INTRODUCTION TO
CORPORATE FINANCE
Laurence Booth • W. Sean Cleary
Prepared by
Ken Hartviksen and Robert Ironside
CHAPTER 5
Time Value of Money
Lecture Agenda
•
•
•
•
•
•
•
•
Learning Objectives
Important Terms
Compounding
Discounting
Annuities and Loans
Perpetuities
Effective Rates of Return
Summary and Conclusions
– Concept Review Questions
– Practice Problems
CHAPTER 5 – Time Value of Money
5-3
Learning Objectives
• Understand the importance of the time value of
money
• Understand the difference between simple interest
and compound interest
• Know how to solve for present value, future value,
time or rate
• Understand annuities and perpetuities
• Know how to construct an amortization table
CHAPTER 5 – Time Value of Money
5-4
Important Chapter Terms
•
•
•
•
•
•
•
Amortize
Annuity
Annuity due
Basis point
Cash flows
Compound interest
Compound interest factor
(CVIF)
• Discount rate
• Discounting
• Effective rate
•
•
•
•
•
•
•
•
•
•
Lessee
Medium of exchange
Mortgage
Ordinary annuities
Perpetuities
Present value interest factor
(PVIF)
Reinvested
Required rate of return
Simple interest
Time value of money
CHAPTER 5 – Time Value of Money
5-5
The Time Value of Money Concept
• Cannot directly compare $1 today with $1 to be
received at some future date
– Money received today can be invested to earn a rate
of return
– Thus $1 today is worth more than $1 to be received at
some future date
• The interest rate or discount rate is the variable
that equates a present value today with a future
value at some later date
CHAPTER 5 – Time Value of Money
5-6
Opportunity Cost
Opportunity cost = Alternative use
– The opportunity cost of money is the interest rate that
would be earned by investing it
– It is the underlying reason for the time value of money
– Money today can be invested to be some greater
amount in the future
– Conversely, if you are promised a cash flow in the
future, it’s present value today is less than what is
promised!
CHAPTER 5 – Time Value of Money
5-7
Choosing from Investment Alternatives
Required Rate of Return or Discount Rate
•
You have three choices:
1. $20,000 received today
2. $31,000 received in 5 years
3. $3,000 per year indefinitely
•
•
To make a decision, you need to know what
interest rate to use
This interest rate is known as your required
rate of return or discount rate.
CHAPTER 5 – Time Value of Money
5-8
Simple Interest
• Simple interest is interest paid or received on
only the initial investment (or principal)
• At the end of the investment period, the
principal plus interest is received
0
1
I1
2
I2
3
I3
CHAPTER 5 – Time Value of Money
n
…
In+P
5-9
Simple Interest
Example
PROBLEM:
Invest $1,000 today for a
five-year term and receive 8
percent annual simple
interest.
SOLUTION:
Annual interest = $1,000 ×
.08 = $80 per year.
Year
1
2
3
4
5
Beginning Amount
$1,000
1,080
1,160
1,240
1,320
Ending Amount
$1,080
1,160
1,240
1,320
$1,400
Value (time n)  P  (n  P  k)
Value5
 $1,000  (5  $1,000  .08)
 $1,000  (5  $80)
 $1,000  $400
 $1,400
CHAPTER 5 – Time Value of Money
5 - 10
Simple Interest
General Formula
[ 5-1]
Value (time n)  P  (n  P  k)
Where:
P = principal invested
n = number of years
k = interest rate
CHAPTER 5 – Time Value of Money
5 - 11
Compound Interest
Compounding (Computing Future Values)
• Simple interest problems are rare; in finance
we are most interested in compound interest
• Compound interest is interest that is earned
on the principal amount invested and on any
accrued interest
CHAPTER 5 – Time Value of Money
5 - 12
Compound Interest
Example
PROBLEM:
Invest $1,000 today for a five-year term and receive 8
percent annual compound interest. How much will the
accumulated value be at time 5?
SOLUTION:
Year
Beginning
Amount
Ending
Amount
1
$1,000.00
$1,080.00
2
1,080.00
1,166.40
FV2  P (1  .08)(1  .08)  P (1  .08) 2  $1,166.40
3
1,166.40
1,259.71
FV3  P (1  .08)(1  .08)(1  .08)  P (1.08) 3  $1,259.71
4
1,259.71
1,360.49
FV4  P (1.08)(1.08)(1.08)(1.08)  P (1.08) 4  $1,360.49
5
1,360.49
1,469.33
FV5  P (1  .08) 5  $1,469.33
Future Value  P  ( 1  k)n
FV1  P (1  .08)1  $1,080
CHAPTER 5 – Time Value of Money
5 - 13
Compound Interest
Example of Interest Earned on Interest
PROBLEM:
Invest $1,000 today for a five-year term and receive 8 percent annual
compound interest.
The Interest earned on Interest Effect:
Interest (year 1) = $1,000 × .08 = $80
Interest (year 2 ) =($1,000 + $80)×.08 = $86.40
Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31
Year
1
2
3
4
5
Beginning Amount
$1,000.00
1,080.00
1,166.40
1,259.71
1,360.49
Ending Amount
$1,080.00
1,166.40
1,259.71
1,360.49
1,469.33
CHAPTER 5 – Time Value of Money
Interest earned
in the year
$80.00
$86.40
$93.31
$100.78
$108.84
5 - 14
Compound Interest
General Formula
[ 5-2]
FVn  PV0( 1  k)
n
Where:
FV= future value
P = principal invested
n = number of years
k = interest rate
( 1  k)n is known as the compound interest factor
CHAPTER 5 – Time Value of Money
5 - 15
Compound Interest
Solution Using a Financial Calculator (TI BA II Plus)
Input the following variables:
→
PMT
; -1,000 →
Press
CPT
(Compute) and then
0
PV
; 10 →
I/Y
; and
5→
N
FV
PMT refers to regular payments
FV is the future value
I/Y is the period interest rate
N is the number of periods
PV is entered with a negative sign to reflect investors must pay money now
to get money in the future.
Answer = $1,610.51
CHAPTER 5 – Time Value of Money
5 - 16
Compound Interest
Simple versus Compound Interest
• Compounding of interest magnifies the returns on
an investment
• Returns are magnified
• The longer they are compounded
• The higher the rate they are compounded
(See Figure 5-1 to compare simple and compound interest effects over time)
CHAPTER 5 – Time Value of Money
5 - 17
Compound Interest
Simple versus Compound Interest
FIGURE 5-1
8,000
7,000
DOLLARS
6,000
5,000
4,000
3,000
2,000
1,000
0
1
2
3
4
5
6
Simple
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Compound
CHAPTER 5 – Time Value of Money
5 - 18
Compound Interest
Compounded Returns over Time for Various Asset Classes
Table 5-1 Ending Wealth of $1,000 Invested From 1938 to 2005 in Various Asset Classes
Government of Canada treasury bills
Government of Canada bonds
Canadian stocks
U.S. stocks
Annual
Arithmetic
Average (%)
Annual
Geometric
Mean (%)
5.20
6.62
11.79
13.15
5.11
6.24
10.60
11.76
Yeark-End Value,
2005 ($)
$29,711
61,404
946,009
1,923,692
So urce: Data are fro m the Canadian Institute o f A ctuaries
CHAPTER 5 – Time Value of Money
5 - 19
Compound Interest
Discounting (Computing Present Values)
[ 5-3]
FVn
1
PV0 
 FVn 
n
(1  k )
( 1  k)n
CHAPTER 5 – Time Value of Money
5 - 20
Computing Present Value
• The present Value is an amount today that equates to
some larger amount in the future
• Example: We know we want $1,000,000 when we
retire 40 years from today. If we can earn a 10%
return on our money, how much should we invest
today?
PV0 =
=
FVn
1+ k 
n
1, 000, 000
1.10 
40
= $22, 094.93
Calculator Approach:
1,000,000
FV
0
PMT
40
N
10
I/Y
CPT PV
22,094.93
CHAPTER 5 – Time Value of Money
5 - 21
Compound Interest
Determining Rates of Return or Holding Periods
FVn  PV (1  k )
CHAPTER 5 – Time Value of Money
n
5 - 22
Calculating the Rate of Return
• If we know the present value, the future value and the
number of time periods, we can calculate the rate of
return we have earned
• For example, suppose we invested $5,000 six years ago
Today, it is worth $10,000. What is the annually
compounded rate
of returned?
n
FVn = PV0 1  k 
1
n
 FVn 
k=
 1
PV
0 

1
6
 10, 000 

1

 5, 000 
 12.25%
Calculator Approach:
10,000
FV
0
PMT
5,000 +/PV
6
N
CPT I/Y
12.25%
CHAPTER 5 – Time Value of Money
5 - 23
Annuities and Perpetuities
Annuities
• Thus far, we have dealt only with single
payments, either today or in the future
• An annuity is a stream of payments that
continues for a finite period of time
• If the payment occurs at the end of the period,
it is an ordinary annuity
• If the payment occurs at the start of the time
period, it is an annuity due
CHAPTER 5 – Time Value of Money
5 - 24
Difference Between Annuity Types
Ordinary Annuity
0
1
2
3
$100
$100
$100
Annuity Due
0
1
2
3
$100
$100
$100
$100
CHAPTER 5 – Time Value of Money
5 - 25
Annuities and Perpetuities
Ordinary Annuities
[ 5-4]
[ 5-5]
( 1  k)n  1
FVn  PMT
k
1

1  (1  k ) n
PV0  PMT 
k


CHAPTER 5 – Time Value of Money





5 - 26
Annuities and Perpetuities
Annuities Due
[ 5-6]
[ 5-7]
 ( 1  k)n  1
FVn  PMT 
 (1  k )
k


1

1

 (1  k ) n
PV0  PMT 
k


CHAPTER 5 – Time Value of Money


 (1  k)


5 - 27
Future Value of an Ordinary Annuity
• Assume that we want to save $2,000 at the end of each
year for the next 10 years. If we can earn 10% on our
investments, how much will we have saved?
 1  k n  1 
FVOrdinary = PMT 



k
Annuity


 1.10 10  1 
 2, 000 



0.10


 $31,874.85
Calculator Approach:
2,000
PMT
0
PV
10
N
10
I/Y
CPT FV
31,874.85
CHAPTER 5 – Time Value of Money
5 - 28
Future Value of an Annuity Due
• Assume that we want to save $2,000 at the beginning
of each year for the next 10 years. If we can earn 10%
on our investments, how much will we have saved?
 1  k n  1 
FVAnnuity = PMT 
 1  k 


k
Due


 1.10 10  1 
 2, 000 
 1.10 
 0.10 


 $35, 062.33
Calculator Approach:
2nd BGN 2nd Set
2,000
PMT
0
PV
10
N
10
I/Y
CPT FV
35,062.33
CHAPTER 5 – Time Value of Money
5 - 29
Relationship Between An Annuity Due and An
Ordinary Annuity
•
•
•
•
The future value of the ordinary annuity is $ 31,874.85
The FV of the annuity due is $ 35,062.33
The interest rate is 10%
Now calculate how much larger the annuity due is
compared to the ordinary annuity
% 
P1  P0
P0
35, 062.33  31,874.85
31,874.85
 10%

CHAPTER 5 – Time Value of Money
5 - 30
Present Value of an Ordinary Annuity
• You have just won a lottery. The Lottery Corporation
gives you two options. You can take $1,000,000 at the
end of each year for 25 years or a lump sum of
$10,000,000 today. If the appropriate discount rate is
10%, what should you do?
 1  1  k  n 
PVOrdinary = PMT 



k
Annuity


 1  1.10 25 
 1, 000, 000 



0.10


 $9, 077, 040.02
Calculator Approach:
1,000,000
PMT
0
FV
25
N
10
I/Y
CPT PV
9,077,040.02
CHAPTER 5 – Time Value of Money
5 - 31
Present Value of an Annuity Due
• Lets continue with the example from the previous page,
but now the Lottery Corporation gives you the option of
taking $1,000,000 at the beginning of each year for 25
years or a lump sum of $10,000,000 today. If the
appropriate discount rate is 10%, what should you do?
 1  1  k  
PVAnnuity = PMT 
 1  k 


k
Due


 1  1.10 25 
 1, 000, 000 
 1.10 


0.10


 $9,984, 744.02
n
Calculator Approach:
CHAPTER 5 – Time Value of Money
2nd BGN 2nd Set
1,000,000
PMT
0
FV
25
N
10
I/Y
CPT PV
9,984,744.02
5 - 32
Annuities and Perpetuities
Perpetuities
• A perpetuity is a stream of cash flows that goes
on forever
• Examples of perpetuities in financial markets
includes:
– Common stock
– Preferred stock
– Consol bonds (bonds with no maturity date)
0
1
2
3
$100
$100
$100
CHAPTER 5 – Time Value of Money

5 - 33
Annuities and Perpetuities
PV of a Perpetuity
PMT
PV0 
k
[ 5-8]
Where:
PV0 = Present value of the perpetuity
PMT = the periodic cash
K = the discount rate
CHAPTER 5 – Time Value of Money
5 - 34
Perpetuity: An Example
• While acting as executor for a distant relative,
you discover a $1,000 Consol Bond issued by
Great Britain in 1814, issued to help fund the
Napoleonic War. If the bond pays annual
interest of 3.0% and other long U.K.
Government bonds are currently paying 5%,
what would each $1,000 Consol Bond sell for in
the market?
CHAPTER 5 – Time Value of Money
5 - 35
Perpetuity: Solution
PMT
PV0 
k
$1, 000  0.03 

0.05
$30

0.05
 $600
CHAPTER 5 – Time Value of Money
5 - 36
Nominal Versus Effective Interest Rates
• So far, we have assumed annual compounding
• When rates are compounded annually, the
quoted rate and the effective rate are equal
• As the number of compounding periods per
year increases, the effective rate will become
larger than the quoted rate
CHAPTER 5 – Time Value of Money
5 - 37
Nominal versus Effective Rates
Determining Effective Annual Rates
• Effective rate for a period is the rate at which a
dollar invested grows over that period
Determining effective annual rate for a given compound interval
[ 5-9]
QR m
k  (1 
) 1
m
CHAPTER 5 – Time Value of Money
5 - 38
Nominal versus Effective Rates
Determining Effective Annual Rates
Determining effective annual rate when compounding is conducted
on a continuous basis
[ 5-10]
k e
QR
1
CHAPTER 5 – Time Value of Money
5 - 39
Nominal versus Effective Rates
Effective Rates for “Any” Period
Determining effective rate for any period, given any quoted rates
[ 5-11]
m
f
QR
k  (1
) -1
m
CHAPTER 5 – Time Value of Money
5 - 40
Calculating the Effective Rate
m
k Effective
 QR 
 1 
 1
m 

Where:
kEffective = Effective annual interest rate
QR = the quoted interest rate
M = the number of compounding periods per year
CHAPTER 5 – Time Value of Money
5 - 41
Example: Effective Rate Calculation
• A bank is offering loans at 6%, compounded monthly.
What is the effective annual interest rate on its loans?
m
k Effective
 QR 
 1 
 1
m 

12
 .06 
 1 
 1
12 

 6.17%
CHAPTER 5 – Time Value of Money
5 - 42
Continuous Compounding
• When compounding occurs continuously, we
calculate the effective annual rate using e, the
base of the natural logarithms (approximately
2.7183)
kEffective  eQR  1
CHAPTER 5 – Time Value of Money
5 - 43
10% Compounded At Various Frequencies
Compounding
Frequency
Effective Annual
Interest Rate
2
10.25%
4
10.3813%
12
10.4713%
52
10.5065%
365
10.5156%
Continuous
10.5171%
CHAPTER 5 – Time Value of Money
5 - 44
Calculating the Rate of Return
• If we know the present value, the future value and the
number of time periods, we can calculate the rate of
return we have earned
• For example, suppose we invested $5,000 six years ago
Today, it is worth $10,000. What is the annually
compounded rate
of returned?
n
FVn = PV0 1  k 
1
n
 FVn 
k=
 1
PV
0 

1
6
 10, 000 

1

 5, 000 
 12.25%
Calculator Approach:
10,000
FV
0
PMT
5,000 +/PV
6
N
CPT I/Y
12.25%
CHAPTER 5 – Time Value of Money
5 - 45
Calculating the Number of Periods
• If we know the present value, the future value and the
rate of return, we can calculate the number of time
periods the money needs to be invested for.
• For example, suppose we invested $25,000 at 8%.
Today, it is worth $40,000. How long has the money
been invested?
FVn = PV0 1  k 
n
 FVn 
ln 

PV0 

n=
ln 1  k 
 40, 000 
ln 
25, 000 



ln 1.08 
 6.11 years
Calculator Approach:
40,000
FV
0
PMT
25,000 +/PV
8.0
I/Y
CPT N
6.11 years
CHAPTER 5 – Time Value of Money
5 - 46
Calculating the Quoted Rate
• If we know the effective annual interest rate, (kEff) and
we know the number of compounding periods, (m) we
can solve for the Quoted Rate, as follows:

QR  1  kEff


1
m

 1 m

CHAPTER 5 – Time Value of Money
5 - 47
When Payment & Compounding Periods Differ
• When the number of payments per year is
different from the number of compounding
periods per year, you must calculate the
interest rate per payment period, using the
following formula
m
 QR  f
kPer  1 
 1
m 
Period

Where:
f = the payment frequency per year
CHAPTER 5 – Time Value of Money
5 - 48
Loan Amortization
• A blended payment loan is repaid in equal
periodic payments
• However, the amount of principal and interest
varies each period
• Assume that we want to calculate an
amortization table showing the amount of
principal and interest paid each period for a
$5,000 loan at 10% repaid in three equal annual
instalments.
CHAPTER 5 – Time Value of Money
5 - 49
Loan Amortization: Solution
• First calculate the annual payments
 1  1  k  n
PVAnnuity  PMT 

k

PVAnnuity
PMT 
 1  1  k  n 




k



5, 000
 1  1.10 3 




0.10


 $2, 010.57




Calculator Approach:
5,000
PV
0
FV
3
N
10
I/Y
CPT PMT
$2,010.57
CHAPTER 5 – Time Value of Money
5 - 50
Amortization Table
Period Principal:
Start of
Period
Payment Interest
Principal
Principal:
End of
Period
1
5,000.00 2010.57 500.00
1,510.57 3,489.43
2
3,489.43 2010.57 348.94
1,661.63 1,827.80
3
1,827.80 2010.57 182.78
1,827.78 0
CHAPTER 5 – Time Value of Money
5 - 51
Calculating the Balance O/S
• At any point in time, the balance outstanding on
the loan (the principal not yet repaid) is the PV
of the loan payments not yet made.
• For example, using the previous example, we
can calculate the balance outstanding at the
end of the first year, as shown on the next slide
CHAPTER 5 – Time Value of Money
5 - 52
Calculating the Balance O/S after the 1st Year
1  1  k  n 
PVt 1  PMT 

k


1  1.10 2 
 2, 010.57 

.10


 $3, 489.42
CHAPTER 5 – Time Value of Money
5 - 53
Loan or Mortgage Arrangements
Mortgages
• Mortgages – a loan involving equal ‘blended’
payments (interest and principal) over a
specified payment period
• Important to distinguish between “term” and
“amortization period”
– Term – the period for which investors can ‘lock in’
at a fixed rate
– Amortization period – the period over which the
loan is to be repaid
CHAPTER 5 – Time Value of Money
5 - 54
Canadian Residential Mortgages
• By law, banks in Canada can only compound the
interest twice per year on a conventional
mortgage, but payments are typically made at
least monthly
• To solve for the payment, you must first
calculate the correct periodic interest rate
CHAPTER 5 – Time Value of Money
5 - 55
Canadian Residential Mortgages
• For example, suppose we want to calculate the
monthly payment on a $100,000 mortgage amortized
over 25 years with a 6% annual interest rate.
• First, calculate the monthly interest rate:
k Per
Period
m
f
 QR 
 1 
 1
m 

2
12
 .06 
 1 
 1
2 

 .004938622 or 0.4938622%
CHAPTER 5 – Time Value of Money
5 - 56
Calculating the Monthly Payment
• Now, calculate the monthly payment on the mortgage
1  1  k  n 
PVt 0  PMT 

k


PVt 0
PMT 
1  1  k  n 


k


100, 000

1  1.004938622 300 


.004938622


 $639.81
Calculator Approach:
100,000
PV
0
FV
300
N
.4938622
I/Y
CPT PMT
$639.81
CHAPTER 5 – Time Value of Money
5 - 57
Summary and Conclusions
In this chapter you have learned:
– To compare cash flows that occur at different points in time
– To determine economically equivalent future values from values
that occur in previous periods through compounding.
– To determine economically equivalent present values from cash
flows that occur in the future through discounting
– To find present value and future values of annuities, and
– To determine effective annual rates of return from quoted
interest rates.
CHAPTER 5 – Time Value of Money
5 - 58
Practice Problem 1
Loan Payments
Your sister has been forced to borrow money to
pay her tuition this year. If she makes annual
payments on the loan at year end for the next
three years, and the loan is for $2,500 at a
simple interest rate of 6 percent, how much will
she pay each year?
CHAPTER 5 – Time Value of Money
5 - 59
Practice Problem 1
Loan Payments
Your sister has been forced to borrow money to
pay her tuition this year. If she makes annual
payments on the loan at year end for the next
three years, and the loan is for $2,500 at a
simple interest rate of 6 percent, how much will
she pay each year?
 $2,500 
PMT  
 ($2,500  .06)

 3 
[ 5-5]
 $833.33  $150.00
 $983.33
CHAPTER 5 – Time Value of Money
5 - 60
Copyright
Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights
reserved. Reproduction or translation of this work beyond that
permitted by Access Copyright (the Canadian copyright licensing
agency) is unlawful. Requests for further information should be
addressed to the Permissions Department, John Wiley & Sons
Canada, Ltd. The purchaser may make back-up copies for his or her
own use only and not for distribution or resale. The author and the
publisher assume no responsibility for errors, omissions, or
damages caused by the use of these files or programs or from the
use of the information contained herein.
CHAPTER 5 – Time Value of Money
5 - 61