A5. NCCI's 2007 hazard group mapping Flashcards
Why the NCCI moved from 17 to 5 limits
- *** ELFs at different pairs of limits were highly correlated across classes
- Limits below $100,000 were too heavily represented in the 17 limits
- The range of limits commonly used for retrospective rating are well represented with as few as 5 limits
- Using only 1 limit would not have been enough to capture the full variability* in XS ratios
Summarize the process used in the NCCI 2007 HG mapping study
- Developed vectors of XS ratios at 5 selected limits for each class
- Stabilized data using credibility weighting of vectors of XS ratios
- Grouped classes with similar vectors of XS ratios using weighted k-means
- Enhanced groupings outliers and validation using PCA
- Determined optimal number of groups (7) using Calinski-Harabasz criterion
- Revised the 7 groupings based on the inputs of an underwriter panel review
step 1: What adjustments did Roberson do to loss data before calculating the XS ratios
- Trended/developped
- because trend/development vary by layer
- need to adjust for trend/development because XS ratios are used to price futur policy - On-level
- need to adjust for futur benefits level
step 2: Credibility formula to weight the XS ratios with the current HG XS ratio
Credibility formula retained by NCCI:
- Z=Min(n/(n+k) *1.5; 1)
- Advantage: gives more weight to larger classes
- Disadvantage: not appropriate for highly skewed distribution
Other options considered:
- using square root( Z=sqrt(n/384)), which gives 95% chance that n is within 10% of its expected value)
- using median instead of avg for k
- excluding med only claims in analysis
- including only serious claims in analysis
- requiring a minimum n for classes used in the calculation of k
Purpose of standardization
WHAT DOES IT PREVENT ?
WHEN APPROPRIATE ?
Purpose:
- ***when vectors of variables have different units and spreads **
- this prevents a variable with large values from exerting undue influence on results
Appropriate: if the spread of values is due to normal random variation
Not appropriate: if the spread of values is due to presence of sub-classes
Standardization of the XS ratios before applying clustering
Standardization retained by NCCI:
-none
Other options considered:
- XS ratio* = (XS ratio - mean)/std dev
- XS ratio* = (XS ratio - min)/(max - min)
Reasons to retain no standardization:
1. the results of the analysis were not sensitive to standardization
- loss of interpretability/common unit of measure
- loss of common range (XS ratio: between 0 and 1, XS ratio*: between -inf to +inf)
**4. loss of information for lower limits because XS ratios at lower limits are actual data/more volatil and XS ratios at higher limits are based on fitted distributions/less volatile
(XS ratio: more weight to lower limits and less weight to higher limits, XS ratio*: less weight to all limits)
Clustering algorithm
Definition:
-method of grouping similar risks based on a target metric
avantage (1)
- minimizes variance withing groups
- maximizes variance between groups
Advantage (2)
- allows the grouping of homogenous risks
- **-achieves greater credibility for each group
Types of clustering algorithms
- Non hierarchical clustering:
-produces optimal clusters that best correlate with expected cost
seeks to create the best k cluster regarless of what the best k-1 cluster were
no subject to parent child constraints
-Disadvantage: the groups might be less intuitive => could result in neighboring groups having significant different rates* - Hierarchical custering:
- any additional cluster is a subset/nested version of an existing cluster
step 3: Measure to calculate distance between vectors of XS ratios in the clustering algorithms
(1) L2/euclidean distance
- Penalyse large deviation between class and HG excess
- give undue weight to outliers
- better if there is a skwed distrivution
(2) L1 distance
- Advantage:
- doest penalyze large deviation more than the sum of many small deviation *
- **minimizes relative error in estimating the excess premium between hg and class
- unit of L1 is the* same as expected loss costs ***( $) while L2 has unit = $^2
- Disadvantage: more difficult to solve
Reason to retain the L2 distance:
- the results of the analysis were not sensitive to the choice of L2 vs L1
- therefore they retained the easiest one to solve
Advantages of k-means (optimality properties)
Equivalent to maximizing the R2 from linear regression: *maximizes the % of total variation explained by the hazard groups
- minimizes the variance within hazard groups => homogeneous HG
- maximizes the variance between hazard groups => well separated HG
step 4 : Advantages of PCA
PCA identifies variables most predictive of the outcome:
- Allows to eliminate other correlated variables to simplify the model
- -Allows to identify outliers and validate clusters separation visually
step 5: Tests statistics used to decide on the optimal # k of hazard groups
- HOW CCC WORKS
- WHY NOT RETAINED
(1) Cubic clustring criterion (or CCC)
- **Compares amount of variance explained by a given set of clusters to that expected when clusters are formed at random based on multidimensional uniform distribution ***
- higher is better
- indicated 9 hazard groups
Reasons to retain (2) Calinski-Harabasz
- Calinski-Harabasz outperforms CCC according to Milligan&Cooper paper
- CCC is less reliable if there is correlation between variables which is the case of XS ratios
- CCC also indicated 7 hazard groups if the small classes were excluded from the analysis
- CCC 9 hazard groups had some crossover between the XS ratios which is not appealing in practice ***
step 6: Underwriter considerations
- similarity * between classes* that were in different hazard groups => makes sense to assign them to the same hazard group
- degree of exposure to accidents in a given class => makes sense to move to a higher hazard group if there is a high degree of exposure, even if not reflected in past experience
- extent to which heavy machinery is used in a given class => makes sense to move to a higher hazard group if there is a lot of heavy machinery, even if not reflected in past experience
Why NCCI defined hazard groups on a country-wide basis, and not by state
- HG is a collection of classes that have similar ELFs over a wide range of limits
- NCCI view is based on homogeneous operations
- Therefore the mix of injuries within a class should not vary between states
ELFs are the same for every class in a given hazard group within a state for a given limit
why the new hazard groups were superior to the prior hazard groups.
- 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. ***
- A more sophisticated method was used to determine the optimal number of hazard groups. 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.
**• 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
