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Commercial Property Size of Loss Distributions. Glenn Meyers Insurance Services Office, Inc. Casualty Actuaries in Reinsurance June 15 , 2000 Boston, Massachusetts. Outline. Data Classification Strategy Amount of Insurance Occupancy Class Mixed Exponential Model

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Commercial property size of loss distributions l.jpg

Commercial Property Size of Loss Distributions

Glenn Meyers

Insurance Services Office, Inc.

Casualty Actuaries in Reinsurance

June 15 , 2000

Boston, Massachusetts


Outline l.jpg
Outline

  • Data

  • Classification Strategy

    • Amount of Insurance

    • Occupancy Class

  • Mixed Exponential Model

    • “Credibility” Considerations

  • Limited Classification Information

  • Program Demonstration

  • Goodness of Fit Tests

  • Comparison with Ludwig Tables


Separate tables for l.jpg
Separate Tables For

  • Commercial Property (AY 1991-95)

  • Sublines

    • BG1 (Fire and Lightning)

    • BG2 (Wind and Hail)

    • SCL (Special Causes of Loss)

  • Coverages

    • Building

    • Contents

    • Building + Contents

    • Building + Contents + Time Element


Exposures l.jpg
Exposures

  • Reported separately for building and contents losses

  • Model is based on combined building and contents exposure

    • Even if time element losses are covered


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Classification Strategy

  • Amount of Insurance

    • Big buildings have larger losses

    • How much larger?

  • Occupancy Class Group

    • Determined by data availability

  • Not used

    • Construction Class

    • Protection Class


Potential credibility problems l.jpg
Potential Credibility Problems

  • Over 600,000 Occurrences

  • 59 AOI Groupings

  • 21 Occupancy Groups

  • The groups could be “grouped” but:

    • Boundary discontinuities

    • We have another approach


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The Mixed Exponential Size of Loss Distribution

  • i’s vary by subline and coverage

  • wi’s vary by AOI and occupancy group in addition to subline and coverage


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The Mixed Exponential Size of Loss Distribution

  • i = mean of the ith exponential distribution

  • For higher i’s, a higher severity class will tend to have higher wi’s.


The fitting strategy for each subline coverage l.jpg
The Fitting Strategyfor each Subline/Coverage

  • Fit a single mixed exponential model to all occurrences

  • Choose the wi’s and i’s that maximize the likelihood of the model.

  • Toss out the wi’s but keep the i’s

  • The wi’s will be determined by the AOI and the occupancy group.




Varying w i s by aoi l.jpg
Varying Wi’s by AOI

Prior expectations

  • Larger AOIs will tend to have higher losses

  • In mixed exponential terminology, the AOI’s will tend to have higher wi’s for the higher i’s.

  • How do we make this happen?


Solution l.jpg
Solution

  • Let W1i’s be the weights for a given AOI.

  • Let W2i’s be the weights for a given higher AOI.

  • Given the W1i’s, determine the W2i’s as follows.


Step 1 choose 0 d 11 1 l.jpg
Step 1Choose 0  d11  1

Shifting the weight from 1st exponential to the 2nd exponential increases the expected claim cost.


Step 2 choose 0 d 12 1 l.jpg
Step 2Choose 0  d12  1

Shifting the weight from 2nd exponential to the 3rd exponential increases the expected claim cost.


Step 3 and 4 similar step 5 choose 0 d 15 1 l.jpg
Step 3 and 4 SimilarStep 5 — Choose 0  d15  1

Shifting the weight from 5th exponential to the last exponential increases the expected claim cost.


Several aoi groups choose w s for lowest aoi group l.jpg
Several AOI GroupsChoose W’s for lowest AOI Group


Then choose d s to construct w s for the 2nd aoi group l.jpg
Then choose d’s toConstruct W’s for the 2nd AOI Group


Then choose d s to construct w s for the 3rd aoi group l.jpg
Then choose d’s toConstruct W’s for the 3rd AOI Group


Then choose d s to construct w s for the 4th aoi group l.jpg
Then choose d’s toConstruct W’s for the 4th AOI Group



Estimating w s for the 1st aoi group and d s for the rest l.jpg
Estimating W’s (for the 1st AOI Group) and d’s (for the rest)

Let:

  • Fk(x) = CDF for kth AOI Group

  • (xh+1, xh) be the hth size of loss group

  • nhk = number of occurrences for h and k

    Then the log-likelihood of data is given by:


Estimating w s for the 1st aoi group and d s for the rest23 l.jpg
Estimating W’s (for the 1st AOI Group) and d’s (for the rest)

  • Choose W’s and d’s to maximize log-likelihood

  • 59 AOI Groups

  • 5 parameters per AOI Group

  • 295 parameters!

    Too many!


Parameter reduction l.jpg
Parameter Reduction rest)

  • Fit W’s for AOI=1, and d’s for AOI=10, 100, 1,000, 10,000, 100,000 and 1,000,000. Note AOI coded in 1,000’s

  • The W’s are obtained by linear interpolation on log(AOI)’s

  • The interpolated W’s go into the log-likelihood function.

  • 35 parameters

-- per occupancy group


On to occupancy groups l.jpg
On to Occupancy Groups rest)

  • LetWbe a set of W’s that is used for all AOI amounts for an occupancy group.

  • Let X be the occurrence size data for all AOI amounts for an occupancy group.

  • Let L[X|W] be the likelihood of Xgiven W i.e. the probability of Xgiven W


There s no theorem like bayes theorem l.jpg
There’s No Theorem rest)Like Bayes’ Theorem

  • Let be n parameter sets.

  • Then, by Bayes’ Theorem:


Bayesian results applied to an aoi and occupancy group l.jpg
Bayesian Results Applied to an AOI and Occupancy Group rest)

  • Let be the ith weight that Wk assigns to the AOI/Occupancy Group.

  • Then the wi‘s for the AOI/Occupancy Group is:


What does bayes theorem give us l.jpg
What Does Bayes’ Theorem Give Us? rest)

  • Before

    • A time consuming search for parameters

    • Credibility problems

  • If we can get suitable Wk’s we can reduce our search to n W’s.

  • If we can assign prior Pr{Wk}’s we can solve the credibility problem.


Finding suitable w k s l.jpg
Finding Suitable rest)Wk’s

  • Select three Occupancy Class Group “Groups”

  • For each “Group”

    • Fit W’s varying by AOI

    • Find W’s corresponding to scale change

      • Scale factors from 0.500 to 2.000 by 0.025

  • 183 Wk’s for each Subline/Coverage



Prior probabilities l.jpg
Prior Probabilities rest)

  • Set:

  • Final formula becomes:

  • Can base update prior on Pr{Wk |X}.


The classification data availability problem l.jpg
The Classification Data Availability Problem rest)

  • Focus on Reinsurance Treaties

    • Primary insurers report data in bulk to reinsurers

    • Property values in building size ranges

    • Some classification, state and deductible information

  • Reinsurers can use ISO demographic information to estimate effect of unreported data.


Database behind psold l.jpg
Database Behind PSOLD rest)

30,000+ records (for each coverage/line combination) containing:

  • Severity model parameters

  • Amount of insurance group

    • 59 AOI groups

  • Occupancy class group

  • State

  • Number of claims applicable to the record


Slide34 l.jpg

Constructing a Size of Loss Distribution Consistent with Available Data Using ISO Demographic Data

  • Select relevant data

  • Selection criteria can include:

    • Occupancy Class Group(s)

    • Amount of Insurance Range(s)

    • State(s)

  • Supply premium for each selection

  • Each state has different occupancy/class demographics


Slide35 l.jpg

Constructing a Size of Loss Distribution for a “Selection”

  • Record output - Layer Average Severity

  • Combine all records in selection:

LASSelection = Wt Average(LASRecords)

Use the record’s claim count as weights


Slide36 l.jpg

Constructing a Size of Loss Distribution for a “Selection”

Where:

i = ith overall weight parameter

wij = ith weight parameter for the jth record

Cj = Claim weight for the jth record


The combined size of loss distribution for several selections l.jpg
The Combined Size of Loss “Selection”Distribution for Several “Selections”

  • Claim Weights for a “selection” are proportional to Premium Claim Severity

  • LASCombined = Wt Average(LASSelection)

  • Using the “selection” total claim weights

  • The definition of a “selection” is flexible


The combined size of loss distribution for several selections38 l.jpg
The Combined Size of Loss “Selection”Distribution for Several “Selections”

  • Calculate i’s for groups for which you have pure premium information.

  • Calculate the average severity for jth group


The combined size of loss distribution for several selections39 l.jpg
The Combined Size of Loss “Selection”Distribution for Several “Selections”

  • Calculate the group claim weights

  • Calculate the weights for the treaty size of loss distribution


The deductible problem l.jpg
The Deductible Problem “Selection”

  • The above discussion dealt with ground up coverage.

  • Most property insurance is sold with a deductible

    • A lot of different deductibles

  • We need a size of loss distribution net of deductibles


Size of loss distributions net of deductibles l.jpg
Size of Loss Distributions “Selection”Net of Deductibles

  • Remove losses below deductible

  • Subtract deductible from loss amount

Relative Frequency


Size of loss distributions net of deductibles42 l.jpg
Size of Loss Distributions “Selection”Net of Deductibles

  • Combine over all deductibles

    LASCombined Post Deductible

    Equals

    Wt Average(LASSpecific Deductible)

  • Weights are the number of claims over each deductible.


Size of loss distributions net of deductibles43 l.jpg
Size of Loss Distributions “Selection”Net of Deductibles

For an exponential distribution:

Net severity

Need only adjust frequency -- i.e. wi’s


Slide44 l.jpg

Adjusting the w “Selection”i’s

  • Dj jth deductible amount

  • ij

  • Wi


Goodness of fit summary l.jpg
Goodness of Fit - Summary “Selection”

  • 16 Tables

  • Fits ranged from good to very good

  • Model LAS was not consistently over or under the empirical LAS for any table

  • Model unlimited average severity

    • Over empirical 8 times

    • Under empirical 8 times


A major departure from traditional property size of loss tabulations l.jpg
A Major Departure from Traditional Property Size of Loss Tabulations

  • Tabulate by dollars of insured value

  • Traditionally, property size of loss distributions have been tabulated by % of insured value.


Fitted average severity against insured value l.jpg
Fitted $ Average Severity Tabulationsagainst Insured Value


Fitted average severity as of insured value l.jpg
Fitted Average Severity as Tabulations% of Insured Value

Blow up this area


Fitted average severity as of insured value49 l.jpg
Fitted Average Severity as Tabulations% of Insured Value

Eventually, assuming that loss distributions based on a percentage of AOI will produce layer costs that are too high.


Psold demonstration l.jpg
PSOLD Demonstration Tabulations

  • No Information

  • Size of Building Information

  • Size + Class Information

  • Size + Class + Location Information


Comparison with ludwig tables l.jpg
Comparison with Ludwig Tables Tabulations

  • Tabulated by % of amount of insurance

  • Organized by occupancy class and amount of insurance

    • Broader AOI classes

    • Broader occupancy classes

  • Fewer occurrances

  • No model

  • A very good paper


Comparison with ludwig tables52 l.jpg
Comparison with Ludwig Tables Tabulations

  • Ludwig — Exhibit 15 (all classes)

  • Matched insured value ranges

  • Obtained % of insured value distributions from PSOLD

    • assuming low end of range

    • assuming high end of range

  • Results on Spreadsheet


What s new for the next review l.jpg
What’s new for the next review? Tabulations

  • Include data through 1998

  • Fewer exclusions of loss information

    • Recall that we excluded claims if exposure and class information were missing.

    • Include claims if we trust the losses and use Bayesian techniques to spread losses to possible class and exposure groups.

  • Include HPR classes