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Fin650:Project Appraisal Lecture 3 Essential Formulae in Project Appraisal

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?

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Fin650:Project Appraisal Lecture 3 Essential Formulae in Project Appraisal

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  1. Fin650:Project 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 notcomparable • 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, thefuture value in one year of some amount X is given by: FV= X(1+i) wherei 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. 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 thediscount factor

  7. 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 /(1+r) +(B /(1+r) 1 +.…….+(B /(1+r) n r 1 n o o o n PVC = (C /(1+r) +(C /(1+r) 1 +.…….+(C /(1+r) r 1 n o o NPV = (B - C )/(1+r) +(B - C )/(1+r) +.…….+(B - C )/(1+r) o 1 n r o o 1 1 n n

  8. 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 1 If NPV = (B - C )(1+r) +(B - C ) +..+..+(B - C )/(1+r) and 1 n - 1 r o o 1 1 n n 3 NPV = (B - C )(1+r) +(B - C )(1+r) +(B - C )(1+r)+(B - C )+...(B - C )/(1+r) 3 2 n - 3 o o 1 1 2 2 3 3 n n r 3 1 Then NPV = (1+r) NPV 2 r r

  9. Examples of Discounting Year 0 1 2 3 4 Net Cash Flow -1000 200 300 350 1440

  10. 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:

  11. 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.

  12. 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.

  13. Calculating 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.

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

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. Alternative Investment Criteria • Net Present Value (NPV) • Benefit-Cost Ratio (BCR) • Pay-out or Pay-back Period • Internal Rate of Return (IRR)

  20. 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 project’s 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.

  21. 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?

  22. 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.

  23. 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.

  24. Net Present Value Criterion (Cont’d) 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.

  25. 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 yearswhen the interest rate is i percent, is called the annuity factor.

  26. Shortcut Methods for Calculating the Present Value of Annuities and Perpetuities 2/2 • Present value of a perpetuity PV = A/i, if i>0 • Present value of an annuity that grows or declines at a constant rate PV(B) = [B1/ (1+g)]x ai0n , i0 = 1-g/1+g if i>g If 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 perpetuity PV(B) = B1/ (1-g), if i>g

  27. 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.

  28. 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

  29. 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,

  30. 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

  31. 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):

  32. Basic Rule 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”

  33. Benefit-Cost Ratio (Cont’d) 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 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

  34. Alternative Investment Criteria 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.

  35. 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.

  36. Alternative Investment Criteria 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

  37. 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

  38. Bt - Ct +300 Time -100 -200 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/(1+1) + -200/(1+1)2 = 0 Solution 2: K = 0%; NPV= -100+300/(1+0)+-200/(1+0)2 = 0

  39. Difficulties With The Internal Rate of Return Criterion (Cont’d) Second difficulty: Projects of different sizes and also strict alternatives Ґ Year 0 1 2 3 ... ... 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% 0 NPV : 600/0.10 - 2,000 = 6,000 - 2,000 = 4,000 A NPV : 4,000/0.10 - 20,000 = 40,000 - 20,000 = 20,000 0 B Hence, NPV > NPV 0 0 B A IRR : 600/K - 2,000 = 0 or K = 0.30 A A A IRR : 4,000/K - 20,000 = 0 or K = 0.20 B B B Hence, K >K A B

  40. 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 5 Project B: Investment costs = 1,000 in year 0 Benefits = 5,200 in year 10 0 5 NPV : - 1,000 + 3,200/(1.08) = 1,177.86 A 10 0 NPV : - 1,000 + 5,200/(1.08) = 1,408.60 B 0 0 Hence, NPV > NPV B A IRR : - 1,000 + 3,200/(1+K ) = 0 which implies that K = 0.262 5 A A A IRR : - 1,000 + 5,200/(1+K ) = 0 which implies that K = 0.179 10 B B B Hence, K >K A B Difficulties With The Internal Rate of Return Criterion (Cont’d)

  41. 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 : - 1,000 + 1,500/(1.08) = 388.88 A 6 5 NPV : - 1,000/(1.08) + 1,600/(1.08) = 327.68 B 0 0 Hence, NPV > NPV A B IRR : - 1,000 + 1,500/(1+K ) = 0 which implies that K = 0.5 A A A 6 IRR : - 1,000/(1+K ) + 1,600/(1+K ) = 0 which implies that K = 0.6 5 B B B B Hence, K >K B A Difficulties With The Internal Rate of Return Criterion (Cont’d)

  42. IRR FOR IRREGULAR CASHFLOWS For Example: Look at a Private BOT Project from the perspective of the Government

  43. 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

  44. 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)

  45. NPV for Three Alternative Projects

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

  47. “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

  48. 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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