FIN 819: Lecture 2

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:

• USD 1 = TRL 1,789,320 Therefore TRL 1bn = USD 558

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

111

1

1CDF

r

CPV

11 )1(

11 r

DF

C1 is the cash in period 1PV is the present value of the cash flow in period 1DF1 is called discount factor for the cash flow in period 1r1 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 =

Year one

$100

PV=?

FIN 819: Lecture 2

Present value for the cash flow at period t

Replacing “1” with “t” allows the formula to be used for cash flows at any point in time

tttt

t CDFr

CPV

)1(

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 =

Year 5

$100

PV=?

FIN 819: Lecture 2

Example 3

What is the present value of $100 received in year 20 if the discount rate is 7%?

• PV=100/(1.07)20 =

Year 20

$100

PV=?

FIN 819: Lecture 2

Example 4You 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 30001 08 2 572 02

( . )$2, .

FIN 819: Lecture 2

Explanation of the discount factor

Discount Factor

ttr

tDF)1(

1

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.

2211

222

111

2121

)1()1(

)()(),(

CDFCDF

rCrC

CPVCPVCCPV

FIN 819: Lecture 2

Present Values of future cash flows

PVs can be added together to evaluate multiple cash flows.

i

N

ii

NN

Nr

C

r

C

DFC

r

CPV

1

)1()1( )1(....2

2

21

1

1

FIN 819: Lecture 2

Example 5

John is given the following set of cash flows and discount rates. What is the PV?

PV=100/(1.1)1 + 100/(1.09)2 =

%101 r

Year one

$100

PV=?1001 C

%92 r1002 C

$100

Year two

FIN 819: Lecture 2

Example 6

John is given the following set of cash flows and discount rates. What is the PV?

• PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 =

503 C

1.01 r

Yr 1

$100

PV=?

1001 C

09.02 r2002 C

$50

Yr 307.03 r Yr 2

$200

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

as follows:

Initial investment(negative cash flows)

Expected cash flows in year one (probably positive)

Expected cash flows in year two (probably positive)

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.

t

T

tt

T

tt

t

t

T

DFCr

CC

CCPVCNPV

010

10

)1(

)(

FIN 819: Lecture 2

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?

• NPV=-50+50/(1.075)+10/(1.08)2 =

%0.810

%5.750

50

22

11

0

rC

rC

C

Yr 0

Yr 1 Yr 2

$10$50-$50

FIN 819: Lecture 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.

000,300000,100000,150

2Year 1Year 0Year

FIN 819: Lecture 2

Present Values

400,18$

900,261000,300873.2

500,93000,100935.1

000,150000,1500.10Value

Present

Flow

Cash

Factor

DiscountPeriod

207.11

07.11

TotalNPV

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

• In a “normal economy” payoffs will be medium• over the next three years, there is a 50% chance of normal

In a “recession” payoffs will be low• over the next 3 years, there is a 30% chance of a recession

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

Normal economy

Recession

-$10 m$8 m $3 m $3 m

-$10 m

-$10 m

$2 m$7 m

$0.9 m$1 m$6 m

$1.5 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• -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36

In the average economy, the NPV is• -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613

In the bust economy, the NPV is • -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87The 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 one-period 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?

PV???

$C $C$C $C $C $C

Yr1 Yr2 Yr3 Yr4 Yr5 Time=infinity

FIN 819: Lecture 2

Perpetuities (continue)

Calculating the PV of the perpetuity could be hard

1

21

)1(

1

)1()1()1(

iir

C

r

C

r

C

r

CPV

FIN 819: Lecture 2

Perpetuities (continue)

To calculate the PV of perpetuities, we can have some math exercise as follows:

rrr

S

SS

S

S

r

1)1/(11

)1/(11

1)1(

1

32

2

1

FIN 819: Lecture 2

Perpetuities (continue)

Calculating the PV of the perpetuity could also be easy if you ask George

rC

SCCr

C

r

C

r

C

r

CPV

i

i

ii

..)1(

1

)1()1()1(

11

21

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 ?

$C $C$C $C $C $C

t+1 t+2 t+3 t+4 T+5 Time=t+infYr0

)1()1()1( 21 r

C

r

C

r

CPV

tt

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 ?

rr

Crr

C

rr

C

rrrr

C

r

C

r

C

r

CPV

tti

it

t

tt

)1(

1.

)1()1(

1

)1(

)1(

1

)1(

1

)1(

1

)1(

)1()1()1(

1

21

21

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”

That is

rr

C

r

VPV

V

ttt

rC

t

)1()1(

FIN 819: Lecture 2

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?

PV???

$C $C$C $C $C $C

Yr1 Yr2 Yr3 Yr4 Yr5 Time=T

FIN 819: Lecture 2

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

rrrC

rr

C

r

CPV

TT )1(

11

)1(

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

71.700,529$07.0*07.1

1

07.0

1*000,50

20

PV

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

Step 2: get the expectation of the PV

Pass up this this wonderful opportunity

mm

mmmmmmPV

818.9$

)08.1(

0.1

)08.1(

0.1)08.1(

0.1202

1$76.0$

0*)13

11(818.9*

13

1][

mm

mmmm

PVE

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

10.774,12$

005.1005.

1

005.

1300Cost Lease 48

Cost

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 Beginningbalance

Interest payment

Principlepayment

Total payment

Ending balance

0

1

2

3

$20,000

13,809

7,154

$1,500 $6,191 $7,691 $13,809

1,036 6,655

537 7,154 7,691 0

7,691 7,154

FIN 819: Lecture 2

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?

grC

r

gC

r

gC

r

gC

r

CPV

ii

i

1

1

21

)1(

)1(

)1(

)1(

)1(

)1(

)1(

FIN 819: Lecture 2

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

)()1(

1.

)1()1(

)1(

)1(

)1(

)1(

)1(

1

)1(

1

)1(

)1(

)1(

)1(

)1(

)1(

1

1

21

21

grr

Cgrr

C

r

g

r

C

r

g

r

g

rr

C

r

gC

r

gC

r

CPV

tti

i

i

t

t

tt

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”

That is

)()1()1( grr

C

r

VPV

V

ttt

grC

t

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

)()1(

)1(1

)()1(

)1(

grr

ggr

Cgrr

gCgr

CPV

T

T

T

T

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:

= present value at time zero = future value in year i

= discount rate during the i years

iittit rPVFV )1(0

0tPV

itr

itFV

itC

FIN 819: Lecture 2

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal?

trillion

FV

979.75$

)08.1(24$ 374

To answer, determine $24 is worth in the year 2003, compounded at 8%.

FYI - The value of Manhattan Island land is FYI - The value of Manhattan Island land is well below this figure.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

11

m

mr

FIN 819: Lecture 2

Compound Interest i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

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

Compound Interest

02468

1012141618

Number of Years

FV

of

$1

10% Simple

10% Compound

FIN 819: Lecture 2

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

%1678.6

1)005.1(rateinterest Effective 12

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 pre-specified 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.

502050018002000-C

582018005002000-B

624,2350005005002000-A

10% @NPVPeriod

PaybackCCCCProject 3210

FIN 819: Lecture 2

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.

0)1(

...1

10

NN

IRR

C

IRR

CCNPV

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

%08.28

0)1(

000,4

)1(

000,2000,4

21

IRR

IRRIRRNPV

FIN 819: Lecture 2

Internal Rate of Return (picture)

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

10 20 30 40 50 60 70 80 90 100

Discount rate (%)

NP

V (

,000

s)

IRR=28%

FIN 819: Lecture 2

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

B

-1,000

+1000

+1,500

-1,500

+50%

+50%

+364

-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%.

Which rate should be used?

150150150150150800000,1

6543210

CCCCCCC

FIN 819: Lecture 2

Pitfall 2 of the IRR rule

The cash flow described above generates NPV=0 at both

-50% and 15.2%.

1000NPV

500

0

-500

-1000

Discount Rate

IRR=15.2%

IRR=-50%

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.

818,1175000,35000,20

182,8100000,20000,10

%10@Project 0

F

E

NPVIRRCC t

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

i

ii Investment

NPV)(PIIndex ity Profitabil

iN

i

i PImoneyTotal

Investment1

WAPI

FIN 819: Lecture 2

Example

We only have $300,000 to invest. Which do we select?

Proj NPV Investment PI

A 230,000 200,000 1.15

B 141,250 125,000 1.13

C 194,250 175,000 1.11

D 162,000 150,000 1.08

FIN 819: Lecture 2

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) = 1.01WAPI (A) = 1.15(200/300) = 0.77WAPI (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

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

Inflation rule

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 rate1+inflation rate

FIN 819: Lecture 2

Inflation

Example - nominal figures

99.429,26$

78.59708741.82=8000x1.034

56.63768487.20=8000x1.033

92.68098240=8000x1.032

73.727280001

10% @ PVFlowCash Year

4

3

2

10.182.87413

10.120.84872

10.18240

1.108000

FIN 819: Lecture 2

Inflation

Example - real figures

Year Cash Flow PV@6.7961%

1 = 7766.99

2 = 7766.99

= 7766.99

= 7766.99

80001.03

7766.991.068

82401.03

8487.201.03

8741.821.03

2

3

4

7272 73

6809 92

3 6376 56

4 5970 78

26 429 99

7766 991 068

7766 991 068

7766 991 068

2

3

4

.

.

.

.

..

..

..

= $ , .

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

Accounting NPV =500

1.10

500

11032

2( . )$41.

FIN 819: Lecture 2

Solution (using cash flows)

500 +1500+2000-FlowCash Free

2000-CostProject

500 $ $1500 IncomeCash

2Year 1Year Today

14.223$)10.1(

500

)10.1(

1500-2000=NPVCash

21

FIN 819: Lecture 2

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 1,000- Cost 600- Depreciation 200- Profit before tax 200- Tax at 35% 70 - Net income 130

Given information above, please use three methods to calculate

Cash flows

FIN 819: Lecture 2

Solution:

Method 1• Cash flows=1000-600-70=330

Method 2• Cash flows =130+200=330

Method 3• 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 1 2 3 4 5 6

Cap Invest

WC

Change in WC

Revenues

Expenses

Depreciation

Pretax Profit

.Tax (35%)

Profit

10 000

1 500 4 075 4 279 4 493 4 717 3 039 0

1 500 2 575 204 214 225 1 678 3 039

15 000 15 750 16 538 17 364 18 233

10 000 10 500 11 025 11 576 12 155

2 000 2 000 2 000 2 000 2 000

3 000 3 250 3 513 3 788 4 078

1 050 1137 1 230 1 326 1 427

1 950 2 113 2

,

, , , , , ,

, , , ,

, , , , ,

, , , , ,

, , , , ,

, , , , ,

, , , , ,

, ,

, , ,283 2 462 2 651

(,000s)

FIN 819: Lecture 2

Cash flows from operations for the first year

Revenues

- Expenses

Depreciation

= Profit before tax

.-Tax @ 35 %

= Net profit

+ Depreciation

= CF from operations

15 000

10 000

2 000

3 000

1 050

1 950

2 000

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

Cap Invest -10,000

Change in WC -1,500 - 2,575 - 204 - 214 - 225 1,678 3,039

CF from Op 3,950 4,113 4,283 4,462 4,651

Net Cash Flow -11,500 1,375 3,909 4,069 4,237 6,329 3,039

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 ProjectCash flow analysis ($1000s)

FIN 819: Lecture 2

IM&C’s Guano Project

NPV using nominal cash flows

$3,519,000or 519,3

20.1

444,3

20.1

110,6

20.1

136,10

20.1

685,10

20.1

205,6

20.1

381,220.1

630,1600,12

76

5432

NPV