Considerations Pertaining to Insurance Rate Dislocations, Rate Transitions, and New Rating Selections: 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 14, pp. 275-281.

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

Problem S5-115-1. Insurance Company Π wishes to implement a rating change but wishes to limit the effect on Territory 3 to a 15% increase. Under the proposed rating changes, without any cap being taken into consideration, the following would be the case:

In Territory 1, premium would increase from $31,135 to $32,241.
In Territory 2, premium would decrease from $43,120 to $41,768.
In Territory 3, premium would increase from $23,012 to $30,128.

What should be the base rate adjustment factor that the company should implement along with the cap in order for the cap to not affect the overall change in rates?

Solution S5-115-1. This problem is based on the discussion in Werner and Modlin, p. 276. We use the formula (Base Rate Adjustment) = 1 + (Premium Above Cap)/(Premium from All Non-Capped Levels). Here, Territories 1 and 2 are uncapped in their changes, and the premium we consider for these territories is the current premium. The sum of current premium for these two territories is 31315 + 43120 = $74,435.

For Territory 3 premium to increase by 15%, the resulting premium would be 23012*1.15 = $26,463.80. Thus, the premium above the cap would be 30128 - 26463.8 = $3,664.2. Our base rate adjustment is thus 1 + 3664.2/74435 = (Base Rate Adjustment) = 1.049226842.

Problem S5-115-2. Insurance Company Π wishes to implement a rating change but wishes to limit the effect on Territory 3 to a 15% increase. Under the proposed rating changes, without any cap being taken into consideration, the following would be the case:

In Territory 1, premium would increase from $31,135 to $32,241.
In Territory 2, premium would decrease from $43,120 to $41,768.
In Territory 3, premium would increase from $23,012 to $30,128.

What should be the differential adjustment factor that the company should apply to the proposed relativity for Territory 3 in order to achieve the desired limitation of the rate increase to 15%?

Solution S5-115-2. This problem is based on the discussion in Werner and Modlin, pp. 276-277. We use the formula

(Differential Adjustment) = (1 + %Cap)/((1 + Uncapped Total Change)*(Base Rate Adjustment).

From Solution S5-115-1, we know that the base rate adjustment is 1.049226842. The percent of the cap for Territory 3 is 15%. The uncapped total change for Territory 3 is 30128/23012 - 1 = 0.309229967.

Thus, (Differential Adjustment) = 1.15/(1.309229967*1.049226842) = (Differential Adjustment) = 0.8371677901.

Problem S5-115-3. An insurer implements the following rating changes to an entirely multiplicative rating plan:

The base rate increases by 40%.
The multiplicative factor for adding a new vehicle to the policy changes from 1.03 to 1.15.
The multiplicative factor for receiving a traffic ticket changes from 1.1 to 1.2.

The insurer implements a premium transition rule such that no insured will receive a premium increase of more than 15% per year, until the full increase aimed at by the rating change is achieved. This cap only applies to the rating changes above, and not to other possible changes pertaining to the policy.

The following applies to Insured E:

Year 1: There are no changes to the policy.
Year 2: Insured E adds a new vehicle to the policy.
Year 3: Insured E receives a traffic ticket.
Year 4: Insured E purchases a stuffed toy llama, which subjects him to the "stuffed toy llama surcharge" factor of 1.04.

For each of the four years, what percentage premium increase will Insured E receive?

Solution S5-115-3. This problem is based on the discussion in Werner and Modlin, p. 278.

In Year 1, only the base rate change of 40% is relevant. This change is capped at 15%, leaving 1.4/1.15 - 1 = 0.2173913043 = 21.73913043% of an increase to be applied over subsequent periods.

In Year 2, the addition of the new vehicle is relevant. The old 1.03 factor would apply irrespective of this rate change, so the only aspect of the change that is subject to the cap is the 1.15/1.03 -1 = 0.116504854 = 11.6504854% that results from this filing. The total change that is subject to capping is thus 1.116504854*1.2173913043 (to account for the base rate change left over from the prior year) = 1.359223301. With the 15% cap being applied, the carried-over change is 1.359223301/1.15 - 1 = 0.1819333051 = 18.19333051%. The actual Year 2 increase will be 1.15*1.03 = 1.1845 → +18.45%.

In Year 3, the traffic ticket is relevant. The old 1.1 factor would apply irrespective of this rate change, so the only aspect of the change that is subject to the cap is the 1.2/1.1 - 1 = 0.0909090909 = 9.09090909% that results from this filing. The total change that is subject to capping is thus 1.0909090909*1.1819333051 (to account for the change left over from the prior year) = 1.289381787. With the 15% cap being applied, the carried-over change is 1.289381787/1.15 - 1 = 0.1212015542 = 12.12015542%. The actual Year 3 increase will be 1.15*1.1 = 1.265 → +26.5%.

In Year 4, the full stuffed toy llama surcharge of 1.04 applies, because it was not affected by the rating changes in question. Moreover, the entire 12.12015542% change, carried over from prior years, falls within the 15% cap for Year 4. Thus, the total Year 4 increase will be 1.04*1.1212015542 = 1.166049616 → +16.6049616%.

Thus, the following changes apply over the four years:

Year 1: +15%
Year 2: +18.45%
Year 3: +26.5%
Year 4: +16.6049616%

Problem S5-115-4. Insurance Company X is initiating a new program, for which it will be primarily using the rates of its competitor, Insurance Company Q, which has a permissible loss ratio of 0.71, with several adjustments. Insurance Company X is willing to take a lower profit percentage (5%, as opposed to Q's 8%), but also has a commission percentage that is higher by 4%. Because Insurance Company X is confident that it has the best personnel in the industry, it estimates that its loss costs will be 10% lower on average. By what factor should Company X adjust Company Q's base rates in developing base rates of its own? Neither company has any fixed expenses or fixed expense provisions in its rates.

Solution S5-115-4. This problem is based on the discussion in Werner and Modlin, pp. 279-280.

First, we consider the expenses and profit, which will change the company's permissible loss ratios. Company X's permissible loss ratio will be lower by 8%-5% = 3%, because it will take a lower profit percentage. However, the permissible loss ratio will also be higher by 4% because of higher commissions. The net difference is 4%-3% = 1%, meaning that Company X's permissible loss ratio will be 0.71 + 0.01 = 0.72, and the adjustment factor for expenses/profit should thus be 0.72/0.71 = 1.014084507. This should be multiplied by 1 - 0.1 = 0.9, which reflects the savings that Company X expects over Company Q on losses. Thus, the appropriate base rate adjustment factor is 1.014084507*0.9 = 0.9126760563.

Problem S5-115-5. Name five elements that a ratemaking actuary may need to communicate with respect to a rate change that affects existing insurance policies.

Solution S5-115-5. This problem is based on the discussion in Werner and Modlin, pp. 280-281.

The following elements may be necessary to communicate with respect to a rate change that affects existing insurance policies:

1. The assumptions involved in proposing the change;
2. The specific selections pertaining to the change;
3. Impact on the company's competitive position, volume, and profitability;
4. Policyholder dislocation as a result of the change;
5. Rates of risk conversion and retention as a result of the change;
6. Comparisons of actual versus expected results (once the rate change has been in effect for some time).

Any five of the above suffice as an answer. Other valid answers may also be possible.

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