**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 52 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:**

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

Past Casualty Actuarial Society exams: 2009 Exam 6.

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

**Problem S6-52-1. Similar to Question 10 from the 2009 CAS Exam 6.** It is estimated that the expected loss rate for an insurance company is $200 per exposure unit. The company has no exposure prior to 2030. You also have the following information by accident year (AY):

**AY 2030** **Exposure Units:** 2340 **Incurred Loss:** 400530 **Incurred Loss Development Factor to Ultimate:** 1.15

**AY 2031** **Exposure Units:** 3000 **Incurred Loss:** 360470 **Incurred Loss Development Factor to Ultimate:** 1.45

**AY 2032** **Exposure Units:** 3560 **Incurred Loss:** 350900 **Incurred Loss Development Factor to Ultimate:** 1.90

**(a)** What is the Bornhuetter-Ferguson estimate of IBNR at December 31, 2032, for all accident years?

**(b)** What is the Cape Cod estimate of IBNR at December 31, 2032, for all accident years?

**(c)** Based on the given information and your information, which method from parts (a) and (b) would produce the more accurate estimate of IBNR? Why?

**Solution S6-52-1.**

**(a)** The Bornhuetter-Ferguson estimate of IBNR does not depend on losses reported to date. For each accident year it is equal to (Expected Loss)*(1 – 1/(LDF to Ultimate). In this case, the expected loss is (Number of Exposures)*(Loss Rate Per Exposure). Thus, the Bornhuetter-Ferguson estimate of IBNR is Σ((Number of Exposures)*(Loss Rate Per Exposure)*(1 – 1/(LDF to Ultimate)) = 2340*200*(1-1/1.15) + 3000*200*(1-1/1.45) + 3560*200*(1-1/1.90) = 584513.5327 = $**584,513.53**.

**(b)** The Cape Cod method first derives an empirical expected loss rate per exposure by first calculating “used-up” exposures equal to (Number of Exposures)*(1/(LDF to Ultimate)) and then obtaining the loss rate as (Sum of Reported Losses)/(Sum of Used-Up Exposures).

The Cape Cod expected loss rate is thus (400530 + 360470 + 350900)/(2340/1.15 + 3000/1.45 + 3560/1.90) = 186.0163256.

The total Cape Cod IBNR is the Bornhuetter-Ferguson IBNR, multiplied by the ratio of the Cape Cod expected loss rate to the given expected loss rate: 584513.5327*186.0163256/200 = 543645.2981 = $**543,645.30**.

**(c)** As a diagnostic, we can calculate the expected loss per exposure for each accident year via the chain ladder method as (Reported Losses)*(LDF)/(Number of Exposures):

**AY 2030:** 400530*1.15/2340 = 196.8416667** AY 2031:** 360470*1.45/3000 = 174.2271667

**AY 2032:**350900*1.90/3560 = 187.2780899

We note that, based on losses reported to date, the a priori loss rate of $200 per exposure is too high. Changes in the recent loss reporting pattern or the nature of losses have reduced this rate, and the Cape Cod method is more responsive to such changes, as compared to the Bornhuetter-Ferguson method, which relies on predetermined expected losses for its IBNR calculation. Thus, the **Cape Cod method** is preferable.

**Problem S6-52-2. Similar to Question 9 from the 2009 CAS Exam 6.** The annual loss ratio trend is +3.0%. You are also given on-level earned premiums for each accident year (AY):

**AY 2055:** On-level premium is 55353. **AY 2056:** On-level premium is 62444. **AY 2057:** On-level premium is 65725.

Cumulative incurred losses are as follows, expressed in the format

(Amount at 12 months, Amount at 24 months, Amount at 36 months), where applicable:

*Cumulative Incurred Losses***AY 2055:** (23400, 34440, 40222)** AY 2056:** (25650, 37000)

**AY 2057:**(28000)

The following development factors to ultimate were selected: **12 months to ultimate:** 1.333 **24 months to ultimate:** 1.155 **36 months to ultimate:** 1.052

**(a)** Use the expected claims technique to find the IBNR for AY 2057 as of December 31, 2057.

**(b)** Identify three situations where it might be desirable to use the expected claims technique.

**Solution S6-52-2.**

**(a)** We want to calculate the expected ultimate loss ratio for every accident year. We develop and trend the most recent known incurred losses (outermost diagonal of the triangle) and divide them by the on-level earned premium for each accident year.

**AY 2055:** Expected loss ratio is 40222*1.052*1.032/ 55353 = 0.8109847493. **AY 2056:** Expected loss ratio is 37000*1.155*1.03/62444 = 0.7049043943. **AY 2057:** Expected loss ratio is 28000*1.333/65725 = 0.5678813237.

To get our total expected loss ratio, we can take the arithmetic mean of the three accident-year expected loss ratios: (0.8109847493 + 0.7049043943 + 0.5678813237)/3 = 0.6945901558.

Expected ultimate losses for AY 2057 are the AY 2057 on-level earned premium, multiplied by the expected loss ratio: 65725*0.6945901558 = 45651.93799. To get IBNR, we subtract already reported losses from expected losses: 45651.93799 – 28000 = 17651.93799 = **17651.94**.

**(b)** The expected claims method might be useful in the following situations:

1. A new book of business for which prior experience is not available;

2. A long-tailed book of business at the early stages of development, where there is too much volatility in reported losses to use methods that rely on them;

3. A book of business subject to recent macroeconomic or regulatory changes which render past data largely irrelevant.

**Problem S6-52-3. Similar to Question 11 from the 2009 CAS Exam 6.** You are given the following information, expressed in the format

(Number at 12 months, Number at 24 months, Number at 36 months) by accident year (AY):

*Cumulative Paid Losses***AY 2044:** (5505, 6666, 7044)** AY 2045:** (5880, 6900)

**(5400)**

AY 2046:

AY 2046:

*Number of Open Claims***AY 2044:** (72, 38, 32)** AY 2045:** (70, 34)

**(66)**

AY 2046:

AY 2046:

*Average Case Reserve***AY 2044:** (100, 194, 202)** AY 2045:** (99, 180)

**(103)**

AY 2046:

AY 2046:

The annual case reserve severity trend is selected to be +2.0%.

The 36-month-to-ultimate incurred loss development factor is selected to be 1.04.

**(a)** What does the Berquist-Sherman case reserve adjustment do, and what is its purpose?

**(b)** Use the Berquist-Sherman case reserve adjustment to arrive at ultimate losses for AY 2046.

**Solution S6-52-3.**

**(a)** The Berquist-Sherman case reserve adjustment takes the most recent (outermost diagonal) known average case reserves and de-trends them by the annual case reserve severity trend to arrive at average case reserve estimates for the same ages of maturity for experience of prior accident years. This done in order to facilitate the assumption of the same case outstanding adequacy for all calendar years as exists in the current calendar year.

**(b)** We first de-trend average case outstanding to get the adjusted average case outstanding:

*Adjusted Average Case Reserve***AY 2044:** (103/1.022, 180/1.02, 202)** AY 2045:** (103/1.02, 180)

**(103)**

AY 2046:

AY 2046:

*Adjusted Average Case Reserve***AY 2044:** (99, 176.47, 202)** AY 2045:** (100.98, 180)

**(103)**

AY 2046:

AY 2046:

Now we can estimate adjusted reported claims for each time period as (Paid Claims) + (Adjusted Average Case Reserve)*(Number of Open Claims):

*Adjusted Reported Claims***AY 2044:** (5505 + 99*72, 6666 + 176.47*38, 7044 + 202*32)** AY 2045:** (5880 + 100.98*70, 6900 + 180*34)

**(5400 + 103*66)**

AY 2046:

AY 2046:

*Adjusted Reported Claims***AY 2044:** (12633, 13371.86, 13508)** AY 2045:** (12948.6, 13020)

**(12198)**

AY 2046:

AY 2046:

Using this information, we can calculate weighted-average age-to-age factors for adjusted reported claims: **Factor for 12 months to 24 months:** (13371.86 + 13020)/(12633 + 12948.6) = 1.031673547 **Factor for 24 months to 36 months:** 13508/13371.86 = 1.010181082 **Factor for 36 months to ultimate****:** 1.04 (given)

**Factor for 12 months to ultimate:** 1.031673547*1.010181082*1.04 = 1.083864183.

Our estimate of ultimate losses for AY 2046 is thus 12198*1.083864183 = 13220.97531 = **13220.98.** (Note that some minor discrepancies with this answer may arise due to rounding.)

**Problem S6-52-4. Similar to Question 12 from the 2009 CAS Exam 6.** You know that the following incremental paid losses occurred by accident year:

AY 2034: Valuation Date: December 31, 2034; Incremental paid loss: 1140

AY 2034: Valuation Date: December 31, 2035; Incremental paid loss: 240

AY 2035: Valuation Date: December 31, 2035; Incremental paid loss: 210

AY 2034: Valuation Date: December 31, 2036; Incremental paid loss: 140

AY 2035: Valuation Date: December 31, 2036; Incremental paid loss: 1240

AY 2036: Valuation Date: December 31, 2036; Incremental paid loss: 1000

You also know that the 36-month-to-ultimate paid loss development factor is 1.03.

**(a)** Develop a triangle of cumulative paid losses on the basis of this information.

**(b)** Use the chain ladder method and volume-weighted-average development factors to estimate the unpaid claim liability for AY 2036 as of December 31, 2036.

**Solution S6-52-4.**

**(a)** We develop our cumulative paid loss triangle by accident year in the format (Number at 12 months, Number at 24 months, Number at 36 months):

*Cumulative Paid Losses***AY 2034:** (1140, 1140+240, 1140+240+140)** AY 2035:** (210, 210+1240)

**(1000)**

AY 2036:

AY 2036:

*Cumulative Paid Losses***AY 2034:** (1140, 1380, 1520)** AY 2035:** (210, 1450)

**(1000)**

AY 2036:

AY 2036:

**(b)** We find the volume-weighted age-to-age factors:

**Factor for 12 months to 24 months:** (1380 + 1450)/(1140 + 210) = 2.096296296 **Factor for 24 months to 36 months:** 1520/1380 = 1.101449275 **Factor for 36 months to ultimate****:** 1.03 (given)

**Factor for 12 months to ultimate:** 2.096296296*1.101449275*1.03 = 2.378232958.

The estimated unpaid claims for AY 2036 are thus 1000*2.378232958 – 1000 = 1378.232958 = **1378.23**.

**Problem S6-52-5. Similar to Question 20 from the 2009 CAS Exam 6.** You have the following information about Primary Insurer Q, which started business on January 1, 2044.

Calendar year 2044 written premium: $6660

Calendar year 2044 policy acquisition cost: $2000

Here is the primary insurer’s balance sheet as of December 31, 2044, *before* any consideration of reinsurance:

*Assets*

Cash: $8000 **Total assets:** $8000

*Liabilities and Surplus*

Unearned premiums: $3350

Loss reserves: $2500 **Total liabilities:** $5850

Surplus:$2150 **Total liabilities and surplus:** $8000

The primary insurer enters into a 60% quota share reinsurance treaty on a risk attaching basis. The reinsurer pays the primary insurer a ceding commission of 25%. There are no claim payments as of December 31, 2044.

**(a)** Create the statutory balance sheet of Primary Insurer Q *net* of reinsurance as of December 31, 2044.

**(b)** What is the change in Primary Insurer Q’s ratio of net written premium to surplus as a result of the reinsurance treaty.

**Solution S6-52-5.**

**(a)** The reinsurer receives 60% of all premiums and losses. This means that the reinsurer gets 0.6*6600 = 3960 of written premiums. But the reinsurer pays 25% of this amount, or 0.25*3960 = 990, to Primary Insurer Q as ceding commission. The resulting amount of the primary insurer’s cash is thus 8000 – 3960 + 990 = 5030.

Unearned premiums are reduced to 0.4*3350 = 1340, and loss reserves are reduced to 0.4*2500 = 1000.

Thus, we have the following statutory balance sheet:

*Assets*

Cash: $5030 **Total assets:** $5030

*Liabilities and Surplus*

Unearned premiums: $1340

Loss reserves: $1000 **Total liabilities:** $2340

Surplus:$2690 **Total liabilities and surplus:** $5030

**(b)** Before the treaty, the written-premium-to-surplus ratio is 6660/2150 = 3.097674419.

The net written premium from the reinsurance treaty is 6660*0.4 = 2664, and so the post-treaty net-written-premium-to-surplus ratio is 2664/2690 = 0.9903345725.

The change in the ratio is 0.9903345725/3.097674419 – 1 = **-68.02973978%.**

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

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