By NEAS - 6/24/2018 6:05:22 PM
MS Module 23: Actuarial risk classification – practice problems
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Reading on discussion forum: Actuarial risk classification
Exercise 23.1: Balance principle multiplicative model
The mean values 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 50 30 Row 1 15 12
Row 2 20 8 Row 2 6 10
Illustration: The cell in row 1 column 1 has a mean of 50 from a sample of 15 observations.
An actuary is setting class relativities for insurance pricing using a multiplicative model balance principle, with
● a base rate of 10
● a starting relativity for column 1 of 1.8
● a starting relativity for column 2 of 1.0
We use the following notation:
B = base rate
r1 = relativity for Row 1
r2 = relativity for Row 2
c1 = relativity for Column 1
c2 = relativity for Column 2
A. What are the observed totals for each cell, row, and column?
B. What are the formulas for each cell, row, and column using base rates and relativities?
C. What is the equation to balance along Row 1?
D. What is the implied relativity for Row 1, given the starting relativities by column?
E. What is the equation to balance along Row 2?
F. What is the implied relativity for Row 2, given the starting relativities by column?
G. What is the equation to balance down Column 1?
H. What is the implied relativity for Column 1, given the computed relativities by row?
I. What is the equation to balance down Column 2?
J. What is the implied relativity for Column 2, given the computed relativities by row?
Part A: The observed totals by cell are
● Row 1, Column 1: 50 × 15 = 750
● Row 1, Column 2: 30 × 12 = 360
● Row 2, Column 1: 20 × 6 = 120
● Row 2, Column 2: 8 × 10 = 80
The table below shows the totals by row and by column:
Column 1 Column 2 Total
Row 1 750 360 1,110
Row 2 120 80 200
Total 870 440 1,310
Part B: The formulas for the mean values by cell are
● Row 1, Column 1: B × r1 × c1
● Row 1, Column 2: B × r1 × c2
● Row 2, Column 1: B × r2 × c1
● Row 2, Column 2: B × r2 × c2
Using obssj,k as the number of observations in Row j and Column k, the totals by cell are
● Row 1, Column 1: B × r1 × c1 × obss1,1
● Row 1, Column 2: B × r1 × c2 × obss1,2
● Row 2, Column 1: B × r2 × c1 × obss2,1
● Row 2, Column 2: B × r2 × c2 × obss2,2
We add the expressions above for the totals by row and by column:
● Row 1: B × r1 × (c1 × obss1,1 + c2 × obss1,2)
● Row 2: B × r2 × (c1 × obss2,1 + c2 × obss2,2)
● Column 1: B × (r1 × obss1,1 + r2 × obss2,1) × c1
● Column 2: B × (r1 × obss1,2 + r2 × obss2,2) × c2
Part C: Using the formula for the Row 1 relativity, the base rate of 10, and the starting relativities of 1.8 for Column 1 and 1.0 for Column 2, we balance the observed and theoretical values to give
750 + 360 = 10 × r1 × (1.80 × 15 + 1.00 × 12)
Part D: The implied relativity for Row 1 is
r1 = (750 + 360) / (10 × (1.80 × 15 + 1.00 × 12) ) = 2.846154
Part E: Using the formula for the Row 2 relativity, the base rate of 10, and the starting relativities of 1.8 for Column 1 and 1.0 for Column 2, we balance the observed and theoretical values to give
120 + 80 = 10 × r2 × (1.80 × 6 + 1.00 × 10)
Part F: The implied relativity for Row 2 is
r2 = (120 + 80) / (10 × (1.80 × 6 + 1.00 × 10) ) = 0.961538
Part G: Using the formula for the Column 1 relativity, the base rate of 10, and the implied relativities of 2.846154 for Row 1 and 0.961538 for Row 2, we balance the observed and theoretical values to give
750 + 120 = 10 × (2.846154 × 15 + 0.961538 × 6) × c1
Part H: The implied relativity for Column 1 is
c1 = (750 + 120) / (10 × (2.846154 × 15 + 0.961538 × 6) ) = 1.795238
Part I: Using the formula for the Column 2 relativity, the base rate of 10, and the implied relativities of 2.846154 for Row 1 and 0.961538 for Row 2, we balance the observed and theoretical values to give
360 + 80 = 10 × (2.846154 × 12 + 0.961538 × 10) × c1
Part J: The implied relativity for Column 1 is
c2 = (360 + 80) / (10 × (2.846154 × 12 + 0.961538 × 10) ) = 1.005272
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