lecture 2 npv & fv

Net Present Value (NPV)To determine net present value we . . .To determine net present value we . . .

Calculate the present value of cash inflows,Calculate the present value of cash outflows,Subtract the present value of the outflows from the

present value of the inflows.

General decision rule . . .General decision rule . . .

Present Value of all costs and benefits of a project.Present Value of all costs and benefits of a project. Concept is similar to Intrinsic Value of a security but subtracts Concept is similar to Intrinsic Value of a security but subtracts

of cost of project.of cost of project.

Net Present ValueNet Present Value

NPV = PV of Inflows - Initial Outlay NPV = PV of Inflows - Initial Outlay

NPV = + + +···+ – IO CF1 (1+ k )

CF2 (1+ k )2

CF3 (1+ k )3

CFn (1+ k )n

CFn = Cash flow at time nk = required rate of returnIO = Initial Investment

Net Present Value (NPV)

0 1 2 3 4

500 500 4,600 10,000(10,000)

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

Find Present Value of Cash Flows…

0 1 2 3 4

500 500 4,600 10,000(10,000)

$500 (1.10)

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

0 1 2 3 4

500 500 4,600 10,000(10,000)

455413

$500 (1.10) 2

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

0 1 2 3 4

500 500 4,600 10,000(10,000)

455413

$4,600 (1.10) 3

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

0 1 2 3 4

500 500 4,600 10,000(10,000)

4133,456

$10,000 (1.10) 4

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

0 1 2 3 4

500 500 4,600 10,000(10,000)

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

4133,456

$11,154

NPV = PV Benefits - PV Costs = $11,154 - $10,000 = $1,154

Accept Projectsince NPV > 0

PV of 3,500 Annuity for 4 years at 10%

0 1 2 3 4

3,500(10,000)

3,500 3,500 3,500

NPV = + + + – 10,000 3,500 (1+ .1 )

3,500 (1+ .1)2

3,500 (1+ .1 )3

3,500 (1+ .1 )4

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

0 1 2 3 4

3,500(10,000)

3,500 3,500 3,500

NPV = + + + – 10,000 3,500 (1+ .1 )

3,500 (1+ .1)2

3,500 (1+ .1 )3

3,500 (1+ .1 )4

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

= 3,500( ) - 10,000 1 .10(1+.10)4

= 11,095 – 10,000 = $1,095$1,095

NPV Decision RulesNPV Decision Rules

ACCEPT B only If projects are If projects are mutually exclusivemutually exclusive, , accept project with higher NPV.accept project with higher NPV.

ACCEPT A & B If projects are independent then accept If projects are independent then accept

all projects with NPV all projects with NPV 0. 0.

Mutually Exclusive:

Means that the acceptance of one project precludes the acceptance of the other projects under consideration. (You may only choose one.)

Net Present Value Profile

Net Present Value Profile:Net Present Value Profile:a graph of the NPV of a project at different

discount rates shows the NPV at 3 different points:a zero discount ratethe normal discount rate (or cost of capital)the IRR

allows an easy way to visualize whether or not an investment should be undertaken

10%5%0

20%15%Cost of Capital

Net Present Value ProfileNet Present Value Profile

Graphs the Net Present Value of the project with different Graphs the Net Present Value of the project with different required ratesrequired rates

NPV(0%) = + + + – 10,000 3,500 (1+ 0 )

3,500 (1+ 0)2

3,500 (1+ 0 )3

3,500 (1+ 0)4

= $4,000

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

10%5%0

NPV(5%) = + + + – 10,000 3,500 (1+ .05 )

3,500 (1+ .05)2

3,500 (1+ .05 )3

3,500 (1+ .05)4

= $2,411

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

Graphs the Net Present Value of the project with Graphs the Net Present Value of the project with different required ratesdifferent required rates

10%5%0

NPV(10%) = + + + – 10,000 3,500 (1+ .10 )

3,500 (1+ .10)2

3,500 (1+ .10 )3

3,500 (1+ .10)4

= $1,095

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

10%5%0

NPV(15%) = + + + – 10,000 3,500 (1+ .15 )

3,500 (1+ .15)2

3,500 (1+ .15 )3

3,500 (1+ .15)4

= – $7.58

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

10%5%0

NPV(20%) = + + + – 10,000 3,500 (1+ .20 )

3,500 (1+ .20)2

3,500 (1+ .20 )3

3,500 (1+ .20)4

= – $939

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

Connect the Points

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

10%5%0

PROJECT A

10%5%0

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

NPV(0%) = + + + – 10,000 500 (1+ 0 )

500 (1+ 0)2

4,600 (1+ 0 )3

10,000 (1+ 0)4

= $5,600

10%5%0

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

NPV(5%) = + + + – 10,000 500 (1+.05)

500 (1+.05)2

4,600 (1+ .05)3

10,000 (1+ .05)4

= $3,130

10%5%0

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

NPV(10%) = + + + – 10,000 500 (1+.10)

500 (1+.10)2

4,600 (1+ .10)3

10,000 (1+ .10)4

= $1,154

10%5%0

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

NPV(15%) = + + + – 10,000 500 (1+.15)

500 (1+.15)2

4,600 (1+ .15)3

10,000 (1+ .15)4

= –$445

10%5%0

PROJECT B

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

Connect the Points

10%5%0

PROJECT B

Net Present Value ProfileNet Present Value Profile Compare NPV of the two projects for different required Compare NPV of the two projects for different required

ratesrates

Crossover point

10%5%0

PROJECT B

Crossover point

For any discount rate < crossover point choose B

Compare NPV of the two projects for different required Compare NPV of the two projects for different required ratesrates

10%5%0

PROJECT B

Crossover point

For any discount rate > crossover point choose A

Compare NPV of the two projects for different required ratesCompare NPV of the two projects for different required rates

PROJECT A

The Internal Rate of Return (IRR)The Internal Rate of Return (IRR)

The internal rate of return is the interest yield promised by an investment project over its useful life.

The internal rate of return is computed by finding the discount rate that will cause the net present value of a project to be zero NPV=0).Requires calculating the interest rate that equates

the cash outflow (cost) with the cash inflows

Definition:The IRR is the discount rate where NPV = 0

The Internal Rate of Return (IRR)The Internal Rate of Return (IRR)ABC Company can purchase a new machine at a

cost of RM104,320 that will save RM20,000 per year in cash operating costs.

The machine has a 10-year life.

Future cash flows are the same every year in this example, so we can calculate the internal rate of return as follows:

Investment required Net annual cash flows

PV factor for theinternal rate of return =

RM 104, 320 RM 20,000

= 5.216

Find the 10-period row, move across until you find the factor 5.216. Look at the top of

the column and you find a rate of 14%14%.

Periods 10% 12% 14%1 0.909 0.893 0.877 2 1.736 1.690 1.647

. . . . . . . . . . . .9 5.759 5.328 4.946 10 6.145 5.650 5.216

Using the present value of an annuity of $1 table . . .

If the internal rate of 14 % return is equal to or greater than the company’s required rate of return, the project

is acceptable.

10%5%0

PROJECT B

Internal Rate of Return Internal Rate of ReturnInternal Rate of Return Internal Rate of Return

Measures the rate of return on a projectMeasures the rate of return on a project

Definition:The IRR is the discount rate in which NPV = 0

NPV = $0

10%5%0

PROJECT B

Measures the rate of return on a projectMeasures the rate of return on a project

Definition:The IRR is the discount rate in which NPV = 0

IRRA 15%

IRRB 14%NPV = $0

Determine the mathematical solution for IRRDetermine the mathematical solution for IRR

0 = NPV = + +···+ – IO CF1 (1+ IRR )

CF2 (1+ IRR )2

CFn (1+ IRR )n

IO = + +···+ CF1 (1+ IRR )

CF2 (1+ IRR )2

CFn (1+ IRR )n

Outflow = PV of Inflows

Solve for Discount Rates

Internal Rate of ReturnInternal Rate of ReturnFor Project BFor Project B

Cannot solve for IRRdirectly, must use Trial & Error

10,000 = + + + 500 (1+ IRR )

500 (1+ IRR )2

10,000 (1+ IRR )4

4,600 (1+ IRR )3

TRY 14%TRY 14%

10,000 = + + + 500 (1+ .14 )

500 (1+ .14)2

10,000 (1+ .14 )4

4,600 (1+ .14 )3

10,000 = 9,849?

PV of Inflows too low, try lower rate

10%5%0

PROJECT B

IRRB 14%

Internal Rate of ReturnInternal Rate of ReturnFor Project BFor Project B

Cannot solve for IRRdirectly, must use Trial & Error

10,000 = + + + 500 (1+ IRR )

500 (1+ IRR )2

10,000 (1+ IRR )4

4,600 (1+ IRR )3

TRY 13%TRY 13%

10,000 = + + + 500 (1+ .13 )

500 (1+ .13)2

10,000 (1+ .13 )4

4,600 (1+ .13 )3

10,000 = 10,155?

13% < IRR < 14%

10%5%0

PROJECT B

IRRB 14%

Decision Rule for Internal Rate of ReturnDecision Rule for Internal Rate of Return

Independent ProjectsAccept Projects with IRR required rate

Mutually Exclusive ProjectsAccept project with highest IRR required rate

Profitability IndexProfitability Index

PI = PV of InflowsInitial Outlay

Very Similar to Net Present ValueVery Similar to Net Present Value

Instead of Subtracting the Initial Outlay from the PVInstead of Subtracting the Initial Outlay from the PVof Inflows, the Profitability Index is the ratio of Initialof Inflows, the Profitability Index is the ratio of InitialOutlay to the PV of Inflows. Outlay to the PV of Inflows.

+ + +···+ CF1 (1+ k )

CF2 (1+ k )2

CF3 (1+ k )3

CFn (1+ k )n

PI =Initial Outlay

Profitability Index for Project BProfitability Index for Project B

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

+ + + 500 (1+ .1 )

500 (1+ .1)2

4,600 (1+ .1 )3

10,000 (1+ .1 )4

10,000PI =

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

+ + + 500 (1+ .1 )

500 (1+ .1)2

4,600 (1+ .1 )3

10,000 (1+ .1 )4

10,000PI =

PI = 11,15410,000

= 1.1154

P R O J E C TTime A B0 (10,000.) (10,000.)1 3,500 5002 3,500 5003 3,500 4,6004 3,500 10,000

+ + + 500 (1+ .1 )

500 (1+ .1)2

4,600 (1+ .1 )3

10,000 (1+ .1 )4

10,000PI =

PI = 11,15410,000

= 1.1154

Profitability Index for Project AProfitability Index for Project A

10,000PI =

PI = 11,095 10,000

= 1.1095

3,500( ) 1 .10(1+.10)4

Profitability Index Decision RulesProfitability Index Decision Rules

Independent ProjectsIndependent ProjectsAccept Project if PI 1

Mutually Exclusive ProjectsMutually Exclusive ProjectsAccept Highest PI 1 Project

Comparison of MethodsComparison of Methods

Project A Project B ChoosePayback < 3 years < 4 years ANPV $1,095 $1,154 BIRR 14.96% 13.50% API 1.1095 1.1154 B

Time Value of MoneyTime Value of MoneyPayback - Does not adjust for timing differences NPV, IRR and PI take into account the time value of

5,000 5,000(10,000)

Project 1

0 1 2 3

5,000 5,000(10,000)

Project 2

10,000

Both Projects haveIdentical Payback

Relevant Cash Flows?Relevant Cash Flows?NPV, IRR and PI use all Cash FlowsPayback method ignores Cash Flows that occur

after the Payback Period.

NPV & PI indicated accept Project B while IRR NPV & PI indicated accept Project B while IRR indicated that Project A should be accepted. Why?indicated that Project A should be accepted. Why?

Reinvestment RateReinvestment RateNPV assumes cash flows are reinvested at the

required rate, k.IRR assumes cash flows are reinvested at IRR.

Reinvestment Rate of k more realistic as most Reinvestment Rate of k more realistic as most projects earn approximately k (due to competition)projects earn approximately k (due to competition)

Conclusion: Conclusion: NPV is the Better Method for project NPV is the Better Method for project evaluationevaluation

0 1 2 3 4 5

(900) 300 400 400 500 600

Summary Problem:Summary Problem:

A project costs $900 and will repay $300, $400, $400, A project costs $900 and will repay $300, $400, $400, $500, and $600 over the first five years, respectively.$500, and $600 over the first five years, respectively.

• Determine the NPV using a discount rate of15%.Determine the NPV using a discount rate of15%.• Find the IRR.Find the IRR.• Add back IO and divide by IO to get PI.Add back IO and divide by IO to get PI.

0 1 2 3 4 5

(900) 300 400 400 500 600

Summary Problem:Summary Problem:

Answers:Answers:

• IRR = 34.37%.IRR = 34.37%.• Using a discount rate of 15%, Using a discount rate of 15%, NPV = $510.52.NPV = $510.52.• PI = 1.57.PI = 1.57.

lecture 2 npv & fv

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