**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 51 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.

**Sources:**

Blanchard, R.S., “Basic Insurance Accounting-Selected Topics,” CAS Study Note, June 2007, pp. 1-20.

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

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

Steeneck, L., “Commutation of Claims,” CAS Study Note, 1998.

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

**Problem S6-51-1. Similar to Question 31 from the 2008 CAS Exam 6.** You have the following information about a book of business subject to a quota share reinsurance treaty:

**Accident Year 2024** **Historical Earned Premium:** 2020 **Reported Losses:** 1414 **On-Level Factor:** 1.231 **Loss Development Factor:** 1.024** Loss Trend Factor:** 1.444

**1.234**

Property Value Inflation Factor:

Property Value Inflation Factor:

**Accident Year 2025** **Historical Earned Premium:** 2222 **Reported Losses:** 1600 **On-Level Factor:** 1.141 **Loss Development Factor:** 1.232** Loss Trend Factor:** 1.255

**1.124**

Property Value Inflation Factor:

Property Value Inflation Factor:

**Accident Year 2026** **Historical Earned Premium:** 2500 **Reported Losses:**2235 **On-Level Factor:** 1.055 **Loss Development Factor:** 1.558** Loss Trend Factor:** 1.088

**1.013**

Property Value Inflation Factor:

Property Value Inflation Factor:

Of these losses, the Accident Year 2026 reported losses include 1200 in catastrophe losses.

The reinsurer also has the following expenses as a percentage of the reinsurance premium:

**Administrative Expenses:** 4% **Brokerage Fees**: 2.5% **Unallocated Expenses:** 0.5%

**(a)** Find the ultimate non-catastrophe loss ratio for these three years of experience.

**(b)** A catastrophe model suggests that a 10% catastrophe load should be added as a percentage of non-catastrophe losses. Calculate the expected ultimate loss ratio for the three-year period.

**(c)** If the reinsurer wishes to achieve a 94% combined ratio on the treaty, should the primary insurer’s request for a 15% ceding commission be accepted? Justify your answer.

**Solution S6-51-1.**

**(a)** We need to adjust the premiums and losses. The losses for each year need to be multiplied by the development factor and the trend factor. The premiums need to be multiplied by the on-level factor and the property value inflation factor (which affects premiums only, since the loss trend factor reflects loss inflation). We do not consider catastrophe losses, so AY 2026 reported losses become 2235 – 1200 = 1035.

Total adjusted losses are 1414*1.024*1.444 + 1600*1.232*1.255 + 1035*1.558*1.088 = 6319.108224.

Total adjusted premiums are 2020*1.231*1.234 + 2222*1.141*1.124 + 2500*1.055*1.013 = 8589.956028.

The ultimate non-catastrophe loss ratio is 6319.108224/8589.956028 = **0.7356391818.**

**(b)** To get the ultimate loss ratio, we multiply the non-catastrophe loss ratio by 1.1: 0.7356391818*1.1 = **0.8092031.**

**(c)** The reinsurer’s combined ratio will be the ultimate loss ratio, plus all of its expense ratios, plus the ceding commission: 0.8092031 + 0.04 + 0.025 + 0.005 + 0.15 = 1.0292031 > 0.94. Since this combined ratio exceeds the reinsurer’s target ratio of 94%, the reinsurer **should not accept** the request to pay a 15% ceding commission.

**Problem S6-51-2. Similar to Question 30 from the 2008 CAS Exam 6.** Primary Insurer X is considering whether to commute a book of liabilities ceded to Reinsurer Y. The following information is known:

Nominal value of liabilities: $353,535

Present value of liabilities: $300,400

Current average IRS discount factor: 0.75

Present value of IRS remainder unwind: $52,424

Marginal tax rate: 35%

**(a)** Calculate the reinsurer’s ambivalence point.

**(b)** Would the reinsurer regard a $260,000 offer to be better or worse than break-even?

(For a discussion of this type of situation, see Steeneck, pp. 15-16.)

**Solution S6-51-2.**

**(a)** The ambivalence point is the point at which the reinsurer would be indifferent between commuting the liability and keeping it on its books.

We calculate the IRS-discounted value of the nominal liabilities: 353535*0.75 = 265151.25.

We calculate the tax on the present value of the unwind: 0.35*52424 = 18348.4.

The *basis* for the ambivalence point is the present value of liabilities, minus the tax on the present value of the unwind: 300400 – 18348.4 = 282051.6.

The ambivalence point can be found via the equation

Ambivalence Point = (Basis for Ambivalence Point – (Nominal IRS-Discounted Liabilities)*(Tax Rate))/(1 – Tax Rate) = (282051.6 – 265151.25*0.35)/0.65 = 291151.7885 = **$291,151.79**.

**(b)** Since the reinsurer holds the liabilities, it would need to pay to commute them. The ambivalence point of $291,151.79 indicates the highest price the reinsurer would be willing to pay. Since $260,000 better than break-even for the reinsurer.

**Problem S6-51-3. Similar to Question 24 from the 2009 CAS Exam 6.** A primary insurer is currently undergoing liquidation due to insolvency. A $800,000 loss is outstanding, and the liquidator has determined that only 70% of the loss will be paid. The primary insurer is party to a per-occurrence excess-of-loss reinsurance treaty of $500,000 in excess of $300,000. If the treaty has an insolvency clause without diminution, how much money would the primary insurer save compared to a situation in which the treaty does not have such a clause.

**Solution S6-51-3.** An insolvency clause without diminution provides that the reinsurer would pay its full obligation under the treaty, irrespective of the primary insurer’s insolvency. For the outstanding $800,000 loss, the reinsurer’s full obligation is $500,000. The liquidator has determined that only 0.7*800000 = $560,000 of the loss needs to be paid, so, with an insolvency clause without diminution, the primary insurer only has to pay the remaining $60,000. Without such a clause, the primary insurer still retains the amount of the adjusted loss below the attachment point, or $300,000, with the reinsurer paying the remaining $260,000. The savings for the primary insurer of having an insolvency clause without diminution is thus $300,000 – $60,000 = **$240,000.**

**Problem S6-51-4. Similar to Question 22 from the 2009 CAS Exam 6.** A reinsurance treaty’s period is from January 1, 2044, to December 31, 2044. You are aware of the following losses:

Loss A: Renewal policy, effective October 2, 2043, date of loss of October 9, 2043, treaty-covered loss amount of $3,440

Loss B: New policy, effective November 24, 2043, date of loss of March 5, 2044, treaty-covered loss amount of $11,000

Loss C: New policy, effective December 6, 2043, date of loss of December 26, 2043, treaty-covered loss amount of $5,656

Loss D: Renewal policy, effective December 15, 2043, date of loss of February 6, 2044, treaty-covered loss amount of $7,000

Loss E: New policy, effective April 5, 2044, date of loss October 17, 2044, treaty-covered loss amount of $2,010

Loss F: Renewal policy, effective June 16, 2044, date of loss March 6, 2045, treaty-covered loss amount of $12,000

**(a)** Define “risks attaching basis” and find the amount of losses covered if this treaty were written on a risks attaching basis.

**(b)** Define “losses occurring basis” and find the amount of losses covered if this treaty were written on a losses occurring basis.

**(c)** Define “policies issued basis” and find the amount of losses covered if this treaty were written on a policies issued basis.

**(d)** Define “in-force policies basis” and find the amount of losses covered if this treaty were written on an in-force policies basis.

**Solution S6-51-4.**

**(a)** *Risks attaching basis:* Only losses on policies written or renewed during the treaty period are covered. This means that only policies written or renewed during 2044 will have losses covered. Thus only losses E and F will be covered, for a total covered amount of $2,010 + $12,000 = **$14,010.**

**(b)** *Losses occurring basis:* Only losses occurring during the treaty period are covered. Thus only losses B, D, and E will be covered, for a total covered amount of $11,000 + $7,000 + $2,010 = **$20,010.**

**(c)** *Policies issued basis:* Only losses on new policies written during the treaty period are covered. Thus only losses on policies written in 2044 are covered. Only loss E qualifies, and the loss amount is **$2,010.**

**(d)** *In-force policies basis:* Only losses on policies already in force at the start of the treaty period are covered. Thus, only losses A, B, C, and D (from policies written or renewed before 2044) are covered, for a total loss amount of $3,440 + $11,000 + $5,656 + $7,000 = **$27,096.**

**Problem S6-51-5. Similar to Question 19 from the 2009 CAS Exam 6.**

**(a)** Define the *bank deposit approach,* the *prospective approach,* and the *retrospective approach* for deposit accounting.

**(b)** Give three general situations for which deposit accounting would be required instead of reinsurance accounting.

(See Blanchard, “Basic Insurance Accounting – Selected Topics”, p. 19.)

**Solution S6-51-5.**

**(a)** Under the **bank deposit approach,** the deposit amount grows over time in accord with a fixed, predetermined interest rate and is reduced by the amount of any withdrawals. Under the **prospective approach,** the deposit’s current value is equal to the value of future payments. Past payments or the amount of the initial deposit do not matter. Under the **retrospective approach,** the deposit amount depends on the initial deposit, past payments, and the current estimate of future payments. The interest rate is set such that the initial deposit is equal to the discounted value of past payments and estimated future payments (Blanchard, “Basic Insurance Accounting – Selected Topics”, p. 19).

**(b)** Deposit accounting would be required instead of reinsurance accounting in the following situations:

1. There is no risk transfer.

2. There is only transfer of timing risk (when a payment is made) but not amount risk (how much will be paid).

3. Many situations of retroactive reinsurance (Blanchard, “Basic Insurance Accounting – Selected Topics”, p. 19).

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

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