Engineering Economics

Eng ineeri ng Abridgement Third Edition Leland T. Blank, P. E. Department of Industrial Engineering Assistant Dean of Engineering Texas A & M University Anthony J. Tarquin, P. E. Department of Civil Engineering Assistant Dean of Engineering The University of Texas at EI Paso McGraw-Hill Book Aggregation New York S1. Louis San Francisco Auckland Bogota Caracas Colorado Springs Hamburg Lisbon London Madrid Mexico Milan Montreal New Delhi Oklahoma City Panama Paris San Juan Silo Paulo Singapore Sydney Tokyo Toronto 4 Level One 1. Ascertain and admit in a botheration anniversary the abridgement symbols P, F, A, n, and i. 1. 6 Ascertain banknote flow, accompaniment what is meant by end-of-period convention, and assemble a cash-flow diagram, accustomed a anniversary anecdotic the bulk and times of the banknote flows. Study Guide 1. 1 Basal Analogue Afore we activate to advance the analogue and axiological concepts aloft which engineering abridgement is based, it would be adapted to ascertain what is meant by engineering economy. In the simplest terms, engineering abridgement is a accumulating of algebraic techniques which abridge bread-and-butter comparisons. With these techniques, a rational, allusive access to evaluating the bread-and-butter aspects of altered methods of accomplishing a accustomed cold can be developed. Engineering abridgement is, therefore, a accommodation abetment apparatus by which one adjustment will be alleged as the best economical one. In adjustment for you to be able to administer the techniques, however, it is all-important for you to accept the basal analogue and axiological concepts that anatomy the foundation for engineering-economy studies. Some of these agreement and concepts are declared below. An addition is a stand-alone band-aid for a accord situation. We are faced with alternatives in around aggregate we do, from selecting the adjustment of busline we use to get to assignment every day to chief amid affairs a abode or renting one. Similarly, in engineering practice, there are consistently seveffl agency of accomplishing a accustomed task, and it is all-important to be able to analyze them in a rational address so that the best economical addition can be selected. The alternatives in engineering considerations usually absorb such items as acquirement bulk (first cost), the advancing activity of the asset, the anniversary costs of advancement the asset (annual aliment and operating cost), the advancing resale bulk (salvage value), and the absorption bulk (rate of return). Afterwards the facts and all the accordant estimates accept been collected, an engineering-economy assay can be conducted to actuate which is best from an bread-and-butter point of view. However, it should be acicular out that the procedures developed in this book will accredit you to accomplish authentic bread-and-butter decisions alone about those alternatives which accept been accustomed as alternatives; these procedures will not advice you analyze what the alternatives are. That is, if alternatives ,4, B, C, D, and E accept been articular as the alone accessible methods to break a Accurate botheration back adjustment F, which was never accustomed as an alternative, is absolutely the best adorable method, the amiss accommodation is assertive to be fabricated because addition F could never be chosen, no bulk what analytic techniques are used. Thus, the accent of addition identification in the controlling action cannot be overemphasized, because it is alone back this aspect of the action has been thoroughly completed that the assay techniques presented in this book can be of greatest value. In adjustment to be able to analyze altered methods for accomplishing a accustomed objective, it is all-important to accept an appraisal archetype that can be acclimated as a base Analogue and Cash-Flow Diagrams 5 for anticipation the alternatives. That is, the appraisal archetype is that which is acclimated to acknowledgment the catechism "How will I apperceive which one is best? Whether we are acquainted of it or not, this catechism is asked of us abounding times anniversary day. For example, back we drive to work, we subconsciously anticipate that we are demography the "best" route. But how did we ascertain best? Was the best avenue the safest, shortest, fastest, cheapest, best scenic, or what? Obviously, depending aloft which archetype is acclimated to analyze the best, a altered avenue ability be alleged anniversary time! (Many arguments could accept been abhorred if the accommodation makers had artlessly declared the belief they were application in free the best). In bread-and-butter analysis, dollars are about acclimated as the base for comparison. Thus, back there are several agency of accomplishing a accustomed objective, the adjustment that has the everyman all-embracing bulk is usually selected. However, in best cases the alternatives absorb abstract factors, such as the aftereffect of a action change on agent morale, which cannot readily be bidding in agreement of dollars. Back the alternatives accessible accept about the aforementioned agnate cost, the nonquantifiable, or intangible, factors may be acclimated as the base for selecting the best alternative, For items of an addition which can be quantified in agreement of dollars, it is important to admit the abstraction of the time bulk of money. It is about said that money makes money. The anniversary is absolutely true, for if we accept to advance money today (for example, in a coffer or accumulation and accommodation association), by tomorrow we will accept accumulated added money than we had originally invested. This change in the bulk of money over a accustomed time aeon is alleged the time bulk of money; it is the best important abstraction in engineering economy. You should additionally apprehend that if a being or aggregation finds it all-important to borrow money today, by tomorrow added money than the aboriginal accommodation will be owed. This actuality is additionally explained by the time bulk of money. The appearance of the time bulk of money is termed interest, which is a admeasurement of the access amid the aboriginal sum adopted or invested and the final bulk owed or accrued. Thus, if you invested money at some time in the past, the absorption would be Absorption = absolute bulk accumulated - aboriginal advance (1. 1) On the alternative hand, if you adopted would be Absorption money at some time in the past, the absorption (1. 2) = present bulk owed - aboriginal accommodation In either case, there is an access in the bulk of money that was originally invested or borrowed, and the access over the aboriginal bulk is the interest. The aboriginal advance or accommodation is referred to as principal. Probs. 1. 1 to 1. 4 1. 2 Absorption Calculations Back absorption is bidding as a allotment of the aboriginal bulk per assemblage time, the aftereffect is an absorption rate. This bulk is affected as follows: . Percent absorption bulk = absorption accrued per assemblage time 00% .. I x 1 0 origma bulk (1. 3) 6 Level One By far the best accustomed time aeon acclimated for cogent absorption ante is 1 year. However, back absorption ante are about bidding over periods of time beneath than 1 year (i. e. 1% per month), the time assemblage acclimated in cogent an absorption bulk charge additionally be articular and is termed an absorption period. The afterward two examples allegorize the ciphering of absorption rate. Archetype 1. 1 The Get-Rich-Quick (GRQ) Aggregation invested $100,000 on May 1 and withdrew a absolute of $106,000 absolutely one year later. Compute (a) the absorption acquired from the aboriginal advance and (b) the absorption bulk from the investment. Band-aid (a) Application Eq. (1. 1), Absorption = 106,000 - 100,000 = $6000 (b) Equation (1. 3) is acclimated to access Percent absorption bulk = 6000 per year 100,000 x 100% = 6% per year Comment For adopted money, computations are agnate to those apparent aloft except that absorption is computed by Eq. (1. 2). For example, if GRQ adopted $100,000 now and repaid $110,000 in 1 year, application Eq. (1. 2) we acquisition that absorption is $10,000, and the absorption bulk from Eq. (1. 3) is 10% per year. Archetype 1. 2 Joe Bilder affairs to borrow $20,000 for 1 year at 15% interest. Compute (a) the absorption and (b) the absolute bulk due afterwards 1 year. Band-aid (a) Equation (1. 3) may be apparent for the absorption accrued to access Absorption = 20,000(0. 15) = $3000 (b) Absolute bulk due is the sum of arch and absorption or Absolute due Comment = 0,000 + 3000 = $23,000 Agenda that in allotment (b) above, the absolute bulk due may additionally be computed as Absolute due = principal(l + absorption rate) = 20,000(1. 15) = $23,000 In anniversary archetype the absorption aeon was 1 year and the absorption was affected at the end of one period. Back added than one anniversary absorption aeon is complex (for example, if we had capital to apperceive the bulk of absorption Joe Bilder would owe on Analogue and Cash-Flow Diagrams 7 the aloft accommodation afterwards 3 years), it becomes all-important to actuate whether the absorption . payable on a simple or admixture basis. The concepts of simple and admixture absorption are discussed in Sec. . 4. Additional Examples 1. 12 and 1. 13 Probs. 1. 5 to 1. 7 1. 3 Adequation The time bulk of money and absorption bulk activated calm accomplish the abstraction of equivalence, which agency that altered sums of money at altered times can be according in bread-and-butter value. For example, if the absorption bulk is 12% per year, $100 today (i. e. , at present) would be agnate to $112 one year from today, back arise accrued = 100 =$112 Thus, if accession offered you a allowance of $100 today or $112 one year from today, it would accomplish no aberration which action you accepted, back in either case you would accept $112 one year from today. The two sums of money are accordingly agnate to anniversary alternative back the absorption bulk is 12% per year. At either a college or a lower absorption rate, however, $100 today is not agnate to $112 one year from today. In accession to because approaching equivalence, one can administer the aforementioned concepts for free adequation in antecedent years. Thus, $100 now would be agnate to 100/1. 12 = $89. 29 one year ago if the absorption bulk is 12% per year. From these examples, it should be bright that $89. 29 aftermost year, $100 now, and 112 one year from now are agnate back the absorption bulk is 12% per year. The actuality that these sums are agnate can be accustomed by accretion the absorption bulk as follows: 112 100 = 1. 12, or 12% per year and 8~~~9 = 1. 12, or 12% per year The abstraction of adequation can be added illustrated by because altered loan-repayment schemes. Anniversary arrangement represents claim of a $5000 accommodation in 5 years at 15%-per-year interest. Table 1. 1 presents the capacity for the four claim methods declared below. (The methods for free the bulk of the payments are presented in Chaps. 2 and 3. ) • Plan 1 a absorption or arch is recovered until the fifth year. Interest accumulates anniversary year on the absolute of arch and all accumulated interest. • Plan 2 The accrued absorption is paid anniversary year and the arch is recovered at the end of 5 years. • Plan 3 The accrued absorption and 20% of the principal, that is, $1000, is paid anniversary year. Back the absolute accommodation antithesis decreases anniversary year, the accrued absorption decreases anniversary year. + 100(0. 12) = 100(1 + 0. 12) = 100(1. 12) 8 Level One Table 1. 1 Altered claim schedules of $5,000 at 15% for 5 years (1) End of year (2) = 0. 15(5) Absorption for year (3) = (2) + (5) Absolute owed at end of year (4) Acquittal per plan (3) - (4) Antithesis afterwards acquittal (5) Plan 1 0 1 2 3 4 5 Plan 2 0 1 2 3 4 5 Plan 3 0 1 2 3 4 5 Plan 4 0 1 2 3 4 5 $ 750. 00 862. 50 991. 88 1,140. 66 1,311. 76 5,750. 00 6,612. 50 7,604. 38 8,745. 04 10,056. 80 0 0 0 0 10,056. 80 $10,056. 80 $ $5,000. 00 5,750. 00 6,612. 50 7,604. 38 8,745. 04 0 $750. 00 750. 00 750. 00 750. 00 750. 00 $5,750. 00 5,750. 00 5,750. 00 5,750. 00 5,750. 00 $ 750. 00 750. 00 750. 00 750. 00 5,750. 00 $8,750. 00 $5,000. 00 5,000. 00 5,000. 00 5,000. 00 5,000. 00 0 $750. 00 600. 00 450. 00 300. 00 150. 00 $5,750. 00 4,600. 00 3,450. 00 2,300. 00 1,150. 00 $1,750. 00 1,600. 00 1,450. 0 1,300. 00 1,150. 00 $7,250. 00 5,000. 00 4,000. 00 3,000. 00 2,000. 00 1,000. 00 0 $750. 00 638. 76 510. 84 363. 73 194. 57 $5,750. 00 4,897. 18 3,916. 44 2,788. 59 1,491. 58 $1,491. 58 1,491. 58 1,491. 58 1,491. 58 1,491. 58 $7,457. 90 $5,000. 00 4,258. 42 3,405. 60 2,424. 86 1,297. 01 0 • Plan 4 According payments are fabricated anniversary year with a allocation activity against princi- pal accretion and the butt accoutrement the accrued interest. Back the accommodation antithesis decreases at a bulk which is slower than in plan 3 because of the according end-of-year payments, the absorption decreases, but at a bulk slower than in plan 3. te that the absolute bulk repaid in anniversary case would be different, alike admitting anniversary claim arrangement would crave absolutely 5 years to accord the loan. The aberration in the absolute amounts repaid can of advance be explained by the time bulk of money, back the bulk of the payments is altered for anniversary plan. With anniversary to equivalence, the table shows that back the absorption bulk is 15% per year, $5000 at time 0 is agnate to $10,056. 80 at the end of year 5 (plan 1), or $750 per year for 4 years and $5750 at the end of year 5 (plan 2), or the abbreviating amounts apparent in years 1 through 5 (plan 3), or $1,491. 8 per year for 5 years (plan 4). Application the formulas developed in Chaps. 2 and 3, we could calmly appearance that if the payments in Analogue and Cash-Flow Diagrams 9 anniversary plan (column 4) were reinvested at 15% per year back received, the absolute bulk of money accessible at the end of year 5 would be $10,056. 80 from anniversary claim plan. Additional Examples 1. 14 and 1. 15 Probs. 1. 8 and 1. 9 1. 4 Simple and Admixture Absorption The concepts of absorption and absorption bulk were alien in Sees. 1. 1 and 1. 2 and ed in Sec. 1. 3 to anniversary for one absorption aeon accomplished and approaching sums of money agnate to a present sum (principal). When added than one absorption aeon is involved, the agreement simple and admixture absorption charge be considered. Simple absorption is affected application the arch only, blank any absorption that was accrued in above-mentioned absorption periods. The absolute absorption can be computed application the affiliation Absorption = (principal)(number of periods)(interest rate) = Pni (1. 4) Archetype 1. 3 If you borrow $1000 for 3 years at 14%-per-year simple interest, how abundant money will you owe at the end of 3 years? Band-aid The absorption for anniversary of the 3 years is = Absorption per year 1000(0. 14) = $140 Absolute absorption for 3 years from Eq. (1. 4) is Absolute absorption = 1000(3)(0. 4)= $420 Finally, the bulk due afterwards 3 years is 1000 + 420 Comment = $1420 The $140 absorption accrued in the aboriginal year and the $140 accrued in the additional year did not acquire interest. The absorption due was affected on the arch only. The after-effects of this accommodation are archival in Table 1. 2. The end-of-year bulk of aught represents th~ present, that is, back the money is borrowed. Agenda that no acquittal is fabricated by the borrower until the end of year 3. Thus, the bulk owed anniversary year increases analogously by $140, back absorption is ample alone on the arch of $1000. Table 1. 2 Simple-interest (1) (2) ciphering (3) (4) (2) + (3) Bulk owed (5) End of year 0 1 2 Bulk adopted $1,000 Absorption Bulk paid 3 $140 140 140 $1,140 1,280 1,420 $ 0 0 1,420 10 Level One In calculations of admixture interest, the absorption for an absorption aeon is affected on the arch additional the absolute bulk of absorption accumulated in antecedent periods. Thus, admixture absorption agency "interest on top of interest" (i. e. , it reflects the aftereffect of the time bulk of money on the absorption too). Archetype 1. 4 If you borrow $1000 at 14%-per-year admixture interest, instead of simple absorption as in the above-mentioned example, compute the absolute bulk due afterwards a 3-year period. Solution The absorption and absolute bulk due for anniversary year is computed as follows: Interest, year 1 = 1000(0. 14) = $140 Absolute bulk due afterwards year 1 = 1000 + 140 = $1140 Interest, year 2 = 1140(0. 14) = $159. 60 Absolute bulk due afterwards year 2 = 1140 + 159. 60 = $1299. 60 Interest, year 3 = 1299. 60(0. 14)= $181. 94 Absolute bulk due afterwards year 3 = 1299. 60 + 181. 94 = $1481. 54 Comment The capacity are apparent in Table 1. 3. The claim arrangement is the aforementioned as that for the simple-interest example; that is, no bulk is repaid until the arch additional all absorption is due at the end of year 3. The time bulk of money is abnormally accustomed in admixture interest. Thus, with admixture interest, the aboriginal $1000 would accrue an added $1481. 54 - $1420 = $61. 54 compared with simple absorption in the 3-year period. If $61. 54 does not accept like a cogent difference, bethink that the alpha bulk actuality was alone $1000. Accomplish these aforementioned calculations for an antecedent bulk of $10 million, and again attending at the admeasurement of the difference! The ability of compounding can added be illustrated through addition absorbing exercise alleged "Pay Now, Play Later". It can be apparent (by application the equations that will be developed in Chap. ) that at an absorption bulk of 12% per year, about $1,000,000 will be accumulated at the end of a 40-year time aeon by either of the Table 1. 3 Compound-interest (1) (2) ciphering (3) (4) = (2) + (3) (5) End of year 0 1 2 3 Bulk adopted $1,000 Absorption Bulk owed $1,140. 00 1,299. 60 1,481. 54 Bulk paid $140. 00 159. 60 181. 94 $ 0 0 1,481. 54 Analogue and Cash-Flow Diagrams 11 - llowing advance schemes: • Plan 1 Advance $2610 anniversary year for the aboriginal 6 years and again annihilation for the abutting 34 years, or • Plan 2 Advance annihilation for the aboriginal 6 years, and again $2600 anniversary year for the abutting 34 years!! 'ote that the absolute advance in plan 1 is $15,660 while the absolute appropriate in plan _ to accrue the aforementioned bulk of money is about six times greater at $88,400. Both the ability of compounding and the acumen of planning for your retirement at he ancient accessible time should be absolutely axiomatic from this example. An absorbing ascertainment pertaining to compound-interest calculations in-olves the admiration of the breadth of time appropriate for a distinct antecedent advance to bifold in value. The alleged aphorism of 72 can be acclimated to appraisal this time. The aphorism i based on the actuality that the time appropriate for an antecedent lump-sum advance to bifold in bulk back absorption is circuitous is about according to 72 disconnected by the absorption bulk that applies. For example, at an absorption bulk of 5% per year, it would booty about 14. 4 years (i. e. , 72/5 = 14. 4) for an antecedent sum of money to bifold in value. (The absolute time appropriate is 14. 3 years, as will be apparent in Chap. 2. ) In Table 1. 4, the times estimated from the aphorism of 72 are compared to the absolute times appropriate for acceleration at assorted absorption ante and, as you can see, absolute acceptable estimates are obtained. Conversely, the absorption bulk that would be appropriate in adjustment for money to bifold in a defined aeon of time could be estimated by adding 72 by the defined time period. Thus, in adjustment for money to bifold in a time aeon of 12 years, an absorption bulk of about 6% per year would be appropriate (i. e. , 72/12 = 6). It should be accessible that for simple-interest situations, the "rule of 100" would apply, except that the answers acquired will consistently be exact. In Chap. 2, formulas are developed which abridge compound-interest calculations. The aforementioned concepts are complex back the absorption aeon is beneath than a year. A altercation of this case is deferred until Chap. 3, however. Back real-world calculations about consistently absorb admixture interest, the absorption ante defined herein accredit to admixture absorption ante unless defined otherwise. Additional Archetype 1. 16 Probs. 1. 10 to 1. 26 Table 1. 4 Acceleration time estimated absolute time from aphorism of 72 against Acceleration lime, no. of periods Absorption rate, % per aeon 1 Estimated from aphorism 72 Absolute 70 35. 3 14. 3 7. 5 2 5 10 20 40 36 14. 4 7. 2 3. 6 1. 8 3. 9 2. 0 12 Level One 1. 5 Symbols and Their Acceptation The algebraic symbols: relations sed in engmeenng abridgement apply the afterward P = bulk or sum of money at a time denoted as the present; dollars, pesos, etc. F A n i = bulk or sum of money at some approaching time; dollars, pesos, etc. = a alternation of consecutive, equal, end-of-period month, dollars per year, etc. amounts of money; dollars per = cardinal of absorption periods; months, years, etc. = absorption bulk per absorption period; percent per month, percent per year, etc. The symbols P and F represent single-time accident values: A occurs at anniversary absorption aeon for a defined cardinal of periods with the aforementioned value. It should be accepted that a present sum P represents a distinct sum of money at some time above-mentioned to a approaching sum or compatible alternation bulk and accordingly does not necessarily accept to be amid at time t = O. Archetype 1. 11 shows a P bulk at a time alternative than t = O. The units of the symbols aid in allegorical their meaning. The present sum P and approaching sum F are bidding in dollars; A is referred to in dollars per absorption period. It is important to agenda actuality that in adjustment for a alternation to be represented by the attribute A, it charge be compatible (i. e. the dollar bulk charge be the aforementioned for anniversary period) and the compatible dollar amounts charge extend through afterwards absorption periods. Both altitude charge abide afore the dollar bulk can be represented by A. Back n is frequently bidding in years or months, A is usually bidding in units of dollars per year or dollars per month, respectively. The compound-interest bulk i is bidding in percent per absorption period, for example, 5% per year. Except area acclaimed otherwise, this bulk applies throughout the absolute n years or n absorption periods. The i bulk is about the minimum adorable bulk of acknowledgment (MARR). All engineering-economy problems charge absorb at atomic four of the symbols listed above, with at atomic three of the ethics known. The afterward four examples allegorize the use of the symbols. Archetype 1. 5. If you borrow $2000 now and charge accord the accommodation additional absorption at a bulk of 12% per year in 5 years, what is the absolute bulk you charge pay? List the ethics of P, F, n, and i. Band-aid In this bearings P and F, but not A, are involved, back all affairs are distinct payments. The ethics are as follows: P = $2000 Archetype 1. 6 i = 12% per year n = 5 years If you borrow $2000 now at 17% per year for 5 years and charge accord the accommodation in according anniversary payments, what will you be appropriate to pay? Actuate the bulk of the symbols involved. Analogue and Cash-Flow Diagrams 13 ~- ution = S2000 = ? per year for 5 years = 17% per year = 5 years - ere is no F bulk involved. - 1 In both examples, the P bulk of $2000 is a cancellation and F or A is a disbursement. appropriately actual to use these symbols in about-face roles, as in the examples below. Archetype 1. 7 T you drop $500 into an anniversary on May 1, 1988, which pays absorption at 17% per year, hat anniversary bulk can you abjure for the afterward 10 years? List the attribute values. Band-aid p = $500 A =? per year i = 17% per year n= 10 years Comment The bulk for the $500 bulk P and cancellation A are accustomed the aforementioned attribute names as before, but they are advised in a altered context. Thus, a P bulk may be a cancellation (Examples 1. 5 and 1. 6) or a bulk (this example). Archetype 1. 8 If you drop $100 into an anniversary anniversary year for 7 years at an absorption bulk of 16% per year, what distinct bulk will you be able to abjure afterwards 7 years? Ascertain the symbols and their roles. Solution In this example, the according anniversary deposits are in a alternation A and the abandonment is a approaching sum, or F value. There is no P bulk here. A = $100 per year for 7 years F =? i = 16% per year n = 7 years Additional Archetype 1. 17 Probs. 1. 27 to 1. 29 14 Level One 1. 6 Cash-Flow Diagrams Every being or aggregation has banknote receipts (income) and banknote disbursements (costs) which action over a accurate time p. These receipts and disbursements in a accustomed time breach are referred to as banknote flow, with absolute banknote flows usually apery receipts and abrogating banknote flows apery disbursements. At any point in time, the net banknote breeze would be represented as Net banknote breeze = receipts - disbursements (1. 5) Back banknote breeze commonly takes abode at common and capricious time intervals aural an absorption period, a simplifying acceptance is fabricated that all banknote breeze occurs at the end of the absorption period. This is accepted as the end-of-period convention. Thus, back several receipts and disbursements action aural a accustomed absorption period, the net banknote breeze is affected to action at the end of the absorption period. However, it should be accepted that although the dollar amounts of F or A are consistently advised to action at the end of the absorption period, this does not beggarly that the end of the aeon is December 31. In the bearings of Archetype 1. 7, back advance took abode on May 1, 1988, the withdrawals will booty abode on May 1, 1989 and anniversary afterwards May 1 for 10 years (the aftermost abandonment will be on May 1, 1998, not 1999). Thus, end of the aeon agency one time aeon from the date of the transaction (whether it be cancellation or disbursement). In the abutting affiliate you will apprentice how to actuate the agnate relations amid P, F, and A ethics at altered times. A cash-flow diagram is artlessly a graphical representation of banknote flows fatigued on a time scale. The diagram should represent the anniversary of the botheration and should accommodate what is accustomed and what is to be found. That is, afterwards the cash-flow diagram has been drawn, an alfresco eyewitness should be able to assignment the botheration by attractive at alone the diagram. Time is advised to be the present and time 1 is the end of time aeon 1. (We will accept that the periods are in years until Chap. . ) The time calibration of Fig. 1. 1 is set up for 5 years. Back it is affected that banknote flows action alone at the end of the year, we will be anxious alone with the times apparent 0, 1, 2, ... , 5. The administration of the arrows on the cash-flow diagram is important to botheration solution. Therefore, in this text, a vertical arrow pointing up will announce a absolute banknote flow. Conversely, an arrow pointing bottomward will announce a abrogating banknote flow. The cash-flow diagram in Fig. 1. 2 illustrates a cancellation (income) at the end of year 1 and a bulk at the end of year 2. It is important that you thoroughly accept the acceptation and architecture of the cash-flow diagram, back it is a admired apparatus in botheration solution. The three examples beneath allegorize the architecture of cash-flow diagrams. ° Bulk 1. 1 A archetypal cash-flow time scale. Year 1 Year 5 r=;:;; r+;:;. I 1 2 Time o I I 3 4 I 5 Analogue and Cash-Flow Diagrams 15 + Bulk 1. 2 Archetype of absolute and abrogating banknote flows. 2 3 Time Archetype 1. 9 Consider the bearings presented in Archetype 1. 5, area P = $2000 is adopted and F is to be activate afterwards 5 years. Construct the cash-flow diagram for this case, bold an absorption bulk of 12% per year. Band-aid Bulk 1. 3 presents the cash-flow diagram. Comment While it is not all-important to use an exact calibration on the cash-flow axes, you will apparently abstain errors afterwards on if you accomplish a accurate diagram. Agenda additionally that the present sum P is a cancellation at year 0 and the approaching sum F is a bulk at the end of year 5. Archetype 1. 10 If you alpha now and accomplish bristles deposits of $1000 per year (A) in a 17%-per-year account, how abundant money will be accumulated (and can be withdrawn) anon afterwards you accept fabricated the aftermost deposit? Construct the cash-flow diagram. Band-aid The banknote flows are apparent in Fig. 1. 4. Back you accept absitively to alpha now, the aboriginal drop is at year 0 and the [lith Comment drop and abandonment action at the end of year 4. Agenda that in this example, the bulk accumulated afterwards the fifth drop is to be computed; thus, the approaching bulk is represented by a catechism mark (i. e. , F = ? ) Bulk 1. 3. Cash-flow diagram for Archetype 1. 9. + P = $2. 000 i = 12% o 2 3 4 5 Year F= ? 16 Bulk 1. 4 Cashflow diagram for Archetype 1. 10. Level One F= ? i = 17"10 2 0 3 4 Year A=$1. 000 Archetype 1. 11 Assume that you appetite to drop an bulk P into an anniversary 2 years from now in adjustment to be able to abjure $400 per year for 5 years starting 3 years from now. Accept that the absorption bulk is 151% per year. Assemble the cash-flow diagram. Bulk 1. 5 presents the banknote flows, area P is to be found. Agenda that the diagram shows what was accustomed and what is to be activate and that a P bulk is not necessarily amid at time t = O. Band-aid Additional Examples 1. 18 to 1. 20 Probs. 1. 30 to 1. 46 Additional Examples Archetype 1. 12 Anniversary the absorption and absolute bulk accrued afterwards 1 year if $2000 is invested at an absorption bulk of 15% per year. Solution Absorption becoming = 2000(0. 15) = $300 Absolute bulk accrued = 2000 + 2000(0. 15) = 2000(1 + 0. 15) = $2300 Bulk 1. 5 Cashflow diagram for Archetype 1. 11. A = $400 o 2 3 4 5 6 7 Year p=? Analogue and Cash-Flow Diagrams 17 Archetype 1. 13 a) Anniversary the bulk of money that charge accept been deposited 1 year ago for you to accept $lOQO now at an absorption bulk of 5% per year. b) Anniversary the absorption that was becoming in the aforementioned time period. Band-aid a) Absolute bulk accrued = aboriginal drop + (original deposit)(interest rate). If X = aboriginal deposit, again 1000 = X + X(0. 5) = X(l + 0. 05) 1000 = 1. 05X 1000 X=-=952. 38 1. 05 Aboriginal drop = $952. 38 (b) By application Eq. (1. 1), we accept Absorption = 1000 - 952. 38 = $47. 62 Archetype 1. 14 Anniversary the bulk of money that charge accept been deposited 1 year ago for the advance to acquire $100 in absorption in 1 year, if the absorption bulk is 6% Per year. Band-aid Let a = a = = absolute bulk accrued and b = aboriginal deposit. Absorption Back a Absorption Absorption b b + b (interest rate), absorption can be bidding as + b (interest rate) b =b = b (interest rate) $100 = b(0. 06) b = 100 = $1666. 67 0. 06 Archetype 1. 5 Accomplish the calculations all-important to appearance which of the statements beneath are accurate and which are false, if the absorption bulk is 5% per year: (a) $98 now is agnate to $105. 60 one year from now. (b) $200 one year accomplished is agnate to $205 now. (c) $3000 now is agnate to $3150 one year from now. (d) $3000 now is agnate to $2887. 14 one year ago. (e) Absorption accumulated in 1 year on an advance of $2000 is $100. Band-aid (a) Absolute bulk accrued = 98(1. 05) = $102. 90 =P $105. 60; accordingly false. Addition way to break this is as follows: Appropriate advance = 105. 60/1. 05 = $100. 57 =P $9? Accordingly false. b) Appropriate advance = 205. 00/1. 05 = $195. 24 =p $200; accordingly false. 18 Level One (e) Absolute bulk accrued = 3000(1. 05) = $3150; accordingly true. (d) Absolute bulk accrued = 2887. 14(1. 05) = $3031. 50 "# $3000; accordingly false. (e) Absorption = 2000(0. 05) = $100; accordingly true. Archetype 1. 16 Anniversary the absolute bulk due afterwards 2 years if $2500 is adopted now and the compoundinterest bulk is 8% per year. Band-aid The after-effects are presented in the table to access a absolute bulk due of $2916. (1) (2) (3) (4) = (2) + (3) (5) End of year Bulk adopted $2,500 Absorption Bulk owed Bulk paid o 1 2 Archetype 1. 17 $200 216 2,700 2,916 $0 2,916 Accept that 6% per year, starting abutting abandoning Band-aid P = you plan to accomplish a lump-sum drop of $5000 now into an anniversary that pays and you plan to abjure an according end-of-year bulk of $1000 for 5 years year. At the end of the sixth year, you plan to abutting your anniversary by the absolute money. Ascertain the engineering-economy symbols involved. $5000 A = $1000 per year for 5 years F = ? at end of year 6 i = 6% per year n = 5 years for A Bulk 1. 6 Cashflow diagram for Archetype 1. 18. $650 $625 $600 $575 $ 550 $525 $500 $625 t -7 -6 -5 -4 -3 -2 -1 t o Year P = $2,500 Analogue and Cash-Flow Diagrams 19 Archetype 1. 1B The Hot-Air Aggregation invested $2500 in a new air compressor 7 years ago. Anniversary assets "-om the compressor was $750. During the aboriginal year, $100 was spent on maintenance, _ bulk that added anniversary year by $25. The aggregation affairs to advertise the compressor for deliver at the end of abutting year for $150. Assemble the cash-flow diagram for the allotment f equipment. The assets and bulk for years - 7 through 1 (next year) are archival low with net banknote breeze computed application Eq. (1. 5). The banknote flows are diagrammed . Fig. 1. 6. Band-aid End of year Net banknote breeze Assets Bulk -7 -6 -5 -4 -3 -2 -1 0 1 Archetype 0 750 750 750 750 750 750 750 750 + 150 $2,500 100 125 150 175 200 225 250 275 $-2,500 650 625 600 575 550 525 500 625 1. 19 Suppose that you appetite to accomplish a drop into your anniversary now such that you can abjure an according anniversary bulk of Ai = $200 per year for the aboriginal 5 years starting 1 year afterwards your drop and a altered anniversary bulk of A2 = $300 per year for the afterward 3 years. How would the cash-flow diagram arise if i is 14! % per year? Band-aid The banknote flows would arise as apparent in Fig. 1. 7. Comment The aboriginal abandonment (positive banknote flow) occurs at the end of year 1, absolutely one year afterwards P is deposited. Figure 1. 7 Cash-flow diagram for two altered A values, Archetype 1. 19. A2 = $300 A, = $200 0 1 2 3 4 i = 14+% 5 6 7 8 Year p=? 20 Level One p=? j = 12% per year Bulk 1. 8 Cash-flow diagram for Archetype 1. 20. F2 1996 1995 A = $50 A = $150 = $50 F, = $900 Archetype 1. 20 If you buy a new television set in 1996 for $900,. advance it for 3 years at a bulk of $50 per year, and again advertise it for $200, diagram your banknote flows and characterization anniversary arrow as P, F, or A with its corresponding dollar bulk so that you can acquisition the distinct bulk in 1995 that would be agnate to all of the banknote flows shown. Assume an absorption bulk of 12% per year. Band-aid Comment Bulk 1. 8 presents the cash-flow diagram. The two $50 abrogating banknote flows anatomy a alternation of two according end-of-year values. As continued as the dollar ethics are according and in two or added afterwards periods, they can be represented by A, behindhand of area they activate or end. However, the $150 absolute banknote breeze in 1999 is a single-occurrence bulk in the approaching and is accordingly labeled an F value. It is possible, however, to appearance all of the alone banknote flows as F values. The diagram could be fatigued as apparent in Fig. . 9. In general, however, if two or added according end-of-period amounts action consecutively, by the analogue in Sec. 105 they should be labeled A ethics because, as is declared in Chap. 2, the use of A ethics back accessible simplifies calculations considerably. Thus, the estimation pictured by the diagram of Fig. 1. 9 is beat and will not about be acclimated added in this text. p=? j = 12% per year F. = $150 1. 9 A banknote breeze for Archetype 1. 20 because all ethics as approaching sums. Bulk 1996 1995 1997 1998 1999 F2 = $50 F3 = $50 F, = $900

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