Net Present Worth, Equivalent Annual Worth, and IRR.

Comparing ProjectsUsing Time Value of

MoneyNet Present Worth, Equivalent Annual

Worth, and IRR

2

Reviewing…NPWReviewing…NPWTwo approaches to handle

differing project lives:Common Multiple Period:

Projects are assumed to be repeated until a common multiple point in time is established.

Study Period: Select a study period for both projects and estimate cash flows to conform to the study period.

3

Equivalent Annual Worth Analysis

Equivalent Annual Worth AnalysisEquivalent Annual Worth (EAW) can be used to compare projects.

Equivalent Annual Cost (EAC) can be used instead of EAW if revenues are not included.

EAC = – EAW

4

Internal Rate of ReturnInternal Rate of Return

Internal Rate of Return (IRR):

The interest rate i* at which NPW = 0

Note: This is the same as finding the roots of a polynomial equation. If there is more than one sign change in the net annual cash flows over the life of the project, then there is more than one internal rate of return (root)!

We may find the IRR by either the manual method we used for the bond yield, or we may use the computer to find the roots by either plots or numerical methods

5

IRR ExampleIRR ExampleProject A costs $10,000 and will last

for 10 years. Annual, end of year revenues will be $3000, and expenses will be $1000. There is no salvage value.

Project B costs $20,000 and will also last for 10 years. Annual revenues will be $4000 with annual expenses of $1500. Salvage value is $14,500.

What is each project’s IRR?

Table Example

i* (P/A, i*, 10) (P/F, i*, 10)

6% 7.3601 .5584

8% 6.7101 .4632

10% 6.1446 .3855

12% 5.6502 .3220

18% 4.4941 .1911

6

7

-10000

-5000

0

5000

10000

15000

20000

25000

0 5 10 15 20

Interest Rate (%)

Ne

t P

res

en

t W

ort

h (

$)

Project A

Project B

ExamplePW(i) = -16 + 3(P/A, i, 5) + 4(P/F, i, 5)

Internal Rate of Return

i PW(i)

12 -2.9210 -2.146 -0.374 0.645 0.12

3,000

50

4,000

16,000

ExamplePW(i) = -16 + 3(P/A, i, 5) + 4(P/F, i, 5)

i* = 5 1/4 %

i* < MARR

Internal Rate of Return

i PW(i)

12 -2.9210 -2.146 -0.374 0.645 0.12

3,000

50

4,000

16,000

Spreadsheet Example

1

2

3

456789

10111213

A B C D E F

ExamplePeriod Cash Flow

0 (16,000)1 3,0002 3,0003 3,0004 3,0005 7,000 MARR = 12.0%

NPV = (2,916) = NPV(E9,C5:C9)+ C4PMT = (809) = -PMT(E9,5,C10)IRR = 5.2% = IRR(C4:C9,E9)

IRR Problems

1,000

4,100

5,580

2,520

n0 1 2 3Consider the following

cash flow diagram. We wish to find the InternalRate-of-Return (IRR).

IRR Problems

1,000

4,100

5,580

2,520

n0 1 2 3Consider the following

cash flow diagram. We wish to find the InternalRate-of-Return (IRR).

PWR(i*) = PWC(i*)

4,100(1+i*)-1 + 2,520(1+i*)-3 = 1,000 + 5,580(1+i*)-2

IRR Problems

Internal Rate ReturnPeriod Cash Flow

0 (1,000)1 4,1002 (5,580)3 2,520

i = NPV =5% $20.41

10% $9.0215% $2.88

20% $0.0025% ($0.96)30% ($0.91)

35% ($0.46)40% $0.0045% $0.2150% $0.0055% ($0.70)60% ($1.95)

NPV vs. Interest

($5)

$0

$5

$10

$15

$20

$25

0% 10% 20% 30% 40% 50% 60%

Interest Rate

Ne

t P

res

en

t V

alu

e

14

Decisions with IRRDecisions with IRR

When applied to project selection among independent projects – i.e. there are enough funds such that any or all of the projects may be selected – then investing based on IRR is easy:

Select all projects with IRR > MARR!

Note: Make sure that there is only one IRR (root)!

15

Decisions with IRRDecisions with IRR

When applied to project selection among mutually exclusive projects – i.e. there is not enough money to do them all – IRR can produce the same result as NPW and EAW.

However, incremental analysis MUST be used!

REASON: IRR is a relative measure of project merit

16

Why Must Incremental Analysis be Used for Competing Projects?


Assume that an MARR of 16% per year is required, and $85 000 is available to invest:

• Project A requires $50 000 upfront to obtain an IRR of 35% per year.

• Project B requires an $85 000 first cost and returns an IRR of 29% per year.

• What could we do with the un-invested money from Project A? ($35 000)

17



It would be reasonable to invest the remaining $35 000 at the MARR:

• Overall IRRA = 50 000(.35) + 35 000(.16)

85 000 = 27.2% per year

• Project B returns an IRR of 29% per year on ALL the money available to invest.

18

Incremental AnalysisIncremental Analysis

A technique or approach that can be used with NPW, EAW, and later with IRR and Cost/Benefit to determine if an incremental expenditure should be made.

Note: If using NPW or IRR, lifetimes must be equal – so use Least Common Multiple or Study Period!

19

Steps of the ProcessSteps of the Process1. Order alternatives from lowest to

highest initial investment.

2. Let Alternative A0 (do nothing) be considered the current best.

3. Consider next Alternative ( j = j+ 1)

4. Determine cash flows for “current best” and Alternative j.

5. Determine incremental cash flows between “current best” and Alternative j.

6. Calculate PW, AW, FW, IRR (or Benefit/Cost) of only the incremental cash flows.

20

Steps of the ProcessSteps of the Process7. If incremental investment yields

NPW, EAW, or a NFW > 0*, then the new “current best” becomes Alternative j.

* (B/C ratio > 1, or IRR > MARR)

8. If there are remaining alternatives, go to Step 3.

9. If all alternatives have been considered, select the “current best” alternative as the preferred alternative.

21

i = 18% Net Cash Flows for Alternatives A0 - A3

End of Year, t

A 0 A 1 A 2 A 3

0 $0 -$10000 -$50,000 -$75,000

1 0 4,500 20,000 20,000

2 0 4,500 20,000 25,000

3 0 4,500 20,000 30,000

4 0 4,500 20,000 35,000

5 0 4,500 20,000 40,000

22

IRR Incremental Analysis

IRR Incremental AnalysisRank projects from lowest to highest

initial cost

Eliminate any projects with IRR < MARR

Starting from the least expensive project to the next most expensive, justify each incremental investment

• IRR HC-LC < MARR Accept LC Project & Reject HC Project

• IRR HC-LC = MARR Indifferent

• IRR HC-LC > MARR Reject LC Project & Accept HC Project

Net Present Worth, Equivalent Annual Worth, and IRR.

Documents

Transcript of Net Present Worth, Equivalent Annual Worth, and IRR.