Transcript Lecture Notes 5 - University of Illinois at Urbana

Math 479
Casualty Actuarial Mathematics
Fall 2014
University of Illinois at Urbana-Champaign
Professor Rick Gorvett
Session 5: Loss Reserving II
September 9, 2014
1
Agenda
• Review of basic loss development technique
and essential metrics / quantities
• Other loss reserving methods
– Expected loss ratio
– Bornhuetter-Ferguson
• Diagnostics
• Reserving issues
2
Review of LDF Technique
• General approach of loss development techniques
– Loss development triangle (cumulative data)
– Age-to-age factors (link ratios)
• Approaches to selecting factors – e.g.,
–
–
–
–
Straight average
Average of last three years
(Middle) 3 of 5 years
Weighted average (by dollars)
– To-ultimate loss development factors
– For each (accident) year: (actual loss) × (LDF) =
estimated ultimate loss
– Reserve = (ultimate loss) – (to-date paid loss)
– IBNR (broad def’n.) = (ult. loss) – (case incurred
loss)
3
Review of LDF Technique (cont.)
• Possible bases:
– Accident Year (AY)
– Policy Year (PY)
– Report Year (RY)
• Possible types of data:
–
–
–
–
–
–
Paid losses (with or without (A)LAE)
Case incurred losses (with or without (A)LAE)
Closed claim counts
Reported / incurred claim counts
Paid loss severity
Incurred loss severity
4
Review of LDF Technique (cont.)
• Data relationships
– Cumulative incurred loss = cumulative paid loss +
“snapshot” case reserve
– Cumulative reported / incurred claim count =
cumulative paid claim count + “snapshot” open
claim count
– Paid severity = cumulative paid loss ÷
cumulative paid claim count
– Reported / incurred severity = cumulative incurred
loss ÷ cumulative reported claim count
5
Metrics and Quantities of Interest
• Paid loss + case reserve = (case) incurred loss
• Ultimate AY or PY loss = paid loss + case
reserve + IBNR
• IBNR (Incurred But Not Reported) loss =
ultimate loss – (case) incurred loss
• Total loss reserve = ultimate loss – paid loss
• Total loss reserve = case reserve + IBNR
• Loss = frequency × average severity
6
Other Basic Loss Reserving Techniques
• Expected loss ratio (ELR) method
– Loss ratio = loss ÷ premium
– Ultimate loss = premium × ELR
– ELR per best estimate (e.g., original loss provision
in premium per ratemaking process)
• Bornhuetter-Ferguson method
– A “middle-ground” between the LDF and ELR
methods
7
Expected Loss Ratio (ELR) Method
Accident
Year
2004
2005
2006
2007
2008
Total
Earned
Premium
500
900
1,200
1,500
1,500
=====
5,600
Projected
Ultimate
ELR
Loss
0.65
0.65
0.65
0.65
0.65
325
585
780
975
975
=====
3,640
Projected
Paid
Loss
Loss @ Reserve
12/08 @ 12/08
380
500
600
500
290
=====
2,270
(55)
85
180
475
685
=====
1,370
8
Bornhuetter-Ferguson (B-F) Method
• Merges the LDF and ELR techniques
• Steps (e.g., for case-incurred B-F):
(1) Determine the to-ultimate LDF
(2) Determine the proportion of ultimate loss
See example
unreported to-date = 1 – {1 / (1)}
(3) Earned premium × ELR = (“a priori”)
expected
ultimate loss
(4) Expected unreported loss = (2) × (3)
(5) B-F ultimate loss = (4) + (case incurred loss todate)
9
B-F Method: Example of Use
of LDF
• “Unpaid”
– Suppose the 12-ult. Paid loss LDF (loss
development factor) is 5.000
– What proportion of the ultimate loss is expected to
have been paid @ 12 months?
• “Unreported”
– Suppose the 24-ult. Incurred LDF is 2.500
– What proportion of the ultimate loss is expected to
have been reported @ 24 months?
10
Bornhuetter-Ferguson Method
Accident
Year
2004
2005
2006
2007
2008
Total
Earned
Premium
500
900
1,200
1,500
1,500
=====
5,600
Projected
Ultimate
ELR
Loss
0.65
0.65
0.65
0.65
0.65
“A priori”
Or “Initial
Expected”
325
585
780
975
975
=====
3,640
11
Bornhuetter-Ferguson Method
(cont.)
Accident
Year
2004
2005
2006
2007
2008
Total
Proportion Proportion
of Ult.
of Ult.
To-Ult
Paid
Unpaid
LDF @ 12/08 @ 12/08
1.000
1.050
1.260
1.890
3.780
1.000
0.952
0.794
0.529
0.265
0.000
0.048
0.206
0.471
0.735
Projected
Projected Ultimate
Reserve
Loss
@ 12/08 @ 12/08
28
161
459
717
=====
1,365
380
528
761
959
1,007
=====
3,635
12
Diagnostics
• Relationship of projected ultimate L/R to ELR
• LDF / payout patterns compared with industry
or other sources
– “Internal vs external”
•
•
•
•
Paid-to-incurred (or –to-reserve) ratios
Closed-to-reported claim count ratios
Average paid claim
Reserve as percentage of ultimate loss
13
Loss Reserving Issues
• Considering multiple results
– LDF vs ELR vs B-F
– Paid vs incurred vs frequency-severity
• Point estimates vs reserve ranges
• Loss reserve discounting
– Regulatory considerations
– Payment patterns
– Interest rates
14
CAS Exam 6, Fall 2005, #10
15
CAS Exam 6, Fall 2005, #10 (cont.)
16
Next Time
• Loss Reserving III
• Loss reserving principles
• CAS Statement of Loss Reserving
Principles
• Accounting issues
• Blanchard study note
17