1

Fin650:Project Appraisal

Lecture 3

Essential Formulae in Project Appraisal

What is “Capital Budgeting” Two big questions: “Yes-No”: Should you invest money today in a

project that gives future payoffs? “Ranking”: How to compare mutually-exclusive

projects? If you have several alternative investments, only one of which you can choose, which should you undertake?

2

Other issues Sunk costs. How should we account for costs

incurred in the past? The cost of foregone opportunities. Salvage values and terminal values. Incorporating taxes into the valuation

decision.

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4

Benefits and Cost Realized at Different Times

Benefits and costs realized in different times are not comparable

Some benefits and costs are recurrent, while some are realized only for a temporary period

Examples: Roads, built now at heavy costs, to generate benefits later, Dams, entail environmental costs long after their economic benefits have lapsed, A life lost now entails cost for at least as long into the future as the person

would have lived

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Discounting Future Benefits and Costs

Basic Concepts:

A. Future Value Analysis

In general, the future value in one year of some amount X is given by:

FV= X(1+i)where i is the annual rate of interest. This is simple compounding

B. Present Value Analysis

In general, if the prevailing interest rate is i, then the present value

of an amount Y received in one year is given by:

Discounting is the opposite of compounding.

i

YPV

1

6

7

Discounting Future Benefits and CostsNet Present Value Analysis

The NPV of a project equals the difference between the present value of benefits, PV(B), and the present value of the costs, PV(C):

NPV = PV(B)-PV(C)Compounding and Discounting Over Multiple Years

Future value over multiple YearsIn general, if an amount, denoted X, is invested for n years and interest is compounded annually at i percent, then the future value is:

FV = X(1+i)n

Present value over multiple yearsIn general, the present value of an amount received in n years, denoted Y, with interest discounted annually at rate i percent, then the present value is:

The term 1/(1+i)n is called the discount factor

ni

YPV

)1(

8

9

Discounting and Alternative Investment Criteria

Basic Concepts:A. Discounting

Recognizes time value of moneya. Funds when invested yield a returnb. Future consumption worth less than present consumption

PVB = (B o/(1+r)

o

+(B 1/(1+r)1+.…….+(Bn /(1+r)

nPVC = (C

o /(1+r) +(C 1/(1+r)1+.…….+(Cn /(1+r)

o

o

r

rNPV = (B o-Co)/(1+r) o+(B 1-C1)/(1+r) 1+.…….+(B n-Cn)/(1+r) n

o n

o

r

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Discounting and Alternative Investment Criteria (Cont’d)

B. Cumulative Values

The calendar year to which all projects are discounted to is important

All mutually exclusive projects need to be compared as of same calendar year

If NPV = (B o-Co)(1+r) 1+(B1-C1) +..+..+(B n-Cn)/(1+r) n-1 and

NPV = (B o-Co)(1+r) 3+(B1-C1)(1+r) 2+(B2-C2)(1+r)+(B 3-C3)+...(B n-Cn)/(1+r) n-3

Then NPV = (1+r) 2 NPV

1r

3r

3r

1r

11

Year 0 1 2 3 4

Net Cash Flow -1000 200 300 350 1440

Examples of Discounting

25.676)1.1(

1440

)1.1(

350

)1.1(

300

1.1

2001000NPV

4320

1.0

88.743)1.1(

1440

)1.1(

350

1.1

300200)1.1(1000NPV

321

1.0

26.818)1.1(

1440

)1.1(

350300)1.1(200)1.1(1000NPV

2122

1.0

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Financial Calculations

The present value of a single sum is:

PV = FV (1 + r)-t

the present value of a dollar to be received at the end of period t, using a discount rate of r.

t

tt

rCFPV

1 )1(

The present value of series of cash flows is:

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Financial Calculations: Cash Flow SeriesA payment series in which cash flows are Equallysized and Equally timed is known as an annuity.There are four types:1. Ordinary annuities; the cash flows occur at the

end of each time period. (Workbook 5.10 and 5.11)

2. Annuities due; the cash flows occur at the start of each time period.

3. Deferred annuities; the first cash flow occurs later than one time period into the future. (Workbook 5.10 and 5.11) 4. Perpetuities; the cash flows begin at the end of the first period, and go on forever.

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Evaluation of Project Cash Flows.

Cash flows occurring within investment projects are assumed to occur regularly, at the end of each year.

Since they are unlikely to be equal, they will not be annuities.

Annuity calculations apply more to loans and other types of financing.

All future flows are discounted to calculate a Net Present Value, NPV; or an Internal Rate of Return, IRR.

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Calculating NPV and IRR With Excel -- Basics.

1. Ensure that the cash flows are recorded with the correct signs: -$, +$, -Tk, +Tk. etc.

2. Make sure that the cash flows are evenly timed: usually at the end of each year.

3. Enter the discount rate as a percentage, not as a decimal: e.g. 15.6%, not 0.156.

4. Check your calculations with a hand held calculator to ensure that the formulae have been correctly set up.

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Calculating NPV and IRR With Excel -- The Excel Worksheet.

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Calculating MIRR and PB With Excel.

Modified Internal Rate of Return – the cash flow cell range is the same as in the IRR, but both the required rate of return, and the re-investment rate, are entered into the formula: MIRR( B6:E6, B13, B14)

Payback – there is no Excel formula . The payback year can be found by inspection of accumulated annual cash flows.

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ARR and Other Evaluations With Excel.

Accounting Rate of Return – there is no Excel formula. Average the annual accounting income by using the ‘AVERAGE’ function, and divide by the chosen asset base.

Other financial calculations – use Excel ‘Help’ to find the appropriate function. Read the help information carefully, and apply the function to a known problem before relying on it in a live worksheet.

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Calculating Financial Functions With Excel -- Worksheet Errors.

Common worksheet errors are: Cash flow cell range wrongly specified. Incorrect entry of interest rates. Wrong NPV, IRR and MIRR formulae. Incorrect cell referencing. Mistyped data values. No worksheet protection.

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Calculating Financial Functions With Excel -- Error Control.

Methods to reduce errors: Use Excel audit and tracking tools. Test the worksheet with known data. Confirm computations by calculator. Visually inspect the coding. Use a team to audit the spreadsheet.

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1. Net Present Value (NPV)2. Benefit-Cost Ratio (BCR)3. Pay-out or Pay-back Period4. Internal Rate of Return (IRR)

Alternative Investment Criteria

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Net Present Value (NPV)

1. The NPV is the algebraic sum of the discounted values of the incremental expected positive and negative net cash flows over a project’s anticipated lifetime.

2. What does net present value mean?

Measures the change in wealth created by the project.

If this sum is equal to zero, then investors can expect to recover their incremental investment and to earn a rate of return on their capital equal to the private cost of funds used to compute the present values.

Investors would be no further ahead with a zero-NPV project than they would have been if they had left the funds in the capital market.

In this case there is no change in wealth.

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First Criterion: Net Present Value (NPV) Use as a decision criterion to answer

following:a. When to reject projects?b. Select project (s) under a budget

constraint?c. Compare mutually exclusive projects?d. How to choose between highly profitable

mutually exclusive projects with different lengths of life?

Alternative Investment Criteria

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Net Present Value Criterion

a. When to Reject Projects?Rule: “Do not accept any project unless it generates a positive net present value when discounted by the opportunity cost of funds”

Examples:Project A: Present Value Costs $1 million, NPV + $70,000Project B: Present Value Costs $5 million, NPV - $50,000Project C: Present Value Costs $2 million, NPV + $100,000Project D: Present Value Costs $3 million, NPV - $25,000Result:Only projects A and C are acceptable. The investor is made worse off if projects B and D are undertaken.

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Net Present Value Criterion (Cont’d)

b. When You Have a Budget Constraint?Rule: “Within the limit of a fixed budget, choose that subset of the available projects which maximizes the net present value”

Example:If budget constraint is $4 million and 4 projects with positive NPV:

Project E: Costs $1 million, NPV + $60,000Project F: Costs $3 million, NPV + $400,000Project G: Costs $2 million, NPV + $150,000Project H: Costs $2 million, NPV + $225,000Result: Combinations FG and FH are impossible, as they cost too much. EG and EH are within the budget, but are dominated by the combination EF, which has a total NPV of $460,000. GH is also possible, but its NPV of $375,000 is not as high as EF.

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c. When You Need to Compare Mutually Exclusive Projects?

Rule: “In a situation where there is no budget constraint but a project must be chosen from mutually exclusive alternatives, we should always choose the alternative that generates the largest net present value”

Example:Assume that we must make a choice between the following three mutually exclusive projects:

Project I: PV costs $1.0 million, NPV $300,000Project J: PV costs $4.0 million, NPV $700,000Projects K: PV costs $1.5 million, NPV $600,000Result:Projects J should be chosen because it has the largest NPV.

Net Present Value Criterion (Cont’d)

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Shortcut Methods for Calculating the Present Value of Annuities and Perpetuities 1/2 Annuities and PerpetuitiesAn annuity is an equal, fixed amount received (or paid) eachyear for a number of years.A perpetuity is an annuity that continues indefinitely. Present value of an annuity

or PV = A x

Where is the annuity factor,

The term , which equals the present value of an annuity of$/Tk. 1 per year for n years when the interest rate is ipercent, is called the annuity factor.

n

tti

APV

1 )1(

nia i

ia

nni

)1(1nia

nia

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Shortcut Methods for Calculating the Present Value of Annuities and Perpetuities 2/2 Present value of a perpetuityPV = A/i, if i>0 Present value of an annuity that grows or declines

at a constant ratePV(B) = [B1/ (1+g)]x ai0

n , i0 = 1-g/1+g

if i>gIf g is small, B1/1+g is approximately equal to B1,

and i0 = 1-g Present value of benefits (or costs) that grow or

decline at a constant rate in perpetuityPV(B) = B1/ (1-g), if i>g

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Long-Lived Projects and Terminal ValuesIt is generally assumed that projects have finite economicLife.For projects with infinite life, we may calculate NPV usingThe formula:

Assumes that the net benefits are constant or grow at a constant rate.Not a very realistic assumption.For most long lived projects, select a relatively short discountingperiod (useful life of the project) and include a terminal value toreflect all subsequent benefits and costs.

Where T(k) denotes the terminal value.

0 )1(ttt

i

NBNPV

)()1(0

kTi

NBNPV

k

ttt

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Alternative Methods for Estimating Terminal Values Terminal Values Based on:

Simple Projections Salvage or Liquidation Value Depreciated Value, economic depreciation Percentage of Initial Constructions Cost

Setting the Terminal Value equal to zeroNote: Accounting depreciation should never be included as a cost (expense) in CBA

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Comparing Projects with Different Time Frames Two Methods for Comparing Projects with Different Time Frames

Rolling Over the Shorter Project Comparison between a cogeneration power plan and a hydroelectric

project Equivalent Annual Net Benefit Method (EANB)EANB of an alternative equals its NPV divided by the annuityfactorThat has the same life as the project

Where is the annuity factor,

nia

NPVEANB

nia i

ia

nni

)1(1

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Real Versus Nominal Currency Constant currency

Use CPI as the deflator If benefits and costs are measured in nominal currency, use

nominal discount rate If benefits and costs are measured in real currency, use real

discount rate To convert a nominal interest rate i, to a real interest rate, r,

with an expected inflation rate, m, use the following equation

If m is small, the real interest rate is approximately equals the Nominal interest rate minus the expected rate of inflation: r = i-m

m

mir

1

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Alternative Investment Criteria: Benefit Cost Ratio

As its name indicates, the benefit-cost ratio (R), or what is sometimes referred to as the profitability index, is the ratio of the PV of the net cash inflows (or economic benefits) to the PV of the net cash outflows (or economic costs):

)CostsEconomicor(OutflowsCashNetofPV

)BenefitsEconomicor(InflowsCashNetofPVR

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If benefit-cost ratio (R) >1, then the project should be undertaken.

Problems?

Sometimes it is not possible to rank projects with the benefit-cost Ratio

Mutually exclusive projects of different sizes

Not necessarily true that if RA>RB, that

project “A” is better than project “B”

Basic Rule

35

Problem:The Benefit-Cost Ratio does not adjust for mutually exclusive projects of different sizes. For example:Project A: PV0of Costs = $5.0 M, PV0 of Benefits = $7.0 M

NPVA = $2.0 M RA = 7/5 = 1.4

Project B: PV0 of Costs = $20.0 M, PV0 of Benefits = $24.0 MNPVB = $4.0 M RB = 24/20 = 1.2

According to the Benefit-Cost Ratio criterion, project A should be chosen over project B because RA>RB, but the NPV of project B is greater than the NPV of project A. So, project B should be chosen

Conclusion: The Benefit-Cost Ratio should not be used to rank projects

Benefit-Cost Ratio (Cont’d)

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Pay-out or Pay-back period

The pay-out period measures the number of years it will take for the undiscounted net benefits (positive net cashflows) to repay the investment.

A more sophisticated version of this rule compares the discounted benefits over a given number of years from the beginning of the project with the discounted investment costs.

An arbitrary limit is set on the maximum number of years allowed and only those investments having enough benefits to offset all investment costs within this period will be acceptable.

Alternative Investment Criteria

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Projects with shortest payback period are

preferred by the criteria

Assumes all benefits that are produced by in

longer life project have an expected value of

zero after the pay-out period.

The criteria may be useful when the project is

subject to high level of political risk.

Pay-Out or Pay-Back Period

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Internal Rate of Return (IRR)

IRR is the discount rate (K) at which the present value of benefits are just equal to the present value of costs for the particular project

Note: the IRR is a mathematical concept, not aneconomic or financial criterion

Alternative Investment Criteria

t

ittt

k

CB

0

0)1(

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Common uses of IRR:

(a) If the IRR is larger than the cost of funds then the

project should be undertaken

(b) Often the IRR is used to rank mutually exclusive

projects. The highest IRR project should be chosen

(c) An advantage of the IRR is that it only uses

information from the project

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First Difficulty: Multiple rates internal rate of return for

Project

Solution 1: K = 100%; NPV= -100 + 300/(1+1) + -200/(1+1)2 = 0

Solution 2: K = 0%; NPV= -100+300/(1+0)+-200/(1+0)2 = 0

Difficulties With the Internal Rate of Return Criterion

+300

Bt - Ct

-200-100

Time

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Second difficulty: Projects of different sizes and also strict alternatives

Year 0 1 2 3 ... ... ҐProject A -2,000 +600 +600 +600 +600 +600 +600

Project B -20,000 +4,000 +4,000 +4,000 +4,000 +4,000 +4,000

NPV and IRR provide different Conclusions:Opportunity cost of funds = 10%

NPV : 600/0.10 - 2,000 = 6,000 - 2,000 = 4,000

NPV : 4,000/0.10 - 20,000 = 40,000 - 20,000 = 20,000

Hence, NPV > NPV

IRRA : 600/K A - 2,000 = 0 or K A = 0.30

IRRB : 4,000/K B - 20,000 = 0 or K B = 0.20

Hence, K A>KB

0B

0A

0B

0A

Difficulties With The Internal Rate of Return Criterion (Cont’d)

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Third difficulty:Projects of different lengths of life and strict alternatives

Opportunity cost of funds = 8%Project A: Investment costs = 1,000 in year 0

Benefits = 3,200 in year 5Project B: Investment costs = 1,000 in year 0

Benefits = 5,200 in year 10

NPV : -1,000 + 3,200/(1.08)5 = 1,177.86NPV : -1,000 + 5,200/(1.08)10= 1,408.60

Hence, NPV > NPV

IRRA : -1,000 + 3,200/(1+KA)5 = 0 which implies that KA = 0.262IRRB : -1,000 + 5,200/(1+KB)10 = 0 which implies that KB = 0.179

Hence, KA>KB

0B

0A

0B

0A

Difficulties With The Internal Rate of Return Criterion (Cont’d)

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Fourth difficulty: Same project but started at different times

Project A: Investment costs = 1,000 in year 0

Benefits = 1,500 in year 1

Project B: Investment costs = 1,000 in year 5

Benefits = 1,600 in year 6

NPV A : -1,000 + 1,500/(1.08) = 388.88

NPV B : -1,000/(1.08) 5 + 1,600/(1.08) 6 = 327.68

Hence, NPV > NPV

IRR A : -1,000 + 1,500/(1+K A) = 0 which implies that KA= 0.5

IRRB : -1,000/(1+K B) 5 + 1,600/(1+K B) 6 = 0 which implies that KB = 0.6

Hence, K B >KA

0B

0A

Difficulties With The Internal Rate of Return Criterion (Cont’d)

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Year 0 1 2 3 4

Project A 1000 1200 800 3600 -8000

IRR A 10%

Compares Project A and Project B ?

Project B 1000 1200 800 3600 -6400

IRR B -2%

Project B is obviously better than A, yet IRR A > IRR B

Project C 1000 1200 800 3600 -4800

IRR C -16%

Project C is obviously better than B, yet IRR B > IRR C

Project D -1000 1200 800 3600 -4800

IRR D 4%

Project D is worse than C, yet IRR D > IRR C

Project E -1325 1200 800 3600 -4800

IRR E 20%

Project E is worse than D, yet IRR E > IRR D

IRR FOR IRREGULAR CASHFLOWSFor Example: Look at a Private BOT Project from the perspective of the

Government

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The Social Discount Rate: Main Issues How much current consumption society is willing to give up

now in order to obtain a given increase in future Consumption?

It is generally accepted that society’s choices, including the choice of weights be based on individuals’ choices

Three unresolved issues Whether market interest rates can be used to represent how

individuals weigh future consumption relative to present consumption?

Whether to include unborn future generation in addition to individuals alive today?

Whether society attaches the same value to a unit of investment as to a unit of consumption

Different assumptions will lead to choice of different discount rate

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Does the Choice of Discount Rate Matter? Generally a low discount rate favors projects with

highest total benefits, irrespective of when they occur, e.g. project C

Increasing the discount rate applies smaller weights to benefits or (costs) that occur further in the future and, therefore, weakens the case for projects with benefit that are back-end loaded (such as project C), strengthens the case for projects with benefit that are front-end loaded (such as project B)

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NPV for Three Alternative ProjectsYear Project A Project B Project C

0 -80,000 -80,000 -80,000

1 25,000 80,000 0

2 25,000 10,000 0

3 25,000 10,000 0

4 25,000 10,000 0

5 25,000 10,000 140,000

Total benefits 45,000 40,000 60,000

NPV (i=2%) 37,838 35,762 46,802

NPV (i=10%) 14,770 21,544 6,929

NPV and IRR The two basic capital budgeting tools Note: We usually prefer NPV to IRR, but IRR is

a handy tool

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“Yes-No” and NPV NPV rule: A project is worthwhile if the NPV

> 0

According to the NPV rule: If NPV > 0, project is worthwhile If NPV < 0, project should not be undertaken

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1 2

0 1 2... 0?

1 1 1

NN

CFCF CFNPV CF

r r r

Technical notes CF0 is usually negative (the project cost)

CF1, CF2, … are usually positive (future payoffs of project)

CF1, CF2, … are expected or anticipated cash flows

r is a discount rate appropriate to the project’s risk

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“Yes-No” and IRR IRR rule: A project is worthwhile if the IRR >

discount rate

According to the IRR rule: If IRR > r, then the project is worthwhile If IRR < r, project should not be undertaken

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1 2

0 1 2... 0

1 1 1

NN

CFCF CFCF

IRR IRR IRR

Basic “Yes-No” example

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123

456789

10111213

A B C

Discount rate 12%

YearProject cash

flow0 -10001 3002 4003 5004 6005 100

NPV 380.68 <-- =B5+NPV($B$2,B6:B10)IRR 26.47% <-- =IRR(B5:B10)

YES-NO WITH NPV AND IRR

This project is worthwhile by both NPV and IRR rules:NPV > 0IRR > discount rate of 12%

Basic “Ranking” example

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“Yes-No”: Both projects are worthwhileNPVA, NPVB > 0IRRA, IRRB > discount rate of 12%

“Ranking”: If you can choose only one project, B is preferred by both NPV and IRRNPVB > NPVA IRRB > IRRA

123456789

10111213

A B C D

Discount rate 12%

Year Project A Project B0 -1000 -8001 200 4202 400 1003 600 3004 300 6005 100 200

NPV 171.92 363.05 <-- =C5+NPV($B$2,C6:C10)IRR 19% 29% <-- =IRR(C5:C10)

RANKING TWO PROJECTS WITH NPV AND IRR

Excel’s NPV function Chapter 2: Excel’s NPV function is really the

present value of future cash flows! To compute the actual NPV, add in the initial

cash flow as shown below:

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Summing up

55

56

123456789

10111213

A B C D

Discount rate 6%

Year Project A Project B0 -500 -5001 100 2502 100 2503 150 2004 200 1005 400 50

NPV 266.60 242.84 <-- =C5+NPV(B2,C6:C10)IRR 19.77% 27.38% <-- =IRR(C5:C10)

NPV AND IRR CAN SOMETIMES GIVE CONFLICTING RANKINGS

In this example:Both A and B are worthwhile by both NPV and IRR criteriaIf discount rate = 6%

A is preferred to B by NPV ruleB preferred to A by IRR rule

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15

16171819202122232425262728293031323334

A B C D E F G

Project ANPV

Project BNPV

0% 450.00 350.00 <-- =$C$5+NPV(A17,$C$6:$C$10)2% 382.57 311.53 <-- =$C$5+NPV(A18,$C$6:$C$10)4% 321.69 275.906% 266.60 242.84

8.5128% 204.58 204.5810% 171.22 183.4912% 129.85 156.7914% 92.08 131.8416% 57.53 108.4718% 25.86 86.5720% -3.22 66.0022% -29.96 46.6624% -54.61 28.4526% -77.36 11.2828% -98.39 -4.9330% -117.87 -20.25

TABLE OF NPVs AND DISCOUNT RATES

-200

-100

0

100

200

300

400

500

0% 5% 10% 15% 20% 25% 30%

Project ANPV

Project BNPV

IRRA is always < IRRB: By IRR rule, B is always preferred to A

For discount rates < 8.5128%: NPVA > NPVB (ranking conflict)

For discount rates > 8.51285: NPVA < NPVB (no ranking conflict)

When IRR and NPV conflict, use NPV Why: IRR gives the rate of return NPV gives the wealth increment

58

1 2

0 1 2Cost ofproject Value today of future

project cash flows

Incremental wealth:How much does the project'snet value add to your wealth?

...1 1 1

NN

CFCF CFNPV CF

r r r

Back to last example:Calculating the crossover point

59

123

456789

10111213

A B C D E

Discount rate 6%

Year Project A Project BProject A - Project B

0 -500 -500 0 <-- =B5-C51 100 250 -150 <-- =B6-C62 100 250 -150 <-- =B7-C73 150 200 -50 <-- =B8-C84 200 100 100 <-- =B9-C95 400 50 350 <-- =B10-C10

NPV 266.60 242.84IRR 19.77% 27.38% 8.5128% <-- =IRR(D5:D10)

CROSSOVER POINT: IRRA = IRRB

compute IRR of differential cash flows

Crossover point is the IRR of the differential cash flows (column D)

60

Essential Formulae -- Summary

1.The Time Value of Money is a cornerstone of finance.

2. The amount, direction and timing of cash flows, and relevant interest rates, must be carefully specified.

3. Knowledge of financial formulae is essential for project evaluation.

4. NPV and IRR are the primary investment evaluation criteria.5. Most financial functions can be automated within Excel.6. Spreadsheet errors are common. Error controls should be

employed.

7.To reduce spreadsheet errors: -document all spreadsheets, keep a list of authors and a history of changes, use comments to guide later users and operators.

8. Financial formulae and spreadsheet operation can be demanding. Seek help when in doubt.