12/04
2000
2001
2002
2003
2004
Ken Fikes, FCAS, MAAA
11,340
11,520
10,500
7,600
5,000
(2)
Dev
Factors
to Ult
1.000
1.050
1.260
1.890
3.780
(3)=(1)*(2)
Estimated
Ultimate
Losses
11,340
12,096
13,230
14,364
18,900
(4)=(3)-(1)
Estimated
Loss
Reserve
576
2,730
6,764
13,900
26
Further Reading
For additional information on Loss Reserving, see the
following references at
www.casact.org/admissions/syllabus/2006/exam6.htm
Wiser, et al., “Loss Reserving,” Foundations of Casualty Actuarial Science (Fourth Edition),
Casualty Actuarial Society, 2001, Chapter 5, pp. 197-285.
Bornhuetter, R.L; and Ferguson, R.E., “The Actuary and IBNR,” PCAS LIX, 1972, pp. 181195. Including discussions of paper: Cooper, W.P., PCAS LX, 1973, pp. 161-164; and
White, H.G., PCAS LX 1973, pp. 165-168.
Brosius, E., “Loss Development Using Credibility,” CAS Study Note, March 1993.
Ken Fikes, FCAS, MAAA
27