Using Trended Present Rates in Finding the Complement of Credibility, Statistical Methods in Multivariate Classification Analysis, and the Challenges of Excess Insurance Ratemaking: 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 following formulas apply when using the method of trended present rates in determining a complement of credibility C:

Formula 95.1: C for the rate = (Present Rate)*(Trend)*(Previously Indicated Loss Cost)/(Loss Cost Implemented with Last Review).

Formula 95.2: C for the indicated rate change = ((Loss Trend)*(1 + Prior % Indication))/((Premium Trend)*(1 + Prior % Rate Change)).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 12, pp. 226-227, 232-233.

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

Problem S5-95-1. An actuary is using the method of trended present rates to find a complement of credibility to the observed loss experience. The present rate for the insurance company in question is $146. The loss cost that was implemented with the last review one year ago was $120, based on an indication of $130. The annual trend by which the rate should be adjusted is 7%. What is the complement of credibility for the rate according to the method of trended present rates?

Solution S5-95-1. We use Formula 95.1: C = (Present Rate)*(Trend)*(Previously Indicated Loss Cost)/(Loss Cost Implemented with Last Review) = 146*1.07¹*(130/120) = 169.23833333 = C = $169.24.

Problem S5-95-2. An actuary is using the method of trended present rates to find a complement of credibility to the observed loss experience. The last rate change by the company occurred 4.5 years ago, where the rates increased by +3%, based on an indication of +6%. The annual premium trend for the company is +5%, while the annual loss trend is +2%. Assume that a new review of the company's rates is being undertaken today, and the actuary wants to find a complement of credibility for the indicated rate change. What is the complement of credibility for the indicated rate change according to the method of trended present rates?

Solution S5-95-2. We use Formula 95.2: C = ((Loss Trend)*(1 + Prior % Indication))/((Premium Trend)*(1 + Prior % Rate Change)). Since 4.5 have elapsed, each annual trend should be taken to the power of 4.5: C = (1.02^4.5*1.06)/(1.05^4.5*1.03) = C = 0.9032699611 - i.e., a 9.67300389% decrease.

Problem S5-95-3.

(a) Name two advantages of the method of trended present rates for finding a complement of credibility.

(b) Name one possible disadvantage of the method of trended present rates for finding a complement of credibility.

Solution S5-95-3. This problem is based on the discussion by Werner and Modlin, p. 227.

(a) The following are advantages of this method:

1. The resulting complement is unbiased due to the unbiased nature of pure trended loss costs.
2. The data for this approach are readily available.
3. The calculations for this approach are straightforward.
4. This approach is relatively easy to communicate.

Any two of the above suffice as answers. Other valid answers may be possible.

(b) A disadvantage of this method might arise if there is overlap between the time periods for the subject experience and for the historical data used in developing the complement of credibility. This would lead rise to the problem of non-independence of the complement of credibility from the subject experience. Another possible disadvantage is that, if the historical data under consideration are sparse, the process variance of the historical loss costs might be too high, and the complement of credibility might not be accurate. Other valid answers may be possible.

Problem S5-95-4. Statistical methods are used with multivariate classification analysis to determine if the multivariate model developed is relevant to the data being analyzed.

(a) Name three kinds of statistical diagnostics that are used for this purpose.

(b) Are the results of multivariate classification analysis typically credibility-weighted with other actuarial estimates?

Solution S5-95-4. This question is based on the discussion by Werner and Modlin, p. 232.

(a) The following statistical diagnostics are often used with multivariate classification analysis:

1. Standard errors of the parameter estimates;
2. Standardized deviance tests (such as the Chi-Square test and the F-test);
3. Consistency of model results over time;
4. Diagnostics that identify the overall appropriateness of the model assumptions (such as deviance residual plots and leverage plots).

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

(b) The results of multivariate classification analysis are not typically credibility-weighted with other actuarial estimates, which tend to be univariate estimates and are thus are not likely to bring added accuracy to the results of a multivariate analysis.

Problem S5-95-5. Briefly describe three problems associated with developing a complement of credibility pertaining to ratemaking for excess insurance (i.e., insurance that only applies to excess layers, below which another insurance policy provides coverage).

Solution S5-95-5. This question is based on the discussion of excess ratemaking by Werner and Modlin, p. 228. The following are some frequent problems:

1. There are very few claims in the excess layers, so loss cost data for losses below the attachment point of the insurance would often need to be considered in developing a complement of credibility.

2. In some excess insurance lines, loss development might be especially slow. This is truest of excess liability insurance.

3. Loss experience when just the excess layers are considered is typically subject to different (mostly higher) inflation than total loss experience when total limits of coverage are considered.

Section 96 of the study guide will address some specific techniques for finding complements of credibility to use in excess ratemaking.

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