Corporate finance chapter 04 time vaue of money

Chapter 4: Time Value of Money Objective Explain the concept of compounding... Example: Interest Rate on a Lump Sum Investment nearest the to % 18... Effective Annual Rates of an APR of

Chapter 4: Time Value of

Money

Objective

Explain the concept of compounding

– Continuing in this manner you will find that

the following amounts will be earnt:

Future Value of a Lump Sum

n

i PV

Example: Future Value of a

1738

$

) 03 0 1

(

* 1500

$

) 1

Present Value of a Lump Sum

n n

n n

i

FV i

FV PV

i

i PV

FV

−

+

= +

(

* )

1 (

: obtain to

) 1

(

by sides

both Divide

) 1

(

*

Example: Present Value of a

2 years Given the

risk, you require a

return of 8% What

is the present value

55281

34293

) 08 0 1

(

000 ,

40

) 1

Solving Lump Sum Cash Flow for Interest Rate

(

) 1

n

n

PV

FV i

PV

FV i

i PV

FV

i PV

FV

Example: Interest Rate on a

Lump Sum Investment

nearest the

(to

% 18 7

071773463

0

1 2

1 2

1 15000 30000

1

10

1

10 10

Review of Logarithms

are used by finance are:

) ln(

) ln(

) ln(

) ln(

)

* ln(

) ln(

0 ,

) ln(

x y

x

y x

y x

x e

x x

e

y

x x

Review of Logarithms

prove from the last ones, and are useful

in finance

) ln(

* ) ln(

) ln(

) ln(

) ln(

) ln(

)

*

* ln(

) ln(

) ln(

) /

ln(

y x

y x

z y

x z

y x

y x

y x

≠ +

+ +

=

−

=

Solving Lump Sum Cash Flow for Number of Periods

( ) ( )( i)( )

PV

FV i

PV

FV n

i n

i PV

FV

i PV

FV

i PV

ln

ln 1

ln ln

1 ln

* )

1 ( ln ln

) 1

(

) 1

(

*

Effective Annual Rates of an APR of 18%

Annual

Percentage

rate

Frequency of Compounding AnnualEffective Rate

The Frequency of

Compounding

compounding increases, so does the annual effective rate

compounding rises to infinity?

1 1

m

k Lim

EFF

The Frequency of Compounding

1 1

1

1

− +

=

+

= +

m

m

EFF m

k

EFF m

k

m

k EFF

m

m

m m

Annual Percentage Rate

Derivation of PV of Annuity Formula: Algebra 1 of 5

( ) ( ) ( ) ( ) n ( ) n

i

pmt i

pmt i

pmt

i

pmt i

pmt PV

+

+ +

+

+ +

+ +

+ +

=

−

1 1

1

1 1

1 3

2 1



Derivation of PV of Annuity Formula: Algebra 2 of 5

1 1

1 1

1

1

1 1

1 {

*

1 3

2 1

n n

i i

i

i i

pmt PV

+

+ +

+

+ +

+ +

+ +

=

−



Derivation of PV of Annuity Formula: Algebra 3 of 5

( ) ( ) ( ) ( ) ( ) 1 }

1 1

1 1

1

1

1 1

1 {

* ) 1

(

* )

1 (

*

1 3

2 1

n n

i i

i

i i

i pmt

i PV

+

+ +

+

+ +

+ +

+ +

+

= +

−



Derivation of PV of Annuity

Formula: Algebra 4 of 5

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )n ( )n ( )n ( )n

n n

n n

i

pmt i

i i

i

i

pmt i

pmt

i i

i i

i

i i

pmt i

PV

+

− +

+ +

+ +

+

+ +

+ +

+ +

=

+

− +

+ +

+ +

+

+ +

+ +

+ +

= +

1

1 1

1 1

1 1

1

1

1 {

* 1

1

*

]} 1

1 1

1 [

1

1 1

1 1

1

1

1 1

1 {

* )

1

(

*

1 2

2

1 0

1 2

2

1 0





+

−

+ +

= +

n

n

n

pmt i

pmt PV

i

pmt pmt

PV i

PV

i

pmt

PV i

pmt i

PV

1 1

*

} 1

1 1

{

*

1

1 )

1 (

*

1

1 1

1

* )

1 (

i

i

pmt PV

1

1 1

*

} 1

1 1

{

*

PV Annuity Formula: Payment

i

i

PV pmt

i i

pmt

i i

pmt PV

*

1 1

*

1

1 1

*

PV Annuity Formula: Number

i PV n

pmt

i

PV i

pmt

i

PV i

n pmt

i

PV i

i pmt

i

PV i

i

pmt PV

n

n

n n

−

−

= +

+

−

= +

* 1

ln

;

* 1

1

* 1

ln 1

ln

*

;

* 1

1

1 1

*

; 1

1

*

Annuity Formula: PV Annuity Due

) 1

{(

*

) 1

(

* } 1

1 {

*

) 1

i

i i

pmt

i

i i

pmt

i PV

=

+ +

−

=

+

=

Derivation of FV of Annuity Formula: Algebra

1 1

* FV

sum) (lump

1

* FV

annuity) (reg.

1

1 1

*

− +

n

n

i i

pmt

i i

i pmt

i PV

i i

pmt PV

FV Annuity Formula: Payment

*

− +

=

− +

i i

pmt FV

FV Annuity Formula: Number

pmt

i

FV i

n i

i pmt

i FV

i i

pmt FV

n

n n

= +

+

= +

− +

=

1 ln

* 1

ln

* 1

ln 1

ln

* 1

ln

1

* 1

1 1

*

Perpetual Annuities / Perpetuities

pmt PV

1

1 1

*

• Let n -> infinity with i > 0:

pmt

PV =

Mortgage: The payment

financial calculator

• The first quantity to determine is the

amount of the loan and the points

500 ,

13

$

03

0

* ) 1 0 1

(

* 500000

$ Points

000 ,

450

$

) 1 0 1

(

* 500000

$ Loan

Calculator Solution

This is the monthly

After Tax Cash Flow

Percent of Interest and Principal