Download - Лекц 4 Special cash flow streams
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