Title: Fin650:Project Appraisal
1 Fin650Project Appraisal Lecture 3 Essential
Formulae in Project Appraisal
2What is Capital Budgeting
 Two big questions
 YesNo Should you invest money today in a
project that gives future payoffs?  Ranking How to compare mutuallyexclusive
projects? If you have several alternative
investments, only one of which you can choose,
which should you undertake?
3Other 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.
4Benefits 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
5Discounting 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(1i)
 where i is the annual rate of interest. This is
simple compounding  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.
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7Discounting 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(1i)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/(1i)n is called the discount factor
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9Discounting and Alternative Investment Criteria
 Basic Concepts
 Discounting
 Recognizes time value of money
 a. Funds when invested yield a return
 b. Future consumption worth less than present
consumption
o
o
PVB (B
/(1r)
(B
/(1r)
1
..(B
/(1r)
n
1
n
r
o
o
n
o
PVC (C
/(1r)
(C
/(1r)
1
..(C
/(1r)
1
n
r
o
o
NPV (B

C
)/(1r)
o
(B

C
)/(1r)
1
..(B

C
)/(1r)
n
o
o
1
1
n
n
r
10Discounting and Alternative Investment Criteria
(Contd)
 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
1
If NPV (B

C
)(1r)
1
(B

C
) ....(B

C
)/(1r)
n

1
and
r
o
o
1
1
n
n
3
NPV (B

C
)(1r)
3
(B

C
)(1r)
2
(B

C
)(1r)(B

C
)...(B

C
)/(1r)
n

3
o
o
1
1
2
2
3
3
n
n
r
3
1
Then NPV (1r)
2
NPV
r
r
11Examples of Discounting
Year 0 1 2 3 4 Net Cash Flow
1000 200 300 350 1440
12Financial 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
13Financial 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
 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.
14Evaluation 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.
15Calculating NPV and IRR With Excel  Basics.
 Ensure that the cash flows are recorded with the
correct signs , , Tk, Tk. etc.  Make sure that the cash flows are evenly timed
usually at the end of each year.  Enter the discount rate as a percentage, not as a
decimal e.g. 15.6, not 0.156.  Check your calculations with a hand held
calculator to ensure that the formulae have been
correctly set up.
16Calculating NPV and IRR With Excel  The Excel
Worksheet.
17Calculating 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
reinvestment rate, are entered into the
formula MIRR( B6E6, B13, B14)
 Payback there is no Excel formula . The
payback year can be found by inspection of
accumulated annual cash flows.
18ARR 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.
19Calculating 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.
20Calculating 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.
21Alternative Investment Criteria
 Net Present Value (NPV)
 BenefitCost Ratio (BCR)
 Payout or Payback Period
 Internal Rate of Return (IRR)
22Net Present Value (NPV)
 The NPV is the algebraic sum of the discounted
values of the incremental expected positive and
negative net cash flows over a projects
anticipated lifetime.  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
zeroNPV 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.
23 Alternative Investment Criteria
 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?
24Net 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.
25Net Present Value Criterion (Contd)
 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.
26Net Present Value Criterion (Contd)
 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.
27Shortcut Methods for Calculating the Present
Value of Annuities and Perpetuities 1/2
 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.
28Shortcut Methods for Calculating the Present
Value of Annuities and Perpetuities 2/2
 Present value of a perpetuity
 PV A/i, if igt0
 Present value of an annuity that grows or
declines at a constant rate  PV(B) B1/ (1g)x ai0n , i0 1g/1g
 if igtg
 If g is small, B1/1g is approximately equal to
B1,  and i0 1g
 Present value of benefits (or costs) that grow or
decline at a constant rate in perpetuity  PV(B) B1/ (1g), if igtg
29LongLived Projects and Terminal Values
 It is generally assumed that projects have finite
economic  Life.
 For projects with infinite life, we may calculate
NPV using  The 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 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.
30Alternative 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
31Comparing 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,
32Real 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 im
33Alternative Investment Criteria Benefit Cost
Ratio
 As its name indicates, the benefitcost 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)
34Basic Rule
 If benefitcost ratio (R) gt1, then the project
should be undertaken.  Problems?
 Sometimes it is not possible to rank projects
with the benefitcost Ratio  Mutually exclusive projects of different sizes
 Not necessarily true that if RAgtRB, that project
A is better than project B
35BenefitCost Ratio (Contd)
 ProblemThe BenefitCost 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 M  NPVB 4.0 M RB 24/20 1.2
 According to the BenefitCost Ratio criterion,
project A should be chosen over project B because
RAgtRB, but the NPV of project B is greater than
the NPV of project A. So, project B should be
chosen  Conclusion The BenefitCost Ratio should not be
used to rank projects
36Payout or Payback period
Alternative Investment Criteria
 The payout 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.
37PayOut or PayBack Period
 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 payout period.  The criteria may be useful when the project is
subject to high level of political risk.
38Internal Rate of Return (IRR)
Alternative Investment Criteria
 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
39 Common uses of IRR
 (a) If the IRR is larger than the cost of funds
then the project should be undertaken  Often the IRR is used to rank mutually exclusive
projects. The highest IRR project should be
chosen  An advantage of the IRR is that it only uses
information from the project
40Difficulties With the Internal Rate of Return
Criterion
 First Difficulty Multiple rates internal rate of
return for  Project
 Solution 1 K 100 NPV 100 300/(11)
200/(11)2 0  Solution 2 K 0 NPV 100300/(10)200
/(10)2 0
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42Difficulties With The Internal Rate of Return
Criterion (Contd)
43Difficulties With The Internal Rate of Return
Criterion (Contd)
44IRR FOR IRREGULAR CASHFLOWS For Example Look at
a Private BOT Project from the perspective of the
Government
Year ? 0 1 2 3 4
Project A 1000 1200 800 3600 8000
IRR A 10
Compares Project A and Project B ? Compares Project A and Project B ? Compares Project A and Project B ? Compares Project A and Project B ? Compares Project A and Project B ? 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 gt IRR B Project B is obviously better than A, yet IRR A gt IRR B Project B is obviously better than A, yet IRR A gt IRR B Project B is obviously better than A, yet IRR A gt IRR B Project B is obviously better than A, yet IRR A gt IRR B Project B is obviously better than A, yet IRR A gt IRR B
Project C 1000 1200 800 3600 4800
IRR C 16
Project C is obviously better than B, yet IRR B gt IRR C Project C is obviously better than B, yet IRR B gt IRR C Project C is obviously better than B, yet IRR B gt IRR C Project C is obviously better than B, yet IRR B gt IRR C Project C is obviously better than B, yet IRR B gt IRR C Project C is obviously better than B, yet IRR B gt IRR C
Project D 1000 1200 800 3600 4800
IRR D 4
Project D is worse than C, yet IRR D gt IRR C Project D is worse than C, yet IRR D gt IRR C Project D is worse than C, yet IRR D gt IRR C Project D is worse than C, yet IRR D gt IRR C Project D is worse than C, yet IRR D gt IRR C Project D is worse than C, yet IRR D gt IRR C
Project E 1325 1200 800 3600 4800
IRR E 20
Project E is worse than D, yet IRR E gt IRR D Project E is worse than D, yet IRR E gt IRR D Project E is worse than D, yet IRR E gt IRR D Project E is worse than D, yet IRR E gt IRR D Project E is worse than D, yet IRR E gt IRR D Project E is worse than D, yet IRR E gt IRR D
45The 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 societys 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
46Does 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 backend
loaded (such as project C), strengthens the case
for projects with benefit that are frontend
loaded (such as project B)
47NPV for Three Alternative Projects
Year 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 (i2) 37,838 35,762 46,802
NPV (i10) 14,770 21,544 6,929
48NPV and IRR
 The two basic capital budgeting tools
 Note We usually prefer NPV to IRR, but IRR is a
handy tool
49YesNo and NPV
 NPV rule A project is worthwhile if the NPV gt 0
 According to the NPV rule
 If NPV gt 0, project is worthwhile
 If NPV lt 0, project should not be undertaken
50Technical 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 projects
risk
51YesNo and IRR
 IRR rule A project is worthwhile if the IRR gt
discount rate  According to the IRR rule
 If IRR gt r, then the project is worthwhile
 If IRR lt r, project should not be undertaken
52Basic YesNo example
 This project is worthwhile by both NPV and IRR
rules  NPV gt 0
 IRR gt discount rate of 12
53Basic Ranking example
 YesNo Both projects are worthwhile
 NPVA, NPVB gt 0
 IRRA, IRRB gt discount rate of 12
 Ranking If you can choose only one project, B
is preferred by both NPV and IRR  NPVB gt NPVA
 IRRB gt IRRA
54Excels NPV function
 Chapter 2 Excels 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
55Summing up
56 In this example
 Both A and B are worthwhile by both NPV and IRR
criteria  If discount rate 6
 A is preferred to B by NPV rule
 B preferred to A by IRR rule
57 IRRA is always lt IRRB By IRR rule, B is always
preferred to A  For discount rates lt 8.5128 NPVA gt NPVB
(ranking conflict)  For discount rates gt 8.51285 NPVA lt NPVB (no
ranking conflict)
58When IRR and NPV conflict, use NPV
 Why IRR gives the rate of return
 NPV gives the wealth increment
59Back to last example Calculating the crossover
point
Crossover point is the IRR of the differential
cash flows (column D)
60Essential 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.