Adjustments to Historical Premium and Written Premium Aggregation for Calendar Years and Policy Years: 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 5, pp. 63-67.

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

Problem S5-31-1. Werner and Modlin (63) discuss three adjustments that are generally made to historical premium figures in order to estimate premium for a future policy period. What are these three adjustments, and why are they important?

Solution S5-31-1.

First adjustment: Bringing the historical premium to the rate level currently in effect.

Why it is important: Any rate changes during or after the historical period to which the premium data apply need to be reflected in the premium figures. Otherwise, the data will not be comparable to one another.

Second adjustment: Develop premium to ultimate levels if premium is still changing.

Why it is important: If current premium data are different from what ultimate premium data are likely to be, the ultimate premium data would be more relevant to premium in the future.

Third adjustment: Project the historical premium to the premium level expected in the future.

Why it is important: This adjustment "accounts for changes in the mix of business that have occurred or are expected to occur after the historical experience period" (Werner and Modlin, p. 63).

Problem S5-31-2. The adjustments discussed in Problem S5-31-1 are required for which general approach to evaluate rate adequacy? For which approach are they not required? (Werner and Modlin, p. 63)

Solution S5-31-2. The adjustments are required for the loss ratio approach, but not for the pure premium approach. Only the loss ratio approach requires an estimation of the premium that will be collected in the future.

Problem S5-31-3. You know the following information on insurance policies, each of which is one year in term length:

Policy T was issued on January 1, 3109, and the policyholder paid a premium of $671.

Policy U was issued on June 1, 3109, and the policyholder paid a premium of $123.

Policy V was issued on August 1, 3109, and the policyholder paid a premium of $870.

Policy W was issued on December 1, 3109, and the policyholder paid a premium of $1300.

Policy X was issued on February 1, 3110, and the policyholder paid a premium of $50.

Policy Y was issued on March 1, 3110, and the policyholder paid a premium of $500.

Policy Z was issued on September 1, 3110, and the policyholder paid a premium of $600.

Assume that each policy was in force until its expiration date.

Find the sum of written premium for the calendar year 3109 as of January 1, 3111.

Solution S5-31-3. Any policy written in 3109 would have all of its premium classified as written premium for 3109. Thus, the premium for policies T, U, V, and W constitutes written premium for the year 3109. This is our answer: 673 + 123 + 870 + 1300 = $2966.

Problem S5-31-4. You know the following information on insurance policies, each of which is one year in term length:

Policy T was issued on January 1, 3109, and the policyholder paid a premium of $671.

Policy U was issued on June 1, 3109, and the policyholder paid a premium of $123.

Policy V was issued on August 1, 3109, and the policyholder paid a premium of $870.

Policy W was issued on December 1, 3109, and the policyholder paid a premium of $1300.

Policy X was issued on February 1, 3110, and the policyholder paid a premium of $50.

Policy Y was issued on March 1, 3110, and the policyholder paid a premium of $500.

Policy Z was issued on September 1, 3110, and the policyholder paid a premium of $600.

Now assume that Policy U was cancelled on March 1, 3110 and that Policy W was cancelled on February 1, 3110. Also, policy Z was cancelled by the policyholder on October 1, 3110. Both policyholders received a full (pro rata) refund of unearned premium.

Find the sum of written premium for the calendar year 3110 as of January 1, 3111.

Solution S5-31-4. The written premium for the calendar year 3110 is the sum of the premiums paid for policies issued in 3110 minus the sum of the unearned premiums for the policies that were cancelled in 3110 (irrespective of when these policies were written) - as these unearned premiums were refunded to the policyholders.

The sum of the premiums paid for policies issued in 3110 is the sum of the premiums for policies X, Y, and Z: 50 + 500 + 600 = $1150.

Policy U was cancelled after 7 months, which means that 5 months of premium - i.e., 123*(5/12) = $51.25 - were unearned.

Policy W was cancelled after 2 months, which means that 10 months of premium - i.e., 1300*(10/12) = $1083.33 - were unearned.

Policy Z was cancelled after 1 month, which means that 11 months of premium - i.e., 600*(11/12) = $550 - were unearned.

Thus, the sum of written premium for the year 3110 is 1150 - 51.25 - 1083.33 - 550 = -$534.58.

Problem S5-31-5. You know the following information on insurance policies, each of which is one year in term length:

Policy T was issued on January 1, 3109, and the policyholder paid a premium of $671.

Policy U was issued on June 1, 3109, and the policyholder paid a premium of $123.

Policy V was issued on August 1, 3109, and the policyholder paid a premium of $870.

Policy W was issued on December 1, 3109, and the policyholder paid a premium of $1300.

Policy X was issued on February 1, 3110, and the policyholder paid a premium of $50.

Policy Y was issued on March 1, 3110, and the policyholder paid a premium of $500.

Policy Z was issued on September 1, 3110, and the policyholder paid a premium of $600.

Now assume that Policy U was cancelled on March 1, 3110 and that Policy W was cancelled on February 1, 3110. Also, policy Z was cancelled by the policyholder on October 1, 3110. Both policyholders received a full (pro rata) refund of unearned premium.

Find the sum of written premium for the policy year 3110 as of January 1, 3111.

Solution S5-31-5. The written premium for the calendar year 3110 is the sum of the premiums paid for policies issued in 3110 minus the sum of the unearned premiums for the policies that were both issued and cancelled in 3110 - as these unearned premiums were refunded to the policyholders.

Important note: For policy year aggregation of written premium, any policies written in years other than 3110 and cancelled in 3110 would not affect written premium in 3110, since policy year aggregation applies any premium adjustments during the term of a policy to the year in which the policy was written.

The sum of the premiums paid for policies issued in 3110 is the sum of the premiums for policies X, Y, and Z: 50 + 500 + 600 = $1150.

Of the policies issued in 3110, only Z was cancelled in 3110. Policy Z was cancelled after 1 month, which means that 11 months of premium - i.e., 600*(11/12) = $550 - were unearned.

Thus, the sum of written premium for the policy year 3110 is 1150 - 550 = $600.

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