**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 5, authored by Mr. Stolyarov. This is Section 79 of the Study Guide. See an index of all sections by following the link in this paragraph.**

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 Adjusted Pure Premium Approach for Specific Rating Variables**

The Pure Premium Approach, discussed in Section 77, can be adjusted via a weighting of exposures by subgroup of the rating variable being considered, such that the outcome is equivalent to that of the Loss Ratio Approach discussed in Section 78.

For instance, the number of earned exposures in each subgroup of Variable A might be multiplied by a subgroup-specific adjustment factor, based on the average relativity for that subgroup with respect to a different rating variable or set of variables (e.g., Variables B, C, etc.).

This is easiest to illustrate via application, as will be done via the problems in this section.

**Source:**

Werner, Geoff and Claudine Modlin. *Basic Ratemaking**.* Casualty Actuarial Society. 2009. Chapter 9, pp. 163-166.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S5-79-1.** 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.

There is another rating variable being used: Variable Q. Exposures in each category of Variable X are not uniformly distributed with respect to Variable Q. There are three categories for Variable Q: g, h, and i.

An actuary makes the following determinations in an attempt to account for the effect of Variable Q:

The exposures in category g should be adjusted by a relativity of 1.02.

The exposures in category h should be adjusted by a relativity of 0.89.

The exposures in category i should be adjusted by a relativity of 0.66.

The following is the distribution of exposures by Variables X and Q:

Category g of Variable Q has 20 exposures in category a, 40 exposures in category b, and 300 exposures in category c.

Category h of Variable Q has 190 exposures in category a, 200 exposures in category b, and 50 exposures in category c.

Category i of Variable Q has 180 exposures in category a, 110 exposures in category b, and 150 exposures in category c.

*Per category of Variable X*, what is the adjustment factor that the actuary needs to apply to the number of exposures to account for the effect of the distribution of exposures with respect to Variable Q? Give an answer for **(a)** category a, **(b)** category b, and **(c)** category c.

**Solution S5-79-1.**

**(a)** Category a of X has 390 total exposures: 20 exposures in category g of Q, 190 exposures in category h of Q, and 180 exposures in category i of Q. The adjustment factor needed is a weighted average of the relativities for each category in Q:

(20*1.02 + 190*0.89 + 180*0.66)/390 = **0.7905128205.**

**(b)** Category b of X has 350 total exposures: 40 exposures in category g of Q, 200 exposures in category h of Q, and 110 exposures in category i of Q. The adjustment factor needed is a weighted average of the relativities for each category in Q:

(40*1.02 + 200*0.89 + 110*0.66)/350 = **0.8325714286.**

**(c)** Category c of X has 500 total exposures: 300 exposures in category g of Q, 50 exposures in category h of Q, and 150 exposures in category i of Q. The adjustment factor needed is a weighted average of the relativities for each category in Q:

(300*1.02 + 50*0.89 + 150*0.66)/500 = **0.899.**

**Problem S5-79-2.** This problem has the same conditions as Problem S5-79-1.

(a) How many adjusted exposures are there for category a?

(b) How many adjusted exposures are there for category b?

(c) How many adjusted exposures are there for category c?

(d) How many adjusted exposures are there overall?

**Solution S5-79-2.** For each category, (Number of adjusted exposures) = (Number of actual exposures)*(Category-specific relativity for adjusting exposures). In Solution S5-79-1, we found, for each category of X, the category-specific relativity for adjusting the exposures.

**(a)** For category a, (Number of adjusted exposures) = 390*0.7905128205= **308.3.**

**(b)** For category b, (Number of adjusted exposures) = 350*0.8325714286= **291.4.**

**(c)** For category b, (Number of adjusted exposures) = 500*0.899 = **449.5.**

**(d)** The overall number of adjusted exposures is the sum of the adjusted exposures per category: 308.3 + 291.4 + 449.5 = **1049.2.**

**Problem S5-79-3.** This problem has the same conditions as Problem S5-79-1.

(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 S5-79-3.**

**(a)** For category a, the indicated pure premium is (Losses and LAE)/(Number of adjusted exposures) = 120125/308.3 = 389.6367175 = **$389.64.**

**(b)** For category b, the indicated pure premium is (Losses and LAE)/(Number of adjusted exposures) = 123012/291.4 = 422.1413864 = **$422.14.**

**(c)** For category c, the indicated pure premium is (Losses and LAE)/(Number of adjusted exposures) = 230234/449.5 = 512.2002225 = **$512.20.**

**(d)** The indicated overall pure premium is (Sum of Losses and LAE)/(Sum of adjusted exposures) = (120125 + 123012 + 230234)/1049.2 = 451.1732749 = **$451.17.**

**Problem S5-79-4.** This problem has the same conditions as Problem S5-79-1.

(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 S5-79-4.** In Solution S5-79-3, we calculated the overall indicated pure premium and the indicated pure premiums by category. For each category, the indicated relativity is equal to (Indicated pure premium by category)/(Overall indicated pure premium).

**(a)** For category a, the indicated relativity is 389.6367175/451.1732749 = **0.8636077073.**

**(b)** For category b, the indicated relativity is 422.1413864/451.1732749 = **0.9356524642.**

**(c)** For category c the indicated relativity is 512.2002225/451.1732749 = **1.135262772.**

**Problem S5-79-5.** The adjusted pure premium approach produces results equivalent to those of the loss ratio approach. If one wanted to obtain such results, in what situations would it be necessary to use the adjusted pure premium approach instead of the loss ratio approach?

**Solution S5-79-5.** This question is based on the discussion in Werner and Modlin, p. 163.

It would be necessary to use the adjusted pure premium approach instead of the loss ratio approach in situations **where it is not possible to obtain premium at current rate levels for every category of the rating variable in question.** The loss ratio approach requires premium to be at current levels, whereas the adjusted pure premium approach does not consider premium directly.

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