**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 48 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:**

Casualty Actuarial Society Enterprise Risk Management Committee, “Overview of Enterprise Risk Management,” Casualty Actuarial Society *Forum*, Summer 2003, Section 3 and Appendix B.

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

Past Casualty Actuarial Society exams: 2008 Exam 6.

Slywotzky, A.J., and Drzik, J., “Countering the Biggest Risk of All,” *Harvard Business Review*, April 2005, Harvard Business School Publishing.

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

**Problem S6-48-1. Similar to Question 36 from the 2008 CAS Exam 6.** You have the following information as of December 31, 2050:

**Earned Premium for Calendar/Accident Year 2047:** 35055 **Earned Premium for Calendar/Accident Year 2048:** 38899 **Earned Premium for Calendar/Accident Year 2049:** 37600 **Earned Premium for Calendar/Accident Year 2050:** 41400

**Adjusted Premium for Calendar/Accident Year 2047:** 34400 **Adjusted Premium for Calendar/Accident Year 2048:** 36011 **Adjusted Premium for Calendar/Accident Year 2049:** 37000 **Adjusted Premium for Calendar/Accident Year 2050:** 40000

**Aggregate Reported Loss for Calendar/Accident Year 2047:** 22222 **Aggregate Reported Loss for Calendar/Accident Year 2048:** 16244 **Aggregate Reported Loss for Calendar/Accident Year 2049:** 12522 **Aggregate Reported Loss for Calendar/Accident Year 2050:** 4040

**Aggregate Loss Report Lag for Calendar/Accident Year 2047:** 0.95 **Aggregate Loss Report Lag for Calendar/Accident Year 2048:** 0.75 **Aggregate Loss Report Lag for Calendar/Accident Year 2049:** 0.60 **Aggregate Loss Report Lag for Calendar/Accident Year 2050:** 0.20

Note that “report lag” here refers to the proportion of losses assumed to be *already reported,* not the proportion assumed to have yet to be reported.

Use the Cape Cod method to calculate IBNR as of December 31, 2050.

**Solution S6-48-1.** We can calculate the expected loss ratio (ELR) as follows:

ELR = (Sum of aggregate reported losses for each year)/(Sum of used-up premiums for each year), where the used-up premium for each year is (Adjusted Premium)*(Aggregate Loss Report Lag).

Thus, ELR = (22222 + 16244 + 12522 + 4040)/(0.95*34400 + 0.75*36011 + 0.60*37000 + 0.20*40000) = 0.6121823486.

By the Cape Cod Method, IBNR = ELR*(Sum of adjusted premiums for each year) – (Sum of aggregate reported losses for each year) =

0.6121823486*(34400 + 36011 + 37000 + 40000) – (22222 + 16244 + 12522 + 4040) = 35214.4122 = **$35,214.41**.

**Problem S6-48-2. Similar to Question 37 from the 2008 CAS Exam 6.** Provide two advantages and disadvantages of using a reinsurer’s own experience, as opposed to using reinsurance industry data, in applying the Stanard-Bühlmann (Cape Cod) reserving technique.

**Solution S6-48-2.** This is a sample answer, and other valid answers are possible.

Advantages:

1. The reinsurer may have a history of rate changes that is not reflected in industry data.

2. The reinsurer may have a unique book of business that consists of a mix of treaty types not adequately represented in industry data.

Disadvantages:

1. The reinsurer’s experience may be sparse and therefore not credible.

2. The reinsurer’s data may be less stable than industry data, and a lot of the volatility in reinsurer data may be “noise” that would not provide a meaningful reserve estimate.

**Problem S6-48-3. Similar to Question 38 from the 2008 CAS Exam 6.** You have the following information about a reinsurance treaty:

The minimum ceding commission is 10% at an 80% loss ratio.

The commission then slides 1:1 to a 30% commission at a 60% loss ratio.

The commission then slides 1:4 to a maximum 40% commission at a 20% loss ratio.

The probability of a loss ratio being less than 20% is 0.3, and the average loss ratio in the range is 15%.

The probability of a loss ratio being between 20% and 60% is 0.5, and the average loss ratio in the range is 45%.

The probability of a loss ratio being between 60% and 80% is 0.15, and the average loss ratio in the range is 72%.

The probability of a loss ratio being greater than 80% is 0.05, and the average loss ratio in the range is 90%.

What is the expected ceding commissions?

**Solution S6-48-3.** While it is tempting to simply calculate the expected loss ratio and then calculate the corresponding ceding commission, there is not a one-to-one correspondence between loss ratio and ceding commission. The correct approach is to calculate the expected ceding commission for each loss ratio first, and then take the probability-weighted average of these expected commissions.

For the loss ratio range less than 20%, the expected ceding commission is 40%.

For the loss ratio range between 20% and 60%, the expected ceding commission is 30% + (1/4)*(60% – 45%) = 33.75%.

For the loss ratio range between 60% and 80 , the expected ceding commission is 10% + (80%-72%) = 18%.

For the loss ratio range greater than 80%, the expected ceding commission is 10%.

Thus, the overall expected ceding commission is 0.3*40% + 0.5*33.75% + 0.15*18% + 0.05*10% = **32.075%.**

**Problem S6-48-4. Similar to Question 39 from the 2008 CAS Exam 6.** A $400,000 in excess of $300,000 property per-occurrence excess-of-loss treaty is effective from January 1, 2044, to December 31, 2044. The annual ground-up loss trend is +2%. The loss development factors applicable to *the treaty layer* of experience are as follows:

**Loss Development Factors for Treaty Layer of Losses**

**For AY 2039:** 1.04** For AY 2040:** 1.23

**For AY 2041:**1.44

**For AY 2042:**1.55

**For AY 2043:**2.03

You are analyzing the following ground-up losses:

Loss 1 occurred on January 1, 2039, and has ground-up amount of $400,000.

Loss 2 occurred on July 1, 2041, and has ground-up amount of $250,000.

Loss 3 occurred on July 1, 2043, and has ground-up amount of $450,000.

What is the total amount of trended ultimate losses in the treaty layer?

**Solution S6-48-4.** First, we consider the trend factor that applies to each loss. The trend period is from the date of the loss to the midpoint of the treaty period – July 1, 2044.

For Loss 1, the trend factor is 1.025.5, and the trended loss is thus 400000*1.025.5 = 446026.781, implying that losses in the treaty layer are 146026.781.

For Loss 2, the trend factor is 1.023, and the trended loss is thus 250000*1.023 = 265302, implying that there are no losses in the treaty layer.

For Loss 3, the trend factor is 1.02, and the trended loss is thus 450000*1.02 = 459000, implying that losses in the treaty layer are 159000.

For Loss 1, we develop losses in the treaty layer to ultimate: 146026.781*1.04 = 151867.8522.

For Loss 3, we develop losses in the treaty layer to ultimate: 159000*2.03 = 322770.

The total amount of trended ultimate losses in the treaty layer is thus 151867.8522 + 322770 = 474637.8522 = **$474,637.85**.

**Problem S6-48-5. Similar to Question 40 from the 2008 CAS Exam 6.** An Enterprise Risk Management approach has identified the following risks for a company:

**Product obsolescence:** Probability of 10%, income impact of 40%. **Customer priority shift:** Probability of 20%, income impact of 8%. **Market stagnation:** Probability of 5%, income impact of 30% **Entrance of a new competitor:** Probability of 15%, income impact of 10%.

**(a)** Which of the above is the greatest risk to the company, according to the given information?

**(b)** Select two pairs of risks that are each correlated, either positively or negatively, and explain why they are correlated.

**(c)** How would the correlations in part (b) affect the selection of the second-greatest risk?

**Solution S6-48-5.**

**(a)** We consider the expected income impact of each risk (Probability*Income impact):

Product obsolescence: 10%*40% = 4%

Customer priority shift: 20%*8% = 1.6%

Market stagnation: 5%*30% = 1.5%

Entrance of a new competitor: 15%*10% = 1.5%

**Product obsolescence** is the greatest risk, with a 4% expected income impact.

**(b)** Two possible correlated pairs are as follows. (Other valid answers are possible.)

1. Product obsolescence and customer priority shift may be positively correlated. A product may become obsolete because customers no longer demand the same kind of functionality, or customers’ priorities may change because more advanced technological possibilities are available for meeting the same needs.

2. Market stagnation and entrance of a new competitor are negatively correlated, since a competitor’s entry suggests that the market is dynamic enough to attract competition.

**(c)** A risk that is positively correlated with the greatest risk is more likely to be the second-greatest risk. Even without any correlation, the customer priority shift risk has the second-highest expected income impact. Considering the correlation, one may have more evidence to suggest that the customer priority shift risk is indeed the second-greatest risk.

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