DISCLAIMER: This publication is intended for EDUCATIONAL purposes only. The information contained
herein is subject to change with no notice, and while a great deal of care has been taken to provide accurate
and current information, UBC, their affiliates, authors, editors and staff (collectively, the "UBC Group") makes
no claims, representations, or warranties as to accuracy, completeness, usefulness or adequacy of any of
the information contained herein. Under no circumstances shall the UBC Group be liable for any losses or
damages whatsoever, whether in contract, tort or otherwise, from the use of, or reliance on, the information
contained herein. Further, the general principles and conclusions presented in this text are subject to local,
provincial, and federal laws and regulations, court cases, and any revisions of the same. This publication is
sold for educational purposes only and is not intended to provide, and does not constitute, legal, accounting,
or other professional advice. Professional advice should be consulted regarding every specific circumstance
before acting on the information presented in these materials.
© Copyright: 2017 by the UBC Real Estate Division, Sauder School of Business, The University of British
Columbia. Printed in Canada. ALL RIGHTS RESERVED. No part of this work covered by the copyright
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©Copyright 2017 by the UBC Real Estate Division
LESSON 1
The Basis of Financial Analysis and Equivalent Interest Rates
Course Bulletins
Remember to check the Course Resources website for course bulletins and note any changes in
your workbook and manual.
Note: Selected readings can be found under Online Readings on your Course Resources website.
Assigned Reading
1.
UBC Real Estate Division. 2017. Foundations of Real Estate Mathematics. Vancouver: UBC Real Estate
Division
Chapter 1: The Basis of Financial Analysis
Chapter 2: Equivalent Interest Rates
Recommended Reading
1.
UBC Real Estate Division. 2017. HP10bII: Introduction to the Calculator and Review of Mortgage
Finance Techniques. Vancouver: UBC Real Estate Division
2.
UBC Real Estate Division. 2017. HP10bII Computer Based Training. Vancouver: UBC Real Estate Division
Learn how to use the HP 10bII calculator by using an interactive training tool. This will show you
the steps on the calculator that are required to solve real estate finance problems.
3.
UBC Real Estate Division. 2017. Math Review Kit. Vancouver: UBC Real Estate Division
4.
UBC Real Estate Division. 2014. CPD: 152 Financial Analysis with Excel. Vancouver: UBC Real Estate
Division
Lesson 1: Introduction to Excel: Orientation and Tools
Lesson 2: Financial Applications of Excel
5.
UBC Real Estate Division. “Introduction to Excel C Screencast Video Series”
A series of 14 videos between 9 and 22 minutes long, providing orientation and instruction on
how to use Excel, found under Software Help on the BUSI 121 Course Resources website.
Sessions 1-6, 9, 10, 14: Orientation, formatting, basic functions
Sessions 12-13: Financial calculations
6.
Dykman, A. “How Savings Accounts Grow from the Magic of Compound Interest”.
www.savingsaccounts.com
An easily understood demonstration of the concept of compound interest using retirement
savings.
7.
Schmidt, R. “What You Should Know About the Time Value of Money”.
www.propertymetrics.com
An in-depth review of the time value of money, helpful for clarifying concepts discussed in the
lesson.
1.1
©Copyright 2017 by the UBC Real Estate Division
Foundations of Real Estate Mathematics
Learning Objectives
Upon completion of this lesson, the student should be able to:
1.
2.
3.
4.
5.
6.
7.
8.
Explain the distinction between simple and compound interest
Convert between nominal and periodic interest rates
Calculate the future and present value of lump sum loans/investments
Calculate the yield of an investment/loan, expressed as a given nominal rate
Calculate the holding period of an investment/loan
Calculate equivalent interest rates using different compounding frequencies
Calculate the effective annual rate from a given nominal rate
Calculate the interest adjustment payment required for early advancement of a mortgage
Instructor’s Comments
The first eight chapters of the course manual cover the mortgage finance aspects of real estate
mathematics. While we will spend a fair amount of time explaining concepts and theories, ultimately
most of the chapters will focus on illustrating solutions to math problems. Much like how a photo can tell
1,000 words, you will learn a lot more from doing the examples than from just reading about them.
Almost all students find the mathematics of mortgage finance challenging; don’t be surprised if you find
you need to read the material more than once. A good strategy:



Read the chapter through at a high level to get a general feel for the concepts covered.
Read it in more detail and do all of the examples presented, calculating each step along with the
illustrations. This will solidify your understanding of the concepts.
Attempt the assignment and refer to the course manual as required.
You will need to use a financial calculator for your exam, so you need to get to know how to use this as
soon as possible. The best way to master a new tool is to work with it. In this course, we will use the
Hewlett Packard (HP) 10bII+ calculator to demonstrate analytical techniques. There are a variety of preprogrammed financial calculators on the market, some of which perform more sophisticated calculations
or have greater programming capacity. Students in this course are welcome to choose other calculators if
they prefer, but it is up to the student to ensure that the alternate calculator will perform all necessary
calculations and to determine its operations.
You will discover that the financial formulas have been programmed into the calculator such that any
financial problem is a function of entering the known variables and computing the value of a single
unknown variable. Since there are six main financial keys, you may simplify this to say that nearly all
financial problems in the course will involve specifying five variables and then calculating the value of
the sixth.
With this in mind, there may actually be a lot less math in this real estate mathematics course than you
might initially think C as the calculator does the heavy lifting, most of your work is really about
“structured problem solving”. Your task is to work through the complications of real estate problems to
figure out what is going on and to specify the values to be entered correctly.
However, remember that the calculator is not some magic conjuring box; it is simply a machine
programmed with simple financial formulas. The calculator provides the algorithms, but you provide the
inputs. And as the computer programming expression says, “GIGO = Garbage In, Garbage Out”. This
powerful computing tool is of no use if you get the inputs wrong.
We cover a lot of fundamental content in the first few mortgage finance chapters. There will be some
new terminology to master as well as the calculator usage. Be sure to read over the material carefully and
practice as much as possible. Be patient with yourself as the material will eventually make sense as the
topics unfold.
1.2
©Copyright 2017 by the UBC Real Estate Division
Lesson 1: The Basis of Financial Analysis and Equivalent Interest Rates
In addition to the calculator, we can also use Microsoft Excel to solve the mathematical concepts
covered in this course. There are considerable resources to assist with learning Excel; however, as with the
calculator, the best way to master the tool is to work with it. Project 1 requires you to use a spreadsheet;
therefore, it is important that you spend time in advance of Project 1 learning how to use this tool. There
are numerous online resources available for using Microsoft Excel. The best way to find a particular topic
is to type exactly what you see into your favourite search engine. The more specific you are, the more
focused the choices will be.
Value Matters: Connecting Theory and Practice
Many readers of this book are studying towards becoming real estate valuation professionals. You
will find that the mathematics covered in this course provide the basis for most of the appraisal
approaches and investment analysis techniques. Reading this book, you may find at times that
the concepts seem too theoretical to be useful in practice. But rest assured, all of the concepts
covered are applicable, so your task as the reader is to approach these topics with an open-mind.
As you proceed through the book, try to step back and thoughtfully consider how the discussion
may apply to your market realities. Talk it over with your classmates in the course discussion
forum and with your colleagues and seek out these links. Hopefully you will find these value
matters boxes helpful in making the connection between theory and practice.
Review and Discussion Questions
1.
Use the internet to find out the interest rates offered on various credit cards. Post your findings on
the BUSI 121 course discussion forum. Convert all the given interest rates to the effective annual rate
and together with the other students determine which credit cards offer the best rates.
2.
Use a search engine to search for terms “compound interest” and “grains of rice” on the internet.
Share your findings with the other students on the BUSI 121 forum. Did you find any interesting
stories demonstrating the power of compound interest?
3.
In Canada, the federal Interest Act requires mortgage interest rates to be quoted either with annual or
semi-annual compounding. Why do you think this provision was adopted by the government?
4.
The industry standard in Canada has become to quote mortgage interest rates with semi-annual
compounding. Why do you think the mortgage lenders adopted semi-annual rather than annual
compounding?
5.
An investor requires the use of borrowed funds for a brief period of time. The investor has a choice of
either borrowing from the neighbourhood lender at a rate of 5% per annum, compounded daily, or
using his credit card to obtain a cash advance on which the interest rate is 5.25% per annum,
compounded monthly.
Which borrowing rate is less costly for the borrower and what are the equivalent rates compounded
annually?
1.3
©Copyright 2017 by the UBC Real Estate Division
Foundations of Real Estate Mathematics
6.
A local investor has purchased a condominium in Maui for $300,000. The investor is now attempting
to arrange a mortgage in the amount of $225,000 and has contacted two sources of funds: a Canadian
lender who has agreed to supply the necessary funds at an interest rate of 6% per annum,
compounded semi-annually, and an American lender who has quoted an interest rate of 6.1% per
annum, compounded monthly.
Assist the investor by finding equivalent interest rates for each source of loans, expressed as
equivalent annual, semi-annual, and monthly rates.
7.
Borrowers are offered a choice of six sources of funds. These six lenders have announced their rates
as follows:
Lender
Nominal Rate
Frequency of
Compounding
A
B
C
D
E
F
13.174622%
13.541565%
13.319794%
14.000000%
13.119348%
13.105178%
12
2
4
1
52
365
Calculate the effective annual rate for each lender.
8.
A lending institute, recently embarrassed by a court case involving the wording of their mortgage
contract, has decided that every loan must include a statement reporting the effective annual interest
rate, the nominal rate with semi-annual compounding, and the equivalent monthly rate. By
including all three expressions of the interest rate, the lender hopes to avoid any misunderstanding
as to the correct rate.
Assist the lending institution by calculating the effective annual rate and equivalent monthly rates
for their Summary Sheet.
Summary Sheet
9.
Nominal Rate with SemiAnnual Compounding (j2)
Effective Rate (ia)
Monthly
Equivalent (imo)
A
10.000%
=?
=?
B
10.125%
=?
=?
C
10.250%
=?
=?
D
10.375%
=?
=?
E
10.500%
=?
=?
F
10.625%
=?
=?
G
10.750%
=?
=?
Why do lenders charge an interest adjustment when funds are advanced prior to the beginning of the
first payment period?
10. Your friend is having difficulty understanding the concept of the time value of money. How might
you help explain in language that is easily understood?
1.4
©Copyright 2017 by the UBC Real Estate Division
Lesson 1: The Basis of Financial Analysis and Equivalent Interest Rates
ASSIGNMENT 1
CHAPTER 1: The Basis of Financial Analysis
CHAPTER 2: Equivalent Interest Rates
SUBMITTING MULTIPLE CHOICE ASSIGNMENTS
Answers to multiple choice questions should be submitted using the Real Estate Division’s website
www.realestate.ubc.ca. See “How to Submit Multiple Choice Assignments” in the Real Estate Division
Student Handbook for more information.
VIEWING ASSIGNMENT ANSWER GUIDES
As soon as your assignment has been submitted on the Real Estate Division’s website, you can immediately
download the answer guide. See your Student Handbook or visit your Course Resources website for more
information on how to download assignment answer guides.
Marks: 1 mark per question.
1.
Jessie has borrowed $50,000 from Tina and will repay the amount in 6 months. Tina is charging an
interest rate of 6% per annum and has specified that simple interest will be used. What is the total
amount payable at the end of the term?
(1)
(2)
(3)
(4)
2.
$55,000
$3,000
$50,000
$51,500
A mortgage investor has identified a short-term investment opportunity offering a simple interest
rate of 3.5% per annum. For each $75,000 invested, the investor will receive $75,143.84 at the end of
the investment term. Calculate the required duration of the term of the investment (in days).
(1)
(2)
(3)
(4)
Approximately 20 days
Approximately 30 days
Approximately 40 days
Approximately 10 days
THE NEXT (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Michael has decided to invest his money in a bakery shop in Kitsilano. He invested $500,000 and he
expects to earn 9% per annum, compounded semi-annually, for 3 years. He will receive no cash flows
during this period.
Assignment 1 continued on the next page
1.5
©Copyright 2017 by the UBC Real Estate Division
Foundations of Real Estate Mathematics
3.
How much will Michael receive at the end of the term on this interest accrual loan, rounded to the
nearest dollar?
(1)
(2)
(3)
(4)
4.
$500,000
$695,508
$847,940
$651,130
If Michael receives $700,000 at the end of 3 years on this interest accrual loan, what rate of interest,
expressed as an annual rate with monthly compounding will he earn?
(1)
(2)
(3)
(4)
9.000000%
10.254277%
11.268319%
8.241254%
THE NEXT (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Duncan Lane recently inherited a small fortune. He has several investment options and is analyzing them
to determine which offers the highest return. His options are as follows:
5.
Amount to Invest
(Today)
Term (Years)
Amount Received at
the end of the term
A
$475,000
1
$600,000
B
$525,000
2
$572,000
C
$450,000
1
$575,000
D
$500,000
2
$626,000
Which one of the following statements is TRUE?
(1)
(2)
(3)
(4)
6.
Investment
The effective annual rate (j1) earned on Investment A is 24.780592%.
The semi-annual rate (isa) earned on Investment B is 4.333313%.
The nominal rate per annum, compounded daily (j365) on Investment C is 22.043371%.
None of the above statements is true; all the statements are false.
Which one of Duncan=s investment options has the highest return?
(1)
(2)
(3)
(4)
A
B
C
D
Assignment 1 continued on the next page
1.6
©Copyright 2017 by the UBC Real Estate Division
Lesson 1: The Basis of Financial Analysis and Equivalent Interest Rates
7.
A loan is written at j2 = 9.5% where payments are to be made at the first of each month and funds are
advanced on June 14th. The first payment is due on August 1st, after the first full calendar month
(July) of the contract. What is the interest adjustment period?
(1)
(2)
(3)
(4)
8.
17 days
18 days
19 days
48 days
The daily periodic rate (id) that is equivalent to 8.5% per annum, compounded quarterly, is:
(1)
(2)
(3)
(4)
8.411912%
0.0432892%
0.0230463%
8.589943%
THE NEXT THREE (3) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Not wanting to invest in RRSPs, Sally and some of her friends each placed $5,000 in long-term savings
accounts at their individual banks. In the table below, the nominal rates and future amounts to be
accumulated in the accounts are given. For questions 9 to 11, calculate how long the funds must be
invested in order to accumulate at least the desired amounts.
9.
Question
Amount Invested
Nominal Rate
Amount
Accumulated
9
$5,000
j1 = 10.5%
$6,105.12
10
$5,000
j2 = 10.5%
$9,724.27
11
$5,000
j4 = 10.75%
$8,498.09
The minimum number of annual compounding periods required to accumulate at least $6,105.12 is:
(1)
(2)
(3)
(4)
1
2
3
4
10. The minimum number of semi-annual compounding periods required to accumulate at least
$9,724.27 is:
(1)
(2)
(3)
(4)
6
7
12
13
Assignment 1 continued on the next page
1.7
©Copyright 2017 by the UBC Real Estate Division
Foundations of Real Estate Mathematics
11. The minimum number of years required to accumulate at least $8,498.09 is:
(1)
(2)
(3)
(4)
6
5
18
20
12. John is considering investing his poker winnings of $3,500 in some bonds bearing an interest rate of
10.5% per annum, compounded daily (j365). His banker has offered him the choice of switching to
other bonds with the following interest rates. Based on interest rates only, which option should John
choose?
(1)
(2)
(3)
(4)
j52 = 11%
j2 = 9%
j4 = 10%
Remain at j365 = 10.5%
13. A land investor owes $450,000 on a loan for a piece of lakefront property. If the interest accrual loan
was financed with a 4-year term at 7.5% per annum, compounded quarterly, what is the original
amount the land investor received?
(1)
(2)
(3)
(4)
$410,230.49
$334,295.53
$605,751.45
$515,664.57
14. The nominal rate per annum, compounded monthly that is equivalent to 5% per annum,
compounded weekly is:
(1)
(2)
(3)
(4)
5.800000%
5.008019%
4.997940%
5.124584%
THE NEXT TWO (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Erica takes out a $60,000 loan at 5% per annum, compounded monthly. The funds are to be advanced on
July 11th. The first monthly payment is due September 1st. Assume it is not a leap year.
15. What is the interest adjustment payment Erica will owe on August 1st if she is advanced the full loan
amount on July 11th?
(1)
(2)
(3)
(4)
$251.50
$172.49
$246.43
$164.60
Assignment 1 continued on the next page
1.8
©Copyright 2017 by the UBC Real Estate Division
Lesson 1: The Basis of Financial Analysis and Equivalent Interest Rates
16. What is the interest adjustment owing on August 1st if Erica is advanced less than $60,000 on July 11th
such that the interest adjustment plus the amount advanced equals $60,000.
(1)
(2)
(3)
(4)
$174.49
$167.21
$180.73
$172.00
17. The interest rate per month (imo) that is equivalent to a daily rate of 0.022456% (ida) is:
(1)
(2)
(3)
(4)
0.269472%
0.552412%
0.685297%
0.895214%
18. Joseph wants to take a trip in four years. If he requires $15,000 at the end of 4 years and earns a rate
of 3% per annum, compounded annually on his investment, how much will he have to invest today?
Joseph will not make any additional payments during the four-year term.
(1)
(2)
(3)
(4)
$13,327.31
$12,225.47
$14,332.42
$11,987.41
19. Ryan recently won $7,777 in a provincial lottery. After buying a new TV, Ryan invested his remaining
$7,000 in Mike’s Motors bonds bearing an interest rate of 2.5% per annum, compounded weekly. His
friend, a business student, has suggested that Ryan instead invest his money in one of the following:



Tyler’s Taters bond at a rate of j1 = 2.5%
Raf’s Roots bond at a rate of j12 = 2.5%
Lee’s Lemonade bond at a rate of j4 = 2.5%
Ryan should choose:
(1)
(2)
(3)
(4)
Tyler’s Taters.
Raf’s Roots.
Lee’s Lemonade.
to remain invested with Mike’s Motors.
20. Which of the following statements is TRUE?
(1) The rate per annum, compounded semi-annually, that is equivalent to 6.75% per annum,
compounded monthly, is 6.656992%.
(2) The quarterly periodic rate of interest (iq) that is equivalent to 7.5% per annum, compounded
semi-annually, is 1.775548%.
(3) The daily periodic rate (id) that is equivalent to 4% per annum, compounded semi-annually, is
0.0108513%.
(4) None of the above
___
20 Total Marks
End of Assignment 1
1.9
©Copyright 2017 by the UBC Real Estate Division
Foundations of Real Estate Mathematics
Planning Ahead
Project 1 requires you to submit written answers to questions based on Lessons 1-6. You should
read ahead to Project 1 to see what is expected. The first question requires you to research the
current interest rates for mortgages and a risk-free security and present your findings graphically.
You should be able to find this information relatively easily using your local paper, the internet, or
by talking to somebody in your local bank. Familiarize yourself with the Excel “Chart Wizard” now in
order to be able to draw the yield curve for the interest rates – for Excel help, refer to the list of
supporting documents in the Recommended Readings.
1.10
©Copyright 2017 by the UBC Real Estate Division