A.5. Robertson

Question 1

Why the NCCI moved from 17 limits to 5 for ELFs

Answer 1

1. ELFs at any pair of excess limits are highly correlated across classes.
2. Limits below $100,000 were heavily represented in the prior 17 limits.
3. They wanted to cover the range of limits commonly used for retrospective rating.

Question 2

General excess ratio formula

Answer 2

Question 3

Normalized excess ratio function for injury type i

Answer 3

Question 4

Injury type weighted excess ratios

Answer 4

Question 5

Credibility weighted class excess ratio vectors (Robertson)

Answer 5

Question 6

Other credibility options considered

Answer 6

Question 7

Measuring distance between vectors

Answer 7

Question 8

Why the NCCI didn’t use standardization

Answer 8

1. The resultant hazard groups using standardization didn’t differ much from not using it.
2. Excess ratios at different limits have a similar unit of measure, which is dollars of excess loss per dollar of total loss. Standardization would have eliminated this common denominator.
3. All excess ratios are between 0 and 1, while standardization could have led to results outside this range.
4. There is a greater range of excess ratios at lower limits, and this is a good thing since it is based on actual data (compared to excess ratios at higher limits being based more on fitted loss distributions). Standardization would have given this real data less weight.

Question 9

Weighted k-means algorithm

Answer 9

Question 10

Hierarchical clustering

Answer 10

Hierarchical clustering means any additional cluster would be a subset of a single existing cluster. Non-hierarchical clustering methods seek the best partition for any given number of clusters.

Question 11

Desirable optimality properties for k-means

Answer 11

K-means maximizes the equivalent of R² from linear regression, which means maximizing the percentage of total variation explained by the hazard groups. This is equivalent to stating that k-means minimizes the within variance and maximizes the between variance, which means the hazard groups will be homogeneous and well separated.

Question 12

Test statistics used to decide # of hazard groups

Answer 12

Question 13

Why the NCCI didn’t rely on the CCC test statistic showing 9 groups

Answer 13

1. Milligan and Cooper found the Calinski and Harabasz statistic outperformed the CCC statistic.
2. The CCC statistic deserves less weight when correlation is present, which was the case.
3. The selection of # of hazard groups ought to be driven by the large classes where most of the experience was concentrated. Using these highly or fully credible classes showed 7 as the optimal number.
4. There was crossover in the excess ratios between hazard groups when using 9 groups, which is something that isn’t appealing in practice.

Question 14

Underwriter considerations

Answer 14

1. Similarity between class codes that were in different groups.
2. Degree of exposure to automobile accidents in a given class.
3. Extent heavy machinery is used in a given class.

Question 15

3 key ideas of hazard group remapping

Answer 15

1. Computing excess ratios by class
2. Sorting classes based on excess ratios
3. Cluster analysis