Increased Limits Ratemaking for Single Limits with Censored 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 11, pp. 191-194.

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

Problem S5-86-1. Insurance Company F's data for losses in calendar year 2040 are censored at the policy limits. That is, if a loss is in excess of the limits of the applicable policy, the excess amount is not known. What is known is the following:
The basic limit offered by the company is $20,000.

There were 500 claims pertaining to policies with limits of $20,000. The total reported losses for these claims were $6,172,500.

There were 813 claims pertaining to policies with limits of $50,000. Of these, 213 claims were under $20,000, and the total loss amount for these claims was $3,104,688, whereas 600 claims were between $20,000 and $50,000, and the total loss amount for these claims was $20,807,400.

There were 135 claims pertaining to policies with limits of $100,000. Of these, 12 claims were under $20,000, and the total loss amount for these claims was $105,528. 49 of these claims were between $20,000 and $50,000, and the total loss amount for these claims was $1,409,289. 74 of these claims were between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108.

What is the limited average severity (LAS) pertaining to the basic limit of $20,000?

Solution S5-86-1. With LAS pertaining to the smallest possible limit, LAS can be calculated in the relatively straightforward fashion demonstrated in Section 85. We use reported losses for the claims up to $20,000, and we censor all the other claims at $20,000.

The 500 + 213 + 12 = 725 claims up to $20,000 constitute $6,172,500 + $3,104,688 + $105,528 = $9,382,716 of reported losses.

The remaining claims will be censored at $20,000; there are 600 + 49 + 74 = 723 such claims. Thus, the total censored loss amount for these is $20,000*723 = $14,460,000.

The numerator of the LAS calculation will thus be $9,382,716 + $14,460,000 = $23,842,716.

The denominator of the LAS calculation is the total number of claims, since all claims are being considered for calculating the LAS pertaining to the smallest possible limit: 725 + 723 = 1448 claims. Thus, LAS(20000) = $23,842,716/1448 = 16465.96409 = $16,465.96.

Problem S5-86-2. Insurance Company F's data for losses in calendar year 2040 are censored at the policy limits. That is, if a loss is in excess of the limits of the applicable policy, the excess amount is not known. What is known is the following:
The basic limit offered by the company is $20,000.

There were 500 claims pertaining to policies with limits of $20,000. The total reported losses for these claims were $6,172,500.

There were 813 claims pertaining to policies with limits of $50,000. Of these, 213 claims were under $20,000, and the total loss amount for these claims was $3,104,688, whereas 600 claims were between $20,000 and $50,000, and the total loss amount for these claims was $20,807,400.

There were 135 claims pertaining to policies with limits of $100,000. Of these, 12 claims were under $20,000, and the total loss amount for these claims was $105,528. 49 of these claims were between $20,000 and $50,000, and the total loss amount for these claims was $1,409,289. 74 of these claims were between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108.

What is the limited average severity (LAS) pertaining to the layer between $20,000 and $50,000, if one considers only the claims for which positive loss amounts can be found within that layer? (Note that this is not the same as LAS pertaining to the limit of $50,000, as it excludes from consideration policies that have a $20,000 limit.)

Solution S5-86-2. We cannot consider claims for policies with a $20,000 limit, since those claims are not within the layer between $20,000 and $50,000. We consider claims for which the size of reported loss is between $20,000 and $50,000; there are 600 + 49 = 649 such claims, with total reported losses of $20,807,400 + $1,409,289 = $22,216,689. From this amount, we subtract the amount of each claim that is under $20,000, i.e., 649*20000 = $12,980,000. The difference is

$22,216,689 - $12,980,000 = $9,236,689. We also need to consider claims in excess of $50,000; Each such claim contributes the maximum possible amount to the layer between $20,000 and $50,000, i.e., $50,000 - $20,000 = $30,000. There are 74 such claims, so their contribution to the excess layer is 74*30000 = $2,220,000. The total numerator for the LAS calculation is thus $9,236,689 + $2,220,000 = $11,456,689. The denominator for the LAS calculation is 649 + 74 = 723 claims. Thus, LAS for the layer between $20,000 and $50,000 is $11,456,689/723 = 15846.04288 = $15,846.04.

Problem S5-86-3. Insurance Company F's data for losses in calendar year 2040 are censored at the policy limits. That is, if a loss is in excess of the limits of the applicable policy, the excess amount is not known. What is known is the following:
The basic limit offered by the company is $20,000.

There were 500 claims pertaining to policies with limits of $20,000. The total reported losses for these claims were $6,172,500.

There were 813 claims pertaining to policies with limits of $50,000. Of these, 213 claims were under $20,000, and the total loss amount for these claims was $3,104,688, whereas 600 claims were between $20,000 and $50,000, and the total loss amount for these claims was $20,807,400.

There were 135 claims pertaining to policies with limits of $100,000. Of these, 12 claims were under $20,000, and the total loss amount for these claims was $105,528. 49 of these claims were between $20,000 and $50,000, and the total loss amount for these claims was $1,409,289. 74 of these claims were between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108.

What is the limited average severity (LAS) pertaining to the limit of $50,000?

Solution S5-86-3. We know from Solution S5-86-1 that LAS(20000) = 16465.96409. We also know from Solution S5-86-2 that LAS(Layer between 20000 and 50000) = 15846.04288. We cannot, however, simply add these two values together to get LAS(50000). Rather, LAS(Layer between 20000 and 50000) needs to be adjusted for the fact that, of the claims pertaining to policies with limits of $50,000 and $100,000, the claims where losses were under $20,000 were not used. 723 claims were used in determining LAS(Layer between 20000 and 50000). However, there were 813 + 135 = 948 total claims pertaining to policies with limits of $50,000 and $100,000. Thus, LAS(Layer between 20000 and 50000) would need to be adjusted by a factor of (723/948); the adjusted LAS for this layer is (723/948)*15846.04288 = 12085.11498; this is the figure we add to LAS(20000). Thus, LAS(50000) = 16465.96409 + 12085.11498 = 28551.07907 = LAS(50000) = $28,551.08.

Problem S5-86-4. Insurance Company F's data for losses in calendar year 2040 are censored at the policy limits. That is, if a loss is in excess of the limits of the applicable policy, the excess amount is not known. What is known is the following:
The basic limit offered by the company is $20,000.

There were 500 claims pertaining to policies with limits of $20,000. The total reported losses for these claims were $6,172,500.

There were 813 claims pertaining to policies with limits of $50,000. Of these, 213 claims were under $20,000, and the total loss amount for these claims was $3,104,688, whereas 600 claims were between $20,000 and $50,000, and the total loss amount for these claims was $20,807,400.

There were 135 claims pertaining to policies with limits of $100,000. Of these, 12 claims were under $20,000, and the total loss amount for these claims was $105,528. 49 of these claims were between $20,000 and $50,000, and the total loss amount for these claims was $1,409,289. 74 of these claims were between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108.

What is the limited average severity (LAS) pertaining to the layer between $50,000 and $100,000, if one considers only the claims for which positive loss amounts can be found within that layer?

Solution S5-86-4. Here, only policies with a limit of $100,000 (the highest possible limit in this scenario) can be used to determine LAS pertaining to the layerbetween $50,000 and $100,000. There were 74 claims between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108. From this, we subtract $50,000 from the amount of each loss:

5331108 - 74*50000 = $1,631,108. This will be the numerator of our LAS calculation. The denominator is the number of claims: 74. Thus, LAS for this layer is $1,631,108/74 = 22042 = $22,042.00.

Problem S5-86-5. Insurance Company F's data for losses in calendar year 2040 are censored at the policy limits. That is, if a loss is in excess of the limits of the applicable policy, the excess amount is not known. What is known is the following:
The basic limit offered by the company is $20,000.

There were 500 claims pertaining to policies with limits of $20,000. The total reported losses for these claims were $6,172,500.

There were 813 claims pertaining to policies with limits of $50,000. Of these, 213 claims were under $20,000, and the total loss amount for these claims was $3,104,688, whereas 600 claims were between $20,000 and $50,000, and the total loss amount for these claims was $20,807,400.

There were 135 claims pertaining to policies with limits of $100,000. Of these, 12 claims were under $20,000, and the total loss amount for these claims was $105,528. 49 of these claims were between $20,000 and $50,000, and the total loss amount for these claims was $1,409,289. 74 of these claims were between $50,000 and $100,000, and the total loss amount for these claims was $5,331,108.

What is the limited average severity (LAS) pertaining to the limit of $100,000.

Solution S5-86-5. From Solution S5-86-3, we know that LAS(50000) = 28551.07907. From Solution S5-86-4, we know that LAS(Layer between 50000 and 100000) = 22042. The latter figure needs to be adjusted to account for the number of claims under $50,000, pertaining to the limit of $100,000. There were 135 total claims pertaining to the limit of $100,000, of which 74 were in excess of $50,000. Our adjustment factor is thus (74/135), and the adjusted LAS(Layer between 50000 and 100000) = (74/135)*22042 = 12082.28148. Therefore, LAS(100000) = 28551.07907 + 12082.28148 = 40633.36055 = LAS(100000) = $40,633.36.

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