Premium Trends, Exposure Trends, One-Step Trending, and Two-Step Trending: 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. 78-87.

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

Problem S5-37-1. Due to the Federal Reserve's enormous expansions of the money supply, the annual rate of inflation in 2010 and every year thereafter is 20%. Assume that there was no inflation in 2009. On February 1, 2010, X Insurance Company increased its rates by 6%. On December 1, 2011, the company decreased its rates by 9%. Earned premium for X Insurance Company's book of business in calendar year 2009 was $90,000. Bring this premium to the premium level which exists as of January 1, 2012.

Solution S5-37-1. Here, to bring the premium on level, we need to consider not just the company's rate changes, but also the inflation trend. Assume a starting premium level index of 1 in 2009. Each year, 20% inflation will cause the premium level index to be multiplied by a factor of 1.2. Here, this should be done twice, to account for inflation in 2010 and 2011. In addition, the company's two rate changes correspond to factors of 1.06 and 1-0.09 = 0.91, by which the premium level index should be multiplied. On January 1, 2012, the new premium level index is 1.2*1.2*1.06*0.91 = 1.389024, which means that the on-level CY 2009 premium is

90000*1.389024 = $125,012.16.

Problem S5-37-2. Company Γ has just purchased Company Δ. Both companies have never changed their rates before and, in their premiums and exposures, have hitherto experienced unchanging underlying conditions. Company Γ's book of business has 4890 historical earned exposures and has a total historical earned premium of $1,860,000. Company Δ's book of business has 9000 historical earned exposures and has a total historical earned premium of $426,000. Assuming that nothing else changes in the future, by what factor will Company Γ's pre-acquisition premium need to be multiplied in order to be brought to post-acquisition levels?

Solution S5-37-2. Prior to acquisition, Company Γ has an average rate of

(Earned Premium)/(Earned Exposures) = 1860000/4890 = 380.3680982. After acquisition of Company Δ, Company Γ has an average rate of (1860000+426000)/(4890+9000) = 164.5788337. The factor by which the average rate changes is thus 164.5788337/380.3680982 = 0.4326830628.

Problem S5-37-3. Actuary A is using Π Insurance Company's written premium and written exposure data to estimate average annual premium trends. All written premium data have been brought to the relevant current rate level. The actuary has the following data:
Year 5361, Quarter 1: Written premium = 34259; Written exposures = 135.

Year 5361, Quarter 2: Written premium = 36269; Written exposures = 145.

Year 5361, Quarter 3: Written premium = 33267; Written exposures = 112.

Year 5361, Quarter 4: Written premium = 42829; Written exposures = 133.

Year 5362, Quarter 1: Written premium = 46201; Written exposures = 137.

Year 5362, Quarter 2: Written premium = 44444; Written exposures = 111.

Year 5362, Quarter 3: Written premium = 47356; Written exposures = 122.

Year 5362, Quarter 4: Written premium = 42219; Written exposures = 138.

Four percentage estimates of annual changes in premium can be estimated from the above data. What are these four changes?

Solution S5-37-3. To estimate the annual premium level changes from quarterly data, it is necessary to divide the average written premium for Quarter X in Year Y by the average written premium for Quarter X in Year (Y-1) - i.e., the same quarter of the previous year. We are given 8 quarters of data, so we can estimate the change from any Quarter X of 5361 to the corresponding Quarter X of 5362. Here, average written premium is

(Written premium)/(Written exposures).

Thus, from Quarter 1 of 5361 to Quarter 1 of 5362, the following was the percent change in average written premium:
(42829/133)/(34259/135) - 1 = 0.2689525413 = 26.89525413%change from Quarter 1 of 5361 to Quarter 1 of 5362.
From Quarter 2 of 5361 to Quarter 2 of 5362, the following was the percent change in average written premium:

(44444/111)/(32629/145) - 1 = 0.779321385 = 77.9321385%change from Quarter 2 of 5361 to Quarter 2 of 5362.

From Quarter 3 of 5361 to Quarter 3 of 5362, the following was the percent change in average written premium:

(47356/122)/(33267/112) - 1 = 0.306831414 = 30.6831414% change from Quarter 3 of 5361 to Quarter 3 of 5362.

From Quarter 4 of 5361 to Quarter 4 of 5362, the following was the percent change in average written premium:

(42219/138)/(42829/133) - 1 = -0.0499585307 = -4.99585307% change from Quarter 4 of 5361 to Quarter 4 of 5362.

Problem S5-37-4. Actuary B uses one-step trending to estimate the premium trend for the book of business of Standardized Insurance Company, which consists entirely of one-year policies. B knows that in calendar year (CY) 2135, the earned premium was 4600 Golden Hexagons (GH). He estimates that the average annual premium growth is 3%. No other relevant changes that would affect the premium level have occurred between 2135 and 2140. B is trying to adjust the CY 2135 earned premium to the level at which it would need to be in order to accurately estimate the rate need for policy year (PY) 2140. What is this adjusted CY 2135 earned premium?

Solution S5-37-4. As discussed by Werner and Modlin, p. 83, the average written date for premium earned in a calendar year is the beginning of the year - i.e., January 1. The average written date for premium earned in a policy year is the middle of the year - i.e., June 30. Between the average written date for premium earned in CY 2135 - January 1, 2135 - and the average written date for premium earned in PY 2140 - June 30 2140 - there are 5.5 years. Thus, the trend factor by which the CY 2135 earned premium would need to be multiplied is 1.03^5.5 = 1.176534687, resulting in an adjusted earned premium of 4600*1.176534687 = 5412.059562 GH.

Problem S5-37-5. Actuary C uses two-step trending to estimate the premium trend for the book of business of Generalized Insurance Company, which consists entirely of one-year policies.

C knows that in calendar year (CY) 2222, the average earned premium was 3160 Golden Hexagons (GH). The average written premium for the latest available quarter, Quarter 3 of 2224, is 3536 GH. Thereafter, C estimates that the premium trend was 4% per year. No other relevant changes that would affect the premium level have occurred between 2222 and 2229. B is trying to adjust the CY 2222 earned premium to the level at which it would need to be in order to accurately estimate the rate need for policy year (PY) 2229. What is this adjusted CY 2222 earned premium?

Solution S5-37-5. Two-step trending involves selecting a current trend factor based on available data and a projected trend factor for time periods for which data are not yet available. The current trend factor applies to the time until Quarter 3 of 2224 and is equal to 3536/3160 = 1.118987342. Now we need to determine the length of time over which the projected premium trend factor would apply. The midpoint of Quarter 3 of a year occurs on August 15, with 4.5 months remaining in the year.

As discussed by Werner and Modlin, p. 83, the average written date for premium earned in a calendar year is the beginning of the year - i.e., January 1. The average written date for premium earned in a policy year is the middle of the year - i.e., June 30. Thus, the average written date for premium earned in PY 2229 is June 30, 2229. From August 15, 2224, to June 30, 2229, there are 4 + 1-1.5/12 = 4.875 years. Thus, the projected premium trend factor is 1.04^4.875 = 1.210702751.

The factor needed to adjust the CY 2222 earned premium to the current rate level is thus 1.118987342*1.210702751 = 1.354761053, leading to the adjusted premium of 1.354761053*3160 = 4281.044927 GH.

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