Corporate finance chapter 04 time vaue of money

Chapter 4: Time Value of
Money

Objective
Explain the concept of compounding
and discounting and to provide
examples of real life
applications
1

Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.


Value of Investing $1
– Continuing in this manner you will find that
the following amounts will be earnt:

1 Year

$1.1

2 Years

$1.21

3 Years

$1.331

4 Years

$1.4641
2


Value of $5 Invested
• More generally, with an investment of $5
at 10% we obtain
1 Year

$5*(1+0.10)

$5.5

2 years

$5.5*(1+0.10)

$6.05

3 years

$6.05*(1+0.10)

$6.655

4 Years $6.655*(1+0.10)
3

$7.3205

Future Value of a Lump Sum
FV = PV * (1 + i )

n

FV with growths from -6% to +6%
Future Value of $1000

3,500

6%

3,000

2,500

4%

2,000

1,500

2%

1,000

0%
-2%
-4%

-6%

500

0
0

2

4

6

8

4

10

Years

12

14

16

18

20

Example: Future Value of a
Lump Sum
• Your bank offers a
CD with an interest
rate of 3% for a 5
year investments.
• You wish to invest
$1,500 for 5 years,
how much will your
investment be
worth?

5

FV = PV * (1 + i )

n

= $1500 * (1 + 0.03) 5
= $1738.1111145
n
i
PV
FV
Result

5
3%

1,500
?
1738.911111


Present Value of a Lump Sum
FV = PV * (1 + i )

n

Divide both sides by (1 + i ) to obtain :
FV
−n
PV =
= FV * (1 + i )
n
(1 + i )
n

6


Example: Present Value of a
Lump Sum
• You have been
offered $40,000 for
your printing
business, payable in
2 years. Given the
risk, you require a

return of 8%. What
is the present value
of the offer?
7

FV
PV =
(1 + i ) n
40,000
=
(1 + 0.08) 2
= 34293.55281
≅ $34,293.55 today


Solving Lump Sum Cash Flow
for Interest Rate
FV = PV * (1 + i ) n
FV
= (1 + i ) n
PV
FV
n
(1 + i ) =
PV
FV
n
i=
−1
PV

8


Example: Interest Rate on a
Lump Sum Investment
• If you invest $15,000
for ten years, you
FV
n
i
=
−1
receive $30,000.
PV
What is your annual
30000
10
=
− 1 = 10 2 − 1 = 2 − 1
return?
15000
1
10

= 0.071773463
= 7.18% (to the nearest basis point)

9

Review of Logarithms
• The basic properties of logarithms that
are used by finance are:

e

= x, x > 0

ln( x )

ln(e ) = x
ln( x * y ) = ln( x) + ln( y )
x

ln( x ) = y ln( x)
y

10


Review of Logarithms
• The following properties are easy to
prove from the last ones, and are useful
in finance

ln( x / y ) = ln( x) − ln( y )
ln( x * y * z ) = ln( x) + ln( y ) + ln( z )
ln( x + y ) ≠ ln( x) * ln( y )
11

Solving Lump Sum Cash Flow
for Number of Periods
FV = PV * (1 + i ) n
FV
= (1 + i ) n
PV
 FV 
n
(
)
ln
=
ln
(
1
+
i
)
= n * ln (1 + i )

 PV 
 FV 
ln

PV  ln ( FV ) − ln ( PV )

n=
=

ln (1 + i )
ln (1 + i )
12


Effective Annual Rates of an
APR of 18%
Annual
Percentage
rate
18

Frequency of
Annual
Compounding Effective Rate
1

18.00

18

2

18.81

18

4

19.25

18

12

19.56

18

52

19.68

18

365

19.72

13


The Frequency of
Compounding
• Note that as the frequency of
compounding increases, so does the
annual effective rate
• What occurs as the frequency of
compounding rises to infinity?
  km m 

EFF = Lim 1 +   − 1 = e k∞ − 1

m →∞ 
m




14


The Frequency of
Compounding
m

 k 
1 + EFF = 1 + m 
m

1
km
m
1+
= (1 + EFF )
m

(

)

k m = m * (1 + EFF ) − 1
1
m

15


The Frequency of
Compounding
Annual
Compounding
Effective Rate Frequency
12

1

Annual
Percentage
Rate
12.00

12

2

11.66

12

4

11.49

12

12

11.39

12

52

11.35

12

365

11.33

12

Infinity

11.33

16

Derivation of PV of Annuity
Formula: Algebra. 1 of 5
pmt
pmt
PV =
+
+
1
2
(1 + i ) (1 + i )
pmt
pmt
pmt
++
+
3
n −1
n
(1 + i )
(1 + i )
(1 + i )
17


Derivation of PV of Annuity
Formula: Algebra. 2 of 5
1
1
PV = pmt *{
+

+
1
2
(1 + i ) (1 + i )
1
1
1
++
+
}
3
n −1
n
(1 + i )
(1 + i )
(1 + i )

18


Derivation of PV of Annuity
Formula: Algebra. 3 of 5
1
1
PV * (1 + i ) = pmt * (1 + i ) *{
+
+
1
2
(1 + i ) (1 + i )

1
1
1
++
+
}
3
n −1
n
(1 + i )
(1 + i )
(1 + i )

19


Derivation of PV of Annuity
Formula: Algebra. 4 of 5
1
1
+
+
0
1
(1 + i ) ( 1 + i )
1
1
1
1
1

++
+
+[
−
]}
2
n−2
n −1
n
n
(1 + i )
(1 + i )
(1 + i )
(1 + i ) (1 + i )
1
1
= pmt *
+ pmt *{
+
0
1
(1 + i )
(1 + i )
1
1
1
1
1
+


+
+
+
}
−
pmt
(1 + i ) 2
(1 + i ) n−2 (1 + i ) n−1 (1 + i ) n
(1 + i ) n

PV * (1 + i ) = pmt *{

20


Derivation of PV of Annuity
Formula: Algebra. 5 of 5
1
1
PV * (1 + i ) = pmt *
+ PV − pmt
0
n
(1 + i )
(1 + i )
1
PV * (1 + i ) + PV = pmt − pmt
n
(1 + i )
1

pmt *{1 −
}
n
pmt 
1 
(
1+ i)

PV =
=
* 1 −
n 
i
i  (1 + i ) 
21


PV of Annuity Formula

pmt *{1 −
PV =

1
}
n
(1 + i )

i

pmt 

1 

=
* 1 −
n 
i  (1 + i ) 

22


PV Annuity Formula: Payment

PV =

pmt 
1 

* 1 −
n 
i  (1 + i ) 

(

pmt
−n
=
* 1 − (1 + i )
i
PV * i
pmt =

−n
1 − (1 + i )

(

23

)

)


PV Annuity Formula: Number
of Payments
(

)

pmt
−n
PV =
* 1 − (1 + i ) ;
i

(1 + i )

−n

(1 + i ) − n

PV * i
−n
= 1 − (1 + i )
pmt

PV * i
 PV * i 
= 1−
; − n * ln(1 + i ) = ln1 −

pmt
pmt 

 PV * i 
ln1 −

pmt 
PV * i

= 1−
; n=−
pmt
ln (1 + i )

24


Annuity Formula: PV Annuity
Due
PVdue = PVreg * (1 + i )

pmt
−n
*{1 − (1 + i ) } * (1 + i )
i
pmt
1− n
=
*{(1 + i ) − (1 + i ) }
i
=

25