A Spiral Approach to Financial Mathematics

592 Pages·2018·2.851 MB·English

by  Nathan Tintle, Nathan Schelhaas, Todd Swanson

A S A
PIRAL PPROACH TO
F M
INANCIAL ATHEMATICS
A S A
PIRAL PPROACH TO
F M
INANCIAL ATHEMATICS
NATHAN TINTLE
Department of Mathematics, Dordt College,
Sioux Center, IA, United States
NATHAN SCHELHAAS
Principal Financial Group, Des Moines, IA, United States
TODD SWANSON
Department of Mathematics, Hope College, Holland,
MI, United States
AcademicPressisanimprintofElsevier
125LondonWall,LondonEC2Y5AS,UnitedKingdom
525BStreet,Suite1650,SanDiego,CA92101-4495,UnitedStates
50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom
Copyrightr2018ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,
electronicormechanical,includingphotocopying,recording,oranyinformationstorageand
retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek
permission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangements
withorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,
canbefoundatourwebsite:www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe
Publisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand
experiencebroadenourunderstanding,changesinresearchmethods,professionalpractices,or
medicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein
evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein.In
usingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyof
others,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,
assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts
liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products,
instructions,orideascontainedinthematerialherein.
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
ISBN:978-0-12-801580-3
ForInformationonallAcademicPresspublications
visitourwebsiteathttps://www.elsevier.com/books-and-journals
Publisher:CandiceJanco
AcquisitionEditor:ScottBentley
EditorialProjectManager:SusanIkeda
ProductionProjectManager:MohanaNatarajan
CoverDesigner:MarkRogers
TypesetbyMPSLimited,Chennai,India
PREFACE
A traditional approach to mathematics and statistics education involves
direct instruction of techniques and tasks for problem solving, presenting
topics in a sequence, which scaffolds students from simpler problems to
more complex problems, with a focus on algorithms, equations, and
computation (http://en.wikipedia.org/wiki/Traditional_mathematics).
Alternative approaches (e.g., reform mathematics) tend to emphasize con-
ceptual understanding, critical thinking, and problem solving. While
debate between the two approaches continues, comparative studies show
that students using these alternative (reform) approaches tend to perform
similarly on basic skills tests but have enhanced conceptual understanding
and problem solving skills (http://www.nctm.org/news/content.aspx?
id512320). This conceptual approach can also be seen in the recently
adopted Guidelines for Assessment and Instruction in Statistics Education
[GAISE (http://www.amstat.org/education/gaise/)], which emphasize six
recommendations for statistics education: (1) emphasize statistical literacy
and develop statistical thinking, (2) use real data, (3) stress conceptual
understanding, rather than mere knowledge of procedures, (4) foster
active learning in the classroom, (5) use technology for developing con-
ceptual understanding and analyzing data, and (6) use assessments to
improve and evaluate student learning.
While some curricula for Calculus, Introductory statistics and K-12
mathematics and statistics have been reenvisioned from the ground up to
embrace this conceptual approach, few other college level mathematics
and statistics courses have been reenvisioned from the ground up, includ-
ing most financial mathematics (FM) courses or other courses taken as
part of an Actuarial Science program. At best, curricula have been
“updated” by sprinkling in some new language and problems on an exist-
ing course framework. In order to attain the benefit of students who have
a deeper conceptual understanding of topics in FM, while simultaneously
preparing students for actuarial certification exams, we will develop a full
course-length textbook, which embraces GAISE-like guidelines (adopted
for FM topics). We state these guidelines here:
(1) Emphasize FM literacy and develop financial math thinking,
(2) Use real financial data and real situations,
xi
xii Preface
(3) Stress conceptual understanding, rather than mere knowledge of
procedures,
(4) Foster active learning in the classroom,
(5) Use technology for developing conceptual understanding and analyz-
ing data and financial situations, and
(6) Use assessments to improve and evaluate student learning.
To do this we have (1) reordered the typical content covered in a FM
course to embrace a spiral approach to educating students about financial
concepts and financial problem solving, (2) integrated exposition, exam-
ples and explorations, (3) integrated use of technology, and (4) used real
data and real situations throughout. The following sections briefly sum-
marize our approach across the four main distinctives:
1. Spiral approach. Typical FM books have the following
characteristics:
a. Chapters are organized by major financial concept (e.g., Time
value of money, annuities, loans, etc.), and within each chapter,
both the large and small concepts are given equal treatment [e.g.,
compound interest, simple interest, continuous (force) and dis-
count are each given a section] even though some concepts (com-
pound interest) will be used more than the others in later chapters.
b. Students are presented with a laundry list of equations in every
chapter, with little sense of priority. Many of the equations are
similar/related, but students do not always make the connections.
To combat these deficiencies, we propose a syllabus, which gently
introduces the major financial concepts in the most realistic manner
possible. For example, topics like loans and refinancing come early, as
does compound interest, while topics like discount and arithmetically
increasing annuities come later. This allows us to spiral over important
financial concepts throughout the text, instead of relegating them to a
single section of a single chapter in the middle or end of the course.
This approach has the benefit of (1) ensuring students see the most
important financial concepts early in the course, (2) that these basic
concepts are reinforced throughout the semester as the concepts are
revisited while learning some of the minor (related) concepts, and (3)
students having a balanced preparation for Exam FM if that is their
goal (read more on student audience at the end of the preface).
Notably, this approach also allows us to judiciously present equa-
tions to students; the most important, fundamental equations and
learning objectives are presented early in the course when they have
Preface xiii
pedagogical value (i.e., they reinforce fundamental understanding of
the financial concepts). Early presentation (Unit 1) of these key for-
mulas also maximizes student exposure, leading to increased retention
of key concepts. More subtle/nuanced and/or less important equa-
tions are relegated until later in the course (Unit 2), when they can be
introduced on top of a solid foundational understanding of the key
concepts and only for those students for whom such material would
be beneficial.
2. Integration of exposition, examples, and explorations. Every
section includes exposition about the topic of that section, at least one
example to illustrate how to apply the ideas and methods presented,
and at least one exploration that students work through to learn more
about the topic and gain experience with applying the topic. We offer
maximal flexibility for instructors to decide on the order in which
they will present sections and components within sections, and what
they will ask students to do in class vs outside of class. For example,
one instructor could ask students to read exposition and examples out-
side of class and spend class time leading students through explora-
tions. Another instructor might present exposition and examples in
class and ask students to work through explorations outside of class. To
facilitate this flexibility, examples and explorations within a section are written
so that neither depends on the other, allowing the instructor to present
either one first and, furthermore, it is not necessary for students to do
both if the student feels confident in their understanding after either
just the example or just the exploration.
3. Easy-to-use technology integrated throughout. Implementing a
conceptual approach to FM requires effective use of technology.
Rather than asking students to learn to use only a financial calculator,
we have designed easy-to-use Excel spreadsheets that allow students to
explore financial concepts and their behavior to provide a comple-
mentary view to merely seeing an equation, while simultaneously
learning an important skill for business practice and personal finance.
BA-II PLUS calculator instructions are not included because they are
easily and freely available online. Actuarial students should make it a
point to practice with/learn the calculator at a time that they think is
best for their own learning style—this could be “along the way” or it
could be “at the end.” On the other hand, Excel is the “coin of the
realm” when it comes to practical, hands-on problem solving. Excel
spreadsheets are provided for most chapters and students are regularly
xiv Preface
asked to go to the Excel spreadsheets in order to explore concepts and
solve problems. Like the calculator, we’ve limited explicit and detailed
Excel instructions since this depends greatly upon the student’s version
of Excel and operating system. Instead, students and instructors should
find plenty of Excel help online. This “find your own technology
help” should help students greatly as they transition to the “real
world” where such independent research and problem solving is
necessary.
4. Real data from real financial situations. We utilize real data from
real financial situations throughout the book. These situations are
taken from a variety of fields of application, to maximize student
interest and to see the vast application of key financial concepts. We
try to balance examples both from personal finance (e.g., student
loans) and corporate decision-making (e.g., protecting a company’s
assets and liabilities from interest rate changes). We’ve also tried to
point out to students places where we are illustrating a topic that we
think is no longer of great practical value given recent advances in
computation for FM. These cases are primarily in Unit 2 where we
have included some topics primarily because they are on the FM
Actuarial certification exam.
WHY THE CHANGES IN CONTENT SEQUENCING?
The traditional approach to teaching FM starts with simple, unreal-
istic financial situations, with an emphasis on formulas and computation.
Alternatively, we start with practical, conceptual problems for which stu-
dents have a good intuition. For example, while an amortized loan has a
challenging formula, the key concepts (e.g., payback must include interest
1 principal; the larger the payment the quicker the loan is paid off) are
relatively straightforward and intuitive. By starting students in a place
where their intuition is correct, students are then asked to explore FM
concepts using a mix of role playing activities and graphical and numeri-
cal examples. The goal is to build on students natural intuition—building
toward complex, but intuitive, formulas, while downplaying the role of
minor corollary level formulas and formulas used only for the sake of
computation—most of which we have tried to relegate to the second
Preface xv
unit of the course. This approach yields students a deeper, more solid
understanding of FM concepts without sacrificing understanding of for-
mulas and ability to solve problems.
WHAT ABOUT CHANGES IN PEDAGOGY?
In addition to changes to the content of the course, we have also
substantially changed our pedagogical approach from passive (e.g., listen-
ing to lectures) to active learning which engages the full range of stu-
dents’ senses. Each chapter contains a number of explorations for the
students to complete, in addition to example-driven exposition of con-
cepts. These materials allow for a variety of instructor-determined
approaches to content delivery including approaches where examples/
concepts are presented first by the instructor, then explored by the stu-
dent or vice versa.
Student explorations involve a variety of tactile learning experiences:
role-playing examples (you be the bank, I will be the borrower) and using
Excel spreadsheets for explorations. The explorations are flexibly designed
to be completed by students working individually, in small or large
groups, and/or inside or outside of class.
Concepts are introduced using compelling examples explained in an
easy-to-understand format that limits technical jargon and focuses on
conceptual understanding. We have also included key idea boxes and
“Think about it” questions to help students understand what they read,
identify core concepts, and be engaged readers. Overall, we advocate uti-
lizing a small amount of instructor-led interactive lectures and discussions,
but mainly focusing on engaging and strengthening different student
learning processes by way of a variety of active, self-discovery learning
experiences for students.
In addition to the explorations and examples, each chapter contains an
extensive set of exercises: (1) rote exercises, (2) conceptual problems, (3)
application problems based on real financial situations and real data, and
(4) sample problems from the Financial Mathematics Actuarial
Certification exam.
xvi Preface
STUDENT AUDIENCE AND HOW TO UTILIZE THE
MATERIALS
Finally, we note that there are two primary target audiences of this
textbook (1) students who are pursuing a potential career as an actuary
andpreparingtotaketheFMexamand(2)studentswishingtogainasolid
understanding of FM for personal and, possibly, professional purposes
(e.g.,smallbusinessowners;CharteredFinancialAnalysts,CFA;accounts).
Students Hoping to Gain a Solid Understanding of Financial
Mathematics Concepts
For students who do not intend to sit for the FM certification exam, the
entirety of Unit 1 (Chapters 1(cid:1)5) should provide a solid mathematical
foundation on which to operate for future personal and professional
financial decision making. The Unit can be covered in a half to full-
semester long course, depending on the amount of time, depth of cover-
age and amount of time devoted to homework exercises in each chapter.
Importantly, we believe that this material is appropriate regardless of
whether or not a student has taken Calculus, and by relying heavily on
technology (Excel, calculators) in Unit 1, it provides a practical founda-
tion without making the course get bogged down on algebra techniques.
That said, for instructors or courses where a secondaryobjective is algebra
practice, all equations are provided and instructors can opt to restrict
technology usage on exams and certain assignments to allow for students
practice their algebra skills.
Students Hoping to Pass Actuarial Certification Exam FM
For students considering the actuarial profession and planning to sit for
the FM certification exam, we believe this course provides a deeper, con-
ceptual understanding of the concepts than most other curricula available.
By spending time in Unit 1 which, may at times, seem simple for some
of these students, students will be ensured to have a very strong, concep-
tual foundation on which to layer the more nuanced, technical and
detailed concepts which come up in Unit 2. Along the way, students will
gain practice with Excel which is an important “resume builder” for their
future profession—no practicing actuary is actually going to use the BA-
II Plus to do financial calculations in their everyday work! Some
Preface xvii
additional practical advice for these students (and their instructors) to pass
the exam is provided in the section titled “To the student” which follows.
TO ALL STUDENTS
This book is about finance, not mathematics. Mathematics books
tend to focus on what can be done mathematically and let the formula-
tions of mathematical proofs act as the roadmap through the curriculum.
Because this book is about finance, we will let financial concepts be the
roadmap through the curriculum, and not focus on concepts just because
we can do them, but because theyare done and are used in financial prac-
tice by actuaries, financial experts and everyday citizens in daily life. With
this in mind, we will
1. Use notation only when necessary, not everywhere it is possible
2. Focus on formulas that are useful in daily practice and help to rein-
force intuitive understanding, not focusing on niche formulas
3. Start exploration of concepts using a mix of intuition, simple mathe-
matics and spreadsheets (Excel)
By the time you finish Unit 1, which is a good stopping place unless
you plan to go on and take the actuarial certification exam, you should
have a solid mathematical foundation on which to operate for future per-
sonal and professional financial decision making, along with a solid set of
Excel experiences you can use when making decisions about loans, sav-
ings and investments. Many people in the world today have little to no
financial acumen. When you finish Unit 1, you will. Use it wisely!
TO THE ACTUARY STUDENT
We’d love to be able to say that exam FM follows this same set of
goals as those described above. We believe that, for the most part, they
do. However, to make sure you are as prepared as possible to succeed on
exam FM, the latter half of the book “fills in the gaps” and “spirals over”
some topics to make sure we explicitly dealt with some of the nuanced
topics of the actuarial exam. Most curricula throw these topics at students