Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 7

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.

Retrospective rating determines the premium for the present time period on the basis of the losses for the present time period.

Let P be the premium in retrospective rating. Then the following formula holds:

Formula 14.1
P = (B + CL*LCF)*TM, where
B = the basic premium - a constant which does not vary depending on losses;
CL = the capped loss; losses are typically capped at a certain value for the purposes of retrospective rating;
LCF = loss conversion factor - by definition, the multiplier which converts loss into premium;
TM = tax multiplier, needed to for the premium to incorporate the premium tax, which is a percentage of premium.

Due to the development of losses, multiple retrospective adjustments ("retro adjustments") are often required to accurately determine the premium.

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

Past Casualty Actuarial Society exams: 2007 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 S6-14-1. Similar to Problem 5 from the 2007 CAS Exam 6. Suppose that an insurance policy is retrospectively rated and the following information is known:

Loss conversion factor = 1.22
Tax multiplier = 1.04
Basic premium = $13,165
Loss at first retro adjustment: $15,530
Capped loss at first retro adjustment: $15,530
Loss at second retro adjustment: $25,010
Capped loss at second retro adjustment: $20,000

What is the ratio of premium development to loss development between the first and second retro adjustments?

Solution S6-14-1. We can already figure out how much loss has developed between the two adjustments: 25010 - 15530 = $9480.

To determine premium development, we need to calculate the retrospectively rated premium at each adjustment, using Formula 14.1: P = (B + CL*LCF)*TM.

At the first retro adjustment, the premium is (13165 + 15530*1.22)*1.04 = 33396.064.
At the second retro adjustment, the premium is (13165 + 20000*1.22)*1.04 = 39067.6.
Premium development is thus 39067.6 - 33396.064 = 5671.536.

The desired ratio is 5671.536/9480 = 0.5982632911.

Problem S6-14-2. Similar to Problem 6 from the 2007 CAS Exam 6. During each particular accident year, the exposure to loss is uniform. You also know that cumulative loss development follows the function D(t) = 1-e^-2t, where t is the time in years from the average accident date. As of December 31, 2055, you know the following:

Incurred losses for accident year (AY) 2052 are $31,531.
Incurred losses for accident year (AY) 2053 are $12,330.

What is the ultimate loss for AY 2052 and AY 2053 combined?

Hint: This problem should be intuitive enough, but be careful about your choice for t!

Solution S6-14-2. The function D(t) simply represents the fraction of the ultimate loss amount that is developed to date. What is t? As defined, t is the time from the average accident date. Since exposure to loss in each year is uniform, the average accident date is the midpoint of the year. Thus, from the midpoint of 2052 to the end of 2055, there are 3.5 years. From the midpoint of 2053 to the end of 2055, there are 2.5 years. The rest is a matter of dividing known incurred losses for each year by the fraction we expect to be already developed; this gives us the ultimate loss amounts for each year, and we then add those together:

31531/(1-e^-2*3.5) + 12330/(1-e^-2*2.5) = 43973.42146 = $43,973.42.

Problem S6-14-3. Similar to Problem 8 from the 2007 CAS Exam 6. The disposal rate for claims measures the proportion of claims from a given report year that are settled within the specified time interval from the report year.

Consider a triangle displaying disposal rates in the following format for each report year:

(Rate at 0-24 months from the report year, rate at 25-48 months, rate from 48 months to ultimate)

Report Year 2028: (0.431, 0.352, 0.217)
Report Year 2029: (0.540, 0.260)
Report Year 2030: (0.410)

Estimate the disposal rate for the group of claims at 25-48 months from report year 2030. Use only in the information from the most recent available calendar year.

Solution S6-14-3. The information from the most recent available calendar year is the information pertaining to 2029 data. This shows that a 0-24 disposal rate of 0.540 corresponds to a 25-48 month disposal rate of 0.260. In 2029, after 24 months, the proportion of unsettled claims was 1 - 0.540 = 0.460. So the proportion of this amount that was settled was 0.260/0.460 = 0.5652173913.

In 2030, after 24 months, the proportion of unsettled claims was 1 - 0.410 = 0.590. If 56.52173913% of 0.590 gets settled 25-48 months from 2030, the desired disposal rate is 0.590*0.5652173913 = 0.3334782609.

Problem S6-14-4. Similar to Problem 9 from the 2007 CAS Exam 6. You know the following:

Earned premium for Accident Year (AY) 2050 was $4000.
Earned premium for AY 2051 was $2600.
Earned premium for AY 2052 was $3600.
Earned premium for AY 2053 was $4500.

You are also given a cumulative reported loss triangle, in the following format for each accident year: (Developed loss at 12 months, developed loss at 24 months, developed loss at 36 months, ultimate loss).

Cumulative reported losses
AY 2050: (1800, 1900, 2500, 3000)
AY 2051: (1660, 1790, 2300)
AY 2052: (1900, 2200)
AY 2053: (2000)

Use the percentage of premium method to estimate the IBNR for accident year 2053. When calculating factors for each accident year, use a simple arithmetic average of the factors.

Solution S6-14-4. As the name suggests, the percentage of premium method assumes that the IBNR is a set percentage of the premium for the accident year in question, where the percentage is equal to the not-yet-developed losses as a fraction of premium.

First, it is helpful to convert our cumulative reported loss triangle into an incremental reported loss triangle:

Incremental reported losses
AY 2050: (1800, 100, 600, 500)
AY 2051: (1660, 130, 510)
AY 2052: (1900, 300)
AY 2053: (2000)

For each accident year, we can calculate the ratios of incremental reported losses to earned premium:

Ratios of incremental reported losses to earned premium:

AY 2050 - at 24 months: 100/4000 = 0.025
AY 2051 - at 24 months: 130/2600 = 0.05
AY 2052 - at 24 months: 300/3600 = 0.08333333333
Arithmetic average - at 24 months: (0.025 + 0.05 + 0.08333333333)/3 = 0.0527777778

AY 2050 - at 36 months: 600/4000 = 0.15
AY 2051 - at 36 months: 510/2600 = 0.1961538462
Arithmetic average - at 36 months: (0.15 + 0.1961538462)/2 = 0.1730769231

AY 2050 - at ultimate: 500/4000 = 0.125 - the only value, so there is no need to take an arithmetic average.

The cumulative ratio of incremental reported losses to earned premium - between 12 months and ultimate - is 0.0527777778 + 0.1730769231 + 0.125 = 0.3508547009.

Thus, the estimated IBNR is the above ratio multiplied by the AY 2053 earned premium: 4500*0.3508547009 = 1578.846154 = $1578.85.

Problem S6-14-5. Similar to Problem 10 from the 2007 CAS Exam 6. Briefly state two kinds of strategic risk - other than new product failure, customer priority shifts, a technology shift, and a one-of-a-kind competitor - for a company that produces physical goods - as discussed in the Slywotzky and Drzik 2005 paper. For each risk you mention, state a mitigation strategy for the risk.

Solution S6-14-5. Three other kinds of risks are mentioned in the Slywotzky and Drzik 2005 paper, and any two suffice as an answer:

Risk: Brand erosion
Mitigation strategy: Redefine the scope of brand investment - focus on quality of product and customer service instead of just marketing.

Risk: Industry margin squeeze
Mitigation strategy: Shift the collaborate/compete ratio: firms whose margins are being squeezed should cooperate and share functions more than previously.

Risk: Market stagnation
Mitigation strategy: Engage in demand innovation - focus on the customer's perspective and how the firm can help the customer beyond just offering a product, i.e., by helping the customer cut costs, improve profitability and reduce capital intensity.

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