present value

present value

Suppose you are going to receive

$

⁡

10

,

000

, to be paid in two payments at the end of the next two years. You have the following two options

options

year 1

year 2

option 1

$

⁡

6

,

000

$

⁡

4

,

000

option 2

$

⁡

4

,

000

$

⁡

6

,

000

W

⁢

h

⁢

i

⁢

c

⁢

h

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

w

⁢

o

⁢

u

⁢

l

⁢

d

⁢

y

⁢

o

⁢

u

⁢

s

⁢

e

⁢

l

⁢

e

⁢

c

⁢

t

⁢

i

⁢

n

⁢

o

⁢

r

⁢

d

⁢

e

⁢

r

⁢

t

⁢

o

⁢

h

⁢

a

⁢

v

⁢

e

⁢

t

⁢

h

⁢

e

⁢

m

⁢

a

⁢

x

⁢

i

⁢

m

⁢

u

⁢

m

⁢

g

⁢

a

⁢

i

⁢

n

⁢

?

⁢

O

⁢

f

⁢

c

⁢

o

⁢

u

⁢

r

⁢

s

⁢

e

,

i

⁢

f

⁢

t

⁢

h

⁢

e

⁢

r

⁢

e

⁢

i

⁢

s

⁢

n

⁢

o

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

,

b

⁢

o

⁢

t

⁢

h

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

s

⁢

a

⁢

r

⁢

e

⁢

e

⁢

q

⁢

u

⁢

a

⁢

l

.

I

⁢

f

⁢

a

⁢

n

⁢

y

⁢

n

⁢

o

⁢

n

-

z

⁢

e

⁢

r

⁢

o

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

r

⁢

a

⁢

t

⁢

e

⁢

s

⁢

a

⁢

r

⁢

e

⁢

i

⁢

n

⁢

v

⁢

o

⁢

l

⁢

v

⁢

e

⁢

d

,

o

⁢

n

⁢

e

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

m

⁢

a

⁢

y

⁢

b

⁢

e

⁢

p

⁢

r

⁢

e

⁢

f

⁢

e

⁢

r

⁢

a

⁢

b

⁢

l

⁢

e

⁢

t

⁢

h

⁢

a

⁢

n

⁢

t

⁢

h

⁢

e

⁢

o

⁢

t

⁢

h

⁢

e

⁢

r

.

B

⁢

y

⁢

c

⁢

a

⁢

l

⁢

c

⁢

u

⁢

l

⁢

a

⁢

t

⁢

i

⁢

n

⁢

g

⁢

t

⁢

h

⁢

e

⁢

present values

⁢

o

⁢

f

⁢

t

⁢

h

⁢

e

⁢

s

⁢

e

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

s

,

o

⁢

n

⁢

e

⁢

m

⁢

a

⁢

y

⁢

b

⁢

e

⁢

a

⁢

b

⁢

l

⁢

e

⁢

t

⁢

o

⁢

c

⁢

o

⁢

m

⁢

p

⁢

a

⁢

r

⁢

e

⁢

t

⁢

h

⁢

e

⁢

`

⁢

`

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

′′

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

⁢

s

⁢

o

⁢

f

⁢

t

⁢

h

⁢

e

⁢

s

⁢

e

⁢

p

⁢

a

⁢

y

⁢

m

⁢

e

⁢

n

⁢

t

⁢

s

⁢

a

⁢

n

⁢

d

⁢

f

⁢

i

⁢

g

⁢

u

⁢

r

⁢

e

⁢

o

⁢

u

⁢

t

⁢

w

⁢

h

⁢

i

⁢

c

⁢

h

⁢

i

⁢

s

⁢

t

⁢

h

⁢

e

⁢

p

⁢

r

⁢

e

⁢

f

⁢

e

⁢

r

⁢

a

⁢

b

⁢

l

⁢

e

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

.

S

⁢

o

⁢

w

⁢

h

⁢

a

⁢

t

⁢

i

⁢

s

⁢

a

⁢

present value

⁢

?

⁢

𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧

.

L

⁢

e

⁢

t

P

b

⁢

e

⁢

t

⁢

h

⁢

e

⁢

a

⁢

m

⁢

o

⁢

u

⁢

n

⁢

t

⁢

o

⁢

f

⁢

a

⁢

p

⁢

a

⁢

y

⁢

m

⁢

e

⁢

n

⁢

t

⁢

a

⁢

t

⁢

s

⁢

o

⁢

m

⁢

e

⁢

t

⁢

i

⁢

m

⁢

e

t¿0

i

⁢

n

⁢

t

⁢

h

⁢

e

⁢

f

⁢

u

⁢

t

⁢

u

⁢

r

⁢

e

.

t

⁢

h

⁢

e

⁢

n

⁢

t

⁢

h

⁢

e

⁢

present value

PV(P)

o

⁢

f

P

i

⁢

s

⁢

s

⁢

i

⁢

m

⁢

p

⁢

l

⁢

y

⁢

t

⁢

h

⁢

e

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

⁢

o

⁢

f

⁢

t

⁢

h

⁢

i

⁢

s

⁢

p

⁢

a

⁢

y

⁢

m

⁢

e

⁢

n

⁢

t

⁢

a

⁢

t

⁢

t

⁢

i

⁢

m

⁢

e

t=0

.

S

p

e

c

i

f

i

c

a

l

l

y

,

i

f

t

h

e

i

n

t

e

r

e

s

t

r

a

t

e

f

r

o

m

0

t

⁢

o

t

i

⁢

s

r

,

t

h

e

n

PV

⁡

(

P

)

=

P

1

+

r

.

I

⁢

n

⁢

o

⁢

t

⁢

h

⁢

e

⁢

r

⁢

w

⁢

o

⁢

r

⁢

d

⁢

s

,

i

⁢

f

⁢

w

⁢

e

⁢

i

⁢

n

⁢

v

⁢

e

⁢

s

⁢

t

PV(P)

t

⁢

o

⁢

d

⁢

a

⁢

y

,

e

⁢

a

⁢

r

⁢

n

⁢

i

⁢

n

⁢

g

⁢

a

⁢

n

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

a

⁢

t

⁢

a

⁢

r

⁢

a

⁢

t

⁢

e

⁢

o

⁢

f

r

b

⁢

e

⁢

t

⁢

w

⁢

e

⁢

e

⁢

n

⁢

t

⁢

i

⁢

m

⁢

e

⁢

s

0

a

⁢

n

⁢

d

t

,

t

h

e

n

a

t

t

i

m

e

t

,

w

e

w

o

u

l

d

h

a

v

e

m

a

d

e

P

.

N

o

w

,

s

u

p

p

o

s

e

i

n

t

h

e

e

x

a

m

p

l

e

a

b

o

v

e

,

b

o

t

h

o

p

t

i

o

n

s

h

a

v

e

a

n

e

f

f

e

c

t

i

v

e

a

n

n

u

a

l

i

n

t

e

r

e

s

t

r

a

t

e

(

http://planetmath.org/InterestRate

)

o

f

5%

c

⁢

o

⁢

m

⁢

p

⁢

o

⁢

u

⁢

n

⁢

d

⁢

e

⁢

d

⁢

a

⁢

n

⁢

n

⁢

u

⁢

a

⁢

l

⁢

l

⁢

y

⁢

(

http://planetmath.org/CompoundInterest

)

,

t

⁢

h

⁢

e

⁢

n

⁢

t

⁢

h

⁢

e

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

⁢

o

⁢

f

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

1

⁢

i

⁢

s

$

⁡

6

,

000

1.05

+

$

⁡

4

,

000

(

1.05

)

2

≈

$

⁡

9

,

342.40

w

⁢

h

⁢

e

⁢

r

⁢

e

⁢

a

⁢

s

⁢

t

⁢

h

⁢

e

⁢

s

⁢

e

⁢

c

⁢

o

⁢

n

⁢

d

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

h

⁢

a

⁢

s

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

$

⁡

4

,

000

1.05

+

$

⁡

6

,

000

(

1.05

)

2

≈

$

⁡

9

,

251.70

C

⁢

l

⁢

e

⁢

a

⁢

r

⁢

l

⁢

y

,

t

⁢

h

⁢

e

⁢

f

⁢

i

⁢

r

⁢

s

⁢

t

⁢

o

⁢

p

⁢

t

⁢

i

⁢

o

⁢

n

⁢

i

⁢

s

⁢

s

⁢

u

⁢

p

⁢

e

⁢

r

⁢

i

⁢

o

⁢

r

⁢

t

⁢

h

⁢

a

⁢

n

⁢

t

⁢

h

⁢

e

⁢

s

⁢

e

⁢

c

⁢

o

⁢

n

⁢

d

⁢

o

⁢

n

⁢

e

.

𝐑𝐞𝐦𝐚𝐫𝐤𝐬

.

•

Of course, the result will be the same if one instead computes the future values of these options, which are the values of the payments at a specific future time

>
t
0

: if payment is valued at

P

at time

0

, its value at some future time

>
t
0

, or its future value is
FV(P)=P(1+r), if

r

is the interest rate from

0

to

t

.

•

An accompanying concept is that of the net present value

NPV

. It is the present value of all the future payments minus the initial investment: suppose an investment

I

is made where an initial amount of

A

is made at time

0

, and payments

P
1

,
…
,

P
n

are returns as a result of this investment. Then
NPV(I)=(PV(P_1)+PV(P_2)+⋯+PV(P_n))-A.

⁢

I

⁢

f

⁢

w

⁢

e

⁢

t

⁢

r

⁢

e

⁢

a

⁢

t

⁢

t

⁢

h

⁢

e

⁢

i

⁢

n

⁢

i

⁢

t

⁢

i

⁢

a

⁢

l

⁢

i

⁢

n

⁢

v

⁢

s

⁢

e

⁢

t

⁢

m

⁢

e

⁢

n

⁢

t

A

a

⁢

s

⁢

a

⁢

`

⁢

`

⁢

n

⁢

e

⁢

g

⁢

a

⁢

t

⁢

i

⁢

v

⁢

e

′′

⁢

r

⁢

e

⁢

t

⁢

u

⁢

r

⁢

n

,

A=-P_0=-PV(P_0)

,

t

h

e

n

t

h

e

n

e

t

p

r

e

s

e

n

t

v

a

l

u

e

o

f

t

h

e

i

n

v

e

s

t

m

e

n

t

c

a

n

b

e

w

r

i

t

t

e

n

NPV

⁡

(

I

)

=

PV

⁡

(

P

0

)

+

PV

⁡

(

P

1

)

+

⋯

+

PV

⁡

(

P

n

)

=

∑

i

=

0

n

PV

⁡

(

P

i

)

.

O

⁢

n

⁢

e

⁢

w

⁢

o

⁢

u

⁢

l

⁢

d

⁢

u

⁢

s

⁢

u

⁢

a

⁢

l

⁢

l

⁢

y

⁢

w

⁢

a

⁢

n

⁢

t

⁢

t

⁢

o

⁢

i

⁢

n

⁢

v

⁢

e

⁢

s

⁢

t

⁢

i

⁢

n

⁢

s

⁢

o

⁢

m

⁢

e

⁢

t

⁢

h

⁢

i

⁢

n

⁢

g

⁢

w

⁢

i

⁢

t

⁢

h

⁢

a

⁢

p

⁢

o

⁢

s

⁢

i

⁢

t

⁢

i

⁢

v

⁢

e

⁢

n

⁢

e

⁢

t

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

.

N

⁢

e

⁢

t

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

⁢

v

⁢

a

⁢

l

⁢

u

⁢

e

⁢

s

⁢

a

⁢

r

⁢

e

⁢

c

⁢

o

⁢

m

⁢

m

⁢

o

⁢

n

⁢

l

⁢

y

⁢

u

⁢

s

⁢

e

⁢

d

⁢

w

⁢

h

⁢

e

⁢

n

⁢

o

⁢

n

⁢

e

⁢

i

⁢

s

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

e

⁢

d

⁢

i

⁢

n

⁢

c

⁢

o

⁢

m

⁢

p

⁢

a

⁢

r

⁢

i

⁢

n

⁢

g

⁢

c

⁢

a

⁢

r

⁢

l

⁢

o

⁢

a

⁢

n

⁢

s

⁢

o

⁢

r

⁢

h

⁢

o

⁢

m

⁢

e

⁢

m

⁢

o

⁢

r

⁢

t

⁢

g

⁢

a

⁢

g

⁢

e

⁢

s

.

Title

present value

Canonical name

PresentValue

Date of creation

2013-03-22 16:40:59

Last modified on

2013-03-22 16:40:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 91B28

Defines

net present value

Defines

future value

Generated on Sat Feb 10 13:38:34 2018 by

LaTeXML