The Loss Ratio Approach for Determining Relativities Pertaining to Specific Insurance Rating Variables: 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.

The Loss Ratio Approach for Specific Rating Variables

Let R1 by a rating variable. Let R1_C,i be the current rating factor (relativity) associated with subgroup i of R1. Let R1_I,i be the indicated rating factor (relativity) associated with subgroup i of R1. Let P^-_C,i be the current average premium for subgroup i of R1. Let B be the base class, on the basis of which the base rate is established. Let P^-_C,B be the current average premium for the base class. Then the following formulas hold:

Formula 78.1:

R1_C,i = (P^-_C,i)/(P^-_C,B)

Formula 78.2:

Indicated Differential Change =
(R1_I,i)/(R1_C,i) = (Loss & LAE Ratio for i)/(Loss & LAE Ratio for B).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 9, pp. 160-163.

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

Problem S5-78-1. The current relativity associated with subgroup z of rating variable Q is 1.23. The current average premium for the base class is $421. For the base class, the average loss and loss adjustment expenses per exposure are $208. For subgroup z, the average loss and loss adjustment expenses per exposure are $230.

(a) What is the current average premium for subgroup z?

(b) Assuming that the current average premium for subgroup z is also the indicated average premium for this subgroup, what is the indicated relativity for subgroup z?

Solution S5-78-1.

(a) We use Formula 78.1: R1_C,i = (P^-_C,i)/(P^-_C,B) → (R1_C,i)*(P^-_C,B) = (P^-_C,i).

Here, P^-_C,B = 421, and R1_C,i = 1.23. Thus, (P^-_C,i) = 421*1.23 = $517.83.

(b) We useFormula 78.2: (R1_I,i)/(R1_C,i) = (Loss & LAE Ratio for i)/(Loss & LAE Ratio for B), rearranging it thus: (R1_I,i) = (R1_C,i)*(Loss & LAE Ratio for i)/(Loss & LAE Ratio for B).

Here, R1_C,i = 1.23. We find (Loss & LAE Ratio for subgroup z) = 230/517.83 = 0.4441612112.

We find (Loss & LAE Ratio for B) = 208/421 = 0.4940617577.

Thus, (R1_I,i) = 1.23*0.4441612112/0.4940617577 = 1.105769231.

Problem S5-78-2.

An insurance company uses rating variable N, with three subgroups: d, e, and f. It analyzes data from calendar year 2090 to determine the indicated relativities for each of these subgroups.

For subgroup d, the earned premium at the current rate level is $43100. The incurred losses & loss adjustment expenses (LAE) are $23000. The current relativity for this subgroup is 0.90.

For subgroup e, the earned premium at the current rate level is $60000. The incurred losses & loss adjustment expenses (LAE) are $50000. The current relativity for this subgroup is 1.25.

For subgroup f, the earned premium at the current rate level is $20000. The incurred losses & loss adjustment expenses (LAE) are $5000. The current relativity for this subgroup is 0.40.

(a) What is the Loss & LAE ratio for subgroup d?

(b) What is the Loss & LAE ratio for subgroup e?

(c) What is the Loss & LAE ratio for subgroup f?

(d) What is the overall Loss & LAE ratio for this book of business?

Solution S5-78-2.

(a) The Loss & LAE ratio for subgroup d is (Losses and LAE)/(Earned Premium) = 23000/43100 = 0.5336426914.

(b) The Loss & LAE ratio for subgroup e is (Losses and LAE)/(Earned Premium) = 50000/60000 = 0.8333333333.

(c) The Loss & LAE ratio for subgroup f is (Losses and LAE)/(Earned Premium) = 5000/20000 = 0.25.

(d) The overall Loss & LAE ratio is (Sum of Losses and LAE)/(Sum of Earned Premium) =

(23000 + 50000 + 5000)/(43100 + 60000 + 20000) = 0.6336311942.

Problem S5-78-3. An insurance company uses rating variable N, with three subgroups: d, e, and f. It analyzes data from calendar year 2090 to determine the indicated relativities for each of these subgroups.

For subgroup d, the earned premium at the current rate level is $43100. The incurred losses & loss adjustment expenses (LAE) are $23000. The current relativity for this subgroup is 0.90.

For subgroup e, the earned premium at the current rate level is $60000. The incurred losses & loss adjustment expenses (LAE) are $50000. The current relativity for this subgroup is 1.25.

For subgroup f, the earned premium at the current rate level is $20000. The incurred losses & loss adjustment expenses (LAE) are $5000. The current relativity for this subgroup is 0.40.

(a) What is the indicated relativity change factor for subgroup d?

(b) What is the indicated relativity change factor for subgroup e?

(c) What is the indicated relativity change factor for subgroup f?

Solution S5-78-3. In Solution S5-78-2, we calculated loss and LAE ratios for each subgroup and overall. For each subgroup, the indicated relativity change factor is equal to

(Loss and LAE ratio for the subgroup)/(Overall loss and LAE ratio).

(a) (Loss and LAE ratio for subgroup d)/(Overall loss and LAE ratio) = 0.5336426914/0.6336311942 = 0.8421976322.

(b) (Loss and LAE ratio for subgroup e)/(Overall loss and LAE ratio) = 0.833333333/0.6336311942 = 1.31517094.

(c) (Loss and LAE ratio for subgroup f)/(Overall loss and LAE ratio) = 0.25/0.6336311942 = 0.3945512821.

Problem S5-78-4. An insurance company uses rating variable N, with three subgroups: d, e, and f. It analyzes data from calendar year 2090 to determine the indicated relativities for each of these subgroups.

For subgroup d, the earned premium at the current rate level is $43100. The incurred losses & loss adjustment expenses (LAE) are $23000. The current relativity for this subgroup is 0.90.

For subgroup e, the earned premium at the current rate level is $60000. The incurred losses & loss adjustment expenses (LAE) are $50000. The current relativity for this subgroup is 1.25.

For subgroup f, the earned premium at the current rate level is $20000. The incurred losses & loss adjustment expenses (LAE) are $5000. The current relativity for this subgroup is 0.40.

(a) What is the indicated relativity for subgroup d?

(b) What is the indicated relativity for subgroup e?

(c) What is the indicated relativity for subgroup f?

Solution S5-78-4. In Solution S5-78-3, we calculated the indicated relativity change factor for each subgroup. For each subgroup, the indicated relativity change is equal to

(Current relativity)*(Indicated relativity change factor).

(a) For subgroup d,
(Indicated relativity) = (Current relativity for d)*(Indicated relativity change factor for d) = 0.90*0.8421976322 = 0.757977869.

(b) For subgroup e,
(Indicated relativity) = (Current relativity for e)*(Indicated relativity change factor for e) = 1.25*1.31517094= 1.643963675.

(c) For subgroup f,
(Indicated relativity) = (Current relativity for f)*(Indicated relativity change factor for f) = 0.40*0.3945512821= 0.1578205128.

Problem S5-78-5.

(a) Why might the loss ratio approach produce indicated relativities closer to the true relativities than those produced by the pure premium approach?

(b) How might the loss ratio approach still produce distortions from the true relativities?

Solution S5-78-5. This problem is based on the discussion in Werner and Modlin, pp. 162-163.

(a) The loss ratio approach might produce indicated relativities closer to the true relativities than those produced by the pure premium approach because, while the pure premium approach assumes that, within each subgroup of the variable being analyzed (e.g., Variable A), there is a uniform distribution of exposures among the other variables (e.g., Variables B, C, etc.)., the loss ratio approach relies on the premium within each subgroup of the variable. This enables some consideration, within each subgroup of Variable A, of the fact that exposures by Variables B, C, etc., are not necessarily uniformly distributed within said subgroup, since differences in other variables may result in differences in premium for each subgroup of A.

(b) The loss ratio approach relies on adjusting the current relativities by a multiplicative factor to arrive at the indicated relativities. If the current relativities vary substantially from the true relativities, there will be greater inaccuracy in the indicated relativities. When the current relativities are the true relativities, however, the loss ratio approach will produce correct results.

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