3-1
Fundamentals
of Corporate
Finance
Second Canadian Edition
prepared by:
Carol Edwards
BA, MBA, CFA
Instructor, Finance
British Columbia Institute of Technology
copyright © 2003 McGraw Hill Ryerson Limited
3-2
Chapter 3
The Time Value of Money
Chapter Outline
Future Values and Compound Interest
 Present Values
 Multiple Cash Flows
 Level Cash Flows: Perpetuities and
Annuities
 Inflation and the Time Value of Money
 Effective Annual Interest Rates

copyright © 2003 McGraw Hill Ryerson Limited
3-3
Introduction
• Money


Problems …
Assume interest rates are 4.3884%.
You have just won a lottery and must choose
between the following two options:
 Receive
a cheque for $150,000 today.
 Receive $10,000 a year for the next 25 years.
KEY QUESTIONS FOR YOU:
Which option gives you the biggest “winnings”
How should you tackle this kind of problem?
copyright © 2003 McGraw Hill Ryerson Limited
3-4
Introduction
• Money

Problems …
As a financial manager you will often have to
compare cash payments which occur at
different dates:
Cash flows now, versus …
… cash flows later.

To make optimal decisions, you must
understand the relationship between a dollar
received (paid) today and a dollar received
(paid) in the future.
copyright © 2003 McGraw Hill Ryerson Limited
3-5
Introduction
• Money

Problems …
As a financial manager you will face two
basic types of cash flow problems:
 Present
Value (PV) problems.
 Future Value (FV) problems.
copyright © 2003 McGraw Hill Ryerson Limited
3-6
Introduction
• Present Value (PV) Problems
 PV problems involve calculating the value
today of future cash flow(s).
 For example:
 Interest
rates are 7%. If I need to have $100,000
saved in 10 years, how much money must I put
aside today to create that cash flow?
 Interest rates are 12%. If I need to create an
income of $5,000 per year for 10 years, how
much money must I put aside today to create that
cash flow?
copyright © 2003 McGraw Hill Ryerson Limited
3-7
Introduction
• Future Value (FV) Problems
 FV problems involve calculating the value an
investment will grow to after earning interest.
 For example:
 Interest
rates are 5%. If I invest $1,000 today,
how much will it be worth in 8 years?
 Interest rates are 10%. If I open an account
and invest $2,500 per year, how much will it be
worth in 12 years?
copyright © 2003 McGraw Hill Ryerson Limited
3-8
Future Values
• Compound


Interest vs Simple Interest
Future value is the amount to which an
investment will grow after earning
interest.
There are two types of interest you may
receive:
 Compound
interest.
 Simple interest.
copyright © 2003 McGraw Hill Ryerson Limited
3-9
Future Values
• Simple Interest
 Simple interest means that interest is
earned only on your original investment:
 No

interest is earned on the interest.
Example:
Assume interest rates are 6%.
 You invest $100 in an account paying
simple interest.
How much will the account be worth in 5
years?
copyright © 2003 McGraw Hill Ryerson Limited
3-10
Future Values
• Simple


Interest
You earn interest only on the amount invested.
Therefore you would earn:
 $100
x 6% = $6.00 per year for 5 years.
Answer – you would have $130 after 5 years:
Balance in your account:
$100
0
$106
1
$112
2
$118
3
$124
4
$130
5
Period (t)
copyright © 2003 McGraw Hill Ryerson Limited
3-11
Future Values
• Compound



Interest
Most financial problems you will deal
with will involve compound interest.
Compound interest means that interest is
earned on interest.
The result: the income you earn would
be higher than it would be with simple
interest.
Can you see why?
copyright © 2003 McGraw Hill Ryerson Limited
3-12
Future Values
• Compound Interest
 Your income would be higher than it would
be with simple interest because you earn
interest on both the original investment and
the interest earned in previous years.
 Try the example again using compound
interest:
 Interest rates are 6%. You invest $100 in an
account paying compound interest. How
much will the account be worth in 5 years?
copyright © 2003 McGraw Hill Ryerson Limited
3-13
Future Values
• Compound

Interest
You earn interest on your interest:
 $100
x 6% = $6.00 the first year.
 $106 x 6% = $6.36 the second year.
 $112.36 x 6% = $6.74 the third year … etc.
After 5 years you would have $133.82 :
Balance in your account:
$100
0
$106
1
$112.36
2
$119.10
3
$126.25 $133.82
4
5
Period (t)
copyright © 2003 McGraw Hill Ryerson Limited
3-14
Future Values
• Formula
for Calculating FV
FV = Investment x (1 + r)t

Try the example again using the formula above:
 Interest
rates are 6%. You invest $100 in an
account paying compound interest. How much will
the account be worth in 5 years?
FV = $100 x (1 + 0.06)5
= $100 x 1.3382
= $133.82
copyright © 2003 McGraw Hill Ryerson Limited
3-15
Present Value
• More Money Problems …
 Assume interest rates are 10%.
 You have just won a lottery and must choose
between the following two options:
 Receive
$1,000,000 today.
 Receive $1,000,000 five years from now.
KEY QUESTIONS FOR YOU:
Which option gives you the biggest “winnings”
How should you tackle this kind of problem?
copyright © 2003 McGraw Hill Ryerson Limited
3-16
Present Value
• More



Money Problems …
This is an example of a present value
problem.
You shouldn’t even have to do a
calculation to get the correct answer.
Obviously the first option is the better
choice!
 You
would want to take the money today so
that you could immediately start earning
interest on your winnings.
copyright © 2003 McGraw Hill Ryerson Limited
3-17
Present Value
• More Money Problems …
 The above example demonstrates a basic
financial principle:
 A dollar
received today is worth more than a dollar
received tomorrow.

The key question is:
 How
much less valuable is a dollar received
tomorrow as versus a dollar received today?

That question is answered by using the
interest rate (also known as the discount rate)
to calculate the PV of the second option.
copyright © 2003 McGraw Hill Ryerson Limited
3-18
Present Value
• Formula
for Calculating PV
PV = Future Value x 1/(1 + r)t


You have been offered $1 million five years
from now. Interest rates are 10%.
What is that worth to you in today’s dollars?
PV = $1.0 million x 1/ (1 + 0.10)5
= $1.0 million x 0.620921
= $620,921
copyright © 2003 McGraw Hill Ryerson Limited
3-19
Present Value
• More


Money Problems …
Thus, you could have $1 million today.
Or you could have the second option, which
equates to $620,921 in today’s dollars.
$1 million now
vs
The equivalent
of $620,921 now
You knew before that the first option was better,
but now you can calculate exactly how much better
off you are:

$379,079
better off!
copyright © 2003 McGraw Hill Ryerson Limited
3-20
Present Value vs Future Value
• PV and FV are related!
 Have you noticed that $620,921 becomes $1
million (and that $1 million requires $620,921) if
you have a time period of 5 years and a discount
rate of 10%?
PV at 10%
$1,000,000
$620,921
FV at 10%
copyright © 2003 McGraw Hill Ryerson Limited
3-21
Present Value vs Future Value
• PV and FV are related!
 $620,921 invested for 5 years at 10% grows to $1
million.
 Or, working it in reverse, if rates are 10%, and you
need $1 million in 5 years, you must put aside
$620,921 right now.
FV = PV x (1 + r)t
PV = FV x 1/(1 + r)t
= $620,921 x (1 + 0.10)5
= $1 million x 1/ (1 + 0.10)5
= $620,921 x 1.61051
= $1 million x 0.620921
= $1 million
= $620,921
copyright © 2003 McGraw Hill Ryerson Limited
3-22
Present Value vs Future Value
• PV
and FV are related!

To calculate the FV of money which is
available now (PV) to be invested for t
years at an interest rate r, multiply the
PV by (1+r)t.

To calculate the PV of a future payment,
run the process in reverse and divide the
FV by (1+r)t.
copyright © 2003 McGraw Hill Ryerson Limited
3-23
Present Value vs Future Value
• PV and FV
 Note that:



are related!
(1+r)t is called the future value factor.
r is called the discount rate
Finding the PV is often called discounting.
copyright © 2003 McGraw Hill Ryerson Limited
3-24
Present Value vs Future Value
• Two


Key Principles for Financial Calculations
Think of the example in which we compared
receiving $1 million today against $1 million
received 5 years from now.
You should see from that example that:
 A dollar
received today is worth more than a dollar
received tomorrow.
Lesson: The value of cash flows received at
different times can never be directly compared.
You must first discount all cash flows to a
common date and then compare them.
copyright © 2003 McGraw Hill Ryerson Limited
3-25
Present Value vs Future Value
• Finding
the Unknown …
FV = PV x (1 + r)t
PV = FV x 1/(1 + r)t
The FV and PV formulas
have many applications.
Note that the variables used
in these two equations are:
FV
PV
r
t
Given any three variables
in the equation, you can
always solve for the
remaining variable!
copyright © 2003 McGraw Hill Ryerson Limited
3-26
Multiple Cash Flows
• Future



Value Calculations
So far, we have looked at problems
involving only a single cash flow.
This is unrealistic – most business
investments will involve multiple cash
flows over time.
We need a method for coping with such
streams of cash flows!
copyright © 2003 McGraw Hill Ryerson Limited
3-27
Multiple Cash Flows
• Future
Value Calculations
EXAMPLE
 Assume interest rates are 8%.
 You make 3 deposits to your bank
account:
 $1,200
today
 $1,400 one year later.
 $1,000 two years later.

How much money will you have in your
account 3 years from now?
copyright © 2003 McGraw Hill Ryerson Limited
3-28
Multiple Cash Flows
• Doing Future Value Calculations
 Calculate what each cash flow will be worth at the
specified future date and add up these future values.
$1,200
0
$1,400
1
$1,000
2
3
FV in Year 3:
$1,080.00 = $1,000 x 1.08
$1,632.96 = $1,400 x (1.08)2
$1,511.65 = $1,200 x (1.08)3
$4,224.61
copyright © 2003 McGraw Hill Ryerson Limited
3-29
Multiple Cash Flows
• Present


Value Calculations
Suppose we need to calculate the PV of
a stream of future cash flows.
We use basically the same procedure as
for working with the FV of multiple cash
flows:
 Calculate
what each cash flow would be
worth today, i.e. get its PV.
 Add up these present values.
copyright © 2003 McGraw Hill Ryerson Limited
3-30
Multiple Cash Flows
• Present Value Calculations
EXAMPLE
 Assume interest rates are 8%.
 You wish to buy a car making three
installments:
 $8,000
today
 $4,000 one year later.
 $4,000 two years later.

How much money would you have to place in
an account today to generate this stream of
cash flows?
copyright © 2003 McGraw Hill Ryerson Limited
3-31
Multiple Cash Flows
• Present Value Calculations
 You would need to place $15,133.06 in an account
today to generate the desired cash flows:
-$8,000
PV today:
0
-$4,000
1
-$4,000
2
$8,000.00
$4,000 / (1.08) = $3,703.30
$4,000 / (1.08)2 = $3,429.36
$15,133.06
copyright © 2003 McGraw Hill Ryerson Limited
3-32
Multiple Cash Flows
• Special


Situations
In the previous examples, we worked with
multiple cash flows of different sizes.
Sometimes we have a situation in which a
series of equal cash flows is involved:
 What
would you pay to own a guaranteed income of
$1,000 per year to be received forever, if interest
rates are 4%?
 Calculate what the value of your account would be if
you were to deposit $2,500 per year for 5 years and
interest rates are 7%?
copyright © 2003 McGraw Hill Ryerson Limited
3-33
Multiple Cash Flows
• Special
Situations
Any sequence of
equally spaced,
level cash flows
is called an annuity.
If the payment stream lasts forever,
it is called a perpetuity.
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3-34
Multiple Cash Flows
• Perpetuities
 The PV of a perpetuity is calculated by dividing the
cash payment by the interest rate:
PV of a perpetuity =

C
r
=
Cash Payment
Interest rate
The interest rate on a perpetuity is calculated by
dividing the cash payment by the PV:
Interest rate on a perpetuity =
C
PV
=
Cash Payment
Present Value
copyright © 2003 McGraw Hill Ryerson Limited
3-35
Multiple Cash Flows
• Perpetuities

We are now ready to answer a question we asked
earlier:

What would you pay to own a guaranteed income of
$1,000 per year to be received forever, if interest rates
are 4%?
PV of a perpetuity =
C
r
=
Cash Payment
Interest rate
=
$1,000
4%
=
$25,000
copyright © 2003 McGraw Hill Ryerson Limited
3-36
Multiple Cash Flows
• PV



of an Annuity: the Long Method
In previous examples, we have worked with
multiple cash flows of different sizes.
Suppose we now need to calculate the PV of a
stream of level future cash flows.
We could use the same procedure as before:
 Calculate
what each cash flow would be worth
today, i.e. get its PV.
 Add up these present values.
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3-37
Multiple Cash Flows
• PV of an Annuity: the Long Method
EXAMPLE
 Assume interest rates are 10%.
 You wish to buy a car making three
installments:
 $4,000
a year from now.
 $4,000 one year later.
 $4,000 two years later.

How much money would you have to place in
an account today to generate this stream of
cash flows?
copyright © 2003 McGraw Hill Ryerson Limited
3-38
Multiple Cash Flows
• PV of an Annuity: the Long Method
 You would need to place $9,947.41 in an account
today to generate the desired cash flows:
-$4,000
PV today:
0
1
-$4,000
2
-4,000
3
$4,000 / (1.10) = $3,636.36
$4,000 / (1.10)2 = $3,305.79
$4,000 / (1.10)3 = $3,005.26
$9,947.41
copyright © 2003 McGraw Hill Ryerson Limited
3-39
Multiple Cash Flows
• PV

of an Annuity: the Short Cut!
We have calculated that we need to put aside
$9,947.41 to fund the following cash flows:
 $4,000
a year from now.
 $4,000 one year later.
 $4,000 two years later.

However, is there an easier way to reach this
answer?
Yes!
When you have level cash flows there
is a short cut you can use …
copyright © 2003 McGraw Hill Ryerson Limited
3-40
Multiple Cash Flows
• PV
of an Annuity: the Short Cut!
PV of an annuity = C x [ 1/r – 1/(r(1 + r)t)]
(Where C = Cash Payment)
PVannuity = $4,000 x [1/0.10 – 1/(0.10 (1 + 0.10)3)]
= $4,000 x 2.48685
= $9,947.41

Using the PV of an annuity calculation, we get
the same answer as before:
Put
aside $9,947.41 to fund the cash flows.
copyright © 2003 McGraw Hill Ryerson Limited
3-41
Multiple Cash Flows
• Calculating the FV of an Annuity
 Suppose interest rates are 10% and you decide to
save $4,000 per year for 20 years. How much will
you have saved for your retirement?
 This is a FV problem.
 We could use the same procedure as we used for
multiple cash flows of different sizes:
 Calculate
what each cash flow would be worth in, 20
years, i.e. get its FV.
 Add up these future values.
Can you see the problem with using this method?
copyright © 2003 McGraw Hill Ryerson Limited
3-42
Multiple Cash Flows
• Calculating

the FV of an Annuity
Calculating the FV this way would mean working
out the FV for 20 separate cash flows ...


Is there an easier way?
Yes!
When you have level cash flows there
is a short cut you can use …
copyright © 2003 McGraw Hill Ryerson Limited
3-43
Multiple Cash Flows
•
FV of an Annuity: the Short Cut!
FV of an annuity = C x [ ((1 + r)t – 1)/r ]
(Where C = Cash Payment)
FVannuity = $4,000 x [ ((1 + 0.10)20 – 1) / 0.10 ]
= $4,000 x 57.27499949
= $229,100

Using the FV of an annuity calculation, we
see that you will have $229,100 in your
account when you retire in 20 years.
copyright © 2003 McGraw Hill Ryerson Limited
3-44
Multiple Cash Flows
• Our


First Question …
You now have all the tools necessary to
answer the very first question we asked!
Give it a try:
 Assume
interest rates are 4.3884%. You have just
won a lottery and must choose between the
following two options:
o
o
Receive a cheque for $150,000 today.
Receive $10,000 a year for the next 25 years.
Which option gives you the biggest “winnings”?
copyright © 2003 McGraw Hill Ryerson Limited
3-45
Multiple Cash Flows
• Our


First Question …
Option 1 is worth $150,000.
To value Option 2, find the PV of $10,000 per year
for 25 years at 4.3884%:
PV of an annuity = C x [ 1/r – 1/(r(1 + r)t )]
PV = $10,000 x [1/0.043884 – 1/0.043884 (1 + .043884)25]
= $10,000 x 15.000
= $150,000
Both options are worth $150,000!
copyright © 2003 McGraw Hill Ryerson Limited
3-46
Multiple Cash Flows
• Cash


Flows Growing at a Constant Rate
What if the cash flows in a financial problem
are not equal, but are instead growing at a
constant rate?
For example:
 Assume
the discount rate is 8%. You are thinking of
buying a condo which generates $12,000 per year
in net cash flow in perpetuity. These cash flows
grow at 3% per year.
What is the maximum price you
should pay for this condo?
copyright © 2003 McGraw Hill Ryerson Limited
3-47
Multiple Cash Flows
• Valuing

Growing Perpetuities
The PV of a growing perpetuity is calculated by
dividing the cash payment by the discount rate
less the growth rate:
PV of a perpetuity =
=
=
C
r-g
=
Cash Payment
Discount Rate – Growth Rate
$12,000
0.08 – 0.03
$240,000
copyright © 2003 McGraw Hill Ryerson Limited
3-48
Multiple Cash Flows
• Cash



Flows Growing at a Constant Rate
In the previous problem we assumed that the
cash flows grew at a constant rate forever.
It may be more reasonable to assume a
constant growth rate for a limited time period.
For example:
 Assume
the condo in the previous problem
generates $12,000 per year in net cash flow for 20
years. These cash flows grow at 3% per year.
Now, what is the maximum price you
should pay for this condo?
copyright © 2003 McGraw Hill Ryerson Limited
3-49
Multiple Cash Flows
• Valuing

Finite Growing Cash Flows
The PV of cash flows which grow at a constant
rate for a limited time period (T) is calculated by:
T
PV of cash flow =
C
r-g
(1- [ (1 + g) ] )
(1 + r)
20
=
$12,000
0.08 – 0.03
(1- [ (1 + 0.03)] )
=
$240,000 * 0.61250
=
$147,000
(1 + 0.08)
copyright © 2003 McGraw Hill Ryerson Limited
3-50
Effective Interest Rates
• EAR




vs APR
So far, we have used annual interest rates
applied to annual cash flows.
But interest can be applied daily, weekly,
monthly, semi-annually – or for any other
convenient time period.
The simplest way to deal with this situation is to
convert to annual rates.
There are two ways you may do this:
 Calculate
the Effective Annual Rate (EAR).
 Calculate the Annual Percentage Rate (APR).
copyright © 2003 McGraw Hill Ryerson Limited
3-51
Effective Interest Rates
• Annual


Percentage Rate (APR)
The annual percentage rate (APR) is an
interest rate that is annualized using simple
interest.
For example:
 Your
credit card charges 1.5% per month. What is
the annual charge?
APR = Quoted rate x Number of Periods Per Year
= 1.5% x 12
= 18%
copyright © 2003 McGraw Hill Ryerson Limited
3-52
Effective Interest Rates
• Effective Annual


Interest Rate (EAR)
The effective annual interest rate (EAR) is an
interest rate that is annualized using
compound interest.
For example:
 Your
credit card charges 1.5% per month. What is
the annual charge?
EAR = (1 + Quoted rate) Number of Periods Per Year
= (1 + 0.015)12
= 19.5%
copyright © 2003 McGraw Hill Ryerson Limited
3-53
Effective Interest Rates
• Calculating

the EAR
Convert the APR to a period rate and then apply the
equation:
(1 + Period Rate) m
Quoted APR: 12% (a)
Compounding
Period
1 year
Semiannually
Quarterly
Monthly
Daily
(m = Number of periods per year)
(b)
(c)
Periods
per Year
(m)
1
2
4
12
365
Period Rate
(= a/b)
12.0000%
6.0000%
3.0000%
1.0000%
0.0329%
EAR
m
= (1+ c) - 1
12.0000%
12.3600%
12.5509%
12.6825%
12.7475%
copyright © 2003 McGraw Hill Ryerson Limited
3-54
Summary of Chapter 3




Future value (FV) is the amount to which an
investment will grow after earning interest.
Find the FV of an investment by multiplying the
t
investment by the future value factor of (1+r)
where t is the time period and r is the discount
rate.
The present value (PV) of a future cash payment
is the amount you would need to invest today to
create that future cash payment.
Find the PV of an investment by multiplying the
future cash payment by the discount factor of
1/(1+r)t.
copyright © 2003 McGraw Hill Ryerson Limited
3-55
Summary of Chapter 3




Apply the formulas for FV and PV to calculate the
value of multiple cash flows.
A level stream of payments which continues
forever is called a perpetuity.
One which continues for a limited number of years
is called an annuity.
You can use the FV and PV formulas to calculate
their value or you can use the shortcut formulas.
 Variations
on the shortcut formulas allow you to
calculate the value of multiple cash flows growing
at a constant rate.
copyright © 2003 McGraw Hill Ryerson Limited
3-56
Summary of Chapter 3




Interest rates for periods of less than one year
are often quoted as annual rates by converting
to either an APR or an EAR.
Annual percentage rates (APR) do not
recognize the effect of compound interest, that
is, they annualize assuming simple interest.
Effective Annual Rates (EAR) annualize using
compound interest.
EAR equals the rate of interest per period
compounded for the number of periods in a
year.
copyright © 2003 McGraw Hill Ryerson Limited