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 8 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.
Formulas for This Section
Formula 8.1
Reported Bornhuetter-Ferguson Method (Friedland, p. 153):
Ultimate Claims = Actual Reported Claims + (Expected Claims)*(% Claims Unreported)
Formula 8.2
Paid Bornhuetter-Ferguson Method (Friedland, p. 153):
Ultimate Claims = Actual Paid Claims + (Expected Claims)*(% Claims Unpaid)
Sources:
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009. Chapter 9, p. 153.
Past Casualty Actuarial Society exams: 2008 Exam 6.
Original Problems and Solutions from The Actuary’s Free Study Guide
Problem S6-8-1. Similar to Problem 2 from the Fall 2008 Exam 6.
You know the following regarding data from policy year 2044:
Premium was $5,000,000.
It is expected that 50% of the loss would emerge after 48 months, and 70% of the loss would emerge after 60 months.
Reported loss as of the end of 2047 was $2,120,000.
The estimate of ultimate loss via the Bornhuetter-Ferguson method is $4,200,000.
(a) What was the expected loss ratio used in the Bornhuetter-Ferguson estimate of ultimate loss?
(b) Use the chain-ladder method to calculate the ultimate loss estimate for policy year 2044.
(c) Use the Bornhuetter-Ferguson method to find the expected 2048 calendar year development for losses from policy year 2044.
Solution S6-8-1.
(a) We use the Reported Bornhuetter-Ferguson method, applying Formula 8.1:
Ultimate Claims = Actual Reported Claims + (Expected Claims)*(% Claims Unreported)
Here, Ultimate Claims = 4,200,000 and Actual Reported Claims = 2,120,000. % Claims Unreported = 50% at the end of 2047, which is the point at which we have reported claim data.
Expected Claims can be expressed as (Premium)*(ELR), where the ELR is the expected loss ratio.
Thus, 4,200,000 = 2,120,000 + 5,000,000*ELR*0.5 →
2,080,000 = 2,500,000*ELR →
ELR = 2,080,000/2,500,000 = ELR = 0.832 = 83.2%.
(b) The chain ladder method takes the latest known reported loss figure and asks, “What percentage of the ultimate reported loss is this figure expected to be?” Here, the latest known reported loss figure is $2,120,000, and this is expected to be 50% of the ultimate loss, so the ultimate loss is 2,120,000/0.5 = $4,240,000.
(c) The development portion of the Bornhuetter-Ferguson method formula is the
(Expected Claims)*(% Claims Unreported) part. Here, we only want to focus on claims expected to emerge in calendar year 2048. Based on our given expected loss percentages, this is 70% – 50% = 20% of all claims. Based on part (a), we can already calculate expected claims to be 5,000,000*ELR = 5,000,000*0.832 = $4,160,000. Of this, 20% is $832,000 – our estimate of development during CY 2048.
Problem S6-8-2. Similar to Problem 5 from the Fall 2008 Exam 6.
(a) If an insurer makes a one-time change in its policy limits applicable to all policies written after Day X, which method of data aggregation would be preferable: policy year or accident year? Why?
(b) When an insurer’s business is growing rapidly within a particular year, which method of data aggregation would be preferable: accident year or accident quarter? Why?
(c) If there is a significant legal decision that changes typical amounts of damages resulting from particular incidents, which method of data aggregation would be preferable: report year or accident year? Why?
(d) What could happen to claim counts so as to make earned exposures a more reliable measure by comparison?
Solution S6-8-2.
(a) If an insurer makes a one-time change in its policy limits applicable to all policies written after Day X, the policy year method would be preferable, because it could separately analyze policies written before the limit change and policies written after the limit change. Accident-year aggregation could mix data on losses occurring in the same period, but pertaining both to policies written before the limit change and policies written after it.
(b) When an insurer’s business is growing rapidly within a particular year, the accident quarter method of aggregation would be preferable, because a growing book of business would be expected to experience growing amounts of losses as well. This means that losses would be more heavily concentrated toward the end of the year, and separating data into accident quarters could segment the periods of greater losses from the periods of smaller losses.
(c) If there is a significant legal decision that changes typical amounts of damages resulting from particular incidents, the report year method of aggregation would be preferable, because claims reported after the decision would be subject to different likely severities than claims reported before the decision, irrespective of when the underlying losses occurred.
(d) Either the definition of what constitutes a claim or the insurer’s claim-handling practices might change in such a way as to make “claim counts” non-comparable across time. In such cases, earned exposures are a more reliable measure.
Problem S6-8-3. Similar to Problem 6 from the Fall 2008 Exam 6.
During Year X, an insurer’s claim-handling practices changed and each claim is now given a significantly lower initial case reserve than previously. However, the claims are also settled faster.
(a) Which of these methods would lead to definite overstatement of losses – the unadjusted reported loss development method or the unadjusted paid loss development method? Why?
(b) What changes unrelated to claims settlement could be responsible for a lowering of the initial case reserve assigned to each claim?
Solution S6-8-3.
(a) Both methods depend on development factors calculated from historical information and assuming that historical patterns of development will continue into the future. The unadjusted reported loss development method, however, will have to work with initial case reserve estimates that are lower than previously. Thus, an application of a historical loss development factor (LDF) to a lower case reserve will result in a lower estimate. However, the effect of faster claims settlement may or may not compensate for this – depending on the degree. If claims are settled significantly faster, and the historical LDF assumes a longer settlement pattern, then the effect of this would be a relative overstatement of losses. With the paid loss development method, however, the focus is only on the settlement pattern, and in this case a decrease in settlement times would produce an overstatement of ultimate losses if historical assumptions are used. So the unadjusted paid loss development method would lead to definite overstatement of losses.
(b) The following changes unrelated to claims settlement could be responsible for a lowering of the initial case reserve assigned to each claim:
1. Increase in deductibles on all policies – reducing the insurer’s potential liability per claim.
2. Decrease in limits on all policies – reducing the insurer’s potential liability per claim.
3. More rigorous underwriting standards – meaning that the insurer expects lower-risk insureds to be accepted into the program.
4. Movement of business to a different geographical area which tends to be populated by lower-risk insureds.
Problem S6-8-4. Similar to Problem 12 from the Fall 2008 Exam 6.
For a workers’ compensation high-deductible policy, the full coverage premium is $120,140, and the full-coverage expected loss ratio is 0.560. The excess ratio (the ratio of losses expected to be above the deductible) is 0.222. The aggregate ratio (the proportion of losses below the deductible for which the insurer may still have to incur expenses, for instance, via administering the policy or collecting the deductible from the insured) is 0.05. For the time period in question, it is known that reported excess losses were $16,000, and the excess loss development factor is 1.11.
(a) Using the loss ratio method to set reserves, what is the estimated ultimate loss for this policy?
(b) Would the loss ratio method be more preferable for a small start-up workers’ compensation insurer or a large, established insurer with plenty of its own credible data? Explain your choice.
Solution S6-8-4.
(a) The loss ratio method for workers’ compensation high-deductible policies does not rely on observed losses. Thus, reported excess losses and their associated development factor are not going to be used. There are two components to the loss ratio method: (1) the expected excess loss and (2) the losses associated with the aggregate ratio.
The expected excess loss component is (Premium)*(Full-Coverage ELR)*(Excess Ratio) = 120140*0.560*0.222 = 14935.8048.
The losses associated with the aggregate ratio are (Premium)*(Full-Coverage ELR)*(1 – Excess Ratio)*(Aggregate Ratio) = 120140*0.560*(1-0.222)*0.05 = 2617.12976.
The total estimated ultimate loss is 14935.8048 + 2617.12976 = 17552.93456 = $17,552.93.
(b) The loss ratio method to set reserves would be preferable for the small insurer which does not have a lot of its own data to estimate ultimate losses. The approach uses industry data and therefore adds credibility. However, for a large, established insurer, the loss ratio method has the drawback of not considering that insurer’s extensive loss experience data. Also, the insurer’s own practices and book of business may differ from those of the overall industry, and thus a reliance on the large insurer’s own data may be more appropriate.
Problem S6-8-5. Similar to Problem 16 from the Fall 2008 Exam 6.
(a) There are two claims of the exact same type. Claim A has an incurred loss amount of $60,000, while Claim B has an incurred loss amount of $6,000. An actuary is estimating unallocated loss adjustment expenses (ULAE) for these claims. He must choose between the dollar-based approach and the count-based approach. Which of these approaches would be likely to give the same estimate for ULAE for both claims?
(b) For each of the two approaches, give a diagnostic that might suggest the desirability of one approach over the other.
Solution S6-8-5.
(a) The count-based approach would be likely to give the same estimate for ULAE for both claims. This is because this approach assumes that ULAE does not correlate with the loss amount and is essentially the same for similar types of claims. The dollar-based approach assumes that ULAE is directly proportional to the loss amount.
(b) If a cost analysis of each claim identifies that the ULAE per claim is close to the same, irrespective of claim size, then the count-based approach can be reliable.
If the ratio of ULAE to paid loss amount is stable across all claims, then the dollar-based approach can be reliable.
See other sections of The Actuary’s Free Study Guide for Exam 6.
