Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 23

Some of the questions here ask for short written answers. This is meant to give the student practice in answering questions of the format that will appear on Exam 6. 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.

Some of the problems in this section were designed to be similar to problems from past versions of Exam 6, offered by the Casualty Actuarial Society. They use original exam questions as their inspiration - and the specific inspiration is cited to give students an opportunity to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Sources:
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009.

Harrison, C.M., Reinsurance Principles and Practices (First Edition), American Institute for Chartered Property Casualty Underwriters/Insurance Institute of America, 2004, Chapters 1, 2, 4, 8, 9, and 10.

Past Casualty Actuarial Society exams: 2008 Exam 6 and 2009 Exam 6.

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

Problem S6-50-1. Similar to Question 28 from the 2008 CAS Exam 6. A primary insurer purchased a $500,000 in excess of $300,000 per-occurrence excess-of-loss reinsurance treaty with an annual aggregate deductible of $600,000. The following occurrences are subject to the treaty:

Occurrence A: Loss of $210,000
Occurrence B: Loss of $700,000
Occurrence C: Loss of $550,000
Occurrence D: Loss of $1,600,000
Occurrence E: Loss of $320,000
Occurrence F: Loss of $480,000

What is the net loss retained by the primary insurer?

Solution S6-50-1. The aggregate deductible of $600,000 means that the primary insurer must pay $600,000 in excess-layer losses before the reinsurer pays anything.

We evaluate each occurrence in turn.

Occurrence A is below the primary insurer's retention, so the primary insurer will be responsible for the entire $210,000, with no contribution to the aggregate deductible.

For Occurrence B, $400,000 is above the primary insurer's retention, and, while the primary insurer will be liable for the entire amount of the loss - $700,000 - because of the aggregate deductible, this deductible will be reduced to 600000 - 400000 = $200,000.

For Occurrence C, $250,000 is above the primary insurer's retention. $200,000 of this amount exhausts the aggregate deductible, meaning that the reinsurer will pay the rest. The primary insurer will be liable for 300000 (retention) + 200000 (rest of deductible) = $500,000.

For Occurrence D, the reinsurer is only liable for the treaty layer of losses, or $500,000. The primary insurer is liable for the remaining $1,100,000.

For Occurrences E and F each, the primary insurer is only liable for the retention of $300,000.

Thus, the total net loss retained by the primary insurer is

210000 + 700000 + 500000 + 1100000 + 300000*2 = $3,110,000.

Problem S6-50-2. Similar to Question 29 from the 2008 CAS Exam 6. You are analyzing a $600,000 excess of $200,000 per-occurrence excess-of-loss reinsurance treaty, with the following occurrences subject to the treaty:

Occurrence G: Loss of $400,000; Incurred ALAE of $50,000
Occurrence H: Loss of $130,000; Incurred ALAE of $200,000
Occurrence I: Loss of $800,000; Incurred ALAE of $90,000
Occurrence J: Loss of $530,000; Incurred ALAE of $400,000
Occurrence K: Loss of $1,000,000; Incurred ALAE of $120,000

(a) What is the total loss amount ceded to the treaty if incurred ALAE is to be considered a part of the loss?

(b) What is the total loss amount ceded to the treaty if incurred ALAE is to be shared pro rata with the loss?

Solution S6-50-2. (a) If incurred ALAE is to be considered a part of the loss, then the losses subject to the treaty are as follows:

Occurrence G: Loss of $450,000 → Ceded amount is $250,000.
Occurrence H: Loss of $330,000 → Ceded amount is $130,000.
Occurrence I: Loss of $890,000 → Ceded amount is $600,000.
Occurrence J: Loss of $930,000 → Ceded amount is $600,000.
Occurrence K: Loss of $1,120,000 → Ceded amount is $600,000.

Total ceded amount: $2,180,000.

(b) If incurred ALAE is to be shared pro rata with loss, this means that the ratio of the treaty-covered loss to the total loss is also the ratio of the treaty-covered ALAE to the total ALAE.

Occurrence G: Loss of $400,000 → Treaty-covered loss of $200,000 → Treaty-covered ALAE of $50,000*(200/400) = $25,000 → Total ceded amount of $225,000.

Occurrence H: Loss of $130,000 → Treaty-covered loss of $0 → Treaty-covered ALAE of $0 → Total ceded amount of $0.

Occurrence I: Loss of $800,000 → Treaty-covered loss of $600,000 → Treaty-covered ALAE of 90000*(600/800) = $67,500 → Total ceded amount of $667,500.

Occurrence J: Loss of $530,000 → Treaty-covered loss of $330,000→ Treaty-covered ALAE of 400000*(330/530) = $249,056.60 → Total ceded amount of $579,056.60.

Occurrence K: Loss of $1,000,000 → Treaty-covered loss of $600,000 → Treaty-covered ALAE of 120000*(600/1000) = $72,000 → Total ceded amount of $672,000.

Total ceded amount for all occurrences: $2,143,556.60.

Problem S6-50-3. Similar to Question 28 from the 2009 CAS Exam 6. Consider the methods of experience rating and exposure rating in reinsurance. Which of these methods can produce the problem of free cover and why?Which of these methods can be applied to avoid it and how?

Solution S6-50-3. Experience rating can produce the problem of free cover if the losses on the basis of which reinsurance rates are determined do not trend into the highest layers of coverage being offered. Exposure rating can avoid this problem when applied to these upper layers. Experience rating can still be used for the lower layers where actual loss data exists. Exposure rating can be used to develop a relativity for the upper layers which is multiplied by the experience-based rate to get the total rate.

Problem S6-50-4. Similar to Question 31 from the 2009 CAS Exam 6.

You have the following information as of December 31, 2022:

Earned Premium for Calendar/Accident Year 2020: 11500
Earned Premium for Calendar/Accident Year 2021: 12024
Earned Premium for Calendar/Accident Year 2022: 14444

Adjusted Premium for Calendar/Accident Year 2020: 11000
Adjusted Premium for Calendar/Accident Year 2021: 12000
Adjusted Premium for Calendar/Accident Year 2022: 13000

Aggregate Reported Loss for Calendar/Accident Year 2020: 9700
Aggregate Reported Loss for Calendar/Accident Year 2021: 6400
Aggregate Reported Loss for Calendar/Accident Year 2022: 4100

Aggregate Loss Report Lag for Calendar/Accident Year 2020: 0.85
Aggregate Loss Report Lag for Calendar/Accident Year 2021: 0.66
Aggregate Loss Report Lag for Calendar/Accident Year 2022: 0.42

Note that "report lag" here refers to the proportion of losses assumed to be already reported, not the proportion assumed to have yet to be reported.

Use the Cape Cod method to calculate IBNR as of December 31, 2022.

Solution S6-50-4. We can calculate the expected loss ratio (ELR) as follows:

ELR = (Sum of aggregate reported losses for each year)/(Sum of used-up premiums for each year), where the used-up premium for each year is (Adjusted Premium)*(Aggregate Loss Report Lag).

Thus, ELR = (9700 + 6400 + 4100)/(11000*0.85 + 12000*0.66 + 13000*0.42) = 0.8886933568.

By the Cape Cod Method, IBNR = ELR*(Sum of adjusted premiums for each year) - (Sum of aggregate reported losses for each year) = 0.8886933568*(11000 + 12000 + 13000) - (9700 + 6400 + 4100) = 11792.96084 = $11,792.96.

Problem S6-50-5. Similar to Question 33 from the 2009 CAS Exam 6. What is aggregate excess-of-loss reinsurance? Name one goal of reinsurance for which aggregate excess-of-loss reinsurance is the best possible option. Name two other goals of reinsurance which aggregate excess-of-loss reinsurance does not fulfill as well.

Solution S6-50-5. Aggregate excess-of-loss reinsurance is a type of reinsurance that pays for all of the primary insurer's losses in a given time period (e.g., a year) once the sum of the primary insurer's losses exceeds the aggregate treaty retention. This type of reinsurance is the best at stabilizing loss experience, since the primary insurer is assured that its losses will not exceed a certain magnitude. It is not as good at providing surplus relief (since the aggregate retention will probably be set so that exceeding it is not probable), or facilitating withdrawal from a market, as the primary insurer still retains losses below the aggregate retention.

See other sections of The Actuary's Free Study Guide for Exam 6.