**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 61 of the Study Guide. See an index of all sections by following the link in this paragraph.**

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.

**Sources:**

Clark, D.R., “Basics of Reinsurance Pricing,” CAS Study Note, 1996.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S6-61-1.** According to Clark (p. 19), why is the actual number of claims during the historical period in question not a good basis for credibility of reinsurance data? What is a superior basis?

**Solution S6-61-1.** The actual number of claims during the historical period in question not a good basis for credibility, because this measure implies that fortuitously worse time periods in terms of loss are necessarily assigned more credibility, thus making the reinsurer’s experience appear worse than it actually is. The number of claims *expected* during the historical period is a better basis for credibility, because it does not bias the data by assigning more credibility to less favorable experience.

**Problem S6-61-2.** Describe the three general categories of casualty per-occurrence excess-of-loss reinsurance treaties (Clark, p. 22).

**Solution S6-61-2.** The three general categories of casualty per-occurrence excess-of-loss reinsurance treaties are as follows (Clark, p. 22):

1. **Working cover:** The attachment point is low and is expected to be penetrated multiple times per treaty period.

2. **Exposed excess:** The excess layer of reinsurance coverage attaches below the policy limits for some of the policies in the book of business. It is possible for the reinsurer to incur a loss on the treaty if losses on enough underlying policies are close to their limits.

3. **Clash cover:** The attachment point is typically above the limits of any one policy, and the treaty is expected to apply only in cases where multiple policies (including policies of different types) are triggered, or extra-contractual obligations or damages in excess of policy limits are required of the primary insurer.

**Problem S6-61-3.** Clark (p. 23) discusses a theoretical problem with regard to capping trended losses at applicable policy limits. Explain the difficulty in determining the right approach to follow on this matter.

**Solution S6-61-3.** It is theoretically desirable to cap trended losses at the limits at which the policy would have been issued, had it been issued during the later time period under consideration. This means that applying the policy’s historical limit may not be appropriate, as insurers generally increase policy limits over time in an inflationary environment. If no capping is used, this assumes that policy limits increase at the rate of inflation, which may also not correspond to reality; this approach would also require an adjustment to the subject premium, or else the loss costs will be overstated (Clark, p. 23).

**Problem S6-61-4.** Clark (p. 25) describes four areas of caution to be exercised with regard to using development data from the Reinsurance Association of America (RAA). Describe these areas.

**Solution S6-61-4.** The following cautions should be taken with regard to development data from the Reinsurance Association of America (RAA) (Clark, p. 25):

1. The RAA data are from various companies, and the reporting lag between the event’s occurrence and the reinsurer’s establishment of a case reserve for it varies by company. Also, the reporting lag for retrocessional business exceeds the reporting lag for ordinary reinsurance.

2. The RAA data are not sufficiently segmented by limits and attachment points, and, even when some segmentation is attempted, the stability of the data is compromised as a result.

3. It is not clear whether the reporting companies consistently adhere to the RAA’s standard of excluding asbestos and environmental claims. Also, some long-term exposure claims pertaining to certain products are not excluded, even though they are not relevant to all reinsurance business.

4. It is not clear whether reporting workers’ compensation reinsurers consistently treat the tabular discount on large claims.

**Problem S6-61-5.**

**(a)** What are two shortcomings of using a single continuous distribution function to model reinsurance losses? (See Clark, p. 36.)

**(b)** Briefly explain how a collective risk model differs from using a single continuous distribution function. (See Clark, pp. 39-40.)

**Solution S6-61-5.**

**(a)** Two shortcomings of using a single continuous distribution function to model reinsurance losses are that (1) the distribution does not permit a scenario where the loss amount is zero (since x = 0 is the smallest independent variable, the cumulative distribution function is zero at x= 0, implying a zero probability) and (2) the impact on the distribution of changing the treaty limits (e.g., the per-occurrence limits) is not easy to determine.

**(b)** A collective risk model employs a probability distribution to model the severity of each loss, and the number of losses also follows its own probability distribution. This enables explicit recognition of both frequency and severity of losses.

**Problem S6-61-6.**

**(a)** According to Clark (p. 40), what assumption do most collective risk models make regarding loss occurrences?

**(b)** Clark (p. 40) distinguishes between “process variance” and “parameter variance”. In a collective risk model, which of these is always reflected by the aggregate distribution, and which might not be?

**Solution S6-61-6.**

**(a)** Most collective risk models assume that loss occurrences are independent of one another. This may or may not be true in reality.

**(b)** “Process variance” is “the random fluctuation of actual results about the expected value” (Clark, p. 40). “Parameter variance” can also be referred to as “model risk” and is uncertainty about whether the model’s own design and parameters are appropriate for describing the situation in question (in Clark’s words, “whether you are in the right model”). The aggregate distribution of a collective risk model always reflects process variance, but not necessarily parameter variance.

**Problem S6-61-7.** You have the following information about a catastrophe excess-of-loss reinsurance treaty for the annual term encompassing the entire year 2022:

Annual premium: $4,000,000

Occurrence limit: $50,000,000

Date of loss: September 1, 2022

Loss amount: $35,000,000

Reinstatement provision: 120%

**(a)** Calculate the reinstatement premium after the loss if the reinstatement provision is pro rata as to amount, but not pro rata as to time.

**(b)** Calculate the reinstatement premium after the loss if the reinstatement provision is pro rata as to amount *and* pro rata as to time.

**(c)** Why are most reinstatement premiums for catastrophe excess-of-loss treaties *not* pro rata as to time? (See Clark, p. 41.)

**Solution S6-61-7.**

**(a)** If the reinstatement provision is pro rata as to amount, but not pro rata as to time, then we only need to consider the fraction of the annual premium corresponding to the proportion of the loss amount to the occurrence limit, multiplied by the percentage in the reinstatement provision.

Reinstatement premium = ($35,000,000/$50,000,000)*$4,000,000*120% = **$3,360,000.**

**(b)** If the reinstatement provision is pro rata as to amount and to time, then a further reduction of the reinstatement premium is needed to account for the time during which the new coverage will be effective – i.e., from September 1, 2022, until the end of 2022, or 4/12 = 1/3 years. Thus, the reinstatement premium is $3,360,000*(1/3) = **$1,120,000.**

**(c)** Most reinstatement premiums for catastrophe excess-of-loss treaties are not pro rata as to time because many catastrophes, such as hurricanes, occur seasonally, so a pro rata approach to time does not take into account the actual exposure to risk during the remainder of the treaty period.

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