Lecture - 6 Time Value of Money - Basic Analysis Part 1

# Lecture - 6 Time Value of Money - Basic Analysis Part 1 -...

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Unformatted text preview: CE 395 Engineering Economics Spring 2005 W. Hitchcock Time Value of Money Analysis Basic Applications Time Value of Money Review of Concepts Present Value Future Value Annuity Uniform Gradient Effective Interest Rate Continuous Compounding Financial Concepts - Equivalence Present Time F Future Time P Time Increments ( n time intervals) Equivalence: If we say we are indifferent to whether I have a sum of money now (P) or the assurance of some other sum of money in the future (F), we say the sums are equivalent F = P(1+i)n P = F(1+i)n Uniform Series of Cash Flows (Annuities) A A A A A A A A A A P Uniform equivalent cash flows occur at the end of each interest period in the cash flow series. P occurs one interest period before the first A. A=P i (1+i)n (1+i)n -1 Uniform Series of Cash Flows (Annuities) A A A A A A A A A A F Uniform equivalent cash flows occur at the end of each interest period in the cash flow series. F occurs at the same point in time as the last A. i A=F (1+i)n -1 Financial Concepts Equivalence Formulas A A A A A A A A A A Time Increments P F = P(1+i)n P = F(1+i)n P=A (1+i)n -1 n F A=P i (1+i)n i (1+i) Series Present Worth F=A (1+i)n -1 i (1+i)n -1 Capital Recovery i (1+i)n -1 Sinking Fund A=F Series Compound Amount Uniform Gradient 3G 4G 5G 6G 7G G 2G First cash flow at the end of the 2nd period There are N-1 cash flows in a gradient series of N intervals Uniform Gradient 3G 4G 5G 6G 7G G 2G 1 (1 + i ) N - 1 F = G - N i i F = G ( F G, i %, N ) F Uniform Gradient 2G 3G 4G 5G 6G 7G G 1 (1 + i ) N - 1 N P = G - N N (1 + i ) i i (1 + i ) P P = G ( P G , i %, N ) Uniform Gradient 3G 4G 5G 6G 7G G 2G A 1 N A = G - N i (1 + i ) - 1 A = G ( A G, i %, N ) Effective and Nominal Interest Rates Nominal interest rate ( r ) is the stated annual rate without consideration for compounding periods less than one year time value relationships over one or more periods. "Effective interest" rate is the proper interest rate for calculating If interest is compounded over time periods less than one year, the effective interest rate per period (quarter, month, day, etc.) is r divided by the # of periods per year. periods per year) the effective annual interest rate is If interest is compounded over periods less than one year ( M determined by the expression: ieffective r = (1 + ) M - 1 M Handling Cash Flows Longer Than the Compounding Period Sometimes the timing of the cash flows does not coincide with the compounding periods. Solve by spreading payments to equivalent period payments or , Determine an effective interest rate for the timing of the cash flows Continuous Compounding Discrete Cash Flow Intervals What happens if the length per compounding period approaches zero? Compounding is said to be continuous. If compound is continuous but cash flows occur over discrete periods (annually, quarterly, etc.) The following relationships hold: rN e rN - 1 e rN - 1 F = P (e ) P = A rN r e (e - 1) F = A e r - 1 P = F (e - rN ) r = nominal rate per period er -1 A = F rN e -1 e rN (e r - 1) A = P rN e -1 Continuous Compounding Continuous Cash Flow Interval If compounding is continuous and cash flows uniformly or continuously over each period we say that the cash flow is continuous. We do no the total cash flowing on a continuous basis over each period. The following time-value relationships hold: e rN - 1 e rN - 1 re rN F = A P = A A = P rN r re rN e -1 r = nominal rate per period N = number of periods r A = F rN e -1 Capital Investments How Do We Decide? Engineering Economics Investment Analysis Capital Investment Capital is available for investment Projected revenues and costs are known. Company would only invest in "profitable ventures" What is a "profitable venture"? Generally the terminology is that a "profitable venture" produces an "acceptable return or rate of return" on invested capital. Necessarily takes into account the time value of money Requires an interest rate for analysis and acceptability assessment (Minimum Attractive Rate of Return) Minimum Attractive Rate of Return (MARR) It is common for companies to establish interest return floor rates for evaluating investment alternatives. Companies may have more than one MARR depending on the categories of investment possibilities. The interest rate depends on many factors: Company purpose, mission, objectives, etc. Available cash for capital investment Cost of additional funds for the company Number of competing investment opportunities within the category Liquidity of the investment (ease of exit) Length of investment Perceived risk (uncertainty) in the investment Idealized Investment Selection Assume that available capital for investment is limited. Companies want to make the most profitable investments. Ideally choices could be ordered most attractive to least attractive. Starting with the most attractive, the company would invest in as many profitable projects as available funds will allow. Finding and Ordering Profitable Alternatives Engineering Economics Our desire is to learn methodology for rationally determining profitable investments and selecting the best ones. We will learn to evaluate individual investments We will learn to compare competing investments in search of the best alternative Basic Economic Evaluation Measures Present Worth or Value of total investment (PW) Future Worth or Value of total investment (FW) Annual Worth or Value of total investment (AW) Internal Rate of Return of total investment (IRR) External Rate of Return of total investment (ERR) Fundamental Skill Requirement Cash Flow Tables and Diagrams Probably the most difficult aspect of economic analysis is the construction of the proforma investment cash flow. Requires uncertain estimates of future unknown events. Requires accurate notation of the timing of all cash inflows and outflows Financial Decision Making We will study basic concepts in financial decision making Requires the use of cash flow models of project alternatives and criteria for evaluating them Basic Criteria: Fixed Input Cost -------- Maximize Output Value Fixed Output Value ---- Minimize Input Cost Neither Fixed ------------ Maximize (Output Input) Project Present Value Analysis Net Present Value If the cash outlays and revenues received are known or estimated for a project over an analysis period (say "N" periods) we can determine the Present Value (PV) of the costs and the Present Value (PV) of the benefits using the formulas provided (may involve several calculations). Present Value of benefits Present Value of Costs equals The Net Present Value of the Project (NPV) The NPV could be positive or negative It should be clear that the Net Present Value depends upon the interest rate utilized Present Value Analysis Criteria Application Case Fixed Input Situation Input Fixed Choice Criteria Maximize PV of benefits Minimize PV of costs or other inputs Maximize NPV Fixed Output Fixed use requirement or benefits Neither the input or other costs or the benefits are fixed Neither Fixed Net Present Value Example The cash outlays and projected benefits for a proposed project that has a five year life are shown in the table below: Year 0 1 2 3 4 5 Investment \$1000 \$2000 0 0 0 0 Net Income 0 0 \$2,500 \$3,000 \$4,000 \$4,000 What is the net present value of the project if the MARR for the company is 12% ? Assume all costs and benefits occur at the end of the interest period (year) Financial Concepts Equivalent Uniform Annual Benefit/Cost The project benefits can be converted to an equivalent uniform annual benefit (EUAB) for the life of the project typically using MARR for "i". Similar covert project costs to an equivalent uniform annual cost (EUAC). If EUAB EUAC > 0, the project is acceptable Annual Cash Flow Analysis Criteria Application Case Fixed Input Fixed Output Situation Input Fixed Fixed use requirement or benefits Neither the input or other costs or the benefits are fixed Choice Criteria Maximize EUAB Minimize EUAC Neither Fixed Maximize EUAB - EUAC ...
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