**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 45 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.

**Bayesian Credibility Formula**

Z = (VHM)/(EVPV + VHM), where Z = credibility percentage, EVPV = expected value of process variance, and VHM = variance of hypothetical means. Generally, EVPV = EY(Var(X│Y)) and VHM = VarY(E(X│Y)) for random variables X and Y.

If we have as our variables Y = losses and X/Y = reporting ratio, then, in the linear approximation to the Bayesian credibility estimate, the following formulas hold:

VHM = VarY(E(X/Y)*Y) and

EVPV = Var(X/Y)*(Var(Y) + E(Y)2).

(More discussion on this subject can be found in Brosius, pp. 13-15.)

**Sources:**

Blanchard, R.S., “Accounting Concepts for the Actuary,” CAS Study Note, June 2003.

Brosius, E., “Loss Development Using Credibility,” CAS Study Note, March 1993.

Financial Accounting Standards Board, “Statement of Financial Accounting Standards No. 5, Accounting for Contingencies” (FAS 5), Paragraphs 1-4, and 8-10.

Friedland, Jacqueline F. *Estimating Unpaid Claims Using Basic Techniques**.* Casualty Actuarial Society. July 2010.

Past Casualty Actuarial Society exams: 2008 Exam 6 and 2009 Exam 6.

Teng, M.T.S.; and Perkins, M.E., “Estimating the Premium Asset on Retrospectively Rated Policies,” *PCAS* LXXXIII, 1996, pp. 611-647.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S6-45-1. Similar to Question 4 from the 2009 CAS Exam 6.** You are given the following information for Insurer Ξ:

*Cumulative Closed Claim Counts***,** expressed in the format

(Number at 12 months, Number at 24 months, Number at 36 months, Number at 48 months): **AY 2056:** (124, 234, 304, 350) **AY 2057:** (150, 225, 320)** AY 2058:** (130, 240)

**AY 2059:**(144)

*Selected Ultimate Claim Counts*** AY 2056:** 380

**AY 2057:**400

**AY 2058:**390

**AY 2059:**410

*Selected Cumulative Disposal Rates*** At 12 Months:** 0.35

**At 24 Months:**0.60

**At 36 Months:**0.78

**At 48 Months:**0.87

*Cumulative Paid Loss ($000)*** AY 2056:** (1000, 2150, 3340, 4400)

**AY 2057:**(1200, 2000, 3600)

**AY 2058:**(1200, 1950)

**AY 2059:**(1240)

Find the following using the disposal-rate frequency-severity technique and a 4% annual severity trend factor:

**(a)** The expected incremental claim counts for the time periods 36-48 months and 48 months to ultimate for accident year 2057;

**(b)** The 36-months-to-ultimate tail severity at 2059 levels;

**(c)** The ultimate losses for accident year 2057.

**Solution S6-45-1.**

**(a)** We use the selected disposal rates to estimate the incremental claim count for 36-48 months:

(Not-Yet-Opened Claims for 2057)*(Disposal rate at 48 months – Disposal rate at 36 months)/(1 – Disposal rate at 36 months) = (400-320)*(0.87-0.78)/(1-0.78) = 32.727272727, which we round to 33. Since 80 claims remain to be opened in total, the claims opened during the time period 48 months to ultimate will be 80 – 33 = 47.

Incremental claim count for 36-48 months: **33**

Incremental claim count for 48 months to ultimate: **47**

**(b)** We consider the incremental severity between 36 and 48 months for AY 2056 as (Incremental Paid Loss)/(Incremental Claim Counts) = (4400 – 3340)*1000/(350-304) = 23043.47826. Since we have no information about paid losses from 48 months to ultimate, this is our best estimate regarding severity from 36 months to ultimate. Now we need to trend this answer to 2059 by using the factor 1.043: 23043.47826*1.043 = 25920.77913 = **$25,920.78.**

**(c)** There are 80 claims remaining to be paid in 2057. The losses that have already been paid are $3,600,000. For the remaining claims, we multiply the claim count by the estimated 2057 36-months-to-ultimate severity, which is the 2056 severity trended by one year: 23043.47826*1.04 = 24923.82609. Our answer is thus 3600000 + 80*24923.82609 = **$5,593,906.09**.

**Problem S6-45-2. Similar to Question 6 from the 2009 CAS Exam 6.** You have the following information on a book of retrospectively rated workers’ compensation policies:

Standard premium: $260,000

Expected loss ratio to standard premium: 70%

Tax multiplier: 1.03

Loss cost factor: 1.25

Basic premium factor: 0.30

*Percentages at Third Retrospective Adjustment*

Loss eliminated by maximum and minimum: 16%

Loss eliminated by accident limit: 10%

Total loss emerged: 55%

*Percentages at Ultimate*

Loss eliminated by maximum and minimum: 24%

Loss eliminated by accident limit: 13%

Total loss emerged: 100%

What is the estimated future premium after the third retrospective adjustment?

**Solution S6-45-2.** The estimated future premium after the third retrospective adjustment is the difference between the estimated future premium at ultimate and the estimated premium up to the third retrospective adjustment.

The basic premium is the same irrespective of loss experience. Since it does not vary among retrospective adjustments, it can be disregarded for the purposes of this calculation. We want to find the components of premium that vary on the basis of losses. We need to convert standard premium to losses, take into account the losses eliminated, and then multiply the result by the loss cost factor and the tax multiplier to have the estimated premium take into account other elements besides losses (e.g., taxes and other expenses).

Estimated non-basic premium up to the third retrospective adjustment:

(Standard Premium)*(Expected Loss Ratio)*(% Loss Emerged)(1 – % Loss Eliminated)*(Loss Cost Factor)*(Tax Multiplier) = 260000*0.7*0.55*(1-0.16-0.1)*1.25*1.03 = 95370.275

Estimated ultimate non-basic premium:

(Standard Premium)*(Expected Loss Ratio)*(% Loss Emerged)*(1 – % Loss Eliminated)*(Loss Cost Factor)*(Tax Multiplier) = 260000*0.7*1*(1-0.24-0.13)*1.25*1.03 = 147624.75.

Thus, the estimated future premium after the third retrospective adjustment is 147624.75 – 95370.275 = 52254.475 = **$52,254.48**.

**Problem S6-45-3. Similar to Question 10 from the 2008 CAS Exam 6.** You are given the following information for an insurer’s book of business for a particular accident year:

Earned premium: $34,350

Reported losses as of 24 months: $14,515

Expected loss ratio: 80%

Coefficient of variation of loss ratio: 0.66

Coefficient of variation of percent of loss reported: 0.88

Expected percent of loss reported at 24 months: 55%

**(a)** What is the linear approximation to the Bayesian credibility estimate, as of 24 months, for the ultimate loss for this accident year?

**(b)** Calculate estimates of ultimate loss at age 24 months using each of the following methods (i) the chain ladder method, (ii) the Bornhuetter-Ferguson method, (iii) the Benktander method.

**(c)** Explain, for this particular situation, how the Benktander method incorporates the idea of credibility.

**Solution S6-45-3.**

**(a)** Let Y denote losses and X/Y denote the reporting pattern.

Then E(Y) = (Expected Loss Ratio)*(Earned Premium) = 0.8*34350 = E(Y) = 27480.

The coefficient of variation (CV) is (Standard Deviation)/(Mean). So SD(Y)/E(Y) = 0.66, and thus SD(Y) = 0.66*E(Y) = 0.66*27480 = 18136.8, and Var(Y) = SD(Y)2 = 18136.82 = 328943514.2.

E(X/Y) is given as 0.55, and SD(X/Y) = CV(X/Y)*E(X/Y) = 0.55*0.88 = 0.484. Thus, Var(X/Y) = 0.4842 = Var(X/Y) = 0.234256.

We recall the formula for the credibility percentage Z: Z = (VHM)/(EVPV + VHM).

We need to find VHM = VarY(E(X/Y)*Y) = Var(0.55Y) = 0.552*Var(Y) = 0.552*328943514.2 = VHM = 99505413.06.

We also need to find EVPV = Var(X/Y)*(Var(Y) + E(Y)2) = 0.234256*(328943514.2 + 274802) = 2539455504.

Thus, Z = (VHM)/(EVPV + VHM) = 99505413.06/(2539455504 + 99505413.06) = Z = 0.2815174416.

This credibility is being assigned to expected ultimate losses based on losses that are already developed, while the complement of credibility is assigned to E(Y), the expected losses.

Here, expected ultimate losses based on losses that are already developed, are (Losses already developed)/(% Losses reported) = 14515/0.55 = 26390.0909090909.

Thus, our estimate is 0.2815174416*26390.0909090909 + (1-0.2815174416)*27480 = 27173.40191 = **$27,173.40**.

**(b)** (i) Using the chain ladder method, the expected loss is (Loss already reported)/(% Loss reported) = 14515/0.55 = 26390.0909090909 = **$26,390.09**.

(ii) Using the Bornhuetter-Ferguson method, one assumes that unreported losses will be (Expected losses)*(1-% Losses already reported), and adds already reported losses to this value:

27480*(1-0.55) + 14515 = **$26,881**.

(iii) The Benktander method is an iteration of the Bornhuetter-Ferguson method, with the Bornhuetter-Ferguson estimate substituted in place of expected losses:

26881*(1-0.55) + 14515 = **$26,611.45**.

**(c)** The Benktander method can be seen as a credibility-weighted estimate in the sense that the percentage of credibility assigned to the chain-ladder estimate is equal to the percent of ultimate losses reported. The value 14515 in part (b)(iii) can be seen as the chain ladder estimate (26390.0909090909) multiplied by the percentage of credibility (55%). The complement of credibility is assigned to the Bornhuetter-Ferguson estimate (26881).

**Problem S6-45-4. Similar to Question 17 from the 2008 CAS Exam 6.**

**(a)** What is the purpose of an *income statement* in financial reports?

**(b)** What is the purpose of a *balance sheet* in financial reports?

**(c)** What is the purpose of a *statement of cash flows* in financial reports?

**Solution S6-45-4.**

**(a)** An income statement shows revenues, expenses, and their difference, net income, of a company during a particular period of time.

**(b)** A balance sheet functions as a snapshot in time of a company’s financial position, showing assets, liabilities, and equity as of a particular date.

**(c)** A statement of cash flows describes how the company got from its position at the beginning of a time period to the end of that time period and shows the company’s sources and uses of cash.

**Problem S6-45-5. Similar to Question 18 from the 2008 CAS Exam 6.** A workers’ compensation insurer is aware that an employee of one of its insureds has had a workplace accident, but a series of medical tests is necessary to determine whether an injury has occurred and how severe it will be. The medical tests will all be paid for by the insurer, but it is not clear what the costs of treating any injury will be. A review of similar accidents has shown that injuries are incurred in most cases, and a probability distribution for the severity of possible injuries has been constructed on the basis of historical data.

**(a)** What is the name for this event in terms of financial accounting standards? (See FAS 5.)

**(b)** What are the two general criteria that this event has met to be an instance of the term from part (a)? How has each criterion been met in the situation in question?

**Solution S6-45-5.**

**(a)** The situation in question is known as a *contingency*. There is uncertainty about whether the workers’ compensation insurer will suffer a loss in terms of paying for the worker’s medical treatments. The uncertainty will be resolved when the medical tests are performed and the results are obtained.

**(b)** The two general criteria for a contingency, per FAS 5, are

1. It is probable that a liability has been incurred or an asset has been impaired as of the date of the financial statement and

2. Reasonable estimation of the amount of the loss is possible.

Criterion 1 has been met here because it is probable that an injury has occurred, per the insurer’s analysis that most similar accidents result in injuries. Criterion 2 has been met because the insurer can estimate the expected loss based on its probability distribution of the severity of the likely injury.

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