**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 60 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 S6-60-1.** What is the paradox of reinsurance pricing described by Clark?

**Solution S6-60-1.** The paradox of reinsurance pricing is that a ceding company will not want to buy a reinsurance contract that can be precisely priced ( Clark , p.1).

**Problem S6-60-2.** Fill in the blanks:

**(a)** If a reinsurance treaty is written on a “risks attaching” basis, use ______ (type of premium) and _______ (type of losses).

**(b)** If a reinsurance treaty is written on a “losses occurring” basis, use ______ (type of premium) and _______ (type of losses).

**Solution S6-60-2.** This question is based on the discussion by Clark , p. 3.

**(a)** If a reinsurance treaty is written on a “risks attaching” basis, use **written premium** and **losses covered by the attaching policies**.

**(b)** If a reinsurance treaty is written on a “losses occurring” basis, use **earned premium** and **accident-year losses.**

**Problem S6-60-3.** In adjusting reinsurance experience to an ultimate level, describe two adjustments to premium and two adjustments to losses that might be made.

**Solution S6-60-3.** This question is based on the discussion by Clark , pp. 3-4.

Two adjustments to premium are (i) on-level factors to bring premiums to current rate levels and (ii) exposure inflation factors if the premium base is inflation-sensitive.

Two adjustments to losses are (i) loss development factors to ultimate and (ii) loss trend factors.

**Problem S6-60-4.** Clark (pp. 4-5) discusses four ways of estimating catastrophe loads for proportional property reinsurance treaties. What are these four ways?

**Solution S6-60-4.** The four ways of estimating catastrophe loads for proportional property reinsurance treaties are as follows (Clark, pp. 4-5):

1. Basing the estimate on the ceding company’s expected distribution of premium by state;

2. Basing the estimate on the average number of times per year that the treaty’s occurrence limit is expected to be exceeded;

3. Basing the estimate on historical catastrophe losses spread evenly over a longer time period, adjusted to current cost and exposure levels;

4. Basing the estimate on the output from a catastrophe simulation model.

**Problem S6-60-5.** In situations where there is a sliding-scale ceding commission that varies on the basis of loss ratio, Clark (pp. 9-10) distinguishes between a “naïve” approach of estimating the expected ceding commission and a more accurate approach. Conceptually describe each and explain why the latter approach is more accurate.

**Solution S6-60-5.** The “naïve” approach works as follows:

1. Calculate the expected loss ratio.

2. The expected ceding commission is the ceding commission corresponding to the expected loss ratio.

The improved approach works as follows:

1. Calculate the ceding commission associated with the average loss ratio for each range in an aggregate loss distribution model.

2. Calculate a probability-weighted average ceding commission, based on the ceding commissions in each range of the aggregate loss distribution model.

The second approach is preferable because it takes into account the possibility (indeed, the likelihood) that the distribution of ceding commissions may differ from the distribution of loss ratios (i.e., a ceding commission may “slide” differently depending on the range of loss ratios being considered). Thus, simply calculating an expected loss ratio fails to capture the impact of loss ratio variation on the ceding commission. Using an aggregate loss distribution at least enables one to account for the differential “slides” of the ceding commission within each selected loss ratio range.

**Problem S6-60-6.** What is a *carryforward provision* for a ceding commission? Give a brief numerical example of how such a provision would work. (See Clark, p. 10)

**Solution S6-60-6.** A carryforward provision is a clause in the reinsurance contract that allows subsequent years’ ceding commissions to be modified by any of the primary insurer’s prior-year loss amounts in excess of the loss ratio corresponding to the minimum ceding commission.

For instance, if the minimum ceding commission is 10%, corresponding to an 80% loss ratio, and the ceding company’s loss ratio in Year X is 85%, then the loss amount corresponding to the excess 5% of losses may be used in calculating the Year (X+1) loss ratio for ceding commission purposes.

**Problem S6-60-7.** Clark (pp. 10-12) describes two ways of addressing carryforward provisions in estimating expected ceding commissions for proportional reinsurance contracts. Conceptually describe each approach and identify a shortcoming of each.

**Solution S6-60-7.** The following are two ways of addressing carryforward provisions in estimating expected ceding commissions for proportional reinsurance contracts:

1. Shift the ceding commission “slide” by the amount of the carryforward from prior years. For instance, if the ceding commission slides 1:1 from 10% at an 80% loss ratio to 30% at a 60% loss ratio, and the carryforward from prior years is +3%, then one could adjust the slide to 1:1 from 10% at an 77% loss ratio to 30% at a 57% loss ratio.

*Shortcoming*: Only carryforward for the current year is addressed; the possibility of carryforward for subsequent years is not taken into account.

2. Look at the long-run circumstances of the contract and, instead of applying the sliding scale to just the current year, apply it to a longer block of years. This allows for reduced aggregate variance in losses.

*Shortcomings* (any one will suffice): (1) The approach assumes that the reinsurance contract will be renewed over many years – which is often not a certainty. (2) There is no way of addressing situations where only ceding commission deficits, but not credits, can be carried forward. (3) There is no unambiguous way of estimating the variation reduction achieved as a result of this approach.

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