The Minimum Bias Procedure for Two Rating Variables and Two Categories Per Variable: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Basic Ratemaking, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Steps in the Minimum Bias Procedure for Two Rating Variables and Two Categories per Variable

The minimum bias procedure is an iterative technique that endeavors to account for possible correlations between multiple rating variables in an insurer's rating structure. Here, we will work with a simple case of the minimum bias procedure, applicable to a multiplicative insurance rating structure.

Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Let B be some assumed base rate.

Then the following are the steps in the Minimum Bias Procedure for determining the relativities pertaining to the categories for these two variables:

1. Select one of the variables - here, variable X. Set category x₂ to have a relativity of 1.

2. Set category x₁ to have a relativity of (Pure premium for x₁)/(Pure premium for x₂).

3. Set up the following equation:

(Pure premium for (x₁, y₁))*(Exposures for (x₁, y₁)) + (Pure premium for (x₂, y₁))*(Exposures for (x₂, y₁)) = B*(Exposures for (x₁, y₁))*(Relativity for x₁)*(Relativity for y₁) + B*(Exposures for (x₂, y₁))*(Relativity for x₂)*(Relativity for y₁).

Solve this equation for (Relativity for y₁).

4. Set up the following equation:

(Pure premium for (x₁, y₂))*(Exposures for (x₁, y₂)) + (Pure premium for (x₂, y₂))*(Exposures for (x₂, y₂)) = B*(Exposures for (x₁, y₂))*(Relativity for x₁)*(Relativity for y₂) + B*(Exposures for (x₂, y₂))*(Relativity for x₂)*(Relativity for y₂).

Solve this equation for (Relativity for y₂).

5. Now it will be necessary to recalculate (Relativity for x₁) and (Relativity for x₂) on the basis of the values of (Relativity for y₁) and (Relativity for y₂) found in steps 3 and 4.

Set up the following equation:

(Pure premium for (x₁, y₁))*(Exposures for (x₁, y₁)) + (Pure premium for (x₁, y₂))*(Exposures for (x₁, y₂)) = B*(Exposures for (x₁, y₁))*(Relativity for x₁)*(Relativity for y₁) + B*(Exposures for (x₁, y₂))*(Relativity for x₁)*(Relativity for y₂).

Solve this equation for (Relativity for x₁).

6. Set up the following equation:

(Pure premium for (x₂, y₁))*(Exposures for (x₂, y₁)) + (Pure premium for (x₂, y₂))*(Exposures for (x₂, y₂)) = B*(Exposures for (x₂, y₁))*(Relativity for x₂)*(Relativity for y₁) + B*(Exposures for (x₂, y₂))*(Relativity for x₂)*(Relativity for y₂).

Solve this equation for (Relativity for x₂).

7. Now it will be necessary to recalculate (Relativity for y₁) and (Relativity for y₂) on the basis of the values of (Relativity for x₁) and (Relativity for x₂) found in steps 5 and 6.

The recalculation of relativities will need to continue until the next iteration produces a result that is substantively identical to that of the previous iteration. This iteration process stabilizes at relativities that reasonably reflect the interaction between the two variables.

8. After the stabilized relativities have been calculated, it is typical to normalize these relativities by setting one category of each variable (here, x₂ for X and y₂ for Y) to equal 1. Then, the other category's relativity (here, the relativities of x₁ for X and y₁ for Y) will be the ratio

(Non-normalized relativity for x₁)/(Non-normalized relativity for x₂) or (Non-normalized relativity for y₁)/(Non-normalized relativity for y₂).

9. If normalization has been done, the base rate B needs to be adjusted as follows:
(New B) = (Original B)*(Non-normalized relativity for x₂)*(Non-normalized relativity for y₂).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 10, pp. 167-171.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S5-80-1. Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Combination (x₁, y₁) has 356 exposures and $430 of pure premium.

Combination (x₁, y₂) has 462 exposures and $221 of pure premium.

Combination (x₂, y₁) has 636 exposures and $500 of pure premium.

Combination (x₂, y₂) has 300 exposures and $800 of pure premium.

The assumed base rate is $200.

Begin a minimum bias procedure as follows: Set category x₂ to have a starting ("seed") relativity of 1. What would be the starting ("seed") relativity for category x₁?

Solution S5-80-1. The starting relativity for category x₁ would be (Pure premium for x₁)/(Pure premium for x₂).

We are given pure premium by combinations of categories for each variable. To figure out

(Pure premium for x₁), we need to perform an exposure-weighting of pure premiums for combination (x₁, y₁) and combination (x₁, y₂):

((Pure premium for (x₁, y₁))*(Exposures for (x₁, y₁)) + (Pure premium for (x₁, y₂))*( Exposures for (x₁, y₂)))/(Total exposures for x₁) =

(430*356 + 221*462)/(356 + 462) = (Pure premium for x₁) = 311.9584352.

A similar exposure-weighting of pure premiums can be performed for combination (x₂, y₁) and combination (x₂, y₂) to get (Pure premium for x₂):

((Pure premium for (x₂, y₁))*(Exposures for (x₂, y₁)) + (Pure premium for (x₂, y₂))*( Exposures for (x₂, y₂)))/(Total exposures for x₁) =

(500*636 + 800*300)/(636 + 300) = (Pure premium for x₂) = 596.1538462.

Our starting relativity for category x₁ is (Pure premium for x₁)/(Pure premium for x₂) = 311.9584352/596.1538462 = 0.5232851171.

Problem S5-80-2. Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Combination (x₁, y₁) has 356 exposures and $430 of pure premium.

Combination (x₁, y₂) has 462 exposures and $221 of pure premium.

Combination (x₂, y₁) has 636 exposures and $500 of pure premium.

Combination (x₂, y₂) has 300 exposures and $800 of pure premium.

The assumed base rate is $200.

Continue the minimum bias procedure from Problem S5-80-1 by determining the first iteration of relativities for y₁ and y₂.

Solution S5-80-2. Per Solution S5-80-1, the seed relativities we will use for x₁ and x₂ are 0.5232851171 and 1, respectively. We apply the following formula:

(Pure premium for (x₁, y₁))*(Exposures for (x₁, y₁)) + (Pure premium for (x₂, y₁))*(Exposures for (x₂, y₁)) = B*(Exposures for (x₁, y₁))*(Relativity for x₁)*(Relativity for y₁) + B*(Exposures for (x₂, y₁))*(Relativity for x₂)*(Relativity for y₁).

Substituting known values, the equation becomes the following:

430*356 + 500*636 =

200*356*0.5232851171*(Relativity for y₁) + 200*636*1*(Relativity for y₁) →

471080 = 164457.9003*(Relativity for y₁) →

471080/164457.9003 = (Relativity for y₁) = 2.864441289.

We now apply the following formula:

(Pure premium for (x₁, y₂))*(Exposures for (x₁, y₂)) + (Pure premium for (x₂, y₂))*(Exposures for (x₂, y₂)) = B*(Exposures for (x₁, y₂))*(Relativity for x₁)*(Relativity for y₂) + B*(Exposures for (x₂, y₂))*(Relativity for x₂)*(Relativity for y₂).

Substituting known values, the equation becomes the following:

221*462 + 800*300 = 200*462*0.5232851171*(Relativity for y₂) + 200*300*(Relativity for y₂) → 342102 = 108351.5448*(Relativity for y₂) → 342102/108351.5448 = (Relativity for y₂) = 3.157333849.

Problem S5-80-3. Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Combination (x₁, y₁) has 356 exposures and $430 of pure premium.

Combination (x₁, y₂) has 462 exposures and $221 of pure premium.

Combination (x₂, y₁) has 636 exposures and $500 of pure premium.

Combination (x₂, y₂) has 300 exposures and $800 of pure premium.

The assumed base rate is $200.

Continue the minimum bias procedure from Problem S5-80-2 by determining the second iteration of relativities for x₁ and x₂.

Solution S5-80-3. Per Solution S5-80-2, the relativities we will use for y₁ and y₂ are 2.864441289and 3.157333849, respectively. We apply the following formula:

(Pure premium for (x₁, y₁))*(Exposures for (x₁, y₁)) + (Pure premium for (x₁, y₂))*(Exposures for (x₁, y₂)) = B*(Exposures for (x₁, y₁))*(Relativity for x₁)*(Relativity for y₁) + B*(Exposures for (x₁, y₂))*(Relativity for x₁)*(Relativity for y₂).

Substituting known values, the equation becomes the following:

430*356 + 221*462 = 200*356*2.864441289*(Relativity for x₁) + 200*462*3.157333849*(Relativity for x₁) →

255182 = 296351.3771*(Relativity for x₁) →

255182/296351.3771 = (Relativity for x₁) = 0.8610791773.

We now apply the following formula:

(Pure premium for (x₂, y₁))*(Exposures for (x₂, y₁)) + (Pure premium for (x₂, y₂))*(Exposures for (x₂, y₂)) = B*(Exposures for (x₂, y₁))*(Relativity for x₂)*(Relativity for y₁) + B*(Exposures for (x₂, y₂))*(Relativity for x₂)*(Relativity for y₂).

Substituting known values, the equation becomes the following:

500*636 + 800*300 = 200*636*2.864441289*(Relativity for x₂) + 200*300*3.157333849*(Relativity for x₂) →

558000 = 553796.9629*(Relativity for x₂) → 558000/553796.9629 = (Relativity for x₂) = 1.007589491.

Problem S5-80-4. Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Combination (x₁, y₁) has 356 exposures and $430 of pure premium.

Combination (x₁, y₂) has 462 exposures and $221 of pure premium.

Combination (x₂, y₁) has 636 exposures and $500 of pure premium.

Combination (x₂, y₂) has 300 exposures and $800 of pure premium.

The assumed base rate is $200.

Suppose that the relativities for x₁ and x₂ and y₁ and y₂ from Solutions S5-80-3 and S5-80-2, respectively, were accepted as the relativities that would be used in the insurer's rating structure - even though further iterations of the minimum bias procedure might produce more accurate relativities. If this decision were made, what would be the normalized relativities, assuming that the second category of each variable were treated as the base category?

Solution S5-80-4. Since x₂ and y₂are our base categories, it follows, by definition, that (Relativity for x₂) = (Relativity for y₂) = 1.

We find (Relativity for x₁) = (Non-normalized relativity for x₁)/(Non-normalized relativity for x₂) = 0.8610791773/1.007589491 = (Relativity for x₁) = 0.8545932494.

We find (Relativity for y₁) = (Non-normalized relativity for y₁)/(Non-normalized relativity for y₂) = 2.864441289/3.157333849 = (Relativity for y₁) = 0.9072342128.

Problem S5-80-5. Assume that there are two rating variables, X and Y, each with two categories, x₁ and x₂ and y₁ and y₂, respectively.

Assume that we also know the pure premium and number of exposures for each of the possible combinations of categories of these variables, i.e., (x₁, y₁), (x₁, y₂), (x₂, y₁), (x₂, y₂).

Combination (x₁, y₁) has 356 exposures and $430 of pure premium.

Combination (x₁, y₂) has 462 exposures and $221 of pure premium.

Combination (x₂, y₁) has 636 exposures and $500 of pure premium.

Combination (x₂, y₂) has 300 exposures and $800 of pure premium.

The assumed base rate is $200.

Suppose that the normalization in Problem S5-80-4 has been done. What would be the adjusted base rate, as a consequence of the normalization?

Solution S5-80-5. The adjusted base rate is (New B) = (Original B)*(Non-normalized relativity for x₂)*(Non-normalized relativity for y₂) = 200*1.007589491*3.157333849 = 636.2592812 = $636.26.

See other sections of The Actuary's Free Study Guide for Exam 5.