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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

Commercial Property Size of Loss Distributions

Glenn Meyers

Insurance Services Office, Inc.

Casualty Actuaries in Reinsurance

June 15 , 2000

Boston, Massachusetts

outline
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
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
Exposures
  • Reported separately for building and contents losses
  • Model is based on combined building and contents exposure
    • Even if time element losses are covered
classification strategy
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
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
the mixed exponential size of loss distribution
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
the mixed exponential size of loss distribution8
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
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
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
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
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
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
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.

estimating w s for the 1st aoi group and d s for the rest
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
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
Parameter Reduction
  • 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
On to Occupancy Groups
  • 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
There’s No Theorem Like Bayes’ Theorem
  • Let be n parameter sets.
  • Then, by Bayes’ Theorem:
bayesian results applied to an aoi and occupancy group
Bayesian Results Applied to an AOI and Occupancy Group
  • 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
What Does Bayes’ Theorem Give Us?
  • 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
Finding Suitable 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
Prior Probabilities
  • Set:
  • Final formula becomes:
  • Can base update prior on Pr{Wk |X}.
the classification data availability problem
The Classification Data Availability Problem
  • 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
Database Behind PSOLD

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

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

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

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
The Combined Size of LossDistribution 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
The Combined Size of LossDistribution 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
The Combined Size of LossDistribution for Several “Selections”
  • Calculate the group claim weights
  • Calculate the weights for the treaty size of loss distribution
the deductible problem
The Deductible Problem
  • 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
Size of Loss Distributions Net of Deductibles
  • Remove losses below deductible
  • Subtract deductible from loss amount

Relative Frequency

size of loss distributions net of deductibles42
Size of Loss Distributions 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
Size of Loss Distributions Net of Deductibles

For an exponential distribution:

Net severity

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

slide44

Adjusting the wi’s

  • Dj jth deductible amount
  • ij
  • Wi
goodness of fit summary
Goodness of Fit - Summary
  • 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
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 as of insured value49
Fitted Average Severity as % of Insured Value

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

psold demonstration
PSOLD Demonstration
  • No Information
  • Size of Building Information
  • Size + Class Information
  • Size + Class + Location Information
comparison with ludwig tables
Comparison with Ludwig Tables
  • 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
Comparison with Ludwig Tables
  • 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
What’s new for the next review?
  • 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
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