The Pure Premium Approach for Determining Relativities Pertaining to Specific Insurance Rating Variables: 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.

The Pure Premium Approach for Specific Rating Variables

Let R1 by a rating variable. Let R1_i be the rating factor (relativity) associated with subgroup i of R1. Let B be the base rate. Then the rate for subgroup i of R1 is as follows:

Formula 77.1: Rate_i = R1_i*B.

Let R1_I,i be the indicated rating factor (relativity) associated with subgroup i of R1. Let Rate_I.i be the indicated rate for subgroup i of R1. Let B_I be the indicated bate rate. Then the following formula holds:

Formula 77.2: R1_I,i= (Rate_I.i)/B_I.

Now we make the following assumptions, per Werner and Modlin, p. 158:

Assumption Set 77.3:

1. There are no fixed expenses;
2. All policies have the same underwriting expenses. The percentage comprised of underwriting expenses (which are all variable) is V;
3. All policies have the same profit provisions. The percentage profit provision selected by the company is Q_T.

If these assumptions are true, then the following formula holds:

Formula 77.4: R1_I,i= (L^- + E^-_L)_i/(L^- + E^-_L)_B.

Here, (L^- + E^-_L)_i is the pure premium (including loss adjustment expenses) for subgroup i of R1, whereas (L^- + E^-_L)_B is the pure premium (including loss adjustment expenses) for the class on the basis of which the base rate was established.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 9, pp. 156-160.

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

Problem S5-77-1. The base rate for an insurance program is $360. It is determined that subgroup q of Variable Ψ is, on an overall basis, 0.6 times as risky as the base class, and a relativity has been assigned to this subgroup accordingly.

(a) What is the rate for subgroup q of Variable Ψ?

(b) If the indicated base rate were $380, what would be the new rate for subgroup q of Variable Ψ?

Solution S5-77-1.

(a) We use Formula 77.1:Rate_i = R1_i*B. Here, R1_i = 0.6, and B = 360, and so our answer is 0.6*360 = $216.

(b) We use Formula 77.2:R1_I,i= (Rate_I.i)/B_I, which is equivalent to Rate_I.i = (R1_I,i)*B_I, where R1_I,i = 0.6 and B_I = 380. Thus, our answer is Rate_I.i = 0.6*380 = $228.

Problem S5-77-2. In this problem, Assumption Set 77.3 holds. Insurance Company Π experienced losses of $36 per exposure and loss adjustment expenses of $12 per exposure in calendar year 2024. In the same year, for every exposure in subgroup z of Variable Ξ, the company experienced losses of $19 per exposure and loss adjustment expenses of $10 per exposure. Based on this information, what is the indicated relativity pertaining to subgroup z of Variable Ξ?

Solution S5-77-2. We use Formula 77.4:R1_I,i= (L^- + E^-_L)_i/(L^- + E^-_L)_B. Here, (L^- + E^-_L)_B = 36 + 12 = 48, whereas (L^- + E^-_L)_i = 19 + 10 = 29. Our answer is thus R1_I,i= 29/48 = 0.6041666667.

Problem S5-77-3. In this problem, Assumption Set 77.3 holds. You are given the following information pertaining to an insurer's book of business in calendar year 2030:
Variable X is being used to classify insureds into three categories: a, b, and c.

There are 390 exposures in category a, and total losses and loss adjustment expenses (LAE) for category a are $120,125.

There are 350 exposures in category b, and total losses and LAE for category b are $123,012.

There are 500 exposures in category c, and total losses and LAE for category c are $230,234.

Assume that the same loss trend and loss development applies to each category.

(a) What is the indicated pure premium for category a?

(b) What is the indicated pure premium for category b?

(c) What is the indicated pure premium for category c?

(d) What is the indicated overall pure premium?

Solution S5-77-3.

(a) For category a, the indicated pure premium is (Losses and LAE)/(Number of exposures) = 120125/390 = 308.0128205 = $308.01.

(b) For category b, the indicated pure premium is (Losses and LAE)/(Number of exposures) = 123012/350 = 351.4628571 = $351.46.

(c) For category c, the indicated pure premium is (Losses and LAE)/(Number of exposures) =

230234/500 = 460.468 = $460.47.

(d) The indicated overall pure premium is (Total losses and LAE)/(Total number of exposures) = (120125 + 123012 + 230234)/(390 + 350 + 500) = 381.7524194 = $381.75.

Problem S5-77-4. In this problem, Assumption Set 77.3 holds. You are given the following information pertaining to an insurer's book of business in calendar year 2030:
Variable X is being used to classify insureds into three categories: a, b, and c.

There are 390 exposures in category a, and total losses and loss adjustment expenses (LAE) for category a are $120,125.

There are 350 exposures in category b, and total losses and LAE for category b are $123,012.

There are 500 exposures in category c, and total losses and LAE for category c are $230,234.

Assume that the same loss trend and loss development applies to each category.

(a) What is the indicated relativity for category a?

(b) What is the indicated relativity for category b?

(c) What is the indicated relativity for category c?

Solution S5-77-4. In Solution S5-77-3, we calculated the overall indicated pure premium and the indicated pure premiums by category. For each category, by Formula 77.4, the indicated relativity is equal to (Indicated pure premium by category)/(Overall indicated pure premium).

(a) The indicated relativity for category a is 308.0128205/381.7524194 = 0.80683921.

(b) The indicated relativity for category b is 351.4628571/381.7524194 = 0.9206565283.

(c) The indicated relativity for category c is 460.468/381.7524194 = 1.206195368.

Problem S5-77-5. Explain how the univariate Pure Premium Method is subject to distortions when one uses it to calculate the relativities associated with subgroups of specific rating variables.

Solution S5-77-5. The univariate Pure Premium Method implicitly assumes that exposure within each subgroup of a particular variable (e.g., Variable A) is uniformly distributed within each subgroup with respect to characteristics of all other variables within the insurer's rating structure (e.g., Variables B, C, etc.). However, if some variables are correlated with others, then performing a univariate Pure Premium Method analysis on each variable would result in "double-counting" (or "multiple-counting") of the effects of certain variables.

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