Editor’s Note: Part I of this series was published in the Journal of Systemics, Cybernetics, and Informatics. This article builds on Part I and includes some material reproduced from that article.
This article was inspired by challenging pedagogic issues related to the Financial Mathematics (FM) Examination,[1] whose content is typically based on a standard introductory course in the theory of interest. Passing the FM Exam is one important step in attaining Society of Actuaries/Casualty Actuarial Society (SOA/CAS) credentialing and is also important for internships, hirings and promotions. The article assumes the reader is familiar with this exam and its content.
Part I explains that “[i]n preparing for the exam, many students who have already completed and passed the course may seek additional help. There are several online educational companies specializing in providing assistance for preparing for these exams. The material in this paper is based on the author’s experience assisting and tutoring for exam preparation at both Towson University, and the help desks of the ACTEX corporation, an online educational company specializing in a rich variety of exam preparation materials for a variety of examinations.[2] These experiences provided the author with valuable background facilitating discovery of broad themes and areas of commonality in the help requested by the students.”[3]
The Problem
As stated in Part I, “The statement of the problem is remarkably simple. These students are motivated, having taken the Theory of Interest Course in order to prepare them for the SOA examination. So, what did they miss? If they passed the course and presumably mastered the material, why are they seeking help with specific problems? What are the broad themes and characteristics of these problems?”[4]
Prior Work
Although Part I of this study[5] explored this problem for the probability examinations, the theme of Parts I and II are the same; the techniques discussed in Part I apply equally to the students and material in this article, which provides further techniques. Part I presented four general issues prompting additional instructional energy:
- Problems requiring multiple techniques (such as doing uniform distributions using geometric area formulas versus integral formulas)
- Excessively many-parameter problems (such as the six-parameter hypergeometric distribution formula)
- Modules with multiple formulas (such as the geometric and exponential distribution)
- Modules whose solution is often better formulated by a tabular algorithm rather than a formula (such as conditional expectation and variance)
Who Is This Article Written For?
Although this study was inspired by the courses and tutorials preparing students for actuarial examinations, it will benefit any instructor in any course—for example, literature and composition courses—by providing immediately implementable, easy-to-learn instructional techniques.
Pedagogical Theory
This study is consistent with and reflects several important constructs from the psychology of pedagogy, which we briefly review here.
Attention
The following discussion is reproduced from Part I: “An established adage is that learning cannot take place without prior attention to the material being learned. The following punchy example found in several memory books amusingly illustrates this:
Example: Prior to the advent of I-phones people ascertained time by wrist watches. Memory books ask, ‘Without looking at your watch, state whether your watch uses Arabic numerals for time (1,2,3,4) or roman numerals (I, II, III).'
Many people don’t know the answer to this question because they never paid attention to learn how their watch displays time. Attention as a prerequisite for learning is well established.”[6]
Interference
A simple example of interference is the contrast between the following two tasks:
Task A: Identify the font type in the following words: red, blue, green
Task B: Identify the font type in the following words: bold, underline, italics
Generalizations of Tasks A and B can be presented to people via a computer screen with each task having several dozen examples (instead of the three shown). We can then compute the average time needed to respond. Task B always takes, on average, longer than Task A. The reason for this is that the meaning of the words in Task B interferes with recognition of the font type. Thus emerges the principle that when interference is present, learning is slower.
Stroop did the original interference study, while later studies showed that interference affects average response time, omission rates (not responding to a particular trial) and error rates.[7]
Illustrative Examples
The rest of this article reviews five specific examples with an emphasis on identifying the distinguishing features of each example as well as how to improve textbook and instructor pedagogy. As already mentioned, the inspiration for the examples comes from the FM syllabus.
Example 1: Money Growth Functions—Commonality and Distinction
The FM syllabus recognizes nine types of functions describing money growth over time of initial deposits. At first blush, this should be easy to master: simply create a list of the nine types and associate each type with its name and the underlying function or formula.
However, certain subtleties arise with such an approach depending on the frequency or number of examples of each method. This is easily fixable: ensure that each of the nine methods is represented by an illustrative problem or homework exercise. This corresponds to the attention pedagogical theme mentioned earlier; a student who is weak in a money growth method may not have had an opportunity to practice it.
Corresponding to the interference pedagogical theme, we note that certain money growth functions allow the use of relative time while others may require the use of absolute time. A simple problem illustrates this:
A stock fund whose growth satisfies a quadratic accumulation function, a(t) = t2 + 1, offers a product to be sold at time t = 1, which pays 1,000 at time t = 2 and 2,000 at time t = 3. Fairly price the product at t = 1.
An incorrect approach using relative time might argue that the 1,000 payment reflects one year of exposure. A correct approach would use absolute time—for example, by discounting everything to time 0.
Pv(1) = 1,000 v(2) + 2,000 v(3), with P price and a(t) v(t) = 1 for all t.
The instructional takeaways from this example for improving instruction in any course are for the instructor (1) to create and present summaries of lists for students and (2) to further embed those lists in a table with features that might interfere with each other.[8]
Example 2: Geometrically Increasing Annuities—How Many Formulas?
Geometrically increasing annuities is a standard topic in the FM syllabus. Instructors, however, must decide—besides teaching the underlying theory—how many formulas, methods or techniques to present. A balance must be struck between memorization and derivations from first principles, neither of which should be excessive.
At one extreme, it is possible to simply teach one technique and formula: equating the price (i.e., present value) of a geometrically increasing annuity with a discounted annuity due evaluated at the real inflation rate. However, for geometrically increasing perpetuity immediates, a quite simple formula exists, multiplying the initial payment by the reciprocal of the difference of the interest and inflation rate.
So, both these formulas should be presented. Then, using first principles, these two basic formulas can be adjusted in each problem for immediate versus due and for deferral.
There is possibly a third formula, though it is debatable whether to let students discover it themselves (on tests!) or to teach it (either via textbooks or the instructor): When interest and inflation rates are equal, the price of an n-year nondeferred annuity due is simply the sum of the payments.
The instructional takeaway from this example for any course is for the instructor to plan and decide prior to teaching, how many basic techniques, methods and formulas should be presented and what should be left to students who must derive the technique, method or formula from first principles.[9]
Example 3: Nested Annuities
Nested annuities is not a standard term found in textbooks; so, we introduce the concept with a typical example: a level-monthly, inflation-increasing, yearly annuity. Here, within each year, the payments are level, while the present value of yearly payments will form a geometrically increasing annuity. Thus, the level annuity is nested within the geometrically increasing annuity.
The author has not found syllabi, textbooks or software that recognizes nested annuities as a separate topic to teach. The argument seems to be that if, say, a level annuity is nested within an inflation adjusted annuity, there is nothing new; rather, this is a problem integrating level and inflation annuities. The fallacy with such an approach is that it doesn’t work! The decisive test is students coming to tutorials. If they see these problems as challenging, then it is our responsibility to identify the problem before they come and add this topic to the syllabus.
Nested annuities wonderfully illustrate interference. There are two annuities: the annuity paying on consecutive periods that is nested within an annuity paying at broader intervals. Each of these annuities has three different and interfering parameters: interest rates, number of periods and annuity type. Once this is made explicit, the interference confusing the student can be confronted and avoided during instruction proper.[10]
The takeaway for arbitrary courses is for instructors to clearly identify up-front methods, techniques and formulas that directly interfere with each other.
Example 4: Forward-Rate Problems—Proper Sequencing of Subtasks
For specificity of exposition, we consider the following very typical problem:
A bond fund has a spot rate for one-year bonds of 3%, 5% and 6% for one-, two- and three-year bonds, respectively. Your company wishes to offer a product one year from now paying $1,000 for two consecutive years. Price this bond.
Many weaker candidates can get a problem like this totally wrong without even achieving partial credit. An examination of their papers shows a typical pattern. Students can successfully calculate the one-year forward rates, f; but then they start using formulas like (1 + f)2, reflecting interference with compound interest problems with which they are more familiar.
We can fix this problem by using the theory of goal-setting, which teaches that if two equally competent teams perform the same project, one may succeed while one fails, and the reason lies in how the project is broken up into subtasks.[11] In our example, we will sequence the solution to the illustrative problem using two subtasks: Task 1 is to construct a spot-forward rate table; Task 2 is to use this table to solve the problem. The subtasking slows the solution process, which facilitates removing the effects of interference. On my classroom examinations, I assign two separate questions to each forward-rate problem, thereby forcing the students to sequence properly, slow down in their solution and ameliorate the effects of interference.
A model solution is as follows, beginning with Table 1 for Task 1:[12]
Table 1
Spot-Forward Rate Table for Subtask 1 in Example 4
For Task 2, we use the equation
Pv(1) = 1,000 v(2) + 1,000 v(3)
where P refers to price, and v refers to the appropriate discount factor.
Plugging into this equation and solving is routine and left to the reader. The key point is that by breaking the task into two subtasks (each of which is a separate credited problem on exams), I have tricked the candidates into slowing down, taking their time and hence avoiding interference. Skillfully breaking up complicated subject matter into several steps in an arbitrary course is a well-known pedagogical improvement technique.
Example 5: Multiparameter Problems
A classic FM multiparameter problem is immunization, which addresses three approaches: full immunization, Reddington immunization and exact asset-liability matching. This type of problem requires consideration of five parameters to solve: the zeroth, first and second derivatives (or their equivalent formulation in terms of duration) of A(t) and L(t) (assets and liabilities); the relative positions of the maturity time of the assets and liabilities; and whether interest rates are assumed level.
However, for greater expositional clarity, to present the underlying technique needed, we use the more familiar issue of deciding on statistical tests in sampling problems (When do you use t-test, normal-test or nonparametric methods?), leaving the application of this technique to immunization problems to the reader.
Several obvious approaches don’t work (as indicated by the parenthetical inserts after them):
- Teach first rules of thumb, such as less than 30 justifies t, and only give full details later. (This doesn’t work because there are two versions in student notes leading to confusion.)
- Teach the entire rule the first time: “Use t-test if all of the following are simultaneously true: the sample size is small, the population variance is unknown and the population is assumed to be normal.” (This doesn’t work because too much is thrown at the students.)
- Align teaching with software package practice. (This doesn’t work because software can be mathematically wrong.[13])
A good solution, applicable to any curriculum, is to use the prime implicant normal form (PINF), which constructs a table resembling a truth table except that empty cells are allowed besides those labeled T and F. The PINF for statistical tests for sampling distributions is presented in Table 2. One presents the table and then more leisurely considers the four cases listed in it. The PINF is superior to disjunctive normal form because it reduces the number of rows (in this case by 50%.)
Table 2
PINF Table for Determining the Statistical Test to Use for a Sampling Distribution
Conclusion
This article reviewed several areas where traditional instruction cannot fully meet student needs in an initial instructional setting. Specific, easy-to-implement techniques have been presented that instructors can immediately use to improve teaching in these areas.
This article is provided for informational and educational purposes only. Neither the Society of Actuaries nor the respective authors’ employers make any endorsement, representation or guarantee with regard to any content, and disclaim any liability in connection with the use or misuse of any information provided herein. This article should not be construed as professional or financial advice. Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries or the respective authors’ employers.
Russell Jay Hendel, Ph.D., ASA, has taught full- and part-time at over half a dozen universities. He currently assists ACTEX learning with production of instructional materials and provides tutorial support. He can be reached at DrRussellJayHendel@gmail.com.
Endnotes
[1] “Financial Mathematics (FM) Exam,” Society of Actuaries, n.d., https://www.soa.org/education/exam-req/edu-exam-fm-detail/.
[2] “Who We Are,” ActexLearning, n.d., https://www.actexlearning.com/about.
[3] As noted later in this article, the current paper continues the same theme applied to the Probability Preliminary Examination originally published in Russell Jay Hendel, “They Learned the Course! Why Then Do They Come to Tutorials?” Journal of Systemics, Cybernetics, and Informatics 23, no. 7 (2025): 183–87, https://doi.org/10.54808/JSCI.23.07.183.
[4] Hendel, “They Learned the Course! Part I,” References 1n, 2n, 3n.
[5] Hendel, “They Learned the Course! Part I,” References 4n, 5n.
[6] Hendel, “They Learned the Course! Part I,” References 4n, 5n, 6n.
[7] Hendel, “They Learned the Course! Part I,” References 4n, 5n.
[8] Russell Jay Hendel, “Dr. Hendel’s FM Lecture Notes v6.2,” 2025, http://www.Rashiyomi.com/Mathematics/FMLectureNotes.pdf, Tables 2.1 and 2.2, 11–13.
[9] Hendel, “Dr. Hendel’s FM Lecture Notes,” 40, 43.
[10] Hendel, “Dr Hendel’s FM Lecture Notes,” 54–56.
[11] Russell Jay Hendel, “Optimal (Sub) Goal Setting for Superior Project Outcomes,” The Stepping Stone, March 2024, https://www.soa.org/sections/leadership-development/leadership-development-newsletter/2024/march/ss-2024-03-hendel/.
[12] Hendel, “Dr. Hendel’s FM Lecture Notes,” Tables 14.5 and 14.12.
[13] Russell Jay Hendel, “A Mathematical-Logic Technique Facilitating Good Teaching,” Journal of Systemics, Cybernetics and Informatics 21, no. 3 (2023): 69–73, https://doi.org/10.54808/JSCI.21.03.69; and Russell Jay Hendel, “Teaching Course Modules Involving Multiple Parameters,” Expanding Horizons, August 2025, https://www.soa.org/sections/education-research/educ-research-newsletter/2025/august/eh-2025-08-hendel/.