**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 5, authored by Mr. Stolyarov. This is Section 96 of the Study Guide. See an index of all sections by following the link in this paragraph.**

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of *Basic Ratemaking*, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. 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.

The following formulas pertain to methods in excess insurance ratemaking for finding the complement of credibility C.

**Method of Increased Limit Factors**

**Formula 96.1:**

C = L-A*(ILAA+L/ILFA – 1)

**Meaning of Variables:**

A = the attachment point of the excess insurance

L = the limit of liability of the excess insurance

ILFA = the increased limit factor associated with attachment point A

ILFA+L = the increased limit factor associated with the sum of attachment point A and the excess insurer’s limit of liability L

L-A = The loss cost capped at the attachment point A.

**Method of Lower Limits Analysis**

**Formula 96.2:**

C = L-d*(ILAA+L – ILFA)/ILFd

**Meaning of Variables:**

A = the attachment point of the excess insurance

L = the limit of liability of the excess insurance

d = A lower limit than the attachment point

L-d = The loss cost capped at the limit d

ILFd = the increased limit factor associated with the lower limit d

ILFA = the increased limit factor associated with attachment point A

ILFA+L = the increased limit factor associated with the sum of attachment point A and the excess insurer’s limit of liability L

**Method of Limits Analysis**

**Formula 96.3:**

C = (LR)*(d≥AΣ(Pd*(ILFmin(d, A+L) – ILFA)/ILFd))

**Meaning of Variables:**

A = the attachment point of the excess insurance

L = the limit of liability of the excess insurance

d = A higher limit than the attachment point

LR = Total loss ratio

Pd = Total premium for policies with limit d

ILFd = the increased limit factor associated with the lower limit d

ILFA+L = the increased limit factor associated with the sum of attachment point A and the excess insurer’s limit of liability L

ILFmin(d, A+L) = the smaller of ILFd and ILFA+L.

**Source:**

Werner, Geoff and Claudine Modlin. *Basic Ratemaking**.* Casualty Actuarial Society. 2009. Chapter 12, pp. 228-232.

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

**Problem S5-96-1.** An excess insurance policy has an attachment point of $200,000, and the excess insurer’s limit of liability under the policy is $100,000. It is known that the loss cost for losses capped at $200,000 is $136. The increased limit factor associated with a ground-up limit of $200,000 is 1.34. The increased limit factor associated with a ground-up limit of $300,000 is 1.59. An actuary seeks to develop a complement of credibility for data pertaining to the excess insurance policy. Find this complement of credibility using the method of increased limit factors.

**Solution S5-96-1.** We apply Formula 96.1: C = L-A*(ILAA+L/ILFA – 1).

Here, A = 200000, L = 100000, so A + L = 300000. L-A = L-200000 =136, and ILFA = ILF200000 = 1.34, while ILFA+L = ILF300000 = 1.59. Our answer is thus 136*(1.59/1.34 – 1) = 25.37313433 = **C = $25.37**.

**Problem S5-96-2.** An excess insurance policy has an attachment point of $200,000, and the excess insurer’s limit of liability under the policy is $100,000. It is known that the loss cost for losses capped at $50,000 is $86. The increased limit factor associated with a ground-up limit of $50,000 is 1.02. The increased limit factor associated with a ground-up limit of $200,000 is 1.34. The increased limit factor associated with a ground-up limit of $300,000 is 1.59. An actuary seeks to develop a complement of credibility for data pertaining to the excess insurance policy. Find this complement of credibility using the method of lower limits analysis.

**Solution S5-96-2.** We apply Formula 96.2: C = L-d*(ILAA+L – ILFA)/ILFd.

Here, d = 50000, A = 200000, L = 100000, so A + L = 300000. L-A = L-200000 =136, and ILFA = ILF200000 = 1.34, while ILFA+L = ILF300000 = 1.59. L-d = L-50000 = 86, and ILFd = ILF50000 = 1.02.

Our answer is thus 86*(1.59 – 1.34)/1.02 = 21.07843137 = **C = $21.08**.

**Problem S5-96-3.** An excess insurance policy has an attachment point of $200,000, and the excess insurer’s limit of liability under the policy is $100,000. Ground-up insurance data is available for policies with limits of $50,000, $150,000, $200,000, $250,000, $300,000, and $700,000.

The loss ratio pertaining to this data is uniformly 80%.

The total premium associated with policies with $50,000 limits is $1,230,132.

The total premium associated with policies with $150,000 limits is $2,888,120.

The total premium associated with policies with $200,000 limits is $1,348,102.

The total premium associated with policies with $250,000 limits is $1,366,776.

The total premium associated with policies with $300,000 limits is $902,319.

The total premium associated with policies with $700,000 limits is $120,211.

You also know all the increased limit factors (ILFs) for each limit:

The ILF for $50,000 is 1.00.

The ILF for $150,000 is 1.85.

The ILF for $200,000 is 2.03.

The ILF for $250,000 is 2.43.

The ILF for $300,000 is 2.76.

The ILF for $700,000 is 3.66.

An actuary is using data about loss amounts from the layer between $200,000 and $300,000 to develop a complement of credibility to the excess insurance data. The method of limits analysis is being used.

(a) For policies with limits of $50,000, what is the percentage of losses in the desired layer?

(b) For policies with limits of $150,000, what is the percentage of losses in the desired layer?

(c) For policies with limits of $200,000, what is the percentage of losses in the desired layer?

(d) For policies with limits of $250,000, what is the percentage of losses in the desired layer?

(e) For policies with limits of $300,000, what is the percentage of losses in the desired layer?

(f) For policies with limits of $700,000, what is the percentage of losses in the desired layer?

**Solution S5-96-3.**

**(a)** Policies with limits of $50,000 cannot have losses in the layer between $200,000 and $300,000, since those losses exceed the policy limits. Thus, the percentage of losses for those policies that are within the layer is **0%.**

**(b)** Policies with limits of $150,000 cannot have losses in the layer between $200,000 and $300,000, since those losses exceed the policy limits. Thus, the percentage of losses for those policies that are within the layer is **0%.**

**(c)** Here, d = $200,000 = A = $200,000, so our percentage, based on Formula 96.3, is

(ILFmin(d, A+L) – ILFA)/ILFd = (ILFd – ILFA)/ILFd, but ILFd = ILFA , so our percentage is **0%.**

**(d)** Here, d = $250,000 > A = $200,000, so our percentage, based on Formula 96.3, is

(ILFmin(d, A+L) – ILFA)/ILFd = (ILFd – ILFA)/ILFd = (2.43 – 2.03)/2.43 = 0.1646090535 = **16.46090535%**

**(e)** Here, d = $300,000 = A + L = $300,000 , so our percentage, based on Formula 96.3, is

(ILFmin(d, A+L) – ILFA)/ILFd = (ILFA+L – ILFA)/ILFd = (2.76 – 2.03)/2.76 = 0.2644927536 = **26.44927536%.**

**(f)** Here, d = $700,000 > A + L = $300,000 , so our percentage, based on Formula 96.3, is

(ILFmin(d, A+L) – ILFA)/ILFd = (ILFA+L – ILFA)/ILFd = (2.76 – 2.03)/3.66 = 0.1994535519 = **19.94535519%**.

**Problem S5-96-4.** An excess insurance policy has an attachment point of $200,000, and the excess insurer’s limit of liability under the policy is $100,000. Ground-up insurance data is available for policies with limits of $50,000, $150,000, $200,000, $250,000, $300,000, and $700,000.

The loss ratio pertaining to this data is uniformly 80%.

The total premium associated with policies with $50,000 limits is $1,230,132.

The total premium associated with policies with $150,000 limits is $2,888,120.

The total premium associated with policies with $200,000 limits is $1,348,102.

The total premium associated with policies with $250,000 limits is $1,366,776.

The total premium associated with policies with $300,000 limits is $902,319.

The total premium associated with policies with $700,000 limits is $120,211.

You also know all the increased limit factors (ILFs) for each limit:

The ILF for $50,000 is 1.00.

The ILF for $150,000 is 1.85.

The ILF for $200,000 is 2.03.

The ILF for $250,000 is 2.43.

The ILF for $300,000 is 2.76.

The ILF for $700,000 is 3.66.

An actuary is using data about loss amounts from the layer between $200,000 and $300,000 to develop a complement of credibility to the excess insurance data. The method of limits analysis is being used.

(a) What is the magnitude of the expected loss, associated with policies with $250,000 limits, that will be in the desired layer?

(b) What is the magnitude of the expected loss, associated with policies with $300,000 limits, that will be in the desired layer?

(c) What is the magnitude of the expected loss, associated with policies with $700,000 limits, that will be in the desired layer?

**Solution S5-96-4.** In Solution S5-96-3, we calculated each of the (ILFmin(d, A+L) – ILFA)/ILFd) components in Formula 96.3, as applicable to this situation. This problem asks us to calculate the LR*Pd*(ILFmin(d, A+L) – ILFA)/ILFd) component for each policy limit amount. To do this, we multiply the corresponding percentage obtained in Solution S5-96-3 by the loss ratio of 0.8 and the total premium for the policies with the limits in question.

**(a)** The portion of losses in the desired layer from policies with $250,000 limits is 0.1646090535. The total premium associated with all policies with such limits is $1,366,776. Thus, the expected loss that will be in the desired layer is 0.8*1366776*0.1646090535 = 179986.963 = **$179,986.96**.

**(b)** The portion of losses in the desired layer from policies with $300,000 limits is 0.2644927536. The total premium associated with all policies with such limits is $902,319. Thus, the expected loss that will be in the desired layer is 0.8*902319*0.2644927536 = 190925.4695 = **$190,925.47**.

**(c)** The portion of losses in the desired layer from policies with $700,000 limits is 0.1994535519. The total premium associated with all policies with such limits is $120,211. Thus, the expected loss that will be in the desired layer is 0.8*120211*0.1994535519 = 19181.20874 = **$19,181.21**.

**Problem S5-96-5.** An excess insurance policy has an attachment point of $200,000, and the excess insurer’s limit of liability under the policy is $100,000. Ground-up insurance data is available for policies with limits of $50,000, $150,000, $200,000, $250,000, $300,000, and $700,000.

The loss ratio pertaining to this data is uniformly 80%.

The total premium associated with policies with $50,000 limits is $1,230,132.

The total premium associated with policies with $150,000 limits is $2,888,120.

The total premium associated with policies with $200,000 limits is $1,348,102.

The total premium associated with policies with $250,000 limits is $1,366,776.

The total premium associated with policies with $300,000 limits is $902,319.

The total premium associated with policies with $700,000 limits is $120,211.

You also know all the increased limit factors (ILFs) for each limit:

The ILF for $50,000 is 1.00.

The ILF for $150,000 is 1.85.

The ILF for $200,000 is 2.03.

The ILF for $250,000 is 2.43.

The ILF for $300,000 is 2.76.

The ILF for $700,000 is 3.66.

An actuary is using data about loss amounts from the layer between $200,000 and $300,000 to develop a complement of credibility to the excess insurance data. The method of limits analysis is being used. What is the complement of credibility using this method?

**Solution S5-96-5.** We apply Formula 96.3:

C = (LR)*(d≥AΣ(Pd*(ILFmin(d, A+L) – ILFA)/ILFd)) = (d≥AΣ(Pd*LR*(ILFmin(d, A+L) – ILFA)/ILFd)).

In Solution S5-96-4, we found each of the individual components Pd*LR*(ILFmin(d, A+L) – ILFA)/ILFd of this summation. These components are 179986.963, 190925.4695, and 19181.20874. Their sum is 390093.6142 = **C = $390,093.61.** Note that this is the complement of credibility *for total losses.*

**See other sections of The Actuary’s Free Study Guide for Exam 5.**