A.5. Robertson - HG Mapping

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  1. Hazard Group
    Collection of WC classifications that have relatively similar expected excess loss factors over a broad range of limits.
  2. Original classification methodology
    • grouped 7 variables (indicative of excess loss potential) into 3 subsets based on correlation
    • ran principal components analysis to determine a single representative variable
    • first component was used to determine the classes
  3. WCIRB classification methodology
    • used 2 statistics to sort classes into HG
    • first statistic = % of claims XS of $150,00 (proxy for large loss potential)
    • second statistic = difference between class loss distribution and average loss distribution
  4. New classification methodology
    • sorted class into HG based on their XS ratios
    • note: a distribution is characterized by its excess ratios so there’s no loss of information 
    • used cluster analysis to group classes with similar XS ratios
    • determined the optimal # of HG
    • compared the new HG assignments with the prior assignments
  5. Selection of loss limits - why 5 instead of the prior 17
    • excess ratios at different limits are highly correlated
    • there were initially too many limits below $100,000
    • 1 limit didn’t capture full variability
    • matches range commonly used for retro-rating
  6. Class excess ratios
    • j = HG, Xi = loss for injury type i, L = limit
    • Si(r) = normalized state excess ratio for injury type i = E[max(Xii - r,0)], r = L/μ
    • Rj(L) = excess ratio = ∑wi,jSi(L/μi,j)
    • wi,j = % loss due to injury i in group j and μi,j = average cost per case for injury i in group j
  7. Credibility
    • previous: z = min[n / n + k) x 1.5, 1], n = # claims in class, k = average # of claims per class
    • this gave 72% of classes a credibility of 100%, which was perceived as an issue
    • considered excluding medical only claims, or including only serious claims
    • considered using the median rather than the mean for k, restricting k to classes with some minimal number of claims, square root method, advanced square root (by injury type)
    • → no alternative was compelling enough to warrant change, so stuck with original formula
  8. Steps of cluster analysis
    • selection of loss limits
    • metrics to evaluate cluster distances
    • standardization
  9. Metrics used
    • Euclidian distance L2 = ‖x - y ‖2 = √∑(xi - yi)2; penalizes large deviations
    • L1 = ‖x - y ‖1 = ∑|xi - yi|; minimizes relative error = PLR x |Rj(L) - Rc(L)|
  10. Standardization
    • applied to prevent a variable with large values from exerting undue influence on the results (each variable has a similar impact on the clusters). However:
    • groups already have a common denominator (limit) which would get filtered out
    • wanted to keep excess ratios between 0 and 1
    • analysis didn’t produce significantly different results with or without standardization
    • without it, excess ratios at lower loss limits have more influence on clusters; not undesirable since they rely on more observed values (rather than fitted distribution)
  11. k-means clustering technique
    • k-means groups classes into k HG so as to minimize ∑∑‖Rc - Ri22 where centroid Ri = 1/|HGi| ∑Rc is average XS ratio vector for the ith HG 
    • k-means algorithm: start with random clusters. Compute centroid of each cluster, and assign each class to the cluster with the closest centroid. Repeat until no class is re-assigned
    • weights: to avoid letting small classes have undue influence, premium-weigh each class
  12. Cooper and Milligan tests to find the best k
    • Calinski and Harabasz: higher values indicates a higher between distance (B) and lower within distance (W); aka Pseudo-F test
    • Cubic Clustering Criterion (CCC): compare variance explained by a given set of clusters to that of random clusters; less reliable when data is elongated (highly correlated variables), which is the case here (excess ratios correlated across limits)
  13. Underwriting criteria used to modify HGs
    • similarity between class codes that were in different groups
    • degree of exposure to automobile accident by class
    • extend heavy machinery is used in a given class
  14. Comparison to old hazard groups
    • new hazard groups have more even distribution of claims & premium by class
    • complement of credibility was prior hazard group (many small classes stayed grouped)
    • new hazard groups show less within variance and more between variance

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A.5. Robertson - HG Mapping
2015-09-05 13:42:25
Robertson Hazard Group Mapping

Robertson NCCI 2007 Hazard Group Mapping
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