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 2 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.
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009. Chapter 14, pp. 329-330.
Past Casualty Actuarial Society exams: 2008 Exam 6 and 2009 Exam 6.
Slywotzky, A.J., and Drzik, J., “Countering the Biggest Risk of All,” Harvard Business Review, April 2005, Harvard Business School Publishing.
Problems and Solutions from The Actuary’s Free Study Guide
Problem S6-2-1. This problem is similar to Problem 1, Part (a), on the 2009 CAS Exam 6.
You are given the following information as of December 31, 2013:
(1) Paid Claims (including Salvage and Subrogation) by Accident Year (AY)
For AY 2011: 44,224
For AY 2012: 52,143
For AY 2013: 80,087
(2) Selected Ultimate Claims (including Salvage and Subrogation)
For AY 2011: 49,500
For AY 2012: 58,700
For AY 2013: 82,420
(3) Ratio of Received Salvage and Subrogation (S&S) to Paid Claims
For AY 2011: 0.151
For AY 2012: 0.176
For AY 2013: 0.210
(4) Development Factor to Ultimate for S&S Ratio
For AY 2011: 1.000
For AY 2012: 1.014
For AY 2013: 1.114
Using the ratio method, estimate the recoverables for Salvage and Subrogation (S&S) for accident years 2011-2013.
Solution S6-2-1. First, we estimate the ultimate S&S for each accident year. This is done by multiplying Ultimate Claims (2) by the S&S ratio (3) and the development factor to ultimate (4).
(5) Ultimate S&S by Accident Year: (5) = (2)*(3)*(4)
For AY 2011: 49,500*0.151*1.000 = 7474.5
For AY 2012: 58,700*0.176*1.014 = 10475.8368
For AY 2013: 82,420*0.210*1.114 = 19281.3348
Then we estimate paid S&S for each accident year. This is done by multiplying actual paid claims (1) by the S&S ratio (2). No development factors apply because we are only estimating what has already been paid.
(6) Paid S&S by Accident Year: (6) = (1)*(2)
For AY 2011: 44,224*0.151 = 6677.824
For AY 2012: 52,143*0.176 = 9177.168
For AY 2013: 80,087*0.210 = 16818.27
The S&S recoverables are the difference between ultimate S&S and paid S&S. They are what remains to be recovered.
(7) S&S Recoverables by Accident Year: (7) = (5) – (6)
For AY 2011: 7474.5 – 6677.824 = 796.676
For AY 2012: 10475.8368 – 9177.168 = 1298.6688
For AY 2013: 19281.3348 – 16818.27 = 2463.0648
Total: 796.676 + 1298.6688 + 2463.0648 = 4558.4096.
Problem S6-2-2. How is the development for salvage recoveries typically different from the development for subrogation recoveries, and why? (See Friedland, p. 329).
Solution S6-2-2. Salvage is associated with property coverages, where the losses are often quickly reported and settled. Thus, the salvage can also be determined much faster.
Subrogation is associated with liability coverages, where the losses can take years to ascertain, and it may take years to determine who is liable and the ultimate claim payout. Also, subrogation recoveries may take years to materialize after the underlying claim is paid, because the insurer still has to pursue the responsible party.
Friedland (p. 329) notes that some subrogation age-to-age factors may be less than 1. This can happen for older claims where the prospect of recovering from the responsible party diminishes over time.
Problem S6-2-3. This problem is similar to Problem 8 on the 2008 CAS Exam 6.
You are analyzing a contract where the premium is paid in full at the start of the contract term. The upfront premium is $3650.
The expected incurred losses occur in the following percentages per year of the contract:
Year 1: 34%
Year 2: 15%
Year 3: 40%
Years 4-14: 1% per year
For the end of each the years 1, 2, 3, and 4, calculate the unearned premium reserve based on (a) the assumption that premium is earned in the same pattern as expected losses, (b) the assumption that premium is earned on a pro rata basis, and (c) the difference between the answers in (a) and (b).
(d) Based on your answer to part (c), explain the problem with applying the approach in part (b) to this situation.
(e) Name three kinds of insurance-related products for which assuming that premium is earned on a pro rata basis would not be appropriate.
Solution S6-2-3. (a) The unearned premium reserve (here, UPR) is equal to
(Total premium)*(1 – Fraction of premium that is earned).
For Year 1, UPR = 3650*(1 – 0.34) = 2409.
For Year 2, UPR = 3650*(1 – 0.34 – 0.15) = 1861.5.
For Year 3, UPR = 3650*(1 – 0.34 – 0.15 – 0.40) = 401.5.
For Year 4, UPR = 3650*(1 – 0.34 – 0.15 – 0.40 – 0.01) = 365.
(b) There are 14 years over which the policy is expected to have losses. Thus, the pro rata method assumes that each year, 1/14th of the premium is earned, leaving the unearned premium reserve to be (Total premium)*(1 – (1/14)*Number of years elapsed).
For Year 1, UPR = 3650*(1 – 1/14) = 3389.286714
For Year 2, UPR = 3650*(1 – 2/14) = 3128.571429
For Year 3, UPR = 3650*(1 – 3/14) = 2867.857143
For Year 4, UPR = 3650*(1 – 4/14) = 2607.142857
(c) These answers are simply the difference between the corresponding values in (a) and (b):
For Year 1: 2409 – 3389.286714 = -980.286714
For Year 2: 1861.5 – 3128.571429 =-1267.071429
For Year 3: 401.5 – 2867.857143 =-2466.357143
For Year 4: 365 – 2607.142857 =-2242.142857
(d) The answers in part (c) can be thought of as the degree to which the pro rata method of estimating earned premium underestimates the true profitability of this product. Most of the losses for this product occur early on – during the first four years. But the pro rata method assumes that the losses occur evenly throughout the 14 years. This might, for instance, lead the company to assume that this product is not profitable and withdraw from offering it, when the product might in fact be a decent revenue source.
(e) The pro rata assumption for earned premium is not appropriate for the following kinds of products:
1. Warranties, where losses typically occur later during the contract term;
2. Policies covering seasonal exposures, such as hurricane risk. More premium should be earned during the season(s) of peak exposure.
3. Aggregate excess insurance policies, which cover losses above a certain attachment point. The attachment point is likely to be reached only later in the policy term, so that is when premium should start to be earned.
Problem S6-2-4. This problem is similar to Problem 42 on the 2008 CAS Exam 6.
What are the four considerations for identifying and assessing a risk, as mentioned by Slywotzky and Drzik (2005)? Briefly explain each consideration in your own words – not in the words of the authors or the CAS solutions.
Solution S6-2-4. The following is a sample answer. Students are encouraged to develop their own phrasings to help internalize the ideas.
1. Severity – What portion of the company’s overall value (measured in shareholders’ equity, market value, earnings, etc.) could be jeopardized by the risk?
2. Probability – How likely is the risk to happen?
3. Timing – Will the risk occur sooner or later? Can the time of occurrence be pinpointed, or can the risk occur at any of a broad range of times?
4. Changing probability over time – Will the passage of time raise, lower, or not affect the chances of the risk occurring?
Problem S6-2-5. This problem is similar to Problem 34 on the 2009 CAS Exam 6.
Explain each of the following strategic risks identified by Slywotzky and Drzik (2005) and state an approach for how a company might overcome such a risk.
(a) Brand erosion
(b) Customer priority shift
(c) New-project failure
(d) Market stagnation
(a) Brand erosion: Company experiences a significant drop in market share either because a sudden event (e.g., a well-publicized product defect or a large accident) has rendered its brand less attractive to consumers or a gradual erosion of the company’s brand occurred because the company failed to innovate and satisfy consumer expectations.
A way to overcome brand erosion is to “redefine the scope of brand investment” to focus on other aspects than mere marketing, such as quality of the product or service being offered.
(b) Customer priority shift: Shifts in the customers’ preferences or a shift in the balance of power toward consumers may reduce the company’s value and profitability.
A way to overcome this problem is through “fast and cheap experimentation”, where the company engages in relatively flexible and inexpensive ways to assess changing consumer tastes and receive feedback from consumers in various segments of the market.
(c) New-project failure: A company suffers because its new venture failed to attract customers or to function as intended, or competitors seized on the idea so quickly that the company could not earn sufficient profits.
The stepping-stone method, which uses a series of projects instead of a single lump-sum project, is a way of countering this risk. Each step in this method can be self-contained but can also lead to further undertakings if successful. The farther along on the stepping-stone approach the company is, the more likely success is, since earlier steps would have addressed and adapted to some of the challenges involved.
(d) Market stagnation: Growth in the market is diminished or halted, and new sources of growth cannot be easily found.
Demand innovation, the focus on the value the company brings to the customer, can help overcome market stagnation. The company can focus on more than just how the product functions but also on how consumers use the product. The company can then provide its consumers with services that would help them employ its product more effectively.
See other sections of The Actuary’s Free Study Guide for Exam 6.