Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 15

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:

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

Past Casualty Actuarial Society exams: 2007 Exam 6.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S6-37-1. Similar to Question 42 from the 2007 CAS Exam 6. If there is a clearly identifiable trend in an insurer's loss ratio experience from one year to another, what aspects of (a) the Bornhuetter-Ferguson development method and (b) the Least-Squares development method would render such methods sub-optimal for developing ultimate loss and unpaid claim estimates?

Solution S6-37-1.

(a) The unreported component of losses under the Bornhuetter-Ferguson development method depends entirely on an expected loss ratio. Unless that expected loss ratio has already been adjusted to reflect the most recent loss ratio trends, there will be an over- or underestimation.

(b) The Least-Squares development method is designed for situations where any changes in loss ratio experience are random. If there is a clear directional trend that the insurer can identify, then this assumption would not hold, and the Least-Squares method would ignore this systematic change in the book of business.

Problem S6-37-2. Similar to Question 43 from the 2007 CAS Exam 6. You are given the following cumulative paid claim data by accident year (AY) for Insurer Λ as of December 31, 2049, expressed in the format

(Amount at 12 months, Amount at 24 months, Amount at 36 months, Amount at 48 months), where applicable.

Cumulative Paid Claims

AY 2046: (2330, 3345, 4010, 4430)
AY 2047: (2402, 3504, 4123)
AY 2048: (2403, 3450)
AY 2049: (2420)

There are two reserving methods used to determine ultimate claim amounts for each accident year. The following are the development factors to ultimate for each method:

12 months to ultimate - Method 1: 1.89
12 months to ultimate - Method 2: 1.93
24 months to ultimate - Method 1: 1.30
24 months to ultimate - Method 2: 1.35
36 months to ultimate - Method 1: 1.10
36 months to ultimate - Method 2: 1.06
48 months to ultimate - Method 1: 1.00
48 months to ultimate - Method 2: 1.00

In calendar year (CY) 2050, the following losses are actually paid out:

For AY 2046: 0
For AY 2047: 332
For AY 2048: 704
For AY 2049: 1022
For AY 2050: 2450
Total: 4508

(a) Use a retrospective test of reserve adequacy to select either Method 1 or Method 2 as the more appropriate reserving method of the two.

(b) How could the bias of the selected method be corrected via an adjustment? Explain any assumptions in your answer.

Solution S6-37-2.

(a) A retrospective test of reserve adequacy would compare the losses actually paid out in CY 2050 to the projections by each of the methods. The losses for AY 2046-2049 paid out in CY 2050 are 4508 - 2450 = 2058.

We now determine the losses projected by Method 1:

For AY 2046: 0, since losses are already at ultimate.
For AY 2047: 4123*(1.10 - 1) = 412.3
For AY 2048: 3450*(1.30/1.10 - 1) = 627.2727272727
For AY 2049: 2420*(1.89/1.30 - 1) = 1098.307692
Total: 2137.915384

Error: 2137.915384/2058 - 1 = 0.0388315784 = Overestimate of circa 3.88%.

We now determine the losses projected by Method 2:

For AY 2046: 0, since losses are already at ultimate.
For AY 2047: 4123*(1.06 - 1) = 247.38
For AY 2048: 3450*(1.35/1.06 - 1) = 943.8679245
For AY 2049: 2420*(1.93/1.35 - 1) = 1039.703704
Total: 2230.951628

Error: 2230.951628/2058 - 1 = 0.084038692 = Overestimate of circa 8.40%.

Method 1 is preferable because it has a lower overall error.

(b) To adjust for the overestimate in Method 1, one could multiply the result by

1/(1 + Error Amount) - in this case, 1/1.0388315784 = 0.962619948. This would bring the overall reserve for CY 2050 to the level of actual losses in CY 2050 and would presumably correct any bias in estimates for subsequent years. This adjustment requires the assumption that Method 1 would continue having the same bias over time, and that the insurer experiences consistent losses and has a stable book of business.

Problem S6-37-3. Similar to Question 44 from the 2007 CAS Exam 6. In accident years (AY) 2023 through 2026, the number of cumulative reported and closed claims for Insurer Σ did not vary by accident year for any particular age of maturity. Cumulative incurred losses and case loss reserves were as follows, expressed in the format

(Amount at 12 months, Amount at 24 months, Amount at 36 months, Amount at 48 months), where applicable. Assume all losses are at ultimate at 48 months.

Cumulative Incurred Losses - Data as of December 31, 2026

For AY 2023: (3033, 4044, 4505, 4606)
For AY 2024: (3185, 4246, 4730)
For AY 2025: (3344, 4459)
For AY 2026: (3511)

Case Loss Reserves - Data as of December 31, 2026

For AY 2023: (1000, 500, 200, 0)
For AY 2024: (1050, 525, 210)
For AY 2025: (1050, 525)
For AY 2026: (1103)

(a) Find the IBNR as of December 31, 2026, using the chain ladder method.

(b) What aspect of this scenario renders the IBNR estimate in part (a) inaccurate? Justify your answer by reference to the given data.

Solution S6-37-3.

(a) We first calculate age-to-age development factors for incurred losses, using the format
(12-24-month factor, 24-36-month factor, 36-48-month factor), where applicable.

Age-to-Age Factors for Incurred Losses
(4044/3033, 4505/4044, 4606/4505)
(4246/3185, 4730/4246)
(4459/3344)

Age-to-Age Factors for Incurred Losses
(1.333, 1.114, 1.022)
(1.333, 1.114)
(1.333)

Our selection for age-to-age factors is made simple in this scenario. We can also select factors to ultimate:
12-month-to-ultimate factor: 1.333*1.114*1.022 = 1.517631164
24-month-to-ultimate factor: 1.114*1.022 = 1.138508
36-month-to-ultimate factor: 1.022

Now we can estimate IBNR by multiplying each still-not-ultimate value on the outermost diagonal of the incurred loss triangle by (the appropriate factor to ultimate - 1) and adding these products:

4730*(1.022 - 1) + 4459*(1.138508 - 1) + 3511*(1.517631164 - 1) = 2539.070189 = IBNR = 2539. (Slight variations on this are possible if rounding was used at different steps of the calculation.)

(b) We consider the incurred loss trend at each age of maturity and compare it to the case reserve trend:

Cumulative Incurred Loss Trend

AY 2023 to AY 2024: (3185/3033, 4246/4044, 4730/4505)
AY 2024 to AY 2025: (3344/3185, 4459/4246)
AY 2025 to AY 2026: (3511/3344)

Cumulative Incurred Loss Trend

AY 2023 to AY 2024: (1.05, 1.05, 1.05)
AY 2024 to AY 2025: (1.05, 1.05)
AY 2025 to AY 2026: (1.05)

Case Reserve Trend

AY 2023 to AY 2024: (1050/1000, 525/500, 210/200)
AY 2024 to AY 2025: (1050/1050, 525/525)
AY 2025 to AY 2026: (1103/1050)

Case Reserve Trend

AY 2023 to AY 2024: (1.05, 1.05, 1.05)
AY 2024 to AY 2025: (1.00, 1.00)
AY 2025 to AY 2026: (1.05)

While incurred losses increased by 5% from AY 2024 to AY 2025, case reserves did not increase at all. The net result of this is reduced case outstanding strength. The chain ladder method assumes constant case outstanding strength. With reduced case outstanding strength and the same loss development factors calculated via the chain ladder method, there will be an underestimation of IBNR.

Problem S6-37-4. Similar to Question 45 from the 2007 CAS Exam 6. There are two excess-of-loss reinsurance treaties. In Treaty A, the primary insurer's retention is $500,000. In Treaty B, the primary insurer's retention is $5,000,000. For which treaty would you expect the excess loss development factors to be higher? Give two reasons justifying your answer.

Solution S6-37-4. One would expect the excess loss development factors to be higher for Treaty B, because it has the higher retention. Two reasons why this happens is (1) larger losses that would exceed the higher retention are more likely to be reported later, since the primary insurer may not expect certain initial claims to develop to such an extent and (2) the smaller claims that do not exceed the retention are likely to be reported sooner. More of the smaller claims would contribute to the excess loss development for a treaty with a smaller retention.

Problem S6-37-5. Similar to Question 46 from the 2007 CAS Exam 6. You have the following information about a particular insurance policy from a well-established book of business:

Premium: 200,000
Expected loss ratio: 80%
Observed loss up to December 31, 2020: 130,000
Age-to-ultimate development factor applicable at December 31, 2020: 1.60

(a) According to the Bornhuetter-Ferguson method, what is the estimated ultimate loss amount for this policy?

(b) In the answer from part (a), what is the percentage credibility assigned to the loss development projection?

(c) What is one possible shortcoming of the Bornhuetter-Ferguson method in this case, and what can be used to mitigate this shortcoming?

Solution S6-37-5.

(a) The Bornhuetter-Ferguson method uses the formula
Ultimate Claims = Actual Reported Claims + (Expected Claims)*(% Claims Unreported).
Here, we know that Actual Reported Claims = 130,000.
We calculate Expected Claims = Premium*(Expected Loss Ratio) = 200000*0.8 = 160,000.
We calculate % Claims Unreported = 1 - 1/1.60 = 0.375 = 37.5%
Thus, Ultimate Claims = 130000 + 160000*0.375 = 190,000.

(b) For the Bornhuetter-Ferguson method, the percentage credibility assigned to the loss development projection is the percentage of claims assumed to be reported at the time as of which the data are being analyzed. This is 1/(Development Factor to Ultimate), which here is 1/1.60 = 0.625 = 62.5%.

(c) The Bornhuetter-Ferguson method relies on a predetermined expected loss ratio that may not take into account recent changes in loss experience. To assign more credibility to the development projection, one could use the Benktander method, which is an iterative application of the Bornhuetter-Ferguson method, using the result from the first application of the Bornhuetter-Ferguson method as the "Expected Claims" component. One could also use the Stanard-Bühlmann (Cape Cod) method, which contains a systematic way of calculating the expected loss ratio.

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