Property Ratemaking Exposure Rating 16 July, 2007 Property Ratemaking Exposure Rating London CARe Seminar Introduction Bio’s Steve White, FCAS MAAA, Guy Carpenter

Property Exposure Rating Types of Exposure Rating Curves

Property Exposure Rating History 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)

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)

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)

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 (Relatively unknown) In work by Bernegger, parameters for this curve that approximate the Swiss Re Y Curves have been produced

Property Exposure Rating First Loss Scale Methodology

First Loss Scales A Quick Review of FLSs 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

First Loss Scales Calculations – Calculating Expected Losses

First Loss Scales Calculations – Expected Claim Count

Using First Loss Scales Other 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 Rating Working with PSOLD

PSOLD Calculations using the Mixed Exponential 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 Class 60 - Amount of Insurance (AOI) 2 - Net of Deductible vs Ground Up 50 – State Deductible Distributions

PSOLD Methodology Interpreting a single policy LAS in an AOI Ranges in 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.

PSOLD Methodology PSOLD LAS Calculations over Single AOI Range (“Dtl”) LAS for an Mixed Exponential For Coverages B, C and B+C PSOLD constrains the LAS Calculation PSOLD has two Ranges of Interest

PSOLD Methodology Alternate LAS Calculations over a Continuous AOI Range (“Grp”) Calculating the LAS over a continuous range adds one more degree of complexity Which simplifies to

PSOLD Methodology Advantages of the Alternate LAS Calculations 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 Methodology Weighting 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

PSOLD Methodology Weighting 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

PSOLD Methodology Additional 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

PSOLD Methodology Stacking 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)

PSOLD+ Methodology New 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)

PSOLD Other 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 Participation Ventilated Layers

Stacking and Participation 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 Participation Stacking Individual Contracts Stacked Contracts 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 Participation Stacking Reinsurance Layer 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 Participation Stacking 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 Participation Partial Participation without Stacking 25% Share 50% Share 100% Share

Stacking and Participation Partial Participation with Stacking 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 25% Share (250k) 50% Share (150k) 50% Share (100k) 100% Share (300k) 100% Share (100k)

MBBEFD Curve Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac

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 The FLS(x) for the MBBEFD curve type where x is the Limit/TIV is as follows: When b=0 or g=1 ProRata When b=1 and g>1 When bg=1 and g>1 else (b>0 and b<>1 and bg <> 1 and g>1)