liberty mutual group - cas.confex.com · liberty mutual insurance reinsurance per risk exposure...
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![Page 1: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/1.jpg)
PEBELS: Policy Exposure Based Excess Loss SmoothingMarquis J. Moehring
Liberty Mutual Group
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Liberty Mutual Insurance
Outline
1. Background2. Goal3. PEBELS Defined4. PEBELS Derived (PPR Generalized)5. Applications6. Summary
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Liberty Mutual Insurance
My Challenge
Strong Regional Focus• State/Program Large Loss Provisions• Low Credibility• High Heterogeneity
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Liberty Mutual Insurance
This Should be Easier
No applicable method in literature• ILFs for Liability• ELFs for Workers Compensation• Nothing for Commercial Property or Homewners!
![Page 5: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/5.jpg)
Liberty Mutual Insurance
Goal of PEBELS
PEBELS = Property Large Loss Exposure Segmentation• Meet my challenge• New applications!• Deceptively difficult
1) No clear limit2) Multiple non-linearities3) Additional nuances4) Practical considerations
![Page 6: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/6.jpg)
Liberty Mutual Insurance
PEBELS Defined
Defined as ������ = �� ∗ ���� ∗ ��
• � ∗ � �� = �( �) = ������������� ���
• ��� = �(��)-� � = �!��"��#��$�( �)%"��&�!
![Page 7: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/7.jpg)
Liberty Mutual Insurance
Exposure Curve
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
G(L
oss
/ In
sure
d V
alu
es)
Loss / Insured Value
c = 5.0
![Page 8: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/8.jpg)
Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 9: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/9.jpg)
Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 10: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/10.jpg)
Liberty Mutual Insurance
Reinsurance Per Risk Exposure Rating
Insured Value Range
($000s)
Midpoint ($000s)
Retention as a % of Insured value
Retention + Limit as a % of Insured
value
Exposure Factor
Subject Premium
Expected Loss Ratio
Expected Primary Losses
Expected Reinsurer
Losses
20-100 60 167% 833% 0% 682,000 65% 443,300 0
100-250 175 57% 286% 26% 161,000 65% 104,650 27,209
250-1,000 625 16% 80% 41% 285,000 65% 185,250 75,953
1,000-2,000 1,500 7% 33% 33% 1,156,000 65% 751,400 247,962
Grand Total 2,284,000 65% 1,484,600 351,124
![Page 11: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/11.jpg)
Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 12: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/12.jpg)
Liberty Mutual Insurance
Per Policy Generalization
Insured Value Range ($000s)
Midpoint ($000s)
Retention as a % of Insured value
Retention + Limit as a % of Insured
value
Exposure Factor
Subject Premium
Expected Loss Ratio
Expected Primary Losses
Expected Reinsurer
Losses
20-100 60 167% 833% 0% 682,000 65% 443,300 0
100-250 175 57% 286% 26% 161,000 65% 104,650 27,209
250-1,000
625 16% 80% 41% 285,000 65% 185,250 75,953
1,000-2,000
1,500 7% 33% 33% 1,156,000 65% 751,400 247,962
Grand Total 2,284,000 65% 1,484,600 351,124
![Page 13: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/13.jpg)
Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via ����
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 14: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/14.jpg)
Liberty Mutual Insurance
Heterogeneity Generalization
Insured Value Range
($000s)
Midpoint ($000s)
Retention as a % of Insured value
Retention + Limit as a % of Insured
value
Exposure Factor
Subject Premium
Expected Loss Ratio
Expected Primary Losses
Expected Reinsurer
Losses
20-100 60 167% 833% 0% 682,000 65% 443,300 0
100-250 175 57% 286% 26% 161,000 65% 104,650 27,209
250-1,000 625 16% 80% 41% 285,000 65% 185,250 75,953
1,000-2,000 1,500 7% 33% 33% 1,156,000 65% 751,400 247,962
Grand Total 2,284,000 65% 1,484,600 351,124
![Page 15: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/15.jpg)
Liberty Mutual Insurance
Heterogeneity Generalization
�'� � = � ∗ ���� ∗ ���
• Expected catastrophe loss• Risk loads• Rate adequacy
State: House
X 65.0%
Y 65.0%
Z 40.0%
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Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ��
4) Historical vs. prospective exposure profiles5) Credibility
![Page 17: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/17.jpg)
Liberty Mutual Insurance
Heterogeneity Generalization
Insured Value Range
($000s)
Midpoint ($000s)
Retention as a % of Insured value
Retention + Limit as a % of Insured
value
Exposure Factor
Subject Premium
Expected Loss Ratio
Expected Primary Losses
Expected Reinsurer
Losses
20-100 60 167% 833% 0% 682,000 65% 443,300 0
100-250 175 57% 286% 26% 161,000 65% 104,650 27,209
250-1,000 625 16% 80% 41% 285,000 65% 185,250 75,953
1,000-2,000 1,500 7% 33% 33% 1,156,000 65% 751,400 247,962
Grand Total 2,284,000 65% 1,484,600 351,124
![Page 18: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/18.jpg)
Liberty Mutual Insurance
Exposure Curve
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
G(L
oss
/ In
sure
d V
alu
es)
Loss / Insured Value
c = 5.0
![Page 19: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/19.jpg)
Liberty Mutual Insurance
Exposure Curve Dispersed: e.g. Estate / University Campus
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
G(L
oss
/ In
sure
d V
alu
es)
Loss / Insured Value
c = 5.0
![Page 20: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/20.jpg)
Liberty Mutual Insurance
Exposure Curve Dispersed: e.g. Estate / University Campus
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
G(L
oss
/ In
sure
d V
alu
es)
Loss / Insured Value
C = 1.5
c = 2.0
c = 3.0
c = 4.0
c = 5.0
Concentrated: e.g. Barn / Minimart
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Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 22: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/22.jpg)
Liberty Mutual Insurance
PEBELS Derived
PEBELS Derived = PPR Generalized• Classic Reinsurance Per Risk Exposure Rating• Generalized to contemplate,
1) Policy level heterogeneity2) Expected loss heterogeneity via � ��
3) Loss process heterogeneity via ���
4) Historical vs. prospective exposure profiles5) Credibility
![Page 23: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/23.jpg)
Liberty Mutual Insurance
Applications
Indications• Motivated PEBELS• Allocate large losses to state and program
– Low credibility– High heterogeneity in underlying exposures
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Liberty Mutual Insurance
Applications
Adjusted Modeled Catastrophe AALs• Traditionally assume AAL linear with IV• This contradicts
– Theory presented– Ludwig’s study of Hurricane Hugo
• Implies bias between Personal & Commercial• Can adjust AALs with PEBELS
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Liberty Mutual Insurance
Applications
Predictive ModelsHypothesize that PEBELS• More predictive of large loss than IV• Most predictive for highly skewed perils• Most predictive in severity/excess models
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Liberty Mutual Insurance
Applications
Revised Property Per Risk Reinsurance Exposure RatingCurrent formulation:
() *+,-./01�2 03�450/6780/9:;0<=01>/+?;0<=�@0
= () ./01�2 06+A0/6780/3�?=+/�<7 ∗
�'� *+,-./01�2 03�450/6780/>/+?;0<=�@0
�'� ./01�2 06+A0/6780/>/+?;0<=�@0
![Page 27: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/27.jpg)
Liberty Mutual Insurance
Applications
Revised Property Per Risk Reinsurance Exposure Rating
Proposed formulation:
() *+,-./01�2 03�450/6780/9:;0<=01>/+?;0<=�@0
= (() ./01�2 06+A0/6780/3�?=+/�<7 ) *
(�'� *+,-./01�2 03�450/6780/
3�?=+/�<7
�'� ./01�2 06+A0/6780/3�?=+/�<7
) ∗ (�'� *+,-./01�2 03�450/6780/
>/+?;0<=�@0
�'� *+,-./01�2 03�450/6780/3�?=+/�<7 .C,,�7 �D01
)
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Liberty Mutual Insurance
Summary
PEBELS = Property Large Loss Exposure Segmentation• Only game in town• Quantifies messy non-linearities• Multiple applications
– Indications– Catastrophe Modeling– Risk Segmentation
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Liberty Mutual Insurance
Historical vs. Prospective
Selecting exposure profile for the application?Prospective (current inforce)
– Catastrophe modeling– Reinsurance quotes
Historical (“earned” over experience period)– Loss ratio ratemaking– Revised per risk reinsurance exposure rating
![Page 31: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/31.jpg)
Liberty Mutual Insurance
Historical vs. Prospective
Loss ratio ratemaking examples1) State in run-off scenario2) State newly entered scenario
Both scenarios lead to skewed state indicationsEven small shifts will distort indications
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Liberty Mutual Insurance
Credibility
Indications exampleLayer experience to maximize credibility
Complements
1) () $G.HI=+$G.JI3�?=+/�<7 ∗
>9K96$L.MNOPQRSQRQOTUQVOPWQXYZ
>9K96$L.[NOP$L.MNUQVOPWQXYZ
2) ]%!���� ∗ ��%"�. ���� ∗ (��%"�^!�!_� �)
![Page 33: Liberty Mutual Group - cas.confex.com · Liberty Mutual Insurance Reinsurance Per Risk Exposure Rating Insured Value Range ($000s) Midpoint ($000s) Retention as a % of Insured value](https://reader033.vdocument.in/reader033/viewer/2022060400/5f0e09327e708231d43d4cf0/html5/thumbnails/33.jpg)
Liberty Mutual Insurance
Appendix
Misc. Topics• Exposure curve considerations• Data limitations and NLE• Methods in common usage