**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 30 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.

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:**

Bouska, A.S., “From Disability Income to Mega-Risks: Policy-Event Based Loss Estimation,” Casualty Actuarial Society *Forum*, Summer 1996, pp. 291-320.

Friedland, Jacqueline F. *Estimating Unpaid Claims Using Basic Techniques**.* Casualty Actuarial Society. July 2009.

Harrison, C.M., *Reinsurance Principles and Practices* (First Edition), American Institute for Chartered Property Casualty Underwriters/Insurance Institute of America, 2004, Chapters 1, 2 (from beginning through page 2.21), 4, 8, 9, and 10.

Past Casualty Actuarial Society exams: 2007 Exam 6.

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

**Problem S6-30-1. Similar to Question 26 from the 2007 CAS Exam 6.**

**(a)** What are the two elements of Policy-Event-Based Loss Estimation (PEBLE)? (See Bouska, p. 293.)

**(b)** Identify three shortcomings of traditional triangular loss analysis with regard to asbestos and pollution claims.

**(c)** In general terms, describe the stepwise operation of the PEBLE process.

**Solution S6-30-1.**

**(a)** The two elements of PEBLE are

1. “A loss event that might give rise to an insurance claim” and

2. “The application of the terms of an individual policy to that loss event in order to determine the insured loss” (Bouska, p. 293).

**(b)** The following are three shortcomings of traditional triangular loss analysis with regard to asbestos and pollution claims (See Bouska, pp. 296-297.)

1. Triangular methods rely for their accuracy on numerous, small loss events that occur with relative regularity. Asbestos and pollution claims can be rare and might often not manifest themselves until years after the insurance policies in question were issued. When they do manifest themselves, they can be immense in severity, and their magnitudes are extremely difficult to predict from prior experience.

2. In pollution/asbestos events, there is often a lack of correlation between the loss occurrence and the accounting for the loss. The original exposure may occur at a particular point in time, but its effects might be latent and develop gradually – with claims tending to follow the manifestation of the effects.

3. Courts have generally allowed all policies between first exposure to the peril and manifestation of the adverse consequences to be triggered. It is therefore not always clear to what year(s) the claims would be assigned.

Other valid answers may be possible.

**(c)** The following can be a description of the stepwise operation of PEBLE:

1. Acquire policy data, including the policy terms and the attributes of the loss exposure;

2. Model the loss event (e.g., a pollution or asbestos claim).

3. Derive an understanding of the losses specific to the exposure from the model of the insured event and the attributes of the loss exposure.

4. Estimate the insured cost from the policy terms and exposure-specific losses.

(See Bouska, p. 301.)

Some variations on this description may be possible.

**Problem S6-30-2. Similar to Question 27 from the 2007 CAS Exam 6.** You have the following data for Insurer Ω:

**Policy 1:** Limit is $350,000; premium is $20,000; loss is $100,000. **Policy 2:** Limit is $45,000; premium is $3,000; loss is $18,000. **Policy 3:** Limit is $600,000; premium is $56,000; loss is $195,000.

**(a)** Under a 30% quota share treaty, how much would the *reinsurer* (i) receive in premium and (ii) be liable for in losses?

**(b)** Under a 11-line surplus share treaty with a retained line of $50,000, how much would the *reinsurer* (i) receive in premium and (ii) be liable for in losses?

**(c)** For the primary insurer, what is one possible disadvantage of surplusshare treaties relative to quota share treaties?

**(d)** For the primary insurer, what is one possible disadvantage of quota share treaties relative to surplusshare treaties?

**Solution S6-30-2.**

**(a)** The percentage in a quota share treaty is always defined relative to the amounts ceded to the reinsurer. The reinsurer thus gets 30% of all premium and 30% of all losses.

(i) Premium: 0.3*(20000 + 3000 + 56000) = **$23,700**.

(ii) Losses: 0.3*(100000 + 18000 + 195000) = **$93,900**.

**(b)** For policies with a limit under $600,000, The primary insurer retains that percentage of a policy’s premium and loss that corresponds to ($50,000)/(Policy limit). The rest is ceded to the reinsurer.

For Policy 1, the reinsurer’s proportion is 1 – 50000/350000 = 6/7.

For Policy 2, the primary insurer retains everything, because the policy limit is under $50,000.

For Policy 3, the reinsurer’s proportion is 1 – 50000/600000 = 11/12.

(i) Premium: (6/7)*20000 + (11/12)*56000 = **$68,476.19**.

(ii) Losses: (6/7)*100000 + (11/12)*195000 =**$264,464.29**.

**(c)** Surplus share treaties generally provide less surplus relief than quota share treaties, as less premium is ceded to the reinsurer.

**(d)** Quota share treaties require the primary insurer to cede X% of every loss exposure, no matter how profitable. Surplus share treaties allow the primary insurer to fully retain its lower-limit, more profitable loss exposures.

For (c) and (d), other valid answers may be possible.

**Problem S6-30-3. Similar to Question 29 from the 2007 CAS Exam 6.** There are two kinds of policies that Primary Insurer Π writes. Policy A has a limit of $200,000, and Policy B has a limit of $600,000. Primary Insurer Π wishes to retain all the risk on policies of type A. For policies of type B, the insurer wishes to retain only at most $300,000 on each policy. Describe the terms of two distinct kinds of reinsurance treaties that would achieve this goal.

**Solution S6-30-3.** This is a sample answer, and any two possibilities will suffice. Other kinds of treaties may be possible.

1. A per-policy excess-of-loss treaty: $400,000 in excess of $200,000, with a co-participation provision of 25%.

2. A one-line surplus share treaty, with the line being $300,000.

3. A variable quota share treaty, with 0% ceded on Policy A risks and 66.666666667% ceded on Policy B risks.

4. A per-policy excess-of-loss treaty: $300,000 in excess of $300,000, with no co-participation provision.

**Problem S6-30-4. Similar to Question 30 from the 2007 CAS Exam 6.** You know the following aggregate loss distribution for a reinsurer under a reinsurance treaty:

**Loss ratio from 0% to 50%:** Expected loss ratio is 30%; probability of loss in this range is 25%. **Loss ratio from 50% to 70%:** Expected loss ratio is 65%; probability of loss in this range is 45%. **Loss ratio from 70% to 90%:** Expected loss ratio is 80%; probability of loss in this range is 15%. **Loss ratio of 90+%:** Expected loss ratio is 105%; probability of loss in this range is 15%.

The reinsurer pays the following sliding-scale commission:

Minimum of 5% for loss ratios of 90% or higher.

Slides via a 1.5:1 ratio from 90% to 70%, to 35% at 70% loss ratio.

Slides via a 0.2:1 ratio from 70% to 40%, to maximum of 41% at 40% loss ratio.

Find the reinsurer’s expected technical ratio.

**Solution S6-30-4.** The technical ratio is (Reinsurer’s loss ratio) + (Reinsurer’s commission ratio).

In the range where expected loss ratio is 30%, the commission ratio is 41%, so the technical ratio is 71%.

In the range where expected loss ratio is 65%, the commission ratio is 35% + 0.2%*5 = 36%, so the technical ratio is 101%.

In the range where expected loss ratio is 80%, the commission ratio is 35% – 1.5%*10 = 20%, so the technical ratio is 100%.

In the range where expected loss ratio is 105%, the commission ratio is 5%, so the commission ratio is 110%.

The expected technical ratio is 0.25*71% + 0.45*101% + 0.15*100% + 0.15*110% = **94.7%.**

**Problem S6-30-5. Similar to Question 50 from the 2007 CAS Exam 6.** You have the following triangle of cumulative closed claim counts per accident year (AY), with age of development being expressed at

(12 months, 24 months, 36 months, 48 months):

**AY 2034** (13000 earned exposures): (100, 150, 175, 200) **AY 2035** (13500 earned exposures): (102, 148, 155) **AY 2036** (14000 earned exposures): (99, 130) **AY 2037** (13000 earned exposures): (80)

**(a)** What operational change might have occurred within the insurance company to explain the data above?

**(b)** How would the operational change in part (a) affect the accuracy of the calculations of ultimate losses on the basis of the corresponding paid loss triangle?

**Solution S6-30-5.**

**(a)** We can construct a triangle of claim counts per earned exposure to spot any differences:

**AY 2034**: (0.00769, 0.01154, 0.01346, 0.01538) **AY 2035**: (0.00756, 0.01096, 0.01148) **AY 2036**: (0.00707, 0.00929) **AY 2037**: (0.00615)

It appears that, over the years 2035-2037, the insurer’s claims department has closed increasingly fewer claims at each age of the experience, as compared to prior years. The decline is particularly evident going from AY 2036 to AY 2037. Perhaps the claims department has become less efficient or has chosen to scrutinize claims more closely.

**(b)** Since claims in more recent time periods are being closed at a slower rate, applying ultimate loss estimates based on the corresponding paid loss triangle, where the diagonals based on the most recent experience will give lower factors, will result in losses from earlier periods being multiplied by smaller development factors, leading to an **underestimate.**

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