Casualty Actuarial Society


REQUEST FOR PROPOSAL: Aggregate Loss Distributions:
Convolutions and Time Dependency

December 18, 1996

1. Casualty Actuarial Society (CAS)

The CAS was organized in 1914 as a professional society for the promotion of actuarial and statistical science as applied to insurance other than life insurance, such as automobile, liability other than automobile, workers compensation, fire, homeowners, commercial multiple peril, and others. Such promotion is accomplished by communication with those affected by insurance, presentation and discussion of papers, attendance at seminars and workshops, collection of a library, research and other means. The membership of the CAS includes over 2700 actuaries employed by insurance companies, industry advisory organizations, national brokers, accounting firms, educational institutions, state insurance departments, the federal government and independent consultants.

2. Committee on Theory of Risk (CTOR)

The Committee on Theory of Risk is charged with developing and demonstrating the utility of specific applications of the theory of risk to various lines of property and casualty insurance.

3. CAS Interest in the Subject

The CAS wishes to enhance the development of tools and models that improve the accuracy of the estimation of aggregate loss distributions for blocks of insurance risks. This is an important step in developing more sophisticated tools for Dynamic Financial Analysis.

4. Request for Proposal; Project and Proposal Requirements

Aggregate loss distributions are probability distributions of the total dollar amount of
loss under one or more insurance policies. They combine the separate effects of the underlying frequency and severity distributions.

Over the past fifteen years, various papers have been published and models developed that enable actuaries to estimate aggregate loss distributions. This list includes papers by Heckman & Meyers, Robertson, Panjer, etc. The issues underlying this RFP are how to combine aggregate loss distributions for separate but correlated classes of business as well as how the distribution changes over time.

The CTOR requests proposals from qualified researchers to produce a research document that solves one or both of the following two problems:

A) Assume a book of business is the union of disjoint classes of business each of which has an aggregate distribution. These distributions may be given in many different ways. Among other ways, they may be specified parametrically, e.g. lognormal with given parameters; they may be given by specifying separate frequency and severity distributions; e.g. pareto severity and negative binomial frequency with given parameters or they may be given as specific discrete distributions. The classes of business are NOT independent. For this project, assume that you are given a correlation matrix (or some other easily obtainable measure of dependency) and that the correlation coefficients vary among different pairs of classes. The problem is how do you calculate the aggregate distribution for the whole book.

The Committee has discussed several approaches that may form the basis for further research. These include:

1) There may be families of distributions (e.g. lognormals or shifted gammas) such that if each separate distribution is a member of these families, a closed form and elegant solution is possible.

2) The individual distributions may all be converted to discrete, piecewise linear, or some other form and then a methodology described to combine them into a single distribution.

3) Simulation techniques. These are probably the methods most widely used today by actuaries to solve this problem. The challenge here would be to create a highly efficient, flexible and sophisticated algorithm.

4) A somewhat different approach to the ones outlined above would postulate that the correlation among the classes are all a result of a number of underlying forces that affect different classes in different ways. For example, one might attribute all correlation to be the result of: economic inflation, judicial climate, tort reform, property catastrophes, health of the economy, and rate level. The individual aggregate distributions conditional on specification of these elements would all be independent. One advantage of this method is that this could be used to deal with the problem of dependencies among distributions that are different at different points of the distribution. An example is the effect a punitive damage cap might have on the aggregate distribution for both general liability and auto liability.

The Committee is looking for ideas that would, in the judgment of the Committee, represent a significant advance. The researcher could produce results along one or more of the above approaches or present a completely new way of viewing and solving the problem. The solution(s) should be broad enough to cover most widely used aggregate loss distributions and should be sophisticated enough to incorporate parameter uncertainty in some form (e.g. uncertainty due to model selection, parameter estimation and/or model extrapolation/projection).

B) Assume you know the aggregate loss distribution at policy inception and you have expected patterns of claims reporting, losses emerging and losses paid and other pertinent information, how do you modify the distribution as the policy matures and more information becomes available? Actuaries have historically dealt with the problem of modifying the expectation conditional on emerged information. This expands the problem to continuously modifying the whole distribution from inception until it decays to a point. One might expect that there are at least two separate states that are important. There is the exposure state. It is during this period that claims can attach to the policy. Once this period is over no new claims can attach. The second state is the discovery or development state. In this state claims that already attached to the policy can become known and their value can begin developing. These two states may have to be treated separately.

Here again, the Committee is looking for ideas that would, in the judgment of the Committee, represent a significant advance. The solution(s) should be broad enough to cover most widely used aggregate loss distributions.

The research for these projects will be performed under contract and funded by the CAS. It is expected that the researcher(s) will work with representatives of the committee on the specific design of the project.

5. Project Requirements

a. Literature Survey: The paper should survey actuarial as well as other technical literature for any research relating to these issues and include a bibliography of such research.

b. Proof of solution: The author would be required to include within the paper proof that the solution is mathematically valid.

c. Practical solution: The solution should be presented in a form that a practicing actuary can implement in a practical manner solely from reading the paper. Inclusion of software that would be available to CAS members would be a plus. At a minimum, an algorithm in some suitable form must be provided.

6. Proposal Requirements

Proposals should include a clear outline of the work that will be performed and the time frame in which it will be performed (including key dates). The more specific the better.

This proposal should be reviewed in conjunction with the attached consulting agreement(s) which defines the terms and conditions under which the work is performed.

The proposal should be accompanied by the resumes of the researcher(s), indicating how their background, education, and experience bear on their qualifications to undertake the research.

Respondents should demonstrate their interest in and familiarity with the application of loss distributions to insurance problems by including a resume (if a firm, of the principal consultant(s) performing or directing the work) showing relevant work/research experience and professional accomplishments (e.g.. papers published).

The contract will be awarded by the CAS to the respondent who, in the judgment of the CTOR and entirely on the basis of his or her written proposal, is best able to perform the work as specified herein. If the CTOR determines that no proposal meets the requirements of the RFP, then no contract will be awarded.

Receipt of proposals will be acknowledged by providing each respondent with a list of all respondents. Respondents not awarded the contract will be so informed shortly thereafter.

Interested researchers should submit their proposals and any questions in writing to:

Isaac Mashitz
Casualty Actuarial Society
1100 N. Glebe Road, Suite 600
Arlington, VA 22201
Phone: (703) 276-3100
Fax: (703) 276-3108

The proposals will be reviewed by members of a subcommittee of CTOR. The current membership of the CTOR includes:

Roger Hayne, Chairperson
Rodney Kreps, Vice Chairperson
John Aquino
Ward M. Brooks
Bob Finger
Louise Francis
Jim Garven
Robin Gillam
Phil Heckman
Stuart Klugman
Isaac Mashitz
Glenn Meyers
Gary Patrik
Richard Roth, Jr.
Gary Venter
William F. Weimer

7. Schedule

January 20, 1997
Deadline for questions (must be written) from researchers regarding the RFP.

February 1, 1997
All written questions together with their answers will be distributed to all proposers.

February 17, 1997
Proposal deadline - end of business day

April 4, 1997
Proposal selection by CTOR

December 1, 1997
Report due

8. Compensation

A maximum total of $40,000 is available to be awarded to one or more reseachers. Payment of award(s) will be contingent upon delivery of an acceptable research product.

9. Presentation, Ownership and Publication of Report
If asked, the researcher(s) agree to be available to present the report at a CAS meeting or seminar. If travel is required, reasonable expenses will be paid in addition to the compensation provided in Section 8.

The CAS intends to copyright the report and to submit it to the CAS Committee on Review of Papers for consideration as a CAS Proceedings paper. Alternately, the CAS may publish the report in the CAS Forum. It is intended that the results of the report can be used freely by any interested party. The research will be considered work-for-hire and all rights thereto belong to the CAS. However, appropriate credit will be given to the researcher(s).

Guidelines for submitting papers for publishing in the CAS Proceedings are as stated in the 1996 CAS Yearbook (pages 255-263), a copy of which is available upon request, or on the CAS Web Site.

10. Consulting Agreement

Aggregate Loss Distributions: Convolutions and Time Dependency

This agreement is made by and between the Casualty Actuarial Society ("CAS") and ___________________ ("CONSULTANT"). The CAS agrees to contract for the services of the CONSULTANT, and the CONSULTANT agrees to provide services under the terms and conditions of this agreement.


The CONSULTANT shall provide consulting services on behalf of the CAS to produce a research document discussing existing actuarial guidance on the subject of risk transfer and the reinsurance industry responses prompted by the risk transfer criteria of certain accounting standards, or related areas the CAS or the Committee on Theory of Risk ("CTOR") may request during the period of performance, as specified in the Request for Proposal, which is hereby incorporated herein by reference.


In full consideration of the consulting services provided hereunder, the CAS agrees to pay CONSULTANT a fixed fee of _____ plus reasonable expenses incurred at the request of the CAS or CTOR, based upon a payment schedule to be agreed upon.


CONSULTANT shall be available for a minimum of ________ and a maximum of ________ days for the period of performance beginning _____ and ending _____. This period of performance shall not be changed without the written authorization of the CAS or CTOR.


Total payment under this contract including reasonable expenses incurred at the request of the CAS or CTOR shall not exceed _____, unless authorized in writing by CTOR or the CAS.


It is understood and agreed that CONSULTANT is an independent contractor in the performance of this Agreement, CONSULTANT is not an agent of CTOR or the CAS, and CONSULTANT is not authorized to act on behalf of CTOR or the CAS.

CONSULTANT shall assume full responsibility for payment of all federal, state and local taxes with respect to performance of the CONSULTANT's obligations under this Agreement.


CONSULTANT warrants to the CAS that he/she is not subject to any obligations, contracts, covenants or restrictions that would prevent him from entering into or carrying out the provisions of this Agreement.


This Agreement may be terminated by either party at any time by giving written notice of such termination to the other party. Upon receipt of such written notice by either party, no further charges will be made under this Agreement. Termination shall not affect the CONSULTANT's obligations under articles IX and X.


CONSULTANT shall indemnify and hold CTOR and the CAS harmless from any and all suits, claims, damages or losses whatsoever, resulting from any act or omission of the CONSULTANT, his employees, agents, and subcontractors in his performance under this Agreement.


CONSULTANT warrants that, to protect the privacy of respondents to surveys conducted, CONSULTANT will collect only the data needed for the purpose of their inquiry and inform each potential survey respondent about the general nature and sponsorship of the inquiry and the intended uses of the data.

CONSULTANT acknowledges that information it obtains in the performance of this Agreement is valuable and confidential. Accordingly, CONSULTANT agrees not to disclose any such information to any person not authorized by CTOR or the CAS to receive it. Upon completion of the work, CONSULTANT shall deliver to CTOR all data, documents, reports, surveys, or other materials prepared by CONSULTANT in his performance under this Agreement.


CONSULTANT will promptly disclose to CTOR all discoveries made and ideas conceived by CONSULTANT in his performance under this agreement. CONSULTANT assigns to the CAS all right and title to such discoveries and ideas, and agrees to execute any and all such documents, including copyright assignments, as the CAS deems necessary to secure it all right, title and interest.


This Agreement may be amended only by a written document, signed by both the CAS and CONSULTANT.


CONSULTANT may not assign this agreement or any right hereunder. Any such attempted assignment shall be void.


This Agreement shall be governed by the laws of ___________ (the jurisdiction of CONSULTANT's principal place of business).

Name __________________________
Title ___________________________
Date ___________________________
Casualty Actuarial Society
Name ___________________________
Title ____________________________
Date ____________________________

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