Special Cash Flow Streams

Common Cash Flow Streams

Perpetuity A constant stream of cash flows that lasts forever.

Growing perpetuity A stream of cash flows that grows at a constant rate

forever.

Annuity A stream of constant cash flows that lasts for a fixed

number of periods.

Growing annuity A stream of cash flows that grows at a constant rate for a

fixed number of periods.

Perpetuity

A constant stream of cash flows that lasts forever.

0

…1

C

2

C

3

C

The formula for the present value of a perpetuity is:

32 )1()1()1( r

C

r

C

r

CPV

r

CPV

Perpetuity: Example

What is the value of a British consol that promises to pay £15 each year, every year forever?

The interest rate is 10-percent.

0

…1

£15

2

£15

3

£15

£15010.

£15PV

Growing PerpetuityA growing stream of cash flows that lasts forever.

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

The formula for the present value of a growing perpetuity is:

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

Growing Perpetuity: Example

The expected dividend next year is $1.30 and dividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value of this promised dividend stream?

0

…1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

00.26$05.10.

30.1$

PV

AnnuityA constant stream of cash flows with a fixed maturity.

The formula for the present value of an annuity is:

Tr

C

r

C

r

C

r

CPV

)1()1()1()1( 32

Trr

CPV

)1(

11

0 1

C

2

C

3

C

T

C

Annuity Intuition

An annuity is valued as the difference between two perpetuities:

one perpetuity that starts at time 1 less a perpetuity that starts at time T + 1

0 1

C

2

C

3

C

T

C

TrrC

r

CPV

)1(

Annuity: Example

If you can afford a $400 monthly car payment, what priced car can you afford if annual interest rates are 7% on 36-month loans?

59.954,12$)1207.1(

11

12/07.

400$36

PV

0 1

$400

2

$400

3

$400

36

$400

What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

 

22.297$09.1

97.323$0

PV

0 1 2 3 4 5

$100 $100 $100 $100$323.97$297.22

97.323$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

11

tt

PV

Growing AnnuityA growing stream of cash flows with a fixed maturity.

The formula for the present value of a growing annuity:

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

1

11

0 1

C

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

PV of Growing AnnuityYou are evaluating an income property that is providing increasing rents. Net rent is received at the end of each year. The first year's rent is expected to be $8,500 and rent is expected to increase 7% each year. Each payment occurs at the end of the year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%?

0 1 2 3 4 5

500,8$

)07.1(500,8$

2)07.1(500,8$

095,9$ 65.731,9$

3)07.1(500,8$

87.412,10$

4)07.1(500,8$

77.141,11$

$34,706.26

Growing AnnuityA retirement plan offers to make payments for 40 years after

retirement with a payment of $20,000 at the end of the first year and an increase in the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?

57.121,265$10.1

03.11

03.10.

000,20$40

PV

0 1

$20,000

2

$20,000×(1.03)

40

$20,000×(1.03)39