Robertson - ELFs for HGs

Question 1

NCCI publishes Excess Loss Factors (ELFs) to assist with

Answer 1

pricing excess policies and some retrospectively rated policies for WC

Question 2

NCCI calculates ELFs at level

Answer 2

hazard group and state level for each possible limit

ELFs are the same for every class in a given hazard group within a state for a given limit

Question 3

hazard groups are defined on

Answer 3

CW basis since NCCI believes that mix of injuries within a class should not vary much by state

Question 4

in 2007, NCCI implemented new hazard group definitions and moved from

Answer 4

4 to 7 hazard groups

Question 5

end goal of new analysis was to produce

Answer 5

new hazard group definitions at CW level which could later be used to produce new ELFs for every limit, state, and hazard group combo

-also required deciding for which limits to calculate ELFs, deciding on # of hazard groups, and deciding which classes go into each hazard group

Question 6

to group classes with similar vectors into potential hazard groups, can use

Answer 6

L2 or Euclidean distance

-this would have advantage in that it minimizes the relative error in estimating excess premium

Question 7

relative error in estimating excess premium for class c with limit L is

Answer 7

PLR*|R_HG(L)-R_c(L|

Question 8

NCCI performed a cluster analysis called ___ to group classes into hazard groups using premium weights

Answer 8

k-means algorithm

Question 9

NCCI used cluster analysis to group classes with similar

Answer 9

vectors of excess ratios as measured by Euclidean distance into hazard groups

Question 10

used __ clustering

Answer 10

non-hierarchical

which meant new hazard groups did not have to be subsets of existing groups and could instead represent best partition for given number of clusters

Question 11

weighted k-means algorithm they used works as follows

Answer 11

1. decide on # of clusters (potential hazard groups) k to target
2. assign classes to k arbitrary groups
3. calculate centroid of excess ratios of each group (essentially weighted excess ratios)
4. for each class, find closest centroid using L2 distance
5. move each class to group with closet centroid
6. if any classes move go back to step 2 and continue

Question 12

k-means clustering is equivalent to

Answer 12

to max R squared statistic in linear regression

it minimizes variance within and maximizes the variance between

which means hazard groups will be homogeneous and well separated

Question 13

can think of R2 formula as

Answer 13

Trace(B)/Trace(T)

or

1-Trace(W)/Trace(T)

Question 14

2 statistics used to decide on #HGs

Answer 14

1. Calinski and Harabasz statistic
2. Cubic Clustering Criterion (CCC) statistic

Question 15

Calinski and Harabasz statistic

Answer 15

-measures between variance of clusters/within variance

[Trace(B)/(k-1)]/[Trace(W)/(n-k)]

n=#classes, k=#HGs

• higher values of this statistic indicate better # of clusters
• test also known as Pseudo-F test since it resembles F-test of regression analysis

*higher statistic indicates better clusters with higher between-cluster variance and lower within-cluster variance

Question 16

Cubic Clustering Criterion (CCC) statistic

Answer 16

compares amount of variance explained by given set of clusters to that expected when clusters are formed at random based on multi-dimensional uniform distribution

• high value indicates better performance
• since CCC is less reliable when data is highly correlated, NCCI gave more weight to #1

Question 17

result of 7 HGs was

Answer 17

best choice under #1 and 2nd best under #2

-to make final decision, NCCI re-ran analysis using only classes with 50% and 100% credibility and both statistics showed 7 as optimal #

Question 18

reasons NCCI decided to not give further consideration to CCC test that showed 9 groups as optimal:

Answer 18

1. found that #1 outperformed #2 statistic
2. CCC deserves less weigh when correlation is present which was case
3. selection of #HGs ought to be driven by large classes where most of experience was concentrated; using highly or fully credible classes showed 7 as optimal #
4. there was crossover in excess ratios between HGs when using 9 groups which is something that is not appealing in practice (somewhat counterintuitive)

Question 19

NCCI sent proposed HGs to underwriters to review and made some changes based on feedback; considered things such as:

Answer 19

1. similarity between class codes that were in different groups
2. degree of exposure to automobile accidents in given class
3. extent heavy machinery is used in given class

Question 20

Robertson sums up by saying remapping of HGs was founded on 3 key ideas:

Answer 20

1. computing excess ratios by class
2. sorting classes based on excess ratios
3. cluster analysis

Question 21

Why did the NCCI move from using 17 limits for ELFs to 5 limits

Answer 21

• ELFs at any pair of excess limits are highly correlated across classes.
• Limits below $100,000 were heavily represented in the prior 17 limits.
• Using 1 limit wouldn’t have captured the full variability in excess ratios.
• The NCCI wanted to cover the range of limits commonly used for rating

Question 22

What is the formula the NCCI used for determining the credibility to apply to the class vectors of excess ratios in the analysis described by Robertson?

Give 5 other credibility options the NCCI considered in the analysis

Answer 22

z=min(n/n+k,1); n=#claims in class; k=avg # claims per class

i. Using the median instead of the average for k.
ii. Excluding Medical Only claims from the analysis.
iii. Including only Serious claims in the analysis.
iv. Requiring a minimum # of claims for classes used in the calculation of k.
v. Various square root rules, such as z =sqrt(n/384), which corresponds to a 95% chance of n being within 10% of its expected value.

Question 23

Provide two reasons why the new hazard groups were superior to the prior hazard groups.

Answer 23

• The excess ratios for the new hazard groups were well separated compared to the prior hazard groups, meaning that there was less overlap in the excess ratios between classes in different hazard groups. • Amoresophisticatedmethodwasusedtodeterminetheoptimalnumberofhazardgroups. The NCCI used the Calinski and Harabasz and CCC statistics to decide on using 7 hazard groups. • A more sophisticated method was used to group classes into hazard groups by using the k-means algorithm. • The new hazard groups had a more even concentration of classes and premium, while previously classes were primarily concentrated in 2 of the 4 hazard groups. • As part of the analysis, NCCI updated their excess ratio calculations, which had not been done since 1993. Thus the 2007 excess ratios used more recent/applicable data. • The NCCI used an underwiter review process for finalizing the new hazard groups, which helped identify issues such as classes in different hazard groups with similar exposures. • The prior analysis used proxy variables to measure excess loss potential, while the new analysis used excess ratios directly. • The new analysis used excess ratios by injury type, while the prior analysis treated all injury types the same

Question 24

Provide two reasons why a class code should be assigned to a different hazard group than the hazard group resulting from the cluster analysis.

Answer 24

• If a class code has very similar exposure to another class code, it may make sense to assign both to the same hazard group. • If a class code has a high degree of exposure to automobile accidents that is not fully reflected in the past experience, it may make sense to move it to a higher risk hazard group. • If a class code has a high degree of heavy machinery usage that is not fully reflected in the past experience, it may make sense to move it to a higher risk hazard group. • If a class code has a high degree of exposure to dangerous substances that is not reflected in the past experience, it may make sense to move it to a higher risk hazard group.