Property Ratemaking - an Advanced Approach Exposure Rating

Steve White, FCAS MAAA, Guy Carpenter

Property Ratemaking - an Advanced ApproachExposure Rating

June 6-7, 2005

Section 1

Property Exposure RatingTypes of Exposure Rating Curves

Advanced Property Exposure Rating 3

Property Loss CurvesHistory

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)

Steve White

Moved


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)

Steve White

Swiss Re description not quite right.


Property Loss Curves Advantages/Disadvantages

Ludwig Curves (Personal and Commercial)

– Based on actual Homeowners and Commercial data, (but uses Hartford small commercial property book – may not be good for large national accts)

– Varies by Construction/Protection Class for HO 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)


Property Loss Curves Advantages/Disadvantages

ISO’s PSOLD+

– First Update to Homeowners Property since Ludwig study

– Varies by amount of insurance, policy form, state, and construction


– (Based on ISO data only, New untested outside of ISO)

MBBEFD

– Loss Distribution from Physics

– Found to be useful for Property Loss Distributions


– (Relatively unknown in the US)

Section 2

Property Exposure RatingFirst Loss Scale Methodology (“FLS”)


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)

E(X), the unlimited average severity of X


First Loss ScalesCalculations - Detailed Policy Limits

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


First Loss ScalesCalculations - Detailed Policy Limits

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


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

Section 3

Property Exposure RatingWorking with PSOLD


PSOLDCalculations

)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


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 minimum of the loss limit and 1.5x(upper bound of the AOI range) for all policies in the range.


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


PSOLD MethodologyAlternate LAS Calculations over a Continuous AOI Range (“Grp”)

Calculating the LAS over a continuous range adds one more degree of complexity

2

)(:**

* LowerMEUpperMELimitALTUpperLimit

AOILASAOILASxLASAOIx

)()(:*LimitMELimitALTLowerLimit xLASxLASAOIx

%)1(* ExposureAdditionalAOIAOIwhere UpperUpper

:**LowerLimitUpper AOIxAOI

**

**

2)(

LowerUpper

LowerLimitLimitMELowerMELimitALT AOIAOI

AOIxxLASAOILASxLAS

**

*

LowerUpper

LimitUpperLimitME AOIAOI

xAOIxLAS

**

**

2)(

LowerUpper

LowerLimitLowerMELimitALT AOIAOI

AOIxAOILASxLAS

**

*

21

LowerUpper

LowerLimitUpperME AOIAOI

AOIxxLAS

Which simplifies to

Limitx

%)1(* ExposureAdditionalAOIAOIwhere LowerLower


Different policy limits within the same AOI range will get different LAS

Smoother transition as you move from one AOI range to the next

Since this impacts the unlimited average severity for the policy, it will change the allocation of losses to the layer for any exposed policy

An additional enhancement would be to adjust the wi’s as you move within an AOI range

PSOLD MethodologyAdvantages of the Alternate LAS Calculations


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


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.

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


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 – 50%

Building+Contents+Business Interruption – Unlimited

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


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


PSOLD+ MethodologyNew Homeowners Curves

Available Shortly

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)

– (Does not include Protection Class)


PSOLDOther Issues/Observations

Limited Credibility for very Large Insured Values

TIV Scale Produced by PSOLD

Gross Limited Average Severities “consistent” across AOI ranges by Net of Deductible curves are not

– Due to different mixture of deductibles

Section 4

Stacking and Participation


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


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


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


Reinsurance Layer


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



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


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

Section 5

MBBEFD


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


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

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Property Ratemaking - an Advanced Approach Exposure Rating