Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 20

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:

Feldblum, S., Discussion of "An Exposure Rating Approach to Pricing Property Excess-of-Loss Reinsurance", PCAS LXXX, 1993, pp. 380-395.

Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2010.

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

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

Problem S6-47-1. Similar to Question 32 from the 2008 CAS Exam 6. A primary insurer has a 40% quota share treaty with one reinsurer, subject to a maximum reinsurance recovery of $200,000 per occurrence, and a $500,000 in excess of $200,000 per-occurrence excess-of-loss treaty with another reinsurer. The excess-of-loss treaty has a co-participation percentage of 10% above the retention.

The following occurrences are subject to the treaty:

Occurrence A has ground-up loss amount $500,000.
Occurrence B has ground-up loss amount $150,000.
Occurrence C has ground-up loss amount $750,000.
Occurrence D has ground-up loss amount $400,000.

(a) What is the primary insurer's total retained loss if neither treaty inures to the benefit of the other?

(b) What is the primary insurer's total retained loss if the quota share treaty inures to the benefit of the excess-of-loss treaty?

(c) What incentive does the co-participation provision in the excess-of-loss treaty give the primary insurer in this case?

Solution S6-47-1. (a) If neither treaty inures to the benefit of the other, then each will pay the full amount that it would pay if no other treaty existed. We can thus consider the primary insurer's total retained loss as (Total ground-up loss) - (Total loss paid by reinsurer under quota share treaty) - (Total loss paid by reinsurer under excess-of-loss treaty).

Total ground-up loss = $500,000 + $150,000 + $750,000 + $400,000 = $1,800,000.

Total loss paid under quota share treaty = 40%*($500,000 + $150,000 + $400,000) + $200,000 (since 40% of $750,000 exceeds the $200,000 maximum recovery) = $620,000.

Total loss paid by reinsurer under excess-of-loss treaty = 0.9*($300,000 + $500,000 + $200,000) = $900,000. Note that there is no recovery for Occurrence B, for which the loss is below the retention. Also, the excess recovery for Occurrence C is limited to 90% of $500,000.

Thus, the total retained loss is $1,800,000 - $620,000 - $900,000 = $280,000.

(b) If the quota share treaty inures to the benefit of the excess-of-loss treaty, then the quota share treaty is applied first to reduce the losses applicable to the excess-of-loss treaty. Thus, the quota share treaty still pays $620,000, as in part (a). However, after the quota share treaty, the applicable loss amounts change as follows:

For Occurrence A: $500,000*0.6 = $300,000
For Occurrence B: $150,000*0.6 = $90,000
For Occurrence C: $750,000 - 200,000 = $550,000
For Occurrence D: $400,000*0.6 = $240,000

Total loss paid by reinsurer under excess-of-loss treaty = 0.9*($100,000 + $350,000 + $40,000) = $441,000.

Thus, the total retained loss is $1,800,000 - $620,000 - $441,000 = $739,000.

(c) The co-participation provision gives the primary insurer the incentive to control even those claim costs which exceed the retention for the treaty, since the primary insurer will be sharing 10% of these costs.

Problem S6-47-2. Similar to Question 33 from the 2008 CAS Exam 6. A primary insurer has a 5-line surplus share reinsurance treaty with a net line of $150,000. You know the following information about policies and losses:

Policy 1: Limit is $60,000 - loss is $25,000.
Policy 2: Limit is $1,000,000 - loss is $680,000.
Policy 3: Limit is $450,000 - loss is $200,000.
Policy 4: Limit is $300,000 - loss is $40,000.

What is the primary insurer's net loss?

Solution S6-47-2. For Policy 1, the limit is less than the primary insurer's line, so the primary insurer retains the entire $25,000 loss.

For Policy 2, we need to keep in mind that the reinsurer's obligation is limited to 5 lines, or $750,000, so, of the total limit of $1,000,000, the primary insurer's obligation extends to the line of $150,000 and (1,000,000 - 750,000 - 150,000) = $100,000, for a total of $250,000 of the limit, and, correspondingly, a quarter of any loss. Thus, the primary insurer's obligation for this loss is 680000*0.25 = $170,000.

For Policy 3, the primary insurer's obligation is a third of any loss, since 150000/450000 = 1/3. Thus, the primary insurer's obligation for this loss is 200000/3 = $66,666.67.

For Policy 4, the primary insurer's obligation is half of any loss, since 150000/300000 = 1/2. Thus, the primary insurer's obligation for this loss is 40000/2 = $20,000.

The primary insurer's total obligation is thus $25,000 + $170,000 + $66,666.67 + $20,000 = $281,666.67.

Problem S6-47-3. Similar to Question 34 from the 2008 CAS Exam 6. According to the discussion of "An Exposure Rating Approach to Pricing Property Excess-of-Loss Reinsurance" by Feldblum (p. 385), why do issues of (a) subjectivity and (b) complexity limit the usefulness of fitting curves to reinsurance data?

Solution S6-47-3. (a) Subjectivity exists in deciding which family of curves to use. In reinsurance analysis, analyzing the "tail" (greater extremes) of losses is particularly important - especially for excess-of-loss reinsurance in high layers. Different families of distributions can be similarly reasonable in estimating "usual" severities of a particular kind of loss but might diverge dramatically in their tail estimates.

(b) Complexity may occur in explaining curve-fitting techniques to representatives of the primary insurer and the reinsurance underwriter, and it is possible to derive different rates from different curve-fitting techniques, making it difficult to select rates on their basis.

Problem S6-47-4. Similar to Question 35 from the 2008 CAS Exam 6. A primary insurer has 20 claims, of which 15 will eventually develop to $80,000, while 5 will develop to $1,500,000.

The primary insurer has an excess-of-loss reinsurance treaty of $1,000,000 in excess of $500,000.

(a) Assuming that the primary insurer establishes a case reserve for each claim using the mode of claim amounts, what will be the primary insurer's IBNR reserve?

(b) What will be the reinsurer's IBNR, following the assumption in part (a)?

(c) Why does the selection of the mode for the case reserve often lead to results of the kind in parts (a) and (b)?

Solution S6-47-4. (a) Reserving at the mode would imply that each claim has a case reserve of $80,000, for a total case reserve of 20*80000 = 1600000. Of the five claims of $1,500,000 each, the primary insurer will only be responsible for the first $500,000 for each, meaning that the IBNR is 5*(500000 - 80000) = $2,100,000.

(b) The reinsurer will be responsible for the $1,000,000 in excess of the retention for each of the five claims of $1,500,000. Thus, the reinsurer's IBNR is 5*1000000 = $5,000,000.

(c) When the primary insurer reserves at the mode, that mode is typically below the primary insurer's retention. The reinsurer does not typically find out about a particular claim until that claim exceeds the retention. If all claims are assumed to be at the mode, then the reinsurer does not find out about them at all until the large losses corresponding to a few claims are actually reported. Thus, all of these claims' amounts in excess of the retention are part of the reinsurer's IBNR. By contrast, the primary insurer also has a case reserve equal to (Number of claims)*(Mode of claim amount), and the primary insurer's IBNR is limited due to the retention per the excess-of-loss treaty.

Problem S6-47-5. Similar to Question 8 from the 2009 CAS Exam 6. You have the following information about cumulative paid losses for an insurer, expressed in the format
(Number at 12 months, Number at 24 months, Number at 36 months, Number at 48 months).

Cumulative Paid Losses
AY 2044: (40000, 60000, 80000, 90000)
AY 2045: (45000, 64000, 86000)
AY 2046: (51000, 69000)
AY 2047: (80000)

You also have the following information about on-level premiums and exposures:

AY 2044: On-level premium: 60000; Exposures: 300; Average premium: 200
AY 2045: On-level premium: 66000; Exposures: 327; Average premium: 201.83
AY 2046: On-level premium: 70000; Exposures: 350; Average premium: 200
AY 2047: On-level premium: 80000; Exposures: 404; Average premium: 198.02

Use two diagnostics to show why it would not be proper to use the paid development method to estimate ultimate losses for AY 2047.

Solution S6-47-5. First we consider the ratio of cumulative paid loss to on-level premium at 12 months of development:

Ratios of Cumulative Paid Loss to On-Level Premium
AY 2044: 40000/60000 = 0.666667
AY 2045: 45000/66000 = 0.681818
AY 2046: 51000/70000 = 0.728571
AY 2047: 80000/80000 = 1

We note the immense increase in the ratio of cumulative paid loss to on-level premium in AY 2047. It is possible that faster claim settlement in part contributed to this, in which case the paid development method would overestimate the claim development factors and thus the ultimate loss for AY 2047.

We can also compare the trend in exposures to the trend in paid losses at 12 months.

Exposure Trend - AY 2044 to AY 2045: 327/300 - 1 = +9%.
Exposure Trend - AY 2045 to AY 2046: 350/327 - 1 = +7.03%
Exposure Trend - AY 2046 to AY 2047: 404/350 - 1 = +15.43%

Paid Loss Trend - AY 2044 to AY 2045: 45000/40000 - 1 = +12.5%
Paid Loss Trend - AY 2045 to AY 2046: 51000/45000 - 1 = +13.33%
Paid Loss Trend - AY 2046 to AY 2047: 80000/51000 - 1 = +56.87%

The paid loss trend is higher than the exposure trend in all cases, but the difference is especially pronounced from AY 2046 to AY 2047. Since average premiums are close to constant, one can infer that a faster settlement rate has been manifested in later accident years, especially in AY 2047. Claim development factors using the chain ladder method will overestimate the ultimate loss in this situation.

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