Special Cash Flow Streams
Common Cash Flow Streams


Perpetuity




Growing perpetuity




A stream of cash flows that grows at a constant rate...
Perpetuity
A constant stream of cash flows that lasts
forever.
C
C
C
0

1

2

…

3

C
C
C
PV =
+
+
+
2
3
(1 + r ) (1 + r ...
Perpetuity: Example
What is the value of a British consol that
promises to pay £15 each year, every year
forever?
The inte...
Growing Perpetuity
A growing stream of cash flows that lasts
forever.
C
C×(1+g) C ×(1+g)2
0

1

2

…

3

C
C × (1 + g ) C ...
Growing Perpetuity: Example
The expected dividend next year is $1.30 and dividends are
expected to grow at 5% forever.
If ...
Annuity
A constant stream of cash flows with a fixed maturity.
C

C

C

C


0

1

2

3

T

C
C
C
C
PV =
+
+
+
2
3
(1 + r...
Annuity Intuition
C

C

C

C


0

1

2

3

An annuity is valued as the difference between
two perpetuities:
one perpetuit...
Annuity: Example
If you can afford a $400 monthly car payment, what
priced car can you afford if annual interest rates are...
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...
Growing Annuity
A growing stream of cash flows with a fixed maturity.
C

C×(1+g) C ×(1+g)2

C×(1+g)T-1


0

1
2
3
T
T −1
...
PV of Growing Annuity
You are evaluating an income property that is providing
increasing rents. Net rent is received at th...
Growing Annuity
A retirement plan offers to make payments for 40 years after
retirement with a payment of $20,000 at the e...
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Лекц 4 Special cash flow streams

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Лекц 4 Special cash flow streams

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Лекц 4 Special cash flow streams

  1. 1. Special Cash Flow Streams
  2. 2. Common Cash Flow Streams  Perpetuity   Growing perpetuity   A stream of cash flows that grows at a constant rate forever. Annuity   A constant stream of cash flows that lasts forever. 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.
  3. 3. Perpetuity A constant stream of cash flows that lasts forever. C C C 0 1 2 … 3 C C C PV = + + + 2 3 (1 + r ) (1 + r ) (1 + r ) The formula for the present value of a perpetuity is: C PV = r
  4. 4. 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. £15 0 £15 £15 1 2 3 £15 PV = = £150 .10 …
  5. 5. Growing Perpetuity A growing stream of cash flows that lasts forever. C C×(1+g) C ×(1+g)2 0 1 2 … 3 C C × (1 + g ) C × (1 + g ) PV = + + + 2 3 (1 + r ) (1 + r ) (1 + r ) 2 The formula for the present value of a growing perpetuity is: C PV = r−g
  6. 6. 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? $1.30 0 1 $1.30×(1.05) $1.30 ×(1.05)2 2 $1.30 PV = = $26.00 .10 − .05 … 3
  7. 7. Annuity A constant stream of cash flows with a fixed maturity. C C C C  0 1 2 3 T C C C C PV = + + + 2 3 (1 + r ) (1 + r ) (1 + r ) (1 + r )T The formula for the present value of an annuity is: C 1  PV = 1 − r  (1 + r )T  
  8. 8. Annuity Intuition C C C C  0 1 2 3 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 C    C r PV = − r (1 + r )T T
  9. 9. 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? $400 $400 $400 $400  0 1 2 3 36  $400  1 PV = = $12,954.59 1 − 36  .07 / 12  (1 + .07 12) 
  10. 10. 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%? 4 $100 $100 $100 $100 $100 PV1 = ∑ = + + + = $323.97 t 1 2 3 4 (1.09) (1.09) (1.09) (1.09) t =1 (1.09) $297.22 0 $323.97 1 $100 2 $323.97 PV = = $297.22 0 1.09 $100 3 $100 4 $100 5
  11. 11. Growing Annuity A growing stream of cash flows with a fixed maturity. C C×(1+g) C ×(1+g)2 C×(1+g)T-1  0 1 2 3 T T −1 C C × (1 + g ) C × (1 + g ) PV = + ++ 2 (1 + r ) (1 + r ) (1 + r )T The formula for the present value of a growing annuity:   1 + g T  C PV =   1 −  r − g   1+ r    
  12. 12. PV of Growing Annuity You 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%? $8,500 × (1.07) $8,500 0 1 $34,706.26 $8,500 × (1.07) 3 $9,095 $9,731.65 $10,412.87 $11,141.77 2 3 $8,500 × (1.07) 2 4 5 $8,500 × (1.07) 4
  13. 13. Growing Annuity A 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 threepercent each year. What is the present value at retirement if the discount rate is 10 percent? $20,000 $20,000×(1.03) 0 1 2  $20,000×(1.03)39 40   1.03 40  $20,000 PV =   = $265,121.57 1 −  .10 −.03  1.10    

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