MS Module 24 Least squares bias function practice exam questions

(The attached PDF file has better formatting.)

The mean value and the number of observations in each cell of a 2 × 2 classification table are

Means    Column 1    Column 2    Observations     Column 1    Column 2
Row 1    71    59    Row 1    11    12
Row 2    36    25    Row 2    14    19


Illustration: The cell in row 1 column 1 has a mean of 71 from a sample of 11 observations.

An actuary is setting class relativities for insurance pricing using a multiplicative model and a least squares bias function with

●    a base rate of 10
●    a starting relativity for column 1 of 1
●    a starting relativity for column 2 of 1.2


Question 24.1: Multiplicative model least squares implied relativity row 1

What is the implied relativity for Row 1, given the starting relativities by column?

Answer 24.1: (71 × 11 × 1.0 + 59 × 12 × 1.2) / (10 × (1.02 × 11 + 1.22 × 12) ) = 5.766

(relativities computed by taking partial derivatives to minimize the sum of the squared errors; see practice problems for the derivation)


Question 24.2: Multiplicative model least squares implied relativity row 2

What is the implied relativity for Row 2, given the starting relativities by column?

Answer 24.2: (36 × 14 × 1.0 + 25 × 19 × 1.2) / (10 × (1.02 × 14 + 1.22 × 19) ) = 2.597


Question 24.3: Multiplicative model least squares implied relativity column 1

What is the implied relativity for Column 1, given the computed relativities by row?

Answer 24.3: (71 × 11 × 5.766 + 36 × 14 × 2.597) / (10 × (5.7662 × 11 + 2.5972 × 14) ) = 1.263


Question 24.4: Multiplicative model least squares implied relativity column 2

What is the implied relativity for Column 2, given the computed relativities by row?

Answer 24.4: (59 × 12 × 5.766 + 25 × 19 × 2.597) / (10 × (5.7662 × 12 + 2.5972 × 19) ) = 1.009