Notes on Engineering Economic Analysis

Transcription

1 College of Engneerng and Computer Scence Mechancal Engneerng Department Mechancal Engneerng 483 lternatve Energy Engneerng II Sprng 200 umber: 7724 Instructor: Larry Caretto otes on Engneerng Economc nalyss Introducton The economc analyss of alternatve energy sources typcally nvolves the comparson of an ntal cost wth a future savngs. For example the decson to pay more money for a vehcle wth a hybrd drve tran s based on a comparson of the hgher ntal prce for the hybrd drvetran wths the future savngs n fuel costs. Smlarly the decson to nstall a photovoltac solar collector on your home balances the ntal cost of the collector aganst the future savngs n electrcty blls. smple way to make such comparsons s to dvde the ntal cost to the expected savngs rate, a calculaton that results n a payback perod. For example, f an ntal cost of $2,000 resulted n a savngs of $500 per year, the payback perod would be computed as ($2,000) / ($500/year) 4 years. n ndvdual could then judge f the payback perod was short enough to justfy the ntal nvestment. One problem wth the payback perod analyss s that t does not account for the tme value of money. The money spent on the ntal alternatve energy technology could be nvested n some other area that would pay a return on the ntal nvestment. If the return from ths nvestment were greater than the savngs from the alternatve energy technology, t would make more economc sense to make ths nvestment nstead of purchasng the alternatve energy technology. The purpose of these notes s to summarze the basc deas of applyng the concept of the tme value of money to the economc analyss of engneerng decson makng. In ths course, we wll apply these des to the economc analyss of dfferent energy technologes. The tme value of money The tme value of money s specfed n terms of an nterest rate,. If an ntal amount of money,, called the present worth or present value (hence the symbol ), s nvested at an nterest rate,, for a tme perod, t, the nvestment wll earn an nterest payment at the end of the tme perod of t. The sum of the ntal nvestment and the nterest payment s called the future worth or future value, F. We ths have the followng equaton. F t ( t) [] ote that the dmensons of nterest rate are /tme. Typcally the nterest rate s expressed as a percentage and one has to be careful to convert the percentage to a decmal fracton before usng t n an equaton. The typcal unt for nterest rate s /year. If the nterest rate s appled for t year, the t term n equaton [] s usually omtted and one wrte the future value equaton as follows. F ( ) [2] Ths equaton assumes that the tme perod s one unt; ths s typcally one year when the nterest rate has unts of /year. However other perods, such as monthly may be used. When usng equaton [2] n place of equaton [] t s mportant to ensure that the /tme unts for the nterest Jacaranda (Engneerng) 3333 Mal Code hone: E-mal: lcaretto@csun.edu 8348 Fax:

2 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 2 rate are the same as the tme unts for the perod. If the perod s one month, then the unts for the nterest rate must be /month. The calculaton of the nterest rate for a dfferent tme unt s smply done by usng the unt converson factor for the tme unts. For example, when applyng equaton [2] to a perod of one quarter (/4 of a year) an annual nterest rate of 6% would be converted to a quarterly nterest rate as (6%/year)( year/4 quarters).5%/quarter. Interest rates converted n ths fashon are called nomnal or base nterest rates to dstngush them from the effectve nterest rates dscussed below. If the value of ( ) from equaton [2] s nvested for a second year t wll earn addtonal nterest of ( ). t the end of the second year the total amount form the ntal prncpal value,, and the two nterest payments can be found as follows. F ( ) ( ) ( )( ) ( ) 2 [3] If we contnue to renvest the total amount (ntal nvestment plus accumulated nterest) each year we wll contnue to earn nterest on the orgnal amount plus the accumulated nterest. Extendng the analyss that led to equaton [3] gves the followng result for the value at the end of years. F ( ) [4] If we solve ths equaton for we can answer the followng queston: how much do we have to nvest at an nterest rate to have a future value of F? F( ) - [5] The fact that ths multyear nvestment earns nterest not only on the ntal nvestment, but also on the nterest earned n earler years s called compoundng. In some cases the compoundng perod can be dfferent from the tme unts used for the nterest rate. For example, some bank savngs accounts provde a quarterly nterest payment. In ths example we can use equaton [] to determne the future value at the end of one quarter by settng t ¼ year. F [ (/4)] [6] (ote that ths result s the same as usng equaton [2] wth a quarterly nterest rate computed as the annual nterest rate dvded by 4 quarters per year.) We can apply equaton [4] to determne the amount that we would have at the end of one year ( 4 quarters). F [ (/4)] 4 ( /4) 4 [7] What nterest rate would be requred to have the same future value at the end of one year wthout the quarterly compoundng? We can fnd ths by rewrtng equaton [2] n terms of an effectve nterest rate, eff, such that F ( eff ); settng ths expresson for F equal to the expresson for F n equaton [7].gves. ( eff ) ( /4) 4 eff ( /4) 4 - [8] We can generalze ths equaton for the case of n c compoundng perods n one year. n c eff [9] nc

3 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 3 Table shows the value of the effectve nterest rate for dfferent nomnal (base) nterest rates and dfferent compoundng perods. Ths shows that the effect of compoundng s greater for hgher nterest rates and dmnshes as the number of compoundng perods becomes large. Table Effectve Interest Rate as a Functon of Base Rates and Compoundng erods Compoun- Effectve nterest rates for base Interest rates n next row dng erods 0.25% 0.50% % 2% 4% 0% 20% % %.00000% % % % % % %.00250% % % % % % %.00334% % % % % % %.00376% % % 0.383% % % %.0040% % % 0.408% % % %.00430% % % % % % 0.503%.0045% 2.080% % % % % 0.507%.00468% % % % % % 0.509%.00476% 2.09% % % % % %.00502% % % 0.57% % Sample roblem: Interest on a credt card s typcally stated as.5% per month. What s the nomnal annual nterest rate? What s the effectve annual nterest rate f the nterest s compounded monthly? Soluton: Snce there are 2 months per year, the nomnal annual nterest rate s smply the product (2 months/year)(.5%/month) 8%/year. The effectve annual nterest rate s found from equaton [9] wth n c 2 compoundng perods and 8%, the nomnal annual nterest rate just found. eff nc nc % Remember that nterest rates are converted to decmal fractons before beng used n equatons. resent worth of a seres of unform payments In the ntroducton we dscussed the tradeoff between an ntal one-tme nvestment and a contnuous seres of future savngs. To analyze ths examne an ntal nvestment wth a present worth,, and a set of equal tme ncrements wth a fxed payment,, at the end of each tme ncrement. 2 We want to fnd the equvalence, consderng the tme value of money, between the ntal nvestment,, and the seres of payments,. Typcally the schedule for the payments wll be monthly or annual payments. If the nterest rate for one tme perod of the payments s, we can determne the present worth of each future payment,, from equaton [5]. The present worth of the frst payment wll be ( ) - ; for the second payment, at the end of two tme perods, the present worth wll be ( ) -2. In general the present worth of the payment at the end of k tme The result for an nfnte number of compoundng perods, called nstantaneous compoundng can be found by takng the lmt as n c approaches nfnty. Ths leads to the followng equaton for nstantaneous compoundng: ( eff ) nstantaneous e. 2 The notaton for each payment n ths seres comes from the use of ths formula for determnng a set of annual payments, sometmes called an annuty. However, any tme nterval can be used for the payments.

4 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 4 perods wll be ( ) -k. To get the present worth of all payments, we have to sum the present worth of all ndvdual payments. Ths gves. k ( ) k [0] ppendx shows that ths summaton can be expressed as the rato, /, n ether of two equvalent forms shown below. ( ) ( ) ( ) [] Ths equaton answers the queston of what present worth s requred to provde a seres of a unform payment,, at the end of perods when the nterest rate per perod s. The answer to the opposte queston, what s the unform payment that we would get from a present worth for tme perods wth an nterest rate per perod of s smply the recprocal of equaton []. ( ) ( ) ( ) [2] key pont to remember for use of equatons [] and [2] s that the nterest rate must be the nterest rate per perod. If the payments are monthly and the nterest rate s gven as an annual nterest rate, the monthly nterest rate must be found to use these equatons. Sample problem: n ndvdual s offered a loan of $0,000 at an annual nterest rate (nomnal) of 9%. The loan s to be pad off n a seres of unform monthly payments over sx years. What s the amount of the monthly payment? Soluton: The perod between payments s one month. The nterest rate per monthly perod s (9%/year)(2 months/year) 0.75%/month. The total number of payments, 6 x We can use equaton [2], multpled by, to determne the payment, month ( ) ( $0,000) ( ) 72 $80.26 month Sample problem contnued: The amount of a loan s called the prncpal. The monthly payment n the prevous problem s used two ways: () the frst part s the payment of nterest (computed as the monthly rate tmes the remanng balance of the loan prncpal). The remander of the monthly payment s used to reduce the prncpal. Determne how much the loan prncpal s reduced durng the frst year of payments. Soluton: For the frst payment, the monthly nterest s the monthly nterest rate tmes the orgnal prncpal of $0,000. Ths s (0.0075)($0,000) $75. The remander of the monthly payment, $ $75 $05.26 s used to reduce the prncpal. fter the frst loan payment the 3 ote that the unts of /month are not used for nterest rate n the ( ) term n the equaton for. Ths s because of the shorthand that we used n gong from equaton [] to from equaton [2]. The calculaton of should really have the term as ( t), where t month. Snce we have dropped ths term, we have not shown the unts for n ths sample calculaton.

5 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 5 remanng prncpal s $0,000 - $05.26 $ For the second month, the nterest payment s (0.0075)( $9,895.74) $74.2. Contnung the calculatons n ths fashon gves the results for the entre loan shown n Table 2. fter the frst year (2 payments) the loan balance s $8, (See the 3 lne n Table 2.) The amount pad towards reducng the prncpal n the frst year s $0, $8, $, rncpal Table 2 mounts ad to Interest and rncpal for Sample roblem mount mount mount mount mount to to rncpal to to rncpal to Interest rncpal Interest rncpal Interest mount to rncpal $0, $75.00 $ $7, $54.33 $ $3, $29.59 $ $9, $74.2 $ $7,7.46 $53.38 $ $3,794.7 $28.46 $ $9, $73.42 $ $6, $52.43 $ $3,642.9 $27.32 $ $9,68.85 $72.6 $ $6, $5.47 $ $3, $26.7 $ $9, $7.8 $ $6, $50.50 $ $3, $25.02 $ $9, $70.99 $ $6, $49.53 $ $3,80.64 $23.85 $ $9, $70.7 $ $6, $48.55 $ $3, $22.68 $ $9, $69.35 $ $6,34.76 $47.56 $ $2, $2.50 $ $9,35.48 $68.52 $ $6, $46.57 $ $2, $20.3 $ $9, $67.68 $ $6, $45.57 $ $2, $9. $6.5 $8,9.6 $66.83 $ $5, $44.56 $ $2, $7.90 $ $8, $65.98 $ $5, $43.54 $ $2, $6.68 $ $8, $65.3 $ $5, $42.5 $ $2, $5.46 $ $8, $64.26 $ $5,530.5 $4.48 $ $, $4.22 $ $8, $63.39 $ $5,39.73 $40.44 $ $,730.0 $2.98 $ $8, $62.52 $ $5,25.9 $39.39 $ $, $.72 $ $8,27.7 $6.63 $ $5,.04 $38.33 $ $,394.9 $0.46 $ $8, $60.74 $ $4,969. $37.27 $ $, $9.8 $ $7, $59.85 $ $4,826.2 $36.20 $ $,053.3 $7.90 $ $7,859.5 $58.94 $ $4, $35.2 $ $ $6.6 $ $7, $58.03 $ $4, $34.03 $ $ $5.30 $ $7,65.60 $57.2 $ $4, $32.93 $ $ $3.99 $ $7, $56.9 $ $4, $3.83 $ $ $2.67 $ $7, $55.26 $ $4, $30.7 $ $78.48 $.34 $78.92 In ths table s the number of the payment; the rncpal column shows the loan balance pror to the payment. The amount to nterest s the perodc nterest rate tmes the prncpal. The payment of ths loan s $80.26; the amount to prncpal s the dfference between the loan payment and the amount to nterest. The new prncpal s the old prncpal mnus the amount to prncpal. The loan nterest s rounded to two decmal places. Because of ths the fnal calculaton s not correct; the fnal amount to prncpal s less than the amount of the prncpal. In practce, the fnal payment would be reduced by $0.44 (from $80.26 to $79.82) to pay off the prncpal exactly. The nformaton n Table 2 becomes mportant when economc analyses are done consderng taxes. Interest pad on a loan s a deductable busness expense. The amount pad for prncpal s not. When consderng the annual ncome statement of a company for preparng tax returns, t s mportant to know how much of a loan payment goes to nterest. Sample problem concluded: What s the total amount of the payments on ths loan. How much goes to nterest?

6 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 6 Soluton: There are 72 payments of $80.25 for a total of $2, Subtractng the ntal loan amount of $0,000 from ths total shows that the total nterest pad s $2, ractcal calculatons Equatons [] and [2] can be readly solved for or. Both of these equatons can be solved for to gve ln ln ( ) [3] However, there s no explct soluton for. Tradtonally engneerng economcs textbooks have ncluded tables of quanttes such as / as a functon of and. Such tables allow or to be determned by nterpolaton. Modern calculators, especally fnancal calculators have equatons for the varous functons consdered here. Spreadsheets, such as Excel, also have formulas for computng the varous terms n these equatons. Fgure s an example of an Excel spreadsheet wth formulas for fnancal calculatons. Ths spreadsheet uses names for cell locatons 5 and dsplays the formulas nstead of the numercal results. Fgure : Excel Spreadsheet wth Fnancal Formulas nnual nterest rate 5% umber of perods per year 2 Interest rate per perod nnual_nterest_rate/umber_of_perods_per_year Total number of perods 360 resent Value erodc payment MT(Interest_rate_per_perod,Total_number_of_perods,-resent_Value) Calculate number of perods ER(Interest_rate_per_perod,erodc_payment,-resent_Value) Calculate nterest rate RTE(Total_number_of_perods,erodc_payment,-resent_Value) In ths spreadsheet the names nnual_nterest_rate, umber_of_perods_per_year, Total_number_of_perods and resent_value have been defned to represent the cells that contan the values 5%, 2, 360, and 00000, respectvely. The names Interest_rate_per_perod represents the cell that contans the formula for defnng the perodc nterest rate, whch s ( 5%/2) n ths example. 6 4 It s possble to obtan an nterest-only loan n whch the monthly payments would be only (.0075)($0,000) $75. However a lump-sum payment of $0,000 would be requred at the end of the loan. Here the total nterest would be $5,400. nother possblty s a smple nterest loan n whch the total nterest would be ($0,000)(0.09/year)(6 years) $5400. The total amount of prncpal plus nterest would be pad accordng to some payment schedule. For example, the total prncpal plus nterest of $5,400could be pad n twelve sem-annual payments of $, cell name n Excel provdes an alternatve way for referrng to a cell locaton. The usual cell reference s a (column)(row) combnaton such as C4. Cell names provde an alternatve way to refer to cell locatons that gves someone revewng the spreadsheet a better dea of what the varables n an equaton represent. 6 ote that percentages n Excel are a formattng opton. The underlyng value s not changed. Thus, the explct converson of percentages to decmal fractons s not requred n Excel.

7 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 7 The functon MT computes the value of from equaton [2]. Excel spreadsheet fnancal formulas nclude the drecton of the payment. Thus the MT functon assumes that the present value and the payment result wll have opposte sgns. 7 To obtan a postve value for the payment result, a mnus sgn s placed before the present value n the MT formula. The cell that contans the formula s gven the name erodc_payment. The functon ER computes the value of, the number of perods, whch could be found from equaton [3]. gan, ths Excel formula uses opposte sgns for and ; to accommodate ths, a mnus sgn s placed n front of the present value. The functon RTE computes the perodc nterest rate,, whch satsfes both equatons [] and [2]. In the actual spreadsheet for ths example the computed payment s $ per month and the computed values of and match the nput values of 360 and 5%/2. What s the nterest rate? The nterest rate used prevously n these notes s easly understood as the nterest that you have to pay on a loan or the nterest that you would receve on a bank savngs account. In some cases the term dscount rate s used. Ths s commonly used for the rate that the Federal Reserve charges for loans to banks. It s also used n bond sales where the ntal prce for a bond wth a certan face value (say $000) s dscounted so that the ntal prce for the bond s less than ts face value. company comparng alternatve project proposals, each wth dfferent cash flows, wll often examne the nternal rate of return. Ths s the equvalent nterest rate that the cash flows for the project would pay f the cash outflows were costs and the cash nflows were profts. When an ndvdual or company s consderng nvestments on dfferent projects wth expected cash flows they wll generally seek a mnmally acceptable rate of retur (MRR) on the nvestment. Regardless of whch term s used nterest rate, dscount rate, nternal rate of return, MRR the formulas derved above wll apply. on-unform payment seres and comparson of alternatves The formulas used so far relate a present value to a sngle future value and an ntal present value to a unform seres of payments. Texts on engneerng economcs also present formulas for a seres of ncreasng (or decreasng payments) where the rate of ncrease (or decrease) s the same and the payment nterval s constant. These notes do not consder such formulas. In the most general economc analyss of engneerng projects, the amount of the cash flows wll not be unform and the tme ntervals may not be constant. general comparson of alternatves can be done by a method known as dscounted cash flows. In ths approach each cash nput or output s converted to an equvalent value at a fxed tme usng equatons [4] or [5]. Typcally ths fxed tme s the start or end of the project so the cost s expressed as a present value at the start of the project or a fnal value at the end of the project. The calculaton of the net present value of a seres of cash flows that occur at regular ntervals (yearly, monthly, weekly, etc.) essentally apples equaton [5] to ths seres of evenly spaced cash flows (wth postve and negatve sgns) to determne the net present value. If the cash flow at the end of perod k s CF k, the net present value, V, s found from the followng equaton when the tme nterval s constant and the nterest rate refers to the annual perod. 7 If s negatve, ndcatng a cash outflow (a loan that you make) then s postve ndcatng a cash nflow (the payments that the borrower returns to you). If s postve, ndcatng a cash nflow (a loan that you receve), then s negatve ndcatng a cash outflow as you make payments on the loan.

8 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 8 k 0 k ( ) Equal tme ntervals V CF k [4] The more general case of unequal cash flows wth uneven tme ncrements can be expressed by the followng equaton where the nterest rate s the daly nterest rate (the annual rate dvded by days/year), d 0 s the date of the ntal cash flow, expressed as a seral date, and d k s the date of the k th cash flow. k 0 d ( ) 0 d k Unequal tme ntervals V CF [5] k daly s usual, s the nterest rate for the perod. The summatons are started at k 0 to consder a cash flow at the start of the frst perod. The Excel functon V(rate, values) has the nterest rate as ts frst argument and the seres of payments from k to k as the remanng arguments. Ths functon does not consder any cash flow at the start of the frst nterval (the k 0 term n equaton [4]). If an Excel spreadsheet has the nterest rate n cell, the ntal (k 0) cash flow n cell 2, and the remanng cash flows n cells 3:2, the net present value, ncludng the k 0 term, would be computed by the spreadsheet formula 2 V(, 3:2). If the ntal cash flow n cell 2 were ncluded n the arguments to the V functon, t would be consdered as the cash flow at the end of the frst tme ncrement. If the net future value s requred, t can be computed from the net present value usng equaton [4]. The Excel functon XV computes the net present value for a seres of uneven cash flows at dfferent dates, where the date ntervals need not be the same. In ths functon, the ntal cash flow s accounted for. If an Excel spreadsheet has the nterest rate n cell, the ntal (k 0) cash flow n cell 2, the remanng cash flows n cells 3:2, and the dates for each cash flow n cells B2:B2 the net present value, ncludng the k 0 term, would be computed by the spreadsheet formula XV(, 2:2, B2:B2). ote the dfference between XV and V: XV ncludes the ntal (k 0) cash flow; V does not. The nternal rate of return (IRR) s the nterest rate that makes the net present value equal to zero for a seres of postve and negatve cash flows. (Typcally these start wth a negatve cash flow to purchase an tem or make some other nvestment, followed by postve cash flows representng ncome from the nvestment.) If we use the net present value from equaton [4], where the cash flows can vary, but the tme nterval s the same, the IRR s the value of that makes V 0 for an arbtrary seres of cash flows. The Excel functon IRR computes the nternal rate of return for a seres of cash flows at equally-spaced tme ntervals. The IRR functon ncludes the k 0 cash flow. For the spreadsheet wth the ntal (k 0) cash flow n cell 2 and the remanng cash flows n cells 3:2, the nternal rate of return would be computed by the functon call IRR(2:2). If we want to compute the IRR for a seres of cash flows at dfferent tmes we can use the Excel functon XIRR. If an Excel spreadsheet has the ntal (k 0) cash flow n cell 2, the remanng cash flows n cells 3:2, and the dates for each cash flow n cells B2:B2 the net present value, ncludng the k 0 term, would be computed by the spreadsheet formula XV(2:2, B2:B2). Both the IRR and XIRR functons have a fnal, optonal argument, whch s an ntal guess for the nterest rate. Ths argument can be used f the results from usng ether of these functons wthout ths optonal argument do not gve correct results. You can check the results of the IRR or XIRR functons by usng, respectvely, the V or XV functons. If the IRR or XIRR functon has the

9 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 9 correct result the V or XV functons should gve zero to wthn roundoff error when used wth the same data set and an nterest rate equal to that found by IRR or XIRR. Effect of nflaton Inflaton (deflaton) n a currency s sad to occur when the purchasng power of that currency decreases (ncreases) over tme. If the purchase prce of an dentcal tem s d at some ntal tme and ncreases to d 2, tme perods later, the nflaton rate for each tme perod, f, assumed constant over perods, s defned as follows: d 2 d ( f) [6] lthough the equaton has the same form as n nterest rate equaton, nflaton does not represent an ncrease n the amount of money avalable. Rather t s a decrease n the purchasng power of the currency. Measures of nflaton n the US are mantaned by the bureau of labor statstcs. The most commons ndces are the consumer prce ndex (CI) and producer prce ndex (I). See the web ste for nformaton on the CI; the ste s the general ste for varous statstcs ncludng the CI, I, and the employment cost ndex. Table 2 shows the average nflaton rate n the CI for all US ctes. The nflaton rate shown for a gven year s the nflaton rate from June of the prevous year to June of the year shown. The cumulatve nflaton rate s the rate from June 996 to June of the year shown. 8 Table 3 - Inflaton n US verage Urban Consumer rce Index Endng n June of Year Shown Year nnual 2.3%.7% 2.0% 3.7% 3.2%.% 2.% 3.3% 2.5% 4.3% 2.7% 5.0% -.4% Cumulatve 2.3% 4.0% 6.% 0.0% 3.6% 4.8% 7.2% 2.% 24.% 29.5% 33.0% 39.6% 37.6% The nflaton rate from June 996 to June 997 was 2.3%; the nflaton rate from June 996 to June 2009 was 37.6%. The negatve nflaton rate from June 2008 to June 2009 ndcates a deflaton n ths perod. In equaton [6] we assumed that the nflaton rate was constant. In order to compute the average (constant) nflaton rate for a perod of years from the total nflaton over the perod we need to understand how an overall nflaton rate s related to a seres of ndvdual rates. ssume that a prce ncreases by 0% then ncreases by 2% agan. What s the total prce ncrease? fter the frst ncrease the new prce s ( 0%). fter the second ncrease the prce becomes ( 0%)( 2%). The total percent ncrease, x, from the two prce ncreases s found by settng ( x) ( 0%)( 2%) so that x 0% 2% (0%)(2%) 23.2%. In general, the total fractonal ncrease n a quantty g total due to a seres of ndvdual ncreases g s gven by the followng equaton: ( g total ) ( g )( g 2 ) ( g ) so that g total ( g ) [7] The product operator, Π, n ths equaton s the multplcaton analog of the summaton operator, Σ. To get the average gan per perod for a total gan g total over perods we set each g n equaton [7] equal to an average growth rate for each perod, g,assumed constant over the tme perods. Ths gves 8 ote that the cumulatve nflaton rate s not smply the sum of the ndvdual nflaton rates. Instead t s computed as shown n the text followng the table.

10 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 0 g total ( g ) ( g ) g ( gtotal ) [8] Sample roblem: What s the average nflaton rate for the years 996 to 2009 from Table 2? Soluton: From Table 2 the total nflaton over ths 3-year perod s 37.6%. Usng equaton [8] (wth the symbol f for nflaton rate n place of g) we fnd the average rate as follows. f 3 ( ) ( 0.376) 2.488% f total How s nflaton consdered n engneerng cost analyses? There are two approaches for dong ths. The frst approach assumes that the nterest rate ncludes the effects of nflaton. In ths case we have the so-called market nterest rate and the dollar amounts are expressed n socalled current dollars (or current Euros or whatever currency s used). The current currency amount s the amount of currency n the actual transacton. n alternatve s to use the concept of constant dollars that corrects for the effects of nflaton. Both of these approaches wll be outlned below. Let s start wth the example of an nvestment of $000 n June 996 that pays a constant nterest rate of 5% compounded annually. From equaton [4] we can compute the value n June 2009 (thrteen years later) as follows: F ($,000)( 0.05) 3 $, What can we buy wth ths amount, compared to the purchasng power of our orgnal nvestment? ccordng to the data n Table 2, t wll cost us 37.6% more to buy the tem n June 2006 than t would have cost us n June 996. Thus our new purchasng power s only $,885.55/( 0.376) $ n terms of the value of the dollar n June 996 when the ntal nvestment was made. We could make the same computaton usng the average nflaton rate of 2.488% just found. In ths case we would compute $,885.55/( ) 3 $ The nterest rate of 5% used n the above example s called the market nterest rate. Ths s the nterest rate that one usually sees offered for nvestments or charged for loans. Ths market nterest rate ncludes the effect of nflaton. person offerng ths rate s wllng to accept the return of the 5% nvestment wth the realzaton that the effect of nflaton wll reduce the purchasng power of the eventual return on the nvestment. How do we dstngush between the market nterest rate and the true nterest rate or actual nterest rate whch gnores the effects of nflaton? Because the true nterest rate, true, gnores the effects of nflaton, we say that the calculatons are n hypothetcal constant dollars ; ths s a currency measure that gnores the effects of nflaton. In the prevous example we saw that a $,000 nvestment n June 996 yelded a constant dollar return (.e. a return of equvalent purchasng power) of $ n June Usng equaton [4] wth these constant dollar amounts we can compute the true nterest rate as follows. $ ($,000)( true ) 3 true [($369.92) / ($,000)] /3 2.45% What s the relatonshp between the true nterest rate, the market nterest rate and the nflaton rate? We prevously used the usual formula that relates present and future values usng the market nterest rate F ( market ) to show that the future value of a $,000 nvestment n June 996 was $, n June We then used the nflaton formula to determne the future value n constant dollars, F const F( f) -, Fnally we defned the true nterest rate by the equaton F const ( true ). We can combne these three equatons as follows.

11 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age F const ( ) ( f ) ( market ) ( f ) F true [9] Consderng only the two (equal) terms multpled by gves ( ) ( market ) ( f ) ( ) market true true [20] Takng the / root of both sdes of ths equaton gve ( true )( f ) true f true f market market true [2] f Ths gves the followng relatonshp between the three rates consdered here: the market nterest rate, the true nterest rate, and the nflaton rate. f f [22] market true ote that market s not equal to the sum of the true nterest rate and the nflaton rate. However, f both of these are small, the market rate wll be approxmately the same as the sum of the true nterest rate and the nflaton rate. Example problem: What s the true nterest rate f the market rate s 5% and the nflaton rate s 2.488%? What would the error be by usng the approxmaton that market true f? Soluton: We can solve equaton [22] for the true nterest rate and substtute the data gven wth the followng result. true f true f market f % Usng the approxmaton that market true f gves true market f 5% % 2.52%. Ths s an error of 0.06 percentage ponts. Comment: We see that the nput data for ths problem are the same as the data that have been used n the examples of ths secton. Thus the true nterest rate computed from equaton [22] has the same value found prevously. Because both true and f are relatvely small the approxmaton gves only a small error. Whch approach true nterest rate and constant dollars or market nterest rate and current dollars should we use n calculatons of alternatve energy economcs? The choce may be made by a regulatory agency. For example, publc utltes, whose proft s regulated by governmental utlty agences, must use standard accountng determned by the agency. For smple problem solvng you should notce that the same equatons are used for market nterest rates, whch do not explctly account for nflaton and present results n current dollars, and for true nterest rates, whch present results n constant dollars. Here s a set of gudelnes for problem solvng: If you have only the true nterest rate, you must do calculatons n constant dollars. If you have only the market nterest rate, you must do calculatons n current dollars.

12 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 2 If you have two of the three quanttes () market nterest rate, (2) true nterest rate, and (3) nflaton rate, you can fnd the thrd from equaton [22]. You can then choose to do the calculatons n ether constant or current dollars. In comparng optons for alternatve energy projects, t s often useful to explctly account for the nflaton rate. Ths s partcularly true when comparng a conventonal fossl fuel opton n whch the fuel nflaton can be a sgnfcant factor wth an alternatve such as solar and wnd n whch the ongong costs, such as mantenance, wll not be as sgnfcant as the ongong costs for the fossl fuel opton. However, t s smpler to use the market nterest rate because ths gves all fgures n current dollars. Death and taxes Well, only taxes wll be consdered here, and only brefly. ll prevous sectons have made no menton of taxes; such analyses are called before-tax analyses. Ths secton wll outlne the consderatons that are requred for an after-tax analyss. We are all famlar wth the taxman who vsts our homes every prl 5 wth the requrement of fllng out forms that show what percentage of our ncome we owe to the government. These are personal ncome taxes. In addton to such taxes, busness enttes pay a corporate ncome tax. The margnal tax rate (the tax rate on the hghest level of taxable ncome) ranges from 34% to 39% for taxable ncomes over $75,000. The computaton of taxable ncome allows deductons for nterest payments and deprecaton. Deprecaton s an accountng tool where ncome s saved on a regular schedule to replace purchased equpment. The rate at whch companes may charge deprecaton s governed by regulatons of the Internal Revenue Servce (IRS). lthough there are several methods of deprecaton, the commonly used method s called the modfed accelerated cost recovery system (MCRS), whch allows a hgh rate of deprecaton n early years. Ths s ntended to encourage companes to purchase new equpment snce the accelerated deprecaton schedule allows then to reduce ther taxable ncome. Interest payments on loans are a partcularly mportant tem n after-tax analyses. Companes can fnance new projects by a combnaton of debt and equty fnancng. Equty fnancng s the use of exstng cash reserves (or the sale of stock by a corporaton); debt fnancng s the borrowng of money, ether as a loan or bond fnancng for large corporatons. Most nvestments nvolve a combnaton of these two. 9 When consderng the return that a company makes on ts nvestments, only the equty fnancng s consdered. The debt fnancng s accounted as a future expense of nterest payments on the debt. s noted above, such nterest payments are tax deductable. Energy ccountng Our man concern here wll be wth the tradeoffs between the nvestment n a new energy technology wth hgher frst costs, but lower operatng costs than conventonal energy technology. We wll be comparng the ncreased purchase prce, an ntal nvestment, aganst the ongong future savngs, that wll be a tme seres of payments. Equaton [2] wll be our man tool for makng ths comparson. We can use ths equaton to determne the present worth of a seres of cost savngs and compare t to the ntal purchase prce. For example, the purchase of a photovoltac solar collector represents an ntal cost. The annual electrcty savngs represents a seres of costs that can be reduced to a present worth usng equaton [2] for a specfed nterest rate (what the purchaser would obtan from nvestng the 9 company s debt-to-equty rato, the rato of long term debt dvded by common shareholder equty s a measure of the rsk of nvestng n the company.

13 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 3 money nstead of buyng the solar system) and expected lfetme for the collector. If the present worth of the savngs s less than the purchase cost, the other nvestment would make more economc sense. Such a calculaton mght better be done wth a real nterest rate, snce the cost of the electrcty saved would be expected to ncrease over tme. Because, the / formula n equaton [2] assumes equal future payments, you could not account for any future ncreased savngs due to nflaton.

14 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 4 ppendx Dervaton of Equatons Used n otes Dervaton of equaton [] for a seres of unform payments We start wth equaton [0] and dvde both sdes of the equaton by unform payment,. ( ) k k [-] We start by wrtng the last term n the summaton as a separate term (and decreasng the upper lmt of the summaton by one to account for ths.) 0 k k ( ) ( ) [-2] ext we extract a common factor of ( ) from the summaton. k ( ) ( ) ( ) k [-3] ow we defne a new summaton ndex, m k. When k, m 2; when k, m. Ths gves the followng result. m ( ) ( ) ( ) m 2 [-4] Fnally we add and subtract the same term, ( ) - from the summaton term. Because ( ) - has the same form as the general term n the summaton for m, the addton of ths term smply adds the m term to the summaton. Ths gves. m ( ) ( ) ( ) ( ) ( ) m 2 m ( ) ( ) ( ) ( ) m [-5] We see that the fnal summaton term n ths equaton s the same as the summaton term n equaton [-] wth the dummy ndex k replaced by the dummy ndex m. Thus 0 Students may not have seen the typcal approach for workng wth equatons for summatons. Gettng a closed form equaton for the summaton s approached by wrtng specfc terms n the sum outsde the sum then gettng a common factor that changes the expresson nsde the summaton. revsed summaton ndex s defned and the summaton s manpulated so that the rght sde of the equaton contans the orgnal sum plus some other terms. The orgnal left sde (whch equals the orgnal sum) s then substtuted for the orgnal sum on the rght sde and the resultng equaton s then solved for the ntal sum. If ths sounds confusng, just regard the dervaton as a demonstraton of ths general approach to the specfc problem consdered here.

15 Engneerng economcs notes ME 483, L. S. Caretto, Sprng 200 age 5 we can replace the second summaton term by ts equvalent term / from equaton [- ]. Dong ths and solvng for / gves. ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) [-6] The frst equaton below s found by solvng equaton [-6] for /; the second equaton s found by multplyng the frst by ( ). ( ) ( ) ( ) [-7] Ether of these equatons may be used to solve for /. The concept of a dummy ndex means that we can use any letter for the summaton ndex and the result wll be the same. For example, x x x x m m k k,