1

FIN 301 Class Notes

Chapter 4: Time Value of Money
The concept of Time Value of Money:
An amount of money received today is worth more than
the same dollar
value received a year from now. Why?
Do you prefer a $100 today or a $100 one year from now?
why?
- Consumption forgone has value
- Investment lost has opportunity cost
- Inflation may increase and purchasing power decrease
Now,
Do you prefer a $100 today or $110 one year from now?
Why?
You will ask yourself one question:
- Do I have any thing better to do with that $100 than
lending it for $10
extra?
- What if I take $100 now and invest it, would I make more
or less than
$110 in one year?
Note:
Two elements are important in valuation of cash flows:
- What interest rate (opportunity rate, discount rate,
required rate of
return) do you want to evaluate the cash flow based on?
- At what time do these the cash flows occur and at what
time do you need
to evaluate them?
2
Time Lines:
� Show the timing of cash flows.
� Tick marks occur at the end of periods, so Time 0 is
today; Time 1 is the
end of the first period (year, month, etc.) or the beginning
of the second
period.
Example 1 : $100 lump sum due in 2 years
Today End of End of
Period 1 Period 2
(1 period (2 periods
form now) form now)
Example 2 : $10 repeated at the end of next three years
(ordinary annuity )
CF0 CF1 CF2 CF3
0123
i%
100
012
i
10 10 10
0123
i
3
Calculations of the value of money problems:
The value of money problems may be solved using
1- Formulas.
2- Interest Factor Tables. (see p.684)
3- Financial Calculators (Basic keys: N, I/Y, PV, PMT, FV).
I use BAII Plus calculator
4- Spreadsheet Software (Basic functions: PV, FV, PMT,
NPER,RATE).
I use Microsoft Excel.
4
FUTUR VALUE OF A SINGLE CASH FLOW
Examples:
• You deposited $1000 today in a saving account at
BancFirst that pays
you 3% interest per year. How much money you will get at
the end of the
first year ?
i=3% FV1
01
$1000
• You lend your friend $500 at 5% interest provided that
she pays you
back the $500 dollars plus interest after 2 years. How
much she should
pay you?
i=5% FV2
012
$500
• You borrowed $10,000 from a bank and you agree to pay
off the loan after
5 years from now and during that period you pay 13%
interest on loan.
$10,000
012345
FV5 i=13%
Present
Value of
Money
Future
Value of
Money
Investment
Compounding
5
Detailed calculation:
Simple example:
Invest $100 now at 5%. How much will you have after a
year?
FV1 = PV + INT
= PV + (PV × i)
= PV × (1 + i)
FV1 = $100 + INT
= $100 + ($100 × .05)
= $100 + $5
= $105
Or
FV1 = $100 × (1+0.05)
= $100 × (1.05)
= $105
6
Another example: Invest $100 at 5% (per year) for 4 years.
Interest added: + $5.00 + $5.25 + $5.51 + $5.79
FV1= 100 × (1.05) = $105
FV2= 105 × (1.05) = $110.25
= 100 × (1.05) × (1.05) = $110.25
= 100 × (1.05)2 = $110.25
FV3= 110.25 × (1.05) = $115.76
= 100 × (1.05) × (1.05) × (1.05)= $115.76
= 100 × (1.05)3 = $115.76
FV4 = $100 × (1.05) × (1.05) × (1.05) × (1.05)
= PV × (1+i) × (1+i) × (1+i) × (1+i)
= PV × (1+i)4
In general, the future value of an initial lump sum is: FVn =
PV × (1+i)n
01234
PV = $100 FV1 = $105 FV2 = $110.25 FV3 = $115.76 FV4 = $121.55
× 1.05 × 1.05 × 1.05 × 1.05
7
To solve for FV, You need
1- Present Value (PV)
2- Interest rate per period (i)
3- Number of periods (n)
Remarks: As PV�, FVn�.
As i�, FVn�.
As n�, FVn�.
1- By Formula 0 (1 )n
FV n = PV + i
2- By Table I 0, ( ) n in FV = PV FVIF
, (1 )n
i n ⇒FVIF = + i
3- By calculator (BAII Plus)
Clean the memory: CLR TVM �
Notes:
- To enter (i) in the calculator, you have to enter it in %
form.
- Use To change the sign of a number.
For example, to enter -100: 100
- To solve the problems in the calculator or excel, PV and
FV cannot have the
same sign. If PV is positive then FV has to be negative.
INPUTS
OUTPUT
N I/Y PMT
FV
3 10 0
133.10
-100
CPT
PV
+/-
2nd FV
+/-
CE/C
8
Example:
Jack deposited $1000 in saving account earning 6%
interest rate.
How much will jack money be worth at the end of 3 years?
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
PV= 1000
PMT= 0
Solution:
By formula: FVn = PV × (1+i)n
FV3 = $1000 × (1+0.06)3
= $1000 ×(1.06)3
= $1000 ×1.191
= $ 1,191
By Table: FVn= PV × FVIFi,n
FV3 = $1000 × FVIF6%,3
= $1000 × 1.191
= $ 1,191
1000
0123
? 6%
9
By calculator:
Clean the memory: CLR TVM �
By Excel:
=FV (0.06, 3, 0,-1000, 0)
INPUTS
OUTPUT
N I/Y PMT
FV
360
1,191.02
-1000
CPT
PV
CE/C 2nd FV
10
PRESENT VALUE OF A SINGLE CASH FLOW
Examples:
• You need $10,000 for your tuition expenses in 5 years
how much should
you deposit today in a saving account that pays 3% per
year?
$10,000
012345
PV0 FV5 i=3%
• One year from now, you agree to receive $1000 for your
car that you sold
today.
How much that $1000 worth today if you use 5% interest
rate?
$1000
0 i=5% 1 FV1
PV0
Present
Value of
Money
Future
Value of
Money
Discounting
11
Detailed calculation
(1 )n
FV n = PV + i
0 (1 )
n
n
PV FV
i
⇒=
+
0
1
n (1 )n PV FV
i
⇒=×
+
Example:
PV4= FV4 = $121.55
PV3= FV4× [1/(1+i)]
= $121.55× [1/(1.05)]
= $115.76
PV2= FV4× [1/(1+i)(1+i)]
= $121.55× [1/(1.05)(1.05)]
= $121.55× [1/(1.05)2]
= $110.25
$100 $105 $110.25 $115.76 = $121.55
÷ 1.05 ÷ 1.05 ÷ 1.05 ÷ 1.05
01234
12
Or
PV2= FV3× [1/ (1+i)]
= $115.76× [1/ (1.05)]
= $110.25
PV1= FV4× [1/(1+i)(1+i) (1+i)]
= $121.55× [1/(1.05)(1.05) (1.05)]
= $121.55× [1/(1.05)3]
= $105
Or
PV1= FV2× [1/ (1+i)]
= $110.25× [1/ (1.05)]
= $105
PV0 = FV4× [1/ (1+i) (1+i) (1+i) (1+i)]
= FV4× [1/(1+i)4]
= $121.55× [1/(1.05)(1.05) (1.05) (1.05)]
= $121.55× [1/(1.05)4]
= $100
In general, the present value of an initial lump sum is: PV0
= FVn× [1/(1+i) n]
13
To solve for PV, You need
4- Future Value (FV)
5- Interest rate per period (i)
6- Number of periods (n)
Remarks: As FVn �, PV�
As i�, PV�
As n�, PV�
1- By Formula 0
1
n (1 )n PV FV
i
=×
+
2- By Table II 0, ( ) n in PV = FV PVIF
,
1
in (1 )n PVIF
i
⇒=
+
3- By calculator (BAII Plus)
Clean the memory: CLR TVM �
INPUTS
OUTPUT
N I/Y PV PMT
PV
3 10 0
-100
133.10
CPT
FV
CE/C 2nd FV
14
Example:
Jack needed a $1191 in 3 years to be off some debt. How
much should jack
put in a saving account that earns 6% today?
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
FV= $1191
PMT= 0
Solution:
By formula: PV0 = FV3 × [1/(1+i) n]
PV0 = $1,191 × [1/(1+0.06) 3]
= $1,191 × [1/(1.06) 3]
= $1,191 × (1/1.191)
= $1,191 × 0.8396
= $1000
By Table: = FVn × PVIFi,n
PV0 = $1,191 × PVIF6%,3
= $1,191 × 0.840
= $ 1000
?
012
6%
3
$1191
15
By calculator:
Clean the memory: CLR TVM �
By Excel:
=PV (0.06, 3, 0, 1191, 0)
INPUTS
OUTPUT
N I/Y PV PMT
PV
360
-1000
1191
CPT
FV
CE/C 2nd FV
16
Solving for the interest rate i
You can buy a security now for $1000 and it will pay you $1,191
three years from
now. What annual rate of return are you earning?
By Formula: 1
PV
i FV
n
1
n − ⎥⎦
⎤
⎢⎣
=⎡
1
1191 3 1
1000
i = ⎡⎢ ⎤⎥ − ⎣ ⎦ = 0.06
By Table: 0 , ( ) n in FV = PV FVIF
,
0
n
in
FVIF FV
PV
⇒=
,3
1191 1.191
i 1000 FVIF = =
From the Table I at n=3 we find that the interest rate that yield 1.191 FVIF is 6%
Or 0 , ( ) n in PV = FV PVIF
0
i ,n
n
PVIF PV
FV
⇒=
,3
1000 0.8396
i 1191 PVIF = =
From the Table II at n=3 we find that the interest rate that yield 0.8396 PVIF is 6%
17
By calculator:
Clean the memory: CLR TVM �
INPUTS
OUTPUT
N PV PV PMT
I/Y
3 -1000 0
5.9995
1191
CPT
FV
CE/C 2nd FV
18
Solving for n:
Your friend deposits $100,000 into an account paying 8%
per year. She wants
to know how long it will take before the interest makes
her a millionaire.
By Formula:
()()
()
ln
1
n Ln FV PV

n
Ln i
−
=
+
$1,000,000 PV $100,000 1 i 1.08 n FV = = + =
ln (1,000,000) ln (100,000)
ln(1.08)
n
−
=
13.82 11.51 30
0.077
years −
==
By Table: 0, ( ) n in FV = PV FVIF
,
0
n
in
FVIF FV
PV
⇒=
8,
1,000,000 10
n 100,000 FVIF = =
From the Table I at i=8 we find that the number of periods that yield 10 FVIF is 30
Or 0, ( ) n in PV = FV PVIF
0
i ,n
n
PVIF PV
FV
⇒=
8,
100,000 0.1
n 1,000,000 PVIF = =
From the Table II at i=8 we find that the number of periods that yield 0.1 PVIF is 30
19
By calculator:
Clean the memory: CLR TVM �
FUTURE VALUE OF ANNUTIES
An annuity is a series of equal payments at fixed
intervals for a specified
number of periods.
PMT = the amount of periodic payment
Ordinary (deferred) annuity: Payments occur at the end of
each period.
Annuity due: Payments occur at the beginning of each
period.
INPUTS
OUTPUT
I/Y PV PMT
N
8 -100,000 0
29.9188
1,000,000
CPT
FV
CE/C 2nd FV
20
Example: Suppose you deposit $100 at the end of each
year into a savings
account paying 5% interest for 3 years. How much will you
have in the
account after 3 years?

( ) 1( ) 1 1 .... n n
2

n FVAN PMT i PMT i PMT − − = + + + + +

(Hard to use this formula)
PM PM
0123
i
PM
Due
Ordinary
PM PM PM
0123
i
0 1 2 3 4 n-1 n
PMT PMT PMT PMT PMT PMT
Time
0123
100 100 100.00
105.00
110.25
$315.25
5%
21
()
,
11
()
n
n
in
i
FVAN PMT
i
PMT FVIFA
⎡+−⎤
=⎢⎥
⎢⎣ ⎥⎦
=
Note: For an annuity due, simply multiply the answer
above by (1+i).
()()
,So (annuity due) ( )(1 ).
11
1
n in
n
FVAND PMT FVIFA i
i
PMT i
i
=+
⎡+−⎤
=⎢⎥+
⎢⎣ ⎥⎦
Annuity:
Future Value Interest
Factor for an Annuity
22
Annuity Due:
23
Remark:
FVIFA = FVIF + FVIF + FVIF
i ,3 i ,2 i ,1 i ,0
To solve for the future value of Annuities, You need:
1-Payemnt or annuity amount (PMT)
2-Interest rate per period (i)
3-Number of periods (n)
1-BY Formula:
(1 ) 1n
n
i
FVAN PMT
i
⎡+−⎤
=⎢⎥
⎢⎣ ⎥⎦
==� Ordinary Annuity
(1 ) 1 ( )
1
n
n
i
FVAND PMT i
i
⎡+−⎤
=⎢⎥+
⎢⎣ ⎥⎦
==� Annuity Due

(1 ) nn FVAND = FVAN + i
2- BY Table III:
,( ) n in FVAN = PMT FVIFA ==� Ordinary Annuity
( ) , ( )1 n in FVAND = PMT FVIFA + i ==� Annuity Due
24
3- BY calculator:
Ordinary Annuity:
1- Clean the memory: CLR TVM�
2- Set payment mode to END of period: BGN �
SET �
3- Make sure you can see END written on the screen then
press
NOTE: If you do not see BGN written on the upper right
side of the screen,
you can skip
Step 2 and 3.
INPUTS
OUTPUT
N I/Y PV PMT
30
315.25
-100
CPT FV
5
CE/C 2nd FV
2nd PMT
2nd ENTER
CE/C
25
Annuity Due:
Clean the memory: CLR TVM �
Set payment mode to BGN of period: BGN �
SET �
Make sure you can see BGN written on the screen then
press
INPUTS
OUTPUT
N I/Y PV PMT
30
331.10
-100
CPT FV
5
CE/C 2nd FV
2nd PMT
2nd ENTER
CE/C
26
Example:
You agree to deposit $500 at the end of every year for 3
years in an investment
fund that earns 6%.
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
PMT=500
PV= 0
FV=?
Solution:
By formula:
(1 ) 1n
n
i
FVAN PMT
i
⎡+−⎤
=⎢⎥
⎢⎣ ⎥⎦
(1 0.06)3 1 500
0.06
⎡+−⎤
=⎢⎥
⎣⎦
500 1.191 1 1,591.80
0.06
⎡−⎤=⎢⎥=⎣⎦
By Table: , ( ) n in FVAN = PMT FVIFA
3 6,3 FVAN = 500(FVIFA )
= 500(3.184) = 1,592
01
$500
2
$500
6%
3
$500
FV=?
27
By calculator:
Clean the memory: CLR TVM�
Make sure you do not see BGN written on the upper right
side of the screen.
By Excel: =FV (0.06, 3, -500, 0, 0)
INPUTS
OUTPUT
N I/Y PV PMT
30
1,591.80
-500
CPT FV
6
CE/C 2nd FV
28
Now assume that you deposit the $500 at the beginning of
the year not at the
end of the year.
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
PMT=500 (beg)
PV= 0
FV=?
Solution:
By formula:
(1 ) 1 ( )
1
n
n
i
FVAND PMT i
i
⎡+−⎤
=⎢⎥+
⎢⎣ ⎥⎦
()
3
1 0.06 1
500 (1 0.06)
0.06
n
FVAND
⎡+−⎤
=⎢⎥+
⎢⎣ ⎥⎦
500 0.191 (1.06) 1,687.30
0.06
= ⎡ ⎤ = ⎢⎣ ⎥⎦
By Table: () , ( )1 n in FVAND = PMT FVIFA + i
() FVAND = 500(FVIFA ) 1+ 0.06
3 6,3

= 500(3.184)(1.06) = 1,687.52
0
$500
1
$500
2
$500
6%
3
FV=?
29
By calculator:
Clean the memory: CLR TVM �
Set payment mode to BGN of period: BGN �
SET �
Make sure you can see BGN written on the screen then
press
By Excel: =FV (0.06, 3, -500, 0, 1)
INPUTS
OUTPUT
N I/Y PV PMT
30
1,687.31
-500
CPT FV
6
CE/C 2nd FV
2nd PMT
2nd ENTER
CE/C
30
PRESENT VALUE OF ANNUTIES
Problem: You have a choice
a) $100 paid to you at the end of each of the next 3 years
or
b) a lump sum today.
i = 5%, since you would invest the money at this rate if
you had it.
How big does the lump sum have to be to make the
choices equally good?
Formula:
()()()
()
( ) i,n
n
n12n
PMT PVIFA
i
1i
11
PMT
1i
.... PMT
1i
PMT
1i
PVA PMT
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
−
=
+
++
+
+
+
=
Present Value Interest Factor
0123
Time
÷1.05 ÷1.052
÷1.053
95.24
90.70
86.38
PVAN3 = 272.32
100 100 100
31
$100(2.7232) $272.32
.05
1.05
11
PVA $100 3
3
==
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
=
Note: For annuities due, simply multiply the answer
above by (1+i)
PVANDn (annuity due) = PMT (PVIFAi,n) (1+i)
To solve for the present value of Annuities, You need:
1-Payemnt or annuity amount (PMT)
2-Interest rate per period (i)
3-Number of periods (n)
1- BY Formula:
()
11
1n
n
i
PVAN PMT
i
⎡−⎤⎢+⎥=⎢⎥
⎢⎥
⎢⎥
⎣⎦
==� Ordinary Annuity
()()
11
1
1
n
n
i
PVAND PMT i
i
⎡−⎤⎢+⎥=⎢⎥+
⎢⎥
⎢⎥
⎣⎦
==� Annuity Due

(1 ) nn PVAND = PVAN + i
32
2- BY Table IV:
PVANn = PMT (PVIFAi,n ) ==� Ordinary Annuity
( ) , ( )1 n in PVAND = PMT PVIFA + i ==� Annuity Due
3- BY calculator:
Ordinary Annuity:
Clean the memory: CLR TVM�
Make sure you do not see BGN written on the upper right
side of the screen.
INPUTS
OUTPUT
N I/Y FV PMT
30
272.32
-100
CPT PV
5
CE/C 2nd FV
33
Annuity Due:
Clean the memory: CLR TVM �
Set payment mode to BGN of period: BGN �
SET �
Make sure you can see BGN written on the screen then
press
INPUTS
OUTPUT
N I/Y FV PMT
30
285.94
-100
CPT PV
5
CE/C 2nd FV
2nd PMT
2nd ENTER
CE/C
34
Example:
You agree to receive $500 at the end of every year for 3
years in an investment
fund that earns 6%.
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
PMT=500
FV= 0
PV=?
Solution:
By formula:
()
11
1
n
n
i
PVAN PMT
i
⎡−⎤⎢+⎥=⎢⎥
⎢⎥
⎢⎥
⎣⎦
( )3
11
1 0.06
500
n 0.06 PVAN

⎡−⎤⎢+⎥=⎢⎥
⎢⎥
⎢⎥
⎣⎦
11
500 1.191
0.06
⎡−⎤⎢⎥
=⎢⎥
⎢⎥
⎣⎦
= $1,336.51
0
PV=?
1
$500
2
$500
6%
3
$500
35
By Table: PVANn = PMT (PVIFAi,n )
3 6,3 PVAN = 500(PVIFA )
= 500(2.673) = 1,336.51
By calculator:
Clean the memory: CLR TVM�
Make sure you do not see BGN written on the upper right
side of the screen.
By Excel: =PV (0.06, 3, -500, 0, 0)
INPUTS
OUTPUT
N I/Y FV PMT
30
1,336.51
-500
CPT PV
6
CE/C 2nd FV
36
Now assume that you receive the $500 at the beginning of
the year not at the
end of the year.
Time line
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
PMT=500 (beg)
FV= 0
PV=?
Solution
By formula:
()()
11
1
1
n
n
i
PVAND PMT i
i
⎡−⎤⎢+⎥=⎢⎥+
⎢⎥
⎢⎥
⎣⎦
( )3
11
1 0.06
500 (1 0.06)
n 0.06 PVAND

⎡−⎤⎢+⎥=⎢⎥+
⎢⎥
⎢⎥
⎣⎦
11
500 1.191 (1.06)
0.06
⎡−⎤⎢⎥
=⎢⎥
⎢⎥
⎣⎦
= 1, 416.70
0
$500
PV=?
1
$500
2
$500
6%
3
37

By Table: PVANDn = PMT(PVIFAi,n )(1+ i)

() PVAND = 500(PVIFA ) 1+ 0.06
3 6,3

= 500(2.673)(1.06) = 1, 416.69
By calculator:
Clean the memory: CLR TVM �
Set payment mode to BGN of period: BGN �
SET �
Make sure you can see BGN written on the screen then
press
By Excel: =PV (0.06, 3, -500, 0, 1)
INPUTS
OUTPUT
N I/Y FV PMT
30
1,416.69
-500
CPT PV
6
CE/C 2nd FV
2nd PMT
2nd ENTER
CE/C
38
Perpetuities
A perpetuity is an annuity that continues forever.
11
(1 )n
n
PVAN PMT i
i
⎡−⎤⎢+⎥=⎢⎥
⎢⎥
⎢⎣ ⎥⎦
()
As n gets very large, 1 0
1 ni
→
+
0
PVPER ( perpetuity) PMT 1 0
i
⎡−⎤=×⎢⎥⎣⎦
PMT 1 PMT
ii
= ×⎛ ⎞ = ⎜ ⎟
⎝⎠
Formula:
0
PVPER PMT
i
=
39
UNEVEN CASH FLOWS
How do we get PV and FV when the periodic payments are
unequal?
Present Value
( ) ( )n
n
2
12
0 1i
.... CF
1i
CF
1i
PV CF CF
+
++
+
+
+
=+
Future Value
( ) ( ) ( )0
n
n1
1
n
0 FV = CF 1+ i + CF 1+ i + ....+ CF 1+ i −
0123
÷1.05 ÷1.052
÷1.053
95.24
45.35
172.77
$313.36
100 50 200
0123
100 50 200.00
52.50
110.25
$362.75
5%
×1.05
×1.052
40
Example:
Present Value of Uneven Cash Flows
41
By Calculator:
Clean the memory:
Input cash flows in the calculator’s CF register:
CF0 = 0 � 0
CF1 = 100 � C01 100 F01 1
CF2 = 200 � C02 200 F02 1
CF3 = 300 � C03 300 F03 1
Press , then the it will ask you to enter the Interest rate (I)
Enter I = 10 � 10
Use to get to the NPV on the screen
When you read NPV on the screen, press
You will get NPV = $481.59 (Here NPV = PV.)
NOTE:
To calculate the future value of uneven cash flows, it is
much easier to start by
calculating the Present value of the cash flows using NPV
function then
calculate the future value using the future value of a
single cash flow rules. The
single cash flow in this case will be the present value.
CF 2nd CE/C
ENTER
ENTER ENTER
ENTER ENTER
ENTER ENTER
NPV
ENTER
CPT
42
Simple and Compound Interest
Simple Interest
� Interest paid on the principal sum only
Compound Interest
� Interest paid on the principal and on interest
Example:
Calculate the future value of $1000 deposited in a saving account
for 3 years earning
6% . Also, calculate the simple interest, the interest on interest,
and the compound
interest.
FV3 = 1000 (1.06) 3
= $1,191.02
Principal = PV = $1000
Compound interest = FV – PV = 1191.02 – 1000 = 191.02
Simple Interest = PV * i * n =1000 * 0.06 * 3 = $180
Interest on interest = Compound interest - Simple Interest
= 191.02 – 180 =
11.02
43
Effect of Compounding over Time
Other Compounding Periods
So far, our problems have used annual compounding. In
practice, interest is
usually compounded more frequently.
44
Example: You invest $100 today at 5% interest for 3 years.
Under annual compounding, the future value is:
()
$115.76
$100(1.1576)
$100(1.05)
FV PV 1 i
3
3
3
=
=
=
=+
What if interest is compounded semi-annually (twice a
year)?
Then the periods on the time line are no longer years, but
half-years!
Time:
$115.97
$100(1.1597)
$100(1.025)
(1 )
n No.of periods 3 2 6
2.5%
2
i Periodic interest rate 5%
6
6
=
=
=
=+
==×=
===
FV
FV PV i n
n
Note: the final value is slightly higher due to more
frequent compounding.
0
2.5%
6 months
PV=100
FV6=?
123456
45
Important: When working any time value problem, make
sure you keep
straight what the relevant periods are!
n = the number of periods
i = the periodic interest rate
From now on:
n = m*n
i = i/m
Where m = 1 for annual compounding
m = 2 for semiannual compounding
m = 4 for quarterly compounding
m = 12 for monthly compounding
m = 52 for weekly compounding
m = 365 for daily compounding
For continuously compounding: (1+i) n =e in
=� FVn = PV (e) in
=� PV = FVn (e) -
6-24
Will the FV of a lump sum be larger or
smaller if compounded more often,
holding the stated I% constant?
�LARGER, as the more frequently compounding
occurs, interest is earned on interest more often.
Annually: FV3 = $100(1.10)3 = $133.10
0 1 2 3 10%
100 133.10
Semiannually: FV6 = $100(1.05)6 = $134.01
0123
5%
456
134.01
0123
100
46
EFFECTIVE INTREST RATE
You have two choices:
1- 11% annual compounded rate of return on CD
2- 10% monthly compounded rate of return on CD
How can you compare these two nominal rates?
A nominal interest rate is just a stated (quoted) rate. An
APR (annual
percentage rate) is a nominal rate.
For every nominal interest rate, there is an effective rate.
The effective annual rate is the interest rate actually
being earned per year.
To compare among different nominal rates or to know
what is the actual rate
that you’re getting on any investment you have to use the
Effective annual
interest rate.
Effective Annual Rate: 1 1
m
eff
ii
m
=⎛+⎞−⎜⎟
⎝⎠
To compare the two rates in the example,
1-
1 1 0.11 1 0.11
eff 1 i = ⎛⎜ + ⎞⎟ − =

⎝⎠
or 11% (Nominal and Effective rates are equal in annual
compounding)
2-
12 1 0.10 1 0.1047
eff 12 i = ⎛⎜ + ⎞⎟ − =
⎝ ⎠ or 10.47 %
You should choose the first investment.
47
To compute effective rate using calculator:
ICONV �
Enter Nominal Rate � NOM 10
Enter compounding frequency per year (m) � C/Y 12
Compute the Effective rate � EFF
Nominal Versus Real Interest Rate
Nominal rate f r is a function of:
� Inflation premium n i :compensation for inflation and
lower
purchasing power.
� Real risk-free rate f r′ : compensation for postponing
consumption.
(1+ rf ) = (1+ rf′ )(1+ i n )
ffnfnr = r ′ + i + r ′i
ffnr ≈ r ′ + i
2nd 2
ENTER
ENTER
CPT
48
Amortized Loans
An amortized loan is repaid in equal payments over its life.
Example: You borrow $10,000 today and will repay the loan in
equal installments at
the end of the next 4 years. How much is your annual payment if
the interest rate is
9%?
Inputs:
The periods are years. (m=1)
n=4
i = 9%
PVAN4 = $10,000
FV =0
PMT = ?
PVAN 4 = PMT (PVIFA9%,4 )
$10,000 = PMT (3.240)
$10,000
3.240
PMT = = $3,087
Time 0 9% 1 2 3
PVA N= $10,000 PMT PMT PMT
4
PMT
49
Interest amount = Beginning balance * i
Principal reduction = annual payment - Interest amount
Ending balance = Beginning balance - Principal reduction
Beginning balance: Start with principal amount and then
equal to previous
year’s ending balance.
As a loan is paid off:
• at the beginning, much of each payment is for interest.
• later on, less of each payment is used for interest, and
more of it is applied
to paying off the principal.