Transforming the Net Present Value for a Comparable One

Transcription

1 'Club of coomics i Miskolc' TMP Vol. 8., Nr. 1., pp Trasformig e Ne Prese Value for a Comparable Oe MÁRIA ILLÉS, P.D. UNIVRSITY PROFSSOR vgilles@ui-miskolc.u SUMMARY Tis sudy examies e aure of e prese value. I defies e ecoomic coe of e e prese value ad maemaically proves a defiiio is correc. Tis ecoomic coe iducs a e e prese values are o comparable. Te sudy sysemaically elimiaes e disorio affecs. Te e prese value rasforms io a special kid of rae, amely, e modified differece bewee e facual ad e reuired rae of reur. Te rakig lis accordig o is rasformed e prese value correspods o e lis accordig o e ieral rae of reur. Tis is a ew cogiio ad a very impora correspodece. Keywords: busiess ecoomics, capial budgeig decisios, e prese value, ieral rae of reur, rakig, ecoomic efficiecy Joural of coomic Lieraure (JL) code: M1 INTRODUCTION, PURPOS AND MTHODS OF RSARCH Te lieraure of decisio preparaio meods is very large ad polycromaic. ve creaig a compreesive picure oly abou e lieraure of capial budgeig decisios is seems o be impassable. Te discussio abou e bes meod of capial budgeig decisios as bee log ad iesive. May wriigs already were bor i e 5s of e las ceury (Alcia, 1955; Solomo, 1956; Bierma Smid, 1957; ad so o). Sice e e discussio is ruig. I e forefro of discussio are e e prese value () ad e ieral rae of reur (IRR) as rival meods. Oe of e peculiariies of e discussio is a icreasigly appear more ad more complicaed refiemes of e basic meods. Oe of e ypical varia of is is e modified ieral rae of reur (MIRR). Oe of e academic supporers of MIRR is Kierulff. (Kireulff, 8.) Tis meod was appearig i e 18 ceury, was ewly rediscovered i e 5s of e las ceury, ad owadays is comig o e fro as well. Te MIRR is calculaed as follows Fuure value of posiive cas flows (wi reivesme rae) MIRR -1, Prese value of egaive cas flows (wi fiace rae) were is e umber of oal life-spa of projec. I would be difficul o defie exacly e ecoomic coe of MIRR. Te e prese value ad e ieral rae of reur are wo well kow caegories of capial budgeig decisios. Durig e decisio preparaio processes of ivesme projecs bo of em ca provide useful iformaio. Te way of approacig ad ierpreaio problems of e wo meods ad eir calculaio resuls as is appears i e lieraure are o cleared up deeply eoug. Terefore a lo of iadeuae explaaios ad coradicios occur i coecio wi em. For example e lieraure does o defie exacly wa does e sum of e e prese value meas. Despie of is i e lieraure exis a srog edecy wi improvemes appearig ime ad agai empasizes e advaages of e prese value meod ad is beer ualiy from e decisiomakig process poi of view. Tis meod is suggesed for examiaio of a ivesme projec s accepabiliy ad o creae a rakig lis as well. I coras, e pracical expers usually prefer e ieral rae of reur meod for bo purposes i a impora par of developed couries. For example Arold - Hope (199: 6-63) cie wo surveys doe i e 7s ad 8s of e las ceury demosraig a e larges Briis compaies (a are oerwise able o pay e bes expers, or o rai specialiss) defiiely rak e ieral rae of reur iger a e e prese value i pracice. Afer is review ey meio a los of America surveys prove a e pracice prefers e ieral rae of reur i e USA as well. Te kowledge of e coe of e iformaio used for decisio makig ca provide pracical aspecs of coosig bewee e wo meods. Reviewig e uesio of ow e meodological process ad e resuls of is mac e logic of e maageme pracice ad e ikig rouie of e decisio makers is reasoable as well. Busiess ecoomics, amog oers, as e fucio of providig meodological aid, meodologically-fouded ideas for ecoomiss i pracice. Probably based o is fucio, e busiess ecoomics lieraure ofe meios e imporace of e professioal clariy ad pracical implicaios of e suggesed meodology (e.g. Garriso, 1988: 71; Arold - Hope, 199: 6; H. Scmale, : 6-65). Ta is, e majoriy of e compay expers ca oly apply e 4

2 Trasformig e Ne Prese Value for a Comparable Oe meodology correcly if ey ca fi meods o eir way of ikig, if ey ca someow coec em o e logic of e ecoomic process. Tis sudy ieds o clarify some basic issues of e e prese value. Te mai uesios of e researc are: applicabiliy area of e meod; ecoomic coe of e idex umber received based o e calculaio; comparabiliy of e e prese values of differe ivesme projecs; purifyig is idex umber from disorig effecs. Te ieral rae of reur is regularly used i e sudy o make e aalysis ad fidigs uambiguous. Te mai meods of e researc are e logical aalysis ad e use of udersadable, relaively o difficul maemaical models. Te provig ways of fidigs are logical ad maemaical processes as well. TH FUNDAMNTAL QUSTIONS OF NT PRSNT VALU AND INTRNAL RAT OF RTURN MTHODS Te meodologies of bo procedures are widely kow i e professio; erefore, is sudy does o explai e meodologies i deail. Tis secio maily serves as sarig poi, ad o draw aeio o some geeral issues. Te Reuired Rae of Reur Tree decades ago e reuired rae of reur was ierpreed i e busiess ecoomics lieraure as e ieres rae, or e compay s average rae of capial profiabiliy. For example Clifo ad Fyffe (1981) also meio ese wo reuiremes i eir book: i e meod of e e prese value e reuired rae of reur is e rae of reur wic is aalogue wi ieres rae, as wrie o page 164. Laer is coe was sligly modified: Te reuired rae ca be e ieres rae or e effecive rae of e compay s ow ivesed capial (p. 39). I 9- years of e las ceury i busiess ecoomics became obvious a deermiaio of e reuired rae of reur is e same as e opporuiy cos ierpreed for capial ad defied by microecoomics. Applyig is approac became geeral i e busiess ecoomics lieraure. Parly parallel wi is process, e cocepio appeared i e fiacial sudies a e reuired rae of reur is o eual o bo mai elemes of capial. Te cos of capial applied o e euiy ad a applied o e deb cao be e same. By is cocepio cocerig e euiy, besides e price of usig e capial, e commodiy marke risk-premium reuiremes also mus be reured; cocerig e deb, owever, a is eoug if e ieres is repaid. Te reuired rae of reur cocerig e euiy is ypically defied as if is were idepede from e level of idebedess. Te mos caracerisic meodological soluio for is is e weiged average cos of capial. Here e reuired yield rae from euiy ad e ieres rae are averaged accordig o e weig rae of euiy ad deb. Tis soluio seems o be e growig red; owever, e cocepio is o scieifically clarified. I e lieraure los of criiues cou wi weiged average cos of capial as well. (.g. Luerma, 7.) Tese criiues are very impora, bu always are o abou e maer of e priciple of a reuired yield rae. Ne Prese Value Meod I eory, e e prese value meod is calculaed so a e discoued sum of all cas ouflows is subraced from e discoued sum of all cas iflows a are associaed wi a ivesme projec. Te calculaio ca also be made wi e ime series of e differece of cas iflows ad ouflows. Te aual differeces are discoued ad summarized. I busiess pracice e logic of e calculaio ca be rasformed: subracio of e discoued sum of e ivesme expeses from e discoued sum of e differece of reveues ad ouof-pocke coss. Te coe of e differece of reveues ad ou-of-pocke coss is very similar o e earigs before ieres, axes, depreciaio, ad amorizaio. Accordig o is meod e ivesme projec is accepable if e e prese value is o less a zero. Ieral Rae of Reur Meod Te ieral rae of reur was earlier kow i busiess ecoomics as e ime-adjused rae of reur (Garriso, 1988: 657). Calculaio of e ieral rae of reur meas searcig for e rae of reur wic makes e cas iflow lie ad e ouflow lie eual o eac oer. I essece, a is e rae a wic e e prese value would be zero. I case of ivesme projecs a ca ave oly oe ieral rae of reur, e ieral rae of reur sows e facual ime-adjused profiabiliy rae of e ivesme. Te accepabiliy crieria of e projec are decided by ow large facual profiabiliy rae is geeraed compared o e reuired rae of reur. Te differece sows ow large surplus raes (or lack of raes) are geeraed compared o e reuired rae of reur. Geerally, is differece eeds o be defied umerically; is becomes visible we wriig e wo raes ex o eac oer. Were e wo raes are eual, a sill meas a ecoomic efficiecy ad reuired profiabiliy are exacly acieved. Applicabiliy I e case of ivesme projecs or aciviies were e yield effec of e decisio ca be calculaed umerically i a relaively obvious way, e mai uesio is weer e geeraed yields are eoug o mee e reuired rae of reur. Te relaively correc soluio of e coice bewee varias excludig eac oer icludes compariso of reur opios as well. A wide rage of ivesme projecs ave o direc yield effec or is cao be measured. very ivesme iem cocerig a office buildig ad euipme of admiisraio ca be classified as suc. Te yields of e idividual elemes 5

3 Mária Illés of e ecological cai of e producio process, or e yield of a give macie or euipme i iself cao be defied eier. For example, i e case of a power pla e cas flows arisig from a sigle urbie cao be calculaed. A is uesio e ecoomic survey ses as is goal o miimize e ime-adjused average aual repayme reuiremes coeced o e give fucio performace. Tis ype of decisios ca be called as fucio-orieed decisios. Te sigifica appearace i busiess ecoomics of fucioorieed capial budgeig decisio meods as a isory loger a alf a ceury. I e fucio-orieed capial budgeig decisios e ime-adjused average amou of aual repayme reuireme gives meodologically correc grouds for compariso. Meodologically, measurable yield effec projecs ca also be divided io wo groups. Te busiess ecoomics lieraure differeiaes bewee ivesme projecs wi orodox ad uorodox cas flow paers (e.g. Arold Hope, 199: 6-63). Te ame ca be differe; for example: ypical ad oypical, coveioal ad o-coveioal, orodox ad uorodox cas flows. Te fiacial lieraure usually does o adle is differeiaio. (Usig is differeiaio is o a geeral pracice i busiess ecoomics lieraure, eier.) A mai caracerisic of e orodox cas flow paers is a e ime series of e differece bewee cas iflows ad cas ouflows sars wi oe or more egaive sig amous, ad from a poi i ime were is differece urs io posiive firs, is posiive sig does o cage. So o oer year will be i e duraio of e projec we e amou of cas ouflow will exceed cas iflow. Tus e cas flow series sars ou wi a egaive sig member or members, ad sig cage occurs oly oce. I e case of orodox cas flow paer projecs, e projec iself creaes ad produces all e yield elemes appearig i e examiaio. So e examiaio of ecoomic efficiecy ca follow e logical uesio arisig before decisio makig, a yields of ivesmes i coecio wi e projec would be eoug o fulfill e reur reuiremes. Ivesme projecs wi orodox cas flow paers ca ave oly oe ieral rae of reur. Tis saeme resuls from a eorem of Ree Descares (Frec scolar ad pilosoper ). From is as follows e ime series of e differece of cas iflows ad ouflows may ave as may ieral rae of reur, as e umber of sig cages. Te basic relaiosip ca be cosidered as geerally kow our days. Havig more a oe ieral rae of reur is e caracerisic of uorodox cas flow paer projecs oly. Because of e fiacial lieraure usually does o adle e disicio of orodox ad uorodox cas flow paers ofe dey expediece of ieral rae of reur (e.g. Brealy - Myers, 199: 76-8), ad ofe says a e ieral rae of reur is wiou meaig (Hill, 8: 36). Tese saemes are valid oly e case of uorodox cas flow paers. Ivesmes belogig o e group of uorodox cas flow paer projecs are ose i wic sig cages occur a leas wice i e cas flow series. (For example e ouflow is larger a e iflow i case of periodical reovaio, a parial rebuildig of radiioal iro furaces, or i e period of remediaio works afer e closig of ope-pi mies.) A mai caracerisic of e majoriy of uorodox cas flow paer projecs is a a par or e wole payback, oce disegaged from e projec, mus laer be reivesed io e same projec. Tis reivesig maer for e ecoomic efficiecy of e projec ow large yields ca be acieved wi amous of moey emporarily uilized i oer areas or projecs. (For a deailed explaaio ad demosraio wi a example, see Illés, 7.) Classical meods auomaically assume a amous emporarily ivesed resul i yield effecs accordig o reuired rae of reur or e ieral rae of reur. We ey are used, e yield effec iside e projec becomes iexricably mixed up wi e yield effec of amous emporarily ivesed beig assumed auomaically by e meod. As e resul of all is, eier e e prese value, or e ieral rae of reur of uorodox cas flow paer projecs provide adeuae iformaio for decisio makig. Tis relaiosip ca be cosidered geerally kow i coecio wi e ieral rae of reur. However, i coras wi e suggesios i e lieraure eve e e prese value cao be cosidered o be clear iformaio i case of uorodox cas flow paer projecs. Tesis 1 Ne prese value ad ieral rae of reur ca oly provide well ierpreed, clear iformaio i e case of ivesme projecs wi orodox cas flow paers. TH CONOMIC FFICINCY BY FOLLOWING UP TH RPAYMNT PROCSS, AND TH CONOMIC CONTNT OF NT PRSNT VALU Te ecoomic efficiecy of ivesmes ca be examied correcly by several meods, o oly by e publised oes. Suc o familiar meod ca be e meod based o followig up e repayme process. Te essece of is meod is calculaig for eac give year uil e ed of e projec e differece bewee e sum of cos of capial o repaid ad e yield. A e ed of e process e oupu is e sum of facual exra yield or lack of a compared o e reuired rae of reur. Te las sep could lead o e e prese value if is sum is discoued back o e zero poi of duraio (Illés, 1997). Te meod iself ivolves sigificaly more seps a e e prese value meod ad reuires more backgroud calculaios. From is regard, is calculaio process cao compee wi e e prese value meod. However, e exeded iformaio backgroud gives e opporuiy o survey ad o follow e repayme process rougou ime. Followig up e repayme process ca serve as a coemplaio suppor for a compay ad ca be simulaig, sice e sraegic visios ad projec performace ca be followed up coiuously rougou e life of e projec. Tis meod is based o e fac a roug discouig back e exra yield (or lack of a) geeraed by e ed of e duraio of e projec leads o e e prese value. Accordig 6

4 Trasformig e Ne Prese Value for a Comparable Oe o is calculaio e ecoomic coe of e e prese value ca be clearly defied. Te e prese value is e discoued sum of e surplus yield (or lack of a) geeraed above e yield reuireme accordig o e reuired rae of reur (Illés, 199: 13-15). Te followig examiaio sows e deducio roug a simple example, ad e is coex proves geerally valid relaiosip applyig o orodox cas flow paer projecs. Demosraig e Follow-up of e Repayme Process Troug a Simple xample Te cas flow series are calculaed as e differece bewee e aual cas iflows ad cas ouflows of a ivesme projec (i order of years; geeraed a e ed of eac year) as follows: - uis 8; + uis 4; + uis 44 ad + uis 194. Te reuired rae of reur is 1%. = =43.8 uis. Te examied projec mees e repayme reuireme; besides surplus yield is geeraed wi a prese value of 43.8 uis. Te developme of e repayme reuiremes roug ime ad e repayme process iself is sow below (amous coued i uis). Te seps of calculaio process: A e ed of Year 1: = A e ed of Year : = A e ed of Year 3: = 61.6 = = 43.8 xplaaio of e calculaio: From all e repayme reuiremes occurrig 4 uis were repaid a e ed of e firs year. A a ime remais uis o reur. Tis uis ad e reuired yield geeraed because is amou coiuig o be locked up, a is 558. mus be reured. A e ed of e secod year 44 uis were repaid from e amou of repayme reuireme uis were o repaid. A e ed of e ird year a yield of 194 uis are geeraed. Tis exceeds e 13.4 uis reuired repayme, ad leaves 61.6 uis. Tis is e surplus compared o e reuired repayme a e ed of e ird year. Te surplus yield (61.6 uis) geeraed a e ed of e ird year is discoued back o ime zero ad euals o 43.8 uis, wic is exacly e same amou as e amou of e e prese value calculaed above. Te relaiosip iroduced based o e example ca be proved o be geerally valid for all orodox cas flow paer projecs. Geeral Demosraio Sarig poi: Te e prese value calculaio varia applied o orodox cas flow paer projecs. 1 - H, (1) i) were 1 = Iiial ivesme. Te ivesme sum occurrig i e zero poi of ime, ad ivesme amous occurrig earlier added up wi reuired rae of reur. = Serial umber of years ( > ). H = Differece bewee cas iflows ad cas ouflows i year, were H > for orodox cas flow paer projecs. = Duraio of e projec, were e ime of ivesme realizaio does o cosiue par of e duraio. i = Reuired rae of reur. From e demosraio poi of view, e coe relaiosip a furer ivesme-like expeses for maiaiig e workig abiliy of a give fixed asse or o resore e asse ca occur durig e workig period of e ivesme projec bears o relevace i is relaiosip. Te demosraio cosiss of wo sages ad several seps a eac sage. Sage oe: Te firs sage of e demosraio describes e repayme process of e cos of capial. Te cos of capial is e sum of e face value repayme reuireme of e capial ad is reuired yield accordig o e reuired rae of reur. Te repayme process describes e umerical defiiio of e cos of capial o ye repaid a idividual pois i ime ad e compariso o e cocerig aual yield. Te amou of e cos of capial o ye repaid a e ed of Year 1: (1 i) H 1 a e ed of Year : (1 i) H 1(1 i) H a e ed of Year 3: { (1 i) H 1(1 i) H (1 i) H 3 } ad so o. Assumig a e saus of e repayme a e ed of Year 3 already sows e paer of e developme of e process roug ime, simplificaio is iroduced o e above iscripio i pareeses. limiaig e curly brackes: i) H i) H i) 1 H 3 limiaig e suare brackes: 3 i) H i) 1 H i) H3 Te umerical defiiio of e repayme saus iscribed for e ed of Year 3 ca be applied furer for e full duraio of e ivesme. Formula () defies umerically e amou of e surplus repayme or lack of a geeraed a e ed of e duraio of e ivesme projec. () -1 - i) H1 i) H i)... H-1 i) H Sage wo of e demosraio: I is sage e ed resul of e repayme process is rasformed io prese value form. Te prese value of e amou of e surplus repayme or lack of a is calculaed for e ed of e duraio i ree seps. Sep oe: sig up e discouig formula [ i) H1 i) H i)... H -1 i) H ] i) (3) Sep wo: perform e discouig process H1 H...H -1 H -1 (1 i) (1 i) (1 i) (1 i) (4) Sep ree: simplificaio of e (4) formula. 7

5 Mária Illés 1 - H 1 (1 i) (5) A sep ree e e prese value calculaio formula iscribed as e sarig poi (1) is arrived. Te proof is complee. Te deducio proved a i e case of orodox cas flow paer projecs e e prese value is e discoued amou of e surplus yield (or lack of a) geeraed above e yield reuireme accordig o e discou rae. Te exisece of e surplus yield ad is amou depeds o e reuired rae of reur, oo. I case of geeraig o lack of yield a e ed of e period, e projec is ecoomically efficie ad accepable. Te advaage of e meod is a e process of e calculaio logically follows e repayme process. Te iformaio coe of e idex umber ca be uie raspare from a pracical poi of view, i coras wi e e prese value wic is someimes mysifyig. Troug discouig e face value of e surplus yield (or lack of a), e e prese value ca also be defied umerically. Tesis I e case of ivesme projecs wi orodox cas flow paers e e prese value sows e sum of e surplus yield above e reuired oe (or lack of a), discoued for prese value. (Tis coe is proved maemaically.) Tesis 3 I e case of ivesme projecs wi orodox cas flow paers e sum of e surplus yield (or lack of a) calculaed for e closig dae of e projec ca also be suggesed for pracical expers as a correc ecoomic idex umber. Te advaage of e meod is a e calculaio process logically follows e repayme process. Te iformaio coe of is idex umber is easily coceivable from a ecoomic poi of view; by discouig is idex umber e e prese value is reaced. COMPARABILITY OF NT PRSNT VALUS Te e prese value as a geeral coe coradicio. Tis coradicio resuls from e fac a e meod oly adles e all-ime exe of e capial ivesme correcly i e maer of e reuired rae of reur. Te surpluses (or lacks) of yield geeraed above e reuiremes are simply discoued ad summarized. From e viewpoi of e meod e sum of e capial ad is presece i e projec cocerig e surpluses ad e lifeime of e projec are irreleva. (Te deducio above clearly proves is.) Followig from all ese, e e prese values of differe ivesme projecs are o suiable for comparig e ecoomic efficiecy of e projecs. Te e prese value coais disorios regardig ree relaiosips. Disorio ffecs 1. Te iiial ivesme reuiremes of e projec ca be differe. Oerwise assumig ucaged codiios, e iiial ivesme of e lesser amou is e more advaageous.. Te duraio of e ivesme projec ca be differe. Oerwise assumig ucaged codiios, sorer duraio is more advaageous. Te re-ivesme of capial ca appe earlier ad e ew yields are us geeraed earlier; i addiio i e case of e oer codiios beig e same e projec bears smaller risk beid e repayme process of sorer period. 3. Te rapidiy of capial payback ca be differe. Oerwise assumig ucaged codiios, faser payback is more advaageous. I e case of slower payback, for example ose a are coceraed a e ed of e lifeime of e projec, e reivesme ca commece laer, affecig a sigifica par of e ivesed capial. Te disorio affecs or eier of em are performed a los of publicaios (for example Keae, 1975, Va Hore Vacowicz, 8). Aloug e demosraio of e above ree disorio effecs makes clear a ivesme projecs are geerally o comparable based o eir e prese values, e lieraure is far from beig omogeeous i is uesio. Some of e sources sugges e compariso based o e prese value wiou ay resrais. For example Muually exclusive projecs: Accep e projec wi e iges posiive. (Brigam Houso, 9:34.) Furermore Projecs ca also be raked accordig o eir. (Hill, 8:36.) I fac, e Fiser s iersecio was bore o e basis of comparabiliy of e prese values as well. Tis iersecio is e rae wic brigs e s of wo ivesmes io eualiy (Fiser, 193). Fiser s iersecio is ofe referred owadays as well (Baker - Powell, 5, Hill, 8, Va Hore Vacowicz, 8. ad so o). Possibiliies for limiaig e Disorio ffecs Tere are lierary sources for suggesio of cerai correcios o elimiae of e disorio effecs. Tese suggesios, owever, oly correc e firs or e secod ou of e ree disorio effecs demosraed above. As I see, ere is o e pursui for a complex correcio. Te elimiaio of e disorio affecs is solvable i may ways. I is sudy e sar-up basis of e applied cleaig meod is e oe-problem-orieed proposals of lieraure. (Supposedly is way of soluio will give a ad o survey ad appreed e relaiosips.) Some sources sugges dividig e e prese value by e iiial ivesme [ ] or e profiabiliy idex. [ 1 ] For example Brealey & Myers (199:115.). 8

6 Trasformig e Ne Prese Value for a Comparable Oe Tis idex umber or e profiabiliy idex oly elimiaes e disorio effec of differeces i iiial ivesmes. Te disorio effecs of e differeces i duraio ad rapidiy of payback sill remai. Te suggesio of e ime-adjused average of e e prese value ofe appears maily i e fiacial lieraure i e las wo decade. (.g. Helfer, 1991: 5-51, Baker - Powell, 5: 6, or Lee, A. C. - Lee, J. C. - Lee, C. F. 9: ) Defiig e average is calculaed i e way a e e prese value is divided by e auiy facor, wic i busiess ecoomics meas muliplyig by e loa repayme facor. (Busiess ecoomics always used e formula of loa repayme facor. Te fiacial lieraure uses e auiy facor, wic is e reciprocal of e previous formula.) Te formula of is ime-adjused average is: (were is e loa repayme facor, wii e reuired rae of reur ad lifeime of e projec). Tis soluio oly elimiaes e disorio effec of e differeces i duraio; e disorio effecs i e differeces bewee capial reuiremes ad rapidiy of payback remai. Aoer sep could be aloug I did o fid ay suggesios for is i e lieraure mergig of e wo meods demosraed above, a is, e umerical defiiio of e ime-adjused average of e e prese value divided by iiial ivesme, a is. Afer elimiaig e disorio effecs of e iiial ivesme ad e duraio a e same ime, e disorio effec would be oly i e rapidiy of capial payback remaiig i e idex umber. (Demosraio of is wi a example ca be foud a e ed of is sudy.) Tis way, owever, e rasformed e prese value is geig closer o e mai iformaio used durig applicaio of e ieral rae of reur meod, a is, ow muc e facual profiabiliy rae differs compared o e reuired oe. Te meodological elaboraio of calculaig e coefficie wic ca measure payback rapidiy seems o be very complicaed. I believe a e calculaio of is coefficie is o ecessary, bu very impora o kow is essece. Fudameal cases: Payback by years is uiform. Ta e coefficie is 1. Payback is uickly. Te bigger cas flows arise a e begiig of e life-spa. I is case e coefficie is bigger, a 1. Payback is slow. Te bigger cas flows arise a e ed of e life-spa. I is case e coefficie is lesser, a 1. We e ird problem, e disorio effec i e differece bewee e rapidiy of payback, is successfully elimiaed, e rasformed idex umber arrives a a correced differece bewee e ieral rae of reur ad e reuired rae of reur. Te Cleased Formula Te rasformed e prese value is a special raedifferece. Tis coe is followed from exibied calculaig procedure. Te rae-caracer as appeared we e e prese value was divided by iiial ivesme. Te maer of raedifferece is followig from a, e e prese value is a surplus yield (or lack of is). Muliplyig is surplus yield rae wi e loa repayme facor i rasforms e ime-adjused average of e prese value rae. Te las sep is e correcio wi e coefficie of payback rapidiy. Tis rae will be e modified differece bewee e ieral rae of reur ad e reuired rae of reur. Tis way cao lead o e accurae differece of wo raes. Tis is comig from e special cleaig meod i wic are mixig e elemes of saic ad dyamic procedures for capial budgeig. Te formula as follows λ r - iε, (6) were, besides e above, = Coefficie of payback rapidiy, r = Ieral rae of reur, = Modifyig facor o e differece of ieral rae of reur ad reuired rae of reur. Tesis 4 From e poi of view of e comparabiliy of decisio varias, e e prese value coais disorios i ree relaiosips. Tese are e iiial ivesme, e duraio ad e rapidiy of capial payback. By sysemaically elimiaig ese disorios e e prese value rasforms io a special kid of rae, amely, e modified differece bewee e facual ad e reuired rae of reur. Reducio of e Formula i e Case of xisig Auiy Terms Accordig o e relaiosip demosraed above, e sysemaic elimiaio of disorio effecs i e e prese value leads o e correced differece bewee e facual ad e reuired rae of reur. Tis is clear ad udersadable we e erms of auiy meod exiss. Two codiios sould be me o apply e auiy meod: 1. Te ivesme sould be processed i a very sor ime (is mus be a ivesme poi).. Te differece bewee reveues ad ou-of-pocke coss sould be e same every year, so e geeraed cas flow size mus be a cosa amou per year. I is case e e prese value meod ca become simpler (as is well kow): - were = Sum of e ivesme were e ivesme process is very sor. Ta is, =. = Differece of reveues ad ou-of-pocke coss i year. = Cosa differece of reveues ad ou-of-pocke coss. Ta is, =, if >. Wi ese wo codiios e ieral rae of reur meod is simpler as well. - ec ad ec 9

7 Mária Illés were ec is a ecical loa repayme facor, usig e facual profiabiliy rae (ieral rae of reur) r isead of reuired rae of reur i. Sarig formula (6): λ r - iε If e yearly payback is cosa ( = ; ad = as was discussed before), e e coefficie of payback rapidiy is 1 (a is =1). Accordig o ese Ta is, r - iε ad Afer rearragig euaio (7): ad - (r-i)ε As oed above or ec - (r-i)ε, (7) ad (8) ad (9) r - iε ec, so ad (1) r - iε r r) i i) - r) 1 i) 1 ad (11) r-iε ad I is special case e rasformed e prese value is e differece bewee e ecical ad e rue loa repayme facor as sow i formula (11). Tis differece by absolue value is smaller a e differece of e facual ad reuired raes of reur because of e ifluece of specific cosrucio of e e prese value divided by iiial ivesme. I e case of exisig auiy erms e modifyig facor ca cou as well. ec ε r - i ad (1) For example if e lifeime of e projec is 1 years, e ieral rae of reur is % ad e reuired rae of reur is 15%, e i e case of exisig auiy erms e rasformed e prese value from formula (11) is: (..15) = =.397. Te 3.9 perceage poi is smaller a e perceage poi of 5, as e differece of e facual ad reuired raes of reur is. Te umerical value of i is case is Tesis 5 I e case of exisig auiy erms e e prese value rasformed o comparable is e differece bewee e ecical ad e rue loa repayme facor, were e ecical repayme facor is e cosa differece of reveues ad ou-of-pocke coss divided by ivesme. RANKING OF INVSTMNT PROJCTS ACCORDING TO CONOMIC FFCTIVNSS I e case of ivesme projecs wi orodox cas flow paers e e prese value ad e ieral rae of reur meods lead o e same resul we selecig e accepable varias, i spie of e fac a eir iformaio coe differs. However, e rakig lis accordig o e wo meods ca differ. Te rakig lis accordig o comparable (rasformed) e prese value correspods o e lis accordig o e ieral rae of reur. Coseuely e iges profiabiliy projec accordig o e ieral rae of reur is also o e firs place i e rakig lis accordig o e e prese value cleared of mai disorio effecs. Rakig by ieral rae of reur is i coformiy of operaioal priciple of e ecoomy, by wic e capial is waderig o e possibiliies of iges profiabiliy (by e give risk). Te everlasig debaed uesio is wic oe is e beer from wo ecoomically efficie ivesme varias a exclude eac oer. Te ivesme of course yields more profi i e projec varia of larger profiabiliy, bu a is o sure e oe wi e larger profiabiliy is o be cose. Te iiial ivesme, e duraio ad e rapidiy of payback sould also be ake io cosideraio, amely ow muc ivesme, ow log ime ad wi wa rapidiy e relaive large profiabiliy akes. Ta ca appe, especially we e less ivesme, sorer life-spa or uicker payback refers o modes yield sum opporuiies, a e somewa less profiable, bu bigger ivesme, loger duraio, slowly payback varia (compared o e larges profiabiliy varia) becomes more advaageous. (Tis ca be see a firs sig; for example, a wi oerwise ucaged codiios a 5 millio uies or -year duraio projec of 4% profiabiliy is more advaageous a oe 5 millio uies or lasig for years avig 5% profiabiliy.) Te relaively iger sum of e prese value or relaively ig rae of surplus yield by e e prese value divided by iiial ivesme ca refer o e laer advaages as well. So e soluio is: seig up a ree-idex umber for rakig were e ieral rae of reur is e primary ad e sum of e e prese value wi e e prese value divided by iiial ivesme appear as secodary rakig idicaors. If all of e ree rakig idicaors sow e same rakig lis, e is will be a e same ime e pracical rakig lis as well. If, owever, e ree idicaors lead o differe rakig liss, e furer aalysis is eeded. 3

8 Trasformig e Ne Prese Value for a Comparable Oe To coiue e seig up of e rakig lis, a is pracical o deermie e criical profiabiliy level (criical ieral rae of reur) of e differece of ivesme sum, or reivesme amous. Tis criical profiabiliy level sould be reaced a e differece of ivesme sum or reivesed amous of e less sum, sorer duraio or uicker payback projec o make e ieral raes of reur of e wo projecs e same. Oly wi is criical profiabiliy rae would bo opios be cosidered eually favorable. If larger profiabiliy caces a e criical profiabiliy rae ca be foresee, e a is favorable o coose e projec wi iger ieral rae of reur; oerwise a is favorable o coose lower capial profiabiliy rae projec (Illés, 1997: ). XAMPL OF A DIFFRNT PAYBACK RAPIDITY AND RANKING Te example below demosraes e relaiosips discussed above usig real umbers. For e sake of is purpose e example disregards e fac a e occurrece of e problem wi suc clariy is o probable i pracice. A compay as a ivesme opio of 35 uis. Te pracical expers worked ou wo ivesme projec varias. Te marke risk of e wo varias is e same, so a uiform 1% reuired rae of reur was assiged o eac of em. Table 1 summarizes e mai daa of e idividual varias. Table 1 Mai Daa of e Two Projec Varias i e xample (i uis) Projec varia 1 Projec varia Year Amou of Ou-of-pocke Amou of Ou-ofpocke coss Reveues Yields Reveues ivesme coss ivesme Yields Decisio preparaio iformaio: 1. Te e prese value of e wo varias is e same (were var. = varia). (var.1) = = 14.3 (var.) = = Sice e e prese value ad e ivesed amou of e wo projecs are e same, e e prese value divided by iiial ivesme ad e profiabiliy idex (PI) will be e same as well ; PI Te ieral rae of reur of e wo varias sows a sigifica differece: Ieral rae of reur (var.1) ~ 13.5%. Ieral rae of reur (var.) ~ 16.5%. Te facual profiabiliy sows a sigifica differece, wic is approximaely 3 perceage pois. Te reaso: e rapidiy of payback is sigificaly iger i e case of e secod varia; coseuely e average locked-up capial is smaller. Mos of e ivesed amou s omial value is repaid a e ed of year oe. 4. For e criical profiabiliy rae of e reivesme amous freed i e secod projec (wi e ieio o esimae rougly) a is pracical o examie wi wa profiabiliy e 4 uis freed a e ed of firs year sould be ivesed o esure a e yield of 5 uis a e ed of year ree. 4 r) 5 ; r) 1.5 ; 1 r ; r.118, a is 11.8% Te 11.8% does o reac e reuired rae of reur, so by a roug esimae e secod varia is e beer oe. For more exac esimaes a is pracical o cosider eac of e aual differeces i yield: 395 r) ; r) ; 1 r 1.1 ; r.1, a is1% Accordig o e more accurae calculaios, coosig e secod varia is more pracical i e case of a reivesme a exceeds e reuired rae of reur by ay small amou. (However cosiderig e smaller risk beid e sorer repayme period, e defiiely e secod varia is beer.) Sould e reuired rae of reur be raised o 15% due o a uexpecedly occurrig iger commodiy marke risk, e e firs varia would drop ou of e group of accepable varias. Is ieral rae of reur would o reac e reuired rae of reur ad is e prese value also would ur egaive. (var.1) = = -13. (var.) = = 4.6 I e case of a 15% reuired rae of reur, e discoued amou of e lack of yield is 13.1 uis for e firs varia, wile e discoued amou of e surplus yield is 4.6 uis for e secod varia. 5. Wi e 1 % reuired rae of reur e dyamic average of e e prese value divided by iiial ivesme will o sow a rakig lis. Cosiderig a e ivesme amou, e prese value ad duraio of e wo projecs are e same, e resulig dyamic average will also be e same

9 Mária Illés Te ime adjused average of e prese value divided by iiial ivesme is 1.7 % i e case of bo projecs. Ta is because of e fac a is sep cao ake io cosideraio e rapidiy of payback. By adlig rapidiy of payback e aual rae of e e prese value would be differe. Te payback rapidiy idexes are: less a 1 of e firs varia ad iger a 1 of e secod varia. So e e prese values cleared of mai disorio effecs are 1.5 ɛ 1 perceage pois ad 4.5 ɛ perceage pois, respecively. Te example sows a e umerical defiiio of e ime adjused average of e e prese value divided by iiial ivesme leads o iformaio a ca be ierpreed i pracice as well (e researcers did o combie e wo seps so far). I addiio, by sysemaically elimiaig e effecs of e e prese value disorig compariso possibiliies, e correced idex umber ge closer ad closer o e differece bewee e ieral rae of reur ad e reuired rae of reur. Tese will o be compleely eual because of e applied examiaio erms. Ackowledgeme Te described work was carried ou as par of e TÁMOP-4../B-1/1-1-8 projec i e framework of e New Hugaria Developme Pla. Te realizaio of is projec is suppored by e uropea Uio, co-fiaced by e uropea Social Fud. RFRNCS ALCHIAN, A. A. (1955): Te rae of ieres, Fiser s rae of reur over coss, ad Keyes ieral rae of reur. Te America coomic Review, 45(5), ARNOLD, J. HOP, T. (199): Accouig for Maageme Decisios, Preice Hall Ieraioal (UK) Ld. BAKR, H. K. - POWLL, G.. (5): Udersadig Fiacial Maageme: A Pracical Guide. Jo Wiley & Sos. USA, UK, Ausralia BIRMAN, H. - SMIDT S. (1957): Capial Budgeig ad e Problem of Reivesig Cas Proceeds, Te Jouralof Busiess, 3, No. 4: BRALY, R. A. - MYRS, S. C. 1(99): Moder vállalai pézügyek. Paem Kf. Budapes. BRIGHAM,. F. - HOUSTON J. F.(9): Fudameals of Fiacial Maageme. Sou-Weser Gegage Learig, Maso, USA COYN, T. J. (1984): Maagerial coomics: Aalysis ad Cases. Busiess Publicaios, Ic. Plao, Texas. FISHR, I. (193): Te Teory of Ieres. Macmilla. GARRISON, R. H. (1988): Maagerial Accouig. Coceps for Plaig, Corol, Decisio Makig. Busiess Publicaios, Ic. Plao, Texas. HLFRT,. A. (1991): Teciues of Fiacial Aalysis. Busiess Oe Irwi. USA. HILL, A. R. (8): Sraegic Fiacial Maageme. ILLÉS M. (199): A gazdaságossági és jövedelmezőségi számíások alapjai. Budapes: Szakszervezeek Gazdaság- és Társadalomkuaó Iézee. ILLÉS M. (1997): Vezeői gazdasága. Kossu Kiadó. ILLÉS M. (7): Scieific problems of moder approac of e prese value. I: Teory, Meodology, Pracice. Uiversiy of Miskolc, Volume 4.Number KAN, S. M. (1975) Ivesme selecio crieria: a examiaio of e eory of e ieral rae of reur ad of e ivesme discou rae uder codiios of uceraiy. PD esis. Uiversiy of Glasgow. KIRULFF, H. (8): MIRR: A beer measure. Busiess Horizos 51, L, A. C. - L, J. C. - L, C. F. (9): Fiacial Aalysis, Plaig, ad Forecasig: Teory ad Applicaio. World Scieific Publisig Co. Pe. Ld. LUHRMAN, T. A A geeral maager s guide o valuaio. Harvard Busiess Review. 75(3): SOLOMON,. (1956).: Te arimeic of ivesme decisios. Te Joural of Busiess, 9(), SCHMALN, H. (): Álaláos üzlei gazdasága. Axel-Spriger Budapes Kiadó. VAN HORN, J.C. VACHOWICZ, J. M. jr. (8): Fudameals of Fiacial Maageme. Pearso ducaio Limied 3