Trending of Fixed Expenses, Reinsurance Costs, and Treatment of Profit in Insurance Ratemaking: 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 7, pp. 132-137.

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

Problem S5-74-1. It is known that the average fixed expense per policy was $56 in calendar year (CY) 2021. The annual inflation rate during the next several years is 32%. It is assumed that the fixed expense is incurred at the onset of the policy. An actuary uses this information to trend the fixed expenses for policy year (PY) 2026. What is the trended fixed expense per policy for PY 2026?

Solution S5-74-1. According to Werner and Modlin, p. 133, "Expenses that are incurred at the inception of the policy should be trended from the average date that the policies were written in the historical period to the average written date in the projection period."

The average written date for a calendar year is the midpoint of that year. For CY 2021, the average written date is therefore June 30, 2021. Likewise, the average written date for a policy year is also the midpoint of the year. In the case of PY 2026, the midpoint is June 30, 2026. This means that the fixed expenses must be trended for five years of inflation, leading the trended fixed expense to be 56*1.32⁵ = 224.4189986 = $224.42.

Problem S5-74-2. It is known that the average fixed expense per policy was $56 in calendar year (CY) 2021. The annual inflation rate during the next several years is 32%. It is assumed that the fixed expense is incurred evenly throughout the policy period. An actuary uses this information to trend the fixed expenses for policy year (PY) 2026. What is the trended fixed expense per policy for PY 2026?

Solution S5-74-2. According to Werner and Modlin, p. 133, "expenses that are incurred evenly throughout the policy period should be trended from the average date the policies were earned in the historical period to the average earned date in the projection period."

The average written date for a calendar year is the midpoint of that year. For CY 2021, the average written date is therefore June 30, 2021. For a policy year, the average earned date is the beginning of the next calendar year. That is, for PY 2026, the average earned date of the policies is January 1, 2027; because, if policies are written evenly throughout the year, half the premium from all the policies will have been earned by that date. This means that the fixed expenses must be trended for 5.5 years of inflation, leading the trended fixed expense to be 56*1.32^5.5 = 257.8366493 = $257.84.

Problem S5-74-3.

(a) With what kinds of reinsurance is it typically unnecessary to consider reinsurance costs in ratemaking analysis? With what kinds of reinsurance is it often proper to consider reinsurance costs?

(b) In general terms, how are projected losses and projected premiums affected by taking reinsurance costs into account?

Solution S5-74-3. This question is based on the discussion of reinsurance costs by Werner and Modlin, p. 134.

(a) In ratemaking analysis, it is typically unnecessary to consider reinsurance costs for proportional reinsurance, where the primary insurer cedes the same fraction of premium and losses to the reinsurer. On the other hand, it is often proper to consider reinsurance costs for non-proportional reinsurance, where the reinsurer agrees to assume responsibility for some predetermined portion of losses, typically in excess of a certain amount. For instance, with catastrophe excess-of-loss reinsurance, the reinsurer might cover a certain proportion of losses in excess of a certain amount from a given occurrence. The primary insurer does not necessarily cede to the reinsurer the same fraction of premium as it cedes of losses.

(b) If reinsurance costs are taken into account, projected losses must be reduced by the amount of "expected non-proportional reinsurance recoveries." Projected premiums must be reduced "by the amount to be ceded to the reinsurer" (Werner and Modlin, p. 134).

Problem S5-74-4. Insurer Q lost $6002 on its investments in the year 2103 and earned an underwriting profit of $6310. Insurer Q collected $16000 in premium and incurred $5500 in losses and $2500 in loss adjustment expenses.

(a) What was Insurer Q's total profit in the year 2103?

(b) What were Insurer Q's underwriting expenses in the year 2103?

Solution S5-74-4.
(a) We use the formula Total Profit = Investment Income + Underwriting Profit.

Here, investment income is negative: -6002, so Total Profit = -6002 + 6310 = $308.

(b) We use the formula UW Profit = Premium - Losses - LAE - UW Expenses, rearranging the formula thus: UW Expenses = Premium - Losses - LAE - UW Profit = 16000 - 5500 - 2500 - 6310 = $1690.

Problem S5-74-5. Insurer Z estimates that 9% of its expenses are variable expenses and 3% are fixed expenses. It also has a target profit provision of 10%.

(a) What is Insurer Z's variable permissible loss ratio (VPLR)?

(b) What is Insurer Z's total permissible loss ratio (PLR)?

Solution S5-74-5.

(a) We use the formula VPLR = 1 - (Variable Expense%) - (Target Profit%) = 1 - 0.09 - 0.10 = 0.81 = 81%.

(b) We use the formula PLR = 1 - (Total Expense%) - (Target Profit%) = 1 - 0.09 - 0.03 - 0.10 = 0.78 = 78%.

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