Discounted Cashflows final.ppt..........

McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Discounted Cash Flow Valuation
Chapter 4

Chapter Outline
4.1 Valuation: The One-Period Case
4.2 The Multiperiod Case
4.3 Compounding Periods
4.4 Simplifications
4.5 What Is a Firm Worth?

Key Concepts and Skills
 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 return on an
investment
 Be able to use a financial calculator to solve
time value problems
 Understand perpetuities and annuities

4.1 The One-Period Case
 If you were to invest $10,000 at 5-percent interest
for one year, your investment would grow to
$10,500.
$500 would be interest ($10,000 × .05)
$10,000 is the principal repayment ($10,000 × 1)
$10,500 is the total due. It can be calculated as:
$10,500 = $10,000×(1.05)
 The total amount due at the end of the investment is
call the Future Value (FV).

Future Value – one period case
 In the one-period case, the formula for FV can
be written as:
FV = C0×(1 + r)n
Where C0 is cash flow today (time zero), and
r is the appropriate interest rate.

Present Value
 If you were to be promised $10,000 due in one year
when interest rates are 5-percent, 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 $10,000 = $9,523.81×(1.05).

Present Value – one period case
 In the one-period case, the formula for PV can
be written as:
r
C
PV


1
1
Where C1 is cash flow at date 1(i.e. year 1), and
r is the appropriate interest rate.

Net Present Value
 The Net Present Value (NPV) of an investment is the
present value of the expected cash flows, less the
cost of the investment.
 Suppose an investment that promises to pay $10,000
in one year is offered for sale for $9,500. Your
interest rate is 5%. Are you going to buy this or not?
NPV = - (Cost) + PV 1
(1 )n
C
r



Net Present Value
81
.
23
$
81
.
523
,
9
$
500
,
9
$
05
.
1
000
,
10
$
500
,
9
$







NPV
NPV
NPV
The present value of the cash inflow is greater than the
cost. In other words, the Net Present Value is positive,
so the investment should be purchased.

Net Present Value
In the one-period case, the formula for NPV can be
written as:
NPV = – Cost + PV
If we had not undertaken the positive NPV project
considered on the last slide, and instead invested our
$9,500 elsewhere at 5 percent, our FV would be less
than the $10,000 the investment promised, and we
would be worse off in FV terms :
FV= $9,500×(1.05) = $9,975 < $10,000

4.2 The Multiperiod Case
 The general formula for the future value of an
investment over many periods can be written
as:
FV = C0×(1 + r)n
Where
C0 is cash flow at date 0,
r is the appropriate interest rate, and
n is the number of periods over which the cash is
invested.

Future Value
 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 = C0×(1 + r)n

Future Value
 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 = C0×(1 + r)n
$5.92 = $1.10×(1.40)5

Future Value and Compounding
 Notice that the dividend in year five, $5.92,
is considerably higher than the sum of the
original dividend plus five increases of 40-
percent on the original $1.10 dividend:
$5.92 > $1.10 + 5×[$1.10×.40] = $3.30
This is due to compounding.

Future Value and Compounding
0 1 2 3 4 5
1 0
.
1
$
3
)
4 0
.
1
(
1 0
.
1
$ 
0 2
.
3
$
)
4 0
.
1
(
1 0
.
1
$ 
5 4
.
1
$
2
)
4 0
.
1
(
1 0
.
1
$ 
1 6
.
2
$
5
)
4 0
.
1
(
1 0
.
1
$ 
9 2
.
5
$
4
)
4 0
.
1
(
1 0
.
1
$ 
2 3
.
4
$

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

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
$ 

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?

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

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?

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%.
About 21.15%.

Multiple UNEVEN 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 (cost), should
you purchase it?

Multiple Cash Flows
0 1 2 3 4
200 400 600 800
Present Value v/s COST →
1
(1 )n
C
PV
r


$1,500 (cost),

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
1
(1 )n
C
PV
r



Present & Future value of Uneven
cashflows
FV =

Exercise
 A small-scale businessman deposits money into his savings
account at the beginning of each year, depending on the business
returns. He deposits $1,000 in the first year, $2,000 in the second
year, $5,000 in the third year, and $7,000 in the fourth year. The
account credits interest at an annual interest rate of 7%. What is
the closest value of the money accumulated in the savings account
at the beginning of year 4?

Solution
 Solution
 The future value of the unequal payments is the
sum of individual accumulations:
 1,000(1.07)3+2,000(1.07)2+5,000(1.07)1+
 7000(1.07)0=15,864.48
 Note: He makes payments at the beginning of
each year.

Exercise
 Your broker calls you and tells you that he has this great investment
opportunity. If you invest $100 today, you will receive $40 in one year
and $75 in two years. If you require a 15% return on investments of this
risk, should you take the investment?

Exercise
 Your broker calls you and tells you that he has this great investment
opportunity. If you invest $100 today, you will receive $40 in one year
and $75 in two years. If you require a 15% return on investments of this
risk, should you take the investment?
 Use the CF keys to compute the value of the investment
 CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1
 NPV; I = 15; CPT NPV = 91.49
 No, the broker is charging more than you would be willing to
pay.

Effective Annual Interest rate EAR
 This is the actual rate paid (or received) after
accounting for 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.

EAR FORMULA
1
m
APR
1
EAR
m









Remember that the APR is the quoted rate, and
m is the number of compounding periods per year

QUESTION 1
 You are looking at two savings accounts. One pays 5.25%,
with daily compounding. The other pays 5.3% with
semiannual compounding. Which account should you use?
 First account:
 EAR =
 Second account:
 EAR =
 Which account should you choose and why?
The higher the effective annual interest rate is, the better it is for
savers/investors, but worse for borrowers.
1
m
APR
1
EAR
m










QUESTION 1
 You are looking at two savings accounts. One pays 5.25%,
with daily compounding. The other pays 5.3% with
semiannual compounding. Which account should you use?
 First account:
 EAR = (1 + .0525/365)365 – 1 = 5.39%
 Second account:
 EAR = (1 + .053/2)2 – 1 = 5.37%
 Which account should you choose and why?
The higher the effective annual interest rate is, the better it is for
savers/investors, but worse for borrowers.

Annual percentage rate APR
 This is the annual rate that is quoted by law
 By definition APR = period rate * number of periods per year
 Consequently, to get the period rate we rearrange the APR
equation:
 Period rate = APR / number of periods per year
 The main difference between APR and EAR is that APR is based on
simple interest, while EAR takes compound interest into account. APR is
most useful for evaluating mortgage and auto loans, while EAR (or APY)
is most effective for evaluating frequently compounding loans such as
credit cards.

Computing APR
 What is the APR if the monthly rate is 0.5%?
 *( ) =
 What is the APR if the semiannual rate is 0.5%?
 *( ) =
 What is the monthly rate if the APR is 12% with
monthly compounding?


Computing APR
 What is the APR if the monthly rate is 0.5%?
 .5*(12) =
 What is the APR if the semiannual rate is 0.5%?
 .5*(2) =
 What is the monthly rate if the APR is 12% with
monthly compounding?
 12 / 12 =

Computing APRs from EARs
 If you have an effective rate, how can you
compute the APR? Rearrange the EAR
equation and you get:





 
 1
-
EAR)
(1
m
APR m
1

QUESTION 2
 Suppose you want to earn an effective rate of 12%
and you are looking at an account that compounds
on a monthly basis. What APR must they pay?





 
 1
-
EAR)
(1
m
APR m
1

QUESTION 2
 Suppose you want to earn an effective rate of 12%
and you are looking at an account that compounds
on a monthly basis. What APR must they pay?
 
11.39%
or
8655152
113
.
1
)
12
.
1
(
12 12
/
1




APR

4.4 Simplifications
 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







 3
2
)
1
(
)
1
(
)
1
( r
C
r
C
r
C
PV
r
C
PV 

Perpetuity: Example
What is the value of a British consol (UK bonds)
that promises to pay £15 every year forever? The
interest rate is 10-percent.
0
…
1
£15
2
£15
3
£15
r
C
PV 

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

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

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

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

Annuity
A constant stream of cash flows with a fixed maturity
0 1
C
2
C
3
C
T
r
C
r
C
r
C
r
C
PV
)
1
(
)
1
(
)
1
(
)
1
( 3
2







 








 T
r
r
C
PV
)
1
(
1
1
T
C

Annuity: Example
If you can afford a $400 monthly car payment, how
much car can you afford if interest rates are 7% on 36-
month loans?
0 1
$400
2
$400
3
$400
59
.
954
,
12
$
)
12
07
.
1
(
1
1
12
/
07
.
400
$
36










PV
36
$400

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

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

Growing Annuity: Example
You are evaluating an income generating property. 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. 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
$

 )
0 7
.
1
(
5 0 0
,
8
$

 2
)
07
.
1
(
500
,
8
$
095
,
9
$ 6 5
.
7 3 1
,
9
$

 3
)
07
.
1
(
500
,
8
$
8 7
.
4 1 2
,
1 0
$

 4
)
07
.
1
(
500
,
8
$
7 7
.
1 4 1
,
1 1
$
$34,706.26

Practice questions

Finding the Number of Payments
 Suppose you borrow $2,000 at 5%, and you are
going to make annual payments of $734.42. How
long before you pay off the loan?
 Sign convention matters!!!
 5 I/Y
 2,000 PV
 -734.42 PMT
 CPT N = 3 years

Finding the Rate
 Suppose you borrow $10,000 from your parents to
buy a car. You agree to pay $207.58 per month for
60 months. What is the monthly interest rate?
 Sign convention matters!!!
 60 N
 10,000 PV
 -207.58 PMT
 CPT I/Y = .75%

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