Download Introduction to Financial Management

Overview of the Present
Value Concept, Investment
Criteria and Free-cash flows
The fundamental of
valuation
FIN 819: Lecture 2
Today’s plan




Review the concept of the time value of money
•
•
•
•
present value (PV)
discount rate (r)
discount factor (DF)
net present value (NPV)
Review of two rules for making investment decisions
•
•
The NPV rule
The rate of return rule
Review the formula for calculating the present value of
•
•
perpetuity with and without growth
annuity with and without growth
Review the concepts about interest compounding
FIN 819: Lecture 2
Today’s plan (continue)



Why do we always argue for the use of the NPV
rule
Examination of two other investment criteria
•
•
Payback rule
IRR rule
Some specific questions in using NPV
•
•
•
•
Sunk costs, opportunity cost
Incremental cash flows and incidental cash flows
Working capital
Inflation, real interest rate and nominal interest rate
FIN 819: Lecture 2
Today’s plan (continue)

How to calculate cash flows in Finance
• Depreciations are not actual cash flows
• Three approaches to calculate cash flows
from operations
FIN 819: Lecture 2
Financial choices

Which would you rather receive today?
• TRL 1,000,000,000 ( one billion Turkish
lira )
• USD 652.72 ( U.S. dollars )


Both payments are absolutely
guaranteed.
What do we do?
FIN 819: Lecture 2
Financial choices

We need to compare “apples to apples” this means we need to get the TRL:USD
exchange rate
From www.bloomberg.com we can see:

Therefore TRL 1bn = USD 558

• USD 1 = TRL 1,789,320
FIN 819: Lecture 2
Financial choices at different
times


Which would you rather receive?
• $1000 today
• $1200 in one year
Both payments have no risk, that is,
• there is 100% probability that you will be paid
• there is 0% probability that you won’t be paid
FIN 819: Lecture 2
Financial choices at different
times (2)




Why is it hard to compare ?
•
•
$1000 today
$1200 in one year
This is not an “apples to apples” comparison. They
have different units
$1000 today is different from $1000 in one year
Why?
•
A cash flow is time-dated money
•
•
It has a money unit such as USD or TRL
It has a date indicating when to receive money
FIN 819: Lecture 2
Present value

In order to have an “apple to apple”
comparison, we convert future payments to the
present values
•
•
•
this is like converting money in TRL to money in USD
Certainly, we can also convert the present value to the
future value to compare payments we get today with
payments we get in the future.
Although these two ways are theoretically the same,
but the present value concept is more important and
has more applications, as to be shown in stock and
bond valuations.
FIN 819: Lecture 2
Present value for the cash flow
at period 1
C1
PV 
 DF1  C1
1  r1
DF1 
1
(1 r1 )1
C1 is the cash in period 1
PV is the present value of the cash flow in period 1
DF1 is called discount factor for the cash flow in period 1
r1 is the discount rate
FIN 819: Lecture 2
Example 1

What is the present value of $100
received in one year (next year) if the
discount rate is 7%?
•
PV=100/(1.07)1 =
$100
PV=?
FIN 819: Lecture 2
Year one
Present value for the cash flow
at period t
PV 

Ct
(1  rt )
t
 DFt  Ct
Replacing “1” with “t” allows the formula
to be used for cash flows at any point in
time
FIN 819: Lecture 2
Example 2

What is the present value of $100
received in year five if the discount rate
is 7%?
•
PV=100/(1.07)5 =
$100
PV=?
FIN 819: Lecture 2
Year 5
Example 3

What is the present value of $100
received in year 20 if the discount rate is
7%?
• PV=100/(1.07)20 =
$100
PV=?
Year 20
FIN 819: Lecture 2
Example 4
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
PV 
3000
( 1.08 ) 2
 $2,572.02
FIN 819: Lecture 2
Explanation of the discount
factor
Discount Factor
DFt 
1
t
(1 rt )
FIN 819: Lecture 2
Example for the discount factor

Given two dollars, one received a year from now
and the other two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%. What is the
present value for each dollar received?

DF1=1.00/(1+0.2)=0.83
DF2=1.00/(1+0.07)2=0.87

FIN 819: Lecture 2
Present value of multiple cash
flows


For a cash flow received in year one and a
cash flow received in year two, different
discount rates may be used.
The present value of these two cash flows is
the sum of the present value of each cash flow,
since two present value have the same unit:
time zero USD.
PV (C1 , C2 )  PV (C1 )  PV (C2 )
 C1 (1  r1 )1  C2 (1  r2 ) 2
 DF1  C1  DF2  C2
FIN 819: Lecture 2
Present Values of future cash
flows

PVs can be added together to evaluate
multiple cash flows.
PV 
C1
(1 r1 )
1

C2
(1 r2 )
2
 .... 
N
  Ci  DFi
i 1
FIN 819: Lecture 2
CN
(1  rN ) N
Example 5

John is given the following set of cash flows
and discount rates. What is the PV?
C1  100
r1  10%
C2  100
r2  9%
$100
PV=?
Year one

$100
PV=100/(1.1)1 + 100/(1.09)2 =
FIN 819: Lecture 2
Year two
Example 6

John is given the following set of cash flows
and discount rates. What is the PV?
C1  100
r1  0.1
C2  200
r2  0.09
C3  50
r3  0.07
$100
PV=?
Yr 1
$200
Yr 2
$50
Yr 3
• PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 =
FIN 819: Lecture 2
Projects

A “project” is a term that is used to describe
the following activity:
• spend some money today
• receive cash flows in the future

A stylized way to draw project cash flows is
Expected cash flows
Expected cash flows
as follows:
in year one (probably positive)
in year two (probably positive)
Initial investment
(negative cash flows)
FIN 819: Lecture 2
Examples of projects




An entrepreneur starts a company:
•
•
initial investment is negative cash outflow.
future net revenue is cash inflow .
An investor buys a share of IBM stock
•
cost is cash outflow; dividends are future cash inflows.
A lottery ticket:
•
•
investment cost: cash outflow of $1
jackpot: cash inflow of $20,000,000 (with some very small
probability…)
Thus projects can range from real investments, to
financial investments, to gambles (the lottery ticket).
FIN 819: Lecture 2
Firms or companies

A firm or company can be regarded as a
set of projects.
• capital budgeting is about choosing the best
projects in real asset investments.

How do we know one project is worth
taking?
FIN 819: Lecture 2
Net present value

A net present value NPV is the sum of
the initial investment (usually made at
time zero) and the PV of expected future
cash flows.
NPV  C0  PV (C1 CT )
T
 C0  
T
Ct
t 1 (1  rt )
t
FIN 819: Lecture 2
  Ct  DFt
t 0
NPV rule


If the NPV of a project is positive, the
firm should go ahead to take this project.
This rule is often called the DCF
approach, because we have to use the
discount rate to calculate the PV of the
future cash flows of a project
FIN 819: Lecture 2
Example 7

Given the data for project A, what is the
NPV?
$50
$10
-$50
C0  50
C1  50
r1  7.5%
C2  10
r2  8.0%
Yr 1
Yr 0
• NPV=-50+50/(1.075)+10/(1.08)2 =
FIN 819: Lecture 2
Yr 2
Example 8
Assume that the cash flows
from the construction and sale
of an office building is as
follows. Given a 7% required
rate of return, create a present
value worksheet and show the
net present value.
Year 0
Year 1
Year 2
 150,000  100,000  300,000
FIN 819: Lecture 2
Present Values
Period
0
1
2
Discount
Factor
1.0
1
1.07  .935
1
 .873
1.07 2
Cash
Present
Flow
Value
 150,000
 150,000
 100,000
 93,500
 300,000
 261,900
NPV  Total  $18,400
FIN 819: Lecture 2
Example 9







John got his MBA from SFSU. When he was interviewed by a
big firm, the interviewer asked him the following question:
A project costs 10 m and produces future cash flows, as shown
in the next slide, where cash flows depend on the state of the
economy.
In a “boom economy” payoffs will be high
•
over the next three years, there is a 20% chance of a boom
•
over the next three years, there is a 50% chance of normal
•
over the next 3 years, there is a 30% chance of a recession
• In a “normal economy” payoffs will be medium
In a “recession” payoffs will be low
In all three states, the discount rate is 8% over all time
horizons.
Tell me whether to take the project or not
FIN 819: Lecture 2
Cash flows diagram in each
state


Boom economy
-$10 m
Normal economy
$8 m
$3 m
$3 m
$7 m
$2 m
$1.5 m
$1 m
$0.9 m
-$10 m
$6 m

Recession
-$10 m
FIN 819: Lecture 2
Example 9 (continues)

The interviewer then asked John:
• Before you tell me the final decision, how do
you calculate the NPV?
• Should you calculate the NPV at each economy or
take the average first and then calculate NPV
• Can your conclusion be generalized to any
situations?
FIN 819: Lecture 2
Calculate the NPV at each
economy

In the boom economy, the NPV is

In the average economy, the NPV is

In the bust economy, the NPV is
• -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36
• -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613
• -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87
The expected NPV is
0.2*2.36+0.5*(-.613)+0.3*(-2.87)=-$0.696
FIN 819: Lecture 2
Calculate the expected cash
flows at each time




At period 1, the expected cash flow is
•
C1=0.2*8+0.5*7+0.3*6=$6.9
At period 2, the expected cash flow is
•
C2=0.2*3+0.5*2+0.3*1=$1.9
At period 3, the expected cash flows is
•
C3=0.2*3+0.5*1.5+0.3*0.9=$1.62
The NPV is
•
•
NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083
=-$0.696
FIN 819: Lecture 2
The rate of return rule for a oneperiod project with negative C0



Another way to decide whether a project (with one
piece of cash flow occurring in the future) should be
taken or not is to compare the rate of return and the
discount rate.
If the rate of return of a project is larger than the
discount rate (the cost of capital, or hurdle rate), the
firm should go ahead to take this project.
The rate of return is defined as the ratio of the profit
to the cost.
FIN 819: Lecture 2
Example



If you invest $30 today in one share of
stock (no dividends), you will get $36
next year. What is the rate of return for
your investment?
Profit=36-30=$6
Rate of return = 6/30=20%
FIN 819: Lecture 2
NPV rule and the rate of return
rule?


What is the relationship between these
two rules?
If there is some relation between these
two rules, can you show formally?
FIN 819: Lecture 2
Perpetuities



We are going to look at the PV of a perpetuity starting one year from
now (please see the cash flow diagram below).
Definition: if a project makes a level, periodic payment into perpetuity,
it is called a perpetuity.
Let’s suppose your friend promises to pay you $1 every year, starting
next year. His future family will continue to pay you and your future
family forever. The discount rate is assumed to be constant at 8.5%.
How much is this promise worth?
$C
$C $C $C
$C
$C
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
FIN 819: Lecture 2
Time=infinity
Perpetuities (continue)

Calculating the PV of the perpetuity could be hard
PV 
C
(1  r )1

C 

C
(1  r ) 2

1
i 1(1  r )
i
FIN 819: Lecture 2
C
(1  r ) 
Perpetuities (continue)

To calculate the PV of perpetuities, we can have
some math exercise as follows:
1

1
1
(1  r )
S    2    
S   2   3     
  S  S

1 /(1  r )
1
S


1   1  1 /(1  r ) r
FIN 819: Lecture 2
Perpetuities (continue)

Calculating the PV of the perpetuity could also be
easy if you ask George
C
C
C
PV 


(1  r )1 (1  r ) 2
(1  r ) 

1

C
C 
 C.    C.S 
i
r
i 1(1  r )
i 1
i
FIN 819: Lecture 2
Calculate the PV of a perpetuity


Consider the perpetuity of one dollar
every period your friend promises to pay
you. The interest rate or discount rate is
8.5%.
Then PV =1/0.085=$11.765, not a big
gift.
FIN 819: Lecture 2
Perpetuity (continue)

What is the PV of a perpetuity of paying $C
every year, starting from year t +1, with a
constant discount rate of r ?
PV 
C
(1  r )
t 1

C
(1  r )
$C
Yr0
t+1
t 2

$C $C $C
t+2
C
(1  r ) 
$C
t+3 t+4 T+5
FIN 819: Lecture 2
$C
Time=t+inf
Perpetuity (continue)

What is the PV of a perpetuity of paying $C
every year, starting from year t +1, with a
constant discount rate of r ?
PV 
C
(1  r )t 1

C
(1  r )t  2

C
(1  r ) 
 1
1
1 



t
1
2

(1  r )  (1  r ) (1  r )
(1  r ) 
C
C

1
C
1
C


. 

t
i
t r
(1  r ) i 1(1  r ) (1  r )
(1  r )t r
FIN 819: Lecture 2
Perpetuity (alternative method)

What is the PV of a perpetuity that pays $C
every year, starting in year t+1, at constant
discount rate “r”?
•

Alternative method: we can think of PV of a perpetuity
starting year t+1. The normal formula gives us the
value AS OF year “t”. We then need to discount this
value to account for periods “1 to t”
Vt  C
r
That is
PV 
Vt
(1  r )t
FIN 819: Lecture 2

C
(1  r )t r
Annuities


Well, a project might not pay you forever.
Instead, consider a project that promises to
pay you $C every year, for the next “T” years.
This is called an annuity.
Can you think of examples of annuities in the
real world?
$C $C $C $C $C
$C
PV
???
Yr1
Yr2
Yr3 Yr4
Yr5
FIN 819: Lecture 2
Time=T
Value the annuity

Think of it as the difference between two
perpetuities
•
•
add the value of a perpetuity starting in yr 1
subtract the value of perpetuity starting in yr
T+1
1

C
C
1

PV  
 C 
 r (1  r )T r 
r (1  r )T r


FIN 819: Lecture 2
Example for annuities

you win the million dollar lottery! but wait,
you will actually get paid $50,000 per
year for the next 20 years if the discount
rate is a constant 7% and the first
payment will be in one year, how much
have you actually won (in PV-terms) ?
FIN 819: Lecture 2
My solution

Using the formula for the annuity
 1

1
PV  50,000 * 


 0.07 1.07 20 * 0.07 
 $529,700 .71
FIN 819: Lecture 2
Lottery example

Paper reports: Today’s JACKPOT =
$20mm !!
• paid in 20 annual equal installments.
• payment are tax-free.
• odds of winning the lottery is 13mm:1

Should you invest $1 for a ticket?
• assume the risk-adjusted discount rate is 8%
FIN 819: Lecture 2
My solution


Should you invest ?
Step1: calculate the PV
1.0mm 1.0mm
1.0mm
PV 


2
(1.08) (1.08)
(1.08) 20

 $9.818 mm
Step 2: get the expectation of the PV
1
1
E[ PV ] 
* 9.818 mm  (1 
)*0
13mm
13mm
 $0.76  $1

Pass up this this wonderful opportunity
FIN 819: Lecture 2
Example
You agree to lease a car for 4 years at $300
per month. You are not required to pay any
money up front or at the end of your
agreement. If your opportunity cost of
capital is 0.5% per month, what is the cost of
the lease?
FIN 819: Lecture 2
Solution
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
FIN 819: Lecture 2
Mortgage-style loans

Suppose you take a $20,000 3-yr car loan with
“mortgage style payments”
•
•

annual payments
interest rate is 7.5%
“Mortgage style” loans have two main
features:
•
•
They require the borrower to make the same payment
every period (in this case, every year)
They are fully amortizing (the loan is completely paid
off by the end of the last period)
FIN 819: Lecture 2
Mortgage-style loans


The best way to deal with mortgage-style loans
is to make a “loan amortization schedule”
The schedule tells both the borrower and
lender exactly:
•
•
•

what the loan balance is each period (in this case year)
how much interest is due each year ? ( 7.5% )
what the total payment is each period (year)
Can you use what you have learned to figure
out this schedule?
FIN 819: Lecture 2
My solution
year
Beginning
balance
Interest
payment
Principle
payment
Total
payment
Ending
balance
0
1
$20,000
$1,500
$6,191
$7,691
$13,809
2
13,809
1,036
6,655
7,691
7,154
3
7,154
537
7,154
FIN 819: Lecture 2
7,691
0
Perpetuities with a growth rate

What is the PV of the perpetuity with a cash flow of C
in the next period and then growing at a rate of g at
very period in the future?
PV 
C
1
(1  r )

C (1  g )
(1  r )
2

 (1  g )i 1
C
C 

i
rg
i 1 (1  r )
FIN 819: Lecture 2
C (1  g )
(1  r )


Perpetuity with growth
(continue)

What is the PV of a perpetuity of paying $C
in year t+1 and then growing with a rate of g
annually, with a constant discount rate of r ?
FIN 819: Lecture 2
Perpetuity with growth
PV 
C
(1  r )t 1

C (1  g )
(1  r )t  2

C (1  g )
(1  r )


(1  g )
1 g
1
C





2
1
t
(1  r ) 
(1  r )  (1  r ) (1  r )
i 1

C
1
C
(1  g )
C

.



(1  r )t r  g (1  r )t (r  g )
(1  r )t i 1 (1  r )i
FIN 819: Lecture 2
Perpetuity with growth
(alternative method)


What is the PV of a perpetuity that pays $C in year
t+1, and then grows at a rate of g, with a constant
discount rate “r”?
• Alternative method: we can think of PV of a
perpetuity with growth starting year t+1. The
normal formula gives us the value AS OF year “t”.
We then need to discount this value to account for
periods “1 to t”
C
Vt 
That is
PV 
Vt
(1  r )
t
rg

C
(1  r )t (r  g )
FIN 819: Lecture 2
Annuity with growth


Well, a project might not pay you forever.
Instead, consider a project that pays you $C
next year and then grows at a rate of g every
year for the next “T” years. This is called an
annuity with growth.
Please figure out the PV of this annuity ?
FIN 819: Lecture 2
Present value of the annuity

Think of it as the difference between two
perpetuities
•
•
add the value of a perpetuity starting in yr 1
subtract the value of perpetuity starting in yr
T+1
T
 1

C
C (1  g )T
(
1

g
)

PV 

 C

 r  g (1  r )T (r  g ) 
r  g (1  r )T (r  g )


FIN 819: Lecture 2
Example


An oil well, if explored, can now produce
100,000 barrels per year. The well will produce
for 18 years more, but production will decline
by 4% per year. Oil prices, however, will
increase by 2% per year. The discount rate is
8%. Suppose that the price of oil now is $14
for barrel.
If the cost of oil exploration is $1.8 million, do
you want to take this project?
FIN 819: Lecture 2
My solution

First, what are the cash flows?
•
•
•

C0=$1.4; C1=1.4*(1+g); C2=1.4*(1+g)2; C3=1.4*(1+g)3;
…., C18=1.4*(1+g)18.
(1+g)=(1+g1)*(1+g2), where g1= -4% and g2=2%.
g=-2.08%
Second, figure out what it is in Finance ?
•
•
•
•
Is it a perpetuity?
Is it a perpetuity with a growth of g?
Is it an annuity?
Is it an annuity with a growth of g?
FIN 819: Lecture 2
My solution (2)

Step 3: Do we have a formula for
calculating the present value of an
annuity with a growth?
• Yes
FIN 819: Lecture 2
My solution (3)

Step 4, get the formula for the present
value of an annuity with a growth rate of
g.
• PV( first perpetuity staring at time 1)=C1/(r-g);
• PV( second starting at time 19)=
C19/((r-g)*(1+r)18)


PV( annuity with a growth)=
(C1/(r-g))*(1-(1+g)18/(1+r)18)
FIN 819: Lecture 2
My solution (4)




g=-2.1%, r=8%
PV( annuity with a growth )=$11.27 m
NPV=1.4+11.27-1.8=$10.87 m
Should you go ahead to invest in this
project?
FIN 819: Lecture 2
Simpler solution




C0=1.4; C1=1.37; C2=1.34, . . .
Since C0*(1+g)=C1 or C1*(1+g)=C2,
Then g=-2.08%
Then we can use the annuity with growth
formula to calculate the NPV.
FIN 819: Lecture 2
Future value

The formula for converting the present value to
future value:
FVt i  PVt 0  (1  rt i )i
PVt 0 = present value at time zero
FVt i = future value in year i
rt i = discount rate during the i years
Ct i
FIN 819: Lecture 2
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal?
To answer, determine $24 is worth in the year 2003,
compounded at 8%.
FV  $24  (1  .08)
 $75.979 trillion
374
FYI - The value of Manhattan Island land is
well below this figure.
FIN 819: Lecture 2
Another question

Suppose that the annual interest rate is
10% and you start to save $10,000 every
year starting next year for 30 years. How
much money will you have 30 years
later? ( the money in your account after
you just put $10,000 in year 30)
FIN 819: Lecture 2
Interest compounding


The interest rate is often quoted as the
simple interest rate, which is called as APR,
the annual percentage rate.
If the interest rate is compounded m times in
each year and the APR is r, the effective
annual interest rate is
m
1  r   1


 m
FIN 819: Lecture 2
Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
FIN 819: Lecture 2
6.000%
18
16
14
12
10
8
6
4
2
0
10% Simple
Number of Years
FIN 819: Lecture 2
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Compound Interest
Example
Suppose you are offered an automobile loan at an APR of
6% per year. What does that mean, and what is the true
rate of interest, given monthly payments?
FIN 819: Lecture 2
Compound Interest
12
Effective interest rate  (1.005)  1
 6.1678%
FIN 819: Lecture 2
Two other investment criteria

In addition to the NPV rule, some
financial managers used to use two
other investment rules to decide which
project to take
• Payback rule
• Internal rate of return (IRR) rule
FIN 819: Lecture 2
What is the payback rule



The payback period
•
The number of years (in integer) it takes before the
cumulative cash flow is equal to or larger than the initial
outlay (investment cost).
The payback rule
•
If the payback period is less than or equal to the prespecified number of periods (2, 3, or 4 years, quite
arbitrary) for a project, the firm should go ahead to take
this project.
This method is clearly flawed.
•
Why?
FIN 819: Lecture 2
Payback (example)
Examine the three projects and note the
mistake we would make if we insisted on only
taking projects with a payback period of 2
years or less.
Project
A
B
C
Payback
C0
C1
C2
C3
Period
- 2000 500 500 5000
3
- 2000 500 1800
0
2
- 2000 1800 500
0
2
FIN 819: Lecture 2
NPV@ 10%
2,624
 58
50
The IRR rule

The rate of return rule revisited
• Consider one-period model
• Rate of return=(C1+C0)/(-C0)
• NPV=C0+C1/(1+ r), r is the discount rate
• If the rate of return is larger than r, we should
go ahead to take the project

Can we extend this rate of return rule in
the general situation in the multi-period
case?
FIN 819: Lecture 2
The IRR rule (continues)


The internal rate of return is a single
discount rate such that the NPV is zero.
If the IRR is larger than the cost of
capital or the discount rate for a project,
the firm should go ahead to take the
project.
FIN 819: Lecture 2
The IRR

That is to solve the following equation to
calculate IRR.
CN
C1
NPV  C0 
 ... 
0
1  IRR
(1  IRR ) N
FIN 819: Lecture 2
Internal Rate of Return
Example
You can purchase a turbo powered machine
tool gadget for $4,000. The investment will
generate $2,000 and $4,000 in cash flows for
two years, respectively. What is the IRR on
this investment?
FIN 819: Lecture 2
Solution
NPV  4,000 
2,000
1
(1  IRR )

IRR  28.08%
FIN 819: Lecture 2
4,000
(1  IRR )
2
0
Internal Rate of Return (picture)
2500
2000
IRR=28%
1000
500
-1000
-1500
-2000
Discount rate (%)
FIN 819: Lecture 2
0
10
90
80
70
60
50
40
30
-500
20
0
10
NPV (,000s)
1500
Pitfall 1 of IRR
Lending or Borrowing?
 With some cash flows (as noted below)
the NPV of the project increases as the
discount rate increases.
 This is contrary to the normal
relationship between NPV and discount
rates.
 Why does this happen?
FIN 819: Lecture 2
Example
Project
C0
C1
IRR
NPV at 10%
A
-1,000
+1,500
+50%
+364
B
+1000
-1,500
+50%
-364
Are these two projects equally attractive?
FIN 819: Lecture 2
Pitfall 2 of the IRR rule


Certain cash flows can generate NPV=0 at two
different discount rates.
The following cash flow generates NPV=0 at
both (-50%) and 15.2%.
C
0
 1, 000

C
1
 800
C
C
2
 150
C
3
 150
4
 150
Which rate should be used?
FIN 819: Lecture 2
C
5
 150
C
6
 150
Pitfall 2 of the IRR rule

The cash flow described above generates NPV=0 at both
-50% and 15.2%.
NPV
1000
IRR=15.2%
500
Discount
Rate
0
-500
IRR=-50%
-1000
FIN 819: Lecture 2
Pitfall 3 of the IRR rule
Mutually Exclusive Projects
 IRR sometimes ignores the magnitude of the
project.
 The following two projects illustrate this
problem.
Project
E
F
C0
10,000
 20,000
Ct
IRR NPV @10%
 20,000 100
 8,182
 35,000 75
11,818
FIN 819: Lecture 2
Pitfall 4 of the IRR rule
Term Structure Assumption
 Discount rates (the cost of capital) can be
different for different periods, which discount
rate should be used to judge whether the IRR
is larger to make investment decisions?
FIN 819: Lecture 2
Profitability Index (PI)



When resources are limited, the
profitability index (PI) provides a tool for
selecting among various project
combinations and alternatives
A set of limited resources and projects
can yield various combinations.
The highest weighted average PI can
indicate what projects to select.
FIN 819: Lecture 2
Profitability Index
NPVi
Profitabil ity Index (PIi ) 
Investment i
N
Investmenti
WAPI  
PIi
i 1Total money
FIN 819: Lecture 2
Example
We only have $300,000 to invest. Which do we
select?
Proj
A
B
C
D
NPV
230,000
141,250
194,250
162,000
Investment
200,000
125,000
175,000
150,000
FIN 819: Lecture 2
PI
1.15
1.13
1.11
1.08
Profitability Index (solution)
Based on the capital constraint, we have
three sets of projects: A, BD and BC.
WAPI (BD) = 1.13(125/300) + 1.08(150/300)
= 1.01
WAPI (A) = 1.15(200/300)
= 0.77
WAPI (BC) = 1.13(125/300) + 1.11(175/300)
= 1.23
FIN 819: Lecture 2
Some points to remember in
calculating cash flows







Incremental cash flows
Include all incidental effects
Do not forget working capital requirements
Forget sunk costs
Include opportunity costs
Depreciation
Financing
FIN 819: Lecture 2
Incremental cash flows



Incremental cash flows are the increased
cash flows due to investment
Do not get confused about the average
cost or total cost?
Do you have examples about
incremental costs?
FIN 819: Lecture 2
Incidental costs


Costs or cash flows are indirectly related
to investment.
Have examples for this?
FIN 819: Lecture 2
Working capital




(Net) working capital is the difference between
a firm’s short-term assets and liabilities.
The principal short-term assets are cash,
accounts receivable, and inventories of raw
materials and finished goods.
The principal short-term liabilities are accounts
payable.
Do you have examples?
FIN 819: Lecture 2
Sunk costs


The sunk cost is past cost and has
nothing to do with your investment
decision
Is your education cost so far at SFSU is
sunk cost?
FIN 819: Lecture 2
Opportunity cost


The cost of a resource may be relevant
to the investment decision even when no
cash changes hands.
Give me an example about the
opportunity cost of studying at SFSU?
FIN 819: Lecture 2
Inflation rule




Be consistent in how you handle inflation!!
Use nominal interest rates to discount
nominal cash flows.
Use real interest rates to discount real
cash flows.
You will get the same results, whether you
use nominal or real figures
FIN 819: Lecture 2
Example
You own a lease that will cost you $8,000 next
year, increasing at 3% a year (the forecasted
inflation rate) for 3 additional years (4 years
total). If discount rates are 10% what is the
present value cost of the lease?
1  real interest rate =
1+ nominal interest rate
1+inflation rate
FIN 819: Lecture 2
Inflation
Example - nominal figures
Year Cash Flow
PV @ 10%
1
8000
2
8000x1.03 = 8240
3
8000x1.03 2 = 8487.20
4
8000x1.03 3 = 8741.82
8000
1.10
8240
1.102
8487.20
1.103
8741.82
1.104
 7272.73
 6809.92
 6376.56
 5970.78
$26,429.99
FIN 819: Lecture 2
Inflation
Example - real figures
Year
1
2
3
4
Cash Flow
8000
1.03 = 7766.99
8240
= 7766.99
1.032
8487.20
= 7766.99
1.033
8741.82
=
7766.99
4
1.03
[email protected]%
7766.99
1.068  7272.73
7766.99
 6809.92
1.0682
7766.99
 6376.56
1.0683
7766.99
4  5970.78
1.068
= $26,429.99
FIN 819: Lecture 2
Depreciation




Depreciation is not the actual cash flow.
Depreciation is just an accounting way of
allocating capital investment
In accounting, depreciation is regarded as the
accounting cash flow
In Finance, depreciation is not the actual cash
flow, but its impact on firms’ tax payment must
be considered, that is, depreciation reduces a
firm’s tax payment.
FIN 819: Lecture 2
Example

A project costs $2,000 and is expected
to last 2 years, producing cash income of
$1,500 and $500 respectively. The cost
of the project can be depreciated at
$1,000 per year. Given a 10% required
return, compare the NPV using cash flow
to the NPV using accounting income.
FIN 819: Lecture 2
Solution (using accounting
profit)
Year 1 Year 2
Cash Income
$1500 $ 500
Depreciation
- $1000 - $1000
Accounting Income + 500 - 500
500  500
Accounting NPV =

 $41.32
2
1.10 (110
. )
FIN 819: Lecture 2
Solution (using cash flows)
Today
Cash Income
Project Cost
Free Cash Flow
- 2000
- 2000
Cash NPV = -2000 
Year 1
Year 2
$1500
$ 500
+ 1500
+ 500
1500
1
(1.10)
500

(1.10)
FIN 819: Lecture 2
2
 $223.14
Question

In the previous example, we assume that
there is no corporate tax. If there is a
corporate tax of 30%, what are the NPVs
for the accounting cash flows and cash
flows in finance?
FIN 819: Lecture 2
Forget about financing


When calculating cash flows from a
project, ignore how the project is
financed.
You can assume that the firm is financed
by issuing only stocks; or the firm has no
debt but just equity.
FIN 819: Lecture 2
How to calculate free cash
flows?



If we are given discount rates and future cash
flows, it is quite straightforward to calculate
NPV to make investment decisions.
Can we use accounting profits calculated in
income statements as free-cash flows?
If yes, why?; if not, how to calculate free cash
flows?
FIN 819: Lecture 2
How to calculate free cash
flows?

Free cash flows = cash flows from
operations + cash flows from the change
in working capital + cash flows from
capital investment and disposal
• We can have three methods to calculate cash
flows from operations, but they are the exactly
same, although they have different forms.
FIN 819: Lecture 2
How to calculate cash flows
from operations?

Method 1
• Cash flows from operations =revenue –cost
(cash expenses) – tax payment

Methods 2
• Cash flows from operations = accounting
profit + depreciation

Method 3
• Cash flows from operations =(revenue –
cost)*(1-tax rate) + depreciation *tax rate
FIN 819: Lecture 2
Example
-
revenue
Cost
Depreciation
Profit before tax
Tax at 35%
Net income
1,000
600
200
200
70
130
Given information above, please use three methods to calculate
Cash flows
FIN 819: Lecture 2
Solution:

Method 1

Method 2

Method 3
• Cash flows=1000-600-70=330
• Cash flows =130+200=330
• Cash flows =(1000-600)*(1-0.35)+200*0.35
=330
FIN 819: Lecture 2
A summary example 1( Blooper)

Now we can apply what we have
learned about how to calculate cash
flows to the Blooper example, whose
information is given in the following
slide.
FIN 819: Lecture 2
Blooper Industries
Year 0
10,000
1
2
3
4
5
6
WC
1,500
4,075
4,279
4,493
4,717
3,039
0
Change in WC
1,500
2,575
204
214
225
 1,678
 3,039
15,000
10,000
15,750
10,500
16,538
11,025
17,364
11,576
18,233
12,155
2,000
3,000
1,050
1,950
2,000
3,250
1137
,
2,113
2,000
3,513
1,230
2,283
2,000
3,788
1,326
2,462
2,000
4,078
1,427
2,651
Cap Invest
Revenues
Expenses
Depreciation
Pretax Profit
.Tax (35%)
Profit
(,000s)
FIN 819: Lecture 2
Cash flows from operations for
the first year
Revenues
15,000
- Expenses
10,000
 Depreciation
2,000
= Profit before tax
3,000
.-Tax @ 35 %
1,050
= Net profit
1,950
+ Depreciation
2,000
= CF from operations 3,950 or $3,950,000
FIN 819: Lecture 2
Blooper Industries
Net Cash Flow (entire project) (,000s)
Year 0
1
2
3
4
5
6
-10,000
-1,500
- 2,575
- 204
- 214
- 225
1,678
3,039
CF from Op
Net Cash Flow -11,500
3,950
1,375
4,113
3,909
4,283
4,069
4,462
4,237
4,651
6,329
3,039
Cap Invest
Change in WC
NPV @ 12% = $3,564,000
FIN 819: Lecture 2
A summary example 2

Now we can apply what we have
learned about how to calculate cash
flows to the IM&C’s Guano Project (in
the textbook), whose information is
given in the following slide.
FIN 819: Lecture 2
IM&C’s Guano Project
Revised projections ($1000s) reflecting inflation
FIN 819: Lecture 2
IM&C’s Guano Project
Cash flow analysis ($1000s)
FIN 819: Lecture 2
IM&C’s Guano Project

NPV using nominal cash flows
1,630 2,381
6,205 10,685 10,136
NPV  12,600 




2
3
4
1.20 1.20
1.20 1.20 1.205
6,110
3,444


 3,519 or $3,519,000
1.206 1.207
FIN 819: Lecture 2