discounted cash flows

1. Managerial Finance
Master of Business Administration
(MBA)
1

2. 4th lecture
Discounted Cash Flow valuation
2

3. Learning Objectives
• Understand the concept of time value of money.
• Differentiate between simple interest and compound interest
• Be able to compute the future value and/or present value of a single cash
flow or series of cash flows
• Be able to compute the present value of a perpetuity.
• Be able to compute the present value of a growing perpetuity.
• Be able to compute the present value and future value of an ordinary
annuity and an annuity due.
• Be able to compute the present value of a growing annuity.

4. Discounted Cash Flow Valuation
4

5. Time Value of Money
Suppose you won a $1,000 prize, and you were given 2 options:
a- either to collect your money today or,
b- to collect your money in a year
which option would you choose?
Time 0 Time 1
Option A
$1,000
Option B
$1,000
5

6. • Receiving money sooner is better, as a dollar today is worth more
than a dollar promised some time in the future.
• One of the reasons for this is money’s ability to earn interest over
time, so a dollar today would grow to more than a dollar later.
• If you were to invest $1,000 today at 10% interest for one year, your
investment would grow to $1,100, which is better than option B
Time 0 Time 1
Option A
$1,000
Option B
$1,000
$1,100
10% interest for 1 year
Time Value of Money
6

7. Option A
• 1,000 is your Present Value (PV) or principal
• 1,100 is your Future Value (FV): The total amount due at the
end of the investment
• Time (t) = 1 period
• Interest rate (r) = 10%
Time 0 Time 1
$1,000 $1,100
10%
Time Value of Money
7

8. Simple vs. Compound Interest
If you were to leave the money for another period, earning a 10% interest,
how much will you have at the end of the second year?
0 2
$1,000 ?
1
$1,100
10%
10%
This depends on the type of interest whether it is simple interest or
compound interest!
8

9. Simple Interest
• Interest is earned only on the original principal (i.e. interest is
not reinvested)
0 2
$1,000 ?
1
$1,100
10%
10%
Interest per year
Total Interest
• FV at the end of year 2 = PV + (PV x r x t)
= 1,000 + (1,000 x 10% x 2) = 1,200
9

10. Compound Interest
• Interest earned on both the initial principal and reinvested
interest (i.e. earning interest on interest)
• Future Value (FV) at the end of year 2 = PV×(1 + r)T
FV = 1,000 (1 + 10%)2 = 1,210
• PV is cash flow at date 0 or C0, r is the appropriate interest rate,
and T is the number of periods over which the cash is invested.
0 2
$1,000 ?
1
$1,100
10%
10%
10

11. This 1,210 consists of:
1. $1,000 principal “You deposited in Year 0”
2. $100 interest on the principal in the first year
3. $100 interest on the principal in the second year
4. $10 interest on interest in the second year “Value of
compounding”
Compound Interest
11

12. Future Value and Compounding
• Notice that the future value of your 1,000 in year 2 using
compound interest is considerably higher than the future value
of the 1,000 in year 2 using simple interest
$1,210 > $1,200
FV (Compound) > FV (Simple)
• Given that everything else is constant; The extra $10 are due to
compounding (earning interest on interest)
12

13. Future Value (FV)
• Suppose a stock currently pays a dividend of $1.10, which is
expected to grow at 40% per year for the next five years.
• What will the dividend be in five years?
FV = PV×(1 + r)T
$5.92 = $1.10×(1.40)5
13

14. Present Value (PV)
• PV is the current value of a future cash flow.
• The formula for PV can be written as:
T
r
FV
PV
)
1
( 

• Where FV is cash flow to be received in the future, r is the appropriate
interest rate, and T is the number of periods over which the cash is
invested.
14

15. Present Value (PV)
If you were to be promised $10,000 due in one year when
interest rates are 5%, your investment would be worth $9,523.81
in today’s dollars.
05
.
1
000
,
10
$
81
.
523
,
9
$ 
The amount that a borrower would need to set
aside today to be able to meet the promised
payment of $10,000 in one year is called the
Present Value (PV).
Note that FV can be calculated as follows;
$10,000 = $9,523.81×(1.05).
15

16. Present Value and Discounting
How much would an investor have to set aside today in order to
have $20,000 five years from now if the current rate is 15%?
0 1 2 3 4 5
$20,000
PV
5
)
15
.
1
(
000
,
20
$
53
.
943
,
9
$ 
16

17. Discounting and Compounding
“Discounting”
finding the present value of one or more future amounts.
“Compounding”
The opposite of Discounting, in which we calculate the future value of
one or more present amounts.
17

18. How Long is the Wait?
If we deposit $5,000 today in an account paying 10%, how long
does it take to grow to $10,000?
T
r
C
FV )
1
(
0 


T
)
10
.
1
(
000
,
5
$
000
,
10
$ 

2
000
,
5
$
000
,
10
$
)
10
.
1
( 

T
)
2
ln(
)
10
.
1
ln( 
T
years
27
.
7
0953
.
0
6931
.
0
)
10
.
1
ln(
)
2
ln(



T
)
1
ln(
ln
r
PV
FV
T








18

19. What Rate Is Enough?
Assume the total cost of a college education will be $50,000
when your child enters college in 12 years. You have $5,000 to
invest today. What rate of interest must you earn on your
investment to cover the cost of your child’s education?
T
r
C
FV )
1
(
0 


12
)
1
(
000
,
5
$
000
,
50
$ r



10
000
,
5
$
000
,
50
$
)
1
( 12


 r
12
1
10
)
1
( 
 r
2115
.
1
2115
.
1
1
10 12
1





r About 21.15%.
1
1








t
PV
FV
r
>>>>> 19

20. Compounding Periods
Interest can be compounded;
• Annually (compounded once/year)
• Semi-annually (compounded 2 times/year)
• Quarterly (compounded 4 times/year)
• Monthly (compounded 12 times/year)
• Weekly (compounded 52 times/year)
• Daily (compounded 365 times/year)
20

21. Compounding Periods
For example, you want to invest $50 for 3 years in one of two
banks, Bank A offers you 12% compounded annually, whereas
Bank B offers you 12% (per year) compounded semi-annually.
How much will your investment grow to in each bank and which
bank will you choose?
Bank A Bank B
93
.
70
$
)
06
.
1
(
50
$
2
12
.
1
50
$ 6
3
2













FV
0 3
$50 ?
1
12%
12%
2
12%
0 3
$50 ?
1 12%
12% 12%
2
6%
6% 6%
6% 6% 6%
  25
.
70
$
12
.
1
50
$
3




FV

22. Compounding Periods
• How did we solve for FV for Bank B in the previous slide?
• Compounding an investment m times a year for T years
provides for future value of wealth:
• r: Stated rate, Quoted Rate, Annual Percentage Rate (APR)
• m: number of times you compound interest per year
• T: number of years
T
m
m
r
C
FV









 1
0

23. Effective Annual Rates of Interest
So, investing at 12.36% compounded annually is the same as investing
at 12% compounded semi-annually.
93
.
70
$
)
1
(
50
$ 3



 EAR
FV
50
$
93
.
70
$
)
1
( 3

 EAR
1236
.
1
50
$
93
.
70
$
3
1









EAR
1
1 








m
m
r
EAR
What is the annual interest rate that yields the same future
value as the 12% rate compounded semi-annually?
23

24. Effective Annual Rate (EAR)
• The interest rate expressed as if it was compounded once per
year (annually).
• This is the actual rate paid (or received) after taking into
consideration the compounding that occurs during the year.
• If you want to compare two alternative investments with
different compounding periods you need to compute the EAR
and use that for comparison.
24

25. Effective Annual Rates of Interest
• Find the Effective Annual Rate (EAR) of an 18% Annual
Percentage Rate (APR), (stated rate, quoted rate) loan that is
compounded monthly.
• What we have is a loan with a monthly interest rate of 1½ %
• This is equivalent to a loan with an annual interest rate of
19.56%.
1956
.
1
)
015
.
1
(
1
12
18
.
1
1
1 12
12




















m
m
r
25

26. Multiple Cash Flows
• Consider an investment that pays $200 one year from now,
with cash flows increasing by $200 per year through year 4.
If the interest rate is 12%, what is the present value of this
stream of cash flows?
• If the issuer offers this investment for $1,500, should you
purchase it?
26

27. PV of Multiple Cash Flows
0 1 2 3 4
200 400 600 800
178.57
318.88
427.07
508.41
1,432.93
Present Value < Cost → Do Not Purchase
27

28. FV of Multiple Cash Flows
What’s the future value of these multiple cash flows at the end of year 4, if r=12%?
0 1 2 3 4
200 400 600 800
2,254.75
672
501.76
280.99
800
28

29. Simplifications
Because many basic finance problems are potentially time-consuming, we search
for simplifications in this section. We provide simplifying formulas for four classes
of cash flow streams:
1. Perpetuity
A stream of constant cash flows that lasts forever
2. Growing perpetuity
A stream of cash flows that grows at a constant rate forever
3. Annuity
A stream of constant cash flows that lasts for a fixed number of periods
4. Growing annuity
A stream of cash flows that grows at a constant rate for a fixed number of
periods
29

30. 1. Perpetuity
A constant stream of cash flows that lasts forever
0
…
1
C
2
C
3
C







 3
2
)
1
(
)
1
(
)
1
( r
C
r
C
r
C
PV
r
C
PV 
30

31. Perpetuity: Example
What is the value of a British consol (bond) that promises to pay
£15 every year forever, if the interest rate is 10-percent?
0
…
1
£15
2
£15
3
£15
£150
10
.
£15


PV
31

32. 2. Growing Perpetuity
A growing stream of cash flows that lasts forever
0
…
1
C1
2
C1×(1+g)
3
C1×(1+g)2











 3
2
1
2
1
1
)
1
(
)
1
(
)
1
(
)
1
(
)
1
( r
g
C
r
g
C
r
C
PV
g
r
C
PV

 1
32

33. 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
33

34. 3. Annuity
• A stream of constant cash flows that lasts for a fixed number
of periods.
• We can calculate APV and AFV
We have 2 types of Annuities
Ordinary Annuity
CFs occur at the end
of the period
Annuity Due
CFs occur at the
beginning of the period
34

35. 3.1 Ordinary Annuity FV
T
r
C
r
C
r
C
r
C
FV )
1
(
)
1
(
)
1
(
)
1
( 3
2







 
0 1
C
2
C
3
C
T
C

 
1
)
1
( 

 T
r
r
C
FV
A constant stream of cash flows with a fixed maturity with CFs occurring at
the end of the period
35

36. 3.1 Ordinary Annuity FV (Example)
Suppose you put $3,000 per year into a bank account that pays
6% interest per year. How much will you have in your account in
30 years?
  56
.
174
,
237
$
1
)
06
.
1
(
06
.
3000 30




FV
36

37. T
r
C
r
C
r
C
r
C
PV
)
1
(
)
1
(
)
1
(
)
1
( 3
2







 








 T
r
r
C
PV
)
1
(
1
1
0 1
C
2
C
3
C
T
C

3.1 Ordinary Annuity PV
A constant stream of cash flows with a fixed maturity with CFs occurring at
the end of the period
37

38. If you can afford a $400 monthly car payment, what is the budget
of the car you can afford if interest rates are 7% on 36-month
loans?
59
.
954
,
12
$
)
12
07
.
1
(
1
1
12
/
07
.
400
$
36










PV
0 1
$400
2
$400
3
$400
36
$400

3.1 Ordinary Annuity PV (Example)
38

39. Important Note
You ALWAYS need to make sure that the cash flows (C), the
interest rate (r), and the time period (t) match.
(r and t should follow C)
39

40. 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%?
97
.
323
$
)
09
.
1
(
100
$
)
09
.
1
(
100
$
)
09
.
1
(
100
$
)
09
.
1
(
100
$
)
09
.
1
(
100
$
4
3
2
1
4
1
1 




 

t
t
PV
22
.
297
$
09
.
1
97
.
327
$
0


PV
0 1 2 3 4 5
$100 $100 $100 $100
$323.97
$297.22
3.1 Special Case: Delayed Ordinary Annuity (Example)
40

41. )
1
(
)
1
(
1
1 r
r
r
C
PV T









   )
1
(
1
)
1
( r
r
r
C
FV T




An annuity with an immediate initial payment (i.e. the cash flow
occurs at the beginning of the period)
0 3
C
1
12%
12%
2
12%
C
C
3.2 Annuity Due
41

42. 4. Growing Annuity
A growing stream of cash flows with a fixed maturity
T
T
r
g
C
r
g
C
r
C
PV
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
1
1
2
1
1


































T
r
g
g
r
C
PV
)
1
(
1
1
1
0 1
C1

2
C1×(1+g)
3
C1×(1+g)2
T
C1×(1+g)T-1
42

43. Growing Annuity (Example)
A defined-benefit retirement plan offers to pay $20,000 per year
for 40 years and increase the annual payment by 3% each year.
What is the present value at retirement if the discount rate is 10%?
57
.
121
,
265
$
10
.
1
03
.
1
1
03
.
10
.
000
,
20
$
40


















PV
0 1
$20,000

2
$20,000×(1.03)
40
$20,000×(1.03)39
43

44. Quick Quiz
• How is the future value of a single cash flow computed?
• How is the present value of a series of cash flows computed.
• What is an EAR, and how is it computed?
• What is a perpetuity? an annuity?
44

45. The End
45