Report of the Joint CAS/SoA
Working Group on Exam 1/Course 1

The Joint CAS/SoA Working Group on Course 1 was established under the direction of the SoA Design Team (Jeffrey Beckley, Chairman) and the CAS Education Task Force (Steven Lehmann, Chairman).

The Joint Working Group was commissioned with the following charge:

To develop an education and examination structure for the first jointly sponsored course based on the given course description. The course structure should reflect the changes being considered for the SoA and the CAS Education and Examination systems. Where issues of specific concern arise which cannot be resolved within the scope provided, such issues should be noted and raised with the SoA Design Team and the CAS Education Task Force for additional direction.

The scope of this task will include the following components:

To identify the primary learning objectives for the course;

To identify appropriate readings for candidates' use in acquiring such skills - where appropriate material does not exist, the material that would need to be commissioned should be defined; and

To identify the appropriate method of testing, articulate how such a method would fulfill the stated objectives, and develop a sample examination.

Learning Objectives

The Working Group was provided with the following course description for Course 1:

Mathematical Foundations of Actuarial Science The purpose of this course is to develop a knowledge of the fundamental mathematical tools for quantitatively assessing risk. The application of these tools to problems encountered in actuarial science is emphasized. A thorough command of calculus and probability topics is assumed. Additionally, a basic knowledge of risk management is assumed.

The learning objectives for Course 1 are to develop a knowledge of fundamental mathematical tools and a knowledge of their application to problems encountered in actuarial science. The tools emphasized on Course 1 are:

• Limits, series, sequences and functions
• Derivatives of single and multivariate functions (maximums, minimums, constrained maximums and minimums, rate of change)
• Integrals of single and multivariate functions
• Vector valued functions (polar coordinates, parameterized curves)
• Bayes Theorem
• Univariate probability distributions (probabilities, moments, variance, mode, percentiles, transformations)
• Multivariate probability distributions (Central Limit Theorem; joint, conditional and marginal distributions [probabilities, moments, variance, covariance])
• Probability distributions: binomial, Poisson, negative binomial, beta, gamma, uniform, normal, pareto, exponential, lognormal, hypergeometric, geometric

Readings

The Working Group recommends that two or three textbooks in common use at the college level be listed as suggested references for calculus and probability. No specific textbook would be considered required reading for the examination.

The Working Group believes that the use of standard college textbooks is preferable to the use of a study note as a background reference. Under the direction of the respective Task Forces the scope of the risk management topic was limited to a ‘study note not exceeding 50 pages in length'. Some members of the Working Group believe that a short study note with definitions and illustrative problems will be necessary to effectively bridge the risk management topics to the mathematics.

The attached Course of Readings lists some textbooks available at this time. The Working Group anticipates that the Course of Readings will be updated to show texts that are in current use at the university level at the time the examination syllabus is released.

Form of the Examination Questions

The Working Group recommends that the Course 1 examination consist of both narrative style questions and non-narrative style questions. Narrative style questions are those which set the mathematics in an application. Non-narrative style questions are those which are strictly mathematical in their presentation.

• Some topics, such as the Fundamental Theorem of Calculus and the definition of the integral, are key to the candidate's understanding of the mathematics upon which Actuarial Science is based. These topics are not easily tested outside of a purely mathematical context. Because they identify an understanding of important concepts, the Working Group believes that the lack of a reasonable application context should not exclude these topics from the exam.
• The Working Group assumed that the candidate would have a limited knowledge of risk management topics. The Working Group assumed that this knowledge would be obtained through selected readings from a college textbook and/or a short study note. The candidate is not expected to take a college course in risk management in order to acquire sufficient knowledge of the topics.
• The Working Group believes that the use of some non-narrative style questions emphasizes that the actuary must be conversant and competent with traditional mathematics.

Length of the Examination

The Working Group recommends that the Course 1 examination be three hours in length.

• A three hour examination is the maximum length acceptable to the CAS and SoA Task Forces.
• The current Course 100 and 110 (Parts 1 and 2) examinations are three hours in length.
• A shorter length will not allow sufficient time for a well-prepared student to attempt a reasonable proportion of the examination.
• The examination should consist of approximately 40 questions.

Examination Method

The Working Group recommends that the Course 1 examination consist entirely of multiple choice questions.

• The number of students potentially taking the examination coupled with the size of the volunteer pool make grading other forms such as essay or short answer, impractical.
• The multiple choice format is compatible with other nationally administered tests with which the candidate is familiar (ACT, SAT, GRE, etc.).
• The multiple choice format on Course 1 helps smooth the transition from the current Course 100 and 110 (Parts 1 and 2) examinations.

Other Issues

• The Working Group believes that the sample examination as presented is a representative examination.
• The Working Group discovered a general absence of existing questions from published sources that are set in a risk management context and contain the appropriate mathematical rigor. This implies that additional sample questions should be developed to assist both the candidates and the item writers.
• The absence of existing appropriate questions may also indicate a potential difficulty in securing enough item writers from the traditional sources used by American College Testing (ACT) for Courses 100 and 110 (Parts 1 and 2) and a greater reliance on academic actuaries or practicing actuaries.
• The Working Group recommends that three tables be included with the examination: a table of integrals, a table of probability distributions and a table of values for the normal distribution.

Nancy Braithwaite, CAS
John Dawson, CAS
Rob Frasca, SoA
Jed Frees, SoA
Glenn Meyers, CAS
Jim Miles, SoA (Chairman)
Jeff Allen, SoA (Staff Liaison)

December 20, 1996
(Revised September 29, 1997) Exam 1/Course 1

Course of Reading
(Revised - September 15, 1998)

The texts listed below are considered representative of the many texts used by colleges and universities in Canada and the United States to cover material on which the candidate may be examined. Earlier or later editions of the texts listed below contain essentially the same material and should be adequate for review purposes.

Calculus

Calculus and Analytic Geometry (Eighth Edition), 1992, by G. B. Thomas, Jr. and R. L. Finney.

Calculus with Analytic Geometry (Sixth Edition), 1992, by E. J. Purcell and D. Varberg.

Probability

Introduction to Mathematical Statistics (Fifth Edition), 1995, by R.V. Hogg and A. T. Craig. Chapters 1, 2, 3, 4.2, 4.3, 4.5.

Introduction to the Theory of Statistics (Third Edition), 1974, by A. M. Mood, F. A. Graybill and D.C. Boes. Chapters I - V.

Risk Management (background readings)

Principles of Risk Management and Insurance (Second Edition), 1981, by C. Arthur Williams, Jr., George L. Head, Ronald L. Horn and G. William Glendenning. Chapters 4 and 5 of Volume I. (This is a CPCU text.)

Risk Management and Insurance (Eighth Edition), 1998, by C. Arthur Williams, Jr., Michael L. Smith and Peter C. Young. Chapters 4, 10 and 12.

Risk Management Topics

The candidate is expected to have a basic understanding of the following risk management topics:

• premium
  
• fair premium (actuarially fair premium)
  
• losses
  
• claims
  
• deductible
  
• loss sharing (loss participation)
  
• loss severity
  
• loss frequency
  
• retention
  
• self insurance
  
• occurrence
• maximum probable loss
  

Topics not included on Exam 1/Course 1 that are presently on
SoA Courses 100 and 110 (CAS Parts 1 and 2)

• advanced differential equations
• directional derivatives and gradients
• linear algebra
• statistics topics including:

tests of statistical hypotheses
confidence intervals
sampling
order statistics
statistical estimators

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