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Title: Fin650:Project Appraisal


1
Fin650Project Appraisal Lecture 3 Essential
Formulae in Project Appraisal
2
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?

3
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.

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

5
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(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.

6
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7
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(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

8
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9
Discounting 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
10
Discounting 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
11
Examples of Discounting
Year 0 1 2 3 4 Net Cash Flow
-1000 200 300 350 1440
12
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

13
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
  • 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.

14
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.

15
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.

16
Calculating NPV and IRR With Excel -- The Excel
Worksheet.
17
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( B6E6, B13, B14)
  • Payback there is no Excel formula . The
    payback year can be found by inspection of
    accumulated annual cash flows.

18
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.

19
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.

20
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.

21
Alternative Investment Criteria
  1. Net Present Value (NPV)
  2. Benefit-Cost Ratio (BCR)
  3. Pay-out or Pay-back Period
  4. Internal Rate of Return (IRR)

22
Net 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
    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.

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?

24
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.

25
Net 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.

26
Net 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.

27
Shortcut 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.

28
Shortcut 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 1-g/1g
  • if igtg
  • If g is small, B1/1g is approximately equal to
    B1,
  • and i0 1-g
  • Present value of benefits (or costs) that grow or
    decline at a constant rate in perpetuity
  • PV(B) B1/ (1-g), if igtg

29
Long-Lived 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.

30
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

31
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,

32
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

33
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)

34
Basic Rule
  • If benefit-cost ratio (R) gt1, 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 RAgtRB, that project
    A is better than project B

35
Benefit-Cost Ratio (Contd)
  • ProblemThe 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 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
    RAgtRB, 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

36
Pay-out or Pay-back period
Alternative Investment Criteria
  • 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.

37
Pay-Out or Pay-Back 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 pay-out period.
  • The criteria may be useful when the project is
    subject to high level of political risk.

38
Internal 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

40
Difficulties 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

41
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42
Difficulties With The Internal Rate of Return
Criterion (Contd)
43
Difficulties With The Internal Rate of Return
Criterion (Contd)
44
IRR 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
45
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 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

46
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)

47
NPV 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
48
NPV and IRR
  • The two basic capital budgeting tools
  • Note We usually prefer NPV to IRR, but IRR is a
    handy tool

49
Yes-No 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

50
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 projects
    risk

51
Yes-No 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

52
Basic Yes-No example
  • This project is worthwhile by both NPV and IRR
    rules
  • NPV gt 0
  • IRR gt discount rate of 12

53
Basic Ranking example
  • Yes-No 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

54
Excels 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

55
Summing 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)

58
When IRR and NPV conflict, use NPV
  • Why IRR gives the rate of return
  • NPV gives the wealth increment

59
Back to last example Calculating the crossover
point
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.
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