**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 54 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:**

Conger, R.F.; and Nolibos, A., “Estimating ULAE Liabilities: Rediscovering and Expanding Kittel’s Approach,” CAS *Forum*, Fall 2003, pp. 94-139, excluding appendices.

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

Ludwig, S.J., “An Exposure Rating Approach to Pricing Property Excess-of-Loss Reinsurance,” *PCAS* LXXVIII, 1991, pp. 110-145.

Past Casualty Actuarial Society exams: 2009 Exam 6.

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

**Problem S6-54-1. Similar to Question 14 from the 2009 CAS Exam 6.** You are given the following information about an insurer’s book of business by calendar year (CY):

**CY 2024:** Paid ULAE: 4444; Paid Loss & ALAE: 24000; Reported Loss & ALAE: 33000; Estimated Ultimate Loss & ALAE on Claims Reported in Calendar Year: 40000

**CY 2025:** Paid ULAE: 3800; Paid Loss & ALAE: 18000; Reported Loss & ALAE: 29000; Estimated Ultimate Loss & ALAE on Claims Reported in Calendar Year: 38000

**CY 2026:** Paid ULAE: 5190; Paid Loss & ALAE: 30000; Reported Loss & ALAE: 34000; Estimated Ultimate Loss & ALAE on Claims Reported in Calendar Year: 42000

You also know the following information as of December 31, 2026, for all accident years combined:

**Case Reserves:** 90990 **IBNR:** 40440 **Ultimate Loss & ALAE:** 120111

**(a)** Find the ULAE reserve as of December 31, 2026, using the classical method.

**(b)** Find the ULAE reserve as of December 31, 2026, using Kittel’s refinement to the classical method.

**(c)** Find the ULAE reserve as of December 31, 2026, using the Conger-Nolibos generalized approach with the Bornhuetter-Ferguson method. Assume that 75% of work is expended when opening a claim, and 25% of the work is expended when maintaining the claim. No work is expended when closing the claim.

**(d)** How does the Conger-Nolibos generalized approach (i) more satisfactorily address growing books of business and (ii) make possible more realistic assumptions regarding the distribution of ULAE over the lifetime with regard to opening, closing, and maintaining claims, as compared to the classical method and Kittel’s refinement?

**Solution S6-54-1.**

**(a)** In the classical method, we first calculate the ratio of paid ULAE to paid claims. We can select as our total ratio the sum of paid ULAE over all the given years, divided by the sum of paid claims (loss and ALAE) over all the given years: (4444 + 3800 + 5190)/(24000 + 18000 + 30000) = 0.186583333.

The classical method’s estimate of the ULAE reserve is (ULAE Ratio)*(0.5*Case Reserve + IBNR) = 0.186583333*(0.5*90990 + 40440) = 16034.03872 = **16034.04**.

**(b)** Using the Kittel refinement, the ULAE ratio is calculated as (Paid ULAE)/(Claims Basis), where in the case of the Kittel refinement, the Claims Basis is the average of reported claims and paid claims. Here, our claims basis for all of the years is ((24000 + 33000)/2 + (18000 + 29000)/2 + (30000 + 34000)/2) = 84000, and our ULAE ratio is (4444 + 3800 + 5190)/84000 = 0.1599285714.

The Kittel refinement preserves the classical method’s estimate of the ULAE reserve: (ULAE Ratio)*(0.5*Case Reserve + IBNR) = 0.1599285714*(0.5*90990 + 40440) = 13743.46178 = **13743.46**.

**(c)** Using the Conger-Nolibos approach, the ULAE ratio is calculated as (Paid ULAE)/(Claims Basis). The total Claims Basis is calculated as follows (assuming no work is expended closing claims):

(% of Work Expended Opening a Claim)*(Estimated Ultimate Loss & ALAE on Claims Reported in Calendar Year) + (% of Work Maintaining a Claim)*(Paid Claims) = 0.75*(40000 + 38000 + 42000) + 0.25*(24000 + 18000 + 30000) = 108000.

The ULAE ratio is thus (4444 + 3800 + 5190)/108000 = 0.124388889.

Applying the Bornhuetter-Ferguson method to the Conger-Nolibos approach, the ULAE Reserve is (ULAE Ratio)*(Ultimate Loss & ALAE – Total Claims Basis) = 0.124388889*(120111 – 108000) = 1506.473835 = **1506.47**.

**(d)** The Conger-Nolibos generalized approach (i) more satisfactorily addresses growing books of business than approaches that depend on comparing paid ULAE solely to paid claims or to the average of paid and reported claims, because the ULAE for a particular calendar year may not match to the paid claims of that calendar year, and the mismatch may be material if the volume of business written is changing. The Conger-Nolibos approach allows more flexibility in its claims basis and ties the claims basis to a more realistic assessment of how ULAE are distributed throughout the life of the claim.

The Conger-Nolibos generalized approach (ii) also relaxes the assumption that 50% of ULAE are spent opening a claim, and the other 50% are spent opening a claim. The generalized approach allows any mathematically legitimate percentage distribution of costs among opening, maintaining, and closing claims.

**Problem S6-54-2. Similar to Question 15 from the 2009 CAS Exam 6.**

**(a)** If it is known that the payment rate for claims slowed down over a particular time period, which of the following methods would it be inappropriate to use, and why?

(i) The Unadjusted Reported Development Method

(ii) The Unadjusted Paid Development Method

(iii) The Both-Case-and-Payment-Rate-Adjusted Reported Development Method

(iv) The Payment-Rate-Adjusted Paid Development Method

**(b)** You are given the following values for Accident Year 2033 as of December 31, 2033:

**Claims Reported:** 55555 **Claims Paid:** 14244 **Earned Premium:** 130555 **Estimated Ultimate Claim Count:** 57 **Open and IBNR Count:** 50 **Ultimate Claims, Using Unadjusted Reported Development Method:** 60000 **Ultimate Claims, Using Unadjusted Paid Development Method:** 52222 **Ultimate Claims, Using Both-Case-and-Payment-Rate-Adjusted Reported Development Method:** 80220 **Ultimate Claims, Using** **Payment-Rate-Adjusted Paid Development Method:** 81232

Using each of the methods that were *not* rejected in part (a), calculate (i) the ultimate claim ratio, (ii) the ultimate severity, and (iii) the unpaid severity for Accident Year 2033.

**(c)** Describe one conclusion that might be drawn from the diagnostic results in part (b).

**Solution S6-54-2.**

**(a)** The **(ii) Unadjusted Paid Development Method** should be rejected, as it does not take into account the change in paid development as a result of the slowed payment rate. As a result, the unadjusted paid development will understate ultimate losses for years after the slowdown of the payment rate.

**(b)**

**Using Unadjusted Reported Development Method** **Claim Ratio:** (Ultimate Claims/Earned Premium) = 60000/130555 = **0.459576424**. **Ultimate Severity:** (Ultimate Claims/Ultimate Claim Count) = 60000/57 = **1052.631579**. ** Unpaid Severity:** (Ultimate Claims – Paid Claims)/(Open & IBNR Count) = (60000-14244)/50 =

**915.12**.

**Using Both-Case-and-Payment-Rate-Adjusted Reported Development Method** **Claim Ratio:** (Ultimate Claims/Earned Premium) = 80220/130555 = **0.614453679**. **Ultimate Severity:** (Ultimate Claims/Ultimate Claim Count) = 80220/57 = **1407.368421**. ** Unpaid Severity:** (Ultimate Claims – Paid Claims)/(Open & IBNR Count) = (80220-14244)/50 =

**1319.52**.

**Using** **Payment-Rate-Adjusted Paid Development Method** **Claim Ratio:** (Ultimate Claims/Earned Premium) = 81232/130555 = **0.622205201**. **Ultimate Severity:** (Ultimate Claims/Ultimate Claim Count) = 81232/57 = **1425.122807**. ** Unpaid Severity:** (Ultimate Claims – Paid Claims)/(Open & IBNR Count) = (81232-14244)/50 =

**1339.76**.

**(c)** It can be seen that the Unadjusted Reported Development Method gives a dramatically lower estimate of the claim ratio and the severities than the other two methods. This might be due to other changes that unadjusted methods do not capture, such as case outstanding adequacy, which would affect a method based on reported claim development, even though payment rates would not.

**Problem S6-54-3. Similar to Question 27 from the 2009 CAS Exam 6.** You are given the following distribution of an insurer’s premium by Coverage A limit:

Limit of $100,000 – Premium of $30 million

Limit of $200,000 – Premium of $20 million

Limit of $400,000 – Premium of $40 million

Limit of $800,000 – Premium of $10 million

You are also given the following Salzmann table of cumulative loss distributions corresponding to total losses as percentages of Coverage A:

**Cumulative Loss Distribution for 10% of Coverage A:** 8% **Cumulative Loss Distribution for 20% of Coverage A:** 13% **Cumulative Loss Distribution for 30% of Coverage A:** 15% **Cumulative Loss Distribution for 40% of Coverage A:** 18% **Cumulative Loss Distribution for 50% of Coverage A:** 24% **Cumulative Loss Distribution for 60% of Coverage A:** 29% **Cumulative Loss Distribution for 70% of Coverage A:** 38% **Cumulative Loss Distribution for 80% of Coverage A:** 44% **Cumulative Loss Distribution for 90% of Coverage A:** 52% **Cumulative Loss Distribution for 100% of Coverage A:** 70% **Cumulative Loss Distribution for 120% of Coverage A:** 75% **Cumulative Loss Distribution for 140% of Coverage A:** 81% **Cumulative Loss Distribution for 160% of Coverage A:** 89% **Cumulative Loss Distribution for 180% of Coverage A:** 96% **Cumulative Loss Distribution for 200% of Coverage A:** 100%

This primary insurer enters into an excess-of-loss reinsurance treaty covering $600,000 in excess of $200,000.

**(a)** What is the percentage of the insurer’s expected losses covered by the reinsurance treaty? Use linear interpolation where needed. **(b)** Salzmann tables consider only which type of peril? **(c)** Salzmann tables consider only which lines of business? **(d)** Salzmann tables consider only which types of coverage (typically)?

**Solution S6-54-3.**

**(a)** We first consider the percentage of expected losses covered by policy limit.

For the limit of $100,000, 0% is covered by the treaty, since losses never exceed the attachment point, which is 200% of the limit.

For the limit of $200,000, only losses in excess of 100% of the limit are covered. This means that only 100%-70% = 30% of losses are covered.

For the limit of $400,000, only losses between 50% of the limit and 200% of the limit are covered. This means that 100%-24% = 76% of losses are covered.

For the limit of $800,000, only losses between 25% of the limit and 100% of the limit are covered. Thus, only 70% – 14% = 56% of losses are covered. (The 14% was obtained by linear interpolation between the cumulative loss distributions for 20% and 30% of Coverage A.)

The overall percentage of losses covered is the average of the percentages above, weighted by the written premium for each coverage limit:

0%*30/100 + 30%*20/100 + 76%*40/100 + 56*10/100 = **42%.**

**(b)** Salzmann tables only consider the **fire** peril.

**(c)** Salzmann tables only consider the **homeowners’ insurance** line of business.

**(d)** Salzmann tables only typically consider **Coverage A** and assume that losses for other coverages are proportional to losses for Coverage A.

**Problem S6-54-4. Similar to Question 29 from the 2009 CAS Exam 6.** You are given the following information regarding a primary insurer’s experience for Accident Year 2028. The primary insurer is party to a quota share reinsurance treaty written on a losses-occurring basis for a term of 12 months. There are no reported catastrophe losses.

Earned premium: $15150

Incurred loss & ALAE: $10120

Premium on-level factor: 1.023

Loss & ALAE development factor: 1.102

Annual loss trend: +4%

Annual premium trend: +7%

Ceding commission: 15% of premium

Brokerage fees: 4% of premium

Administrative expenses: 6% of premium

ALAE: 10% of loss

Catastrophe load: 8% of non-catastrophe loss and ALAE

ULAE: 5% of total loss and ALAE.

For the 2030 treaty renewal period, calculate the projected combined ratio, on the basis of the experience for Accident Year 2028.

**Solution S6-54-4.** Because the treaty renewal period will occur two years after Accident Year 2028, our trend factors will be (1 + Annual Trend)2.

We conduct the necessary adjustments to premium:

(Earned premium)*(Premium on-level factor)*(Premium trend factor) = 15150*1.023*1.072 = 17744.17541.

We conduct the necessary adjustments to losses and ALAE:

(Incurred loss & ALAE)*(Loss & ALAE development factor)*(Loss trend factor) = 10120*1.102*1.042 = 12062.26278.

Our adjusted non-catastrophe loss ratio is 12062.26278/17744.17541 = 0.6797871699.

We multiply this by the catastrophe load factor: 0.6797871699*1.08 = 0.7341701435.

We multiply this by the ULAE factor: 0.7341701435*1.05 = 0.7708786507. This is our adjusted loss, ALAE, and ULAE ratio, including catastrophes.

We ignore the ALAE factor, because ALAE is already considered in the existing loss figures.

Now we add to our adjusted ratio the various reinsurer expenses and ceding commission, thereby getting our combined ratio: 0.7708786507 + 0.15 + 0.04 + 0.06 = **1.020878651.**

**Problem S6-54-5. Similar to Question 32 from the 2009 CAS Exam 6.** Suppose you are faced with a large set of reinsurance data from various sources.

**(a)** Identify three ways in which you might partition the data. **(b)** Identify four considerations you might examine in deciding how to partition the data.

**Solution S6-54-5.** The following is a sample answer, and other valid answers are possible.

**(a)** One might partition the data

1. By reinsurance treaty type;

2. By type of underlying primary coverage;

3. By type of exposure being insured (e.g., reinsurance for policies written on manufacturing businesses might be treated separately from reinsurance for policies written on service businesses, if enough credible data exist).

**(b)** One might examine the following considerations:

1. How similar or different the reporting patterns are for the various categories of data into which partitions are contemplated;

2. How much data would exist in each category post-partition, and whether that is a credible amount of data;

3. How similar or different the underlying insured risks are;

4. How the treaty terms affect the reinsurer’s exposure to loss.

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