Conversations - Spring 2001

Can Actuarial Science Survive More Finance?
by James D. Broffitt and Elias S. W. Shiu

Department of Statistics and Actuarial Science
College of Liberal Arts and Sciences
The University of Iowa
Iowa City, IA 52242-1409

The April 2001 issue of The Actuary, pages 10-13, contains an interview with SOA President Rob Brown, in which he discussed a proposed designation known as "QRS" (quantitative risk specialist). The Editor of Conversations has kindly invited us to comment on its potential impact on actuarial programs housed in mathematical sciences departments in the U.S.A.

We have since heard that QRS has been modified to QRA, which we assume to stand for quantitative risk analyst. As QRA is still in the development stage, understandably little detail was given in the interview. Perhaps the following quote sheds the most light on how QRA might influence academic actuarial programs:

"Courses 1 and 2 might remain relatively unchanged. But we would have to reshuffle Courses 3 and 4 so as to keep in the QRS subjects such as core statistics, statistical models and modeling, stochastic processes and simulation. But we would have a later exam to cover and test purely actuarial techniques, such as multiple decrement theory, pension mathematics, and credibility theory. We would then probably have what would look like a new Course 4 that would cover some of our existing Course 6, which is investments and asset liability management."

From this paragraph, we infer that there would be another "preliminary" exam on finance, covering topics in investments and asset/liability management, and that core actuarial topics, such as life contingencies, risk theory and credibility, would be relegated to later exams. Our first reaction upon reading the above is that, if we are to continue teaching core actuarial topics such as life contingencies, there will be little to no room in our actuarial programs for more finance or other courses. Our second reaction is that, even if the students are willing and able to take additional courses as an extra load, such courses may not be available to them in our business college.

Our second reaction is easy to explain. The new Course 2 has already created much hardship
for our actuarial students and has perhaps discouraged some of them, since it is difficult for
them to take the course "corporate finance" in our business college. The corporate finance
course in our business college has another finance course as a pre-requisite, which in turn
has two accounting courses (and two economics courses) as pre-requisites. Furthermore,
since our students are not business majors, they are not allowed to register for high-demand
business courses until the third day of classes, because the business college wants to first
accommodate its own students. Adding more finance courses, which are high-demand courses,
to the syllabus of the earlier actuarial exams will exacerbate an already difficult situation.
Accessibility to finance courses may not be a problem for actuarial programs housed in
business schools, but it is for those in a mathematical sciences department. Indeed, the
current Course 2 seems to be a deterrent to students studying in colleges that do not offer
finance courses.

Our first reaction is due to simple arithmetic. The University of Iowa actuarial programs are designed to thoroughly educate students on the material upon which the current Courses 1 through 4 are based. The requirement for an Iowa B.S. degree is 124 semester hours. We require our actuarial students to take 19 hours of calculus and linear algebra, 12 hours of mathematical statistics, and 4 hours of computer science. To prepare for Course 2, the students need 3 hours of compound interest, 4 hours of introductory microeconomics, 4 hours of introductory macroeconomics, 3 hours of intermediate microeconomics (price theory), and 12 hours of finance and accounting from the business college. (We still remember that, when the SOA introduced the new syllabus, we were told that the new Course 2 would only require the students to take three courses on economics, half a course on compound interest, and one course on finance.) The courses offered by our department annually for Courses 3 and 4 are as follows:

Introductory actuarial science (3 hours)
Life contingences (8 hours)
Stochastic processes (3 hours)
Risk theory (4 hours)
Regression and time series (3 hours)
Survival analysis (3 hours)
Credibility and loss distributions (4 hours)

University education in the U.S.A., unlike many other countries, is based on the idea of a liberal or broad-based education. Undergraduate students are required to take courses from diverse areas, such as foreign languages, history, social sciences, physical and natural sciences, literature, fine arts, and so on. Hence, there are another 39 hours of non-actuarial courses that the undergraduate students may need to take. The total number of hours listed so far is 128, which is 4 more than the required 124. We shall attempt to alleviate the problem by teaching corporate finance ourselves, even though we know that this is a poor educational policy. Students should learn corporate finance from scholars in corporate finance, but we seem to have no choice.

The situation is even more critical for our graduate students who usually have undergraduate degrees in mathematics, but have not previously studied finance or economics. Even though we require the graduate students to take 37 semester hours of courses (the graduate colleges requirement is only 30 semester hours), there is not enough room for the many courses needed for Course 2. Adding more finance will make the situation worse.

Our programs seem successful in helping students pass the professional exams. For example, in November 2000, 16 out of 22 Iowa students passed Course 3 (73%), and in May 2001, 14 out of 21 passed Course 4 (67%). We have little quarrel with finance being on the professional exam syllabus; however, adding more finance to the early exams would not be attractive for our students. We would want to continue educating our students in core actuarial subjects such as life contingencies, risk theory, credibility, and loss distributions. If the QRA concept is adopted, we hope that the exams will be structured in such a way that finance, economics, etc., will not be used as pre-requisites for the core actuarial exams.

The remaining paragraphs are not directly related to QRA, but they do relate to issues in actuarial education.

An argument for QRA seems to be that it would attract more students to become actuaries. We have heard the argument that the current Course 2 would be an attractor examination for business students. We were skeptical then, and we remain skeptical now. Given that the current Courses 3 and 4 are so highly mathematical, it is doubtful that students from a typical undergraduate business program in the U.S.A. will have sufficient mathematical maturity to master Courses 3 and 4.

Indeed, the new Courses 1 and 2 may be discouraging students who are studying in small liberal arts colleges from entering our profession. Many American students choose to study in small liberal arts colleges, which do not have business schools, and hence, they may not have access to corporate finance courses to prepare for Course 2. Also, in these colleges, a probability course of the level for Course 1 is likely to be a final-year course. Such students may not be prepared to write any actuarial examinations until graduation. One may suggest that these students attend graduate schools to study actuarial science. This is not what we observe. We received about 60 applications for our M.S. program last year; only a handful of these were domestic students. For business majors considering graduate study, the choice between an M.S. degree in actuarial science with the prospect of eight professional examinations or an M.B.A. degree with no other exams seems obvious.

A goal of the SOA should be to attract the best and brightest students to our profession. QRA is unlikely to help accomplish this goal. In the U.S.A., a large proportion of the best and brightest students attend private universities. Harvard, Yale, Stanford, M.I.T., Chicago, Cornell, etc., only have graduate schools of business. Indeed, Princeton and Caltech do not even have business schools. Where would undergraduate students in these world-class universities take enough finance and other business courses to satisfy the QRA requirements? Obviously, we do not want a system that excludes a large proportion of the best and brightest undergraduate students in the U.S.A. Dont we want many future actuaries to have studied with some of the most brilliant professors in the world? With the current Courses 3 and 4 being so difficult, our profession needs students from these top universities more than ever. We would suggest that persuading some of these universities to start an actuarial program would be a major accomplishment for the SOA.

We find that most of our actuarial graduates work for pension consulting firms and life insurance companies. Undoubtedly, learning more about finance and other business topics are good for them. On the other hand, it seems that classical actuarial topics, such as life contingencies, are de-emphasized in the current syllabus. For example, gross premiums, modified reserve methods, pension valuation, etc., are not even in Course 3, which is the examination that includes life contingencies. It seems to us that employers of our actuarial students would prefer the students to understand more about such basic actuarial topics than the Modigliani-Miller Theorem in finance.

Our next comment is a question: "Will we get enough actuarial professors in the future?" We certainly do not have enough now. The qualification for an actuarial professor is normally a Ph.D. degree plus an A.S.A., A.C.A.S. or better. The old A.S.A. was somewhat equivalent to the first four current examinations. This was not too burdensome for a Ph.D. with a strong mathematical background. However, the new A.S.A. requires six examinations, the last two requiring much memorization. The current salary for new assistant professors in finance at major American research universities is around U.S. $120,000. Knowing the effort needed to qualify as an A.S.A., a young Ph.D. in mathematical sciences nowadays would probably find it more attractive to move to finance. If the SOA is interested in enhancing actuarial education in universities, helping raise the salary levels of actuarial professors would be a step in the right direction. We heard that a Canadian business school has recently received a donation from a major Canadian life insurer for an endowed chair in finance. It would be a wonderful development for actuarial education if the SOA would spend its efforts in persuading this and other insurers to endow chairs in actuarial science.

We end this essay with a plea to the SOA. American insurers and consulting firms want to hire American actuarial students. American students attend American universities, which have constraints different from those in Britain, Australia, or even Canada. Many of the best and brightest American students attend universities without actuarial programs or undergraduate business colleges, but our profession must try to attract some of these students. When changing the actuarial syllabus, please do keep the American actuarial programs in mind, and try to understand what these programs can do and cannot do. Designing an exam system that will require students to attend graduate schools will attract very few American students; we are stating this even though our M.S. program is a potential beneficiary of such a system. Also, in many universities, it is not a simple matter to add or change courses. Such changes may take months or even years, and require approval from many departmental and university committees. Academic actuarial programs had to make many changes to accommodate the new syllabus of 2000. None are looking forward to treading through the bureaucratic hurdles for another set of syllabus changes for 2005. Finally, we would point out that we started our actuarial careers as professors of elegant subjects such as life contingencies, graduation and risk theory, and we would rather not retire as prep course teachers for economics and finance.

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Can Actuarial Science Survive More Finance?
A Response
by Stuart A. Klugman

College of Business and Public Administration
Drake University
Des Moines, IA 50311

Let me begin with some disclaimers. As noted above, Drakes actuarial program is in a business school and we have excellent cooperation with our finance and economics faculty. From my 14 years on the faculty at The University of Iowa (Iowa), I also know that the problems Jim and Elias have with their business school are long-standing and are unlikely to be resolved any time soon. However, as a profession and a Society we should do what is right, not what is feasible at some institutions.

So, what is right? The Society of Actuaries Mission and Vision statement contains the following statement: "The vision of the Society of Actuaries is for actuaries to be recognized as the leading professionals in the modeling and management of financial risk and contingent events." Jim, Elias, and I all earned our actuarial qualifications by mastering the modeling of contingent events. Both the vision statement and the direction of our industry (no longer insurance, but financial services) indicate that we can survive only with more finance.

Then what is the implication for undergraduate actuarial science programs in the United States? And will QRA help or hurt? At no time have we fully prepared our students to do actuarial work. Prior to 2000 life was easy because all early exams (100 series) were mathematical/statistical and we all got our students as far along that path as we could. The current system makes life less simple by introducing finance and economics at an earlier point. (Keep in mind that almost all the topics on current Course 2 and those additional finance topics proposed for QRA were on the pre-2000 syllabus, just with higher numbers. Perhaps their essay should have been titled "Can University Actuarial Programs Survive the Movement of Finance Topics to Lower Numbered Exams?") My guess is that every school has to make some compromises. Iowa has chosen to fully educate students in the mathematical portion of the syllabus (Courses 1, 3, and 4) and does so by offering 63 hours of courses. There is then not enough time for students to also master the Course 2 material.

At Drake, we have only 41 hours devoted by Courses 1, 3, and 4. The reason is that as an accredited business school, 50% of a students hours must be outside the College. All courses labeled as actuarial science and all but 6 hours of courses labeled statistics are considered to be business classes. On the other hand, of the 8 courses required for Course 2 (two accounting, two finance, three economics, one interest theory), five of them are required of all business majors. Thus preparing for Course 2 adds only three courses to their curriculum. The result is that within the 124 hour limit, we can fully prepare students for the first two exams and get them close on the next two.

Regardless of the constraints of ones institution, there just isnt enough time in four years to fully prepare students for all four courses. Each of us must decide where the emphasis should be.

My view is that QRA will make things better, not worse. The current thinking (I am a member of the task force that is working on this, but do not speak for the task force) is that actuarial qualification (i.e., to ASA) will involve the demonstration of competency in the following areas (with correspondence to current or former exams noted):

1. Probability and Statistics (old 110)
2. Economics (current 2)
3. Corporate finance (current 2)
4. Statistical methods (old 120, current 4, but less mathematical)
5. Investment and assets (current 6)
6. Modeling (various parts of current 3 and 4, but in a risk, rather than insurance, context)*
7. Professionalism
8. Actuarial mathematics and advanced theory of interest (old 140, 150, plus)*
9. Actuarial practice (current 5 and 7)

*The division between modeling and actuarial mathematics has yet to be established. One view is that the modeling course provides the tools to read Actuarial Mathematics but no solutions in an insurance context. Topic 8 will be the traditional treatment (bringing back some of the dropped elements from old 150) and may also include topics such as credibility.

As currently envisioned, QRA is likely to comprise the first 7 items. Traditional actuarial education in university (pre-2000) covered topics 1, 4, 6, and 8. Topics 7 and 9 were never covered. This leaves the three topics that Jim and Elias lament being part of actuarial education, those being 2, 3, and 5. Yet all are essential for good actuarial work, and all are likely to be included in the International Actuarial Associations minimal syllabus for member organizations. As noted before, the only real change is the timing of these subjects.

Better yet, it is unlikely these topics will be numbered, so students will not know that they are missing an "early" exam. Any undergraduate program that can successfully prepare students for five of the above subjects will be worthy of a students time. Should Iowa choose to specialize in topics 1, 4, 6, and 8 plus one more, the faculty will get to teach what they like and are good at, and the students will emerge as attractive candidates for the job market. Furthermore, these are the most challenging topics. Leaving the others for self-study may be appropriate.

Jim and Elias address a number of other issues, but I have chosen to concentrate on QRA and finance. While I do not fully agree with their statements on these other issues, they are less pressing. Finally, the Societys Task Force on Education and Qualification 2005 expects to release a report to the membership in August or September. The report will provide more information on what the Task Force is thinking, and I would invite you all to read the report, and provide us with your input and ideas. We very much want to hear from each of you, whether you support or disagree with the proposals of the Task Force. We appreciate the importance of your contribution to actuarial education, and would encourage your participation in the discussion of its future.

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Its Time to Start Studying for Course 2
by Bonnie Averbach

Director of the Program in Actuarial Science
Temple University
baverbac@sbm.temple.edu

Its time to start studying for Course 2. Theres lots of material to cover. Here is a strategy that you might try.

The Order of Study:

1. The TI BA 35 solar calculator is a must. (You may also bring other SOA acceptable calculators to the exam.)
   
2. Study the Theory of Interest before the Corporate Finance. Many of the problems in finance involve some of the methods of theory of interest.
   
3. Study Microeconomics before Macroeconomics. The macro study note is not easy to understand, so it pays to concentrate on the micro and then learn as much macro as you can.
   
4. You can study the Theory of Interest and Micro at the same time. (When you tire of doing interest problems, switch to the Micro material - and vice versa.)

How to Study:

1. As you cover the material, (whether in a classroom setting or on your own) learn the terminology and basic principles, and work on as many exam type problems and questions, pertaining to what you are studying, as you can. These questions can be found in some of the study manuals and in the recent exams that are posted on the Web.
   
2. Keep the recent exams (Y2K sample, May and November 2000 and May 2001 exams) on hand. After you have covered several sections of material, see if you can identify and solve problems that pertain to the material that you have studied. These exams and solutions can be obtained from the SOA web site at Multiple Choice/Essay Examinations or www.casact.org.
   
3. As in all of the actuarial exams, you must know what you know very well so that you can quickly identify which questions you can do quickly, which you will attempt, and which you will skip.

Study Materials:
The best types of study materials give you insight into the nature of the material as well as exam type problems (and solutions) relating to the material. It is important to learn terminology and understand basic principles for all topics.

1. The Calculator: Obtain the TI BA 35 solar calculator and learn how to use it while studying Theory of Interest. You can save lots of time if you use it efficiently. (Complete instructions as well as calculator tricks are given in the Averbach and Vance study manual for Theory of Interest.)

(You may also use the other SOA-acceptable calculators. You may want to store values in three different storages as in the TI 30 XII-a solar.)

2. Theory of Interest: Basic principles are important. A time line with payments placed correctly will enable you to quickly set up the equation that will enable you to solve the problem. In recent exams, there has been emphasis on varying payments (including increasing, decreasing, and percentage - increasing) and changing rates.

Basic text: Theory of Interest by Kellison

The Course 2 Theory of Interest Study Manual by Averbach and Vance is a self- contained manual with many examples, exercises, and practice tests (all with complete solutions). Calculator tricks are included. (This manual is used as the text in the Theory of Interest class at Temple University.)

3. Corporate Finance: Terminology and basic principles are important. Carefully go through the mathematical methods that are discussed.

Basic text: Principles of Corporate Finance by Brealey and Myers

The Exam Preparation Manual for Exam 2 Corporate Finance by Weiss is a self-contained manual that gives insight into the material with exercises and complete solutions pertaining to each chapter. (This manual is used as text in the Corporate Finance class for Actuarial Science majors at Temple University.)

4. Microeconomics: The Price Theory text and the accompanying study guide by Landsburg are well written and relatively easy to understand. The study guide has many tests that help in the understanding of the material.

Consider the CSM manual. It has a good set of problems and solutions for all of the topics in the exam. The Economics section (by Sorrentino) of the Actex manual is also good.

5. Macroeconomics: The Wachtel study note is difficult to read.

(See study manuals for microeconomics.)

(Mary Weiss is in the process of writing a study manual for macro- and microeconomics. She will be class testing it in spring 2002.)

How to Organize a Review or College Prep Class:

1. The students must study the material before coming to class.
   
2. Divide the number of sessions into three equal units: Interest, Finance, and Economics.
   
3. There is so much material to cover that it may be impractical to cover it all in one semester. So for a review, consider covering the problems from the post Y2K exams. Categorize the problems from the Y2K Sample test, the May and November 2000 exams by topic, following the order of the given text material. The students should try these problems before the class. These problems should be thoroughly discussed in class. Continue this method throughout the semester.
   
4. Toward the end of the semester, students should work on the May 2001 exam as a whole.

SOA Staff Note: Additional Information on Course 2

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Lens Focus

For this issue of Conversations, we asked two experienced actuarial educators to provide their perspectives on preparing students for the Course 1 and Course 2 exams. Professor Mark Maxwell, a colleague of mine at Robert Morris College, has experienced significant success in training students in small college actuarial programs, both here and at Maryville University of St. Louis. He shares some thoughts about getting students ready for Course 1 during the first two years of college.

Bonnie Averbach, Director of the Actuarial Science Program at Temple University, has wide experience conducting exam preparation seminars and as a study manual author. She addresses approaches to Course 2 preparation.

The concept of a Quantitative Risk Specialist (QRS) pre-actuarial designation has prompted a great deal of informal discussion among educators and students, especially in the Internet forums of the professional societies. Professors James Broffitt and Elias Shiu of the University of Iowa have written an article for this issue exploring some of the ramifications of this initiative for actuarial educators.

While this notion of a new designation, apparently now transmuted to QRA (Quantitative Risk Analyst), has been circulating among the SOA inner circles for some time, it would not surprise me if many educators have at best only a hazy knowledge of its meaning and its provenance. Here is a brief synopsis, pieced together from the few public sources I could find.

Several years ago, Howard Bolnick, as president of the SOA, brought together a number of ideas for expanding the role of actuaries in the business community. Dubbed the Big Tent, his proposals have been brought forward by a succession of SOA presidents, each of whom puts his own stamp on matters.

The current SOA President, Robert L. Brown, has championed the formulation of a new designation, QRS/QRA, that would certify competency in a body of quantitatively oriented business knowledge that could serve as a core for a wide range of financial careers. These include financial engineers, risk professionals (under GARP), security analysts (under CFA), and so forth, as well as actuaries. If the QRS/QRA is enacted, the actuarial exams would change once again, most likely with Courses 1 4 modified to test only topics that are not specifically actuarial in nature (finance, economics, statistics, and stochastic modeling). Topics like life contingencies, loss distributions, credibility, and so forth, would be deferred to later actuarial exams.

The advantages of this new approach, according to Rob Brown, are :

• The new designation will have broad appeal and will act as a new magnet for the actuarial profession in particular
• The skill set being tested is universal, and the new designation would be international in scope and application
• The QRS/QRA will be widely viewed as "value added," and will be appealing to employers beyond the traditional insurance and actuarial consulting industries
• Universities will be more likely to sponsor QRS/QRA programs at the undergraduate and the MBA levels since the enrollment numbers will far exceed those of specifically actuarial programs.

The original idea for this seems to have grown out of the SOA Proposed Strategic Plan, a document available for perusal and comment at the SOA Web site. While QRS/QRA is not specifically mentioned, it is sort of adumbrated in the concluding section on strategic initiatives, wherein Strategy 5 states, "Offer certification of a variety of accomplishments while preserving the designations of FSA and ASA for those demonstrating commensurate depth and breadth." One of the associated bullet points is, "A basic education curriculum is established and recognized broadly by other actuarial organizations as well as by other risk professionals."

In addition, there are two SOA committees, the Committee on Strategic Planning, and the Education and Qualifications 2005 Task Force, that have had and/or continue to have a hand in bringing this proposal to fruition. I am aware of at least two widely available documents in which Dr. Brown elaborates on these matters, the 2001 SOA Presidential Address, and his interview in the April, 2001 issue of The Actuary.

Obviously, there are many aspects of the QRS/QRA initiative that will directly and significantly affect actuarial science programs at U.S. schools, large and small alike. Professors Broffitt and Shiu touch on a number of these, in particular, the displacement of actuarial mathematics with finance in the exam syllabi, the proliferation of required courses, and the growing difficulty of recruiting students as well as actuarial science professors.

There is certainly merit to the QRS/QRA proposal, and some of the most prominent actuarial educators in North America are solidly behind it. But, as with any new initiative, there will be unanticipated consequences as well as resistance, both passive and active, from those educators not involved in the decision-making. It would behoove the SOA to develop better channels of communication with rank-and-file actuarial educators, many of whom are not SOA members or do not consider the SOA their primary professional organization. There are undoubtedly many mathematics and other professors who promote the actuarial profession to their students, but are not really attuned to SOA issues. They would be surprised to learn that what has traditionally been a "mathematics" profession is evolving into something rather different.

In any event, there is much to ponder, and I would encourage readers of Conversations to provide some feedback with observations and insights that could start an ongoing dialog.

Professor Leonard A. (Len) Asimow, Ph.D., A.S.A.
Program Director, Actuarial Science
Robert Morris College
Massey Hall 3rd Floor
881 Narrows Run Road
Moon Township, PA 15108
Office: 412-299-2455
Fax: 412-262-8494
asimow@robert-morris.edu

SOA Note: Stuart Klugman's response (Can Actuarial Science Survive More Finance? A Response) to the issues presented by Professors Broffitt and Shiu was a late addition to this issue. The SOA wanted the opportunity to present another perspective in this edition of Conversations.

As Len concludes in his Focus, there is much to ponder. Readers are encouraged to continue the dialog. In the next issue of Conversations, the editor plans to incorporate as many viewpoints as possible.

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Preparing for Course 1: Mathematical Foundations of Actuarial Science
by Professor Mark Maxwell, PhD
Robert Morris College
Maxwell@robert-morris.edu

The Characteristics of a Successful Student
I have found no single attribute of a student that will accurately predict success on the early actuarial examinations. The attrition rate even for capable students is remarkable, and earning a professional actuarial designation requires very high levels of both native ability and perseverance.

At Robert Morris College, we strive to help all our actuarial science majors pass at least one examination. To that end, we look to develop ability, maturity, desire, and preparation. We hedge on ability by limiting the program to those students scoring at or above the 90^th percentile on the mathematics portion of the standardized college entrance exams. We know that the incoming students are skilled. Thus, if we faculty do our job well, we will enhance the other facets of success in the courses we teach to prepare students for the Course 1 exam.

We know that it takes maturity, both mathematical and personal, for students to succeed. One notion on the part of students that we work to modify is the misplaced stress on class grades. Students need to be committed to learning (truly understanding) the material in order to pass the first actuarial exam. This is at a level well beyond what is generally required for a grade of "A" in a typical college class. Thus, we try to get students to focus on their comprehension of the underlying mathematics. The course grades will take care of themselves.

One attribute that students who pass actuarial examinations have in common is commitment. We cannot teach desire, but we do try to create an environment among our students that encourages and fosters certain goals. When younger students hear about the summer internship experiences of our older students, they become extremely motivated.

It is important for our students to be focused from the time they arrive at Robert Morris College. To that end, the department sponsors many social events for our students. We want freshmen to meet with the seniors early, to get excited about internship possibilities, and to learn about the dedication required to pass Society examinations. As faculty, we inform students about the opportunities that will be afforded them with one exam passed. And we do all we can to help our students. However, success in our program, and as young actuaries, comes down to wanting to do well and preparing. Once a student decides to make the investment of time and effort, then it is all about preparation. Courses should be designed with the ultimate goal in mind. In addition, there are some fine study manuals (brief reviews follow) that can add another layer of preparation for our students.

The Composition of a Successful Faculty
One cannot optimally help students without knowing what the exam is like. Faculty need not be specialists in actuarial education, but it certainly helps if calculus and statistics instructors have reviewed past examinations and try to keep the topics, concepts, and style of the exams in mind when teaching courses. For example, faculty may be doing a disservice to students by allowing notes, formula tables, or "artificially intelligent" calculators on all classroom examinations. This is a difficult issue in classes that mix actuarial science majors with engineering and other students.

The best single teaching technique is to expect a lot from the students and to hold them to a high standard. Making everything easy by only assigning the routine problems and not challenging a students intellect is a grave disservice¹. In my experience, all of the students not just those preparing for the actuarial exams will learn and retain more if we pose challenging problems, and do not allow notes or formula sheets on exams.

Ideally, the faculty teaching calculus to actuarial students will have a grasp of all of the relevant subjects, including probability and risk theory. I believe most Course 1 examination questions can be classified as probability problems that are worded in terms of insurance that you must use calculus to solve. The ideal faculty member knows how all of the material relates to the examination, what material is critical for actuarial exam success, and how to motivate students.

One of the difficult pedagogical issues is to know which material needs to be memorized and which concepts need to be understood. I commend the societies in making the effort to stress concepts over procedures on their more recent exams. This, however, makes teaching a little more complex. A student cannot do well, nor should he do well, by simply memorizing a list of formulas. As conceptual understanding grows, the formulas become naturally embedded in the minds grasp of the mathematical structure. This is the level of "internalizing" of the material that subsumes strict memorization and leads to fluency. Therefore, it is my goal to reduce to zero the number of formulas that need to be memorized. I am not quite there.

The problems my students find most challenging involve risk theory. It is best to begin working on these problems at the end of the first term of Mathematical Statistics I, and to make risk theory a major topic in Mathematical Statistics II. That is where we can really focus on specific Course 1 problems. I like to assign past examination problems in homework and on examinations.

It is possible to leave all of the specifically actuarial topics and the study of past exam problems to a Course 1 exam preparation course, which follows the calculus and mathematical statistics courses. But experience teaches that this may be too little, too late, and that integrating the exam preparation techniques into the regular courses is far more effective. It is heart wrenching to see how much harder a junior needs to work if she needs to re-learn (or truly learn for the first time) calculus. In addition, if students are trying to pass Course 1, they certainly are not focused on the current Course 2 materials being taught to them. But we understand that each student is unique, and this maturity and focus varies by students.

Teaching Students to Think
While this is not the correct forum for a discussion on the American education system, it is important to note that the direction mathematics education has taken in the last few decades (reliance on technology, graphing calculators, de-emphasis on rote learning) does not prepare students for the types of questions that they see on SOA examinations. For example, many students have calculators that can symbolically integrate and differentiate. On Society examinations, these basic skills must be performed quickly, accurately, and without the use of technology.

The classic comic strip cartoon describes Hell as a library of story problems. Parents, students, and many teachers at all levels often cringe at the idea of solving story problems. So students must be encouraged to concentrate on the translation of the words into a mathematical model (often, a single equation on the practice exams). If students are willing to think hard about each step and the logical connections, they will gradually improve and progress to more difficult problems. Again, practice is the key. The difficult part is not solving story problems; rather it is keeping students on task and getting them to carefully think through all of the model solutions, step by step.

We try to present various solution methods in class. The ability to use alternative methods helps, as students have different learning styles. Also, solving a problem several ways can add to the general understanding of the underlying concept. Rarely is there merely one solution method. Depth of understanding can be measured by being able to solve problems using alternative methods. I think that it is wonderful for a student to solve a difficult problem in a direct but clumsy and time-consuming way. Then they can see the beauty and functionality of solving certain problems with more elegant methods. The ability to approach a problem using a variety of methods is what makes good classroom instruction superior to a study manual or videotape. Besides, it keeps me employed. But in the end, success comes down to students working many practice problems by themselves.

Subsection on Review Course Material:

ASM Study Manual for Course 1/Exam 1, 2001 Edition, by G.V. Ramanathan

It is designed as a comprehensive review manual for students who have had the standard Calculus sequence and a first course in Probability. It has problems throughout (former SOA questions are denoted) and solutions in an appendix. It concludes with a practice examination and solutions.

Solutions by an Actuary and Probability Solutions by an Actuary, by Steven Shawcross

There are some warm-up exercises, but the primary design is by sample exams, with solutions. It can be described more accurately as a review of how to solve problems rather than a review of the material. A student would need a background in calculus and probability to get the most out of this study material. My students and I appreciated the style that Shawcross employs. He words his solutions as though he were talking to you, the student. He often solves problems in several different ways, from the traditional to the slick and clever. While reviewing the manual, I was pleased to see that Mr. Shawcross solutions and approaches were the same that I employ in class. This manual is a nice supplement for students who want some step-by-step approaches to solving problems that are easy to understand. A student can pick up on several significant problem types and special tricks, aided by a comfortable writing style.

Actex Study Manual for Course 1, by Samuel A. Broverman

This is a classically designed study manual for students who have had the standard Calculus sequence and a first course in Probability. A review of the topics is presented initially, designed around rules, results, and formulas. The middle section offers problem sets followed by solutions partitioned by topic. Finally there are three practice examinations for students to work.

The best reference depends on the level and type of candidate sitting for Course 1. Each manual has its distinct style and advantage, and I think the ideal study manual is a combination of the best parts of all that I reviewed, with many more practice examinations (without solutions) added. We currently have access to a sample exam and four prior Course 1 examinations. Within a short time, faculty and students will have quite a few examinations to supplement primary textbooks and study manuals.

Conclusion:
I was once so brash as to think that I could move mountains, make the world a better place, and train my students to pass an actuarial examination on calculus and probability on their first try. I am older now.

I believe that the recipe to pass an actuarial exam is straightforward and has been known for a long time. Start with a bright student and have faculty who can present the material properly. Once the student decides to apply herself, she needs to practice on plenty of similar problems. Access to detailed solutions from manuals, other students, and faculty, can provide some added insights, but must be used sparingly. Students must try to solve problems independently and without aid, just like they must while sitting for the exam. Ideally, a group of other actuarial students will create a supportive environment, and the faculty can help to overcome common pitfalls and focus on the important concepts and methods.

I also believe that if we help students prepare properly, they will also learn how to prepare by themselves for future examinations. The strongest students will have passed two or more examinations by the time they graduate and leave college. Most will be content with one. And then they are alone, with only memories to guide their study.

SOA Staff Note: Additional Information on Course 1

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¹This advice was given to me by one of the best mathematics teachers I know, Dr. Lynda Danielson of Albertson College of Idaho.