Aggregation of Written, Earned, and In-Force Exposures for Insurance Policies of Unequal Terms and the "15th of the Month" Rule: 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.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 4, pp. 57-61.

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

Problem S5-30-1. 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 G has a term of 1 year and was issued on August 1, 3047.

Each policy has the same exposure base at risk per unit of time, and one year is associated with one exposure unit on each policy.

For each policy, determine the number of written exposure units, as of December 31, 3049, associated with calendar years 3046, 3047, 3048, and 3049.

Solution S5-30-1. Written exposures are always assigned to the year in which the policy is issued (written) - irrespective of whether the policy term is entirely within that year. Thus, the policies that were written in 3046 will have all of their written exposure units in 3046 - and policies that were written in 3047 will have all of their written exposure units in 3047. Since 1 year of each policy corresponds to one exposure unit, n months of each policy corresponds to n/12 exposure units. No policies above were written in 3048 or 3049, so no written exposure units apply to those years. Therefore, the following are the answers:
Policy A has 36/12 exposure units = 3 written exposure units in 3046 for Policy A.

Policy B has 5/12 exposure units = 0.416666667 written exposure units in 3046 for Policy B.

Policy C has 12/12 exposure units = 1 written exposure unit in 3047 for Policy C.

Policy D has 9/12 exposure units = 0.75 written exposure units in 3047 for Policy D.

Policy E has 9/12 exposure units = 0.75 written exposure units in 3047 for Policy E.

Policy F has 12/12 exposure units = 1 written exposure unit in 3047 for Policy F.

Policy G has 12/12 exposure units = 1 written exposure unit in 3047 for Policy G.

Problem S5-30-2. 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 G has a term of 1 year and was issued on August 1, 3047.

Each policy has the same exposure base at risk per unit of time, and one year is associated with one exposure unit on each policy.

For each policy, determine the number of earned exposure units, as of December 31, 3049, associated with calendar years 3046, 3047, 3048, and 3049.

Solution S5-30-2.

In this situation, earned exposure units for a policy in a year are calculated by taking the number of months the policy will be in effect during that year and dividing that number by 12.

Policy A will expire on October 1, 3049. It thus lasts through the entirety of 3047 and 3048 and has one exposure unit in each of these years. It lasts for 3 months of 3046 and 9 months of 3049. Therefore, Policy A has 0.25 earned exposure units in 3046, 1 earned exposure unit in 3047, 1 earned exposure unit in 3048, and 0.75 earned exposure units in 3049.

Policy B will expire on March 1, 3047. It therefore lasts for 3 months of 3046 and 2 months of 3047. Therefore, Policy B has 0.25 earned exposure units in 3046 and 0.166666667 earned exposure units in 3047.

Policy C will expire on January 1, 3048. The entirety of its one-year term is in 3047, so Policy C has 1 earned exposure unit in 3047.

Policy D will expire on January 1, 3048. The entirety of its 9-month term is in 3047, so Policy D has 0.75 earned exposure units in 3047.

Policy E will expire on April 1, 3048. 6 months of its term are in 3047, and 3 months are in 3048. Thus, Policy E has 0.50 earned exposure units in 3047 and 0.25 earned exposure units in 3048.

Policy F will expire on July 1, 3048. 6 months of its term are in 3047, and 6 months are in 3048. Thus, Policy F has 0.50 earned exposure units in 3047 and 0.50 earned exposure units in 3048.

Policy G will expire on August 1, 3048. 5 months of its term are in 3047, and 7 months are in 3048. Thus, Policy G has 0.416666666667 earned exposure units in 3047 and 0.58333333333 earned exposure units in 3048.

Problem S5-30-3. 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 G has a term of 1 year and was issued on August 1, 3047.

Each policy has the same exposure base at risk per unit of time, and one year is associated with one exposure unit on each policy.

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 S5-30-3. 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. G is not in force, because G will only get written 1 month later on August 1, 3047.

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

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

For each policy that is in force, the full-term number of exposures is equal to the in-force exposures.
The full-term number of exposures for each policy is the same as its number of written exposures. We can find this number for each policy in Solution S5-30-1.

For policies A, C, D, E, and F, the sum of full-term exposure units is 3 + 1 + 0.75 + 0.75 + 1 = 6.5.

For policies A and G, the sum of full-term exposure units is 3 + 1 = 4.

The desired difference is 6.5 - 4 = 2.5.

Problem S5-30-4. There were 34 insurance policies written during the month of June 4450, 31 policies written during July 4450, and 90 policies written during August 4450. The "15^th of the month" rule (described in Werner and Modlin, p. 59) is being used to estimate the issuance dates of each policy. Assume that each policy has a term of one year and the same exposure base at risk, so that each policy has one exposure unit per year. Also assume that no other insurance policies were issued. How many in-force exposures are there as of August 4, 4450?

Solution S5-30-4. The "15^th of the month" rule assumes that a policy written in a given month was issued on the 15^th day of that month. Thus, none of the August exposures contribute to the in-force exposures on August 4, since all of the August policies are assumed to be written on August 15. Hence, only the June and July exposures comprise the in-force exposures on August 4, and our answer is 34 + 31 = 65 in-force exposure units.

Problem S5-30-5. There were 34 insurance policies written during the month of June 4450, 31 policies written during July 4450, and 90 policies written during August 4450. The "15^th of the month" rule (described in Werner and Modlin, p. 59) is being used to estimate the issuance dates of each policy. Assume that each policy has a term of one year and the same exposure base at risk, so that each policy has one exposure unit per year. Also assume that no other insurance policies were issued. Furthermore, assume that, due to calendar reforms promulgated in the year 4356, each month has exactly 30 days. How many earned exposures are there as of October 1, 4450?

Solution S5-30-5. The "15^th of the month" rule assumes that a policy written in a given month was issued on the 15^th day of that month. During the latter half of that month, 1/24 of the exposure is earned. So the June policies each have 1/24 earned exposures from June, 1/12 from July, 1/12 from August, and 1/12 from September, for a total of 7/24 earned exposures per policy. The July policies have 7/24 - 1/12 = 5/24 earned exposures per policy. The August policies have 5/24 - 1/12 = 3/24 earned exposures per policy.

Thus, the total number of earned exposures as of October 1, 4450, is

(7/24)*34 + (5/24)*31 + (3/24)*90 = 27.625 earned exposures.

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