FINS5511
Corporate Finance
Dr. Ian Kwan
i.kwan@unsw.edu.au
Acknowledgement of Country
I would like to show my respects and acknowledge
the Bedegal people who are the Traditional
Custodians of the Land, of Elders past, present,
and emerging from which these meetings originate.
Week 01
Introduction & Financial Mathematics
FINS5511 Corporate Finance
Dr. Ian Kwan
i.kwan@unsw.edu.au
Text: Berk and De Marzo (2020) Chapters 1, 3, 4, and 5 – relevant parts as indicated
41. Introduction: The Corporate Finance Story
2. Assessments
3. The Corporation & Financial Manager
4. Financial Mathematics
5. Financial Mathematics & Excel Functions
6. Solutions to Practice Problems
Week 01 Contents of Lecture
1. Introduction: The Corporate Finance Story
6In finance, there are two important questions:
11
ൌ ෍ ௜1 ൅ ௜ െ ଴௡
௜ୀଵ
TO INVEST OR NOT TO INVEST? That is the question!
How do you make money?
BUY LOWSELL HIGH How are future cash flows determined?
Are future CF certain or uncertain?
If uncertain, how does NPV handle it?
How is discount rate determined?
What factors change it?
What assumptions are made?
If NPV > 0, then INVEST
These questions make … The Story of Corporate Finance!
Our two main questions ….
Our answers are ….
12
How the Story of Corporate Finance will be explained:
Cash
Flows
Discount
rate
Financial Mathematics
Cash Flow Estimation
and Forecasting
Investment
decision rules
Risk and Return
Cost of capital
Capital structure
Week 02
Week 01
Weeks 04 and 05
Week 03
Week 06
Week 07
Week 08
Types
of CF
Debt
Equity
ൌ ෍ ௜1 ൅ ௜ െ ଴௡
௜ୀଵ
Important: As you take this course, clarify how to connect the different parts of the story!
Week 09:
Payout policy
Week 10:
Team Assignment
Presentations
13
Corporate Finance Story is explained at different levels
Cash
Flows
Discount
rate
(Risk & Return)
Project level CF
ൌ ෍ ௜1 ൅ ௜ െ ଴௡
௜ୀଵ
Security level CF
Project level
discount rate
Portfolio level
discount rate
CF that affect only a small part of the
company E.g. replacing production
machines in a manufacturing plant
CF that affect a big part of the company
E.g. introducing new product that
changes the company strategy
Rate depends on the risk of the
project compared with the average
risk of the company
Rate depends on the risk of the company
compared with other companies that
have the same types of risks,
i.e. portfolio view
Important: As you take this course, be aware of which level of the story you are referring to!
2. Assessments
15
Assessments
Assessments Total
100%
Individual
70%
Group
30%
Online Assessments 60%
• Homework Assignments (W6 and W11) 10%
• Mid-term Quiz (W7) 25%
• Final Examination (Exam period) 25%
Team Assignment 40%
• Group component (W5 and W9/10) 30%
• Individual component (W11) 10%
See Course Outline
and the detailed
Assessment Guide
in Moodle
16
Assessments aimed at Career Development
The Assessments of FINS5511
aim to:
• Consolidate
• Integrate
• Deepen & Widen
• Make more practical
• Give you a complete picture
of Finance, especially
Corporate Finance
• So you can major in Finance.
Review the main concepts
Lots of practical examples
Use Excel & FactSet
Work in teams
Team Assignment Stock pitch
How?
FINS5512 Financial Markets and Institutions
FINS5513 Investments and Portfolio Selection
FINS5514 Capital Budgeting & Financial Decisions
FINS5516 International Corporate Finance
FINS5535 Derivatives and Risk Management
FINS5538 Takeovers, Restructuring …etc.
Finance Career Oriented  Finance Career Showcase
17
Assessments and Learning Outcomes
CLO Description Assessments
CLO1 Analyse using investment decision rules the financial value of
corporate securities and projects
Open Exercises, Mid-term, Final
Exam
CLO2 Evaluate the fundamental value drivers of corporate securities
and projects in terms of business nature, risk and return trade-
offs, capital structure, cost of capital, and payout policies
Open Exercises, Mid-term, Final
Exam
CLO3 Construct Excel financial models to forecast cash flows for
recommendations for investment proposals
Open Exercises, Mid-term, Final
Exam, Team Assignment
CLO4 Collaborate in teams to solve complex problems and
communicate investment decisions
Team Assignment
CLO5 Analyse individually and in teams the investment
recommendations and evaluate their worth
Team Assignment
Course Learning Outcomes
18
Team Assignment - Stock Pitch BUY/SELL Recommendation
Working in
Teams of 4-5
students
Choose a
publicly listed
stock
Analyse the stock
• Financials
• Strategy
• Competition
• Industry
• Technology, etc
Value the stock
using all the
financial tools and
models
Recommend to
investors to BUY,
SELL, or HOLD the
stock
Make a video of your
recommendation
Watch the videos of
other teams’
recommendations
and decide
whether you (as an
investor) should
buy or sell their
stock.
The top three teams will be invited to present to a Portfolio Manager at Aware Super.
Lecture time dedicated to help you
19
UNSW Finance Career Showcase & Team Assignment
UNSW Finance Career Showcase Stock Pitch Competition
More information about the Showcase found in a dedicated Moodle site:
https://moodle.telt.unsw.edu.au/course/view.php?id=66135
Sign up with Self-Enrolment Key: Showcase
The main idea of the Showcase is: TO CONNECT UNSW STUDENTS WITH THE FINANCE INDUSTRY
Working in Teams
of 3-4 students
Analyse the stock
of a public firm
• Financials
• Strategy
• Competition
• Industry
• Technology etc
Value the stock
using financial
tools and models
Teams present their
recommendation to
a panel of finance
industry investment
professionals
Get judges’ feedback
and comments on
how to improve
Recommend to
BUY or SELL the
stock
Network with judges
Invited to submit CV
for potential
internships, jobs, etc.
Win cash prizes
Invited to exclusive
finance career
seminars
Main outcomes
20
Team Assignment >> Showcase >> Int’l Case Competitions
FINS2615 Team
Assignment
• Basic preparation in stock pitching
• Develop fundamentals in finance
• Develop team skills
Finance Career
Showcase
• Undergo competitive selection process
• Get connected with the finance industry
• Kick-start finance career
International
Case Comps
• I choose the best students to represent UNSW
• They compete against all the global major universities
• Demo skills to investment banks and consulting firms
The Team Assignment prepares
you for the UNSW Finance Career
Showcase.
Getting into the Showcase is a
competitive selection process.
From the best of the Showcase, I
choose candidates to train for
international finance case
competitions.
How does all this build your career in finance?
Poll 1
3. The Corporation & Financial Manager
22
Four Types of Firms
1. Sole Proprietorships
2. Partnerships
3. Limited liability companies
4. Corporations
Each business type has its strengths and weaknesses
23
Sole Proprietorships
Features
• Owned by one person who keeps all the profits
• Owner has unlimited the liability
• Most businesses in Australia are sole
proprietorships
• Most businesses start out as sole proprietorships
• The owner can hire employees
Examples
• Solo trades people: electricians, plumbers,
carpenters,
• Solo professionals: accountants, doctors,
solicitors, barristers, surgeons
• Very small businesses
Strengths
• Simple and easy to start
• Little regulation
• Profits are taxed as personal income
• Owner keeps all the profit
Weaknesses
• Life of business limited by life of the owner
• Difficulty raising capital/ borrowing money (limited asset collateral)
• Owner has unlimited liability (if something goes wrong with
business, owner can lose personal wealth)
• Transfer of ownership not possible except by selling entire business
24
Partnerships
Features
• Same as sole proprietorship except owned by
more than one person
• All partners have unlimited liability
• Partnership ends on the death or withdrawal of
any partner unless there are buyout provisions
Examples
• Groups of trades people: electricians, plumbers,
carpenters
• Groups of professionals: accountants, doctors,
solicitors, barristers, surgeons
• Medium-sized businesses (e.g., 50+ employees)
Strengths
• Simplicity
• Good for reputation-based firms, e.g., group of doctors,
accountants, solicitors, whose business depends on the
professional prestige of the partners
• Partners share the profit according to their contribution
• Profits are taxed as personal income for partners,
i.e., partners pay personal tax on the share of the
partnership profits received
Weaknesses
• Unlimited liability of partners – partners must
protect their reputation
25
Limited Liability Companies
Features
• LLC exists in the USA, Europe, Latin America. In
these countries, it is a partnership whose
partners’ liability is limited.
• LLC does not exist in Australia. The closest to an
LLC is a Proprietary Limited (PTY LTD) company.
ASIC Definition of Company and PTY LTD
A company is an entity that has a separate legal existence from its
owners. The owners of the company are known as members or
shareholders. Its legal status gives a company the same rights as a
natural person, which means that a company can incur debt, sue and be
sued.
Small business owners often use a type of company structure called a
proprietary limited company. Generally, members will not be personally
liable for the debts of the company in their capacity as a member.
Income generated by a company attracts a company tax rate.
ASIC: Australian Securities & Investments Commission
https://asic.gov.au/for-business/small-business/starting-a-small-
business-company/
26
Corporations
Features
• A corporation is a legal entity with similar rights as a person: can
own property, borrow money, pay taxes, sue and be sued, etc.
• Ownership is divided into shares, which can be traded (buy/sell)
without dissolving the firm. Shares are also called equity.
• Shareholders are not liable for the debt obligations of the
corporation. Their liability is limited by the worth of their shares.
• The managers of the corporation can be different from the owners.
Examples
• Publicly listed companies. i.e.,
traded on the stock market
• Large private companies with many
shareholders, e.g., Dell Computers
Strengths
• Owners have limited liability
• Separation of ownership and management
• Easy transfer of ownership, and transferring
ownership does not dissolve the corporation
• In principle, a firm has unlimited life
Weaknesses
• Subject to double taxation:
• Profits of firm incur corporate tax
• Profits distributed to shareholders incur personal tax
• Separation of ownership and management lead to agency
problems or conflicts of interest (discussed in Week 8 & 9)
27
Australian corporations & tax implications
• “Classical” tax system
– Corporate profits taxed at corporate tax rate
– What is leftover is distributed to
shareholders, who pay personal tax on this
income
– This is a “double taxation system”
• Australian “imputation” tax system
– In July 1987, Australia effectively abandoned
the classical “double taxation” system by
introducing the imputation tax system.
– Shareholders are now treated as partners in
the firm. It means the firm’s profits are
treated as personal income for shareholders,
who only pay personal tax.
– Corporate taxation is effectively collecting
taxes on behalf of individual shareholders.
• Example of Classical Tax System:
– Corporate Profit: $1000
– Corporate tax 30%: - $300
– Distribute to shareholder: $700
– Shareholder personal tax 45%: - $315
– Shareholder left with: $385
• Example of Imputation Tax System:
– Corporate Profit: $1000
– Corporate tax 30%: - $300
– Distribute to shareholder: $700
– Shareholder franking credits: $300
– Grossed up personal income: $1000
– Shareholder personal tax 45%: - $450
– Shareholder left with: $550
Effective tax
rate = 61.5%
Effective tax
rate = 45%
Added back as
franking credits
• Students are not expected to calculate imputation tax.
The above example is only for illustration.
• Students are expected to know the difference between the
classical tax and Australian Imputation Tax systems
28
Role of the financial manager
• The financial manager is the Chief Financial
Officer (CFO) who oversees the roles of the
Treasurer and Controller
• The role of Treasurer is the “traditional finance
function”, which includes:
– Capital budgeting
– Risk management
– Credit management
• The role of Controller is to maintain internal
control, including:
– Accounting function
– Tax
– Information systems
Source: BDeM (2020), Figure 1.2
29
Role of the financial manager
• Q: What is overall objective of the financial manager?
• A: To maximize the value of the firm!
• To maximize firm value, the financial manager makes three types of decisions:
1. Investment decisions
2. Financing decisions
3. Cash management decisions
• These take into account other factors, including:
– Finding the best deal from suppliers
– Operating lawfully and ethically
– Treating workers justly and fairly
– Being environmentally friendly
– Acting in the interest of shareholders
Q. Which is the most important of the
three decisions? And why?
30
Financial manager – investment decisions
• INVESTMENT DECISIONS ARE THE MOST IMPORTANT DECISIONS. Why?
• Investment decisions are the reason why the firm exists – they define who the firm is!
– Why does Apple exist? Alibaba? Tencent?
• Investment decisions are where firm value is created or destroyed because the decision to:
– Accept good projects commits, for example, the firm’s money and resources
– Reject bad projects prevents, for example, bad decisions from having long-term effects
– Discontinue bad projects shields, for example, the firm from further losses or incompatibility with overall strategy, etc.
• Investment decisions involve:
– Large cash and resource commitments
– Long-term resource commitments
– Irreversible decisions
– High risk of failure
31
Financial manager – financing decisions
• After making the investment decision, projects need to be financed.
Where does the money come from?
• Financing comes from three sources:
1. Profit from operations: retaining the profits of your company to fund new investments, i.e. retained earnings
2. Issuing debt: borrowing funds from lenders, e.g. bank loans, corporate bonds, debentures
3. Issuing equity: selling ownership of the firm to investors, e.g. selling shares
• Each source of finance has advantages and disadvantages
(See Weeks 2-5)
32
Financial manager – Cash Management Decisions
• The financial manager must ensure the firm has enough cash on hand to pay its
financial obligations as they come due.
• If the firm defaults or misses its debt repayment obligations, then:
– Creditors can sue for payment
– Creditors could seize collateral assets and sell them to recover capital
– Firm can be sued and forced into bankruptcy
– Suppliers who hear about it may stop doing business with the firm (opportunity cost)
• Examples:
– A loan interest payment is due next Monday  Is there enough money to pay it?
– A customer will not pay for 3 months  How to manage in the meantime?
– The firm has extra cash / idle resources Where can it be invested for the short-term?
4. Financial Mathematics
34
Main formula
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
35
Basic Concept: Time Value of Money (TVM)
Which would you prefer? Why?
A: $1 million today, or
B: $1 million 1 year from today?
The earlier we receive a cash flow, the
more valuable it is compared to receiving
the same amount later: “A dollar today is
worth more than a dollar in the future.”
The difference in value between $1m
today and $1m in the future is called the
Time Value of Money (TVM).
Because of TVM, A > B, even though we
receive the same amount of cash.
Which would you prefer? Why?
A: $1 million today, or
C: $100K/month for 10 months
In both A and C, the same amount
of cash is received.
But in C, we only receive a small
amount today compared with all
of A today.
So because of TVM, A > C.
A is received earlier than C.
Which would you prefer: B or C?
B: $1 million 10 months from today?
C: $100K/month for 10 months
In both B and C, the same
amount of cash is received.
Because of TVM, C > B.
C is received earlier than B.
Conclusion: A > C > B
The earlier we receive the same cash flows, the more valuable they are.
36
TVM: Interest & Interest Rate Factor (1+r%)
0 Year
$1
1
$1.10
Interest rate = 10%
Interest rate factor converts actual cash flow
received at fixed time into values at different times
$1 $1 + $1*10%
= $1(1+10%)
= $1.10
0 Year
$1
1
Interest rate = r%
଴
$1 + $1*r% = $1(1+r%)
In general, “take a cash flow and
multiply by an interest rate factor”
ଵ ൌ ଴ ൅ ଴. % ൌ ଴ሺ1 ൅ %ሻ
ଵ ൌ ଴ሺ1 ൅ %ሻ In words: Cash flow at end of period equals
cash flow at start of period multiplied
by the interest rate factor (1+r%)
Principal + Interest Principal + Interest
Conclusion:
Principal Principal
conversion
conversion conversion
conversion
ଵ is the cash flow at
the end of period 1
଴ is the starting
cash flow at t=0, or
start of period 1.
Interest rate factor
Interest rate factor
Interest rate factor
37
TVM: Compound interest valuation
$100Compound interest
valuation method
௡ ൌ ଴ 1 ൅ ௡
0
1
2
3
n
…
.
1 ൅ 10% ൌ .
. 1 ൅ 10% ൌ 100 1.1 ଵ 1.1 ଵ ൌ ሺ.ሻ
. 1 ൅ 10% ൌ 100 1.1 ଶ 1.1 ଵ ൌ ሺ.ሻ
. ି 1 ൅ 10% ൌ 100 1.1 ௡ିଵ 1.1 ଵ ൌ .
Multiply original principal by interest
rate factor (1+r%)
Multiply new principal by interest
rate factor (1+r%)
Multiply new principal by interest
rate factor (1+r%)
Interest rate
ൌ 10%
Multiply new principal by interest
rate factor (1+r%)
…
.
“New principal” includes original principal plus
accumulated interest of previous period.
଴
ଵ
ଶ
ଷ
௡
“New principal” includes original principal plus
accumulated interest of previous 2 periods.
“New principal” includes original principal plus
accumulated interest of previous 3 periods.
38
TVM: One cash flow has multiple values
0 1 2 3
$100 $100 1 ൅ 10% ଵ
ൌ 110
Compound interest (CI)
valuation method
௡ ൌ ଴ 1 ൅ ௡ n
….
$100 1 ൅ 10% ଶ
ൌ 121 $100 1 ൅ 10% ଷൌ 133.1 $100 1 ൅ 10% ௡
…. has multiple values
The same diagram as previous slide, drawn horizontally:
Interest rate factor 1 ൅ ௡ “converts
cash flows to different values over time”
One cash flow…
Distinguish between CASH FLOWS versus VALUES
• They are REAL or ACTUAL
• Paid at a certain moment in time
• Cannot be easily changed
• They are THEORETICAL or CALCULATED
• Calculated for any moment in time
• Depend on changing market conditions
39
Single Cash Flow: compounding vs. discounting
$100 1൅ 10% ଵ
ൌ 110 $100 1 ൅ 10% ଶൌ 121 $100 1 ൅ 10% ଷൌ 133.1
A single cash flow of $100 is received at t=3. What are the values at other points in time?
$100 1൅ 10% ି
ൌ 82.6446 $100 1൅ 10% ିൌ 90.9090
Compound
once into the
future
Compound
twice into the
future
Compound
three times
into the future
Discount
once back to
the present
Discount
twice back to
the present
Discount three
times back to
the present
$100 1൅ 10% ି
ൌ 75.1315
“Discounting” means “moving back to the
present”: negative n exponent
଴ ൌ ௡ 1 ൅ ି௡ “Compounding” means “moving forward to the future”: positive n exponent௡ ൌ ଴ 1 ൅ ௡
….….
3 4 5 6
$100
n….0 1 2-1
This is the language used to express the financial thinking of this subject!!
40
What’s the difference between
Present Value & Future Value?
Short answer: NOTHING! Same formula calculated differently.
Long answer:
• We always start with a REAL CASH FLOW at some point in time.
• If we move the cash flow…
• …forward, then we are finding the future value – “future” means forward relative to the time of the cash flow.
• …backward, then we are finding the present value – “present” means backward relative to the time of the cash flow.
• Each REAL CASH FLOW has many different values depending on the time before or after the cash flow.
• Since the values are calculated by the same formula, there is no conceptual difference between PV and FV.
ൌ ௡ ൌ ଴ 1 ൅ ௡ ൌ ଴ ൌ ௡ 1 ൅ ି௡
When -n < 0: moving CF backwards
When n > 0: moving CF forwards
41
Example 1: Present/ Future Value of a cash flow
Axo Corp. expects to receive $100M payout in 4 years’ time. If the interest rate
is 5% and does not change, what is the value of the payout if received:
(a) …now?
(b) …next year?
(c) …6 years from now?
3 4 5 6
$100
n….0 1 2-1
(a) 100M 1 ൅ 5% ିସ ൌ 82.27M
(b) 100M 1 ൅ 5% ିଷ ൌ 86.38M
(c) 100M 1 ൅ 5% ାଶ ൌ 110.25MACTUAL CF
-4 relative to when
the CF is expected.
This is present value
(wrt actual CF) -3 relative to when
the CF is expected.
This is present value
(wrt actual CF)
+2 relative to when
the CF is expected.
This is future value
(wrt actual CF)
CF expected
at t=4
means “with respect to”
4 years awayNow Next year 6 years from now
42
Time, compounds, periods, rate per period
Definitions
Time of deal = T years
No. of Periods in one year = m periods/year or m compounds/year
No. of Periods of deal = n total periods or n total compounds
Compounding frequency = annual, semi-annual, monthly, daily,… etc.
= m=1, m=2, m=12, m=365
Examples
ൌ ∗
, ൌ

Deal length
T years
Compound
frequency
m compound/year or
periods/ year
ൌ ∗
total periods
APR
(rate given)
Rate/period
= r = APR/m
3 Annual m = 1 3 * 1 = 3 10% 10%/1 = 10%
3.5 Semi-annual m = 2 3.5*2 = 7 12% 12%/2 = 6%
4.75 Monthly m = 12 4.75*12 = 57 15% 15%/12
So far time periods were “years”. But “time periods” can be anything … months, days, hours, seconds!
Interest rates are not always a “rate per year”, but can be “rate per any period”, e.g., rate/month, rate/day, rate/hour, etc.
Annual Percentage Rate = APR, interest rate/year (usually the given rate)
Rate per period = r = APR/m, interest rate/period or rate/compound
Number of periods in the deal
Rate per period
43
APR=10%, m=1, r = 10%/1;
Extended period n = (1 + 3/12)*1 = 1.25
final amt = 1000 1 ൅ ଴.଴ଽ
ସ
ଵ଴ 1 ൅ ଴.ଵ଴
ଵ
ଵ.ଶହ
ൌ 1249.203426 1 ൅ ଴.ଵ଴
ଵ
ଵ.ଶହ
ൌ 1441.19316
Example 2: Non-annual compounding
You invest $1000:
• For 2.5 years at 9% compounded quarterly in the initial period
• And reinvest for 1 year and 3 months at 10% per year in the extended period
How much do you have at the end?
APR=9%, m=4, r = 9%/4;
Initial period n = 2.5*4 = 10; initial amt = 1000 1 ൅ ଴.଴ଽ
ସ
ଵ଴
ൌ 1249.203426
௡ ൌ ଴ 1 ൅ ௡଴ ൌ ௡ 1 ൅ ି௡
Important! in all the formula is
rate per period! ൌ ஺௉ோ
௠ ൌ ∗ ൌ 2.5 ∗ 4 ൌ 10
Think:
“Take initial amount, reinvest
it for 1.25 years at a new rate”
ൌ ∗
, ൌ

Why m=1? Assume m=1 unless the wording implies something else
Why m=4? “Compounded quarterly”
Why 3/12? 3 months / 12 months in a year.
Note: It is okay to have decimal exponents!
Initial period
Extended period
APR is usually the given rate!
Why write ଴.ଵ଴
ଵ
? Purely to illustrate my
financial thinking! Rate/Period = ஺௉ோ
௠
.
44
Practice 1: Compound interest
1. Jennifer deposits $1000 in a bank account that earns
compound interest. How much would she have after 10 years
(a) at 10% compounded semi-annually?
(b) at 10% compounded daily?
Soln
(a) APR=…….. m=…… r=………….. n=………
(b) APR=…….. m=…… r=………….. n=………
௡ ൌ ଴ 1 ൅ ௡଴ ൌ ௡ 1 ൅ ି௡(Two practice problems: 1 and 2)
45
Practice 2: Moving single cash flows
Rabbo Corp. expects its customer to pay $83 million in 6 months from now, but it needs the money now to pay other expenses.
a) If the interest rate offered by its bank is 5.4%, what amount could it borrow now?
b) If the customer delays payment to 9 months from now, what would Rabbo receive if it charged a penalty interest rate of 7.5%?
(a) Find Present Value
(b) Find Future Value
Soln
1.5 years0 0.5 1
1.5 years0 0.5 1
$83M
$83M
௡ ൌ ଴ 1 ൅ ௡଴ ൌ ௡ 1 ൅ ି௡
APR = …….%, m=……., r=…………
APR = …….%, m=……., r=…………
46
Where are we?
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
Done
47
Combine multiple CF into single present value
Can I just add these three cash flows?
i.e. $100 + $200 + $300 = $600?
ANSWER: NO! Each cash flow is at a
different time. Different TVM.
How can we add them together? ANSWER: Move to the same point in time.
3 4
$300
0 1 2
$100 $200
3 40 1 2
$100
3 40 1 2
$200
3 4
$300
0 1 2
100 1.1 ିଵ
Interest rate ൌ 10%
200 1.1 ିଶ
300 1.1 ିଷ
1) Split into 3 single cash flows
2) Move each CF to the same point in time.
We say “discount back to present”.
3) Now we can add them together:
଴ ൌ 100 1.1 ିଵ ൅ 200 1.1 ିଶ ൅ 300 1.1 ିଷ ൌ 481.5928
481.59
Conclusion: We can only add CFs with the same TVM
48
Combine multiple CF into single value and move
Idea: After combining multiple cash flows into a single value, we can then move the
value to wherever we want.
3 4
$300
0 1 2
$100 $200
3 40 1 2481.5928
଴ ൌ 100 1.1 ିଵ ൅ 200 1.1 ିଶ ൅ 300 1.1 ିଷ ൌ 481.5928
481.59281. We combine multiple cash
flows into a single value
2. Move it to wherever we
want, say compound into the
future.
Interest rate ൌ 10%
49
Combine multiple CF into single future value
Can we move the cash flows to the
future and add them?
ANSWER: YES! As long as values are at the
same point in time, we can add them.
$300
30 1 2
$100 $200
1) Split into 3 single cash flows
2) Move each CF to the same point in time.
In this case, we compound to the future.
3) Now we can add them together:
ଷ ൌ 100 1.1 ଶ ൅ 200 1.1 ଵ ൅ 300 1.1 ଴ ൌ 641
$300
30 1 2
30 1 2
$200
30 1 2
$100
200 1.1 ଵ
100 1.1 ଶ
ൌ 300 1.1 ଴
Interest rate ൌ 10%What is the value at t=3?
641
Conclusion: We can only add CFs with the same TVM
50
Combine multiple CF into single value and move
$300
30 1 2
$100 $200
Interest rate ൌ 10%
$641
30 1 2
ଷ ൌ 100 1.1 ଶ ൅ 200 1.1 ଵ ൅ 300 1.1 ଴
ൌ 641
Idea: After combining multiple cash flows into a single value, we can then move the
value to wherever we want.
1. We combined multiple cash
flows into a single value
2. Move it to wherever we
want, say discount to present.
641
51
Combine multiple CF into single value and move
$641
30 1 2
Important conclusion: It does not matter where we combine multiple cash
flows into a single value because moving them gives the same result!
30 1 2481.5928
$300
30 1 2
$100 $200
1. We started with these
multiple cash flows.
2. We combined them in
the present and moved
them to the future.
3. We also combined them
in the future and moved
them to the present.
52
Once again….
What’s the difference between
Present Value & Future Value?
Short answer: NOTHING!
As long as you use the formula consistently!
ൌ ௡ ൌ ଴ 1 ൅ ௡ ൌ ଴ ൌ ௡ 1 ൅ ି௡
When -n < 0: moving CF backwards
When n > 0: moving CF forwards
53
Practice 3: combining and moving cash flows
Jonathan’s business has an outstanding loan. He has promised to pay it back in instalments of
$5 000, $18 000, $25 000, and $28 000 after 1, 2, 3, and 4 years. If the interest rate is 6%
compounded semi-annually:
(a) What much would he have to pay if he paid off the loan in full now?
(b) What much would he have to pay if he paid nothing until end of 4 years?
Soln
we can then move the value to wherever we want. +25+18+5 +28
3 40 1 2 5APR=……. m=…….. r=………..
54
Where are we?
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
We’ve looked at Single Cash Flows.
We now look at Multiple CF with infinite number of CFs AND Multiple CF with limited number of CFs.
55
Perpetuity
• A multiple cash flow
• Each cash flow is the same size
• Paid forever, i.e. perpetual AND infinite
• Ordinary perpetuity makes payments at
the end of each period (see below).
• Perpetuity due makes payments at the
start of each period.
Annuity
• A multiple cash flow
• Each cash flow is the same size
• A limited number of payments.
• Ordinary annuity makes payments at the end
of each period (see below).
• Annuity due makes payments at the start of
each period.
What are Perpetuities & Annuities?
Main
differences
∞0 1 2
PMT PMT PMT
…
…
…
… PMT PMT PMT…
n0 1 2 … n+1
0
forever will stop
56
(Ordinary) Perpetuity (Ordinary) Annuity
Formula for PV of Perpetuity & PV of Annuity
∞0 1 2
PMT PMT PMT
…
…
…
…
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡PMT PMT PMT…
n+10 1 2 … n
0
(n payments paid at end of period)
ൌ


1 െ 11 ൅ ௡
ൌ 1

െ
1
1 ൅ ௡
These are other ways of
writing the same thing
above. If your
mathematics is weak,
make sure you
understand that these
are all the same!
Different textbooks will
have different ways of
writing this formula.
57
(Ordinary) Perpetuity (Ordinary) Annuity
What do these formula mean?
∞0 1 2
PMT PMT PMT
…
…
…
… PMT PMT PMT…
n+10 1 2 … n
0
(n payments paid at end of period)
Another way to think financially is:
The formula “takes all the PMTs, wraps them up into
a ball, and finds their total present value at the start”
Another way to think financially is:
The formula “takes all the PMTs, wraps them up into
a ball, and finds their total present value at the start”
ൌ
1 ൅ ൅ 1 ൅ ଶ ൅ 1 ൅ ଷ ൅⋯ ∞
ൌ

ൌ
1 ൅ ൅ 1 ൅ ଶ ൅ 1 ൅ ଷ ൅⋯൅ 1 ൅ ௡
ൌ

1 െ 1 ൅ ି௡PV(Perp) is a simple formula for an infinite sum
of future cash payments
PV(Annu) is similar to
PV(Perp)
IMPORTANT: Notice the standard pattern of PMTs! They start in t=1, but we value at t=0!!
58
Types of Perpetuities & Annuities under study
(Ordinary) Perpetuities
• PV of perpetuity
• PV of perpetuity with delayed payments
• PV of growing perpetuity
• (FV of all of the above)
(Ordinary) Annuities
• PV of annuity
• PV of annuity with delayed payments
• PV of growing annuities
• (FV of all of the above)
Note:
When we say
“perpetuity” we mean
“ordinary perpetuity”,
which means all the
cash flows are at the
end of the period.
Same thing for when
we say “annuity”, we
mean “ordinary
annuity”, where the
cash flows are at the
end of the period.
We show the above using the formula!
Q: Does it look like a lot????
ANS: Not at all!! … Once you think financially, these problems
are all the same but with small and important differences.
Make sure you notice these differences!!!
Use the formulae correctly!!
59
To be SUCCESSFUL in this part of the course …
Combining multiple cash flows into
a single cash flow and moving it
to the right time.
…use formulas correctly by ...
Following the standard
patterns of cash flows.1
2
And by …
Back
60
Perpetuities – present value
∞0 1 2
100 100 100
…
…
…
…
Example. Tracy has inherited from her aunt a
perpetuity that pays $100 every year. If the current
interest rate is 5%, what is its current market value?
PV at t=0 is: ଴ ൌ
௉ெ்
௥
ൌ
ଵ଴଴
଴.଴ହ ൌ 2000
Answer:
The current market value (present value) of the
perpetuity is $2000.
The standard pattern: put a ring around the cash
flows and find value in NW corner! (NW=North-West)
NOTE: The perpetuity formula converts a multiple cash flow
problem into a single cash flow at t=0.
Apply the formula
APR=5% m=1 r=5%/1 PMT=100
m is frequency of compounding
Since the payments are “every year”
ሺ ሻ ൌ

Follow the standard pattern!!
It is a visual aid to reduce common mistakes later!!
Combining multiple CF into a single CF!!
2
1
Annual Percentage Rate Unless otherwise stated, the cash
flows are at the end of the year
How to be
successful
61
Perpetuities – delayed payments
Example. Rana has inherited from her uncle a
perpetuity that pays $100 every year with the first
payment starting in 5 years’ time (t=5). If the current
interest rate is 5%, what is its current market value?
PV at t=4: ସ ൌ
ଵ଴଴
଴.଴ହ ൌ 2000Answer:
The current market value (present value) of the
perpetuity is $1,645.40.
∞0 1 …2 3 4 5 6 7 8 n
100 … …100 100 100 100
PV at t=0: ଴ ൌ 2000 1 ൅ 5% ିସ ൌ 1645.404
NOTE: The delayed cash flows started at t=5, but we first
discounted the perpetuity to t=4 following the standard pattern
and then moved the single cash flow to t=0.
Just follow the standard pattern!
APR=5% m=1 r=5%/1
Since the payments are “every year”
ሺ ሻ ൌ

Combining multiple CF into a single CF!!
Moving single CF to right time!!2
1
WARNING: 40% of students ignore my advice of “putting a ring around the CF and valuing in the
NW corner”, i.e. using the standard pattern. They think that because the CFs start at t=5, they will
evaluate the PV of the annuity using the formula at t=5 and then move it back 5 periods to t=0.
WRONG! WRONG! WRONG! By using the standard pattern, you will evaluate the CFs correctly at
t=4 and then then bring them back to t=0 correctly. Don’t be one of these 40% of students!!
How to be
successful
62
PV at t=0: ଴ ൌ
௉ெ்
௥
ሾ1 െ 1 ൅ ି௡ሿ
ൌ
ଵ଴଴
଴.଴ସ/ଵଶ ሾ1 െ 1 ൅ ଴.଴ସଵଶ ି଺ሿ ൌ 593.0617586
FV at t=12: ଵଶ ൌ ଴ 1 ൅ ௡
ൌ 593.0617586 1 ൅ ଴.଴ସ
ଵଶ
ଵଶ ൌ 617.2240
Annuities – PV and FV
months0 1 …2 3 4 5 6 7 8 n
100100 100 100 100 100
Example. Jackson has negotiated to pay off the
remaining value of his mobile with 6 equal monthly
payments of $100. If the current interest rate is 4%
compounded monthly,
(a) What would he have to pay now to pay it off in full?
(b) What would he have to pay in full in one year if he
paid nothing until then?
Answer:
(a) Pay in full now (present value): $593.06
(b) Pay in full in one year (future value): $617.22
The standard pattern: put a ring around CF and value in the NW corner!
months0 1 …2 3 4 5 6 7 8 12
593.0617586
APR=4% m=12 r = 4%/12 n=6
Since the payments are “monthly”
(a)
(b)
ሺ ሻ ൌ

1 െ 1 ൅ ି௡
Combining multiple CF into a single CF!!
Moving single CF to right time!!
2
1
63
Annuities – delayed payments
XX X X X
Example. Blanca has a student loan currently worth
$10,000 but won’t start paying it off until she graduates.
Realistically, she won’t start paying it off until t=4 years
from now and when she does so, it will take her 5 years
to do so. Interest rates are 3%.
(a) When she starts paying what will the equal annual
payments be?
(b) Ideally, if she starts paying 2 years earlier, what will
they be?
Answer:
(a) If she starts paying in 4 years, payments= $2,386.02
(b) If she starts paying in 2 years, payments= $2,249.05
PV at t=0: ଴ ൌ
௉ெ்
௥
ሾ1 െ 1 ൅ ି௡ሿ
10000 ൌ ௑
଴.଴ଷ 1 െ 1.03ିହ 1.03 ିଷ
0 1 2 3 4 5 6 7 8 109
Current Value
of loan at t=0
Value of annuity
at t=3
Moves annuity
to t=0
⇒ X ൌ 2386.019
10000 ൌ 0.03 1 െ 1.03ିହ 1.03 ିଵ
0 1 2 3 4 5 6 7 8 10
XX X X X
9
⇒ X ൌ 2249.052
APR=3% m=1 r = 3%/1 n=5 PMT=X (unknown)
Solving
(a)
(b)
ሺ ሻ ൌ

1 െ 1 ൅ ି௡
Thought process: Somehow, the value of the annuity of Xs equals
the value of the loan at the same point in time, usually t=0.
10000
Combining multiple CF into a single CF!!
Moving single CF to right time!!
Combining multiple CF into a single CF!!
Moving single CF to right time!!
2
1
Note (2nd warning): cash flows start at t=4. But they are valued at t=3! By
putting a ring around the cash flows and valuing them in the NW corner,
you will get it correct! 40% of students who ignore this visual aid will get it
wrong!!! Again, don’t be one of them! You have been warned AGAIN! 
How to be
successful
64 Follow this advice and you will DRAMATICALLY increase your success rate!
Remember: To be SUCCESSFUL in this part of the course …
Combining multiple cash flows into
a single cash flow and moving it
to the right time.
…use formulas correctly by ...
Following the standard
patterns of cash flows.1
2
And by …
65
Where are we?
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
We’ve looked at Single Cash Flows.
We’ve look at Multiple CF with infinite number of CFs AND We’ve looked at Multiple CF with limited number of CFs.
Let’s now look at Multiple CF with infinite growing CFs AND also Multiple CF with limited number of growing CFs.
66
Growing Perpetuities
3 4 n n+1 ∞….0 1 2
PMT PMT
∗ 1 ൅ ଵ PMT∗ 1 ൅ ଶ0 PMT∗ 1 ൅ ଷ PMT∗ 1 ൅ ௡ିଵ PMT∗ 1 ൅ ௡
….
….
ሺ ሻ ൌ
െ
A growing perpetuity is stream of regular cash flows PMT that grow at a constant rate g forever.
The rate of interest is r. [Note Standard Pattern: the first PMT has no (1+g) factor.]
Notes: 1. ൐ always  if not, the model does not work! There is something wrong!!
2. If ൐ 0, then growing perpetuity; if ൏ 0, then diminishing perpetuity.
3. PMT is always the first cash flow in the infinite stream of growing cash flows.
67
“Thinking financially”, what does the formula mean?
3 4 n n+1 ∞….0 1 2
PMT PMT
∗ 1 ൅ ଵ PMT∗ 1 ൅ ଶ0 PMT∗ 1 ൅ ଷ PMT∗ 1 ൅ ௡ିଵ PMT∗ 1 ൅ ௡
….
….
ሺ ሻ ൌ
െ
The formula means “take all the growing PMT cash flows, wraps them up
into a ball, and finds their total present value at the start”
. ൌ
െ
ൌ
1 ൅ ൅ ሺ1 ൅ ሻ1 ൅ ଶ ൅ 1 ൅ ଶ1 ൅ ଷ ൅ 1 ൅ ଷ1 ൅ ସ ൅⋯∞
A simple formula for what look like complicated
growing cash flows … notice the standard pattern!!!
…when you
calculate this…
“Thinking financially”
means that ….
…you are thinking that you are calculating this!!
It’s not just a formula, but a way of thinking!!
68
Growing Perpetuities
Example. An account pays a monthly cash flow of $2 at the end of the month indefinitely. The cash flow
grows at 0.5% each month. If the APR is 8%, what is the present value of these cash flows?
2 2 1.005 ଵ
ൌ 2.01 2 1.005 ଶൌ 2.020050 2 1.005 ଷൌ 2.03015 2 1.005 ସൌ 2.04030 2 1.005 ௡ିଵ ….….
3 4 5 n ∞….0 1 2 ….
ANSWER:
PV(growing perpetuity) ൌ ௉ெ்
௥ି௚
ൌ
ଶ
଴.଴଼/ଵଶି଴.଴଴ହ ൌ 1200 For comparison: PV(ordinary perpetuity) ൌ ௉ெ்௥ ൌ ଶ଴.଴଼/ଵଶ ൌ 300
Follow the standard pattern: put a ring around the cash flows and find the value in the NW corner!
APR = 8% m=12 r=8%/12 g=0.005
ሺ ሻ ൌ
െ
PMT is the first cash flow in the perpetually growing cash flow stream.
Why 12? “Cash flows are monthly”, so the APR given must also be compounded monthly. Unless told otherwise, m is implied by the cash flow frequency.
69
Growing Annuities
3 4 n n+1 ∞….0 1 2
PMT PMT
∗ 1 ൅ ଵ PMT∗ 1 ൅ ଶ0 PMT∗ 1 ൅ ଷ PMT∗ 1 ൅ ௡ିଵ
….
0
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
A growing annuity is stream of regular cash flows that grow at a constant rate g,
but the stream stops after n periods. The rate of interest is r.
0
Notes:
1. For growing annuities (unlike growing perpetuities), there is no restriction on the relationship
between and , i.e. ൐ or ൏ are okay.
2. If ൐ 0, then growing annuity; if ൏ 0, then diminishing annuity
3. According to the formula, (a) there are n cash flows, (b) PMT is always the first cash flow, and
(c) the growth factor is applied n-1 times.
70
Growing Annuities
Example. An account pays annual cash flows at the end of each year and grow at 3% each year. If last year’s
payment was $200, what is the present value of the next five years’ payments if the interest rate is 7%?
200 1.03 ଵ
ൌ 206 200 1.03 ଶൌ 212.18200 200 1.03 ଷൌ 218.5454 200 1.03 ସൌ 225.1017 0
3 4 5 6 ∞….0 1 2
ANSWER:
PV(growing annuity) ൌ ௉ெ்
௥ି௚
1 െ ଵା௚
ଵା௥
௡
ൌ
200 1.037% െ 3% ሺ1 െ 1.031.07 ହሻ ൌ 893.2861
0
For comparison:
PV(ordinary annuity) ൌ ଶ଴଴
଻% 1 െ 1.07ିହ
ൌ 820.04
Follow standard pattern: a ring around
cash flows, value in NW corner!
APR = 7% m=1 r=7%/1 g=0.03
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
According to the standard formula for
growing annuity, THIS is now the new
PMT. So use it correctly in the formula!!
n refers to the number
of periods in the deal
or cash flows.
i.e. 5 cash flows in this
case.
200 1.03 ହ
ൌ 231.8548
Important Question:
This CF was paid at the end of last year.
Do we count it in our calculation?
ANS: No! We only count future Cash
Flows and value them at the present!!
Past payments are not counted!!
2
1
How to be
successful
Common error: Write “200”,
forgetting to grow by (1+g)
Constant CF or g = 0
71
Practice 4: Perpetuities and Annuities
Perpetuity. If the interest rate is 6%, what is
the current price of a perpetuity that pays $50
every month starting in 5 months?
Annuity. If the interest rate is 5%, what is the
current price of an annuity that pays $15 every
day starting in 30 days for 2 years?
Soln
APR= …….. m= …….. r= ………. APR=……. m=……… r=……………. n=……………………
∞0 1 …2 3 4 5 6 7 8 n
50 50 50 50 50 15 15 151515
∞0 … …29 30 … 759 760 n31
0 0
(Three practice problems: 4, 5, 6)
72
Practice 5: Growing Perpetuities
Ginny wants to sell a perpetuity that pays quarterly cash flows that grow at 0.3% per payment. If the cash
flows don’t start until half a year from now, the current APR is 7%, and the current selling price is $560,
what are the values of the first 3 payments?
APR = ………… m=……….. r=……………. g=…………..
X X 1.003 ଵ0 X 1.003 ଶ X 1.003 ଷ X 1.003 ௡ିଶ ….….3 4 5 n ∞….0 1 2 …. quarters
Soln
73
Practice 6: Delayed Annuities
Jake will start to receive $2000 in exactly two years from now and the same amount thereafter every
half-year for the next 10 years (a total of 20 payments). If a loan shark is willing to pay 6% for this
annuity, how much would he be able to get for it?
APR = ……… m=…….. r=……. n=……
….
4 5 22 23 n….0 1 3 …. Half-years….
….
Soln
74
Where are we?
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
We’ve looked at Single Cash Flows.
We’ve look at Multiple CF with infinite number of CFs AND We’ve looked at Multiple CF with limited number of CFs.
We’ve looked at Multiple CF with infinite growing CFs AND also Multiple CF with limited number of growing CFs.
Let’s now look at…..
75
When to convert between rates?
1. When we have an APR with one compounding frequency but need another APR with a different
compounding frequency:
• This is the most common situation.
• Example 1: We have APR=10% with m=4 but we need an equivalent APR with m=1.
• Example 2: Bank A offers APR=6% with m=2 and Bank B offers APR=5.5% with m=12.
How do we compare two equivalent but different APRs?
2. When Cash flows are one frequency, but interest rate is another frequency
• Situation: mismatching CF and Rate frequencies – tricky! You need to be aware!
• Example 3:
Monthly cash payments, i.e. m=12.
Bank interest rate: APR=10% with m=4
Mismatch of frequencies
76
Examples: Converting rates
1. For an interest rate of 13% with quarterly
compounding, what is the effective annual rate (EAR)?
Given: APR=13% m=4 r= 13%/4
Want: newAPR=?? m=1 r=newAPR/1
Equation:
1 ൅ 1 ଵ ൌ 1 ൅ 0.134 ସ
⇒ ൌ ൌ 1 ൅ ଴.ଵଷ
ସ
ସ
െ 1 ൌ 13.64759%
2. What is the APR with monthly compounding for an
effective rate (EAR) of 15%?
Given: 15% m=1 r = 15%/1
Want: newAPR m=12 r= newAPR/12
Equation:1 ൅ 12 ଵଶ ൌ 1 ൅ 0.151 ଵ
⇒ ൌ 1.15 ଵଵଶ െ 1 ∗ 12 ൌ 14.0579%
Definition: EAR is an APR with m=1
Compounds over 1 year
Rate per period
Effective annual rate when m=1
Note: This is a
power of 1/12
“Thinking financially” means the two rates are equivalent
if the same principal (e.g. $1) of both investments earns
the same interest over the same time period (e.g. 1 year)
despite having different compounding frequencies.
Compounds over 1 year
77
More Examples: Converting rates
3. For an interest rate of 13% with quarterly compounding,
what is the rate with monthly compounding?
Given: APR=13% m=4 r= 13%/4
Want: newAPR=?? m=12 r=newAPR/12
Equation:
1 ൅ 12 ଵଶ ൌ 1 ൅ 0.134 ସ
⇒ ൌ ൌ ሾ 1 ൅ ଴.ଵଷ
ସ
ସ/ଵଶ
െ1ሿ ∗ 12 ൌ 12.86166%
4. What is the APR with weekly compounding for an APR
of 15% compounded every four months?
Given: 15% m=3 r = 15%/3
Want: newAPR m=52 r= newAPR/52
Equation:1 ൅ 52 ହଶ ൌ 1 ൅ 0.153 ଷ
⇒ ൌ 1 ൅ 0.153 ଷହଶ െ 1 ∗ 52 ൌ 14.65767%
Compounds over 1 year
Rate per period
Compounds over 1 year
Rate per period
“compounded every 4 months” means 3 compounds in 1 year
78
Example: Mismatching CF and Rate frequencies
Jack receives monthly payments of $5000 at the end of each month into a bank account. His bank offers him an interest
rate of 3.5% compounded daily. How much will he have after 12 months assuming he does not touch the money?
Cash flow frequency: m =12 (monthly)
Rate compounding freq: m = 365 (daily)
Mismatch of
frequencies
First problem: Have an APR=3.5% with m=365, but Want a newAPR with m=12, i.e. r = newAPR/12 (rate per period)
1 ൅ 12 ଵଶ ൌ 1 ൅ 3.5%365 ଷ଺ହ ⇒ 12 ൌ 0.002920 …
ൌ
FV(Ord Anni) ൌ PV(Ord Anni)ൈ 1 ൅ ௡
ൌ


1 െ 1 ൅ ି௡ 1 ൅ ௡
Note 2: In this formula, r is the rate per
period, newAPR/12.
Note 1: the … mean there are more
decimals available. Use ALL of them in
the calculator. Warning: Don’t make the
error of rounding too early!!
ൌ
5000.002920 … 1 െ 1.002920 … ିଵଶ 1.002920 … ାଵଶ
ൌ 5000 12.19466 … ൌ 60 973.3047
IMPORTANT: we must convert the rate compounding frequency to match the cash
flow payment frequency, not the other way around.
Why? Cash flow frequency is usually fixed and not under our control, while
converting the rate compounding frequency is easy mathematics.
Want: Have:
Second problem: What is the FV of these cash flows?
What if we did not convert the rate
and used:
ൌ 3.5%
FV(Ord Anni)
ൌ


1 െ 1 ൅ ି௡ 1 ൅ ௡
ൌ
5000.035 1 െ 1.035 ିଵଶ 1.035 ଵଶ
ൌ 5000 14.60196 …
ൌ 73 009.8081 Incorrect!
79
Practice 8: Converting Rates
Roger is paying his student loan with annual payments of $5000 at a rate of 6%.
If he changes to either (a) monthly payments or (b) weekly payments, what
equivalent rate would he pay?
Soln
(b) Weekly compounded rate(a) Monthly compounded rate
(Two practice problems: 8 and 9)
80
Practice 9: Converting Rates
Rene is told that if she invests $1000 now, she will receive $1500 in 1 year and 7 months time.
What is her investment return if it were based on: (a) annual compounds (b) monthly compounds.
Soln
(a) APR, m=1 (b) APR, m=12
81
Main formula
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
5. Financial mathematics & Excel functions
Download the Excel file “Week 01 Intro & Debt Valuation”
Open file and follow along with the examples
83
Main Excel Functions
Functions we will look at briefly now:
FV: Future Value of an annuity
PV: Present Value of an annuity
PMT: Calculates regular payments
IRR: Calculate internal rate of return
Functions we will look at later:
XIRR
PRICE
NPV: Calculates NPV of regular cash flows
XNPV: Calculates NPV of cash flows with dates
84
Excel function: =PV(…) and =FV(…)
=PV(r, nper, pmt, [fv], [type]) =FV(r, nper, pmt, [pv], [type])
Rate per
period
Number of periods
or payments
Cash flow
Payment
Default = 0
Future Value Cash Flow
=0 (Default) means ordinary annuity;
=1 means annuity due
Default = 0
Present Value Cash Flow
Important conclusions:
• Follow the standard CF pattern
• Combine CF at the right time
ൌ

1 െ 1 ൅ ି௡The PV formula solves: 1
2
Live Demonstration in Excel: PV FV Demo
85
Excel function: =PMT(…)
Can you contribute to the online forum? Find GOOD online videos that explain how to use these EXCEL functions.
=PMT(r, nper, pv, [fv], [type])
Rate per
period
Number of periods
or payments
Loan principal at
present, ie. pv
Loan value at the end, i.e.
residual value of loan, fv
=0 (Default) means ordinary annuity;
=1 means annuity due
Live Demonstration in Excel: PMT Demo
Important conclusions:
• Use consistent signs for the CF direction
• CF in  +
• CF out  -
• Make sure you use the correct r, rate per period.
• r = APR/m
ൌ

1 െ 1 ൅ ି௡ ൅ 1 ൅ ௡
ൌ
௉௏ି
ி௏
ଵା௥ ೙ .௥
ଵି ଵା௥ ష೙
The formula is solving for PMT in the equation:
The type of problem =PMT() solves is, for example, what equal
monthly payment do you need to make on a loan of $10,000
received today and repaid over 5 months with a final payment of
$2,000? The current interest rate is 12% compounded monthly?
86
Main formula
ሺ ሻ ൌ

ሺ ሻ ൌ

1 െ 1 ൅ ି௡
ሺ ሻ ൌ
െ
ሺ ሻ ൌ
െ
1 െ 1 ൅ 1 ൅ ௡
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Present Value: ଴ ൌ ௡ 1 ൅ ି௡ Future Value: ௡ ൌ ଴ 1 ൅ ௡Single Cash Flows
Multiple Cash Flows
Perpetuities Annuities
1 ൅ 1 ଵ ൌ 1 ൅ ௠
Converting Rates
6. Solutions to Practice Problems
88
Practice 1: Compound interest Soln
1. Jennifer deposits $1000 in a bank account that earns
compound interest. How much would she have after 10 years
(a) at 10% compounded semi-annually?
(b) at 10% compounded daily?
Back
1000 1 ൅ 10%2 ଵ଴ൈଶ ൌ 1000 1.05 ଶ଴ ൌ 2653.30
1000 1 ൅ 10%365 ଵ଴ൈଷ଺ହ ൌ 2717.91
௡ ൌ ଴ 1 ൅ ௡଴ ൌ ௡ 1 ൅ ି௡
(a) APR=…….. m=…… r=………….. n=………
(b) APR=…….. m=…… r=………….. n=………
10% 2 10%/2 10*2
10% 365 10%/365 10*365
Thought process: “Start with 1000, grow by interest
rate factor of (1+5%), compound 20 times”
Thought process: “Start with 1000, grow by interest
rate factor of (1+10%/365), compound 3650 times”
89
APR = …….%, m=……., r=…………
6 months = 0.5 years. 9 months = 0.75 years.
Difference = 0.25 years increase (3 months)
Amount received in future = 83 1 ൅ 7.5% ା଴.ଶହ ൌ 84.51
It would receive $84.51 Million in 9 months.
6 months = 0.5 years.
Loan amount received now = 83 1 ൅ 0.054/1 ି଴.ହ ൌ 80.8459
It could borrow $80.8459 Million now and pay back $83M in 6 months
Practice 2: Moving single cash flows Soln
$83M
1.5 years0 0.5 1
5.4%
$83M
1.5 years0 0.5 1
7.5%
0.75
Back
Rabbo Corp. expects its customer to pay $83 million in 6 months from now, but it needs the money now to pay other expenses.
a) If the interest rate offered by its bank is 5.4%, what amount could it borrow now?
b) If the customer delays payment to 9 months from now, what would Rabbo receive if it charged a penalty interest rate of 7.5%?
Move back half a period
= 6 months or 0.5 years
Move forward a quarter
of a period = 3 months
௡ ൌ ଴ 1 ൅ ௡଴ ൌ ௡ 1 ൅ ି௡
(a) Find Present Value
(b) Find Future Value APR = …….%, m=……., r=…………
5.4 1 5.4%/1
7.5 1 7.5%/1
Common error: It is incorrect to write
83 1 ൅ ଴.଴ହସ
ଶ
ିଵ
ൌ 80.8179.
Why? The APR has m=1, not m=2!!
Important Conclusion: In this problem, get correct, so you can get correct!
90
Practice 3: combining and moving cash flows Soln
Jonathan’s business has an outstanding loan. He has promised to pay it back in instalments of
$5 000, $18 000, $25 000, and $28 000 after 1, 2, 3, and 4 years. If the interest rate is 6%
compounded semi-annually:
(a) What much would he have to pay if he paid off the loan in full now?
(b) What much would he have to pay if he paid nothing until end of 4 years?
+25+18+5 +28
3 40 1 2 5
(a) ൌ 5ሺ1.03ሻିଶ ൅ 18ሺ1.03ሻିସ ൅ 25ሺ1.03ሻି଺ ൅ 28ሺ1.03ሻିଶ
ൌ
51.03ଶ ൅ 181.03ସ ൅ 251.03଺ ൅ 281.03଼ ൌ 63.74631138
(b) Either:
ൌ 63.74631138 1.03 ଼ ൌ 80.75192006
Or:
ൌ 5 1.03 ଺ ൅ 18 1.03 ସ ൅ 25 1.03 ଶ ൅ 28 ൌ 80.75192006
Answers
(a) $63,746.31
(b) $80,751.92
Back
1 year = 2 semi-
annual periods
3 years = 6 semi-
annual periods
2 years = 4 semi-
annual periods
4 years = 8 semi-
annual periods
APR=……. m=…….. r=………..6% 2 6%/2=0.03
This way of financial thinking is “combine
into single CF and move to the future”
This way of financial thinking is “move each
single CF and combine them in the future”
91
Practice 4: Perpetuities and Annuities Soln
Perpetuity. If the interest rate is 6%, what is
the current price of a perpetuity that pays $50
every month starting in 5 months?
Annuity. If the interest rate is 5%, what is the
current price of an annuity that pays $15 every
day starting in 30 days for 2 years?
Back
APR=6% m=12 r=6%/12 APR=5% m=365 r=5%/365 n=2*365=730
Ordinary perpetuity. ൌ ௉ெ்
௥
∞0 1 …2 3 4 5 6 7 8 n
50 50 50 50 50
ൌ
500.0612 1 ൅ 0.0612
ିସ
ൌ 9802.4752
Value of
perpetuity at
t=4
Move value to t=0
Ordinary annuity. ൌ ௉ெ்
௥
ሾ1 െ 1 ൅ ି௡ሿ
∞0 … …29 30 … 759 760 n31
15 15 15 0 0
ൌ
155% 365⁄ ሾ1 െ 1 ൅ 0.05365 ି଻ଷ଴ሿ 1 ൅ 0.05365 ିଶଽ ൌ 10378.31606
Value of annuity
at t=29
Move value
to t=0
1515
ANS: $10,378.32ANS: $9,802.48
92
Practice 5: Growing Perpetuities Soln
Ginny wants to sell a perpetuity that pays quarterly cash flows that grow at 0.3% per payment. If the cash
flows don’t start until half a year from now, the current APR is 7%, and the current selling price is $560,
what are the values of the first 3 payments?
APR = 7% m=4 r=7%/4 g=0.003
X X 1.003 ଵ0 X 1.003 ଶ X 1.003 ଷ X 1.003 ௡ିଶ ….….3 4 5 n ∞….0 1 2 …. quarters
cash flows don’t start until half a year from now = 2 quarters560 ൌ X0.074 െ 0.003 1 ൅ 0.074
ିଵ
⇒ ൌ 8.2621
PV ൌ
െ
Value of
perpetuity at
t=1
Move value to
t=0 (i.e. back
one period = -1)
Price of
perpetuity
at t=0
Answer:
1st payment = $8.26 = X
2nd payment = $8.29 = X(1.003)
3rd payment = $8.31 = X(1.003)2
Back
Following the standard pattern means valuing the perpetuity at the correct time!
93
Practice 6: Delayed Annuities Soln
Jake will start to receive $2000 in exactly two years from now and the same amount thereafter every
half-year for the next 10 years (a total of 20 payments). If a loan shark is willing to pay 6% for this
annuity, how much would he be able to get for it?
APR = 6% m=2 r=6%/2 n=20
0 2000 2000 2000 2000 0….4 5 22 23 n….0 1 3 …. Half-years….0
PV(ordinary annuity) ൌ ௉ெ்
௥
ሾ1 െ 1 ൅ ି௡ሿ 20th payment
cash flows don’t start until 2
year from now = 4 half-years
ଷ ൌ
20000.06 2⁄ ሾ1 െ 1 ൅ 0.062 ିଶ଴ሿ ൌ 29754.9497
଴ ൌ ଷ 1 ൅ ିଷ ൌ 29754.9497 1 ൅ 0.062 ିଷ ൌ 27229.9940 ANS: $27,229.99 Back
94
Practice 8: Converting Rates Soln
Roger is paying his student loan with annual payments of $5000 at a rate of 6%.
If he changes to either (a) monthly payments or (b) weekly payments, what
equivalent rate would he pay?
1 ൅ 12 ଵଶ ൌ 1 ൅ 6%1 ଵ
ൌ 1 ൅ 6%1 ଵ/ଵଶ െ 1 ൈ 12 ൌ 5.84%
(b) Weekly compounded rate
ൌ 1 ൅ 6%1 ଵ/ହଶ െ 1 ൈ 52 ൌ 5.83%
Compounds over 1 year Compounds over 1 year
, m=12
, m=52
Back
6% m=1 is equivalent to 5.84% m=12
6% m=1 is equivalent to 5.83% m=52
1 ൅ 52 ହଶ ൌ 1 ൅ 6%1 ଵ
(a) Monthly compounded rate
95
Practice 9: Converting Rates Soln
Rene is told that if she invests $1000 now, she will receive $1500 in 1 year and 7 months time.
What is her investment return if it were based on: (a) annual compounds (b) monthly compounds.
1000 1 ൅ 1 ሺଵା ଻ଵଶሻ ൌ 1500
ൌ
15001000 ଵା ଻ଵଶ షభ െ 1
ൌ 29.19%
(a) APR, m=1
1000 1 ൅ 12 ሺଵଶା଻ሻ ൌ 1500
ൌ
15001000 ଵଶା଻ షభ െ 1 ൈ 12
ൌ 25.88%
(b) APR, m=12
Invest $1000 for 1 year and 7 months and get $1500 Invest $1000 for 12 month plus 7 months and get $1500
Back
Note: These are equivalent APRs but with
different frequencies of compounding
with m=1 with m=12