Comparisons of Various Types of Loss Trends in Insurance and Simple Methods of Estimating Unallocated Loss Adjustment Expenses: 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 6, pp. 115-120.

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

Problem S5-46-1. Which of the following statements are true if there is a positive loss severity trend? More than one answer may be correct:
(a) The basic limits trend is always greater than or equal to the total limits trend.
(b) The basic limits trend is always greater than or equal to the excess losses trend.
(c) The total limits trend is always greater than or equal to the basic limits trend.
(d) The total limits trend is always greater than or equal to the excess losses trend.
(e) The excess losses trend is always greater than or equal to the basic limits trend.
(f) The excess losses trend is always greater than or equal to the total limits trend.

Solution S5-46-1. This question is based on the discussion in Werner and Modlin, p. 115, where the following inequality is given for cases of positive severity trend:
Basic Limits Trend ≤ Total Limits Trend ≤ Excess Losses Trend.

Thus, the following answers are correct:
(c) The total limits trend is always greater than or equal to the basic limits trend.
(e) The excess losses trend is always greater than or equal to the basic limits trend.
(f) The excess losses trend is always greater than or equal to the total limits trend.

Problem S5-46-2. Which of the following statements are true if there is a negative loss severity trend? More than one answer may be correct:
(a) The basic limits trend is always greater than or equal to the total limits trend.
(b) The basic limits trend is always greater than or equal to the excess losses trend.
(c) The total limits trend is always greater than or equal to the basic limits trend.
(d) The total limits trend is always greater than or equal to the excess losses trend.
(e) The excess losses trend is always greater than or equal to the basic limits trend.
(f) The excess losses trend is always greater than or equal to the total limits trend.

Solution S5-46-2. This question is based on the discussion in Werner and Modlin, p. 115, where the following inequality is given for cases of negative severity trend:
Excess Losses Trend ≤ Total Limits Trend ≤ Basic Limits Trend.

Thus, the following answers are correct:
(a) The basic limits trend is always greater than or equal to the total limits trend.
(b) The basic limits trend is always greater than or equal to the excess losses trend.
(d) The total limits trend is always greater than or equal to the excess losses trend.

Problem S5-46-3. According to Werner and Modlin, p. 115, what three types of trends are considered in the projection of pure premiums into the forecast period when an actuary is developing a pure premium rate level indication?

Solution S5-46-3. The three trends are as follows, as stated in Werner and Modlin, p. 115:
1. Changes in the likelihood of a claim happening;
2. Changes in the average cost of claims;
3. Changes in the level of exposure.

Problem S5-46-4. An actuary is aware of the following loss and loss adjustment expense data for calendar years 4310, 4311, 4312, and 4313. Recall that ALAE stands for allocated loss adjustment expenses, and ULAE stands for unallocated loss adjustment expenses.

CY 4310: Paid Loss and ALAE: 4120 Golden Hexagons (GH); Paid ULAE: 456 GH.
CY 4311: Paid Loss and ALAE: 3333 GH; Paid ULAE: 900 GH.
CY 4312: Paid Loss and ALAE: 5640 GH; Paid ULAE: 1230 GH.
CY 4313: Paid Loss and ALAE: 4210 GH; Paid ULAE: 426 GH.

Find the ULAE ratio for each year and the ULAE Factor that could be derived from considering the data above and applied via multiplication to paid loss and ALAE data for subsequent years.

Solution S5-46-4. For each year, the ULAE ratio is (Paid ULAE)/(Paid Loss and ALAE)

For CY 4310, the ULAE ratio is 456/4120 = 0.1106796117.

For CY 4311, the ULAE ratio is 900/3333 = 0.2700270027.

For CY 4312, the ULAE ratio is 1230/5640 = 0.2180851064.

For CY 4313, the ULAE ratio is 426/4210 = 0.1011876485.

The total ULAE ratio is (Sum of all Paid ULAE)/(Sum of all Paid Loss and ALAE) =

(456 + 900 + 1230 + 426)/(4120 + 3333 + 5640 + 4210) = 0.17407386.

The ULAE factor is (1 + (Total ULAE Ratio)) = 1.17407386.

Problem S5-46-5. An actuary is aware of the following loss and loss adjustment expense data for calendar years 4310, 4311, 4312, and 4313. Recall that ALAE stands for allocated loss adjustment expenses, and ULAE stands for unallocated loss adjustment expenses.

CY 4310: Paid Loss and ALAE: 4120 Golden Hexagons (GH); Paid ULAE: 456 GH.
CY 4311: Paid Loss and ALAE: 3333 GH; Paid ULAE: 900 GH.
CY 4312: Paid Loss and ALAE: 5640 GH; Paid ULAE: 1230 GH.
CY 4313: Paid Loss and ALAE: 4210 GH; Paid ULAE: 426 GH.

You also know that paid losses and ALAE in CY 4314 were 7387 GH.

(a) Using a ULAE Factor derived from considering the data above and applied via multiplication to paid loss and ALAE data for subsequent years, find the estimated total ULAE pertaining to CY 4314.

(b) According to the discussion in Werner and Modlin, p. 119, what inherent assumption must be made about ULAE to use this method?

(c) Why might actual unallocated loss adjustment expenses for CY 4314 be higher than the amount that is estimated via this method? Hint: Look at the magnitude of the losses and ALAE for CY 4314 and compare them to the figures from earlier years.

Solution S5-46-5.

(a) In Solution S5-46-4, we found that the ULAE factor is 1.17407386 and the ULAE ratio is 0.17407386. To find just the estimated ULAE in CY 4314, we multiply the paid losses and ALAE for that year by the ULAE ratio. We get 0.17407386*7387 = 1285.883604 GH.

(b) According to the discussion in Werner and Modlin, p. 119, this method "inherently assumes that ULAE trend and develop at the same rate as loss plus ALAE."

(c) The paid losses and ALAE for CY 4314 appear to be substantially higher than for earlier years, suggesting that a catastrophic event may have occurred. Werner and Modlin, p. 119, discuss this possibility and comment that it is possible for unallocated loss adjustment expenses to be unusually high (in excess of the average ratio that would be observed in the absence of catastrophes) for catastrophic losses - for instance, because the company might need to establish temporary offices in the area affected by the catastrophe.

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