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 24 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.
Casualty Actuarial Society Enterprise Risk Management Committee, “Overview of Enterprise Risk Management,” Casualty Actuarial Society Forum, Summer 2003, Section 3 and Appendix B.
Casualty Actuarial Society Valuation, Finance, and Investments Committee, “Accounting Rule Guidance Statement of Financial Accounting Standards No. 113-Considerations in Risk Transfer Testing” Casualty Actuarial Society Forum, Fall 2002, pp. 305-338, excluding Sections 7 and 8.
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009.
Past Casualty Actuarial Society exams: 2007 Exam 6.
Patrik, G.S., “Reinsurance,” Foundations of Casualty Actuarial Science (Fourth Edition), Casualty Actuarial Society, 2001, Chapter 7, pp. 434-464 (section on Reinsurance Loss Reserving).
Original Problems and Solutions from The Actuary’s Free Study Guide
Problem S6-24-1. Similar to Question 11 from the 2007 CAS Exam 6. For each of the following pairs of variables, explain why examining them via an integrated approach in a Dynamic Financial Analysis (DFA) model is superior to using a silo approach.
(a) Underwriting profit and investment profit
(b) Loss ratio and mix of business in terms of new versus renewal business
Solution S6-24-1. In general terms, an integrated approach examines the interrelationships among variables, while a silo approach considers variables in isolation. A good resource on questions of this type is the CAS “Overview of Enterprise Risk Management”.
The following are sample answers, and other valid explanations are possible.
(a) Underwriting profit and investment profit both contribute to the overall profitability of the insurer. The insurer may be able to afford to sustain slight losses in a particular year if its investments are performing well and can make up the difference. On the other hand, if the stock market is in decline and bonds are not yielding much income, the insurer may need to raise its rates and/or implement stricter underwriting standards in order to remain profitable overall.
(b) The insurer’s loss experience may vary between its established customers and its new customers. If the loss ratio increases, this may be a sign of loosening underwriting standards. On the other hand, if the loss ratio increases, this may be because the insurer has been able to attract more favorable loss exposures to its book of business.
Problem S6-24-2. Similar to Question 12 from the 2007 CAS Exam 6.
The CAS “Overview of Enterprise Risk Management” discusses seven steps of enterprise risk management (ERM).
(a) The first step of ERM is to establish the context. What three kinds of context are typically examined as part of this step?
(b) The second step of ERM is to identify risks. What two broad categories of events is it important to identify here?
(c) The third step of ERM is to analyze/quantify risks. Name three kinds of possible analysis for this step.
(d) What three interrelated activities are part of the fourth step of ERM: integrating risks?
(e) Identify and briefly describe the fifth step of ERM.
(f) Identify the sixth step of ERM and some of the approaches that could be used to enact it.
(g) What is the seventh and last step of ERM, and how does it feed back into the first step?
Solution S6-24-2. Note that slight variations on these answers may also be acceptable.
(a) The three kinds of context are
1. External context: Relationship of the enterprise with its environment, including customers, shareholders, and other stakeholders. Evaluation of the enterprise’s strengths, weaknesses, opportunities, and threats (SWOT).
2. Internal context: The enterprise’s strategies, objectives, oversight/governance, and key performance indicators.
3. Risk management context: The risks relevant to the enterprise and the degree of coordination in risk management within the enterprise, including the adoption of common risk metrics.
(“Overview of Enterprise Risk Management”, p. 112)
(b) It is important to identify both (i) material threats to the enterprise’s objectives and (ii) areas/events the enterprise could exploit for competitive advantage (“Overview of Enterprise Risk Management”, p. 112).
(c) Three kinds of possible risk analysis are
1. Sensitivity analysis: How sensitive is a particular outcome to changes in the factors being analyzed?
2. Scenario analysis: What would happen under a particular set of conditions?
3. Simulation analysis: Examining a large number of simulated situations to see what the possibilities are.
(“Overview of Enterprise Risk Management”, p. 113)
(d) The integration of risks involves
1. Aggregation of the probability distributions for various risks
2. Taking into account the correlations among various risks and portfolio effects (which reduce overall variation where multiple assets or risks are involved)
3. Expression of results in terms of impact on the enterprise’s key performance indicators
(“Overview of Enterprise Risk Management”, p. 113)
(e) The fifth step of ERM is to assess/prioritize risks to determine how much each risk contributes to the overall risk profile of the enterprise and to establish what the most important decisions will be with regard to the totality of risks facing the enterprise (“Overview of Enterprise Risk Management”, p. 113).
(f) The sixth step of ERM is to treat/exploit risks. This can be done by (as the name implies) exploiting a risk (turning it into an opportunity), or retaining/financing, transferring, or reducing the risk (“Overview of Enterprise Risk Management”, p. 114).
(g) The last step of ERM is to monitor and review risks and risk management strategies. The risk management implemented by the enterprises interacts with and contributes to the context for subsequent time periods. Thus, in monitoring and reviewing the current state of affairs with regard to risk management, the enterprise helps to establish a context (step 1 of ERM) for subsequent risk management (“Overview of Enterprise Risk Management”, p. 114).
Problem S6-24-3. Similar to Question 15 from the 2007 CAS Exam 6. A reinsurance treaty has a premium of $5 million, paid in full upfront. The losses under the treaty are paid in a lump sum after 4 years, and the annual discount rate is 6%. The 90th percentile of losses is $5.5 million, and the 95th percentile of losses is $10 million.
(a) Does this treaty pass the 10-10 rule to qualify for reinsurance accounting?
(b) Should this treaty qualify for reinsurance accounting? Justify your answer.
Solution S6-24-3. This question is based on the discussion in the CAS “Accounting Rule Guidance Statement of Financial Accounting Standards No. 113-Considerations in Risk Transfer Testing”.
(a) The 10-10 rule is an (essentially arbitrary) rule of thumb that a 10% probability of a 10% loss is a “reasonable probability of a significant loss” for the reinsurer, which would qualify for reinsurance accounting. Here, the 90th percentile of losses is $5.5 million, but, since losses are paid after four years, the discounted value is (5.5 million)/1.064 = $4,356,515.15, which is less than the reinsurance premium of $5 million. Thus, the reinsurer’s expected loss ratio at the 90th percentile is even less than 100%: $4,356,515.15/($5 million) = circa 87.13%. The treaty does not pass the 10-10 rule.
(b) Consider the discounted loss at the 95th percentile: (10 million)/1.064 = $7,920,936.63. The reinsurer’s expected loss ratio is thus $7,920,936.63/($5 million) =circa 158.42%. This is much more than a 10% loss and can be reasonably considered significant. Also 5% is a non-negligible probability, so per FAS 113, it is justified to conclude that there is a “reasonable probability of a significant loss” for the reinsurer, and this treaty should qualify for reinsurance accounting.
Problem S6-24-4. Similar to Question 21 from the 2007 CAS Exam 6. What do the Stanard-Bühlmann and Bornhuetter-Ferguson methods have in common? How are they different, in essential terms?
Solution S6-24-4. Both methods rely on the formula
Ultimate Losses = Reported Losses + (% Losses Unreported)*(Expected Losses).
In the Bornhuetter-Ferguson Method, the expected loss ratio is estimated judgmentally. Losses are compared to earned premium that is not brought to the present rate levels.
In the Stanard- Bühlmann Method, adjusted premium is used instead of earned premium; adjusted premium is earned premium adjusted to current rate levels. Also, the expected loss ratio is estimated on the basis of reported claim experience from the overall time period being examined.
(See Friedland, pp. 174-175.)
Problem S6-24-5. Similar to Question 25 from the 2007 CAS Exam 6. Identify four possible problems with industry aggregates of reinsurance data as currently exist in the United States.
Solution S6-24-5. The following are possible problems with industry aggregates of reinsurance data:
1. Different reinsurance policies have different limits and attachment points, and the aggregate data are not separated by limits/attachment points.
2. Various reinsurers may choose to include or omit data about asbestos, pollution, or other environmental hazards in the numbers they submit.
3. Reinsurance contracts are unique, so the data are generally not comparable across reinsurers; there is no homogeneity on the basis of which the law of large numbers could work (Patrik, pp. 436-437).
4. Low claim frequency and extreme report lags may contribute to large fluctuations in the data (Patrik, p. 437).
See other sections of The Actuary’s Free Study Guide for Exam 6.