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Commercial Property Size of Loss Distributions

Commercial Property Size of Loss Distributions

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Commercial Property Size of Loss Distributions

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  1. Commercial Property Size of Loss Distributions Glenn Meyers Insurance Services Office, Inc. Casualty Actuaries in Reinsurance June 15 , 2000 Boston, Massachusetts

  2. 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

  3. 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

  4. Exposures • Reported separately for building and contents losses • Model is based on combined building and contents exposure • Even if time element losses are covered

  5. 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

  6. 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

  7. 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

  8. 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.

  9. 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.

  10. Back to the Credibility Problem

  11. Back to the Credibility Problem

  12. 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?

  13. 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.

  14. Step 1Choose 0  d11  1 Shifting the weight from 1st exponential to the 2nd exponential increases the expected claim cost.

  15. Step 2Choose 0  d12  1 Shifting the weight from 2nd exponential to the 3rd exponential increases the expected claim cost.

  16. 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.

  17. Several AOI GroupsChoose W’s for lowest AOI Group

  18. Then choose d’s toConstruct W’s for the 2nd AOI Group

  19. Then choose d’s toConstruct W’s for the 3rd AOI Group

  20. Then choose d’s toConstruct W’s for the 4th AOI Group

  21. Continue choosing d’s and constructing W’s until the end.

  22. 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:

  23. 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!

  24. 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

  25. 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

  26. There’s No Theorem Like Bayes’ Theorem • Let be n parameter sets. • Then, by Bayes’ Theorem:

  27. 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:

  28. 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.

  29. 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

  30. Graph of Log-Likelihoods

  31. Prior Probabilities • Set: • Final formula becomes: • Can base update prior on Pr{Wk |X}.

  32. 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.

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. The Combined Size of LossDistribution for Several “Selections” • Calculate the group claim weights • Calculate the weights for the treaty size of loss distribution

  40. 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

  41. Size of Loss Distributions Net of Deductibles • Remove losses below deductible • Subtract deductible from loss amount Relative Frequency

  42. 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.

  43. Size of Loss Distributions Net of Deductibles For an exponential distribution: Net severity Need only adjust frequency -- i.e. wi’s

  44. Adjusting the wi’s • Dj jth deductible amount • ij • Wi

  45. 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

  46. 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.

  47. Fitted $ Average Severity against Insured Value

  48. Fitted Average Severity as % of Insured Value Blow up this area

  49. 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.

  50. PSOLD Demonstration • No Information • Size of Building Information • Size + Class Information • Size + Class + Location Information