A.5. Robertson  HG Mapping
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Hazard Group
Collection of WC classifications that have relatively similar expected excess loss factors over a broad range of limits.

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

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

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

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 retrorating

Class excess ratios
 j = HG, X_{i} = loss for injury type i, L = limit
 S_{i}(r) = normalized state excess ratio for injury type i = E[max(X_{i}/μ_{i}  r,0)], r = L/μ
 R_{j}(L) = excess ratio = ∑w_{i,j}S_{i}(L/μ_{i,j})
 w_{i,j} = % loss due to injury i in group j and μ_{i,j} = average cost per case for injury i in group j

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

Steps of cluster analysis
 selection of loss limits
 metrics to evaluate cluster distances
 standardization

Metrics used
 Euclidian distance L^{2} = ‖x  y ‖_{2} = √∑(x_{i}  y_{i})^{2;} penalizes large deviations
 L^{1} = ‖x  y ‖_{1} = ∑x_{i}  y_{i}; minimizes relative error = PLR x R_{j}(L)  R_{c}(L)

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)

kmeans clustering technique
 kmeans groups classes into k HG so as to minimize ∑∑‖R_{c}  R_{i} ‖_{2}^{2} where centroid R_{i} = 1/HG_{i} ∑R_{c} is average XS ratio vector for the i^{th} HG
 kmeans 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 reassigned
 weights: to avoid letting small classes have undue influence, premiumweigh each class

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 PseudoF 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)

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

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