Essential formulae in project appraisal 2

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Fin650:Project Appraisal

Lecture 3 Essential Formulae in Project Appraisal

some are realized only for a temporary period

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

Moving Money Through

Time

annual interest rates

in time is ‘compounded’, whilst money moved

backward in time is ‘discounted’

Financial Calculations

 Time value calculations in capital budgeting usually assume that interest is annually compounded

flows’: the symbol is:

 Ct is Cash flow at end of period t

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

 The present value of series of cash flows is:

Financial Calculations:

Cash Flow Series

A payment series in which cash flows are Equally

sized 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

Evaluation of Project Cash Flows.

are assumed to occur regularly, at the end of

each year

 Since they are unlikely to be equal, they will not

be annuities

other types of financing

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

Return, IRR

Decision Making With

Cash Flow Evaluations

 If the Net Present Value is positive, then the

project should be accepted The project will

increase the present wealth of the firm by the

NPV amount

 If the IRR is greater than the required rate of

return, then the project should be accepted The IRR is a relative measure, and does not measure

an increase in the firm’s wealth

Calculating NPV and IRR With Excel Basics.

correct signs: -$, +$, -Tk, +Tk etc

usually at the end of each year

decimal: e.g 15.6%, not 0.156

to ensure that the formulae have been correctly set up

Calculating NPV and IRR With Excel The Excel

Worksheet

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

ARR and Other

Evaluations With Excel.

Average the annual accounting income by using the

‘AVERAGE’ function, and divide by the chosen asset

Calculating Financial

Functions With Excel

Worksheet Errors.

Common worksheet errors are:

 Cash flow cell range wrongly specified

 Incorrect entry of interest rates

 Incorrect cell referencing

Calculating Financial

Functions With Excel

Error Control.

Methods to reduce errors:

 Use Excel audit and tracking tools

 Visually inspect the coding

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.

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

Y PV



 1

18

Discounting Future Benefits and

Costs

Net 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 Years

In 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 years

In 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 thei n discount factor

Y PV

) 1 ( 



20

Discounting and Alternative Investment Criteria

Basic Concepts:

A Discounting

 Recognizes time value of money

a Funds when invested yield a return

b Future consumption worth less than present

o

r

 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)+ (Bn-Cn)/(1+r) n-3

Then NPV = (1+r) 2 NPV

1 r 3

r

3

r 1r

1.1(

1440)

1.1(

350)

1.1(

3001

.1

2001000

NPV00.1    2  3  4 

88

743)

1.1(

1440)

1.1(

3501

.1

300200

)1.1(1000NPV01.1     2  3 

26

818)

1.1(

1440)

1.1(

350300

)1.1(200)

1.1(1000

Alternative Investment

Criteria

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.

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

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 + $70,000

Project B: Present Value Costs $5 million, NPV - $50,000

Project C: Present Value Costs $2 million, NPV + $100,000

Project D: Present Value Costs $3 million, NPV - $25,000

Result:

Only projects A and C are acceptable The investor is made worse off if projects B and D are undertaken.

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 + $60,000

Project F: Costs $3 million, NPV + $400,000

Project G: Costs $2 million, NPV + $150,000

Project H: Costs $2 million, NPV + $225,000

Result:

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.

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 $300,000

Project J: PV costs $4.0 million, NPV $700,000

Projects K: PV costs $1.5 million, NPV $600,000

Result:

Projects J should be chosen because it has the largest NPV

Net Present Value Criterion (Cont’d)

Shortcut Methods for Calculating the Present Value of Annuities

 Annuities and Perpetuities

An annuity is an equal, fixed amount received (or paid) each

year 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 i

percent, is called the annuity factor

1 ( 1 )

n i

a

i

i a

n n

a

n i

a

 Present value of benefits (or costs) that grow or

decline at a constant rate in perpetuity

PV(B) = B1/ (1-g), if i>g

For most long lived projects, select a relatively short discounting

period (useful life of the project) and include a terminal value to

reflect all subsequent benefits and costs.

Where T(k) denotes the terminal value.

)

( )

1 (

0

k

T i

NB NPV

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 zero

Note: Accounting depreciation should never be included as

a cost (expense) in CBA

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 annuity

factor

That has the same life as the project

Where is the annuity factor,

n i

a

NPV EANB 

n i

a

i

i a

n n

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

m

i r





1

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):

) Costs Economic

or ( Outflows Cash

Net of

PV

) Benefits Economic

or ( Inflows Cash

Net of

PV

R 

 Mutually exclusive projects of different sizes

 Not necessarily true that if RA>RB, that

project “A” is better than project “B”

Basic Rule

Problem:The Benefit-Cost Ratio does not adjust for mutually exclusive

projects of different sizes For example:

Project A: PV 0 of Costs = $5.0 M, PV 0 of Benefits = $7.0 M

NPV A = $2.0 M R A = 7/5 = 1.4 Project B: PV 0 of Costs = $20.0 M, PV 0 of Benefits = $24.0 M

NPVB = $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)

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

 The criteria may be useful when the project is subject to high level of political risk.

Pay-Out or Pay-Back

Period

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 an

economic or financial criterion

0

0 )

1 (

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

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

Second difficulty: Projects of different sizes and also strict alternatives

Project 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/KA - 2,000 = 0 or K A = 0.30

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

Hence, KA>KB

0 B

0 A

0 B

0 A

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

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

NPV : -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= 0.262

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

Hence, KA>KB

0 B

0 A

0

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

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

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

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

Hence, NPV > NPV

IRRA : -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

0 B

0 A

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

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

IRR FOR IRREGULAR CASHFLOWS

For Example: Look at a Private BOT Project from the perspective of the

Government