Trends Pertaining to Total Limits Losses, Basic Limits Losses, and Excess Losses: Practice Questions and Solutions

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.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 6, pp. 114.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S5-45-1. Losses on each claim are censored at a basic limit of $56,000. The following losses occurred on specific claims:

i) $67,000
ii) $34,000
iii) $51,000
iv) $152,000
v) $356,000

Every total limits loss is subject to a +15% severity trend.

Determine the basic limits losses and the excess losses for each of the claims above.

Solution S5-45-1. The basic limits losses are the losses that are capped at the basic limit (here, at $56,000). The excess losses on any claim are that portion of losses that exceeds the basic limit. For any loss amount X > 56000, the excess losses are X - 56000. We thus have the following answer:

i) Basic limits loss: $56,000; Excess loss: $11,000;
ii) Basic limits loss: $34,000; Excess loss: $0;
iii) Basic limits loss: $51,000; Excess loss: $0;
iv) Basic limits loss: $56,000; Excess loss: $96,000;
v) Basic limits loss: $56,000; Excess loss: $300,000.

Problem S5-45-2. Losses on each claim are censored at a basic limit of $56,000. The following losses occurred on specific claims:

i) $67,000
ii) $34,000
iii) $51,000
iv) $152,000
v) $356,000

Every total limits loss is subject to a +15% severity trend.

Determine the trend (percentage change) applicable to the basic limits losses for each of the claims above.

Solution S5-45-2. The basic limits losses will only change if the initial total limits losses are below the basic limit. Only claims ii) and iii) are below the basic limit. Thus, for claims i), iv), and v), the trend applicable to the basic limits losses is 0%. The loss amount for claim ii) will increase to 34000*1.15 = $39,100. This is still below the basic limit, so the entire +15% change applies to the basic limits loss. The loss amount for claim iii) will increase to 51000*1.15 = $58,650, which would be capped at the basic limit of $56,000 - leaving an excess loss of $2,650. Thus, the change in the basic limits loss pertaining to claim iii) is 100*(56000/51000 - 1) = +9.803921569%. Our answers are as follows:

i) 0%
ii) +15%
iii) +9.803921569%
iv) 0%
v) 0%

Problem S5-45-3. Losses on each claim are censored at a basic limit of $56,000. The following losses occurred on specific claims:

i) $67,000
ii) $34,000
iii) $51,000
iv) $152,000
v) $356,000

Every total limits loss is subject to a +15% severity trend.

Determine the trend (percentage change) applicable to the excess losses for each of the claims above.

Solution S5-45-3. The loss amount for claim ii) will increase to 34000*1.15 = $39,100. This is still below the basic limit, so there are no excess losses before or after the change. For claim iii), the excess loss changes from $0 to 51000*1.15 - 56000 = $2,650. This is not quantifiable in percentage terms; all we can say is that there is now an excess loss, where there was not one before. For claim i), the original excess loss, per Solution S5-45-1, was $11,000. The new excess loss is 67000*1.15 - 56000 = $21,050, implying a percentage change of 100*(21050/11000 - 1) = +91.3636363636%. For claim iv), the original excess loss, per Solution S5-45-1, was $96,000. The new excess loss is 152000*1.15 - 56000 = $118,800, implying a percentage change of 100*(118800/96000 - 1) = +23.75%. For claim v), the original excess loss, per Solution S5-45-1, was $300,000. The new excess loss is 356000*1.15 - 56000 = $353,400, implying a percentage change of 100*(353400/300000 - 1) = +17.8%. Our answers are as follows:

i) +91.3636363636%
ii) No excess losses exist before or after the total limits loss trend is considered.
iii) Excess losses exist after the total limits loss trend is considered.
iv) +23.75%
v) +17.8%

Problem S5-45-4. Losses on each claim are censored at a basic limit of $56,000. The following losses occurred on specific claims:

i) $67,000
ii) $34,000
iii) $51,000
iv) $152,000
v) $356,000

Every total limits loss is subject to a +15% severity trend.

Determine the trend (percentage change) applicable to the basic limits losses for the aggregation of theclaimsabove. (It is possible to assume, for instance, that these claims represent the claims in a small insurer's entire book of business.)

Solution S5-45-4. From Solution S5-45-1, we have the following information before the application of the +15% severity trend to total limits losses:

i) Basic limits loss: $56,000;
ii) Basic limits loss: $34,000;
iii) Basic limits loss: $51,000;
iv) Basic limits loss: $56,000;
v) Basic limits loss: $56,000;

The total basic limits losses here are the sum of the above: $253,000.

On the basis of Solution S5-45-2, we have the following information after the application of the +15% severity trend to total limits losses:

i) Basic limits loss: $56,000;
ii) Basic limits loss: $39,100;
iii) Basic limits loss: $56,000;
iv) Basic limits loss: $56,000;
v) Basic limits loss: $56,000;

The total basic limits losses here are the sum of the above: $263,100.

The percentage change is therefore 100*(263100/253000 - 1) = +3.99209486%.

Problem S5-45-5. Losses on each claim are censored at a basic limit of $56,000. The following losses occurred on specific claims:

i) $67,000
ii) $34,000
iii) $51,000
iv) $152,000
v) $356,000

Every total limits loss is subject to a +15% severity trend.

Determine the trend (percentage change) applicable to the excess losses for the aggregation of theclaimsabove.

Solution S5-45-5. From Solution S5-45-1, we have the following information before the application of the +15% severity trend to total limits losses:

i) Excess loss: $21,050;
ii) Excess loss: $0;
iii) Excess loss: $2,650;
iv) Excess loss: $96,000;
v) Excess loss: $300,000.

The total excess losses here are the sum of the above: $407,000.

On the basis of Solution S5-45-3, we have the following information after the application of the +15% severity trend to total limits losses:

i) Excess loss: $11,000;
ii) Excess loss: $0;
iii) Excess loss: $0;
iv) Excess loss: $118,800;
v) Excess loss: $353,400.

The total excess losses here are the sum of the above: $483,200.

The percentage change is therefore 100*(483200/407000 - 1) = +18.72235872%.

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