Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 3

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.

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

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

The following conditions apply to Problems S6-6-1 and S6-6-2:

Insurer Z has an excess-of-loss reinsurance treaty that provides $500,000 in excess of $100,000 per occurrence. The following losses occurred during the term of the treaty:

Loss A: $136,000 in incurred loss, $130,000 in loss adjustment expenses

Loss B: $416,000 in incurred loss, $800,000 in loss adjustment expenses

Loss C: $700,000 in incurred loss, $20,000 in loss adjustment expenses

Problem S6-6-1. Similar to Problem 26(a) from the Fall 2009 Exam 6. If the loss adjustment expenses are included in the limit of coverage, what is the reinsurer's total obligation for the three losses?

Solution S6-6-1. Since LAE are included in the limit, the total loss amount per occurrence is the sum of incurred loss and LAE for that occurrence.

Loss A: Total loss amount is 136000 + 130000 = $266,000, of which the reinsurer covers $166,000.

Loss B: Total loss amount is 416000 + 800000 = $1,216,000, of which the reinsurer covers up to its limit of $500,000.

Loss C: Total loss amount is 700000 + 20000 = $720,000, of which the reinsurer covers up to its limit of $500,000.

The reinsurer's total obligation is thus 166000 + 500000 + 500000 = $1,166,000.

Problem S6-6-2. Similar to Problem 26(b) from the Fall 2009 Exam 6. If the loss adjustment expenses are shared on a pro-rata basis in addition to the incurred loss amounts, what is the reinsurer's total obligation for the three losses?

Solution S6-6-2. The proportion of the LAE that the reinsurer would pay on a pro rata basis is the ratio (Incurred loss amount covered by reinsurer)/(Total incurred loss amount).

Loss A: Incurred loss amount is $136,000, of which the reinsurer would cover $36,000, so the amount of LAE the reinsurer covers is 130000*(36000/136000) = $34,411.76, meaning that the reinsurer covers a total of $70,411.76 for Loss A.

Loss B: Incurred loss amount is $416,000, of which the reinsurer would cover $316,000, so the amount of LAE the reinsurer covers is 800000*(316000/416000) = $607,692.31, meaning that the reinsurer covers a total of $923,692.31 for Loss B.

Loss C: Incurred loss amount is $700,000, of which the reinsurer would cover $500,000, so the amount of LAE the reinsurer covers is 20000*(500000/700000) = $14,285.71, meaning that the reinsurer covers a total of $514,285.71 for Loss C.

The reinsurer's total obligation is thus 70411.76 + 923692.31 + 514285.71 = $1,508.389.78.

Note that it is not always the case that pro-rata sharing of LAE results in less reinsurance coverage than including LAE in the limit!

The following conditions apply to Problems S6-6-3 and S6-6-4:

You are examining a proportional reinsurance treaty (where the primary insurer and reinsurer share losses equally), with one modification: a loss corridor provision that requires the primary insurer to assume responsibility for 90% of the losses for the loss ratio layer between 50% and 80%.

You know the following:

For the loss ratio layer 0%-20%: The average loss ratio is 15%, and the probability of the loss ratio being in this layer is 11%.

For the loss ratio layer 20%-50%: The average loss ratio is 34%, and the probability of the loss ratio being in this layer is 30%.

For the loss ratio layer 50%-80%: The average loss ratio is 75%, and the probability of the loss ratio being in this layer is 40%.

For the loss ratio layer 80+%: The average loss ratio is 98%, and the probability of the loss ratio being in this layer is 19%.

Problem S6-6-3. Similar to Problem 30(a) from the Fall 2009 Exam 6. If no loss ratio corridor is applied, what is the expected loss ratio for the primary insurer?

Solution S6-6-3. Since losses are shared evenly between the primary insurer and the reinsurer, the expected loss ratio is the same as the weighted-average loss ratio for these data:
0.15*0.11 + 0.34*0.30 + 0.75*0.40 + 0.98*0.19 = 0.6047 = 60.47%.

Problem S6-6-4. Similar to Problem 30(b) from the Fall 2009 Exam 6. If the loss ratio corridor is applied, what is the expected loss ratio for the primary insurer?

Solution S6-6-4. The primary insurer is obligated to retain 90% of the losses for the loss ratio layer between 50% and 80%.

For this layer, the average loss ratio percentage in the corridor is (75% - 50%) = 25%, and the primary insurer will retain 90% of this, or 22.5%.

For the 80+ layer, the average loss ratio percentage in the corridor is (80% - 50%) = 30%, and the primary insurer will retain 90% of this, or 27%.

The expected retained loss ratio is obtained by weighting the above retained loss ratios by the probability of each layer occurring: 0.225*0.40 + 0.27*0.19 = 0.1413 = 14.13%. This is the additional loss ratio percentage that the corridor requires the primary insurer to retain, so the total loss ratio percentage retained is 60.47% + 14.13% = 74.6%.

Problem S6-6-5. Similar to Problem 1 from the Fall 2008 Exam 6. Based on the CAS Statement of Principles Regarding Property and casualty Loss and Loss Adjustment Expense Reserves, explain how (a) settlement patterns and (b) frequency/severity considerations could affect the determination of whether or not data in a given set are homogeneous.

Solution S6-6-5. (a) Settlement patterns - how long it takes for reported claims to settle - influence the level of reserve uncertainty. Claims that take longer to settle, such as bodily injury liability claims, have more reserve uncertainty, and the ultimate claim amount can vary considerably from the initial estimate. It may therefore be appropriate to analyze claims with long settlement patterns separately from claims with short settlement patterns, lest the actuary understate development on these claims.

(b) Frequency/severity considerations: Claims with high frequency and low severity tend to be subject to more accurate reserve estimates than claims with low frequency and high severity. The latter type of claim will often necessitate greater analysis and may therefore need to be examined separately.

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