**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 31 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: 2007 Exam 6.

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

**Problem S6-31-1. Similar to Question 31 from the 2007 CAS Exam 6.** An excess-of-loss reinsurance treaty has a retention of $600,000 and covers losses $300,000 in excess of that retention. The losses of the experience periods subject to the treaty have been trended to current levels, and the trended ground-up values are as follows:

Year 2045 losses: Loss A: $444,603; Loss B: $747,000; Loss C: $1,000,235.

Year 2046 losses: Loss D: $556,033; Loss E: $800,430.

Year 2047 losses: Loss F: $994,100; Loss G: $650,000.

The loss development factors for excess losses subject to the treaty are as follows:

For year 2045 losses: 1.02

For year 2046 losses: 1.15

For year 2047 losses: 1.50

The on-level trended premiums subject to the treaty for each year are as follows:

Year 2045: $1,670,000

Year 2046: $2,505,200

Year 2047: $1,802,400

Determine the experience rating loss cost for this treaty.

**Solution S6-31-1.** The experience rating loss cost for this treaty is essentially

(Trended On-Level Losses Covered by the Treaty)/(Trended On-Level Premium Subject to the Treaty).

The premiums are already trended and brought to current rate levels. Losses are already trended, but we need to find the excess losses subject to the treaty and develop those losses.

Year 2045 losses: Loss A is below the primary insurer’s retention, and the treaty covers $147,000 of Loss B and $300,000 of Loss C (which exceeds the treaty limit) – for a total of $447,000. We develop this amount: 447000*1.02 = $455,490.

Year 2046 losses: Loss D is below the primary insurer’s retention, and the treaty covers $200,430 of Loss E. We develop this amount: 200430*1.15 = $230,494.50.

Year 2047 losses: The treaty covers $300,000 of Loss F (which exceeds the treaty limit) and $50,000 of Loss G – for a total of $350,000. We develop this amount: 350000*1.50 = $525,000.

The experience rating loss cost for this treaty is thus

(455490 + 230494.5 + 525000)/(1670000 + 2505200 + 1802400) = **0.2025870751.**

**Problem S6-31-2. Similar to Question 32 from the 2007 CAS Exam 6.**

A $500,000 excess of $500,000 reinsurance treaty covers losses from the perils of falling anvils, carnivorous rabbits, and meteorites. Falling anvils constitute 50% of all losses, carnivorous rabbits constitute 30%, and meteorites constitute 20%. You know the following cumulative loss distribution for falling anvils:

**Loss is 25% of coverage limit:** Falling anvil cumulative distribution is at 70%. **Loss is 50% of coverage limit:** Falling anvil cumulative distribution is at 80%. **Loss is 75% of coverage limit:** Falling anvil cumulative distribution is at 87%. **Loss is 100% of coverage limit:** Falling anvil cumulative distribution is at 100%.

The policies covered by the treaty either have ground-up limits of $666,666.67, or $1,000,000.

Policies with $666,666.67 limits have an associated direct premium of $1,200,000.

Policies with $1,000,000 limits have an associated direct premium of $800,000.

The ceding insurer’s expected loss ratio, excluding allocated loss adjustment expenses (ALAE) is 64%. The ratio of ALAE to loss is 15%. The ceding insurer’s rates are inadequate by 10%, and the reinsurer’s profits and expenses are 8% of the reinsurance premium.

The indicated exposure premium for the meteorite cause of loss is $230,000.

The indicated exposure rate for the entire treaty is 16%. What is the exposure rate for losses due to carnivorous rabbits?

**Solution S6-31-2.** We can calculate the exposure rates for falling anvils and meteorites and deduce the exposure rate for carnivorous rabbits.

We start with meteorites. The exposure rate is

(Amount needed to pay for losses and expenses)/(Total direct premium).

The total direct premium is $1,200,000 + $800,000 = $2,000,000.

For meteorites, the indicated exposure premium must be modified by multiplication by the following:

(Loss Ratio)*(1 + Ratio of ALAE to Loss)*(1/(1- Rate Inadequacy))*(1/(1-Reinsurer Profit and Expense)) = 0.64*1.15*(1/(1-0.1))*(1/(1-0.08)) = 0.8888888889 = 8/9. We can call this the modification factor for purposes of easier reference later on.

For the meteorite peril, the amount needed to pay for losses and expenses is 230000*(8/9) = 204444.44444, and the exposure rate is thus 204444.44444/2000000 = 0.10222222222.

Now we find the exposure rate for falling anvils. We consider how much of the direct premium would be part of the exposure premium by looking at the cumulative loss distribution.

For the $666,666.67 limit, 75% (500000/666666.67) of the limit – and this 87% of the loss amounts – would be below the primary insurer’s retention. So only 13% of losses would be above the retention, which implies that 13% of the direct premium should be considered for this limit.

For the $1,000,000 limit, 50% of the limit – and this 80% of the loss amounts – would be below the primary insurer’s retention. So 20% of the direct premium should be considered for this limit.

The exposure premium for falling anvils is thus 0.13*1200000 + 0.2*800000 = $316,000.

We multiply this by our modification factor: 316000*(8/9) = 280888.8888888, and the exposure rate for falling anvils is thus 280888.8888888/2000000 = 0.1404444444.

We now find the exposure rate for carnivorous rabbits (ERCR) by setting up the following equation in consideration of the total exposure rate of 0.16:

0.16 = 0.5*0.1404444444 + 0.2*0.10222222222 + 0.3*ERCR →

ERCR = (0.16 – 0.5*0.1404444444 – 0.2*0.10222222222)/0.3 = **0.231111111111 = 23.111111111111%**.

**Problem S6-31-3. Similar to Question 33 from the 2007 CAS Exam 6.** You are given the following triangles for accident years (AY) 2034 through 2036, where data is expressed in the format (Value at 12 months, Value at 24 months, Value at 36 months), where applicable.

**Average Case Reserve per Open Claim**

AY 2034: (230, 320, 400)

AY 2035: (260, 370)

AY 2036: (320)

**Number of Open Claims**

AY 2034: (110, 80, 20)

AY 2035: (140, 70)

AY 2036: (150)

**Cumulative Paid Losses**

AY 2034: (13000, 18900, 28000)

AY 2035: (14000, 17000)

AY 2036: (18210)

The annual severity trend is +5%. Develop the Berquist-Sherman triangle of adjusted incurred losses for this scenario.

**Solution S6-31-3.** The Berquist-Sherman triangle of adjusted incurred losses is developed by adjusting the case reserve estimates and de-trending the values at the latest known (outermost) diagonal by the severity trend so as to arrive at the rest of the case reserve triangle:

**Adjusted Average Case Reserve per Open Claim**

AY 2034: (320/1.052, 370/1.05, 400)

AY 2035: (320/1.05, 370)

AY 2036: (320)

**Adjusted Average Case Reserve per Open Claim**

AY 2034: (290.2494331, 352.3809524, 400)

AY 2035: (304.7619048, 370)

AY 2036: (320)

Then the adjusted incurred loss for each time period is equal to

Paid Losses + (Average Case Reserve per Open Claim)*(Number of Open Claims).

**Adjusted Incurred Losses**

AY 2034: (13000 + 290.2494331*110, 18900 + 352.3809524*80, 28000 + 400*20)

AY 2035: (14000 + 304.7619048*140, 17000 + 370*70)

AY 2036: (18210 + 320*150)

Our answer is **Adjusted Incurred Losses**

AY 2034: (44927.44, 47090.48, 36000)

AY 2035: (56666.67, 42900)

AY 2036: (66210)

**Problem S6-31-4. Similar to Question 38 from the 2007 CAS Exam 6.**

**(a)** How would the Berquist-Sherman approach be superior to the chain ladder approach in the event of case reserve strengthening by the insurer?

**(b)** How would the Berquist-Sherman approach be superior to the chain ladder approach in the event of a changing claim settlement rate?

**(c)** If insureds are purchasing lower policy limits than before, why would it be preferable to switch from accident-year data aggregation to policy-year data aggregation?

**Solution S6-31-4.**

**(a)** In the event of case reserve strengthening by the insurer, the chain ladder method, with development factors based in part on prior experience under lower case reserves, would overstate ultimate loss results. The Berquist-Sherman approach can mitigate this by adjusting previous, lower case reserves to the level of reserve adequacy that currently exists. This is done by de-trending the most recent case reserves instead of using historical values prior to the reserve strengthening.

**(b)** A changing claim settlement rate could result in the chain ladder method either overstating (if the settlement rate increases) or understating (if the settlement rate decreases) ultimate losses. The Berquist-Sherman approach applies the current claim settlement rate to historical closed claims, thereby mitigating any overstatement or understatement.

**(c)** If insureds are purchasing lower policy limits than before, analysis using the chain ladder method and accident-year aggregation will understate the ultimate losses – in essentially the inverse fashion of what would happen under strengthening case reserves. Accident-year loss data combine losses from policies written in previous years with higher limits and policies written in later years with lower limits, whereas policy-year data are segregated by the year in which policies were written, meaning that there will not be a mix of losses from policies from years with higher limits and years with lower limits. This allows for trending of each policy year’s data by any policy limit change that has been observed.

**Problem S6-31-5. Similar to Question 41 from the 2007 CAS Exam 6.** A large-deductible policy has the following ground-up loss amounts:

1 loss of $600,000

1 loss of $450,000

1 loss of $350,000

2 losses of $100,000

2 losses of $40,000

The per-loss deductible is $400,000, and there is no aggregate deductible. The ultimate loss development factor (ULDF) for ground-up losses is 1.5.

**(a)** Calculate the ULDF (i) solely for losses retained by the insured and (ii) solely for losses transferred to the insurer.

**(b)** If there were an aggregate deductible of $1,550,000, calculate the ULDF (i) solely for losses retained by the insured and (ii) solely for losses transferred to the insurer.

**Solution S6-31-5.**

**(a)** We first find the total ultimate loss amounts by multiplying each of the ground-up initial loss amounts of 1.5. We get the following:

1 loss of $900,000

1 loss of $675,000

1 loss of $525,000

2 losses of $150,000

2 losses of $60,000

(i) The insured’s obligation for the initial losses is

$400,000

+$400,000

+$350,000

+$100,000*2

+$ 40,000*2 = $1,430,000.

The insured’s obligation for the ultimate losses is

$400,000

+$400,000

+$400,000

+$150,000*2

+$ 60,000*2 = $1,620,000.

The insured’s ULDF is thus $1,620,000/$1,430,000 = **1.132867133**.

(ii) The total initial losses are $1,680,000. Since the insured’s obligation is $1,430,000, the insurer’s obligation is the rest: $250,000.

The total ultimate losses are $2,520,000. Since the insured’s obligation is $1,620,000, the insurer’s obligation is the rest: $900,000.

The insurer’s ULDF is thus $900,000/$250,000 = **3.6.**

**(b)** (i) If an aggregate deductible of $1,550,000 existed, the insured’s ultimate obligation would be reduced to $1,550,000, resulting in the insured’s ULDF being $1,550,000/$1,430,000 = **1.083916084**.

(ii) Since the insured’s ultimate obligation is $1,550,000, the insurer’s ultimate obligation is $2,520,000 – $1,550,000 = $970,000. The insurer’s ULDF is thus $970,000/$250,000 = **3.88.**

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