aagemet Studies, ar.-apr. 2019, Vol. 7, No. 2, 87-95 doi: 10.17265/2328-2185/2019.02.001 D DAVID PUBLISHING Aggregate Capital Tied-up by Ivestmet Projects The Possibility of a Simple Estimatio ária Illés Uiversity of iskolc, iskolc, Hugary The aggregate capital eeds are a ew busiess ecoomics category which provides a ew aspect to evaluate ivestmet projects. The literature does ot deal with this category as the project s total fiacial resource requiremet. It is the total capital tied-up for the project i its lifetime. For calculatio of it, the yearly capital tie-ups are beig added together. Based o this, it ca be examied the total capital amout, which results i a give et preset value, or the total capital amout, which operates accordig to the give rate of profitability. The paper iterprets the category, presets its relatioship with the iterest rate, ad also presets the method of calculatio based o model editig. I the case of the iteral rate of retur, the estimatio may be greatly simplified. Istead of determiig the yearly amouts ad summatio of these, the estimatio ca be carried out also with a simple divisio of two data. The paper demostrates the possibility of simplificatio ad shows a example to preset the iterrelatios of data. Keywords: et preset value (NPV), iteral rate of retur (IRR), capital tied-up, retur requiremet, yield structure Purpose ad ethod This paper deals with iterpretatio ad quatificatio of aggregate capital eeds. It is a ew busiess ecoomics category ad is useful iformatio for ivestmet project evaluatios. The mai purposes are as follows: To itroduce the aggregate capital eeds as a fiacial resource tied-up by the project. To outlie the geeral model of the calculatio method alogside the flexible iterpretatio of the yield structure. To explai ad to prove a possibility of a very simple calculatio of this idicator for the case of iteral rate of retur (IRR). The mai research methods are cotet aalysis, methodological aalysis, ad model costructio. Both logical ad mathematical processes are used to provide proof. The aalyses ad the fidigs are all cocerig for ivestmet projects with orthodox cash flow patter. (These cash flow patters are amed also ormal, regular, typical, ad covetioal). The poit is that the series of the differece betwee aual reveues ad aual expeditures start with egative amout or amouts ad the sig of these differeces chages oly oce. This arrowig of the scope of the examiatio is ecessary because of the fact that the et preset value (NPV) ad the IRR method lead to a real result oly i ária Illés, Ph.D., professor, Departmet of Busiess Ecoomics, Uiversity of iskolc, iskolc, Hugary. Correspodece cocerig this article should be addressed to ária Illés, Departmet of Busiess Ecoomics. H-3515 iskolc-egyetemváros, Hugary.
88 AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS this ivestmet project circle. I the case of uorthodox cash flow patter, both methods lead to false results. It should be emphasized: Ulike the geeral perceptio of the literature, the NPV also cotais false iformatio i the case of uorthodox cash flow patters (Illés, 2014; 2016a). The paper uses that busiess ecoomics perspective, i which the busiess ecoomics fuctios as a idepedet disciplie ad ot ay of the braches of microecoomics (Illés, 2016b). This approach iterprets ad maages the ecoomic iformatio ad the database itself accordig to the real ecoomic coditios, furthermore uses basically corporate termiology. Aggregate Capital Needs as a New Busiess Ecoomics Category The idicator of aggregate capital eeds shows all the capital tied-ups of a ivestmet project. This is oe of the importat ecoomic features of the ivestmet project as it shows the sum of fiacial resource eeds. For this reaso, it should be listed amog the most importat iformatio of these projects. Best of the author s kowledge, the aggregate capital eeds do ot occur i this cotext either i busiess ecoomics or busiess practice literature. This iformatio is useful especially whe usig the et preset value (NPV) or the iteral rate of retur (IRR) methods. The category is iterpretable also from a accoutig poit of view accordig to the accoutig database. The backgroud mechaism of the NPV ad IRR method i a correct way takes ito accout the yearly capital tie-ups as the retur requiremet, ad the yearly capital tie-ups are the basis for chargig the profit requiremet or for profitability calculatio accordig to the IRR. These methodological solutios are ot visible o the surface. The mai questio of busiess efficiecy is whether the retur requiremet is met. This questio, of course, ca be aswered without kowig the aggregate capital eeds. The methodology treats the total capital tied-up as the o-iterestig iformatio. The iformatio o the capital tied-ups remais i the backgroud. I the case of some rakig creatig tools, whe it would be justified to take ito accout the aggregate capital eeds, for this purpose, there is used the amout of iitial ivestmet. Solomo (1956, p. 127) said, Whe comparig two projects requirig differet outlays, it is ecessary to compare preset value per dollar of outlay rather tha the absolute preset value of the projects. I the quoted text, outlay is the ivestmet. This cocept is still alive eve today. It ca be see maily i textbooks. For example, Damodara s (2010) rakig creatig tool is the NPV divided by iitial ivestmet (i this book, this idicator is amed profitability idex). The same idex is icluded i Garriso, Noree, ad Brewer s (2009) book, as the project profitability idex. The geerally used profitability idex (that is the summed preset value of yearly capital yields divided by the iitial ivestmet) ca be rearraged as [1 + NPV divided by the iitial ivestmet]. This is suggested as a rakig-formig idicator for example by Brealey ad yers (1988), Albrecht, J. Stice, E. Stice, ad Swai, 2008), ad Watso ad Head (2009). The above two methods lead to the same rakig. The idicator of aggregate capital eeds is formed by three factors. These are the iitial ivestmet, the rapidity of the retur, ad the payback period or the lifetime (i case of NPV is the payback period; i case of IRR is the lifetime). The higher the iitial ivestmet, slower the retur, ad loger the payback period is (or the lifetime), the greater the total capital tied-up by the project is. Highlightig oe of the three factors caot lead to a clear ecoomic result. I order to quatify the aggregate capital eeds, the capital tied-up must be determied for each year ad the summed these. The yearly capital tie-ups are that parts of the iitial ivestmet, which are ot yet retured
AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS 89 util a give year. These parts are gradually decreasig i the case of a cotiuous capital retur. The aggregate capital eeds take ito accout simultaeously all of the yearly existig capital tie-ups ad cosequetly also the iitial ivestmet ad the tied-up time. Accordig to this method, the aggregate capital eeds show such cotet as if this amout would be ivested for oe year. Cosequetly, aggregate capital eeds of differet projects are comparable. This coceptio gives a ew viewpoit to the ivestmet project evaluatio. I this way, it is possible to examie how much is the total capital tied-up that result i the give NPV, or how much is the total capital tied-up that operates with the give IRR. I this case, NPV is more favorable; the smaller aggregate capital eeds resulted i the higher NPV are, the higher discouted surplus profit is. By cotrast, for IRR, it is preferable that the larger aggregate capital eeds operate with the higher IRR, that is, with the higher profitability rate (Illés, 2014). A project with a low capital tied-up ca result i low profits eve by relatively high profitability. With the questio of how much capital of a ivestmet project has ot bee recovered util the ed of a give year, that is, how much is the capital tied-up i the give year is dealt with several publicatios. I these works, the quatificatio happes clearly o the IRR basis. To the author s best kowledge, the first cosiderable work o this subject is Bouldig s (1935) article. Accordig to today s wordig, he maaged to split the yearly yields to capital face value retur ad to profit, with simultaeously quatifyig the yet ot recovered ivestmet for every year. He solved the problem with a detour method; to this, he relies o a kid of quasi market-value. As a cosequece of this problem-solvig method, that is, usig this sort of market-value, may misuderstadigs have arise. Noteworthy the Crea s (2005) work presets i the laguage of busiess ecoomics, how is formed the structure of the yearly yield i the retur process whe is used the IRR. For this, aturally, there was required to quatify the yearly capital tie-ups as well. The demostratio of these cotet relatioships happes with a great detail ad through examples. The article several times refers to the similarity which is valid durig the repaymet of the loa. The similarity is as follows: I the case of a loa, oe part of the repaymet is the iterest paymet ad the rest as the other part is loa capital repaymet (The iterest cocers to the upaid capital i all cases). Haze (2003) ad agi (2010) quatified the preset value (i.e., the discouted amout) of the yearly capital tie-ups. However, this discouted amout is used i other topics as a kid of calculatio aid, ad ot i the sese as basic iformatio of the fiacial resource eeds of a project. For this latter case, the discoutig would be a defiitely icorrect solutio (The explaatio is later). Basic Relatioships Breakdow of Yields I the case of orthodox cash flow patters, the yield is the positive differece betwee the aual reveues ad aual expeditures. The yield ca cover the iitial ivestmet ad the profit requiremet. This relatioship is ideed similar to bak ledig trasactios whe the repaymet icludes the loa face value ad the iterest. However, i the case of loa repaymet, the accoutig officially separates the yearly amouts to the two compoets. Repaymet of the omial value of the loa is cosidered to be capital movemet ; the iterest costitutes a cost. The structure of the aual equal repaymets chages from year to year. The mai reaso for this chagig is that the amout of yet o-repaid capital is steadily decliig. As the yearly capital tied-up is
90 AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS decreased, the yearly iterest sum also is decreasig. Cosequetly, i the costat repaymet amout, the capital part is gettig bigger. The aual yields of the projects geerally are variable amouts, but the structural chages of these are iflueced by similar relatioships. The yield ca be examied from differet poits of view, accordig to differet compoets. The examiatio purpose determies the yield structure to be examied. For example, although the yield is ot a accoutig category, from a accoutig poit of view, the yield structure cosists of two parts: amortizatio (that is, capital moey) ad pre-tax profits. I this poit of view, the aggregate capital eeds are the sum of the row of book values of the assets. The yield of the assets which are described for zero is profit fully. I the case of busiess efficiecy examiatio, the matter of separatig ito appropriate cotet elemets depeds o ot oly the examiatio purpose but also the examiatio method. Whe the NPV method is used, the yield may cosist of three compoets: profit accordig to the required rate of retur; ivestmet face value retur; surplus profit. Before meetig the retur requiremets, the yield cosists of two parts: The first oe is a profit part accordig to the required rate of retur ad the secod is the ivestmet recoverig. I that year, whe is fulfilled the retur requiremet, the yield ca cosist of all the three compoets. After the fulfillmet of the capital-based retur requiremets, the cotet of the resultig yield is evidetly surplus profit. Accordig to these, i a give year s yield, surplus profit may oly arise if beforehad is fulfilled the capital-based retur requiremets, i.e., if the omial value of the iitial ivestmet ad the profit requiremet there are covered. (Whe the NPV is egative, of course, there is ot geeratig ay surplus profit, ad if the project is loss-makig, the eve profits o will be). From the perspective of the IRR method, the yield cosists of two compoets all alog of the duratio: profit accordig to the IRR; ivestmet face value retur. This follows from the method of the IRR. Sice the IRR icludes all profits i the form of a ratio, i this case, there is o surplus profit. Lik Betwee Aggregate Capital Needs ad Iterest Rate I the case of a give iitial ivestmet ad a give yield row, the amout of aually recovered capital is determied by the profit eeded accordig to the iterest rate. The higher the iterest rate is, the smaller the part for capital reimbursemet remais. With a lower yearly recovered capital, the iitial ivestmet will be recovered over a loger period of time ad aggregate capital eeds will be higher. Uder otherwise uchaged coditios, the smallest aggregate capital eeds come at the case of zero iterest rate. I this situatio, the total amouts of the aual yields ca be iterpreted as a capital recoverig, thus the capital tied-up time is shorter. After recoverig of iitial ivestmet, the ecoomic cotet of yields is profit (These coectios are also explaied through some examples i Illés, 2014). The IRR is the highest iterest rate, with which all the iitial ivestmet ad the profit accordig to the iterest rate ca be retured. I the case of the efficiet ivestmet project, this iterest rate geerates the highest aggregate capital eeds. If the iterest rate used i the calculatio is higher tha the IRR, the retur requiremet is ot met.
AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS 91 The atter of Discoutig Accordig to the above, i order to estimate the capital retur of a give year, as the first step the profit requiremet, or the profit accordig to IRR must be set out. If the yield is higher tha the profit requiremet or the calculated profit with IRR respectively, the the differece couts as a ivestmet reimbursemet i the examied year (if for a give year, the yield is lower tha the profit requiremet, the differece must be added to the capital tied-up). The recovered capital of the previous year reduces the capital tied-up for the give year. So, the aually tied-up amouts ca be calculated step-by-step for every year of the pay off period or of the lifetime respectively. The aggregate capital eeds are the amout of the aual capital tie-ups (as it metioed above). It is a very importat relatioship that the profit accordig to the cocered iterest rate has bee take ito accout i the process of quatifyig the aual capital tie-ups. Therefore, durig the summatio, already it should ot be discouted. I the case of discoutig the aual capital tie-ups, there would be takig ito accout the profit requiremet or the profit accordig to IRR respectively twice, ad accordig to this, the aggregate capital eeds would be less tha the factual oe. As a supplemetary explaatio, the similarity to the loa trasactio may also be refereced. After the yearly loa repaymets havig bee separated to face value repaymets ad iterests, the simple sum of the yearly face value repaymets must be equal to the sum of the borrowed capital. I this case, the yearly pricipal repaymets ca be summed up without discoutig, ad like this oe, the aual capital tie-ups as well. odels for Calculatig Aggregate Capital Needs Costructio of a Calculatio odel for the Case NPV ethod The startig relatioships: (1) It follows from the essece of the orthodox cash flow patter that after the iitial ivestmet beig paid, the project does ot require aother capital ivolvemet. (2) The pay off period is z year. Durig this time, the iitial ivestmet (E 0 ) ad the profit accordig to the required rate of retur will be recovered (whe the project is efficiet). (3) Durig the year t, the amout of the capital tied-up is that part of the iitial ivestmet, which is ot recovered util the ed of the previous year: E t-1. (4) I the 0 < t < z years, the yield cosists of two parts: the retur of the required profit ad the capital face value retur. (5) I the year z, the yield ca cosist of three cotet parts. The third oe is the surplus profit. (6) I the year t, the profit requiremet is E t-1 i. (7) I the 1 < t z years, the sum of the yearly capital recoverig ca be calculated as the differece betwee the yield ad the profit requiremet: H t E t-1 i. Symbols: E 0 = iitial ivestmet occurrig at the zero poit of time; E t-1 = the capital tied-up durig the t-th year; H t = the yield (that is the differece betwee the reveues ad the expeditures) i year t, where the value of H t is always positive by the terms of orthodox cash flow patter projects;
92 AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS t = serial umber of years (0 < t); i = required rate of retur; z = the last serial umber of the year of the dyamic payback period. The yearly capital tie-ups: I the first year of operatio (t = 1), the full amout of iitial ivestmet is tied-up i the project (E 0 ). For the secod year, the first year capital tied-up is decreased with the capital retured i the first year: E 1 0 1 0 E -(H - E i) For the third year, the capital tied-up is decreased with the capital retured durig the secod year s operatio: E -(H - E i) E2 1 2 1 Durig 1 < t z years, the amout of capital tied-up is Et-1 t-2 t-1 t-2 E -(H - E i ) t 2...z (1) Based o all these, i the case of NPV, the idex umber of aggregate capital eeds (E AN ) is as follows: E (2) AN E t-1 t1 This idex umber icludes the effect of iitial ivestmet, the pay of period, ad the rapidity of capital payback. Basic odel Costructio for the Case of IRR ethod The IRR shows that the total omial value of the profit how may percet profit abilities mea. Retur of the capital face value lasts util the ed of the duratio. I this case, there is a profit calculated accordig to IRR ad ot a profit requiremet. Cosequetly, i this case, the relatioships for NPV are modified as follows (r = IRR): All alog of the duratio, the yield cosists of profit calculated accordig to the IRR ad ivestmet face value retur. I the year t, the calculated profit accordig to IRR is E t-1 r I each year of the total lifetime, the amout of capital retur is the differece betwee the yield ad the calculated profit. After the first year, util the ed of the lifetime, to each year, a calculatio is eeded to determie the capital tie-ups accordig to the followig formula: Et-1 t-2 t-1 t-2 z E -(H - E r) t 2... (3) ( = lifetime of the project.) The aggregate capital eeds by the IRR (E AI ) is described i Formula (4). E E (4) AI t1 Simplificatio Possibility for the Case of the IRR ethod I the case of IRR, the calculatio of the aggregate capital eeds ca be sigificatly simplified. This possibility is because the total profit sum calculated accordig to the IRR ad the true total profit sum geerated t-1
AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS 93 durig the project lifetime is the same. This follows from the cotet backgroud of the NPV formula ad the solutio of it to zero (Illés, 2014). As a result of the calculatio method of aggregate capital eeds, the face value of the profits geerated durig the whole lifetime of the project is allocated for each year accordig to the IRR. Cosequetly, the profits calculated at the omial value for each year of the project operatio are as follows: rt = profit i year t calculated with the IRR. Summig up: t1 1 The calculated ad the real total profit sum is the same: t1 rt Et- r (5) rt E r (6) rt t1 ( = the differece betwee the total sum of the omial value of sales reveues ad the total sum of the omial value of all expeditures). After rearragig Formula (8): t1 E t1 r t1 r EAI (9) It follows from the Formula (9) that the aggregate capital eeds ca be calculated also as the divisio of the profits sum with the IRR. r (7) (8) E (10) Accordig to the above, i the case of the IRR, relatively legthy calculatios ca be iterchageable by a simple divisio. Furthermore usig the Formula (10), it is easy to verify the correctess of the aalytical calculatio of the aggregate capital eeds for the case of IRR. Overview Through a Example The purpose of the example presetatio is iterpretig the mai relatioships through umbers. The example: There are two project variats ad oe of them ca be chose. Project A requires a iitial ivestmet of 200 uits; Project B requires 500 uits. Both projects have a lifetime of three years, ad both have a IRR of 18%. Risks are already hadled through appropriate correctio of reveue rows. The required rate of retur is 10%. Accordig to the IRR of 18%, the profitability of both projects is good. The reveue ad expediture rows as ecessary startig data for the calculatio are show i Table 1. The detailed calculatio materials are i Table 2. Project B s iitial ivestmet is twice as large as the Project A s oe. The lifetime ad the IRR are the same. However, despite the smaller iitial ivestmet, the total omial profit sum for Project A is higher (see Table 1, last row). Behid these the differet retur rapidity is. The capital tied-up of Project A is larger because of its slower retur. It is very importat iformatio, that i Project A, the higher capital tied-up operates with 18% profitability, that is, more capital works with the same high profitability as Project B. AI
94 AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS ethodological relatioship: multiplicatio of aggregate capital eeds ad the IRR really results i the whole amout of omial profit: 5,873 0.18 = 1,057 ad 5,472 0.18 = 985 The data i Table 2 also show how much computatio is eeded for a aalytical determiatio of the aggregate capital eeds. Accordig to the above, this calculatio ca be cosiderably simplified. The last row i Table 3 shows the simplified calculatio. The profit sum divided by IRR quatifies the aggregate capital eeds ideed. The two amouts are the same as the result of the aalytical calculatio i the last row of Table 2. Table 1 The Reveue ad Expediture Rows ad the Nomial Profit Sums of the Two Project Variats * Project A Project B Year Reveues Reveues Expeditures Sales reveues Expeditures Sales reveues expeditures expeditures 0 2,000 0-2,000 5,000 0-5,000 1 8,400 8,800 400 3,400 8,900 5,500 2 8,400 8,800 400 7,500 7,900 400 3 4,000 6,257 2,257 7,800 7,885 85 Nomial sums 22,800 23,857 Profit = 1,057 23,700 24,685 Profit = 985 Note. * Dimesio of amouts is uit. Table 2 Aalytical Calculatio of the Aggregate Capital Needs of the Two Project Variats * Year Capital tied-up Project A Project B The yield ad its structure The yield ad its structure (r = 0.18) Capital (r = 0.18) tied-up Yield Profit Recovered capital Yield Profit Recovered capital 1 2,000 400 360 40 5,000 5,500 900 4,600 2 1,960 400 353 47 400 400 72 328 3 1,913 2,257 344 1,913 72 85 13 72 E AI = 5,873 3,057 1,057 2,000 E AI = 5,472 5,985 985 5,000 Note. * Dimesio of amouts is uit. Table 3 Quick Estimatio of Aggregate Capital Needs for the Two Project Variats * Appellatio Project A Project B The total sum of the omial value of profits 1,057 985 IRR 0.18 0.18 1,057 Aggregate capital eeds 985 0.18 0.18 Note. * Dimesio of amouts is uit. Of course, the further profitability opportuities ca also effect o the decisio (There are 3,000 uits of differece betwee the two iitial ivestmets. At the ed of the first year 5,500 uits exits from Project B.) Coclusios The iitial ivestmet is a iadequate feature of the project s fiacial resource eeds. The ratios of aggregate capital eeds may differ sigificatly from the ratios of the iitial ivestmets. Computer programs
AGGREGATE CAPITAL TIED-UP BY INVESTENT PROJECTS 95 ca be edited for calculatio of aggregate capital eeds, but for the case of IRR, it ca be easily calculated without it also. The simplified calculatio ca be easily uderstood by practitioers as well. The aggregate capital eeds are oe of the key features of the project ad also are importat iformatio i the rakig process. Refereces Albrecht, W., Stice, J., Stice, E., & Swai,. (2008). Accoutig: Cocepts ad applicatios (10th ed.). aso: Thomso South-Wester. Baker, H. K., & Powell, G. E. (2005). Uderstadig fiacial maagemet: A practical guide. USA, UK, Australia: Joh Wiley & Sos. Bouldig, K. E. (1935). The theory of a sigle ivestmet. The Quarterly Joural of Ecoomics, 49(3), 475-494. Brealey, R. A., & yers, S. C. (1988). Priciples of corporate fiace (3rd ed.). New York, St. Louis, Sa Fracisco: cgraw-hill Publishig Compay. Crea,. J. (2005). Revealig the true meaig of the IRR via profilig the IRR ad defiig the ERR. Joural of Real Estate Portfolio aagemet, 11(3), 323-330. Damodara, A. (2010). Applied corporate fiace. Hoboke: Joh Wiley & Sos Ic. Garriso, R. H., Noree, E. W., & Brewer, P. C. (2009). aagerial accoutig (11th ed.). New Delhi: Tata cgraw-hill Editio. Haze, G. B. (2003). A ew perspective o multiple iteral rates of retur. The Egieerig Ecoomist, 48(1), 31-51. Illés,. (2014) Fisher s rate ad aggregate capital eeds i ivestmet decisios. Theory ethodology Practice: Club of Ecoomics i iskolc, 10(1), 21-32. Illés,. (2016a). The real reivestmet rate assumptio as a hidde pitfall. Theory ethodology Practice: Club of Ecoomics i iskolc, 12(1), 47-60. Illés,. (2016b). Practical usability problems i busiess ecoomics. Advaces i Ecoomics ad Busiess, 4(11), 617-633. Kight, F. H. (1935). The theory of ivestmet oce more: r. Bouldig ad the Austrias. The Quarterly Joural of Ecoomics, 50(1), 36-67. agi, C. A. (2010). Average iteral rate of retur ad ivestmet decisios: A ew perspective. The Egieerig Ecoomist, 55(2), 150-180. Solomo, E. (1956). Source the arithmetic of capital-budgetig decisios. The Joural of Busiess, 29(2), 124-129. Watso, D., & Head, A. (2009). Corporate fiace: Priciples ad practice. Harlow: Pearso Educatio Limited.