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Course: MIME 310, Fall 2009School: McGill
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McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria &quot;How do you make a million? You start with $900 000.&quot;, Stephen Lewis Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is...
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McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria "How do you make a million? You start with $900 000.", Stephen Lewis Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is economically justified? Non-discounted cash flow criteria Total Cash Flow Accounting Rate of Return Payback Period Discounted cash flow criteria Discounted Payback Period Net Present Value Present Value Ratio (Profitability Index) Equivalent Annual Value Benefit-Cost Ratio Internal Rate of Return Project ranking Section 3: Examples 1 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Introduction Aspects to be considered in project evaluation and decisionmaking: Benefits and costs occur over time Only cash items should be considered (no deferred income or expenses) Decisions are generally irreversible Estimates are uncertain; therefore, there is economic risk associated with capital investment projects There is a need for a minimum acceptable return on investment to guarantee the creation of wealth Cost of capital Opportunity cost of money (alternate consideration) The selection of projects is based on the objective of maximisation of investor wealth 2 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Introduction The overall objective of an investor may be stated as follows: Accumulate as much wealth as possible, as rapidly as possible, using the least amount of capital as possible, obtained from the lowest cost sources as possible. Discounted Cash Flow (DCF) methods are evaluation techniques that recognize these aspects. However, They cannot deal with non-monetary aspects The emphasis placed on short-term costs and/or benefits... this presents difficulties when dealing with long-term issues such as conservation, pollution and education 3 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is economically justified? Non-discounted cash flow criteria Total Cash Flow Accounting Rate of Return Payback Period Discounted cash flow criteria Discounted Payback Period Net Present Value Present Value Ratio (Profitability Index) Equivalent Annual Value Benefit-Cost Ratio Internal Rate of Return Project ranking Section 3: Examples 4 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Cash Flow Cash Flow DCF evaluation techniques are based on the concept of cash flow. CF = Inflows of cash - Outflows of cash (per period of time) Inflow (benefits) Sales (or reduction in costs, i.e. benefits) Disposal of assets Capital Expenditures Operating Expenses Tax Payments Rehabilitation expenditures Outflows (costs) 5 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Cash Flow Determination of Cash Flow in a Profit-based Tax System Revenue - Operating expenses = Net income before allowances and taxes (EBDT) - Depreciation (for tax purposes) - Other non-cash tax allowances Operating = Taxable income Profit - Taxes = Net Income CF = Revenue - Operating Expenses - Taxes - Capital Expenditure CF = Net Income + Depreciation + Other Non-cash Tax Allowances - Capital Expenditure 6 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Cash Flow Cash Flow Projects are described by their cash flows over time, i.e., their Time Distribution of Cash Flows or Cash Flow Profile The end-of-period convention is used to establish a time distribution of cash flows Therefore, cash flows range from times 0 to n, n being the project life The cash flow at time 0 is a lump-sum amount, whereas all others occur over a period of time There is NO period 0 in a time distribution of cash flows 7 McGill Example Period Time 0 1 2 3 4 5 6 7 8 9 10 Totals Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Cash Flow Capital Expenditure 1 500 5 000 6 000 500 -1 000* 12 000 Revenue 8 000 8 000 8 000 8 000 8 000 8 000 8 000 8 000 64 000 Operating Expenses 3 000 3 000 3 000 3 000 3 000 3 000 3 000 3 000 24 000 Tax Payments 0 0 0 0 1 000 1 000 1 500 2 000 2 500 2 500 10 500 CASH FLOW -1 500 -5 000 -6 000 5 000 5 000 4 000 4 000 3 000 3 000 2 500 3 500 17 500 * Benefits received from the disposal of equipment and the recovery of operating funds included in the original capital expenditures Pre-production (construction) period: 2 periods Production period: 8 periods Project life: 10 periods 8 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 1: Cash Flow Time Distribution of Cash Flows Periods of time 1 | 0 2 | 1 3 | 2 ... | 3 | 4 | 5 ... | 9 | 10 Time Points in time Pre-Production Period Production Period 9 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is economically justified? Non-discounted cash flow criteria Accounting Rate of Return Total Cash Flow Payback Period Discounted cash flow criteria Discounted Payback Period Net Present Value Present Value Ratio (Profitability Index) Equivalent Annual Value Benefit-Cost Ratio Internal Rate of Return Project ranking Section 3: Examples 10 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Accounting Rate of Return (ARR) Also referred to as accounting "Return on Investment" (ROI) ARR = Average Annual Net (After-tax) Income Average Capital Investment Equivalent to accounting "Return on Total Assets", which is determined on an annual basis Net Income Total Assets Average over year Project Selection Project with higher ARR is preferred Disadvantages/Weaknesses Based on accounting elements, not cash flow Ignores time value of money and timing of cash flows 11 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria The Meaning of Average Investment If investment does not lose value over time Average Initial investment Terminal book value i.e. Salvage Value Start End 12 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria The Meaning of Average Investment If investment does lose value over time Average Initial investment Assume linear loss in value Terminal book value i.e. Salvage Value Start End 13 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Example Accounting Rate of Return Project Initial Investment Terminal Book Value Year 1 2 3 4 5 6 Total Life Average Net Income Average Book Value ARR A 10 000 0 B 10 000 0 Net After-tax Income 1 500 1 500 1 500 1 500 1 500 1 500 9 000 6 1 500 5 000 C 10 000 2 000 200 1 500 1 500 2 000 2 300 7 500 5 1 500 5 000 1 500 1 500 1 500 1 500 3 500* 9 500 5 1 900 6 000 1500 / 5000 = 30% 1500 / 5000 = 30% 1900 / 6000= 31.6% 14 * Terminal book value included as salvage value McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Total Cash Flow (TCF) CFt n t=0 In our cash flow example, TCF = 17 500 Project Selection Project should have a positive total cash flow Disadvantages/Weaknesses Ignores time value of money, timing of cash flows and project life Does not measure investment efficiency CF 15 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Payback Period (PP) [D lai de r cup ration] Time required to recover initial investment from production period cash flows, measured from the beginning of production. In our example, initial investment: 1500 + 5000 + 6000 = 12 500 Production Production Period Year Cash Flows 1 2 3 5000 5000 4000 Cumulative Production Period Cash Flows 5 000 10 000 (less) (less) 2+... 14 000 (more) Payback Period: 2 + 12 500 - 10 000 = 2.6 years 4000 or 3 - 14 000 - 12 500 = 2.6 years 4000 CF 16 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Payback Period Graphical Representation +CFs Initial Investment (12 500) Payback period | | 1 | 2 | 3 17 Start of production McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Example Payback Period Weaknesses Project Period Time 0 1 2 3 4 5 6 Payback Period (yrs) CFs -600 100 100 200 200 400 400 4.0 600 250 A Cum. CFs -600 -500 -400 -200 0 400 800 B CFs -600 250 250 250 2.4 CFs -200 -400 100 150 200 250 3.6 C Cum. CFs -200 -600 -500 -350 -150 100 - 4 + (150) - 1 250 According to the payback period, project B should be selected, but from simple inspection, project A is financially superior. 18 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Non-discounted Cash Flow Criteria Payback Period Project Selection Project with shortest payback period is preferred Disadvantages/Weaknesses Ignores time value of money and timing of cash flows Ignores cash flows beyond payback period Does not measure investment efficiency 19 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is economically justified? Non-discounted cash flow criteria Total Cash Flow Accounting Rate of Return Payback Period Discounted cash flow criteria Discounted Payback Period Net Present Value Present Value Ratio (Profitability Index) Equivalent Annual Value Benefit-Cost Ratio Internal Rate of Return Project ranking Section 3: Examples 20 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Discounted Payback Period Identical to payback period, but based on discounted cash flows. In our cash flow example, Period Time 0 Cash Flow Present Value @ 10% -1500 -5000 -6000 5000 5000 4000 4000 -1500 -4545 -4959 3757 3415 2484 2258 3+... 11 004 1 2 3 4 5 6 CF Cumulative + Cash Flows 3 757 (less) 7 172 (less) 9 656 (less) 11 914 (more) 21 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Discounted payback period: 3 + (11 004 - 9656) = 3.6 yrs 2258 or 4 - (11 914 - 11 004) = 3.6 yrs 2258 Project Selection Project with shortest discounted payback period is preferred Disadvantages/Weaknesses ignores cash flows beyond discounted payback period does not measure investment efficiency 22 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Net Present Value (NPV) [Valeur actuelle nette (VAN)] Net Present Worth (NPW) [ CFt / (1 + i)t ] Cost of Capital Opportunity Cost MARR n t=0 Discount rate i MARR: Minimum acceptable return on investment 23 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Our Example Net Present Value Period Time 0 1 2 3 4 5 6 7 8 9 10 Total Cash Flow -1500 -5000 -6000 5000 5000 4000 4000 3000 3000 2500 3500 17 500 Present Value @ 10% -1500 -4545 -4959 3757 3415 2484 2258 1539 1400 1060 1349 NPV @ 10%: 6258 NPV @ 5%: 10 770 CF Cost of capital as a monetary value 17 500 6258 = 11 242 17 500 10 770 = 6730 24 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Net Present Value With the BA II Plus Use Cash Flow worksheet [CF] [2nd] [CLR WORK] Enter -1500 in CF0 Enter -5000 in C01 and 1 in F01 Enter -6000 in C02 and 1 in F02 Enter 5000 in C03 and 2 in F03 Enter 4000 in C04 and 2 in F04 Enter 3000 in C05 and 2 in F05 Enter 2500 in C06 and 1 in F06 Enter 3500 in C07 and 1 in F07 Note that when some of the cash flow frequencies are greater than 1, the `XX' in CXX no longer represents the time at which a cash flow occurs. [2nd] [QUIT] [NPV] Enter 10% for I [ ] [CPT] Result: 6257.73 [ ] Enter 5% for I [ ] [CPT] Result: 10 770.37 CF 25 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Net Present Value Project Selection Accept project if NPV is greater than or equal to 0 (implementing a project with a NPV of zero expands the business without creating additional wealth) Characteristics Measures absolute wealth creating capacity of a project Does not measure investment efficiency, i.e. largest NPV does not necessarily indicate most efficient investment 26 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Present Value Ratio (PVR) PVR = NPV | Discounted Preproduction Period CFs | In our cash flow example, PVR @ 10%: 6 258 = 0.57 11 004 Discounted investment Project Selection Accept project if PVR greater than or equal to 0 Characteristics Measures investment efficiency Useful ranking tool in situations in which funds are limited 27 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Profitability Index (PI) Variant of the present value ratio (PI = 1 + PVR) PI = Discounted Production Period CFs | Discounted Preproduction Period CFs | PI @ 10%: 17 262 = 1.57 11 004 Project Selection Accept project if PI greater than or equal to 1 Characteristics Measures investment efficiency Useful ranking tool in situations in which funds are limited 28 In our cash flow example, McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Equivalent Annual Value (EAV) Convert NPV into annualized value (i.e. ordinary annuity) EAR Equivalent annual return (+ or -) EAC Equivalent annual cost Also used for comparing projects on a cost basis 29 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Equivalent Annual Value In our cash flow example, EAR or EAR = 6258 (F/P,10%,2) (A/P,10%,8) = 6258 (1.2100) (0.1874) = 1419 Over productive life = NPV (A/P,10%,10) = 6258 (0.1627) = 1018 Over total life 30 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Equivalent Annual Value Project Selection Accept project if EAR greater than or equal to 0, or project with lowest EAC Characteristics Does not measure investment efficiency Assumption of indefinite requirement for project comparison 31 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Equivalent Annual Value Example Comparing projects on a cost basis Machine Investment ($) Operating Expenses ($/year) Life (years) Salvage Value ($) A 10 000 2 000 5 2 200 B 12 000 1 700 6 2 500 A: 10 000 (A/P,10%,5) + 2000 - 2200 (A/F,10%,5) = $4278 B: 12 000 (A/P,10%,6) + 1700 - 2500 (A/F,10%,6) = $4131 32 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Benefit Cost Ratio (BCR or B/C) [Rapport b n fices-co ts] BCR = PW (Benefits) PW (Costs) In our example, REV SV BCR: = 1.213 35 272 + 386 11 004 + 257 + 13 227 + 4912 CAPEX CAPEX OC TX Can be subtracted from CAPEX in denominator as well. In this case, BCR: 35 272 = 1.216 29 014 Note: All values discounted to time 0 CF 33 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Project Selection If BCR greater than or equal to 1, project is acceptable Characteristics Used in the analysis of government projects, in which the benefits do not accrue to those bearing the costs In such analyses, requires the use of a "social discount rate" (i.e. opportunity cost to society) instead of the cost of capital Magnitude of BCR changes with respect to how certain components are classified, e.g. a reduction in cost to the government as opposed to a reduction in benefits to society. Thus, the BCR should not be used as a project ranking tool. 34 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Example Construction of an Elevated Highway through Town Construction Cost $50 million over a 2-year period (uniform) Maintenance Cost $2.5 million/year Benefits: Less travel time for commuters $3 million/year Less gasoline consumption $5 million/year Analysis Period 20 years Discount Rate 5% 35 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria PW of Benefits: 8 (P/A, 5%, 20) (P/F, 5%, 2) = 90.4 12.4622 0.9070 PW of Costs: 25 (P/A, 5%, 2) + 2.5 (P/A, 5%, 20) (P/F, 5%, 2) = 74.8 1.8594 12.4622 0.9070 BCR: 90.4 / 74.8 = 1.21 Construction is justified 36 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return (IRR) [Taux de rendement interne (TRI)] Discount rate to apply to the time distribution of cash flows to produce a Net Present Value equal to 0, i.e., IRR = r, such that n t=0 [ CFt / (1 + r)t ] = 0 Note: The designation "Internal Rate of Return" is very often simplified to "Rate of Return" (ROR). Another designation sometimes used is "DCF Rate of Return" (DCFROR). 37 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria In our cash flow example, NPV at NPV 17 500 5% 10% 15% 20% 25% = 10 770 = 6 258 = 3 164 = 1 003 = -529 23% 10 770 6258 Rate of Return | 0 | 5 | 10 | 15 3164 | 20 1003 - 529 | | 30 | 35 38 Discount rate (%) McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria For more precision, NPV at 23% = 23 24% = -262 By linear interpolation, 23% + 1% NPV 23 = 23.1% 23 [23 - (-262)] Rate of Return: 23.1% 0 | 23 | 24 Discount rate (%) There is no need to use linear interpolation when a financial calculator is available. 39 - 262 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return With the BA II Plus Use Cash Flow worksheet [CF] [2nd] [CLR WORK] Enter -1500 in CF0 Enter -5000 in C01 and 1 in F01 Enter -6000 in C02 and 1 in F02 Enter 5000 in C03 and 2 in F03 Enter 4000 in C04 and 2 in F04 Enter 3000 in C05 and 2 in F05 Enter 2500 in C06 and 1 in F06 Enter 3500 in C07 and 1 in F07 [2nd] [QUIT] [IRR] [CPT] Result: 23.07% 40 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return Project Selection If IRR MARR, then accept project If IRR < MARR, then reject project Note: The MARR is sometimes referred to as a "Hurdle Rate". 41 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return The meaning of rate of return: Return on investment, similar to interest rate received on bank deposit Measure of investment efficiency Greatest cost of capital that can be incurred without financial loss to the investor Special case if initial investment is at time 0 and subsequent positive cash flows are constant over the entire life of the project: the TVM keys on the BA II Plus can be used to find the IRR in this case. 42 McGill Example Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria $30 000 0 | | 1 | 2 | 3 | 4 | 5 | 6 Time $100 000 With the BA II Plus [2nd] [CLR TVM] Enter 5 in N Enter -100 000 in PV Enter 30 000 in PMT [CPT] I/Y Result: 15.24% 43 NPV = -100 000 + 30 000 (P/A, r, 5) = 0 (P/A, r, 5) = 3.3333 At 15%, (P/A) = 3.3522 16%, (P/A) = 3.2743 r=15.2% by linear interpolation McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return Problem of Multiple Rates of Return Summation of discounted CFs is a polynomial expression The IRR being a root of this expression, there may be several roots Descartes' Rule There may be as many real positive roots as there are changes in sign in the polynomial expression (when ordered in increasing or decreasing order of powers) Cash Flow Time 44 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Internal Rate of Return Problem of Multiple Rates of Return Cash Flow 0 Time NPV 3 real roots 0 Discount rate 45 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Financial Functions in EXCEL NPV (discount rate, range of CFs) Discount rate (i) must be specified in decimal Cash flow in first cell of range is assumed to be at t=1 unless beginning-of-period option is on If there is a cash flow at t=0, there are 3 possible options: leave CF0 out of cell range and add to NPV function result [i.e. NPV(discount rate, CF1 to n) + CF0] leave CF0 in cell range and turn beginning-of-period option on leave CF0 in cell range and multiply NPV function result by (1 + i) [i.e. NPV(discount rate, CF0 to n) (1 + discount rate)] IRR (range of CFs, discount rate) Discount rate is an initial guess; use cost of capital CF0 must be left in cell range 46 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Discounted Cash Flow Criteria Financial Functions in EXCEL PV (discount rate, n, payment) PMT (discount rate, n, present value) FV (discount rate, n, payment) RATE (n, payment, present value) NPER (discount rate, payment, present value) See EXCEL help to learn about other useful functions. (P/A, i, n) (A/P, i, n) (F/A, i, n) (i/P, A, n) (n/P, A, i) 47 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whether a project is economically justified? Non-discounted cash flow criteria Total Cash Flow Accounting Rate of Return Payback Period Discounted cash flow criteria Discounted Payback Period Net Present Value Present Value Ratio (Profitability Index) Equivalent Annual Value Benefit-Cost Ratio Rate of Return Project ranking Section 3: Examples 48 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking Use of DCF Criteria for Ranking and Selecting Projects Under conditions of limited funds, problems arise when selecting among independent1 projects with different investment requirements and/or different lives. As well, the selection of one project among a series of mutually exclusive2 projects must be handled correctly. 1 Unrelated projects any combination may be selected within a given budget constraint. The selection of one project excludes the selection of any other. 2 49 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking Example Projects with Different Investment Requirements Project Life (yrs) Capital Expenditure (t=0) NPV @ 10% A 5 1000 400 B 5 300 200 Based on NPV alone, project A is preferable. Based on PVR, (0.4 for A, and 0.666 for B), project B is preferable. 50 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking The ultimate decision depends on assumptions regarding the 700 unit difference in investment, and the availability of funds and other projects... What amount of funds is available? If there is not enough, can more be obtained, and at what cost? Are there other projects available? The usual objective is to maximize wealth on available funds. For instance, if the budget available is $1000 and only projects A and B are available, select A to maximize overall NPV. However, if three projects of type B are available, select these to maximize overall NPV. 51 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking Example Projects with Different Lives Project Life (years) Capital Expenditure (t=0) Annual Benefits Salvage Value NPV@ 10% A 3 5000 3500 1000 4456 B 4 5000 3000 700 4988 Based on NPV alone, project B is preferable. But what happens in year 4 if project A is selected? Are funds left idle or are they invested elsewhere? 52 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking The standard approach in this situation is to consider the projects over a common analysis period... 12 years for instance. 4456 A | 0 | 3 | 6 | 9 | 12 Four identical three-year projects, one after the other NPVA = 4456 + 4456 (P/F,10%,3) + 4456 (P/F,10%,6) + 4456 (P/F,10%,9) = 12 209 * 4988 B | 0 | 4 | 8 | 12 Three identical four-year projects, one after the other NPVB = 4988 + 4988 (P/F,10%,4) + 4988 (P/F,10%,8) = 10 722 53 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking Example Mutually Exclusive Projects Project Time 0 1 2 3 4 5 TCF NPV @ 10% IRR (%) A -12 000 5 000 5 000 5 000 5 000 6 000 14 000 7 575 31.9 B -20 000 8 000 8 000 8 000 8 000 10 500 22 500 11 879 30.4 B-A -8 000 3 000 3 000 3 000 3 000 4 500 8 500 4 304 28.2 B preferred B preferred Incremental Cash Flows Only one project may be selected... Which is preferable? 54 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking Why are NPVs and IRRs in conflict? 25 000 NET PRESENT VALUE ($) 20 000 Project B 15 000 NPVB @ 10% NPVA @ 10% 10 000 Project A 5 000 Cross- over rate NPVA = NPVB IRRB-A IRRA 0 IRRB -5 000 -10 000 0 5 10 NPVB > NPVA 15 20 25 NPVA > NPVB 30 35 40 45 DISCOUNT RATE (% ) 55 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Section 2: Investment Criteria Project Ranking In summary, for mutually exclusive projects, NPV criterion can be applied directly project B is preferable Internal Rate of Return criterion must not be used directly it must be applied in combination with an incremental analysis approach. Incremental IRR of 28.2% of B over A indicates that B is preferable. The same rule applies for all measures of investment efficiency (i.e. PVR, PI and BCR). Assumption made that enough funds are available... 56 McGill Faculty of Engineering MIME 310 <a href="/keyword/engineering-economy/" >engineering economy</a> Chapter 6 Project Evaluation Criteria Section 1: Introduction Cash Flow A formal definition Section 2: Investment Criteria How do we judge whethe...
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Attribution, Learned Helplessness, DepressionNovember 24, 2008Attributions: Understanding the Behavior of Others Kelley's (1965) Attribution Theory Person Dimension: Consensus Situation Dimension: Consistency Entity (Stimulus) Dimension: Disti
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Mathematical Reasoning and Problem Solving Syllabus Mathematical Reasoning Mathematical language and symbols. Definitions. Theorems: hypothesis and conclusion, converse and contrapositive. Proofs: direct proof and proof by contradiction.
East Los Angeles College - HOMEPAGEMA - 191
MATH191: Problem Sheet 8Due Thursday 4th December1. Let f (x) be dened by f (x) = x2 + 2xx x2if x 0 if x > 0.Dierentiate f (x) for x = 0, 2. Find and classify any stationary points, determine any zeros of f (x), and any horizontal and vertic
East Los Angeles College - HOMEPAGEMA - 191
MATH191: Problem Sheet 5Due Thursday 6th November1. Dierentiate the following functions. This is a no mercy question: you will get no marks for an incorrect answer, no matter how minor the mistake. There is no need to simplify your answers and if
East Los Angeles College - HOMEPAGEMA - 191
MATH191: Problem Sheet 2Due Thursday 16th October1. Determine the inverse function f 1 (x) of the rational function f (x) = 2x . x+12. State whether each of the following functions is increasing, decreasing, or neither (on their maximal domains)
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Problem Sheet 6: QuantifiersDue Friday 6 March 1. Determine whether each of the following statements is true or false. For those which are true, explain briefly why (for example, if the statement starts x, give a value of x; if it starts x,
East Los Angeles College - HOMEPAGEMA - 104
MATH104: Mathematical Reasoning and Problem Solving. MATH104a Mathematical Reasoning Professor Mary Rees Weeks 1-6 (until 6 March) MATH104b Problem Solving Dr R. Nair Weeks 7-12 (from 9 March) Lectures Monday 2pm, Tuesday 3pm, Friday 11am Room 02
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Problem Sheet 2: The language of mathematicsDue Friday 6th FebruaryName:You should hand in this problem sheet with your solutions written in. If you modify your answers, please make sure that its clear what your nal answer is. Feel free t
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Problem Sheet 3: DefinitionsDue Friday 13th February Please note that marks will be awarded for the mathematical structure and clarity of your solutions. 1. Show directly from the definitions that a) f (x) = 1 - 4x is injective; b) f (x) = c
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Problem Sheet 4: TheoremsDue Friday 20th February 1. Identify the context, hypothesis, and conclusion of the following: Theorem 1 Let x be a real number. If x2 > 1 then x > 1 or x < -1. State its converse and contrapositive. Is the converse
East Los Angeles College - HOMEPAGEMA - 104
MATH104 Problem Sheet 1: IntroductoryDue Friday 30th January 1. Kelloggs new promotional campaign announces that Every pack of Kelloggs cornakes contains a pink plastic dinosaur. Assume that their claim is true (they have destroyed all old stock, th
East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 4 - Solutions sin x for x < 1. To make f ( x ) = differentiable at x = , we first arrange for it to be mx + c for x continuous. The left limit at x = is sin = 0 ; the right limit and the value are m + c . So we ne
Wilfrid Laurier - CPSC - 503
The MessageHow to Give PresentationsPrepare yourselfknow your message know your audience & venue practice, practice, practiceTypical presentationstop-down structure keep it simple use media effectivelySaul GreenbergUniversity of Calgary
East Los Angeles College - MATH - 101
MATH101 January 2008 Note: there is no bookworkin this paper. All questions are similar to ones set in homework, class tests and past papers. SECTION A1. Sketch the function in each interval, noting which end points are open and which are closed: [
East Los Angeles College - MATH - 101
MATH101 - Foundation Module I - CalculusModule NotesWe hope you enjoy this mathematics module. It is the first of three `Foundation Modules' which are taken by all year one students following any of our Mathematics-based degree programmes. This one
East Los Angeles College - MATH - 101
MATH101 CLASS TEST solutions 20071. (a) For y = |x + 2|, the natural domain is (-, ) and the range is [0, ). [2 marks](b) For y = 1/(x - 2)2 the natural domain is (-, 2) (2, ) and the range is (0, ). [2 marks][2 marks] 2. We have 1 cos = =
East Los Angeles College - MATH - 101
MATH101 CLASS TEST November 2007 1. Write down the natural domain and range of each of the following functions and sketch their graphs: (a) y = |x + 2|, 2. Find the general solution of 1 cos = . 2 [4 marks] 3. Find the inverse function f -1 (x) of
East Los Angeles College - MATH - 101
MATH101 Tutorial Solution Sheet 91.(i) y = (ln x)2 , so dy ln x =2 . dx x (ii) Integrating by parts: (ln x)2 dx = 1.(ln x)2 dx = x(ln x)2 xd(ln x)2 ln x dx .= x(ln x)2 Integrating by parts again:x 2 ln x/x dx = x(ln x)2 2(ln x)2 dx = x(
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 10 1. Herex-6 , h = (2 - 1)/10 = 0.1 . x2 - 3x Construct a table for both Trapezium (T) and Simpson (S) rules: y = f (x) =i x_i y_i c_i(T) c_i(S) -0 1 2.5 1 1 1 1.1 2.3444976 2 4 2 1.2 2.2222222 2 2 3 1.3 2.1266968
East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 1 Solutions 1(a) y = ( x - 3) 3 domain (- , ) range (- , ) not periodic1 (b) y = 3 cos 2 xy 34domain (- , ) range [- 3, 3]x-33 cos 2 x = 3 cos(2 x + 2 ) = 3 cos 2( x + ) periodic with period 1 (c) y = x
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 81. If f (x) = e(x1) + 1, f (x) = 2(x 1)e(x1) = 0 So only one extremum at most. f (x) = 2e(x1) + 4(x 1)2 e(x1) = 22 2 22whenx = 1 only.atx = 1.Hence sketch:so x = 1 (where y = 2) is a maximum and f
East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 3 - Solutions1.(Box instead of filled point)not continuous at x=-1: lim f ( x) = 0 = f (-1) lim f ( x) = 2 - +x -1x 0x -1continuous at x=0:lim f ( x) = 0 = f (0)not continuous at x=1: f (1) undefined con
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 31. Sketch the function in each interval, noting which end points are open and which are closed:From the denition, and the graph we see that the function is continuous at x = 1 , 0, 1 and 3 but not at x = 2 where
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 51. (a) Can rewrite the equation as: (x 1)2 y 2 + 2 =1 22 1 which is an ellipse centred at (x, y, ) = (1, 0). Note that at x = 0, 4y 2 = 3 so y = 3 ; at x = 1, y = 1 ; at y = 0, x 1 = 2, so 4 x = 1 or 3 . Use impl
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 5 - Solutions 1. Take a, b>0 dy/dx=0dy/dx=0x 2 / a 2 + y 2 / b 2 = 1; 2 x / a 2 + ( 2 y dy dx) / b 2 = 0; dy dx = ( xb 2 ) /( ya 2 ) dy dx = 0 for x = 0 so at (0, b) & (0,b)x 2 / a 2 y 2 / b 2 = 1; 2 x / a 2 ( 2 y
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 1 1. (a) Start with the graph of y = x3 which is an odd function going through the origin at which point it has zero slope. The graph of y = (2 x)3 is the same but shifted by +2 along the x-axis. The resulting functio
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
MATH101: Foundation Module I Some theorems related to continuity and dierentiability 1 Intermediate Value TheoremIf f (x) is continuous on the interval [a, b] then it takes every value between f (a) and f (b). i.e. if y0 is between f (a) and f (b),
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 6 - Solutions1 2 miles* Staten Island x y 6 miles *BloomingdalesTime T = ( x / 20) + ( y / 50) . There is a constraint expressed via Pythagoras as 2 2 + (6 - y ) 2 = x 2 and so1 1 we can write T = 100 (5 x + 2 y ) =
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 91.(i) y = (ln x)3 , so 3(ln x)2 dy = . dx x (ii) Integrating by parts twice (writing (ln x)n = 1 (ln x)n ) : I = (ln x)3 dx = 1.(ln x)3 dx 1 dx x x 2 ln x ln x dx .= x(ln x)3 - 3 (ln x)2 x = x(ln x)3 - 3 x(
East Los Angeles College - MATH - 101
MATH101 Homework Solution Sheet 4 1. For f to be dierentiable it must also be continuous. So, for continuity at x = 0 we require that the left and right limits of f should be the same and equal to f (0). For dierentiability, the left and right deriva
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 6 - 2008 These are for discussion during the weekly problems class. (1) Phoebe and Monica are on Staten Island, but have promised to meet Rachel in Bloomingdales. They can take a boat to any point on the shore and then
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 3 - 2008 These are for discussion during the weekly problems class. (1) Draw the graph of the function f dened by: 0 for x 1, |2x| for 1 < x < 0, f (x) = x2 for 0 x < 1, 2 x + 4x 2 for 1 < x 3, 1/(x 3) for
East Los Angeles College - MATH - 101
MATH101 HOMEWORK SHEET 9 - 2008 Hand in your solutions to your MATH101 tutor by 5pm on Monday 8 December. (1) (i) Differentiate (ln x)3 .(ii) By writing (ln x)3 = 1.(ln x)3 and integrating by parts three times find (ln x)3 dx . (iii) Verify your an
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
MATH101 HOMEWORK SHEET 6 - 2008 Hand in your solutions to your MATH101 tutor by 5pm on Monday 10 November. (1) A small rectangular plot of farmland with an area of 216 m2 is to be enclosed by a fence and divided into two equal equal plots by another
McGill - ECSE - 221
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East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 8 - 2008 These are for discussion during the weekly problems class. (1) Show that the function f (x) = ex - x has a local extremum and determine its nature. Sketch the graph of the function. (2) Find the local extrema a
East Los Angeles College - MATH - 101
MATH101 HOMEWORK SHEET 8 - 2008 Hand in your solutions to your MATH101 tutor by 5pm on Monday 1 December. (1) Establish whether the function f (x) = e(x1) + 1 has any local extrema and, if so, determine their nature. Sketch the graph of the function,
McGill - ECSE - 110234458
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Sveriges lantbruksuniversitet - BUS - 312
Pitfalls with IRR Lending vs Borrowing Calculatethe IRR and NPV for the projects below:Cash Flows in DollarsProject: J KC0 -100 +100C1 +150 -150IRR NPV @ 6% 50% + $36.4 50% - $36.4Both projects have the same IRR . but Project J contri
BYU - ECE - 777
BeamformingA brief introductionBrian D. Jeffs Associate Professor Dept. of Electrical and Computer Engineering Brigham Young UniversityMarch 2006ReferencesBarry D. Van Veen and Kevin Buckley, "Beamforming: A Versatile Approach to Spatial Filte
Sveriges lantbruksuniversitet - LING - 19972
PROFILE OF STUDENTS IN SFU COURSES COURSE: LING 100-3 E01 LOCATION: DOW TITLE: CMNS & LANGUAGE SECTION TYPE: LEC SEMESTER: 1997-2 ENROL: 10
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 2 - 2008 These are for discussion during the weekly problems class. (1) The function h is defined by h(x) = Ax + 2 , x-1 x = 1.where A is a constant, (A = -2). Find the corresponding inverse function h-1 (x). Find a v
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 7 - 2008 These are for discussion during the weekly problems class. (1) Show that the equation of the tangent to the curve y = cos x at the point (a, cos a) can be written y = cos a (x a) sin a. Hence derive a linear
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 1 - 2008 These are for discussion during the weekly problems class. (1) Sketch the graphs and state the natural domain and range of the following functions: (a) y = (x 3)3 (b) y = 3 cos 2x (c) y = |x 1| (d) y = |x| 1
East Los Angeles College - MATH - 101
MATH101 Tutorial Problem Sheet 5 - 2008 These are for discussion during the weekly problems class. (1) Sketch the following curves: (a) x2 y2 + 2 =1 a2 b (an ellipse) (b) x2 y2 - 2 =1 a2 b (a hyperbola),(where a, b are constants) indicating (in the
East Los Angeles College - MATH - 101
M101 Tutorial Problem Sheet 2 - Solutions 1.y = ( Ax + 2) ( x 1); ( x 1) y = Ax + 2; xy y = Ax + 2; x( y A) = y + 2; x = ( y + 2) ( y A); h 1 ( x) = ( x + 2) ( x A) domain (, A) ( A, ) range (-,1) (1, )h 1 ( x) = h( x) is ( x + 2) ( x-A) =