A Fictional Commercial Insurance Experience Rating Plan: 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 15, pp. 283-288.

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

All of the questions in this section apply to the following hypothetical example:

Insurance Company Σ writes commercial general liability insurance and employs an experience rating plan based on the following formula:

CD = Z*(AER-EER)/EER, where
CD = credit or debit percentage;
Z = credibility assigned to the data of the risk;
AER = actual experience ratio;
EER = expected experience ratio.

Manuel's Manual Manual Manufacturing, a company, owned by Manuel, that employs workers to write study manuals by hand, is the insured under this program. The credibility of its data is 32%, and the expected experience ratio is 0.723. The historical loss and loss adjustment expense trend has been calculated to be an increase of 5% per year. Also, it is expected that approximately 20% of all losses that occur in a given year are reported in that year. Every subsequent year, another 20% of those losses become reported, until 100% of the losses are reported. Experience from the past two policy periods is used to determine any credits or debits applicable to the policy. The policy is written on an occurrence basis, and the current policy period is the entire year of 2102. For this year, the manual premium is $43,140, subject to any experience rating adjustments.

All historical data are evaluated as of January 1, 2102, and all losses for a given year are assumed to occur at a point in time on January 1 of that year (for instance, all 2101 losses are assumed to occur on January 1, 2101).

The following information is known:

During the years 2100 and 2101, Manuel's Manual Manual Manufacturing has had total reported actual losses and allocated loss adjustment expenses (ALAE) of $210,120.

The annual loss and ALAE cost underlying the insurer's current manual rates is $220,150.

Problem S5-116-1. What are the total "company subject loss and ALAE costs" - i.e., the total manual loss and ALAE costs, as applicable to the historical experience period?

Solution S5-116-1. This question is based on the discussion in Werner and Modlin, pp. 286-288. Loss and ALAE costs within the historical period must be detrended, i.e., adjusted for the reverse of the loss and ALAE trend that applies when going forward in time. The annual trend factor here is 1.05, so the annual detrend factor is 1/1.05. Thus, the manual loss and ALAE costs applicable to 2101 are 220150*(1/1.05), and the manual loss and ALAE costs applicable to 2100 are 220150*(1/1.05²). Together, the "company subject loss and ALAE costs" are

220150*(1/1.05 + 1/1.05²) = 409349.2063 = $409,349.21.

Problem S5-116-2. What are the total expected unreported losses and ALAE for the historical experience period, as of the start of 2102?

Solution S5-116-2. This question is based on the discussion in Werner and Modlin, pp. 286-288.

If 20% of losses/ALAE for a given year are reported in the first year, and 20% are reported in each year thereafter, then, as of the start of 2102, 80% of year 2101 losses/ALAE are unreported, and 60% of year 2100 losses/ALAE are unreported.

The way to calculate the expected unreported losses is via the following formula:
(Expected Unreported Losses) = (Detrended Expected Loss & ALAE Costs)*(Expected Experience Ratio)*(Expected Percentage of Losses Unreported).

For 2100, (Expected Unreported Losses) = (220150*(1/1.05²))*0.723*0.60 = 86622.28571.

For 2101, (Expected Unreported Losses) = (220150*(1/1.05))*0.723*0.80 = 121271.2.

Thus, the total expected unreported losses are 86622.28571 + 121271.2 = 207893.4857 = $207,893.49.

Problem S5-116-3. What are the projected ultimate losses and ALAE for the historical experience period?

Solution S5-116-3. This question is based on the discussion in Werner and Modlin, pp. 286-288.

The projected ultimate losses and ALAE for the historical experience period are the actual reported losses and ALAE, plus the expected unreported losses and ALAE.

The actual reported losses and ALAE are given as $210,120.

The expected unreported losses and ALAE are $207,893.49, based on Solution S5-116-2.

The projected ultimate losses and ALAE for the historical experience period are thus
$210,120 + $207,893.49 = $418,013.49.

Problem S5-116-4. What is the company's actual experience ratio (AER) for the historical period?

Solution S5-116-4. This question is based on the discussion in Werner and Modlin, pp. 286-288.

AER here is calculated as (Projected Ultimate Losses and ALAE)/("Company Subject" Loss and ALAE Costs). From Solution S5-116-1, the "Company Subject" Loss and ALAE Costs are $409,349.21, whereas the Projected Ultimate Losses and ALAE are $418,013.49. Thus, AER = 418013.49/409349.21 = AER = 1.021165976.

Problem S5-116-5. How much actual premium will the company pay for coverage in the year 2102?

Solution S5-116-5. This question is based on the discussion in Werner and Modlin, pp. 286-288.

Now that we found AER to be 1.021165976 in Solution S5-116-4, we can calculate the credit or debit using the formula CD = Z*(AER-EER)/EER. Here, Z = 0.32, and EER = 0.723. Thus, CD = 0.32*(1.021165973 - 0.723)/0.723 = 0.1319683436. This is the premium debit - i.e., the percentage by which the actual premium will be in excess of the manual premium of $43,140. The actual premium is thus 43140*1.1319683436 = 48833.11434 = $48,833.11.

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