MultiVariate Analytics

The Balance Principle Function finds the Class Relativity Factors that minimize the sum of all of the Weighted Errors.

An Error is the difference between an Observed Relativity Factor and Indicated Relativity Factor.
The Errors are weighted for Earned Car Years, and totaled.
The absolute value of the total is always lower than what the Least Squares Function and the X-Squared Function yield. (See "0.00" in the example.)

The Least Squares Function finds the Class Relativity Factors that minimize the sum of all of the Squared Errors.

A Squared Error is the difference between an Observed Relativity Factor and an Indicated Relativity Factor, squared.
The Squared Errors are weighted for Earned Car Years, and totaled.
The total is always lower than what the Balance Principle Function and the X-Squared Function yield. (See "41.46" in the example.)

The X-Squared Function finds the Class Relativity Factors that minimize the sum of all the Relative Squared Errors.

A Relative Squared Error is the difference between an Observed Relativity Factor and an Indicated Relativity Factor, squared and divided by the Indicated Relativity Factor.
The Relative Squared Errors are weighted for Earned Car Years, and totaled.
The total is always lower than what the Balance Principle Function and the Least Squares Function yield. (See "23.85" in the example.)

Observed Relativity Factors are computed by multiplying the Current Relativity Factors for each cell by the Relative Loss & ALAE Ratio for the cell.

Relative Loss & ALAE Ratios are computed by dividing each multi-dimensional cell's Loss & ALAE Ratio by the Loss & ALAE Ratio in the base cell.
If all the Loss & ALAE Ratios in all of the cells were equal, no adjustment in the Current Relativity Factors would be indicated. To the degree they differ, adjustments are in order. High Loss & ALAE Ratios indicate a need for higher relativity factors; and vice versa for low ratios.
The use of Loss & ALAE Ratios instead of Loss Costs compensates for an uneven distribution of business along other classification dimensions. It eliminates the potentially distorting effects of the other classification dimensions that are not being analyzed in the Minimum Bias Functions.

An implicit Credibility component exists in each Minimum Bias Function since Earned Car Years are used to weight the errors in each multi-dimensional cell.

If the Earned Car Years in a multi-dimensional cell are low, Credibility should be applied to the Minimum Bias Function indication.
The Credibility component embedded in a Minimum Bias Function measures the relative relationship of the Earned Car Years amongst the multi-dimensional cells.
A second Credibility algorithm should be exercised to address the absolute value of the Earned Car Years or Claim Count in each cell. eRateMaker^® provides such an algorithm where the Minimum Bias Function Indicated Class Relativities can be credibility weighted with the Current Class Relativities.

Driver Class	Territory	Earned Car Years	Trended Premium at Current Rate Level	Ultimate Loss & ALAE at Future Cost Level	Loss & ALAE Ratio	Relative Loss & ALAE Ratio	Current Relativity Factor	Observed Relativity Factor	Indicated Balance Principle Factor	Indicated Least Squares Factor	Indicated X-Squared Factor
Male	Urban	400	$288,000	$201,600	70.0%	1.17	6.00	7.00	7.12	7.03	7.11
Male	Rural	200	$44,000	$39,600	90.0%	1.50	2.00	3.00	2.76	2.84	2.79
Female	Urban	300	$90,000	$72,000	80.0%	1.33	3.00	4.00	3.84	3.92	3.86
Female	Rural	100	$9,000	$5,400	60.0%	1.00	1.00	1.00	1.49	1.58	1.52

		Errors (Indicated - Observed)			Errors Weighted for Earned Car Years per Balance Principle Criterion (Totals are Absolute Values)			Errors Squared & Weighted for Earned Car Years per Least Squares Criterion			Errors Squared & Divided by Indicated Factor & Weighted for Earned Car Years per X-Squared Criterion
Driver Class	Territory	Balance Principle	Least Squares	X-Squared	Balance Principle	Least Squares	X-Squared	Balance Principle	Least Squares	X-Squared	Balance Principle	Least Squares	X-Squared
Male	Urban	0.12	0.03	0.11	48.59	13.10	43.20	5.90	0.43	4.67	0.83	0.06	0.66
Male	Rural	-0.24	-0.16	-0.21	-48.59	-32.47	-41.34	11.80	5.27	8.55	4.28	1.86	3.06
Female	Urban	-0.16	-0.08	-0.14	-48.59	-23.50	-42.11	7.87	1.84	5.91	2.05	0.47	1.53
Female	Rural	0.49	0.58	0.52	48.59	58.23	51.68	23.61	33.91	26.70	15.89	21.43	17.61
				Totals	*0.00*	15.37	11.43	49.18	*41.46*	45.83	23.05	23.82	*22.85*

MultiVariate Analyses using Minimum Bias Functions

Balance Principle

Least Squares

X-Squared

Maximum Likelihood for a Normal Density Function

Maximum Likelihood for an Exponential Density Function

Printed
Exhibit 003.4