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MER439- Design of Thermal Fluid Systems Engineering Economics Lecture 2- Using Factors Professor Anderson Spring 2012

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Some Definitions Capital: Invested money and resources Interest: The return on capital Nominal IR: the interest rate per year without adjusting for the number of compounding periods Effective IR: the interest rate per year adjusting for the number of compounding periods

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Different sums of money at different times can be equal in economic value. i.e. $100 today with i = 6% is equivalent to $106 in one year. Equivalence depends on the interest rate! Equivalence occurs when different cash flows at different times are equal in economic value at a given interest rate. Equivalence

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Cash Flow Diagrams: An Important Tool Income time Initial Capital Cost Replacement Costs Operating & Maintenance Costs Salvage Costs - Arrows up represent income or profits or payoffs - Arrows down represent costs or investments or loans - The x axis represents time, most typically in years

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Time Value of Money If $4500 is invested today for 12 years at 15% interest rate, determine the accumulated amount. Draw this. F = P(1+i) n P =Present Value (in dollars) F = Future Value (in dollars) $4500 F t=0 t=12 n = 12, i = 15%

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Factors Single Payment Compound Amount Factor (future worth) (F/P, i%, n) : Single Payment Present Worth Factor (P/F, i%, n): n is in years if the i eff is used.

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Example - Factors How much inheritance to be received 20 years from now is equivalent to receiving $10,000 now? The interest rate is 8% per year compounded each 6-months.

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Uniform Series (Annuity) An Annuity is a series of equal amount money transactions occurring at equal time periods Ordinary Annuity - one that occurs at the end of each time period Uniform Series Present Worth Factor Capital Recovery Factor

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Annuities Can Relate an Annuity to a future value: Uniform Series Compound Amount Factor Uniform Series Sinking Fund Factor

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Annuity Example How much money can you borrow now if you agree to repay the loan in 10 end of year payments of $3000, starting one year from now at an interest rate of 18% per year?

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Factors Fortunately these factors are tabulated… And Excel has nice built in functions to calculate them too….

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Spreadsheet Function P = PV(i,N,A,F,Type) F = FV(i,N,A,P,Type) i = RATE(N,A,P,F,Type,guess) Where, i = interest rate, N = number of interest periods, A = uniform amount, P = present sum of money, F = future sum of money, Type = 0 means end- of-period cash payments, Type = 1 means beginning-of-period payments, guess is a guess value of the interest rate

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Gradient Factors Engineering Economic problems frequently involve disbursements or receipts that increase or decrease each year (i.e. equipment maintenance) If the increase is the same every year this is called a uniform arithmetic gradient.

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Gradient Factors Present Value @ time zero The Uniform amount of increase each period is the gradient amount The amount in the initial year is called a base amount, and it doesnt need to equal the gradient amount

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Gradient Factors To get the Gradient Factors we subtract off the base amount, and start things in year (period) 2: P G = Present worth of the gradient starting in year 2… This is what is calculated by P/G factor. P T (total) = P G +P A P A comes from using the P/A factor on an annuity equal to the base amount.

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P G /G and A G /G P/G = factor to convert a gradient series to a present worth. A/G = factor to convert a gradient series to an equivalent uniform annual series.

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Gradient Example Find the PW of an income series with a cash flow in Year 1 of $1200 which increases by $300 per year through year 11. Use i = 15%

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Review of Factors Using the tables.. Single Payment factors (P/F), (F/P) Uniform Series factors (P/A), (F/A) Gradients (A/G), (P/G)

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Unknown Interest Rates and Years Unknown Interest rate: -i.e. F = $20K, P = $10K, n = 9 i = ? -Or A = $1770, n = 10, P = $10K i =? Unknown Years – sometimes want to determine the number of years it will take for an investment to pay off ( n is unknown) -A = $100, P = $2000, i = 2% n = ?

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Unknown interest example If you would like to retire with $1million 30 years from now, and you plan to save $6000 per year every year until then, what interest rate must your savings earn in order to get you that million?

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Use of Multiple Factors Many cash flow situations do not fit the single factor equations. It is often necessary to combine equations Example? What is P for a series of $100 payments starting 4 years from now? 1 2 3 4 5 6 7 8 9 10 11 12 13 $100 P = ? years

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Use of Multiple Factors Several Methods: 1. Use P/F of each payment 2. F/P of each and then multiply by P/F 3. Get F =A (F/A, i,10), then P = F (F/P,i,13) 4. Get P 3 = A(P/A,I,10) and P 0 = P 3 (P/F,i,3) 1 2 3 4 5 6 7 8 9 10 11 12 13 $100 P = ? years

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Use of Multiple Factors Step for solving problems like this: 1. Draw Cash Flow Diagram. 2. Locate P or F on the diagram. 3. Determine n by renumbering if necessary. 4. use factors to convert all cash flows to equivalent values at P or F.

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Multiple Factors: Example A woman deposited $700 per year for 8 years. Starting in the ninth year she increased her deposits to $1200 per year for 5 more years. How much money did she have in her account immediately after she made her last deposit ?

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Eng Econ Practice Problems Check Website for Practice Problems…Remember you ALL have a quiz on Engineering Econ on Monday, not just the economists!

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Interest Formulas – Equal Payment Series

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