Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 5

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:
Financial Accounting Standards Board, "Statement of Financial Accounting Standards No. 60, Accounting and Reporting by Insurance Enterprises," pp. 7, 9.

Past Casualty Actuarial Society exams: 2009 Exam 6.

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

Problem S6-10-1. Similar to Problem 5 from the Fall 2009 Exam 6.

A five-year warranty contract has its premium of $500 paid upfront. It is expected that, each year, the loss on the contract will be 20% higher than the previous year. What is the unearned premium reserve for one such contract at the end of its fourth year - under a deferral-matching accounting paradigm?

Solution S6-10-1. Under a deferral-matching accounting paradigm, the amount of premium earned is proportional to the amount of loss exposure already experienced.

Let x be the proportion of losses expected in the first year. Then we have the following:

First-year losses: x
Second-year losses: 1.2x
Third-year losses. 1.2²*x
Fourth-year losses. 1.2³*x
Fifth-year losses. 1.2⁴*x

The sum of these values must equal 1, so

(1 + 1.2 + 1.2² + 1.2³ + 1.2⁴)x = 1 → x = 1/(1 + 1.2 + 1.2² + 1.2³ + 1.2⁴) = 0.1343797033

After four years, the proportion of expected loss exposure that has not yet taken place is 1.2⁴*x = 1.2⁴*0.1343797033 = 0.2786497527. This is also the proportion of premium unearned. Thus, the unearned premium reserve is 500*0.2786497527 = 139.3248764 = $139.32.

The following information applies to Problems S6-10-2 and S6-10-3.

You are aware of the following information for claims pertaining to accident year (AY) 2033. As of December 31, 2034, reported losses were $130. It is expected that ultimate AY 2033 losses will be $200. Based on many years of data, cumulative development factors have also been selected in the following manner:

12-months-to-ultimate factor: 1.850
24-months-to-ultimate factor: 1.628
36-months-to-ultimate factor: 1.374

As of December 31, 2035, AY 2033 reported losses are $166.

Problem S6-10-2. Similar to Problem 16(a) from the Fall 2009 Exam 6. Calculate the difference between (i)actual AY 2033 reported losses in calendar year (CY) 2035 and (ii) expected AY 2033 reported losses in CY 2035, based on the data and assumptions given.

Solution S6-10-2.

(a) The actual AY 2033 reported losses in CY 2035 are 166 - 130 = $36.
We find the expected reported losses as follows.
The expected losses yet-to-be-reported (all the way to ultimate) are 200 - 130 = $70.
We need to determine what fraction of this yet-to-be-reported amount is expected to be reported in CY 2035.

The 24-months-to-ultimate factor is 1.628, meaning that, as of the end of 2034, 1/1.628 of the loss is expected to have emerged. The loss yet to emerge is thus (1 - 1/1.628) of the total expected amount.

The 36-months-to-ultimate factor is 1.374, meaning that, as of the end of 2035, 1/1.374
of the loss is expected to have emerged.

During 2035, the proportion of the total expected loss that will emerge is thus
(1/1.374 - 1/1.628).

Thus, the fraction of yet-to-be-reported losses assigned to CY 2035 is
(1/1.374 - 1/1.628)/(1 - 1/1.628) = 0.2943657924.

The expected reported losses for CY 2035 are therefore 70*0.2943657924 = 20.60560547.

The desired (actual - expected) difference is thus 36 - 20.60560547 = 15.39439453 = $15.39.

Problem S6-10-3. Similar to Problems 16(b) and 16(c) from the Fall 2009 Exam 6.

(a) Using linear interpolation of the development pattern provided, what are the expected losses emerged between January 1, 2035, and September 30, 2035?

(b) Will the answer in part (a) overestimate or underestimate the projection? Explain your answer.

Solution S6-10-3.

(a) The expected losses yet-to-be-reported (all the way to ultimate) are 200 - 130 = $70.

The 24-months-to-ultimate factor is 1.628.

The 36-months-to-ultimate factor is 1.374.

We want to find, using linear interpolation, the 33-months-to-ultimate factor.

Thus value is 9/12 of the way between 1.628 and 1.374:

1.628 - (9/12)(1.628 - 1.374) = 1.4375.

Thus, the fraction of yet-to-be-reported losses assigned to the first 9 months of CY 2035 is (1/1.4375 - 1/1.628)/(1 - 1/1.628) = 0.1325217391, meaning that the expected losses are 0.1325217391*70 = 9.276521739 = $9.28.

(b) Even a visual comparison of the answer in part (a) to the answer in Solution S6-10-2 suggests that the linear interpolation approach would underestimate the projection. A real-world reason for this is that development tends to occur at a decreasing rate, with more development occurring earlier. Linear interpolation, however, presumes that development occurs at a uniform rate. The interpolated development factor thus overstates the true factor, leading to an understated estimate for the amount of development occurring up to the time in question.

Problem S6-10-4. Similar to Problem 18 from the Fall 2009 Exam 6.

Fill in the blanks in accordance with Financial Accounting Standard (FAS) #60:

(a) A liability for unpaid claims shall be accrued _________ (at what time?). (FAS #60, p. 7)

(b) Salvage and subrogation shall be evaluated ________ (how?) and ________ (added to or deducted from?) ______ (what?). (FAS #60, p. 9)

(c) Claim adjustment expenses shall be accrued when ________ (what?) is accrued? (FAS #60, p. 9)

Solution S6-10-4.

(a) A liability for unpaid claims shall be accrued when insured events occur (FAS #60, p. 7).

(b) Salvage and subrogation shall be evaluated in terms of their estimated realizable value and deducted from the liability for unpaid claims (FAS #60, p. 9).

(c) Claim adjustment expenses shall be accrued when the related liability for unpaid claims is accrued (FAS #60, p. 9).

Problem S6-10-5. Similar to Problem 23 from the Fall 2009 Exam 6.

In the context of reinsurance, what is surplus relief and what kind of reinsurance product typically provides it? Would an insurer with a shrinking, stable, or growing book of business have a greater need for surplus relief and why?

Solution S6-10-5. Surplus relief occurs when a reinsurer pays a ceding commission on the business it assumes from a primary insurer. This compensates the primary insurer for a fact that, when it acquires new business, statutory accounting principles require immediate recognition of acquisition costs but deferred recognition of premium (which must be recognized as earned over time). The ceding commissions can be recognized as earned immediately. Typically, pro rata reinsurance, where the primary insurer and the reinsurer share proportionally in the loss on the ceded policies,provides surplus relief. An insurer with a growing book of business has the greatest need of surplus relief, as its immediate acquisition expenses would be the highest.

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