Present and future value formulae for uneven cash flow Based on performance of a Business

Transcription

1 Advces i Mgemet & Applied Ecoomics, vol., o., 20, ISSN: (prit versio), (olie) Itertiol Scietific Press, 20 Preset d future vlue formule for ueve csh flow Bsed o performce of Busiess Ameh Tefer Tessem Abstrct Busiess firm's icome is ot costt, or fied from period to period becuse of this the firm s csh iflow or out flow is ueve. The decisio of firm either to ivest or to borrow from creditors bsed o ueve csh i-flow eed to hve future or preset vlue predictio formul. The problem to fid future d preset vlue formule for ueve csh flow styed usolved for log periods. However, o this pper it wted to show future d preset vlue of ueve csh flow predictio formule bsed o the performce rte ( P ) of busiess. The Performce rte (P ) is percetge by which the curret performce, ecoomic vlue dded (EVA ), of the busiess eceeds the previous performce. Therefore, the firm csh out flows either for ivestmet or for repymet of the borrowed lo growth ccordig to the performce rte (p) of the firm. Keywords: preset vlue formul, future vlue formul, performce rte, Rte of growth. Costructio d busiess bk s.c., P.O. bo 2042, Addis Abb, Ethiopi, e-mil:mbet22002@yhoo.com. Article Ifo: Revised: Mrch 7, 20. Published olie: My 3, 20

2 94 Preset d future vlue formule for ueve csh flow Itroductio Moey does hve differet vlue t differet periods. The moey we owe ow does hve more vlue th we eed to owe it i the future. The fcts behid this ide re:. we prefer to use moey more productively ow so s to get rel retur i the future 2. Durig ifltio period the moey we use ow does hve more purchsig power th we use the moey i the future 3. At lst sice the future is full of risk d ucertity we prefer curret cosumptio to future cosumptio. This pheomeo is lso referred to s time preferece of moey which is epressed by iterest rte ([]-pp.44-46, [6]-pp , [8]-pp.9-20). A iterest rte is defied s the price pid by borrower to bk/ creditor for the use of moey/ resources durig the give time period. A borrower from bk pys the pricipl d percetge of the pricipl per uit of time. A moey depositor ito the bk ccout will receive i the ed of iterest period the pricipl deposit mout d percetge of the pricipl per uit of time, which is the iterest mout. Compoudig d discoutig techiques re the two methods by which time vlue of moey clculted. The future d preset vlues of uity formule developed i such techique vlid oly for equl mout of periodic csh flow. The formule do hve the followig shortcomigs. The formule do ot clculte csh flow which growth ccordig to the icrese of the project s performce. 2. The formule do cosider project performce to be sme throughout the project life, which is impossible. A project whose performce does ot chge ccordig to the mrket situtio throughout the project life my be out of the mrket.

3 Ameh Tefer Tessem The formule do ot cosider the future ifltio period where the project csh iflow will be reduced. 4. The formule do ot cosider fleibility of deposit to bk or fleibility of pymet to creditors. 5. The preset vlue formul does t cosider low csh flow t iitil stge. At iitil stge of ivestmet where there is high burde of ivestmet cost, high epese for dvertisemet of the busiess products, there might be o csh iflow for the firm. Becuse of this, the set repymet usig preset vlue formul by creditors without cosiderig the busiess firm s iitil ivestmet cost becomes burde for borrower. 6. The repymet set for borrowers by usig uity formul goes decresig from high burde to low becuse the repymet mout i the ed of the lo period comes dow to smll mout d the borrower gets relief ([6], pp.05). This is meigless sice the borrower should hve got relief of lo pymet t the begiig of lo period where the project csh iflow is very low. 7. The csh flow we re goig to clculte by preset vlue formul does t hve y future risk premium though the iterest rte does hve d this cretes mismtch betwee the csh flow d the iterest rte. As hve see bove preset d future vlue formule fil to clculte ueve csh flow. I the ivestmet decisio of firm, oe could receive ueve of csh iflow, which does ot eble him/her to put o decisio merely for there is o y formul to clculte ueve csh ivestmet shortly. Becuse of this, we compelled to use lborious techique ([6]- pp.274, [7]-pp.25, [9]-pp.3-9) by which we re goig to clculte ech periods csh flow to get the totl sum. O this pper, the uthor wt to show how the sum of ueve csh flow either for ivestmet or for repymet of borrowed lo from bk bsed o performce of the busiess firm c be determied.

4 96 Preset d future vlue formule for ueve csh flow 2 Prelimiry ote Although, Trditiol ccoutig mesures of performce of firm re profit, erig per shre(eps), retur o ivestmet (ROI),free csh flow (FCF),cpitl productivity (KP), lbor productivity (LP) d retur o cpitl employed (ROCE) ech of ech igores csh d cost of cpitl so s to geerte the trget profit ([3]-pp.326, [0]-pp.44). Ruther, the best mesurig tool of performce of busiess firm is ecoomic vlue dded which provides the moey vlue creted for ivestor i give period of time by weightig the profit geerted gist the cost of cpitl employed so s to geerte tht profit, [4]-pp.477. From this, sice EVA cosiders the mout of cpitl ivested, the retur ered o cpitl d the cost of cpitl (WACC) reflectig the risk djusted required rte of retur, it is thought to hve ll chrcteristics of the mesure though it is vlid oly for short period of time. Furthermore, sice EVA (Ecoomic Vlue Added) hs cosidered s mesure of both performce d vlue of busiess firm, it ssumes s wy to determie the vlue creted ecess bove the required retur for the shreholder of the busiess firm. The firm cretes welth for the shreholder whe the reveue of the firm eceeds over the cost of doig busiess d the cost of cpitl. A busiess firm cretes vlue for its shreholders o the bses of positive EVA rther th simply mkig ccoutig profits. The positive mgitude of EVA idictes s the busiess firm is improvig its et csh retur o ivested cpitl. The icremet of EVA of the firm from yer to yer will result icrese of the mrket vlue of the firm, []. The eistece of ccoutig iformtio of firm for sigle ccoutig period helps the mger to grsp the bsic kow how of the firm performce i tht ccoutig period. A mger who hs good eperiece of the firm performce helps i fcilittig to predict future performce of firm bsig o the pst ficil sttemet such s icome sttemet d blce sheet. The vilbility of pst tred records help to clculte d predict progressive performce rte of the firm so s to determie progressive csh iflow o the firm ivestmet retur

5 Ameh Tefer Tessem 97 d the firm progressive bk lo repymet, ([2]-pp.64-65, []-pp ). Sice the firm's performce rte hs ssumed to be progressive, the csh flow of the firm ssumes to grow from period to period. Performce rte is percetge by which the curret performce of firm (i this cse, EVA) growth from the previous. 3 Mi results A ivestor who is goig to borrow some fied mout of moey will get ito commitmet to py the lo repymet mout t every period s per the cotrctul greemet betwee the ivestor d creditor. The lo repymet mout pre-clculted d set by the creditors ssumig to py the period iterest mout d portio of the pricipl by the ecess bove the iterest mout. Ech of the et periods repymet mout is the product of the precedig repymet mout d of the curret project s performce rte. At iitil repymet period, the repymet mout ssumed to be smll fied mout without performce rte. However, the et fter the first period repymet mout growth ccordig to the growth of busiess performce. Assume tht the first smll repymet mout to be d the et fter the first repymet cosequetly be clculted s ( P i)( P2 i)...( P 2 i) P Such tht Repymet Period 2 3 Repymet mout ( P i) ( P i) P 2 ( P i)( P i)...( P i) P 2 2

6 98 Preset d future vlue formule for ueve csh flow where P = Performce rte, y rbitrry figure greter th zero, which c be mesured s the percetge by which the ( ) th period EVA growth from the th period EVA, i.e., P EVA EVA =. EVA The reso why the uthor preferred to put performce rte i such wy is to sve the csh flow from u-ecessry eggertio becuse the rtio is very smll though the th period performce rte is greter th/or less th the ( ) th period performce rte. Here EVA for periods is to me projected et icome for lo periods o the firm projected icome sttemet. i = Bk iterest rte which is dded to the performce rte sice the th period repymet should coti portio of the pricipl mout d of the period iterest mout, [2]-pp.444. It is betwee 0 d = Lo period The curret csh vlue of the future csh flow of these lo repymets epressed s follow Result : ( ) ( Pm i) ( i ) i m= ( P i) ( P i) P2 ( P i)( P2 i) P3 = ( i ) ( i) ( i) ( i) ( P i)( P2 i)( P3 i)...( P 2 i) P ( i) Aother decisio of busiess firm is decisio to ivest csh ow i order to receive csh, goods or services i the future period. Here lso let represet the csh ow ivestig for ech period s

7 Ameh Tefer Tessem 99 Period 2 3 Ivestig Csh ( W T) ( p )( W T ) 2 2 ( p )( p )...( p )( p )( W T ) where W = P, T = ( P )( i) for ( P )( i) betwee 0 d, (i.e., 0 < ( P )( i) < ), T = is the firm performce other th bk s deposit iterest. The deposit iterest rte should ot be icluded ito the project performce sice it is lwys less th the bk ledig iterest rte. i = Bk iterest rte which is betwee 0 d. = ivestig period. W T = A portio of performce rte by which the ecess bove cosumptio sved ito bk ccout. Therefore, the future vlue of ivestig csh ow represeted s follow Result 2: ( i ) ( pm )( i) i m= = ( i ) ( W T)( i ) ( p ) ( W T )( i ) ( p )( p )...( p )( W T )( i ) Advtges The pper deeply focus o how busiess firms determie their csh iflow or out flow bsed o their ecoomic profit (or ecoomic vlue dded).as reveled by the formule, the first pymet, which is ecess bove cosumptio, which c be ssiged for pymet of debt or for svig ito bk ccout is very smll mout. The et fter the first csh flow mouts progress or growth

8 00 Preset d future vlue formule for ueve csh flow log with the busiess firm s performce rte. Sice this performce rte shows the reltive level of growth of oe firm, curret to lst ecoomic profit, it embrces ll ctivities of the firm. EVA reflects et of the csh geerted d the csh ivested by the busiess firm. As EVA fluctutes from period to period, the et csh left to the busiess firm lso fluctutes from period to period. This fluctutio of the firm s csh iflow or out flow hs ectly reflected by the busiess performce rte ( P ). Ficig orgs, like bks, use ordiry uity formul to determie lo cpcity s well s lo repymet of the borrowers. Sice the formul does ot coti y mesure of the performce of the borrowig org, Most of the busiess lo re see gettig ito operformig lo ctegory, lettig other thigs beig costt. Becuse of this Bks lwys ly their ow rules d regultios to miimize operformig los t hd but could ot rech t coclusive solutio for log periods. Uless otherwise there eisted stiff cotrol to collect the disbursed lo, the icremet of bd debt from period to period will ectly hrm the ecoomic coditio of the coutry s whole. However, the preset vlue formul stted by this pper clcultes the projected fluctutig repymet mout log with the performce of the borrowig org bsed o the rel csh o hd, which is ecess bove the cosumptio. The preset vlue formul gives dvtge to borrowers sice it relieve borrowers sufferig from pyig huge mout t the beggig periods. The firm t iitil stge ssumed t high estblishmet cost d evetully fter coverig the ivestmet cost the lo repymet ssumes to rise ccordig to the ivestmet performce rte. The repymets clculted d set for ech lo periods by creditors/bks i such wy will beefit them i collectio of lo iterest sice t the beggig yers (periods) the pricipl lo mout hs ot much ffected by the smll repymets comprig to the et periods repymets. Ech of the

9 Ameh Tefer Tessem 0 periods repymet mouts ssumed to cler first the iterest mout d the goes to the pricipl by the ecess bove the iterest mout. O the other hd, future vlue formul reled depositors for they c deposit ech periods ecess bove of their cosumptio. A ewly etrt busiess firm ito mrket does hve little mout of csh t iitil stge d evetully fter coverig the estblishmet cost the liquidity mout o hd grows ccordigly. The growth of the csh iflow of the busiess firm depeds o its performce. Whe performce of the busiess improves, its profit d csh iflows icrese i the sme wy. As result of this, the depositor c deposit the ecess bove his/her busiess cosumptio strtig by smll mout of moey, which c evetully icrese from period to period to higher mout ccordig to the busiess performce. 3.. Cost The formule highly depeded o the performce of the firm, Ecoomic vlue dded (EVA).Sice ll busiess firms do ot use EVA s mesure of performce, they re obliged to go bck d reclculte their EVA first before they hve rrived t decisio to use the formule. Theorem 3. Let for ll elemets of turl umbers such tht p d re rbitrry rel umbers. If the Cf th term csh out flow c be epressed 2 ( pr i) p r= = the the sum of ech csh out flow s preset vlue c clculte s follow ( Pm i) ( i ) ( i) m= ( P i) ( P i) P2 ( P i)( P2 i) P3 = ( i ) ( i) ( i) ( i) ( P i)( P2 i)( P3 i)...( P 2 i) P ( i)

10 02 Preset d future vlue formule for ueve csh flow where i is bk iterest rte which is betwee zero d oe ( 0< i < ). Theorem 3.2 Let for ll elemets of turl umbers such tht p d re rbitrry rel umbers. If the th term csh i flow be epressed 2 = ( pr )( W T )( i r= Cf ) the the sum of ech csh i flow future vlue c clculte s follow ( i ) ( pm )( i) i m= = ( i ) ( W T)( i ) ( p ) ( W T )( i ) ( p )( p )...( p )( W T )( i ) 2 2 where W = P, T = ( P )( i) for ( P )( i) betwee 0 d, (i.e., 0 < ( P )( i) < ), P = performce rte i.e. such tht p or p d, T is the firm performce other th bk s deposit iterest. The deposit iterest rte should ot be icluded ito the project performce sice it is lwys less th the bk ledig iterest rte, i = Bk iterest rte which is betwee 0 d, = ivestig period, W T = A portio of performce rte by which the ecess bove cosumptio should be sved ito bk ccout. Proof. Let the th period csh flow ( ), performce rte p ) d the discout ( rte () be epressed by y figure, which c mthemticlly be epressed s R ( R el umbers) for ll N (turl umbers), such tht,, p R. The the sum of csh flow for periods c be eplied s follow This c eqully epressed s m = 2 m=

11 Ameh Tefer Tessem m =, for 0 2 m= Split the terms s = ( 0) (( 2 2 ) ( )) (( )) ( 2 )) (( ) ( )) Collectig like terms i the brckets we get tht 2 = ( ) ( 2 2 )( ) ( 2 3 )( ) ( ) 2 = ( )( ) ( )(... ) 2 3 ( )( ) ( ) 2 = ( ) 2 ( ) ( ) ( 2 ) 2 2 ( = ) Multiply both sides by, for m= 2 m = 2 ( ) ( ) Sice the formul of geometric progressio t rtio= c be epressed s = ([2]-pp.59), it follows tht = ( ) 2( ) ( ) ( 2 ) Equivletly this c epress s

12 04 Preset d future vlue formule for ueve csh flow = ( ) ( ) ( 2 ) ( 2 ) Let = 0, the it follows tht m= 2 m = ( ) ( 2 ) 3 ( 2 3 ) ( 2 ) From this epressio let s ssume tht the csh flow of project for ech period c be defied s This c epli s =, = p, 2 = p ( ), = p ( ),, = p ( ) 2 or c be writte s =, = p ( ), 2 = p ( )( p ), 3 2 = p ( )( p )( p ),, = p ( )( p )( p )...( p ) 2 2 = p ( ) 2 3 () Such tht P or P & P 0 = d puttig this i (), we hve

13 Ameh Tefer Tessem = P ( 2) ( 2) P2 4 ( 2 3) ( 2 3) P3 ( 2 3 ) ( 2 ) P = ( ) (( p ) ) ( 2)( ( P2 ) ) 4 ( 2 3) (( P3 ) ) ( 2 3 ) (( P ) ) = ( ) (( p ) ) ( 2)( ( P2 ) ) 4 ( 2 3) (( P3 ) ) ( 2 3 ) (( P ) ) Multiplyig both sides of the equtio by, we obti m= m= m = ( ( p ) ) ( 2) ((p2 ) ) 4 ( 2 3)(( p3 ) ) 2 ( ) ( ) ( p ) m = p (( ) ) ( )((p ) ) ( ) ( ) ( ) ( p ) if the bk iterest rte (i) epressed s ( ) ( p ) (2) 2 = d puttig this i (2) d i multiplyig both sides of the equtio by ( i ) we hve the followig

14 06 Preset d future vlue formule for ueve csh flow ( p i) ( 2)( p2 i) ( 2 3)( p3 i) i = ( i ) ( i ) ( i ) ( i ) m m= ( 2 )( p i) ( i ) (3) Suppose tht i the bove equtio (3) P = P i, d ddig both sides of the equtio we obti the followig preset vlue formul ( i ) ( Pm i) ( i ) ( i) m= ( P i) ( P i) P2 ( P i)( P2 i) P3 = ( i ) ( i) ( i) ( i) ( P i)( P2 i)( P3 i)...( P 2 i) P ( i) The iitil csh flow (), i the bove formul, growth progressively from oe period s to other by multiplyig with progressive performce rte of the period. Ech of the periods csh flow cotis iterest rte tht resists ifltio d risk which might be hppeed i the future period. Furthermore ech of the period csh flow cotis the period performce rte tht c move log with the stregth of the busiess which c led the csh flow highly voltile. Agi let cosider equtio (2), by puttig firs = ( i ) d the deductig ( i ) 2 from both sides, d the fter ddig both side of the equtio ( i ), we hve the followig future vlue formul [ m]( i ) ( i ) i m= 2 3 ( i ) ( W T)( i ) ( 2)( W2 T2)( i ) ( 2 )( W T )( i ) This c be writte s =

15 Ameh Tefer Tessem 07 ( i ) ( pm )( i) i m= = ( i ) ( W T)( i ) ( p )( W T )( i ) ( p )( p )...( p )( W T )( i ) 2 2 where W = P, T = ( P )( i) for ( P )( i) betwee 0 d, (i.e., 0 < ( P )( i) < ), P = performce rte i.e. such tht p or p d T = is the firm performce other th bk s deposit iterest. The deposit iterest rte should ot be icluded ito the project performce sice it is lwys less th the bk ledig iterest rte. i = Bk iterest rte which is betwee 0 d. = ivestig period W T = A portio of performce rte by which the ecess bove cosumptio should be sved ito bk ccout. 4 Coclusio Most busiess org uses EVA s mesure of both vlue d performce. The mi objective of bk is cceptig deposit from the society d ledig the collected deposit mout to the people who met liquidity shortge. The lo officer of the bk ought to provide strog relistic fesibility study regrdig the project to idetify the eligibility of the perso to be borrower bsed o the projected et csh iflow of the busiess. The mi problem styed for log is tht projected csh flow of the busiess igores future risk of iterest rte fluctutio d future ifltio. Becuse of these shortcomigs, the lo repymet set by the bk disregrdig ifltio d risk i iterest rte fluctutio will become burde to the borrower whe this fctor hppes i the rel mrket. Becuse of this, the borrowers fil to climb their lo repymet

16 08 Preset d future vlue formule for ueve csh flow obligtio withi the give lo life d this result the bk to fil for collectig the disbursed lo mout properly. The icrese of ucollectble los from period to period leds the bk to meet liquidity problem, which will lso levy shdow to the coutry ecoomy s whole. However, the bk liquidity problem will hve solutio if the bk hs hbit of keepig reserve portio of et csh o hd to fill bck the shortge of liquidity. The future vlue formul eble people, who hve o ecess csh o hd for ivestmet, to decide ow svig portio of icome ccordig to their erigs growth from period to period so s to relize their drem fter some fied period. This lso ecourges the svig hbits of people those who hve low icome d those who re slried, people who get their icome from their employmet t fied period itervl such s mothly or ully. Geerlly, the formule c esily help creditors, ivestors d persos who hve low icome i order to fcilitte their decisio shortly d ccurtely. They c lso help to estimte either the preset or the future csh flow of firm more perfectly. Refereces [] Aswth Dmodr, Corporte fice, 2 d editio, Joh Wiley & Sos, Ic., USA, , 200. [2] Aleder Hmilto Istitute, Ficil mgemet hd book, globl books d subscriptio services, New Delhi, , 998. [3] R.A. Breley d S.C. Myers, Priciples of corporte fice, McGrw-Hill Ic., New York, , [4] Coli Firer, S.A. Ross, R.W. Westerfield d B.D. Jord, Fudmetls of Corporte Fice, 3 rd editio, McGrw-Hill Compies, Lodo, [5] Costructio d Busiess Bk SC, [6] I M Pdey, Ficil Mgemet, 8 th editio, Viks publishig house Pvt Ltd, New Delhi, , 999.

17 Ameh Tefer Tessem 09 [7] A.N. Mosich, Itermedite ccoutig, 6 th editio, McGrw-Hill, Ic., U.S.A, , 963. [8] D.R. Myddelto, The Essece of Ficil Mgemet, Pretice Hll Itertiol Limited, Uited Kigdom(UK), 9-29, 995. [9] Chdr Prs, Fudmetls of ficil mgemet, 3 rd editio, Tt McGrw-Hill, Publishig Compy Limited, New Delhi, 3. 8., 993. [0] Proctor Ry, Mgeril Accoutig For Busiess Decisios, 2 d editio, Perso Eductio Limited, Egld, , [] R.F. Reilly d R.P. Schweihs, Hd Book of Advced Busiess Vlutio, McGrw-Hill Compies, Ic., New York, , [2] M.A. Sobel d N. Lerer, College lgebr, 4 th editio, Pretice-Hll, Ic., New Jersey, , 995.