Engineering Economics

Eng ineeri ng Distribution 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 Magnitude Assembly 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. Bound and avow in a gist announcement the distribution sorts P, F, A, n, and i. 1. 6 Bound coin issue, aver what is averaget by end-of-conclusion treaty, and frame a coin-issue diagram, consecrated a announcement describing the whole and conclusions of the coin issues. Study Guide 1. 1 Basic Terminology Anteriorly we originate to enlarge the terminology and leading concepts upon which engineering distribution is grounded, it would be expend to bound what is averaget by engineering distribution. In the humblest stipulations, engineering distribution is a collation of despicablesituate techniques which disencumber economic congruousitys. With these techniques, a reasonable, significanceful adit to evaluating the economic countenances of opposed manages of accomplishing a consecrated concrete can be enlargeed. Engineering distribution is, hence, a conclusion livelihood machine by which one manage gain be separated as the most superfluous one. In manage for you to be telling to engage the techniques, nevertheless, it is needful for you to apprehend the basic terminology and leading concepts that contrive the substratum for engineering-distribution studies. Some of these stipulations and concepts are descriptive underneath. An choice is a stand-alone key for a confer standing. We are faced behind a while choices in virtually anything we do, from selecting the manage of transportation we use to get to labor complete day to deciding among buying a scion or renting one. Similarly, in engineering exercitation, tshort are constantly seveffl ways of accomplishing a consecrated job, and it is needful to be telling to parallel them in a reasontelling contrive so that the most superfluous choice can be chosen. The choices in engineering considerations usually grasp such items as donation absorb (highest absorb), the anticipated separateality of the asset, the every-year absorbs of concealing the asset (annual livelihood and liberal absorb), the anticipated resale appreciate (salvage appreciate), and the share reprimand (reprimand of reappear). Behind the grounds and all the applictelling prizes entertain been composed, an engineering-distribution sunderition can be conducted to separateize which is best from an economic apex of light. However, it should be apexed out that the procedures enlargeed in this magnitude gain entelling you to compel accureprimand economic conclusions singly about those choices which entertain been avowd as choices; these procedures gain not aid you substantiate what the choices are. That is, if choices ,4, B, C, D, and E entertain been identified as the singly potential manages to rereexplain a Feature gist when manage F, which was never avowd as an choice, is veritably the most pleasant manage, the evil-doing conclusion is infallible to be made consequently choice F could never be separated, no substance what analytical techniques are used. Thus, the significance of choice identification in the conclusion-making manner cannot be poiseemphasized, consequently it is singly when this countenance of the manner has been wholely completed that the sunderition techniques confer-uponed in this magnitude can be of highest appreciate. In manage to be telling to parallel opposed manages for accomplishing a consecrated concrete, it is needful to entertain an evaluation proof that can be used as a premise Terminology and Cash-Flow Diagrams 5 for judging the choices. That is, the evaluation proof is that which is used to reply the scrutiny "How gain I conceive which one is best? Whether we are assured of it or not, this scrutiny is asked of us numerous conclusions each day. For specimen, when we solicit to labor, we subconsciously opine that we are commencement the "best" way. But how did we bound best? Was the best way the safest, shortest, fastest, cheapest, most sumptuous, or what? Obviously, depending upon which proof is used to substantiate the best, a opposed way faculty be chosen each conclusion! (Many arguments could entertain been deserted if the conclusion compelrs had barely averd the criteria they were using in determining the best). In economic sunderition, dollars are openly used as the premise for congruousity. Thus, when tshort are uncertain ways of accomplishing a consecrated concrete, the manage that has the smallest poiseall absorb is usually chosen. However, in most subjects the choices grasp imperceptible smoothtors, such as the consequence of a manner qualify on habituateee morale, which cannot preparedly be explicit in stipulations of dollars. When the choices availtelling entertain ceasely the corresponding equipollent absorb, the nonquantifiable, or imperceptible, smoothtors may be used as the premise for selecting the best choice, For items of an choice which can be quantified in stipulations of dollars, it is leading to avow the concept of the conclusion appreciate of specie. It is usually said that specie compels specie. The announcement is verily gentleman, for if we choice to endue specie today (for specimen, in a bank or savings and hypothecation fellowship), by tomorrow we gain entertain convened departed specie than we had anciently endueed. This qualify in the whole of specie poise a consecrated conclusion conclusion is named the conclusion appreciate of specie; it is the most leading concept in engineering distribution. You should as-well produce that if a separate or assembly furnishs it needful to attribute specie today, by tomorrow departed specie than the ancient hypothecation gain be appropriate. This smootht is as-well explained by the conclusion appreciate of specie. The verification of the conclusion appreciate of specie is termed share, which is a mete of the extension among the ancient sum borrappropriate or endueed and the latest whole appropriate or accrued. Thus, if you endueed specie at some conclusion in the departed, the share would be Share = whole whole convened - ancient enduement (1. 1) On the other compositionman, if you borrappropriate would be Share specie at some conclusion in the departed, the share (1. 2) = confer-upon whole appropriate - ancient hypothecation In either subject, tshort is an extension in the whole of specie that was anciently endueed or ascititious, and the extension poise the ancient whole is the share. The ancient enduement or hypothecation is attributered to as chief. Probs. 1. 1 to 1. 4 1. 2 Share Calculations When share is explicit as a percentage of the ancient whole per item conclusion, the product is an share reprimand. This reprimand is conducive as follows: . Percent share reprimand = share accrued per item conclusion 00% .. I x 1 0 origma whole (1. 3) 6 Level One By far the most despicserviceserviceefficient conclusion conclusion used for expressing share reprimands is 1 year. However, departed share reprimands are usually explicit poise conclusions of conclusion shorter than 1 year (i. e. 1% per month), the conclusion item used in expressing an share reprimand must as-well be identified and is termed an share conclusion. The behindcited two specimens illustreprimand the deduction of share reprimand. Specimen 1. 1 The Get-Rich-Quick (GRQ) Assembly endueed $100,000 on May 1 and behind a whiledrew a whole of $106,000 correspondently one year behind. Compute (a) the share gained from the ancient enduement and (b) the share reprimand from the enduement. Key (a) Using Eq. (1. 1), Share = 106,000 - 100,000 = $6000 (b) Equation (1. 3) is used to gain Percent share reprimand = 6000 per year 100,000 x 100% = 6% per year Comment For borrappropriate specie, deductions are congruous to those demonstrationn aggravatehead bar that share is computed by Eq. (1. 2). For specimen, if GRQ borrappropriate $100,000 now and reremunerated $110,000 in 1 year, using Eq. (1. 2) we furnish that share is $10,000, and the share reprimand from Eq. (1. 3) is 10% per year. Specimen 1. 2 Joe Bilder drawings to attribute $20,000 for 1 year at 15% share. Compute (a) the share and (b) the whole whole due behind 1 year. Key (a) Equation (1. 3) may be resolved for the share accrued to gain Share = 20,000(0. 15) = $3000 (b) Whole whole due is the sum of chief and share or Whole due Comment = 0,000 + 3000 = $23,000 Silence that in sunder (b) aggravatehead, the whole whole due may as-well be computed as Whole due = chief(l + share reprimand) = 20,000(1. 15) = $23,000 In each specimen the share conclusion was 1 year and the share was conducive at the end of one conclusion. When departed than one every-year share conclusion is graspd (for specimen, if we had scantinessed to conceive the whole of share Joe Bilder would owe on Terminology and Cash-Flow Diagrams 7 the aggravatehead hypothecation behind 3 years), it becomes needful to separateize whether the share . paytelling on a humble or unification premise. The concepts of humble and unification share are discussed in Sec. . 4. Additional Examples 1. 12 and 1. 13 Probs. 1. 5 to 1. 7 1. 3 Equivalence The conclusion appreciate of specie and share reprimand utilized simultaneously genereprimand the concept of equivalence, which media that opposed sums of specie at opposed conclusions can be similar in economic appreciate. For specimen, if the share reprimand is 12% per year, $100 today (i. e. , at confer-upon) would be equipollent to $112 one year from today, departed soar accrued = 100 =$112 Thus, if someone exhibited you a alms of $100 today or $112 one year from today, it would compel no dissent which exhibit you penny, departed in either subject you would entertain $112 one year from today. The two sums of specie are hence equipollent to each other when the share reprimand is 12% per year. At either a remarkserviceserviceefficient or a inferior share reprimand, nevertheless, $100 today is not equipollent to $112 one year from today. In separateization to accordingly forthcoming equivalence, one can engage the corresponding concepts for determining equivalence in contriveer years. Thus, $100 now would be equipollent to 100/1. 12 = $89. 29 one year ago if the share reprimand is 12% per year. From these specimens, it should be transparent that $89. 29 latest year, $100 now, and 112 one year from now are equipollent when the share reprimand is 12% per year. The smootht that these sums are equipollent can be systematic by computing the share reprimand as follows: 112 100 = 1. 12, or 12% per year and 8~~~9 = 1. 12, or 12% per year The concept of equivalence can be exalt exemplifyd by accordingly opposed hypothecation-retribution theorys. Each theory resembles retribution of a $5000 hypothecation in 5 years at 15%-per-year share. Ttelling 1. 1 confer-upons the details for the lewd retribution manages descriptive underneath. (The manages for determining the whole of the reimbursements are confer-uponed in Chaps. 2 and 3. ) • Drawing 1 a share or chief is recovered until the fifth year. Interest convenes each year on the whole of chief and all convened share. • Drawing 2 The accrued share is remunerated each year and the chief is recovered at the end of 5 years. • Drawing 3 The accrued share and 20% of the chief, that is, $1000, is remunerated each year. Departed the retaining hypothecation poise decreases each year, the accrued share decreases each year. + 100(0. 12) = 100(1 + 0. 12) = 100(1. 12) 8 Level One Ttelling 1. 1 Opposed retribution schedules of $5,000 at 15% for 5 years (1) End of year (2) = 0. 15(5) Share for year (3) = (2) + (5) Whole appropriate at end of year (4) Reimbursement per drawing (3) - (4) Poise behind reimbursement (5) Drawing 1 0 1 2 3 4 5 Drawing 2 0 1 2 3 4 5 Drawing 3 0 1 2 3 4 5 Drawing 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 • Drawing 4 Similar reimbursements are made each year behind a while a concern going inland princi- pal replacement and the leavings covering the accrued share. Departed the hypothecation poise decreases at a reprimand which is sinferior than in drawing 3 consequently of the similar end-of-year reimbursements, the share decreases, but at a reprimand sinferior than in drawing 3. te that the whole whole reremunerated in each subject would be opposed, smooth though each retribution theory would straight correspondently 5 years to fund the hypothecation. The dissent in the whole wholes reremunerated can of conduct be explained by the conclusion appreciate of specie, departed the whole of the reimbursements is opposed for each drawing. Behind a while honor to equivalence, the ttelling demonstrations that when the share reprimand is 15% per year, $5000 at conclusion 0 is equipollent to $10,056. 80 at the end of year 5 (drawing 1), or $750 per year for 4 years and $5750 at the end of year 5 (drawing 2), or the decreasing wholes demonstrationn in years 1 through 5 (drawing 3), or $1,491. 8 per year for 5 years (drawing 4). Using the contriveulas enlargeed in Chaps. 2 and 3, we could abundantly demonstration that if the reimbursements in Terminology and Cash-Flow Diagrams 9 each drawing (shaft 4) were reinvested at 15% per year when ordinary, the whole whole of specie availtelling at the end of year 5 would be $10,056. 80 from each retribution drawing. Additional Examples 1. 14 and 1. 15 Probs. 1. 8 and 1. 9 1. 4 Humble and Unification Share The concepts of share and share reprimand were introduced in Sees. 1. 1 and 1. 2 and ed in Sec. 1. 3 to apportion for one share conclusion departed and forthcoming sums of specie equipollent to a confer-upon sum (principal). When departed than one share conclusion is graspd, the stipulations humble and unification share must be considered. Humble share is conducive using the chief singly, ignoring any share that was accrued in antecedent share conclusions. The whole share can be computed using the fitness Share = (principal)(prize of conclusions)(share reprimand) = Pni (1. 4) Specimen 1. 3 If you attribute $1000 for 3 years at 14%-per-year humble share, how abundant specie gain you owe at the end of 3 years? Key The share for each of the 3 years is = Share per year 1000(0. 14) = $140 Whole share for 3 years from Eq. (1. 4) is Whole share = 1000(3)(0. 4)= $420 Finally, the whole due behind 3 years is 1000 + 420 Comment = $1420 The $140 share accrued in the highest year and the $140 accrued in the relieve year did not realize share. The share due was conducive on the chief singly. The products of this hypothecation are tabulated in Ttelling 1. 2. The end-of-year condition of naught resembles th~ confer-upon, that is, when the specie is ascititious. Silence that no reimbursement is made by the attributeer until the end of year 3. Thus, the whole appropriate each year extensions unintermittently by $140, departed share is conditiond singly on the chief of $1000. Ttelling 1. 2 Simple-share (1) (2) deduction (3) (4) (2) + (3) Whole appropriate (5) End of year 0 1 2 Whole borrappropriate $1,000 Share Whole remunerated 3 $140 140 140 $1,140 1,280 1,420 $ 0 0 1,420 10 Level One In calculations of unification share, the share for an share conclusion is conducive on the chief plus the whole whole of share convened in contriveer conclusions. Thus, unification share media "share on top of share" (i. e. , it reflects the consequence of the conclusion appreciate of specie on the share too). Specimen 1. 4 If you attribute $1000 at 14%-per-year unification share, instead of humble share as in the antecedent specimen, compute the whole whole due behind a 3-year conclusion. Solution The share and whole whole due for each year is computed as follows: Interest, year 1 = 1000(0. 14) = $140 Whole whole due behind year 1 = 1000 + 140 = $1140 Interest, year 2 = 1140(0. 14) = $159. 60 Whole whole due behind year 2 = 1140 + 159. 60 = $1299. 60 Interest, year 3 = 1299. 60(0. 14)= $181. 94 Whole whole due behind year 3 = 1299. 60 + 181. 94 = $1481. 54 Comment The details are demonstrationn in Ttelling 1. 3. The retribution theory is the corresponding as that for the humble-share specimen; that is, no whole is reremunerated until the chief plus all share is due at the end of year 3. The conclusion appreciate of specie is distinctly avowd in unification share. Thus, behind a while unification share, the ancient $1000 would convene an extra $1481. 54 - $1420 = $61. 54 paralleld behind a while humble share in the 3-year conclusion. If $61. 54 does not show approve a suggestive dissent, retain that the originatening whole short was singly $1000. Compel these corresponding calculations for an modescold whole of $10 darling, and then observe at the bigness of the dissent! The potentiality of unificationing can exalt be exemplifyd through another shareing use named "Pay Now, Play Later". It can be demonstrationn (by using the equations that gain be enlargeed in Chap. ) that at an share reprimand of 12% per year, ceasely $1,000,000 gain be convened at the end of a 40-year conclusion conclusion by either of the Ttelling 1. 3 Compound-share (1) (2) deduction (3) (4) = (2) + (3) (5) End of year 0 1 2 3 Whole borrappropriate $1,000 Share Whole appropriate $1,140. 00 1,299. 60 1,481. 54 Whole remunerated $140. 00 159. 60 181. 94 $ 0 0 1,481. 54 Terminology and Cash-Flow Diagrams 11 - llowing enduement theorys: • Drawing 1 Endue $2610 each year for the highest 6 years and then rush for the instant 34 years, or • Drawing 2 Endue rush for the highest 6 years, and then $2600 each year for the instant 34 years!! 'ote that the whole enduement in drawing 1 is $15,660 while the whole straightd in drawing _ to convene the corresponding whole of specie is ceasely six conclusions main at $88,400. Twain the potentiality of unificationing and the erudition of drawingning for your retreat at he chief potential conclusion should be fullly manifest from this specimen. An shareing notice pertaining to unification-share calculations in-olves the repute of the protraction of conclusion straightd for a uncombined modescold enduement to envelop in appreciate. The so-named administration of 72 can be used to prize this conclusion. The administration i grounded on the smootht that the conclusion straightd for an modescold lump-sum enduement to envelop in appreciate when share is unificationed is ceasely similar to 72 divided by the share reprimand that applies. For specimen, at an share reprimand of 5% per year, it would follow ceasely 14. 4 years (i. e. , 72/5 = 14. 4) for an modescold sum of specie to envelop in appreciate. (The exposed conclusion straightd is 14. 3 years, as gain be demonstrationn in Chap. 2. ) In Ttelling 1. 4, the conclusions prized from the administration of 72 are paralleld to the exposed conclusions straightd for doubling at uncertain share reprimands and, as you can see, very good-tempered-tempered prizes are gained. Conversely, the share reprimand that would be straightd in manage for specie to envelop in a decisive conclusion of conclusion could be prized by dividing 72 by the decisive conclusion conclusion. Thus, in manage for specie to envelop in a conclusion conclusion of 12 years, an share reprimand of ceasely 6% per year would be straightd (i. e. , 72/12 = 6). It should be manifest that for humble-share standings, the "administration of 100" would engage, bar that the replys gained gain constantly be straight. In Chap. 2, contriveulas are enlargeed which disencumber unification-share calculations. The corresponding concepts are graspd when the share conclusion is cease than a year. A discourse of this subject is amplifyed until Chap. 3, nevertheless. Departed real-world calculations ceasely constantly grasp unification share, the share reprimands decisive shortin attribute to unification share reprimands uncease decisive inadequately. Additional Specimen 1. 16 Probs. 1. 10 to 1. 26 Ttelling 1. 4 Doubling conclusion prized exposed conclusion from administration of 72 versus Doubling lime, no. of conclusions Share reprimand, % per conclusion 1 Estimated from administration 72 Exposed 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 Significance The despicablesituate sorts: fitnesss sed in engmeenng distribution habituate the behindcited P = appreciate or sum of specie at a conclusion deexalted as the confer-upon; dollars, pesos, etc. F A n i = appreciate or sum of specie at some forthcoming conclusion; dollars, pesos, etc. = a rotation of continuous, similar, end-of-conclusion month, dollars per year, etc. wholes of specie; dollars per = prize of share conclusions; months, years, etc. = share reprimand per share conclusion; percent per month, percent per year, etc. The sorts P and F reconfer-upon uncombined-conclusion adventure appreciates: A arises at each share conclusion for a decisive prize of conclusions behind a while the corresponding appreciate. It should be silent that a confer-upon sum P resembles a uncombined sum of specie at some conclusion earlier to a forthcoming sum or unicontrive rotation whole and hence does not necessarily entertain to be located at conclusion t = O. Specimen 1. 11 demonstrations a P appreciate at a conclusion other than t = O. The items of the sorts aid in clarifying their significance. The confer-upon sum P and forthcoming sum F are explicit in dollars; A is attributered to in dollars per share conclusion. It is leading to silence short that in manage for a rotation to be resembleed by the sort A, it must be unicontrive (i. e. the dollar appreciate must be the corresponding for each conclusion) and the unicontrive dollar wholes must amplify through continuous share conclusions. Twain stipulations must consist anteriorly the dollar appreciate can be resembleed by A. Departed n is commsingly explicit in years or months, A is usually explicit in items of dollars per year or dollars per month, honorively. The unification-share reprimand i is explicit in percent per share conclusion, for specimen, 5% per year. Bar wshort silenced inadequately, this reprimand applies throughout the full n years or n share conclusions. The i appreciate is usually the minimum pleasant reprimand of reshow (MARR). All engineering-distribution gists must grasp at lowest lewd of the sorts listed aggravatehead, behind a while at lowest three of the appreciates conceiven. The behindcited lewd specimens illustreprimand the use of the sorts. Specimen 1. 5. If you attribute $2000 now and must fund the hypothecation plus share at a reprimand of 12% per year in 5 years, what is the whole whole you must pay? List the appreciates of P, F, n, and i. Key In this standing P and F, but not A, are graspd, departed all proceedings are uncombined reimbursements. The appreciates are as follows: P = $2000 Specimen 1. 6 i = 12% per year n = 5 years If you attribute $2000 now at 17% per year for 5 years and must fund the hypothecation in similar every-year reimbursements, what gain you be straightd to pay? Individualize the appreciate of the sorts graspd. Terminology and Cash-Flow Diagrams 13 ~- ution = S2000 = ? per year for 5 years = 17% per year = 5 years - ere is no F appreciate graspd. - 1 In twain specimens, the P appreciate of $2000 is a reception and F or A is a expenditure. similarly punish to use these sorts in counterposition roles, as in the specimens underneath. Specimen 1. 7 T you safety $500 into an wholeity on May 1, 1988, which pays share at 17% per year, hat annual whole can you behind a whiledraw for the behindcited 10 years? List the sort appreciates. Key p = $500 A =? per year i = 17% per year n= 10 years Comment The appreciate for the $500 expenditure P and reception A are consecrated the corresponding sort names as anteriorly, but they are considered in a opposed treatment. Thus, a P appreciate may be a reception (Examples 1. 5 and 1. 6) or a expenditure (this specimen). Specimen 1. 8 If you safety $100 into an wholeity each year for 7 years at an share reprimand of 16% per year, what uncombined whole gain you be telling to behind a whiledraw behind 7 years? Bound the sorts and their roles. Solution In this specimen, the similar annual safetys are in a rotation A and the behind a whiledrawal is a forthcoming sum, or F appreciate. Tshort is no P appreciate short. A = $100 per year for 7 years F =? i = 16% per year n = 7 years Additional Specimen 1. 17 Probs. 1. 27 to 1. 29 14 Level One 1. 6 Cash-Flow Diagrams Complete separate or assembly has coin receptions (income) and coin expenditures (costs) which arise poise a sundericular conclusion p. These receptions and expenditures in a consecrated conclusion averagespan are attributered to as coin issue, behind a while decisive coin issues usually resembleing receptions and disclaiming coin issues resembleing expenditures. At any apex in conclusion, the net coin issue would be resembleed as Net coin issue = receptions - expenditures (1. 5) Departed coin issue normally follows situate at usual and varying conclusion averagetimes behind a whilein an share conclusion, a disencumbering self-assertion is made that all coin issue arises at the end of the share conclusion. This is conceiven as the end-of-conclusion treaty. Thus, when uncertain receptions and expenditures arise behind a whilein a consecrated share conclusion, the net coin issue is antecedent to arise at the end of the share conclusion. However, it should be silent that although the dollar wholes of F or A are constantly considered to arise at the end of the share conclusion, this does not average that the end of the conclusion is December 31. In the standing of Specimen 1. 7, departed enduement took situate on May 1, 1988, the behind a whiledrawals gain follow situate on May 1, 1989 and each subsequent May 1 for 10 years (the latest behind a whiledrawal gain be on May 1, 1998, not 1999). Thus, end of the conclusion media one conclusion conclusion from the conclusion of the proceeding (whether it be reception or expenditure). In the instant section you gain lrealize how to separateize the equipollent fitnesss among P, F, and A appreciates at opposed conclusions. A coin-issue diagram is barely a graphical resembleation of coin issues drawn on a conclusion layer. The diagram should reconfer-upon the announcement of the gist and should grasp what is consecrated and what is to be fix. That is, behind the coin-issue diagram has been drawn, an beyond witness should be telling to labor the gist by observeing at singly the diagram. Conclusion is considered to be the confer-upon and conclusion 1 is the end of conclusion conclusion 1. (We gain presume that the conclusions are in years until Chap. . ) The conclusion layer of Fig. 1. 1 is set up for 5 years. Departed it is antecedent that coin issues arise singly at the end of the year, we gain be careful singly behind a while the conclusions notefficient 0, 1, 2, ... , 5. The bearing of the arrows on the coin-issue diagram is leading to gist key. Therefore, in this quotation, a perpendicular arrow apexing up gain denote a decisive coin issue. Conversely, an arrow apexing down gain denote a disclaiming coin issue. The coin-issue diagram in Fig. 1. 2 exemplifys a reception (income) at the end of year 1 and a expenditure at the end of year 2. It is leading that you wholely apprehend the significance and frameion of the coin-issue diagram, departed it is a valutelling machine in gist key. The three specimens underneath illustreprimand the frameion of coin-issue diagrams. ° Condition 1. 1 A regular coin-issue conclusion layer. Year 1 Year 5 r=;:;; r+;:;. I 1 2 Conclusion o I I 3 4 I 5 Terminology and Cash-Flow Diagrams 15 + Condition 1. 2 Specimen of decisive and disclaiming coin issues. 2 3 Conclusion Specimen 1. 9 Consider the standing confer-uponed in Specimen 1. 5, wshort P = $2000 is borrappropriate and F is to be fix behind 5 years. Construct the coin-issue diagram for this subject, inflated an share reprimand of 12% per year. Key Condition 1. 3 confer-upons the coin-issue diagram. Comment While it is not needful to use an straight layer on the coin-issue axes, you gain probably desert errors behind on if you compel a handy diagram. Silence as-well that the confer-upon sum P is a reception at year 0 and the forthcoming sum F is a expenditure at the end of year 5. Specimen 1. 10 If you initiate now and compel five safetys of $1000 per year (A) in a 17%-per-year wholeity, how abundant specie gain be convened (and can be behind a whiledrawn) instantly behind you entertain made the latest safety? Construct the coin-issue diagram. Key The coin issues are demonstrationn in Fig. 1. 4. Departed you entertain unwavering to initiate now, the highest safety is at year 0 and the [lith Comment safety and behind a whiledrawal arise at the end of year 4. Silence that in this specimen, the whole convened behind the fifth safety is to be computed; thus, the forthcoming whole is resembleed by a scrutiny symptom (i. e. , F = ? ) Condition 1. 3. Cash-issue diagram for Specimen 1. 9. + P = $2. 000 i = 12% o 2 3 4 5 Year F= ? 16 Condition 1. 4 Cashissue diagram for Specimen 1. 10. Level One F= ? i = 17"10 2 0 3 4 Year A=$1. 000 Specimen 1. 11 Assume that you scantiness to safety an whole P into an wholeity 2 years from now in manage to be telling to behind a whiledraw $400 per year for 5 years initiateing 3 years from now. Presume that the share reprimand is 151% per year. Frame the coin-issue diagram. Condition 1. 5 confer-upons the coin issues, wshort P is to be fix. Silence that the diagram demonstrations what was consecrated and what is to be fix and that a P appreciate is not necessarily located at conclusion t = O. Key Additional Examples 1. 18 to 1. 20 Probs. 1. 30 to 1. 46 Additional Examples Specimen 1. 12 Apportion the share and whole whole accrued behind 1 year if $2000 is endueed at an share reprimand of 15% per year. Solution Share realizeed = 2000(0. 15) = $300 Whole whole accrued = 2000 + 2000(0. 15) = 2000(1 + 0. 15) = $2300 Condition 1. 5 Cashissue diagram for Specimen 1. 11. A = $400 o 2 3 4 5 6 7 Year p=? Terminology and Cash-Flow Diagrams 17 Specimen 1. 13 a) Apportion the whole of specie that must entertain been safetyed 1 year ago for you to entertain $lOQO now at an share reprimand of 5% per year. b) Apportion the share that was realizeed in the corresponding conclusion conclusion. Key a) Whole whole accrued = ancient safety + (ancient safety)(share reprimand). If X = ancient safety, then 1000 = X + X(0. 5) = X(l + 0. 05) 1000 = 1. 05X 1000 X=-=952. 38 1. 05 Ancient safety = $952. 38 (b) By using Eq. (1. 1), we entertain Share = 1000 - 952. 38 = $47. 62 Specimen 1. 14 Apportion the whole of specie that must entertain been safetyed 1 year ago for the enduement to realize $100 in share in 1 year, if the share reprimand is 6% Per year. Key Let a = a = = whole whole accrued and b = ancient safety. Share Departed a Share Profit b b + b (share reprimand), share can be explicit as + b (share reprimand) b =b = b (share reprimand) $100 = b(0. 06) b = 100 = $1666. 67 0. 06 Specimen 1. 5 Compel the calculations needful to demonstration which of the announcements underneath are gentleman and which are spurious, if the share reprimand is 5% per year: (a) $98 now is equipollent to $105. 60 one year from now. (b) $200 one year departed is equipollent to $205 now. (c) $3000 now is equipollent to $3150 one year from now. (d) $3000 now is equipollent to $2887. 14 one year ago. (e) Share convened in 1 year on an enduement of $2000 is $100. Key (a) Whole whole accrued = 98(1. 05) = $102. 90 =P $105. 60; hence spurious. Another way to rereexplain this is as follows: Required enduement = 105. 60/1. 05 = $100. 57 =P $9? Hence spurious. b) Required enduement = 205. 00/1. 05 = $195. 24 =p $200; hence spurious. 18 Level One (e) Whole whole accrued = 3000(1. 05) = $3150; hence gentleman. (d) Whole whole accrued = 2887. 14(1. 05) = $3031. 50 "# $3000; hence spurious. (e) Share = 2000(0. 05) = $100; hence gentleman. Specimen 1. 16 Apportion the whole whole due behind 2 years if $2500 is borrappropriate now and the unificationshare reprimand is 8% per year. Key The products are confer-uponed in the ttelling to gain a whole whole due of $2916. (1) (2) (3) (4) = (2) + (3) (5) End of year Whole borrappropriate $2,500 Share Whole appropriate Whole remunerated o 1 2 Specimen 1. 17 $200 216 2,700 2,916 $0 2,916 Presume that 6% per year, initiateing instant behind a whiledrawing Key P = you drawing to compel a lump-sum safety of $5000 now into an wholeity that pays and you drawing to behind a whiledraw an similar end-of-year whole of $1000 for 5 years year. At the end of the sixth year, you drawing to cease your wholeity by the retaining specie. Bound the engineering-distribution sorts graspd. $5000 A = $1000 per year for 5 years F = ? at end of year 6 i = 6% per year n = 5 years for A Condition 1. 6 Cashissue diagram for Specimen 1. 18. $650 $625 $600 $575 $ 550 $525 $500 $625 t -7 -6 -5 -4 -3 -2 -1 t o Year P = $2,500 Terminology and Cash-Flow Diagrams 19 Specimen 1. 1B The Hot-Air Assembly endueed $2500 in a new air compressor 7 years ago. Annual pay "-om the compressor was $750. During the highest year, $100 was departed on livelihood, _ absorb that extensiond each year by $25. The assembly drawings to dispose-of the compressor for salvage at the end of instant year for $150. Frame the coin-issue diagram for the element f equipment. The pay and absorb for years - 7 through 1 (instant year) are tabulated low behind a while net coin issue computed using Eq. (1. 5). The coin issues are diagrammed . Fig. 1. 6. Key End of year Net coin issue Pay Absorb -7 -6 -5 -4 -3 -2 -1 0 1 Specimen 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 scantiness to compel a safety into your wholeity now such that you can behind a whiledraw an similar annual whole of Ai = $200 per year for the highest 5 years initiateing 1 year behind your safety and a opposed annual whole of A2 = $300 per year for the behindcited 3 years. How would the coin-issue diagram show if i is 14! % per year? Key The coin issues would show as demonstrationn in Fig. 1. 7. Comment The highest behind a whiledrawal (decisive coin issue) arises at the end of year 1, correspondently one year behind P is safetyed. Figure 1. 7 Cash-issue diagram for two opposed A appreciates, Specimen 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 Condition 1. 8 Cash-issue diagram for Specimen 1. 20. F2 1996 1995 A = $50 A = $150 = $50 F, = $900 Specimen 1. 20 If you buy a new television set in 1996 for $900,. conceal it for 3 years at a absorb of $50 per year, and then dispose-of it for $200, diagram your coin issues and imprint each arrow as P, F, or A behind a while its honorive dollar appreciate so that you can furnish the uncombined whole in 1995 that would be equipollent to all of the coin issues demonstrationn. Assume an share reprimand of 12% per year. Key Comment Condition 1. 8 confer-upons the coin-issue diagram. The two $50 disclaiming coin issues contrive a rotation of two similar end-of-year appreciates. As desire as the dollar appreciates are similar and in two or departed continuous conclusions, they can be resembleed by A, regardcease of wshort they originate or end. However, the $150 decisive coin issue in 1999 is a uncombined-adventure appreciate in the forthcoming and is hence imprinted an F appreciate. It is potential, nevertheless, to light all of the separate coin issues as F appreciates. The diagram could be drawn as demonstrationn in Fig. . 9. In open, nevertheless, if two or departed similar end-of-conclusion wholes arise continuously, by the restriction in Sec. 105 they should be imprinted A appreciates consequently, as is descriptive in Chap. 2, the use of A appreciates when potential simplifies calculations considerably. Thus, the exposition pictured by the diagram of Fig. 1. 9 is shock and gain not openly be used exalt in this quotation. p=? j = 12% per year F. = $150 1. 9 A coin issue for Specimen 1. 20 accordingly all appreciates as forthcoming sums. Condition 1996 1995 1997 1998 1999 F2 = $50 F3 = $50 F, = $900