steve white, fcas maaa, guy carpenter property ratemaking exposure rating 16 july, 2007 london care...
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Steve White, FCAS MAAA, Guy Carpenter
Property RatemakingProperty RatemakingExposure RatingExposure Rating
16 July, 200716 July, 2007
London CARe SeminarLondon CARe Seminar
Property Exposure RatingProperty Exposure RatingTypes of Exposure Rating CurvesTypes of Exposure Rating Curves
3
Property Exposure RatingHistory
Lloyds
Salzmann (1960 INA Homeowners data)
Reinsurer Curves (Swiss Re, Munich Re, etc)
Ludwig (1984-1988 Homeowners and Small Commercial data)
ISO’s PSOLD (Recent Commercial data)
ISO’s PSOLD+ (Recent Homeowners data)
MBBEFD (Astin paper by Stephan Bernegger)
4
Property Loss Curves Advantages/(Disadvantages)
Lloyds Curves– (Very old data)– (Does not vary by amount of insurance or occupancy class)– (Underlying data is largely unknown (marine losses? WWII Fires?))
Salzmann (Personal Property)– Based on actual Homeowners data– Varies by Construction/Protection Class– (Very old data – from 1960)– (Does not vary by amount of insurance)– (Building losses only and Fire losses only)– (Salzmann recommends NOT using them, only meant as an example)
Reinsurer Curves (Swiss Re, Munich, Skandia, etc)– Documented study (some curves) on personal & commercial reinsurance
business– (Old data)– (Does not vary by amount of insurance or occupancy class)
5
Property Loss Curves Advantages/Disadvantages
Ludwig Curves (Personal and Commercial)
– Based on actual Homeowners and Commercial data, (but uses Hartford homeowners and small commercial property book – may not be good for large national accts)
– Varies by Construction/Protection Class for Homeowners and Occupancy Class for Commercial
– Includes all property coverages and perils
– (Old data: 1984 - 1988)
ISO’s PSOLD
– Recent Data – updated every 2 years
– Varies by amount of insurance, occupancy class, state, coverage, and peril
– Continuous Distribution (no need for Interpolation)
– (Based on ISO data only)
6
Property Loss Curves Advantages/Disadvantages
ISO’s PSOLD+ (Homeowners)
– First Update to Homeowners Property since Ludwig study
– Varies by amount of insurance, policy form, state, and construction
– Continuous Distribution (no need for Interpolation)
– (Based on ISO data (US) only, New untested outside of ISO)
MBBEFD
– Loss Distribution from Physics
– Found to be useful for Property Loss Distributions
– Continuous Distribution (no need for Interpolation)
– (Relatively unknown)
– In work by Bernegger, parameters for this curve that approximate the Swiss Re Y Curves have been produced
Property Exposure RatingProperty Exposure RatingFirst Loss Scale MethodologyFirst Loss Scale Methodology
8
First Loss ScalesA Quick Review of FLSs
)(
)(1
)(1
)(1
)(
)()( 0
0
0
XE
dxxF
dxxF
dxxF
XE
XLASXFLS
XX
0)(
)(1)(1
)(1
)( 0
XE
XF
dX
dxxFXE
d
dX
XILFd
X
0)(
)()(2
2
XE
Xf
dX
XILFd
The FLS is also non-decreasing (non-negative 1st derivative), similar to ILF
The FLS is also non-decreasing at a decreasing rate (non-positive 2nd derivative)(many commonly used FLSs fail this part of the test)
E(X), the unlimited average severity of X
9
First Loss ScalesCalculations – Calculating Expected Losses
TIVPolDed
FLSTIV
PolLmtFLS
TIVPolDedtLayPolLmtMin
FLSTIV
PolDedtLayLayLmtPolLmtMinFLS
Exposed
)Re,()Re,(
%
Layer Limit 1,000,000 Layer Retention 1,000,000
000,000,3000,50
000,000,3000,000,3
000,000,3000,050,1
000,000,3000,050,2
%
FLSFLS
FLSFLS
Exposed
Policy Limits Profile Information Loss&ALAE
Limit Sir/Ded Prem. Ratio TIV
3,000,000 50,000 100,000 60.00% 3,000,000
%7113.19
%70.32%100
%65.80%12.93
%6667.1%100
%35%3333.68%
FLSFLS
FLSFLSExposed
Entry ActualRatio FLS
0.0% 0.00%1.0% 32.70%1.7% 36.70%5.0% 48.68%
10.0% 58.71%20.0% 70.41%30.0% 77.78%35.0% 80.65%40.0% 83.14%50.0% 87.32%60.0% 90.72%68.3% 93.12%70.0% 93.57%80.0% 96.00%90.0% 98.13%
100.0% 100.00%
827,11%60000,100%7113.19Pr%7113.19 LRemssesExpectedLo
10
First Loss ScalesCalculations – Expected Claim Count
Layer Limit 1,000,000 Layer Retention 1,000,000
Policy Limits Profile Information Loss&ALAE
Limit Sir/Ded Prem. Ratio TIV
3,000,000 50,000 100,000 60.00% 3,000,000
0168.0000,60%70.32%100
%428335%80.6452007842499%80.6451829
LayerCount
LRemium
TIV
PolDedFLS
TIV
PolLmtFLS
TIV
PolDedtLayPolLmtMinFLS
TIV
PolDedtLayPolLmtMinFLS
LayerCount
Pr
)Re,()Re,(
000,60
%6667.1%100
%35%000033.35
1
%60000,100
000,000,3000,50
000,000,3000,000,3
000,000,3000,050,1
000,000,3001,050,1
FLSFLS
FLSFLSFLSFLS
FLSFLS
LayerCount
Entry ActualRatio FLS
0.0% 0.00%1.0% 32.70%1.7% 36.70%
35.000000% 80.64518298%35.000033% 80.64520074%
100.0% 100.00%
608,7020168.0/827.11/ LayerCountssesExpectedLoAveSev
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Using First Loss ScalesOther Issues/Observations
Curves do not vary by Insured Value
Need to match peril and type of policy
Interpolation between points on the common first loss scales
A Tabular First Loss Scale is not adequate for calculating expected claim counts
Requires experience and judgment to know which First Loss Scale is appropriate to use
Property Exposure RatingProperty Exposure RatingWorking with PSOLDWorking with PSOLD
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PSOLDCalculations using the Mixed Exponential
)1(
)Re,()Re,(%
AddlCovgPolLmtGrPolLmt
GrPolLmtLAS
tLayGrLayLmtMinLAStLayLayLmtGrPolLmtMinLASExposed
LRem
GrPolLmtLAStLayGrPolLmtMinLAStLayGrPolLmtMinLAS
LayerCountPr
)Re,(Re,(
Lags
i
x
iiMEiewxLAS
#
1
1)(
15., 10 ipsoldi
wi varies by: 2 - Coverage (B+C, B+C+I)(B only, C only dropped in 2004)4 - Peril (BG1, BG2, Special Causes, All)
22 - Occupancy Class60 - Amount of Insurance (AOI)2 - Net of Deductible vs Ground Up50 – State Deductible Distributions
14
PSOLD MethodologyInterpreting a single policy LAS in an AOI Ranges in PSOLD
Loss vs AOI
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10
(000's)s
(000's)AOI Range
Lo
ss L
imit
Addtl Covg PS
Loss Lmt PS
Loss vs AOI
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10
(000's)s
(000's)AOI Range
Lo
ss
Lim
it
Addtl Covg Alt
Loss Lmt Alt
PSOLD Alternate
Is the movement from one AOI range to the next a step Function or a smooth progression?
Consider three policies, two within a single AOI range and the third in the next highest AOI range but close in value to the second policy
Should two different policy limits within a single AOI range have the same LAS or should the difference in policy limits be reflected?
PSOLD currently calculates the LAS at a single point, the upper bound of the AOI range for all policies in the range.
15
PSOLD MethodologyPSOLD LAS Calculations over Single AOI Range (“Dtl”)
PSOLD has two Ranges of Interest
** )(: UpperMELimitPSOLDUpperLimit AOILASxLASAOIx
)()(:*LimitMELimitPSOLDUpperLimit xLASxLASAOIx
Lags
i
x
iiMEiewxLAS
#
1
1)(
LAS for an Mixed Exponential
%)1(* ExposureAdditionalAOIAOIwhere UpperUpper
For Coverages B, C and B+C PSOLD constrains the LAS Calculation Limitx
18
PSOLD MethodologyWeighting between AOI Ranges
• If a range of Insured values spans more than one AOI Range. You need to combine the results of the Individual AOI ranges
• In PSOLD any AOI group included within the range will be given full weight
• An improvement would be to only Include an AOI range in proportion To the percentage that the range is Covered
• This was fixed in the 2004 ISO PSOLD Software
Loss vs AOI
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10
(000's)s
(000's)AOI Range
Lo
ss
Lim
it
Addtl Covg Alt
Loss Lmt Alt
19
PSOLD MethodologyWeighting between Occupancy Classes
In PSOLD, when using more than one Occupancy class on a single policy group, the relative weight assigned to each occupancy class is based on the occupancy counts in the underlying industry data base. (this can be compensated for by entering a separate limits profile for each occupancy class)
An improvement would be to allow the user to define the weights between the occupancy classes so that you can more accurately reflect the individual ceding companies exposure
20
PSOLD MethodologyAdditional Exposure Percentage
PSOLD uses the following additional exposure percentage
Building Only – 50% (no longer produced in PSOLD 2004)
Contents Only – 50% (no longer produced in PSOLD 2004)
Building+Contents Only – 200% (50% 2004 and prior)
Building+Contents+Business Interruption – 200% (Unlimited – 2004 and prior)
These percentages are based on ISO claims experience
You may want to select a different percentage due to any of the following
Stacking of Excess Policies – you do not want the policies to overlap
Margin Clause – contractually limits exposure greater than the limit
Company Experience
Judgment
21
PSOLD MethodologyStacking and Participation
Additional consideration when dealing with stacking an participation
The selected AOI group should be based on a full value on the insured risk (same AOI group as if the risk was fully covered by a single policy
All stacked policies should have the same AOI group
When stacking, assume additional coverage % is zero or the policies will overlap
Support for stacked policies was added in 2006 (but I have not worked with this feature)
22
PSOLD+ MethodologyNew Homeowners Curves
New in 2005
Newest update of Homeowners Curves since Ludwig
Curves vary by– Insured Value (values don’t go as high as the commercial curves)– State (excludes TX)– Policy Form (Homeowners, Condo, both)– Construction (Brick, Frame, both)– Protection Class (Protected, Unprotected, both)
23
PSOLDOther Issues/Observations
Limited Credibility for very Large Insured Values
TIV Scale Produced by PSOLD
Gross Limited Average Severities may not be consistent across AOI ranges for the Net of Deductible curves due to different mixture of deductibles
Curves based on US Exposure in US Dollars– If rating US business – No Problem– For non-US business – Consider adjusting the scale of the curves
using an international construction cost index– If you adjust the scale of the model you must adjust BOTH the
means of the mixed exponential AND the AOI ranges
Stacking and ParticipationStacking and ParticipationVentilated LayersVentilated Layers
25
Participation allows you to correctly model the situation where a contract only covers a proportional share of the underlying loss.
It is most common in a subscription type market like Lloyds, but it is also useful for modeling some facultative business.
Example
Assume the following:– Write 25% participation on a $1M Contract.– You reinsure a 200K xs 200K layer
In order to get a loss that will expose the Reinsurance Cover– You must have a loss to the primary contract greater than 800K (200K / 25%) – The largest loss you can have exposing the layer is 250K (25% of 1M) or 50K to
the layer– Actually, you would take 25% of losses ceded to an 800K xs 800K reinsurance
layer. But since the primary policy is $1M, it is effectively 25% of 200k xs 800k.
Stacking and ParticipationParticipation
26
Stacking is where an insurer issues multiple excess contracts covering the same underlying risk
Assume someone writes a series of policies covering the same risk, 100K x 100K (Yellow), 300K x 200K (Blue), 500K x 500K (Red) and 1M x 1M (Green)
If all are written at the same level of participation then effectively it is the same as a single 1.9M xs 100K (Purple) policy with the given participation
In practice, not all contracts are at the same participation and not all contract are written (can be thought of as participation=0%, this is sometimes called ventilation)
Stacking and ParticipationStacking
Individual Contracts
StackedContracts
27
Now Assume there is a 500K x 500K reinsurance contract covering these contracts
If the contracts are assumed to be independent, then you would only cover the 500K x 500K layer on the 1M x 1M policy. No other policy would expose.
If the contracts are assumed to be stacked, then you would cover the 500K x 500K layer on the 1.9M x 100K policy.
There can be significantly greater exposure to the Reinsurance Contract under the stacked assumption
Stacking and ParticipationStacking
Reinsurance Layer
28
Stacking is Generally thought of as an International Issue, but…
Stacking can be used in the Facultative Markets
Stacking can be used to model Umbrella written over a company’s own underlying policies
Stacking is commonly used in combination with participation in a subscription market like Lloyds
Stacking and ParticipationStacking
29
Layer: 300k xs 200k - no stacking
Limits Profile Rescaled RescaledPolicy SIR/ Treaty Limit TreatyLimit Retention Participation (Capped) Retention
100,000 100,000 100.0% 0 200,000300,000 200,000 100.0% 100,000 200,000500,000 500,000 50.0% 100,000 400,000
1,000,000 1,000,000 25.0% 200,000 800,000
"Our share" of the layer would be Participation x Capped Treaty Limit
25% Share
50% Share
100% Share
Stacking and ParticipationPartial Participation without Stacking
30
Stacking and ParticipationPartial Participation with Stacking
25% Share (250k)
50% Share (150k)
100% Share (300k)
100% Share (100k)
50% Share (100k)
Assume someone writes a series of policies covering the same risk, 100K x 100K (Yellow), 300K x 200K (Blue), 500K x 500K (Red) and 1M x 1M (Green).
– Your participation on each is: 100K xs 100K (100%), 300K xs 200K (100%), 500K xs 500K (50%), 1M xs 1M (25%)
– These policies are stacked
– You reinsure a 500K xs 500K layer
In order to get a loss that will expose the Reinsurance Cover
– You must have a loss to the excess contracts greater than 600K (100K / 100% + 300K / 100% + 100K / 50%)
– The largest loss you can have exposing the layer is 900K (100K * 100% + 300K * 100% + 500K * 50% + 1M * 25%) or 400K to the layer
MBBEFD CurveMBBEFD CurveMaxwell-Boltzmann, Bose-Einstein, Maxwell-Boltzmann, Bose-Einstein, Fermi-DiracFermi-Dirac
32
MBBEFD Curves
Contains the Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac distributions.
Curves used in Physics but found to be useful for Property Severity curves
1999 ASTIN Paper by Stephan Bernegger of Swiss Re
Two parameter distribution. In Paper a single parameter version is presented where both parameters are defined as function of a new parameter c. Many of the Swiss Re Y scales are reasonably approximated using values for c.
Fits many of the common first loss scales reasonably well
33
MBBEFD Curves
b
bxFLS
x
1
1)(
g
xgxFLS
ln
11ln)(
xxFLS )(
The FLS(x) for the MBBEFD curve type where x is the Limit/TIV is as follows:
When b=1 and g>1
When bg=1 and g>1
else (b>0 and b<>1 and bg <> 1 and g>1) gb
bbgbbg
xFLS
x
ln
1)1()1(
ln
)(
When b=0 or g=1 ProRata