where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return


 James McBride
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1 EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The meas by which a project s aual et cash flows ca be calculated were discussed ad illustrated. The cash flow complexities with regard to taxes, depreciatio, capital gais, ivestmet credit, ad salvage values were also icorporated ito the cash flow computatio. Our earlier discussio did ot, however, suggest a meas for evaluatig alterative ivestmet projects, or did it provide the agribusiess maager with a fiite criterio by which a sigle project could be judged acceptable or uacceptable. This issue shall attempt to address each of the latter two maagemet eeds. It presumes first, that the aual et cash flows attributed to each alterative capital ivestmet project have already bee computed. It presumes secod, that the agribusiess maager is capable of idetifyig his firm s opportuity cost of capital, i.e., the true rate of retur associated with ivestig capital i alterative projects. Iteral Rate of Retur Oe of the most commo criteria by which alterative ivestmet projects are compared ad/or judged acceptable is kow as the iteral rate of retur. I techical terms, the IRR is that iterest rate (aual average) that reders the sum of a ivestmet s aual et cash flows whe discouted to time period zero, equal to zero. Expressed mathematically, the IRR is that iterest rate which satisfies the followig equatio: where: = T = X ( + i) X = et cash flow i year T = umber of years of cash flow i ivestmet's life = the year i which the cash flow X occurs i = IRR = the iteral rate of retur To simplify what appears to may to costitute a rather complex formula, let s cosider a illustrious capital ivestmet project which costs $, ad geerates a aual et cash flow of $ for each of the subsequet six years (see Table 6). TABLE 6 Net Aual Cash Flows ($) Ed of Year Net Cash Flows , By applyig this stream of cash flows to our formula ad solvig for i we obtai:
2 =, ( + i) ( +i) ( +i) 2 6 =,+ PAi,6 where PA = preset worth factor, PAi,6 = = i = 2% as determied from a discrete compoudig table where PA = ad =6 I this illustratio, of course, our aalysis has bee simplified by the fact that our cash flows are discrete ad uiform. More commoly, however, agribusiess ivestmets will geerate aual et cash flows of uequal amouts such as those show i Table 7. I such cases the applicatio of our formula becomes more complex ad we are forced to approximate the IRR by a trailaderror procedure. Typically, we would select two iterest rates such that the discouted cash flows sum to a positive ad egative value (see Table 8) ad the approximate the IRR through liear iterpolatio as show below. Quite obviously this trialaderror process ca be both difficult ad time cosumig. Oe eed ot be frighteed, however, as may electroic had calculators ow have builti programs for makig such IRR calculatios directly. TABLE 7 Net Aual Cash Flows ($) Ed of Year Net Cash Flows ,, 2 2, 3 4, 4 4, 5 4, Accept or Reject Criteria Now that the IRR for the capital ivestmet project has bee determied, how is it used ad what does it mea? As we oted i the March issue, most agribusiess maagers have some geeral coceptio of their true cost of capital ad/or their opportuity cost of capital. Give either or both of these cocepts, that maager also has some policy with regard to a miimum acceptable rate of retur (MARR), i.e., that rate of retur o ivestmet below which the maager or the firm is uwillig to cosider as a attractive project. Give these miimum criteria, if IRR exceeds the MARR, the project is judged attractive or acceptable; if this is ot the case, the project is rejected. If alterative capital ivestmet projects are beig assessed, quite obviously they may ow be preferetially raked by maagemet i accordace with the level of IRR geerated by each. Returig to our earlier illustratio (Table 6) we ca further expad o the true meaig of the IRR. I Table 9 we have demostrated a alterative meaig of IRR = 2%, where IRR is show to be the retur o urecovered capital (allowig for the full recovery of ivested capital over the ivestmet s life). I this cocept, the agribusiess firm is loaig $, to the ivestmet project ad askig for a 2% rate of iterest o those moies remaiig urecovered at the ed of each of the six years of active life. I this case the iteral rate of retur is that retur geerated iterally by the project as a result of its et cash flows. 2
3 Ed of Year Net Cash Flow TABLE 8 Selectig IRR Bouds Discouted Cash Flows PF,2% PF,5% $,. $,.. $,., ,.7972, , ,.78 2, , , , , , ,988.8 Sum Iterpolated IRR = 2+ 3 = 2.5% TABLE 9 IRR as Urecovered Capital ($) (Step 3) (Step ) (Step2) Retur o Ed of Year Net Cash Flow Urecovered Capital Urecovered Capital (2%) Capital Recovered $, ,,+,7 = 8,993.2(,) = 2, 2, =,7 2 8,993+,28= 7,785.2( 8,993) =,799,799=,28 3 7,785+,45 = 6,335.2( 7,785) =,557,557 =,45 4 6,335+,74= 4,595.2( 6,335) =,267,267 =,74 5 4,595+ 2,88= 2,57.2( 4,595) = 99 99= 2,88 6 2,57+ 2,56= *.2( 2,57) = 5 5= 2,56 *=roudig error Observig year i Table 9 we see that as of the ed of that period the firm expects a retur of $2, (2%) for the project s oeyear use of the $, loaed it by the firm. Sice the project actually returs $, this reduces the balace of urecovered capital by $,7 such that the firm s ivestmet i the project durig the secod year is $8,993, upo which a retur of $,799 (2%) is expected durig the secod year. Agai, $ is actually retured, reducig the balace of the firm s urecovered capital from $8,993 to $7,785 for employmet by the project durig year 3. 3
4 Uderstadig of Table 9 may be aided by reviewig each row of the table via steps 3, i order. For those readers who ejoy the mathematical rigor associated with IRR computatios, it should be oted that for certai types of cash flows, IRR either caot be determied or the computatio results i multiple solutios. For example, whe all cash flows associated with a capital ivestmet have the same sig, IRR caot be determied. Coversely, whe the sigs associated with a sequece of aual et cash flows chage more ofte tha oce, a algebraic pheomeo kow as Descarte s Rule of Sigs dictates that there may exist a multiple solutio to the IRR computatio. Whe this situatio arises, there does exist a alterate method for determiig IRR. However, such matters fall beyod the iteded scope of this paper. Net Preset Value A secod commo criterio by which alterative capital ivestmet projects may be compared or judged acceptable is kow as the et preset value. Agai, i techical terms the NPV is the sum of the aual et cash flows associated with a ivestmet project discouted to time zero at the MARR. Mathematically it is expressed as: NPV = T = X ( + k ) where k = MARR Returig oce agai to our earlier example i Table 6 we fid that for a MARR of %: NPV =, ( +.) ( +.) ( +.) ( PA ) =,+,.,6 =, = $3, uacceptable otherwise. Similarly, alterative ivestmets may be preferetially raked i accordace with the magitude of the NPV geerated. A better uderstadig of the true meaig of NPV is facilitated by Table. You will ote the similarity betwee Table ad Table 9. They are similar i computatioal base except that Table 9 uses a IRR of 2% while Table uses a MARR of %. As show below, all capital is fully recovered i the fifth year of the ivestmet period, with a $2,252 surplus remaiig i the fifth year, plus the full cash flow of $ i the sixth year. If these surplus cash flows are discouted to time zero the result is: 2,252 ( PF,.,5) + ( PF,.,6) 2,252 ((.629) + (.5645) = $3,95 This $3,95 is idetical (except for roudig error) to that solutio obtaied from the formula computatio above. Table shows that a NPV > implies that all the capital is recovered over the life of the project (or a shorter period), or a retur (MARR) is received each year o the urecovered capital, ad a surplus or bous (NPV) is also received. If NPV =, this implies that MARR = IRR. IRR ad NPV Relatioship By ow the reader must woder whether or ot the use of IRR or NPV would result i the same maagemet decisio regardig the acceptability of the capital ivestmet project. I fact, for those sigle project assessmets comprised of cash flows with o more tha oe sig chage, either method will produce the same accept/reject decisio. The IRR ad NPV relatioship is diagramatically show i Figure. Give a series of et cash flows, the ivestmet project is judged acceptable if NPV, ad 4
5 TABLE NPVas Urecovered Capital ($) (Step 3) (Step ) (Step2) Ed of Year Net Cash Flow Urecovered Capital Retur o Urecovered Capital Capital Recovered $, ,  ,+ 2,7 = 7,993.(,) =,,= 2,7 2 7,993+ 2,28= 5,785.( 7,993) = = 2,28 3 5,785+ 2,428= 3,357.( 5,785) = = 2, ,357+ 2,67= 686.( 3,357) = = 2, ,938= 2,252.( 686 ) =.69 69= 2, The PayBack Period Criteria Referrig to Figure, it ca be see that if i is set NPV i Figure IRR as the MARR, the NPV is greater tha zero. I this situatio, IRR is greater tha MARR. Cosequetly, the cash flows are deemed to be acceptable uder both NPV ad IRR criteria. If i 2 is set as the MARR, the NPV is egative ad IRR is less tha MARR. Hece, the cash flows are judged uacceptable by both criteria. Where NPV is set as, the IRR = MARR ad cash flows are acceptable. i i 2 i Oe fial criterio is sometimes applied i maagemet s evaluatio of a capital ivestmet project. This criterio is referred to as the payback period ad refers to the umber of years required for the cash flows to completely recover the origial ivestmet. Expressed mathematically, the payback period () is: = X = Usig those data provided i Table, we would calculate that the payback period lies somewhere betwee the third ad fourth years, e.g.: Third year =, 5,+ 7,+ 7, = $, Fourth year =, 5,+ 7,+ 7,+ 7, = $6, 5
6 Usig liear iterpolatio, the payback period ca be approximated as: 3, 4 6,, = 3+ = 3.4years 7, PayBack Limitatios Used aloe as a criterio by which to judge the acceptace or rejectio of a series of cash flows, the payback period method has true limitatios. It is ot difficult to geerate illustrative cash flows where the IRR ad NPV criterio dictate a maagemet decisio opposite that geerated by the payback period. This coflict results because the payback period igores the magitude of cash flows followig the poit of full recovery. Neither does the payback criterio ackowledge i ay way the time value of moey. Obviously for a small agribusiess firm, the time required to recover its origial ivestmet is a importat cosideratio. However, ay attempt to base the acceptability of a series of cash flows o the payback period criterio aloe could lead to faulty decisios. TABLE Cash Flows PayBack ($) Ed of Year Net Cash Flows , 5, 2 7, 3 7, 4 7, 5 7, 6 7, 7 7, 6
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