Solving the Puzzle:
The Hybrid Reinsurance Pricing Method
A Practitioner’s Guide
John Buchanan - Platinum Reinsurance
CAS Ratemaking Seminar – REI 3
March 8, 2007
CAS RM 2007 – The Hybrid Reinsurance Pricing Method
1
Agenda
• Overriding Assumptions
• Recap Traditional Methods
– Experience Rating
– Exposure Rating
– Credibility Weighting
• Hybrid: Experience / Exposure Method
– Highlight differences between traditional
methods
• Testing Default Parameters
• Advanced Topics for Solving the Puzzle
2
Overriding Assumptions
of Hybrid Experience / Exposure Method
• With perfect modeling and data the results under the
experience and exposure methods will be identical.
• In practice,
– if the model and parameter selections for both
experience and exposure methods are proper and
relevant,
– then the results from these methods will be similar,
– except for credibility and random variations.
• Lower layer experience helps predict higher less
credible layers.
• Frequency is a more stable indicator than total burn
estimates.
3
Traditional Methods
Experience
• Relevant parameter
defaults/overrides for:
– LDFs (excess layers)
– Trends (severity, frequency,
exposure)
– Rate changes
– LOB/HzdGrp indicators
• Adjust for historical
changes in:
– Policy limits
– Exposure differences
o Careful “as-if”
Exposure
• Relevant parameters
defaults/overrides for:
–
–
–
–
ILFs (or ELFs, PropSOLD)
Direct loss ratios (on-level)
ALAE loads
Policy profile (LOB, HzdGrp)
o Limit/subLOB allocations
• Adjust for expected
changes in:
– Rating year policy limits
– Rating year exposures
expected to be written
4
Classical Credibility Weighting
• Estimate separate Experience and Exposure burns
• Select credibility weights using combination of:
– Formulaic Approach
• Expected # of Claims / Variability
• Exposure ROL (or burn on line)
– Questionnaire Approach
• Apriori Neutral vs. Experience vs. Exposure
• Patrik/Mashitz paper
– Judgment
• Need to check that burn patterns make sense
– i.e. higher layer ROL < lower ROL
– similar to Miccolis ILF consistency test
5
Classical Credibility Weighting
Credibility weights
judgmentally selected
6
Basic Steps of The Hybrid Method
Step 1: Estimate Experience burns & counts
–
–
–
–
Select base attachment points/layers above the reporting data threshold
Estimate total excess burns using projection factors
Estimate excess counts using frequency trends, claim count LDFs
Calculate implied severities
Step 2: Estimate Exposure burns & counts
– Use same attachment points/layers as Experience
– Estimate total burns and bifurcate between counts, average severities
Step 3: Calculate Experience/Exposure frequency ratio by
attachment point
– Estimate overall averages using number of claims/variability
Step 4: Review frequency ratio patterns
– Adjust experience or exposure models if needed and re-estimate burns (!!)
– Select indicated experience/exposure frequency ratio(s)
Step 5: Similarly review excess severities and/or excess burns
Step 6: Combine Hybrid frequency/severity results
– Using experience adjusted exposure frequencies and severities
Step 7: Determine overall weight to give Hybrid
7
Estimation of Hybrid Counts
Recap Steps 1 to 4
A: Select base attachment points above data
threshold
– Example: threshold=150k; reins layers=500x500k, 1x1mm
– Select 200k, 250k, 350k, 500k, 750k, 1mm attachment points
B: Calculate experience counts
– At lower attachment points, year by year patterns should be
variable about some mean
– For example, if upward trend, then perhaps:
• Overdeveloping or trending later years
C: Calculate exposure counts for comparison
D: Review experience/exposure frequency patterns
– Should be relatively stable until credibility runs out
– Double back to methods if not
– Select frequency ratios to estimate Hybrid counts
8
Step 1a: Experience Counts and Burns
Sublayer $150,000 xs 350,000
Developed &
Untrended
Trended
Accident Count to
Untrended
Count to
Year
Layer
Loss to Layer
Layer
1998
4
409,404
7.0
1999
3
316,512
6.0
2000
5
246,404
8.0
2001
9
405,795
13.0
2002
4
241,151
10.0
2003
6
484,214
7.0
2004
7
760,191
11.6
2005
9
619,885
9.7
2006
3
182,765
5.5
Total/Avg
54
3,823,028
85.8
150,000
xs
350,000
Estimated
Frequency
0.100
0.083
0.106
0.162
0.117
0.080
0.131
0.109
0.059
Avg. freq:
0.109
SPI: 111,000,000
Est. # claims xs 350k attachment:
12.05
Developed &
Trended
Estimated
Loss to
Layer
Burn
613,334
0.88%
519,936
0.71%
722,678
0.95%
1,114,097
1.38%
531,613
0.62%
670,475
0.76%
943,398
1.07%
891,644
1.01%
392,994
0.42%
7,260,658
Avg. burn:
Ult loss:
Avg sev:
0.89%
984,586
81,695
9
Step 1b: Review Experience Counts
Year Variability: >350,000 Attachment
Apparently random pattern
around selection of #=12.05
Note: Claim counts are on-leveled
10
Step 1c: Review Experience Counts
Year Variability: >1,000,000 Attachment
Credibility runs out;
indication is #=.36
11
Step 1-Recap: Estimation of Experience
Burns, Counts and Implied Severities
Layer
(Limit xs Retention)
50,000
100,000
150,000
500,000
250,000
1,000,000
xs
xs
xs
xs
xs
xs
200,000
250,000
350,000
500,000
750,000
1,000,000
Experience - Traditional Burning Cost Method
Indicated
Ultimate Loss Excess Claim
Implied
Experience
(USD)
Severity
Counts
Burn (%)
1.19%
1,322,008
27.05
48,874
1.52%
1,691,358
24.54
68,919
0.89%
984,586
12.05
81,695
0.41%
456,121
2.69
169,751
0.09%
95,024
0.54
176,822
0.03%
30,874
0.36
86,177
To be compared to
exposure counts
12
Step 2: Estimation of Exposure Burns
Bifurcated Between Counts and Severities
Indicated
Exposure
Burn (%)
Layer
Exposure Method
Benchmark
Ultimate Loss
Benchmark
Excess Claim
(USD)
Severity
Counts
50,000
100,000
150,000
xs
xs
xs
200,000
250,000
350,000
1.51%
1.92%
1.33%
1,671,633
2,134,498
1,481,529
38.05
29.80
15.34
43,937
71,616
96,588
500,000
xs
500,000
1.54%
1,709,680
6.00
285,088
250,000
xs
750,000
0.27%
296,553
1.90
156,416
1,000,000
xs
1,000,000
0.27%
304,773
0.77
398,338
12.05 exper / 15.34 expos
= 78.6%
13
Step 3: Calculate Experience/Exposure
Frequency Ratios and Base Layer Weights
Layer
(Limit xs Retention)
50,000
100,000
150,000
500,000
250,000
1,000,000
xs
xs
xs
xs
xs
xs
Total
200,000
250,000
350,000
500,000
750,000
1,000,000
Indicated
Exper/Expos
Freq Ratio
Base Layer
Weights
Devt/Trended #
of Claims
Actual # of
Claims
[A7/B7]
[f/13]
[f/Burn Analysis]
[f/Burn Analysis]
71.1%
82.3%
78.6%
44.8%
28.3%
46.8%
75.1%
39.9%
36.5%
18.1%
4.5%
0.6%
0.4%
100.0%
189.36
173.45
85.79
21.34
3.05
2.05
475.05
178.00
129.00
54.00
11.00
2.00
0.00
374.00
12.05 exper / 15.34 expos
= 78.6%
14
Step 4a: Review Exper/Expos Frequencies
Attachment Point Pattern: 200k…1mm
Frequency of Excess Claims by Attachment Point
By Projection Method
40.0
38.0
Number of Excess Claims
35.0
29.8
30.0
27.0
24.5
25.0
20.0
15.3
15.0
Expos and Exper count ratios relatively
consistent through 350k- IF experience
very credible, then perhaps pressure to
reduce exposure L/R; check out spikes
12.1
10.0
6.0
5.0
2.7
1.9
0.5
0.8
0.4
0.0
200,000
250,000
350,000
500,000
Attachment Point
Exposure
Exper -TBC
# of Claims Expected in Rating Year
Att Pt.
Exposure
Exper -TBC
Hybrid
200,000
250,000
350,000
500,000
750,000
1,000,000
38.0
27.0
30.4
29.8
24.5
23.8
15.3
12.1
12.3
6.0
2.7
4.8
1.9
0.5
1.5
0.8
0.4
0.6
750,000
1,000,000
Hybrid
Exper/
Expos
Ratio
71.1%
82.3%
78.6%
44.8%
28.3%
46.8%
15
Step 4-Recap: Select Exper/Expos Frequency Ratio
For Hybrid Claim Count Estimate
Exposure Method
Indicated
Benchmark
Exposure Burn Excess Claim
(%)
Counts
Layer
50,000
100,000
150,000
500,000
250,000
1,000,000
xs
xs
xs
xs
xs
xs
Total
200,000
250,000
350,000
500,000
750,000
1,000,000
1.51%
1.92%
1.33%
1.54%
0.27%
0.27%
38.05
29.80
15.34
6.00
1.90
0.77
1.81%
Important Selection
Hybrid Method
Indicated
Exper/Expos
Freq Ratio
71.1%
82.3%
78.6%
44.8%
28.3%
46.8%
75.1%
Selected
Selected Excess
Exper/Expos
Claim Counts
Freq Ratio
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%
30.44
23.84
12.27
4.80
1.52
0.61
6.00 expos x 80.0%
16
Step 5: Selected Severity
Layer
(Limit xs Retention)
50,000
100,000
150,000
500,000
250,000
1,000,000
xs
xs
xs
xs
xs
xs
200,000
250,000
350,000
500,000
750,000
1,000,000
Exper - TBC
Exposure
Implied
Severity
Benchmark
Severity
48,874
68,919
81,695
169,751
176,822
86,177
43,937
71,616
96,588
285,088
156,416
398,338
Hybrid
Weight to
Selected
Experience Severity (Wtd)
Severity
100.0%
48,874
100.0%
68,919
85.0%
84,218
22.5%
259,137
5.0%
157,436
2.5%
390,534
Unrealistic experience severity
17
Step 6: Selected Overall Hybrid Burn
Layer
(Limit xs Retention)
50,000
100,000
150,000
500,000
250,000
1,000,000
xs 200,000
xs 250,000
xs 350,000
xs 500,000
xs 750,000
xs 1,000,000
Total
Exposure
Indicated
Exposure
Burn (%)
Exper - TBC
Indicated
Experience
Burn (%)
1.51%
1.92%
1.33%
1.54%
0.27%
0.27%
1.81%
2,014,454
1.19%
1.52%
0.89%
0.41%
0.09%
0.03%
0.44%
486,996
Total $Rept:
# claims
Data Threshold:
# Years
Eff # Years
SPI: 111,000,000
57,801,368
183
150,000
9
6.7
Excess
Claim
Counts
30.44
23.84
12.27
4.80
1.52
0.61
Hybrid Method
Selected
Severity
Hybrid
(Wtd)
Burn (%)
48,874
68,919
84,218
259,137
156,926
390,534
1.34%
1.48%
0.93%
1.12%
0.21%
0.22%
1.34%
Selected
Ultimate
Loss
1,487,569
1,643,296
1,033,439
1,243,242
238,016
239,042
1,482,284
Hybrid: Experience adjusted
Exposure count & severity…
100% credibility to burn??
18
Steps 1-7: Bringing it All Together
A. Experience Method - Traditional Burning Cost (USD)
C. Experience / Exposure Indicated and Selected Ratios
Subject Premium: 111,000,000
1
2
3
5
6
7
8
10
11
12
13
14
15
Implied
Severity
Indicated
Exper/Expos
Freq Ratio
Selected
Exper/Expos
Freq Ratio
Base Layer
Weights
Devt/Trended
# of Claims
Actual # of
Claims
Weight to
Experience
Severity
[6/7]
[A7/B7]
48,874
68,919
81,695
169,751
176,822
86,177
71.1%
82.3%
78.6%
44.8%
28.3%
46.8%
75.1%
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%
80.0%
39.9%
36.5%
18.1%
4.5%
0.6%
0.4%
100.0%
189.4
173.4
85.8
21.3
3.1
2.1
475.0
178
129
54
11
2
0
374
100.0%
100.0%
85.0%
22.5%
5.0%
2.5%
12
13
Experience Method - TBC
Layer
(Limit xs Retention)
Indicated
Experience
Burn (%)
Ultimate Loss Excess Claim
(USD)
Counts
[5xSPI]
1
2
3
4
5
6
50,000
100,000
150,000
500,000
250,000
1,000,000
xs
xs
xs
xs
xs
xs
200,000
250,000
350,000
500,000
750,000
1,000,000
Total
1.19%
1.52%
0.89%
0.41%
0.09%
0.03%
1,322,008
1,691,358
984,586
456,121
95,024
30,874
0.44%
486,996
27.05
24.54
12.05
2.69
0.54
0.36
2.69
181,241
[f/ 13]
80.0%
B. Exposure Method (USD)
1
2
3
Layer
(Limit xs Retention)
D. Hybrid Method (USD)
5
Indicated
Exposure
Burn (%)
6
7
Exposure Method
Benchmark
Indicated
Ultimate Loss Excess Claim
(USD)
Counts
[5xSPI]
8
10
Benchmark
Severity
Selected Excess
Claim Counts
11
Hybrid Method
Selected
Selected
Hybrid
Severity (Wtd)
Burn (%)
Selected
Ultimate Loss
[6/7]
[B7xC11]
[f/ A8,B8,C15]
[13/SPI]
[10x11]
1
2
3
50,000
100,000
150,000
xs
xs
xs
200,000
250,000
350,000
1.51%
1.92%
1.33%
1,671,633
2,134,498
1,481,529
38.05
29.80
15.34
43,937
71,616
96,588
30.44
23.84
12.27
48,874
68,919
84,218
1,487,569
1,643,296
1,033,439
4
500,000
xs
500,000
1.54%
1,709,680
6.00
285,088
4.80
259,137
5
250,000
xs
750,000
0.27%
296,553
1.90
156,416
1.52
157,436
6
1,000,000
xs
1,000,000
0.27%
304,773
398,338
0.61
390,534
1.81%
2,014,454
0.77
6.00
1.34%
1.48%
0.93%
1.12%
0.22%
0.22%
1.34%
1,482,284
Total
335,909
1,243,242
238,790
239,042
19
Benefits of Hybrid Method
• One of main benefits is questioning Experience
and Exposure Selections
– To the extent credible results don’t line up, this provides
pressure to the various default parameters
– For example, there would be downward pressure on
default exposure ILF curves or loss ratios if
• Exposure consistently higher than experience, and
• Credible experience and experience rating factors
• A well constructed Hybrid method can sometimes
be given 100% weight if credible
• Can review account by account, and aggregate
across accounts to evaluate pressure on industry
defaults
20
Test of Default Parameters
• Aggregate across “similar” accounts to evaluate
pressure on industry defaults
– May want to re-rate accounts using e.g. default rate
changes, ILFs, premium allocations, LDFs, trends, etc.
• Each individual observation represents a
cedant/attachment point exper/expos ratio
• Review dispersion of results and overall trend
– E.g. if weighted and/or fitted exper/expos ratios are well
below 100% (or e.g. 90% if give some underwriter credit)
then perhaps default L/Rs overall are too high (or
conversely LDFs or trends too light)
– If trend is up when going from e.g. 100k to 10mm att pt,
then perhaps expos curve is predicting well at lower
points but is underestimating upper points
21
Test of Default Parameters (cont.)
• Before making overall judgments, must
consider
– UW contract selectivity (contracts seen vs. written),
– Sample size (# of cedants/years),
– Impact “as-if” data (either current or historical)
– Survivor bias
– Systematic bias in models
– “Lucky”
22
Test of Default Rating Factors – Example 1
Well below 100%,
pressure to reduce expos
params or increase exper
params…but credible??
23
Test of Default Rating Factors – Example 2
Exposure curve too light
with higher attachment
points?
24
Appendix - More advanced techniques for
Solving the Puzzle
• Inspecting Experience/Exposure differences
Exper/
Expos
Ratio
Layers
Ideal Situation
- No noticeable slope to ratio of Experience/Exposure
- Random fluctuation around mean
25
Appendix - More advanced techniques for
Solving the Puzzle
• Pressure Indicators –years (or layers)
Burn
Burn
Years
Upward slope pressure indicators:
- Not enough trend
- Too much LDF
- Too much later year rate change
- Too much earlier year rate change
…
From forthcoming paper
Years
Downward slope pressure indicators:
- Too much trend
- Not enough LDF
- Not enough later year rate change
- Not enough earlier year rate change
…
- THE HYBRID REINSURANCE PRICING METHOD:
A PRACTITIONER’S GUIDE
26