Calculation of Loss Elimination Ratios Using Observed Data: 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. 197-199.

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

Problem S5-88-1. You have the following data from Insurance Company G during Calendar Year 2032:
There were 346 claims with losses under $200; the total ground-up losses for these claims were $48,786.

There were 800 claims with losses between $200 and $400; the total ground-up losses for these claims were $264,800.

There were 436 claims will losses between $400 and $800; the total ground-up losses for these claims were $242,852.

There were 600 claims with losses over $800; the total ground-up losses for these claims were $930,000.

Based on these observed data, what would be the total losses eliminated if a deductible of $400 were applied to all policies?

Solution S5-88-1. With a $400 deductible, all losses of a size under $400 would be eliminated completely. This corresponds to 48786 + 264800 = $313,586 eliminated. Also, for losses in excess of $400, $400 per loss would be eliminated. This applies to 436 + 600 = 1036 claims; thus, 400*1036 = $414,400 would be eliminated in this way. The total amount eliminated would be $313,586 + $414,400 = $727,986.

Problem S5-88-2. You have the following data from Insurance Company G during Calendar Year 2032:
There were 346 claims with losses under $200; the total ground-up losses for these claims were $48,786.

There were 800 claims with losses between $200 and $400; the total ground-up losses for these claims were $264,800.

There were 436 claims will losses between $400 and $800; the total ground-up losses for these claims were $242,852.

There were 600 claims with losses over $800; the total ground-up losses for these claims were $930,000.

Based on these observed data, what would be the loss elimination ratio (LER) if a deductible of $400 were applied to all policies?

Solution S5-88-2. LER = (Losses eliminated)/(Total losses). In Solution S5-88-1, we found that (Losses eliminated) = $727,986. Total ground-up losses are 48786 + 264800 + 242852 + 930000 = $1,486,438. Thus, LER(400) = $727,986/$1,486,438 = LER(400) = 0.4897520112.

Problem S5-88-3. Which of the following data can be used to determine the loss elimination ratio associated with moving from a $1000 deductible to a $2500 deductible? More than one answer may be possible.

(a) Data from policies with no deductible;
(b) Data from policies with a $100 deductible;
(c) Data from policies with a $500 deductible;
(d) Data from policies with a $1000 deductible;
(e) Data from policies with a $2000 deductible;
(f) Data from policies with a $2500 deductible;
(g) Data from policies with a $5000 deductible;
(h) Data from policies with a $10000 deductible.

Solution S5-88-3. This question is based on the discussion in Werner and Modlin, p. 198. Data from policies with deductibles lower than the deductibles in question can be used to determine loss elimination ratios. Here, only data from policies with $1000 or lower deductibles can be used. Thus, the following data can be used:

(a) Data from policies with no deductible;
(b) Data from policies with a $100 deductible;
(c) Data from policies with a $500 deductible;
(d) Data from policies with a $1000 deductible.

Problem S5-88-4. You are given the following loss data from Insurance Company T in calendar year 2046. Assume that each individual loss exceeds $1000.

The company had 142 claims on policies with no deductible, for which the reported losses were $315,132.

The company had 411 claims on policies with a $500 deductible, for which the reported losses were $600,500.

The company had 126 claims on policies with a $1000 deductible, for which the reported losses were $431,123.

The company had 80 claims on policies with a $2000 deductible, for which the reported losses were $86,320.

Based on the observed data, what would be the total amount of losses eliminated by moving from a $500 deductible to a $1000 deductible?

Solution S5-88-4. Only policies with no deductible or a deductible of $500 can be analyzed, because only for these policies the loss amounts between $500 and $1000 for each individual claim are known. Because each individual loss exceeds $1000, the losses eliminated for the 142 + 411 = 553 claims on policies with a deductible of $500 or less would be (1000 - 500)*553 = $276,500 as a result of moving from a deductible of $500 to a deductible of $1000.

Problem S5-88-5. You are given the following loss data from Insurance Company T in calendar year 2046. Assume that each individual loss exceeds $1000.

The company had 142 claims on policies with no deductible, for which the reported losses were $315,132.

The company had 411 claims on policies with a $500 deductible, for which the reported losses were $600,500.

The company had 126 claims on policies with a $1000 deductible, for which the reported losses were $431,123.

The company had 80 claims on policies with a $2000 deductible, for which the reported losses were $86,320.

Based on the observed data, what would be the loss elimination ratio associated with moving from a $500 deductible to a $1000 deductible?

Solution S5-88-5. Again, we can only analyze policies with no deductible or a deductible of $500. LER = (Losses eliminated)/(Total losses). In Solution S5-88-4, we found that (Losses eliminated) = $276,500. Total losses on the claims for policies with deductibles of $500 or under were $315,132 + $600,500 = $915,632. Thus, our loss elimination ratio is $276,500/$915,632 = LER = 0.3019772136.

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