Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 11

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:

Past Casualty Actuarial Society exams: 2007 Exam 6.

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

The following conditions apply to Problems S6-22-1 and S6-22-2. You know the following information about a reinsurance treaty:

The reinsurer's pro rata share is 60%.
The primary insurer's pro rata share is 40%.
The reinsurer also pays the primary insurer a ceding commission of 30%.
The reinsurer's expense ratio is 5%.
The reinsurer's expected loss ratio is 75%.
The premium subject to the reinsurance treaty is $2,500,000.
The loss subject to the reinsurance treaty is $500,000.

There is also a profit-sharing commission clause which states that, if the reinsurer's profit exceeds the expected amount, the primary insurer will receive 20% of the profit in excess of what was expected.

Problem S6-22-1. Similar to Question 23(a) from the 2007 CAS Exam 6. Before the application of the profit-sharing commission, what is the reinsurer's profit.

Solution S6-22-1.

Add reinsurer's share of premium: $2,500,000*0.60 = $1,500,000.
Subtract reinsurer's ceding commission: $2,500,000*0.30 = $750,000.
Subtract reinsurer's share of losses: $500,000*0.60 = $300,000.
Subtract reinsurer's expenses: $2,500,000*0.05 = $125,000.
Reinsurer's profit = $1,500,000 - $750,000 - $300,000 - $125,000 = $325,000.

Problem S6-22-2. Similar to Question 23(b) from the 2007 CAS Exam 6. What is the amount of the profit-sharing commission the reinsurer would pay the primary insurer?

Solution S6-22-2. First, we calculate the amount of the reinsurer's expected profit. This is the premium ceded to the reinsurer, multiplied by (1 - Expected Loss Ratio - Expense Ratio - Ceding Commission Ratio), i.e., $1,500,000*(1 - 0.75 - 0.05 - 0.30) = -$150,000. So the reinsurer can actually expect a loss of $150,000. The difference between the actual profit of $325,000 and this amount is $475,000, so the reinsurer must pay 20% of this, or $95,000 as a profit-sharing commission to the primary insurer.

The following conditions apply to Problems S6-22-3 through S6-22-5. You know the following information about a reinsurance treaty:
The treaty covers 70% of the loss layer $5 million in excess of $5 million.

There is a claim subject to the treaty, with $8 million in losses and $2 million in allocated loss adjustment expenses (ALAE).

Problem S6-22-3. Similar to Questions 24(a)(i) and 24(b)(i) from the 2007 CAS Exam 6. Using the "pro rata in addition" method, calculate (a) the primary insurer's retained combined losses and ALAE and (b) the combined losses and ALAE ceded to the reinsurer.

Solution S6-22-3.

(a) To use the "pro rata in addition" method, we need to first determine how the loss is divided. The same proportions of loss that apply to each party will also apply with respect to ALAE.

The loss is $8 million, of which the primary insurer retains the first $5 million and also retains 30% of the rest, or 0.3*3 million = $900,000, meaning that the primary insurer retains $5,900,000 of $8 million, or 5.9/8 of the total loss amount. The ALAE retained by the primary insurer is thus also (2 million)*5.9/8 = $1,262,500, meaning that the primary insurer's total amount retained is $1,262,500 + $5,900,000 = $7,162,500.

(b) Since the combined (loss + ALAE) amount is $10 million, the reinsurer gets what the primary insurer does not retain, i.e., $10,000,000 - $7,162,500 = $2,837,500.

Problem S6-22-4. Similar to Questions 24(a)(ii) and 24(b)(ii) from the 2007 CAS Exam 6. Using the "included in the limit" method, calculate (a) the primary insurer's retained combined losses and ALAE and (b) the combined losses and ALAE ceded to the reinsurer.

Solution S6-22-4.

(a) The "included in the limit" method treats the ALAE as part of the loss amount, and so the aggregate amount under consideration here is $10 million, of which the primary insurer must pay $5 million and then 30% of the other $5 million, for a total of 1.3*(5 million) = $6,500,000.

(b) Since the combined (loss + ALAE) amount is $10 million, the reinsurer gets what the primary insurer does not retain, i.e., $10,000,000 - $6,500,000 = $3,500,000.

Problem S6-22-5. Similar to Questions 24(c) from the 2007 CAS Exam 6. In the situation described in the given conditions, the "included in the limit" method is more advantageous to the primary insurer than the "pro rata in addition" method. By tweaking just one item in the given conditions, develop a scenario where the "included in the limit" method is less advantageous to the primary insurer.

Solution S6-22-5. Many answers are possible here. The following is a sample answer:

Change the claim being subject to the treaty to having $10 million in losses instead of $8 million. ALAE remain at $2 million.

We demonstrate that the primary insurer's losses will be greater under the "included in the limit" method.

"Pro rata in addition method": For just the losses, the primary insurer must pay $5 million and then 30% of the other $5 million, for a total of 1.3*(5 million) = $6,500,000. This means that the primary insurer is responsible for 65% of the loss amount - and, by implication, 65% of the ALAE amount, or an additional $1.3 million, for a total of $7,800,000.

"Included in the limit method": The aggregate amount under consideration is $12 million. The reinsurer's responsibility is still only for 70% of the second $5 million, leaving the primary insurer responsible for (12 million - 0.7*(5 million)) = $8,500,000 > $7,800,000.

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