3-1
Chapter 3
Time Value of
Money
3-2
The Time Value of Money
? The Interest Rate
? Simple Interest
? Compound Interest
? Amortizing A Loan
3-3
Obviously,$10,000 today.
You already recognize that there is
TIME VALUE TO MONEY!!
The Interest Rate
Which would you prefer -- $10,000
today or $10,000 in 5 years?
3-4
TIME allows you the opportunity to
postpone consumption and earn
INTEREST.
Why TIME?
Why is TIME such an important
element in your decision?
3-5
Types of Interest
?Compound Interest
Interest paid (earned) on any previous
interest earned,as well as on the
principal borrowed (lent).
?Simple Interest
Interest paid (earned) on only the original
amount,or principal borrowed (lent).
3-6
Simple Interest Formula
Formula SI = P0(i)(n)
SI,Simple Interest
P0,Deposit today (t=0)
i,Interest Rate per Period
n,Number of Time Periods
3-7
?SI = P0(i)(n)
= $1,000(.07)(2)
= $140
Simple Interest Example
?Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years,What is the accumulated
interest at the end of the 2nd year?
3-8
FV = P0 + SI
= $1,000 + $140
= $1,140
?Future Value is the value at some future
time of a present amount of money,or a
series of payments,evaluated at a given
interest rate.
Simple Interest (FV)
?What is the Future Value (FV) of the
deposit?
3-9
The Present Value is simply the
$1,000 you originally deposited,
That is the value today!
?Present Value is the current value of a
future amount of money,or a series of
payments,evaluated at a given interest
rate.
Simple Interest (PV)
?What is the Present Value (PV) of the
previous problem?
3-10
0
5000
10000
15000
20000
1s t Y ea r 10 th
Y ea r
20 th
Y ea r
30 th
Y ea r
F u t u re V a l u e o f a S i n g l e $ 1,0 0 0 D e p o s i t
10 % Simple
Interest
7% C ompound
Interest
10 % C ompound
Interest
Why Compound Interest?
Fu
tu
re
Val
ue
(U
.S.
D
ol
lar
s)
3-11
Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
Future Value
Single Deposit
(Graphic)
0 1 2
$1,000
FV2
7%
3-12
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your
$1,000 deposit over the first year.
This is the same interest you would
earn under simple interest.
Future Value
Single Deposit
(Formula)
3-13
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
FV2 = FV1 (1+i)1
= P0 (1+i)(1+i) = $1,000(1.07)(1.07)
= P0 (1+i)2 = $1,000(1.07)2
= $1,144.90
You earned an EXTRA $4.90 in Year 2 with
compound over simple interest,
Future Value
Single Deposit
(Formula)
3-14
FV1 = P0(1+i)1
FV2 = P0(1+i)2
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
General Future
Value Formula
etc.
3-15
FVIFi,n is found on Table I at End
of Book or on the Card Insert.
Valuation Using Table I
Pe riod 6% 7% 8%
1 1,06 0 1,07 0 1,08 0
2 1,12 4 1,14 5 1,16 6
3 1,19 1 1,22 5 1,26 0
4 1,26 2 1,31 1 1,36 0
5 1,33 8 1,40 3 1,46 9
3-16
FV2 = $1,000 (FVIF7%,2)
= $1,000 (1.145)
= $1,145 [Due to Rounding]
Using Future Value Tables
Pe riod 6% 7% 8%
1 1,06 0 1,07 0 1,08 0
2 1,12 4 1,14 5 1,16 6
3 1,19 1 1,22 5 1,26 0
4 1,26 2 1,31 1 1,36 0
5 1,33 8 1,40 3 1,46 9
3-17
Julie Miller wants to know how large her
$10,000 deposit will become at a
compound interest rate of 10% for 5 years.
Story Problem Example
0 1 2 3 4 5
$10,000
FV5
10%
3-18
?Calculation based on Table I:
FV5 = $10,000 (FVIF10%,5)
= $10,000 (1.611)
= $16,110 [Due to Rounding]
Story Problem Solution
? Calculation based on general formula:
FVn = P0 (1+i)n
FV5 = $10,000 (1+ 0.10)5
= $16,105.10
3-19
We will use the Rule-of-72.
Double Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
3-20
Approx,Years to Double = 72 / i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]
The Rule-of-72
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
3-21
Assume that you need $1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7%.
0 1 2
$1,000
7%
PV1PV0
Present Value
Single Deposit
(Graphic)
3-22
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2
= FV2 / (1+i)2 = $873.44
Present Value
Single Deposit
(Formula)
0 1 2
$1,000
7%
PV0
3-23
PV0 = FV1 / (1+i)1
PV0 = FV2 / (1+i)2
General Present Value Formula:
PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
General Present
Value Formula
etc.
3-24
PVIFi,n is found on Table II at End
of Book or on the Card Insert.
Valuation Using Table II
Pe rio d 6% 7% 8%
1,9 43,9 35,9 26
2,8 90,8 73,8 57
3,8 40,8 16,7 94
4,7 92,7 63,7 35
5,7 47,7 13,6 81
3-25
PV2 = $1,000 (PVIF7%,2)
= $1,000 (.873)
= $873 [Due to Rounding]
Using Present Value Tables
Pe rio d 6% 7% 8%
1,9 43,9 35,9 26
2,8 90,8 73,8 57
3,8 40,8 16,7 94
4,7 92,7 63,7 35
5,7 47,7 13,6 81
3-26
Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000 in 5 years at a discount
rate of 10%.
Story Problem Example
0 1 2 3 4 5
$10,000
PV0
10%
3-27
? Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $10,000 / (1+ 0.10)5
= $6,209.21
? Calculation based on Table I:
PV0 = $10,000 (PVIF10%,5)
= $10,000 (.621)
= $6,210.00 [Due to Rounding]
Story Problem Solution
3-28
Types of Annuities
?Ordinary Annuity,Payments or receipts
occur at the end of each period.
?Annuity Due,Payments or receipts
occur at the beginning of each period.
?An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
3-29
Examples of Annuities
? Student Loan Payments
? Car Loan Payments
? Insurance Premiums
? Mortgage Payments
? Retirement Savings
3-30
Parts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
End of
Year 1
(Annuity Due)
Beginning of
Year 1
Today Equal Cash Flows
Each 1 Year Apart
(Annuity Due)
End of
Year 1
3-31
FVAn = R(1+i)n-1 + R(1+i)n-2 +
..,+ R(1+i)1 + R(1+i)0
Overview of an
Ordinary Annuity --
FVA
R R R
0 1 2 n n+1
FVAn
R,Periodic
Cash Flow
End of Year
i%,,,
3-32
FVA3 = $1,000(1.07)2 +
$1,000(1.07)1 + $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215
Example of an
Ordinary Annuity --
FVA
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
End of Year
7%
$1,070
$1,145
3-33
FVAn = R (FVIFAi%,n)
FVA3 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215
Valuation Using Table III
Pe riod 6% 7% 8%
1 1,00 0 1,00 0 1,00 0
2 2,06 0 2,07 0 2,08 0
3 3,18 4 3,21 5 3,24 6
4 4,37 5 4,44 0 4,50 6
5 5,63 7 5,75 1 5,86 7
3-34
FVADn = R(1+i)n + R(1+i)n-1 +
..,+ R(1+i)2 + R(1+i)1
= FVAn (1+i)
Overview of an
Annuity Due -- FVAD
R R R
0 1 2 n n+1
FVADn
R,Periodic
Cash Flow
Beginning of Year
i%,,,
3-35
FVAD3 = $1,000(1.07)3 +
$1,000(1.07)2 + $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440
Example of an
Annuity Due -- FVAD
$1,000 $1,000 $1,000 $1,070
0 1 2 3 4
FVAD3 = $3,440
Beginning of Year
7%
$1,225
$1,145
3-36
FVADn = R (FVIFAi%,n)(1+i)
FVAD3 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) =
$3,440
Valuation Using Table III
Pe riod 6% 7% 8%
1 1,00 0 1,00 0 1,00 0
2 2,06 0 2,07 0 2,08 0
3 3,18 4 3,21 5 3,24 6
4 4,37 5 4,44 0 4,50 6
5 5,63 7 5,75 1 5,86 7
3-37
PVAn = R/(1+i)1 + R/(1+i)2
+,.,+ R/(1+i)n
Overview of an
Ordinary Annuity --
PVA
R R R
0 1 2 n n+1
PVAn
R,Periodic
Cash Flow
End of Year
i%,,,
3-38
PVA3 = $1,000/(1.07)1 +
$1,000/(1.07)2 +
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32
Example of an
Ordinary Annuity --
PVA
$1,000 $1,000 $1,000
0 1 2 3 4
$2,624.32 = PVA3
End of Year
7%
$934.58
$873.44
$816.30
3-39
PVAn = R (PVIFAi%,n)
PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624
Valuation Using Table IV
Pe riod 6% 7% 8%
1 0,94 3 0,93 5 0,92 6
2 1,83 3 1,80 8 1,78 3
3 2,67 3 2,62 4 2,57 7
4 3,46 5 3,38 7 3,31 2
5 4,21 2 4,10 0 3,99 3
3-40
PVADn = R/(1+i)0 + R/(1+i)1 +,.,+ R/(1+i)n-1
= PVAn (1+i)
Overview of an
Annuity Due -- PVAD
R R R
0 1 2 n n+1
PVADn
R,Periodic
Cash Flow
Beginning of Year
i%,,,
3-41
PVADn = $1,000/(1.07)2 + $1,000/(1.07)1 +
$1,000/(1.07)0 = $2,808.02
Example of an
Annuity Due -- PVAD
$1,000.00 $1,000 $1,000
0 1 2 3 4
PVADn=$2,808.02
Beginning of Year
7%
$ 934.58
$ 873.44
3-42
PVADn = R (PVIFAi%,n)(1+i)
PVAD3 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) =
$2,808
Valuation Using Table IV
Pe riod 6% 7% 8%
1 0,94 3 0,93 5 0,92 6
2 1,83 3 1,80 8 1,78 3
3 2,67 3 2,62 4 2,57 7
4 3,46 5 3,38 7 3,31 2
5 4,21 2 4,10 0 3,99 3
3-43
1,Read Problem Thoroughly
2,Determine if it is a PV or FV Problem
3,Create a Time Line
4,Put Cash Flows and Arrows on Time Line
5,Determine if Solution involves a Single
CF,Annuity Stream(s),or Mixed Flow
6,Solve the Problem
Steps to Solve Time Value
of Money Problems
3-44
Julie Miller will receive the set of cash
flows below,What is the Present Value
at a discount rate of 10%.
Mixed Flows Example
0 1 2 3 4 5
$600 $600 $400 $400 $100
PV0
10%
3-45
1,Solve a piece-at-a-time by
discounting each piece back to t=0.
2,Solve a group-at-a-time by first
breaking problem into annuity
group streams and single cash
flow groups,Then discount
each group back to t=0.
How to Solve?
3-46
Piece-At-A-Time
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%
$545.45
$495.87
$300.53
$273.21
$ 62.09
$1677.15 = PV0 of the Mixed Flow
3-47
Group-At-A-Time?(#1)
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10
3-48
Group-At-A-Time? (#2)
0 1 2 3 4
$400 $400 $400 $400
PV0 equals
$1677.30.
0 1 2
$200 $200
0 1 2 3 4 5
$100
$1,268.00
$347.20
$62.10
Plus
Plus
3-49
General Formula:
FVn = PV0(1 + [i/m])mn
n,Number of Years
m,Compounding Periods per Year
i,Annual Interest Rate
FVn,m,FV at the end of Year n
PV0,PV of the Cash Flow today
Frequency of
Compounding
3-50
Julie Miller has $1,000 to invest for 2
Years at an annual interest rate of
12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2)
= 1,254.40
Semi FV2 = 1,000(1+ [.12/2])(2)(2)
= 1,262.48
Impact of Frequency
3-51
Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)
= 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2)
= 1,269.73
Daily FV2 = 1,000(1+[.12/365])(365)(2)
= 1,271.20
Impact of Frequency
3-52
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.
(1 + [ i / m ] )m - 1
Effective Annual
Interest Rate
3-53
Basket Wonders (BW) has a $1,000
CD at the bank,The interest rate
is 6% compounded quarterly for 1
year,What is the Effective Annual
Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1
= 1.0614 - 1 =,0614 or 6.14%!
BWs Effective
Annual Interest Rate
3-54
1,Calculate the payment per period.
2,Determine the interest in Period t.
(Loan Balance at t-1) x (i% / m)
3,Compute principal payment in Period t.
(Payment - Interest from Step 2)
4,Determine ending balance in Period t.
(Balance - principal payment from Step 3)
5,Start again at Step 2 and repeat.
Steps to Amortizing a Loan
3-55
Julie Miller is borrowing $10,000 at an
annual interest rate of 12%,Amortize the
loan if annual payments are made for 5
years.
Step 1,Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
Amortizing a Loan Example
3-56
Amortizing a Loan Example
En d o f
Year
Paym ent In tere st Prin cipal En d in g
Balan ce
0 --- --- --- $10,000
1 $2,7 74 $1,2 00 $1,5 74 8,42 6
2 2,77 4 1,01 1 1,76 3 6,66 3
3 2,77 4 800 1,97 4 4,68 9
4 2,77 4 563 2,21 1 2,47 8
5 2,77 5 297 2,47 8 0
$13,871 $3,8 71 $10,000
[Last Payment Slightly Higher Due to Rounding]
3-57
Usefulness of Amortization
2,Debt Outstanding -- The
quantity of outstanding debt
may be used in day-to-day
activities of the firm.
1,Interest Expense -- Interest
expenses may reduce taxable
income of the firm.
Chapter 3
Time Value of
Money
3-2
The Time Value of Money
? The Interest Rate
? Simple Interest
? Compound Interest
? Amortizing A Loan
3-3
Obviously,$10,000 today.
You already recognize that there is
TIME VALUE TO MONEY!!
The Interest Rate
Which would you prefer -- $10,000
today or $10,000 in 5 years?
3-4
TIME allows you the opportunity to
postpone consumption and earn
INTEREST.
Why TIME?
Why is TIME such an important
element in your decision?
3-5
Types of Interest
?Compound Interest
Interest paid (earned) on any previous
interest earned,as well as on the
principal borrowed (lent).
?Simple Interest
Interest paid (earned) on only the original
amount,or principal borrowed (lent).
3-6
Simple Interest Formula
Formula SI = P0(i)(n)
SI,Simple Interest
P0,Deposit today (t=0)
i,Interest Rate per Period
n,Number of Time Periods
3-7
?SI = P0(i)(n)
= $1,000(.07)(2)
= $140
Simple Interest Example
?Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years,What is the accumulated
interest at the end of the 2nd year?
3-8
FV = P0 + SI
= $1,000 + $140
= $1,140
?Future Value is the value at some future
time of a present amount of money,or a
series of payments,evaluated at a given
interest rate.
Simple Interest (FV)
?What is the Future Value (FV) of the
deposit?
3-9
The Present Value is simply the
$1,000 you originally deposited,
That is the value today!
?Present Value is the current value of a
future amount of money,or a series of
payments,evaluated at a given interest
rate.
Simple Interest (PV)
?What is the Present Value (PV) of the
previous problem?
3-10
0
5000
10000
15000
20000
1s t Y ea r 10 th
Y ea r
20 th
Y ea r
30 th
Y ea r
F u t u re V a l u e o f a S i n g l e $ 1,0 0 0 D e p o s i t
10 % Simple
Interest
7% C ompound
Interest
10 % C ompound
Interest
Why Compound Interest?
Fu
tu
re
Val
ue
(U
.S.
D
ol
lar
s)
3-11
Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
Future Value
Single Deposit
(Graphic)
0 1 2
$1,000
FV2
7%
3-12
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your
$1,000 deposit over the first year.
This is the same interest you would
earn under simple interest.
Future Value
Single Deposit
(Formula)
3-13
FV1 = P0 (1+i)1 = $1,000 (1.07)
= $1,070
FV2 = FV1 (1+i)1
= P0 (1+i)(1+i) = $1,000(1.07)(1.07)
= P0 (1+i)2 = $1,000(1.07)2
= $1,144.90
You earned an EXTRA $4.90 in Year 2 with
compound over simple interest,
Future Value
Single Deposit
(Formula)
3-14
FV1 = P0(1+i)1
FV2 = P0(1+i)2
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
General Future
Value Formula
etc.
3-15
FVIFi,n is found on Table I at End
of Book or on the Card Insert.
Valuation Using Table I
Pe riod 6% 7% 8%
1 1,06 0 1,07 0 1,08 0
2 1,12 4 1,14 5 1,16 6
3 1,19 1 1,22 5 1,26 0
4 1,26 2 1,31 1 1,36 0
5 1,33 8 1,40 3 1,46 9
3-16
FV2 = $1,000 (FVIF7%,2)
= $1,000 (1.145)
= $1,145 [Due to Rounding]
Using Future Value Tables
Pe riod 6% 7% 8%
1 1,06 0 1,07 0 1,08 0
2 1,12 4 1,14 5 1,16 6
3 1,19 1 1,22 5 1,26 0
4 1,26 2 1,31 1 1,36 0
5 1,33 8 1,40 3 1,46 9
3-17
Julie Miller wants to know how large her
$10,000 deposit will become at a
compound interest rate of 10% for 5 years.
Story Problem Example
0 1 2 3 4 5
$10,000
FV5
10%
3-18
?Calculation based on Table I:
FV5 = $10,000 (FVIF10%,5)
= $10,000 (1.611)
= $16,110 [Due to Rounding]
Story Problem Solution
? Calculation based on general formula:
FVn = P0 (1+i)n
FV5 = $10,000 (1+ 0.10)5
= $16,105.10
3-19
We will use the Rule-of-72.
Double Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
3-20
Approx,Years to Double = 72 / i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]
The Rule-of-72
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
3-21
Assume that you need $1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7%.
0 1 2
$1,000
7%
PV1PV0
Present Value
Single Deposit
(Graphic)
3-22
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2
= FV2 / (1+i)2 = $873.44
Present Value
Single Deposit
(Formula)
0 1 2
$1,000
7%
PV0
3-23
PV0 = FV1 / (1+i)1
PV0 = FV2 / (1+i)2
General Present Value Formula:
PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
General Present
Value Formula
etc.
3-24
PVIFi,n is found on Table II at End
of Book or on the Card Insert.
Valuation Using Table II
Pe rio d 6% 7% 8%
1,9 43,9 35,9 26
2,8 90,8 73,8 57
3,8 40,8 16,7 94
4,7 92,7 63,7 35
5,7 47,7 13,6 81
3-25
PV2 = $1,000 (PVIF7%,2)
= $1,000 (.873)
= $873 [Due to Rounding]
Using Present Value Tables
Pe rio d 6% 7% 8%
1,9 43,9 35,9 26
2,8 90,8 73,8 57
3,8 40,8 16,7 94
4,7 92,7 63,7 35
5,7 47,7 13,6 81
3-26
Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000 in 5 years at a discount
rate of 10%.
Story Problem Example
0 1 2 3 4 5
$10,000
PV0
10%
3-27
? Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $10,000 / (1+ 0.10)5
= $6,209.21
? Calculation based on Table I:
PV0 = $10,000 (PVIF10%,5)
= $10,000 (.621)
= $6,210.00 [Due to Rounding]
Story Problem Solution
3-28
Types of Annuities
?Ordinary Annuity,Payments or receipts
occur at the end of each period.
?Annuity Due,Payments or receipts
occur at the beginning of each period.
?An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
3-29
Examples of Annuities
? Student Loan Payments
? Car Loan Payments
? Insurance Premiums
? Mortgage Payments
? Retirement Savings
3-30
Parts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
End of
Year 1
(Annuity Due)
Beginning of
Year 1
Today Equal Cash Flows
Each 1 Year Apart
(Annuity Due)
End of
Year 1
3-31
FVAn = R(1+i)n-1 + R(1+i)n-2 +
..,+ R(1+i)1 + R(1+i)0
Overview of an
Ordinary Annuity --
FVA
R R R
0 1 2 n n+1
FVAn
R,Periodic
Cash Flow
End of Year
i%,,,
3-32
FVA3 = $1,000(1.07)2 +
$1,000(1.07)1 + $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215
Example of an
Ordinary Annuity --
FVA
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
End of Year
7%
$1,070
$1,145
3-33
FVAn = R (FVIFAi%,n)
FVA3 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215
Valuation Using Table III
Pe riod 6% 7% 8%
1 1,00 0 1,00 0 1,00 0
2 2,06 0 2,07 0 2,08 0
3 3,18 4 3,21 5 3,24 6
4 4,37 5 4,44 0 4,50 6
5 5,63 7 5,75 1 5,86 7
3-34
FVADn = R(1+i)n + R(1+i)n-1 +
..,+ R(1+i)2 + R(1+i)1
= FVAn (1+i)
Overview of an
Annuity Due -- FVAD
R R R
0 1 2 n n+1
FVADn
R,Periodic
Cash Flow
Beginning of Year
i%,,,
3-35
FVAD3 = $1,000(1.07)3 +
$1,000(1.07)2 + $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440
Example of an
Annuity Due -- FVAD
$1,000 $1,000 $1,000 $1,070
0 1 2 3 4
FVAD3 = $3,440
Beginning of Year
7%
$1,225
$1,145
3-36
FVADn = R (FVIFAi%,n)(1+i)
FVAD3 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) =
$3,440
Valuation Using Table III
Pe riod 6% 7% 8%
1 1,00 0 1,00 0 1,00 0
2 2,06 0 2,07 0 2,08 0
3 3,18 4 3,21 5 3,24 6
4 4,37 5 4,44 0 4,50 6
5 5,63 7 5,75 1 5,86 7
3-37
PVAn = R/(1+i)1 + R/(1+i)2
+,.,+ R/(1+i)n
Overview of an
Ordinary Annuity --
PVA
R R R
0 1 2 n n+1
PVAn
R,Periodic
Cash Flow
End of Year
i%,,,
3-38
PVA3 = $1,000/(1.07)1 +
$1,000/(1.07)2 +
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32
Example of an
Ordinary Annuity --
PVA
$1,000 $1,000 $1,000
0 1 2 3 4
$2,624.32 = PVA3
End of Year
7%
$934.58
$873.44
$816.30
3-39
PVAn = R (PVIFAi%,n)
PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624
Valuation Using Table IV
Pe riod 6% 7% 8%
1 0,94 3 0,93 5 0,92 6
2 1,83 3 1,80 8 1,78 3
3 2,67 3 2,62 4 2,57 7
4 3,46 5 3,38 7 3,31 2
5 4,21 2 4,10 0 3,99 3
3-40
PVADn = R/(1+i)0 + R/(1+i)1 +,.,+ R/(1+i)n-1
= PVAn (1+i)
Overview of an
Annuity Due -- PVAD
R R R
0 1 2 n n+1
PVADn
R,Periodic
Cash Flow
Beginning of Year
i%,,,
3-41
PVADn = $1,000/(1.07)2 + $1,000/(1.07)1 +
$1,000/(1.07)0 = $2,808.02
Example of an
Annuity Due -- PVAD
$1,000.00 $1,000 $1,000
0 1 2 3 4
PVADn=$2,808.02
Beginning of Year
7%
$ 934.58
$ 873.44
3-42
PVADn = R (PVIFAi%,n)(1+i)
PVAD3 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) =
$2,808
Valuation Using Table IV
Pe riod 6% 7% 8%
1 0,94 3 0,93 5 0,92 6
2 1,83 3 1,80 8 1,78 3
3 2,67 3 2,62 4 2,57 7
4 3,46 5 3,38 7 3,31 2
5 4,21 2 4,10 0 3,99 3
3-43
1,Read Problem Thoroughly
2,Determine if it is a PV or FV Problem
3,Create a Time Line
4,Put Cash Flows and Arrows on Time Line
5,Determine if Solution involves a Single
CF,Annuity Stream(s),or Mixed Flow
6,Solve the Problem
Steps to Solve Time Value
of Money Problems
3-44
Julie Miller will receive the set of cash
flows below,What is the Present Value
at a discount rate of 10%.
Mixed Flows Example
0 1 2 3 4 5
$600 $600 $400 $400 $100
PV0
10%
3-45
1,Solve a piece-at-a-time by
discounting each piece back to t=0.
2,Solve a group-at-a-time by first
breaking problem into annuity
group streams and single cash
flow groups,Then discount
each group back to t=0.
How to Solve?
3-46
Piece-At-A-Time
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%
$545.45
$495.87
$300.53
$273.21
$ 62.09
$1677.15 = PV0 of the Mixed Flow
3-47
Group-At-A-Time?(#1)
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10
3-48
Group-At-A-Time? (#2)
0 1 2 3 4
$400 $400 $400 $400
PV0 equals
$1677.30.
0 1 2
$200 $200
0 1 2 3 4 5
$100
$1,268.00
$347.20
$62.10
Plus
Plus
3-49
General Formula:
FVn = PV0(1 + [i/m])mn
n,Number of Years
m,Compounding Periods per Year
i,Annual Interest Rate
FVn,m,FV at the end of Year n
PV0,PV of the Cash Flow today
Frequency of
Compounding
3-50
Julie Miller has $1,000 to invest for 2
Years at an annual interest rate of
12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2)
= 1,254.40
Semi FV2 = 1,000(1+ [.12/2])(2)(2)
= 1,262.48
Impact of Frequency
3-51
Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)
= 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2)
= 1,269.73
Daily FV2 = 1,000(1+[.12/365])(365)(2)
= 1,271.20
Impact of Frequency
3-52
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.
(1 + [ i / m ] )m - 1
Effective Annual
Interest Rate
3-53
Basket Wonders (BW) has a $1,000
CD at the bank,The interest rate
is 6% compounded quarterly for 1
year,What is the Effective Annual
Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1
= 1.0614 - 1 =,0614 or 6.14%!
BWs Effective
Annual Interest Rate
3-54
1,Calculate the payment per period.
2,Determine the interest in Period t.
(Loan Balance at t-1) x (i% / m)
3,Compute principal payment in Period t.
(Payment - Interest from Step 2)
4,Determine ending balance in Period t.
(Balance - principal payment from Step 3)
5,Start again at Step 2 and repeat.
Steps to Amortizing a Loan
3-55
Julie Miller is borrowing $10,000 at an
annual interest rate of 12%,Amortize the
loan if annual payments are made for 5
years.
Step 1,Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
Amortizing a Loan Example
3-56
Amortizing a Loan Example
En d o f
Year
Paym ent In tere st Prin cipal En d in g
Balan ce
0 --- --- --- $10,000
1 $2,7 74 $1,2 00 $1,5 74 8,42 6
2 2,77 4 1,01 1 1,76 3 6,66 3
3 2,77 4 800 1,97 4 4,68 9
4 2,77 4 563 2,21 1 2,47 8
5 2,77 5 297 2,47 8 0
$13,871 $3,8 71 $10,000
[Last Payment Slightly Higher Due to Rounding]
3-57
Usefulness of Amortization
2,Debt Outstanding -- The
quantity of outstanding debt
may be used in day-to-day
activities of the firm.
1,Interest Expense -- Interest
expenses may reduce taxable
income of the firm.
