Exposure Bases, Methods of Aggregating Exposures, and In-Force Exposures: 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.

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.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 4, pp. 50-57.

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

Problem S5-17-1. Historically, exposure base A has been used for a particular line of insurance. However, because of advances in technology, exposure base B - which more directly corresponds to expected loss - is now practical to use. However, there may still be some negative side effects due transitioning from exposure base A to exposure B. Werner and Modlin (50) discuss three of these side effects. What are they?

Solution S5-17-1. The three negative side effects discussed are as follows:

1. Individual insureds may encounter large premium changes due to the change in exposure base.

2. A change in the rating algorithm will be required due to the change in exposure base. This may require significant efforts to alter the rating manuals and systems.

3. Ratemaking analysis typically relies on multiple years of data. If the exposure base is changed now, data from the past will need to be adjusted to the new standard in order to make the data before the change comparable to the data after the change.

Note that none of these side effects is necessarily a reason not to transition to a new exposure base. They are simply the likely costs of such a transition and should be compared to the benefits.

Problem S5-17-2. Which of the following statements about composite rating are true? More than one statement may be correct.

(a) Composite rating is typically used for small personal risks.
(b) Composite rating is typically used for small commercial risks.
(c) Composite rating is typically used for large commercial risks.
(d) Composite rating is typically used when there is a single, easily measurable exposure base.
(e) Composite rating is typically used when there are multiple perils being covered, and a different exposure base is used for each aspect of coverage.

(f) Composite rating often involves the use of a proxy measure for overall change in exposure.
(g) Loss-rated composite rating is based on the individual insured's historical loss experience.
(h) Loss-rated composite rating uses rigorously standardized rating algorithms.

Solution S5-17-2. The following answers are correct:

(c) Composite rating is typically used for large commercial risks.
(e) Composite rating is typically used when there are multiple perils being covered, and a different exposure base is used for each aspect of coverage.
(f) Composite rating often involves the use of a proxy measure for overall change in exposure.
(g) Loss-rated composite rating is based on the individual insured's historical loss experience.

Small personal and commercial risks typically do not have complicated exposure bases and so do not require composite rating. Thus, (a) and (b) are incorrect. Answer (d) is incorrect, because composite rating makes no sense if there is only one exposure base that is easy to measure directly. Answer (h) is incorrect, because composite rating typically does not utilize any standard rating algorithms.

Problem S5-17-3. According to Werner and Modlin (51-52), there are two methods of aggregating exposures. What are they? Which of these methods can assign a single earned exposure to multiple time periods?

Solution S5-17-3. The two methods of aggregating exposures are the calendar year or calendar-accident year method (note that these two methods are different for aggregating loss data, but the same for aggregating exposures) and the policy year method. Of these, the calendar year method is the one that can assign a single earned exposure to multiple time periods. For instance, a policy written for a term of one year in September 2030 would have some of the earned exposure assigned to calendar year 2030, and the rest assigned to calendar year 2031. By definition, a policy is in force only during the policy year, so the policy year method always assigns the entirety of the earned exposure to the policy year in question.

Problem S5-17-4. A policy of insurance is issued on May 1, 2045. The term of the policy is one year, and it is assumed that one unit of exposure corresponds to the full policy term. Also assume that the probability of a claim is evenly distributed throughout the term of the policy. Use both the calendar year method and the policy year method to determine how many earned exposure units are allocated to each year by each method.

Solution S5-17-4. The policy is in force from May 1, 2045, to May 1, 2046. This means that, under the calendar year method, the policy is in force for 8 out of 12 months in 2045, and for 4 of 12 months in 2046. Since a full year corresponds to 1 unit of exposure, the calendar year method assigns 8/12 = 2/3 units of earned exposure to calendar year 2045 and 4/12 = 1/3 units of earned exposure to calendar year 2046. For the policy year method, the entirety of the earned exposure is always allocated to the year in which the policy was written, so 1 year of earned exposure is allocated to policy year 2045.

Problem S5-17-5. There are 7 policies of insurance, with the following terms and dates of issuance:
Policy A has a term of 3 years and was issued on October 1, 3046.

Policy B has a term of 5 months and was issued on October 1, 3046.

Policy C has a term of 1 year and was issued on January 1, 3047.

Policy D has a term of 9 months and was issued on April 1, 3047.

Policy E has a term of 9 months and was issued on July 1, 3047.

Policy F has a term of 1 year and was issued on July 1, 3047.

Policy H has a term of 1 year and was issued on August 1, 3047.

Each policy has one unit of exposure associated with it. Policies written for shorter time periods have a proportionally higher amount of the relevant exposure base at risk..

Calculate the difference between the number of in-force exposures on July 1, 3047 and the number of in-force exposures on July 1, 3048.

Solution S5-17-5. Since each policy corresponds to a unit of exposure, the number of in-force exposures at a given time will correspond to the number of policies that are written at or before that time and have an expiration date after that time.

On July 1, 3047, the following policies are in force: A, C, D, E, F.

B is not in force, because B expired on March 1, 3047. H is not in force, because H will only get written 1 month later on August 1, 3047.

On July 1, 3048, the following policies are in force: A, H.

C, D, E, and F have all expired on or before July 1, 3048, so they are no longer in force.

Thus, the desired difference is 5-2 = 3.

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