# 4. Time Value of Money - Contemporary Financial Management ```Contemporary Financial Management
Chapter 4:
Time Value of Money
&copy; 2004 by Nelson, a division of Thomson Canada Limited
Introduction
● This chapter introduces the concepts and skills
necessary to understand the time value of money
and its applications.
&copy; 2004 by Nelson, a division of Thomson Canada Limited
2
Payment of Interest
● Interest is the cost of money
● Interest may be calculated as:
● Simple interest
● Compound interest
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3
Simple Interest
● Interest paid only on the initial principal
Example: \$1,000 is invested to earn 6% per year,
simple interest.
0
1
2
3
-\$1,000
\$60
\$60
\$60
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4
Compound Interest
● Interest paid on both the initial principal and on
interest that has been paid &amp; reinvested.
Example: \$1,000 invested to earn 6% per year,
compounded annually.
0
1
2
3
-\$1,000
\$60.00
\$63.60
\$67.42
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5
Future Value
● The value of an investment at a point in the
future, given some rate of return.
Simple Interest
Compound Interest
FVn = PV0+(PV0  i  n)
FVn = PV0(1 + i)n
FV = future value
PV = present value
i = interest rate
n = number of periods
FV = future value
PV = present value
i = interest rate
n = number of periods
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6
Future Value: Simple Interest
Example: You invest \$1,000 for three years at
6% simple interest per year.
6%
0
1
6%
2
6%
3
-\$1,000
FV3 = PV0+(PV0  i  n)
= \$1,000  \$1,000  0.06  3 
= \$1,180.00
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7
Future Value: Compound Interest
Example: You invest \$1,000 for three years at
6%, compounded annually.
0
6%
1
6%
2
6%
3
-\$1,000
FV3 = PV0 (1 + i)n
= \$1,000 1  0.06 
3
= \$1,191.02
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8
Future Value: Compound Interest
● Future values can be calculated using a table
method, whereby “future value interest factors”
(FVIF) are provided.
● See Table 4.1 (page 135)
FVn = PV0(FVIFi,n ), where: FVIFi,n = 1+i
n
FV = future value
PV = present value
FVIF = future value interest factor
i = interest rate
n = number of periods
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9
Future Value: Compound Interest
Example: You invest \$1,000 for three years at
6% compounded annually.
Table 4.1 Excerpt: FVIFs for \$1
End of Period (n)
5%
6%
8%
2
3
4
1.102
1.158
1.216
1.124
1.191
1.262
1.166
1.260
1.360
FV3 = PV0 (FVIF6%,3 )
=\$1,000(1.191) =\$1,191.00
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10
Present Value
● What a future sum of money is worth today, given
a particular interest (or discount) rate.
PV0 
FVn
1+i
n
FV = future value
PV = present value
i = interest (or discount) rate
n = number of periods
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11
Present Value
Example: You will receive \$1,000 in three years.
If the discount rate is 6%, what is the present
value?
0
6%
1
6%
2
6%
3
\$1,000
PV0 
FV3
1+i
n

\$1,000
1  0.06 
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3
 \$839.62
12
Present Value
● Present values can be calculated using a table
method, whereby “present value interest factors”
(PVIF) are provided.
● See Table 4.2 (page 139)
PV0 = FVn(PVIFi,n ), where: PVIFi,n =
1
1+i
n
FV = future value
PV = present value
PVIF = present value interest factor
i = interest rate
n = number of periods
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13
Present Value
Example: What is the present value of \$1,000 to
be received in three years, given a discount rate
of 6%?
Table 4.2 Excerpt: PVIFs for \$1
End of Period (n)
5%
6%
8%
2
3
4
0.907
0.864
0.823
0.890
0.840
0.792
0.857
0.794
0.735
PV0 = FV3(FVIF6%,3 )
=\$1,000(0.840) =\$840.00
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14
A Note of Caution
● Note that the algebraic solution to the present
value problem gave an answer of 839.62
● The table method gave an answer of \$840.
Caution:
If more accuracy is required, use algebra!
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15
Annuities
● The payment or receipt of an equal cash flow
per period, for a specified number of periods.
Examples: mortgages, car leases, retirement
income.
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Annuities
● Ordinary annuity: cash flows occur at the end
of each period
Example: 3-year, \$100 ordinary annuity
0
1
2
3
\$100
\$100
\$100
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Annuities
● Annuity Due: cash flows occur at the beginning
of each period
Example: 3-year, \$100 annuity due
0
1
2
\$100
\$100
\$100
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3
18
Difference Between Annuity Types
Ordinary Annuity
0
1
2
3
\$100
\$100
\$100
Annuity Due
0
1
2
3
\$100
\$100
\$100
\$100
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Annuities: Future Value
● Future value of an annuity - sum of the future
values of all individual cash flows.
0
1
2
3
\$100
\$100
\$100
FV
FV
FV
FV of Annuity
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20
Annuities: Future Value – Algebra
● Future value of an ordinary annuity
 1+in -1 
FVOrdinary= PMT 



i
Annuity


FV = future value of the annuity
PMT = equal periodic cash flow
i = the (annually compounded) interest rate
n = number of years
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21
Annuities: Future Value
Example: What is the future value of a three
year ordinary annuity with a cash flow of \$100
per year, earning 6%?
 1+in -1 
FVOrdinary= PMT 



i
Annuity


 1.06 3  1 
 100 



.
06


 \$318.36
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Annuities: Future Value – Algebra
● Future value of an annuity due:
 1+in -1 
FVAnnuity= PMT 
 1 + i


i
Due


FV = future value of the annuity
PMT = equal periodic cash flow
i = the (annually compounded) interest rate
n = number of years
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23
Annuities: Future Value – Algebra
Example: What is the future value of a three
year annuity due with a cash flow of \$100 per
year, earning 6%?
 1+in -1 
FVAnnuity= PMT 
 1+i


i
Due


 1.06 3  1 
 100 
 1.06 


.
06


 \$337.46
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24
Annuities: Future Value – Table
● The future value of an ordinary annuity can be
calculated using Table 4.3 (p. 145), where
“future value of an ordinary annuity interest
factors” (FVIFA) are provided.
FVANn = PMT(FVIFAi,n ), where:
1  i
n
FVIFAi,n =
1
i
PMT = equal periodic cash flow
i = the (annually compounded) interest rate
n = number of periods
FVAN = future value (ordinary annuity)
FVIFA = future value interest factor
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25
Ordinary Annuity: Future Value
Example: What is the future value of a 3-year
\$100 ordinary annuity if the cash flows are
invested at 6%, compounded annually?
Table 4.3 Excerpt: FVIFA for \$1 per period
End of Period (n)
5%
6%
10%
2
3
4
2.050
3.152
4.310
2.060
3.184
4.375
2.100
3.310
4.641
FVANn = PMT(FVIFAi,n )
=\$100 3.184   \$318.40
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26
Annuity Due: Future Value
● Calculated using Table 4.3 (p. 145), where
FVIFAs are found. Ordinary annuity formula is
adjusted to reflect one extra period of interest.
FVANDn = PMT FVIFAi,n 1  i , where:
n
1  i  1

FVIFAi,n =
i
PMT = equal periodic cash flow
i = the (annually compounded) interest rate
n = number of periods
FVAND = future value (annuity due)
FVIFA = future value interest factor
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27
Annuity Due: Future Value
Example: What is the future value of a 3-year
\$100 annuity due if the cash flows are invested
at 6% compounded annually?
Table 4.3 Excerpt: FVIFA for \$1 per period
End of Period (n)
5%
6%
10%
2
3
4
2.050
3.152
4.310
2.060
3.184
4.375
2.100
3.310
4.641
FVANDn = PMT FVIFAi,n 1  i 
 \$100 3.184(1.06)  \$337.50
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Annuities: Present Value
● The present value of an annuity is the sum of
the present values of all individual cash flows.
0
1
2
3
PV
PV
PV
PV of Annuity
\$100
\$100
\$100
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Annuities: Present Value – Algebra
● Present value of an ordinary annuity
 1- 1+i-n 
PVOrdinary= PMT 



i
Annuity


PV = present value of the annuity
PMT = equal periodic cash flow
i = the (annually compounded) interest or discount rate
n = number of years
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30
Annuities: Present Value – Algebra
Example: What is the present value of a three
year, \$100 ordinary annuity, given a discount
rate of 6%?
 1- 1+i-n 
PVOrdinary=PMT 



i
Annuity


 1 - 1.06 -3 
 100 



.
06


 \$267.30
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31
Annuities: Present Value – Algebra
● Present value of an annuity due:
 1- 1+i-n 
PVAnnuity= PMT 
 1+i


i
Due


PV = present value of the annuity
PMT = equal periodic cash flow
i = the (annually compounded) interest or discount rate
n = number of years
&copy; 2004 by Nelson, a division of Thomson Canada Limited
32
Annuities: Present Value – Algebra
Example: What is the present value of a three
year, \$100 annuity due, given a discount rate
of 6%?
 1- 1+i-n 
PVAnnuity= PMT 
 1+i


i
Due


 1  1.06  3
 100 

.06

 \$283.34
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
 1.06 


33
Annuities: Present Value – Table
● The present value of an ordinary annuity can be
calculated using Table 4.4 (p. 149), where
“present value of an ordinary annuity interest
factors” (PVIFA) are found.
PVAN0 = PMT(PVIFAi,n ), where:
PVIFAi,n
1- 1+i-n 

= 
i
PMT = cash flow
i = the (annually compounded) interest or discount rate
n = number of periods
PVAN = present value (ordinary annuity)
PVIFA = present value interest factor
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34
Annuities: Present Value – Table
Example: What is the present value of a 3-year
\$100 ordinary annuity if current interest rates
are 6% compounded annually?
Table 4.4 Excerpt: PVIFA for \$1 per period
End of Period (n)
5%
6%
10%
2
3
4
1.859
2.723
3.546
1.833
2.673
3.465
1.736
2.487
3.170
PVAN0 = PMT(PVIFAi,n )
=\$100 2.673  \$267.30
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35
Annuities: Present Value – Table
● Calculated using Table 4.4 (p. 149), where
PVIFAs are found. Present value of ordinary
annuity formula is modified to account for one
less period of interest.
PVAND0 = PMT PVIFAi,n(1  i)
PVIFAi,n
1- 1+in 

= 
i
PMT = cash flow
i = the (annually compounded) interest or discount rate
n = number of periods
PVAND = present value (annuity due)
PVIFA = present value interest factor
&copy; 2004 by Nelson, a division of Thomson Canada Limited
36
Annuities: Present Value – Table
Example: What is the present value of a 3-year
\$100 annuity due if current interest rates are
6% compounded annually?
Table 4.4 Excerpt: PVIFA for \$1 per period
End of Period (n)
5%
6%
10%
2
3
4
1.859
2.723
3.546
1.833
2.673
3.465
1.736
2.487
3.170
PVAND0 = PMT PVIFAi,n(1  i)
 \$100 2.673 1.06    \$283.34
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37
Other Uses of Annuity Formulas
● Sinking Fund Problems: calculating the annuity
payment that must be received or invested each
year to produce a future value.
Ordinary Annuity
FVANn
PMT=
FVIFAi,n
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Annuity Due
FVANn
PMT=
FVIFAi,n 1  i
38
Other Uses of Annuity Formulas
● Loan Amortization and Capital Recovery
Problems: calculating the payments necessary
to pay off, or amortize, a loan.
PVAN0
PMT=
PVIFAi,n
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39
Perpetuities
● Financial instrument that pays an equal cash flow
per period into the indefinite future (i.e. to
infinity).
Example: dividend stream on common and
preferred stock
0
1
2
3
4
\$60
\$60
\$60
\$60
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
40
Perpetuities
● Present value of a perpetuity equals the sum of
the present values of each cash flow.
● Equal to a simple function of the cash flow (PMT)
and interest rate (i).
PMT
PVPER 0  
n
(1+i)
t 1

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PMT
PVPER 0 
i
41
Perpetuities
Example: What is the present value of a \$100
perpetuity, given a discount rate of 8%
compounded annually?
PMT \$100
PVPER 0 

 \$1,250.00
i
0.08
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42
More Frequent Compounding
● Nominal Interest Rate: the annual percentage
interest rate, often referred to as the Annual
Percentage Rate (APR).
Example: 12% compounded semi-annually
12%
0
-\$1,000
6%
0.5
6%
\$60.00
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1
\$63.60
6%
1.5
\$67.42
43
More Frequent Compounding
● Increased interest payment frequency requires
future and present value formulas to be
adjusted to account for the number of
compounding periods per year (m).
Future Value
Present Value
mn
inom 

FVn  PV0 1 

m


&copy; 2004 by Nelson, a division of Thomson Canada Limited
PV0 
FVn
mn
inom 

1+ m 


44
More Frequent Compounding
Example: What is a \$1,000 investment worth in
five years if it earns 8% interest, compounded
quarterly?
mn
inom 

FVn  PV0 1 

m


(4)(5)
0.08 

 \$1,000 1 

4


 \$1, 485.95
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45
More Frequent Compounding
Example: How much do you have to invest
today in order to have \$10,000 in 20 years, if
you can earn 10% interest, compounded
monthly?
PV0 

FVn
mn
inom 

1+ m 


\$10,000
(12)(20)
0.10 

1+ 12 


&copy; 2004 by Nelson, a division of Thomson Canada Limited
 \$1,364.62
46
Impact of Compounding Frequency
\$1,000 Invested at Different
10% Nominal Rates for One Year
\$1,106
\$1,105
\$1,104
\$1,103
\$1,102
\$1,101
\$1,100
\$1,099
\$1,098
\$1,097
Annual
SemiAnnual
Quarterly Monthly
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Daily
47
Effective Annual Rate (EAR)
● The annually compounded interest rate that is
identical to some nominal rate, compounded
“m” times per year.
m
ieff
inom 

 1+
1

m 

ieff  effective annual rate
inom  nominal interest rate
m = compounding frequency per year
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48
Effective Annual Rate (EAR)
● EAR provides a common basis for comparing
investment alternatives.
Example: Would you prefer an investment
offering 6.12%, compounded quarterly or one
offering 6.10%, compounded monthly?
m
ieff
inom 

 1+
1

m 

m
ieff
4
0.0612 

 1+
1

4


 6.262%
&copy; 2004 by Nelson, a division of Thomson Canada Limited
inom 

 1+
1

m 

12
0.061 

 1+

12


 6.273%
1
49
Major Points
● The time value of money underlies the valuation
of almost all real &amp; financial assets
● Present value – what something is worth today
● Future value – the dollar value of something in
the future
● Investors should be indifferent between:
● Receiving a present value today
● Receiving a future value tomorrow
● A lump sum today or in the future
● An annuity
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50
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