Ranking of Mutually Exclusive Investment Projects

Transcription

1 Faculy of Maageme Techology Workig Paper Series Rakig of Muually Exclusive Ivesme Projecs How Cash Flow Differeces ca solve he Rakig Problem by Chrisia Kalhoefer Workig Paper No. 3 November 27

2 Rakig of Muually Exclusive Ivesme Projecs How cash flow differeces ca solve he rakig problem by Chrisia Kalhoefer November 27 bsrac The discussio abou he bes mehod o be used i capial budgeig has bee log ad iesive. Differeces bewee Ne Prese Value ad Ieral Rae of Reur seem o cause everlasig problems, while especially he Ieral Rae of Reur ofe is egleced as a appropriae measure. famous example of he problems caused by he differe approaches is he rakig of muually exclusive projecs. The followig paper is preseig a easy explaaio, wihou iroducig ew ad more complicaed measures, bu by simply explaiig he aure of ad differeces bewee Ne Prese Value ad Ieral Rae of Reur. JEL classificaio G3; G Keywords Capial budgeig; ieral rae of reur; e prese value; rakig problem; icremeal approach; reivesme rae Faculy of Maageme Techology Germa Uiversiy i Cairo l Tagamoa l Khames 835 New Cairo Ciy Egyp chrisia.kalhoefer@guc.edu.eg

3 . Iroducio The rakig of muually exclusive ivesme projec is ieresig for pracice as well as i a heoreical poi of view ad has bee broadly discussed sice may years. The mai cocer is he poeial differece bewee he assessme resuls derived from eiher Ne Prese Value (NPV) or Ieral Rae of Reur (IRR). These differeces bewee Ne Prese Value ad Ieral Rae of Reur seem o cause everlasig problems, while especially he Ieral Rae of Reur usually is egleced as a appropriae measure. While several aemps have bee made o deal wih his problem, i seems as if previous soluios did come o he core of he problem. Proposed suggesios usually eiher ry o solve he problem by fidig a ew ierpreaio of he IRR or by creaig highly sophisicaed mahemaical add-os for or modificaios of he IRR i order o ge he same rakig iformaio as he NPV. From a mahemaical poi of view, he reasos for he differe resuls are he cash flow differeces bewee differe ivesme projecs. The major quesio whe applyig ivesme appraisal echiques o rak muually exclusive projecs is how o deal wih hese differeces. This paper is preseig a explaaio for hese rakig differeces, wihou iroducig ew ad more complicaed measures, bu by aalyzig he aure of ad differeces bewee Ne Prese Value ad Ieral Rae of Reur. The aim of his paper is o explai how a correc usage of boh ivesme appraisals leads o ideical resuls, ad i should clearly be meioed hese resuls should o be misudersood as a pleadig o use he IRR as ivesme appraisal ool. The weakesses of he IRR are sill valid, bu he rakig differeces are excepioal o a weakess he IRR ca be blamed for. pplyig his approach, i will be show ha here are o differe resuls bewee he wo mehods if hey are applied correcly. I addiio, he commo explaaio of differe reivesme assumpios of he wo mehods will be cofued. Furhermore, he superioriy of he Ne Prese Value approach will be cofirmed. 2. The Rakig Problem i Capial udgeig The mos impora ad broadly used approaches i capial budgeig are he Ne Prese Value (NPV) ad he Ieral Rae of Reur (IRR) (Tag ad Tag 23, Graham ad Harvey 2, Drury ad Tayles 997, aco 977).

4 2 The NPV is he sum of he discoued cash flows of he projec. = ( + i) NPV = () where NPV = Ne Prese Value = ime idex = ed of mauriy = cash flow a i = ieres rae ivesme should be acceped if he NPV is exceedig zero. Therefore, he ieres rae used o discou he cash flows is he hurdle rae for he decisio. The IRR is he ieres rae ha, whe applied o he NPV formula, gives a NPV of zero. I oher words, i is he ieres rae where he sum of he discoued cash flows equals he iiial ivesme (Crea 25, 324, Schuck 995, 4). = ( + r) = NPV = (2) where r = ieral rae of reur The IRR shows he effecive reur of he ierally ivesed capial. ivesme should be uderake if he IRR exceeds he applied ieres rae i, because i his case i will have a posiive e prese value (Shapiro 25, 22). Therefore, for he appraisal of a sigle ivesme, boh measures are equivale: if he NPV of a cash flow is posiive, he IRR will also exceed he ieres rae i. I should be meioed ha he previous saeme should o be misudersood as a recommedaio o use he IRR for ivesme decisios, as he problems of he IRR are well kow, especially () i some special cases he meioed equaliy bewee IRR ad NPV migh o be rue, ad (2) he IRR ca oly be used if a reasoable rae of reur ca be calculaed. Due o he aure of he IRR as a roo of a polyomial, i case of some special cash flow paers his may o be possible (row 26, 95). lhough hese problems are well kow, i seems as if decisio makers sill prefer he IRR over he NPV, maybe because i is easier or more commo o udersad ad commuicae. While he echical problems (see e.g. row 26 for a summary) as well as differeces

5 3 bewee NPV ad IRR whe judgig a sigle projec occur oly i special cases, he IRR is probably sill a acceped ivesed appraisal measure. The usage of he IRR is sigificaly more error-proe if he ivesor has o decide bewee muually exclusive ivesme projecs. Therefore, i he followig he focus will o be o he echical problems wih IRR, bu o he well kow rakig problem. I his case, differe rakig resuls are possible while usig he wo approaches. Therefore, he recommedaio of which projec o choose is o clear. Reasos ad cosequeces have bee broadly discussed, ad several proposals have bee made o overcome hese icosisecies. The soluios ofe creae more complicaed measures ha should be used i addiio or aleraively o he IRR (e.g. Crea 25, arey ad Daielso 24, Lefly 23) or aemp o give a differe ierpreaio of he measures (Tag ad Tag 23). This paper will show ha all his is o ecessary if boh measures are applied correcly. Oe of he mai argumes used o explai he differe resuls is called reivesme assumpio. I saes he geerally correc fidig ha differeces bewee iermediae cash flows have o be recogized. The specific argume is ha his is doe hrough a assumpio abou heir reivesme rae ad a broad discussio abou his opic sared i he 95ies (e.g. Solomo 956, ierma ad Smid 957) ad is sill ogoig. For he discussed ivesme appraisal echiques he commo udersadig is ha he NPV assumes a reivesme of iermediae cash flows a he cos of capial while i case of he IRR he cash flow reivesme rae equals he IRR (Shapiro 25, 26, Va Hore ad Wachowicz 25, 329, Pogue 24, 4, Lefley 23, 8). lhough he reivesme argume is commo o explai differe resuls bewee IRR ad NPV, i is based o a wrog ierpreaio. To begi wih a mahemaical aalysis of he formulas preseed above, i ca be see ha o reivesme assumpio is ecessary o calculae NPV ad IRR (Dudley 972, 97). Therefore, he reaso for he differe resuls is somehig else, amely a icoseque applicaio of he wo mehods: if he ivesme projecs have differe cash flow paers, boh of hem are o direcly comparable, a leas o by a udiffereiaed applicaio of he ivesme appraisals, because his may lead o a misierpreaio of he resuls. Isead, o make a proper decisio bewee muually exclusive projecs, a ivesor has o cosider wha will happe o he cash flow differeces i bewee. Differeces may exis i he iiial ivesme, imig ad amou of he followig cash flows, ad he mauriy of he projecs (Va Hore ad Wachowicz 25, 326).

6 4 If, for example, wo muually exclusive ivesme projecs have o be compared, he firs oe wih a iiial ivesme of $, ad he secod wih a iiial ivesme of $ 9,, he ivesor mus cosider wha happes o he $, differece. This is of course rue for all cash flow differeces over he ime. I is possible ha he ivesme projec origially beig rejeced becomes he superior oe if hese cash flow differeces are recogized i a appropriae way. The compariso of differe cash flow paers (P) by heir differeces is kow as icremeal approach (Hajdasiski 24, Hajdasiski 997, Fisher 93), bu he implicaios of his approach have o bee sressed eough i he earlier lieraure. The usabiliy of his approach will be show i he followig secios. I will be clarified ha o addiioal assumpios or measures are ecessary if ivesors fully udersad he saemes of NPV ad IRR. I will also be poied ou, ha, if applied ad ierpreed properly, boh measures do o deliver differe rakig iformaio. 3. Explici Hadlig of Cash Flow Differeces he Special Case ssumig wo ivesme projecs ad, he differeces bewee he cash flow paers ca be mahemaical described by, or respecively. I Germa lieraure, he resulig cash flow paer is usually amed differece ivesme, abbreviaed. Therefore, he cash flow paer of he ca be calculaed as follows: = (3) The ivesor mus ake io cosideraio wha is goig o happe o his differece ivesme. To explai he rakig problem wih muually exclusive projecs, assume wo ivesme projecs ad, boh wih a P showig oe iiial cash ouflow, followed by several cash iflows. This assumpio assures ha o mahemaical problems occur while calculaig he IRR. s usual, all cash flows are accumulaed a yeared. Furhermore, he iiial cash ouflow of is assumed o be smaller ha he oe ; ad he calculaio of NPV ad IRR will resul i a higher NPV for, while he IRR of is higher. Recommedaio is o choose if he ivesor is referrig o NPV, or o choose if he ivesor refers o he IRR. Therefore, he decisio which ivesme o choose is o clear. The followig Table shows a umerical example ha fis o his descripio. The NPV has bee calculaed a i = %.

7 5 Ivesme 2 3 NPV % IRR % % Table : Exemplary daa for muually exclusive ivesme projecs s ivesme is reporig he lower NPV, icludig he io he cosideraios migh resul i plus he givig a higher NPV ha. Wih referece o he above descripio, he ca be calculaed as. Wih he assumpio abou he iiial ivesmes ( beig less expesive ha ), heir differece ( ) will be egaive. This simply sresses he ierpreaio of a differece ivesme as a cash flow paer sarig wih a cash ouflow; however, his is o a ecessary codiio for he followig argumes. Ivesme = Table 2: Derivaio of he differece ivesme The cash flow represes eiher a ivesme (if egaive) or a credi ake (if posiive). Therefore, he cash flows will have fiacial effecs i laer years, where eiher a ivesme will give some reur or a credi has o be paid back. I is assumed ha boh aleraives are cosruced as zero coupo bods wih a cerai ieres rae z. The fiacial effecs of he are icluded i he calculaio by compoudig he releva cash flows wih he ieres rae z o he las year. The resuls are show i Table 3. / Ivesme cash flow +... = = = = = =... = = = + + = ( + z) ( + z) Table 3: Cash Flow icludig

8 6 Icludig he leads o ideical cash flows of ad ( + ) for all <. I he las year, he cash flow of ( + ) equals he cash flow of plus he sum of he compouded cash flows of. oe should be made cocerig he algebraic sig of he compouded cash flows i =. I mus be saed as mius, because if he origial cash flow has bee egaive (represeig a ivesme), he laer oe mus be posiive (represeig he posiive reur of he ivesme). I case of a origially posiive cash flow (represeig a credi ake by he ivesor), he las year s cash flow mus be egaive o idicae he payback of he credi. impora resul of he previous cosideraios is ha as all cash flows for < are ideical boh ivesme aleraives ca be compared simply by comparig he las year s cash flows. The projec wih he higher cash flow will be seleced. This resul highlighs he imporace of he, because he cash flow i = is depede o he ieres rae z used o compoud he cash flows. To ge a idea abou his rae of reur, i will be ieresig o ivesigae he rae where boh ivesmes are equal. pparely, his will be he case if he fuure value (FV), expressed as sum of he compouded cash flows, equals zero, because i his case + + =. To fulfill his requireme, z mus be he IRR of he cash flow. This special ieres rae is called Fisher s rae of iersecio f (Va Hore ad Wachowicz 25, 328). The followig formula shows his effec mahemaically. = ( + ) f = = FV (4) I he example, f is 3.2%. The kowledge of f has a impora cosequece: as soo as z is exceedig f, he combiaio of ad will resul i a higher las year s cash flow; herefore he combiaio of ad will be superior o. The exemplary calculaio i Table 4 uses z = 5%. s 5% exceeds 3.2%, he combiaio of ad is beer ha, a fac ha ca easily be recogized by a compariso of he wo cash flow paers. They are ideical excep for he hird year, where he cash flow of ( + ) is higher ha he oe of. Ivesme 2 3 NPV % IRR % % + 5% % Table 4: Exemplary calculaio of ad wih z = 5% I should be meioed ha, up o his poi, oe of he ypical ivesme appraisal echique is ecessary o ge his resul. I is ohig bu a simple compariso of cash flows. p-

9 7 parely, if he ivesor evaluaes he wo described cash flows ad ( + ), will have he lower NPV ad he lower IRR. These raios are show i Table 4 for he example, where boh measures are higher for he combiaio of ad. There is o discrepacy bewee NPV ad IRR. Wih respec o his clear fidig, differe appraisal resuls offered by NPV ad IRR are eve more asoishig. The reaso will be explaied i he las chaper. 4. Explaaio of Rakig Differeces he Commo pplicaio of Ivesme ppraisals The previous aalysis has show ha ay ivesor explicily kowig or assumig a ivesme rae z ad properly applyig i o he as described above will o face he problem of differe resuls wih NPV ad IRR. The ivesme rae z ca also direcly be used o fid he beer aleraive: if z is exceedig f, he ivesor should choose he projec wih he lower NPV ogeher wih he, as he combiaio of boh will have a higher NPV (ad IRR!) ha he aleraive projec. u o assume a appropriae z migh be difficul or impossible for a ivesor ad i is o he purpose of his paper o eer he discussio abou he deermiaio of a appropriae rae (see e.g. Meyer 979). Therefore, he ex sep will be o aalyze he siuaio wihou a explici assumpio. This aalysis is especially impora as i cosiders he ypical applicaio of NPV ad IRR, i.e. ivesors usually do assume a explici z whe workig wih NPV ad IRR. Geerally speakig, as log as he ivesor does assume a explici rae of reur for he, he simple applicaio of NPV ad IRR calculaio will resul i a implici assumpio, while he problem is ha hese assumpios differ bewee he wo mehods. I oher words: he mahemaical ierpreaio of NPV ad IRR is rue oly if z is applied i a specific way. I should be meioed agai ha he ivesme rae z is o he commoly discussed reivesme rae for iermediae cash flows, bu oly ecessary for he differece ivesme wihi he rakig decisio. To udersad he mahemaical cosequeces of he implici assumpios, i has o be prove uder which circumsaces NPV ad IRR deliver he correc mahemaical resul whe icludig he. Sarig wih he NPV, wihou a explici assumpio of z he resul of he calculaio of he NPV is correc oly if z = i. pplyig his o he previously explaied formula, he FV of he ca be wrie like

10 8 ( ) ( + i) FV = (5) = The effec of his assumpio ca be recogized by calculaig he NPV agai. s he cash flow of plus equals he cash flow of a ay ime excep for =, he NPV of ( + ) equals he NPV of plus he discoued FV of he. NPV + ( + i) = NPV + FV (6) The NPV of he is calculaed by discouig he Fuure Value as show i equaio (5). NPV = = = = = = = NPV ( ) ( + i) ( + i) ( ) ( + i) ( + i) ( ) ( + i) NPV Fially, we ge + NPV = NPV + NPV NPV = NPV (7) Wihou ay explici assumpios whe calculaig he NPV o compare differe cash flow paers, implicily a rae of reur of i is assumed for he differece ivesme (agai: o for a reivesme of he cash flows hemselves!), because oherwise he calculaed resul would o be correc. s a resul, i ca be see ha a rakig decisio based o he NPV will be correc, because a ivesme rae of i for he does chage he calculaed NPV. The followig Table 5 shows he resuls for he previous example. gai, boh ivesme appraisals will resul i he same rakig if applied correcly, i.e. o he complee cash flow. Ivesme 2 3 NPV % IRR % % + % % Table 5: Resuls of he implici assumpio of he NPV The assumpio of z = i maches he commo prerequisie of a perfec capial marke. Therefore, a ivesor o willig or o able o assume a reasoable ivesme rae z is doig fie wih he applicaio of he NPV as a appraisal echique. I ca be see ha he correc ap-

11 9 plicaio will lead o a lower IRR for he aleraive wih he lower NPV, so here are o differe sigals bewee he wo mehods. For he IRR, he explaaio is quie similar. The mahemaical iformaio of he IRR, applied o he oal cash flow (projec ad ) is correc oly if he ivesme rae z equals he IRR of he projec. The followig equaio describes his effec for he combiaio of ad. C = = + = = = = = ( + r ) + ( ) ( + r ) ( + r ) ( + r ) + ( + r ) + ( + r ) ( + r ) = = = The equaio shows ha he resul of he discouig ca be zero oly if he IRR of is used as z. Therefore, i is obvious ha he ivesme wih he higher IRR has o be preferred, as he same higher IRR will be applied for he. u as before, applyig he NPV o he complee cash flow will give he same rakig as he IRR, so he correc applicaio of boh NPV ad IRR will lead o he same resuls. Correc meas usig he complee cash flow, i.e. icludig he. The applicaio of hese fidigs o he example daa ca be see i Table 6. Ivesme 2 3 NPV % IRR % % + 7.5% % Table 6: Resuls of he implici assumpios of IRR lhough he NPV/IRR discrepacy migh be solved by he correc ierpreaio of he umbers, he applicaio of he IRR mehod sill ows i s broadly discussed problems. Wihi he rakig problem, he mos crucial fac is, as several differe cash flows usually lead o several differe IRRs, cosequely several differe ivesme raes z are applied o judge ivesme projecs. This is o raioal, because as a resul he reur of he would be direcly depede o he origial cash flow. Furhermore, he reur of fuure ivesme opporuiies will usually o be relaed wih oday s ivesme projecs IRR. I addiio, he applicaio of a ieres rae oher ha i for he differece ivesme does follow he assumpios of a perfec capial marke. Therefore, if he ivesor is o able or willig o se-

12 lec a appropriae rae of reur for he differece ivesme, i is sill recommeded o use he NPV for ivesme decisios. To summarize he aalysis for he exemplary daa, boh releva cash flows for ivesme are meioed i Table 7. I ca agai be recogized ha he origial mahemaical resuls of NPV ad IRR (see Table ) are derived from differe assumpios regardig he cash flows, which is he reaso for he differe rakig recommedaio whe compared o ivesme. Ivesme 2 3 NPV % IRR + % % + 7.5% % Table 7: Compariso bewee NPV ad IRR cash flows From he previous aalysis he reaso for differe resuls bewee IRR ad NPV i he rakig problem is easy o address. Wihou a explici assumpio he mahemaical resuls of boh appraisals are correc oly if specific ieres raes are assumed for he : i for he NPV ad r for he IRR. These differe ieres raes lead o differe cash flows, so a he ed boh echiques will o assess ideical cash flows. pparely, uder hese codiios he resuls ca be differe. The problem ca obviously be avoided if boh appraisals are applied o ideical cash flows as described i his paper. 5. Coclusio The broadly discussed rakig problem is a resul of a isufficie applicaio of he ivesme appraisal echiques. For mahemaical reasos, he resuls of he calculaios are correc oly if cerai assumpios are made cocerig he ivesme rae for he differece ivesme. s hese assumpios are o ideical, boh mehods will fially assess differe cash flows. Therefore, he resuls ca be differe. To avoid his problem, boh ivesme appraisal echiques should be applied o ideical cash flows. I his case, differe rakigs are o loger exise. These fidigs of course do o preve he ivesor o make a decisio cocerig he differece ivesme. s log as he is o able or willig o choose a appropriae explici rae of reur for he differece ivesme, he will sill have o choose bewee e prese value ad ieral rae of reur. lhough he rakig problem ca be avoided whe applyig he mehods correcly, as he assumpio cocerig z whe applyig IRR is quesioable ad he oher well-kow problems of he IRR sill exis, he usage of he

13 NPV is recommeded if o explici choice for he reur of he differece ivesme ca be made. Refereces aco, P.W. (977): The Evaluaio of Muually Exclusive Ivesmes. Fiacial Maageme, 6, No. 2: arey, L.D. Jr. ad Daielso, M.G. (24): Rakig Muually Exclusive Projecs: The Role of Duraio. The Egieerig Ecoomis 49: 43 6 ierma, H. ad Smid S. (957): Capial udgeig ad he Problem of Reivesig Cash Proceeds, The Joural of usiess, 3, No. 4: row, R.J. (26): Sis of he IRR, Joural of Real Esae Porfolio Maageme 2: Copelad, T.E., Weso, J.F., ad Shasri, K. (25): Fiacial Theory ad Corporae Policy, Fourh Ieraioal Ediio, oso e al. Crea, M.J. (25): Revealig he True Meaig of he IRR via Profilig he IRR ad Defiig he ERR, Joural of Real Esae Porfolio Maageme, : Drury, C. ad Tayles, M. (997): The misapplicaio of capial ivesme appraisal echiques, Maageme Decisio, 35/2: Dudley, C.L. Jr. (972): Noe o Reivesme ssumpios i Choosig bewee Ne Prese Value ad Ieral Rae of Reur, The Joural of Fiace, 27, No. 4: Fisher, I. (93): The Theory of Ieres, Macmilla Graham, J.R. ad Harvey, C.R. (2): The heory ad pracice of corporae fiace: Evidece from he field, Joural of Fiacial Ecoomics, 6: Hajdasiski, M.M. (997): Techical Noe Commes o Usig Heurisics o Evaluae Projecs: The Case of Rakig Projecs by IRR, The Egieerig Ecoomis, 42: Hajdasiski, M.M. (24): Techical Noe The Ieral Rae of Reur (IRR) as a Fiacial Idicaor, The Egieerig Ecoomis, 49: Lefley, F. (23): The hird way, Fiacial Maageme, Ocober 23: 8 9

14 2 Meyer, R.L. (979): Noe o Capial udgeig Techiques ad he Reivesme Rae, The Joural of Fiace, 34, No. 5: Pogue, M. (24): Ivesme appraisal: a ew approach. Maagerial udiig Joural, 9: Schuck, E. (995): Reivesme rae risk aalysis: a comme. Joural of Propery Evaluaio ad Ivesme, 3: 39 5 Shapiro,.C. (25): Capial udgeig ad Ivesme alysis, Upper Saddle River Solomo, E. (956): The rihmeic of Capial budgeig Decisios, Joural of usiess XXIX, No. 2: Tag, S.L. ad Tag H.J. (23): Techical Noe: The Variable Fiacial Idicaor IRR ad he Cosa Ecoomic Idicaor NPV. The Egieerig Ecoomis 48: Va Hore, J.C. ad Wachowicz, J.M. jr. (25): Fudameals of Fiacial Maageme, welfh ediio, Harlow e al.