CORPORATE VALUATION MODEL IN A STOCHASTIC FRAMEWORK

Transcription

1 CORPORATE VALUATION MODEL IN A STOCHASTIC FRAMEWORK Barbara Dömöör, Péer Juhász, PhD, CFA Deparmen of Finance Corvinus Universiy of Budapes 1093, Budapes, Hungary barbara.domoor@uni-corvinus.hu KEYWORDS Corporae valuaion, Business modeling, Financial simulaion ABSTRACT 1 In his paper we presen a corporae valuaion model, which inegraes he differen approaches of various fields of finance in a single sochasic framework. We consruc a sylized company, which has is explici producion, invenory, expor and impor aciviy. In our simulaion we analyze he affec of he fuure volailiy of hree sochasic facors (producion, inflaion and exchange rae) and heir correlaions on he disribuion of he curren value of he firm. Furhermore we invesigae he added value of he possibiliy of early erminaion of he projec. MOTIVATION In corporae finance he immanen uncerainy of financial processes is reaed generally by aking he expeced value of a random variable he elemens of he fuure cash-flow and he risk of he realizaion appears in he increased discoun facor. 2. As a consequence of his soluion, on one hand he cash-flow of a firm is prediced on he base of he expeced value of is projecs and is financing, on he oher hand he sock price movemen is described wih a disribuion, preending as if he risk of he equiy was generaed by he capial markes (he rading) only, and no even parly by he corporae operaion iself. Moreover he classical presen value calculaion neglecs he sochasics of he applied financial processes like inflaion, money marke reurns and currency raes. In he following analysis we presen a corporae valuaion model, which uses he disribuion insead of he expeced value of he sochasic processes. In our calculaions he values of he sochasic facors are generaed as a realizaion of correlaed Io processes. So he inpu parameers of he valuaion model come from 1 The paper was suppored by he European Union and cofinanced by he European Social Fund in he framework of TÁMOP-4.2.2/B-10/ projec. 2 However he Gaussian disribuion emerges when modeling he sock reurns, and consequenly he lognormaliy of sock prices is acceped, he corporae finance deals wih consan figures. he random walk model used in derivaive asse pricing in he financial markes. Neverheless his Mone Carlo simulaion is no an alernaive of he sensiiviy, or scenario analysis. In case of sensiiviy analysis our aim is o idenify he facors he changes of which have a criical effec on he resuls (being eiher a disribuion or an expeced value), and which because of his need o be forecased very carefully. The scenario analysis however invesigaes some special join oucomes of he random variables. This kind of analysis can be carried ou in our framework as well. The srucure of he paper is as follows. The nex session inroduces he heoreical background and hen we presen he assumpions of our model. The second par of he paper conains he resuls of he simulaions: firs we analyze he effec of he volailiy of one sochasic facor on he disribuion of he curren value of he firm, hen we model he corporae value aking all 3 facors volaile and finally he real opion of erminaing he projec early is valued. THEORETICAL BACKGROUND Financial modeling uses several models o describe he ime depending sochasic variables. The classic model assumes ha he change in a sochasic parameer (marke price) is independen of is pas price movemens (Markov propery, due o marke efficiency). The change in he price during a ime of d adds up from a deerminisic and a sochasic par, where he random par is normally disribued. If his change is coninuous boh in ime and value, he process is called Io-process: ds = x( S, ) d + σ ( S, ) (1) dw Where ds is he change of process S during a period of d, he lengh of which approaches o zero and x and σ are variables depending on S and he ime (); dw sands for he change of a Wiener process 3 during he same period, ha is a normally disribued random variable wih a mean of zero and variance of. In case of marke raded asses, like foreign exchange raes, normaliy is a consequence of he marke 3 See deails in Hull (2009) and Medvegyev, Száz (2010). Proceedings 26h European Conference on Modelling and Simulaion ECMS Klaus G. Troizsch, Michael Möhring, Ulf Lozmann (Ediors) ISBN: / ISBN: (CD)

2 efficiency, namely ha only a per definiion random new informaion can affec he acual price. We agreed o use he same process (wih a sligh modificaion) for he process of inflaion and producion. In he simulaion he price can change only in discree ime, ha s why we used he discree version of equaion (1): S = x( S, ) + σ ( S, ) (2) w This one-leer difference (he change in ime is considered no in limi) has severe mahemaical consequences, which exceeds he conen of our paper. The mos imporan consequence of he dicreizaion of he coninuous model is he fac, ha he risk canno be perfecly eliminaed excep for he binomial approach. In all oher discree cases (rinomial model, ec) he marke is incomplee (Medvegyev, Száz 2007). The random figures used for he simulaion are calculaed by weighing independen normal variables, where he weighs are generaed from he Cholesky facorizaion of he correlaion marix. The mehod is deailed among ohers in Bau and Trefehen (1997). THE MODEL The model we used for our calculaions is no very much differen o he sandard presen value calculus. In order o be able o analyze more complex problems in he fuure we buil an MS Excel model ha is more deailed and is parameers can be se more flexibly. So beside balance shees and income saemens i also conains a business cash flow and a discouned cash flow based business valuaion model. The explici period is 20 years long, in he erminal value he invesed capial can no generae a reurn in excess of he required level, so he value of he firm equals he invesed capial. The firm produces only one ype of produc ha is sold boh locally and for expor. The invesed asses and working capial needed for he producion are invesed a ime zero. The useful lifeime of he invesed asses is 10 years, he firm uses a linear depreciaion and amorizaion model. In year 10 we need o replace he invesed asses and he model exends unil his replacemen is compleely used up again. In his paper as some financial heories do - we assume, ha financing is always available for good projecs, so our model-company is financed exclusively from equiy, we disregard he quesion of capial srucure. Boh selling and purchasing prices and he cos of replacemen for invesed asses are affeced by he inflaion rae of he corresponding currency. Ye here is no difference made beween consumer and indusrial price index. Among he expenses of he firm we have boh fix (independen of he quaniy produced) and variable iems. The urnover days of invenory, receivables and payables can be se year-by-year individually, hough in he models presened here hose are all se o 30 days meaning hose depend only upon he sales. Our model calculaes he ax due always based on he given years earnings so no deferred ax exiss. We also ook care of money no needed for he operaion. If he exra money can no be redrawn as dividend because he maximum of ha is he annual afer ax earning, owners decrease capial (buy back shares). Based on he cash flow of he 20 years prediced we deermine he value of he firm (projec) for he sar of each year. For ha we apply he APV mehod where he sum of he free cash flow (FCFF) and he ax shield (TS) generaed by he annual ineres paymens on deb is o be discouned by he operaive cos of capial (r A ) Wih a formula: FCFF + TS ICTV V 0 = (3) (1 + r ) 20 i i + i= 1 i 20 (1 + r ) j= 1 Aj j= 1 where V sands for he value of he firm. As erminal value we used he invesed capial value for he end of he 20 h year in line wih our assumpion ha afer ha year achieved and required reurn will equal. (For a deailed descripion of his model see Koller, Goedhar and Wessels (2010).) In our model hree sochasic processes appear. Probabiliy variable describes for each of he 20 years he demand (and so he produced quaniy) of he good (Q), he local inflaion (Π) and he foreign exchange rae (S). The variables in conras o he classic Mone Carlo simulaion are no independen from each oher bu raher he correlaion beween hem can be se by parameers. The expeced amoun of annual quaniy sold is given. The source of uncerainy here is a random variable following a normal disribuion wih an expeced value of zero and a se sandard deviaion. This variable gives he difference in percenages beween he acual and he expeced annual quaniy. The risk originaing from his is somewha limied: because of long erm conracs he realized value may no be lower han b percen (60%) of he expeced and due o he maximum capaciy of he machines i may no exceed a percen (140%). (The expeced value is 1 million pieces.) Q = max(min( 1+ σ w ; a); b) (4) For he focus of our invesigaion we picked he uncerainy emerging associaed wih he local inflaion whils he change in he foreign price level is deal wih as an exogenous, well predicable variable. The changes of he local ineres raes follow exacly he paern of he local inflaion (in oher words he local real ineres rae is consan) while he exchange rae (saring form 1 euro = 300 HUF) is linked o he inflaion differences only as a rend wih some volailiy around ha. We assume he foreign inflaion o be consan and modes (2 percenage) compared o he local level. q q Aj

3 Π = σ w (5) I So while in case of he inflaion and he producion he expeced value has an unchanged mean hough he full ime horizon of he model in case of he exchange rae we have a non-consan mean. This is because he mean of he laer for a given year is he heoreical rae calculaed from he previous year s rae based on he inflaion differences. S = I domesic foreign ( Π Π ) S + σ S S w (6) 0 0 S Effec of he volailiy of producion The increasing volailiy of he producion corresponding o he relevan heory increases he exen of he disribuion, and so he probabiliy of negaive oucomes, bu i reduces he expeced value as well. Figure 1 depics he probabiliy disribuions of he ne corporae value in ime 0, assuming differen volailiy of he annually produced quaniy. Table 1 summarizes he saisics of he disribuions. Figure 1: Disribuion of corporae value as a funcion of producion volailiy () The correlaion marix of he random pars of he sochasic processes ( w i ) is predeermined and consan in ime. Firs our model calculaes five housand realizaions for each combinaions of saring parameers and hen deermines he base saisics for he disribuion. We recorded no only he ne presen value of he projec (ha is he difference beween he inrinsic value and he invesed capial) bu also he value of he firm and he presen value of ax paid o he sae and he probabiliy of bankrupcy. The saring poin of our calculus is very similar o a formal paper of Száz [2007] on he Ho-Lee model. Tha model examined he long erm equilibrium raios using a sochasic ineres rae. In our curren model he sochasic inflaion deermines he change in ineres raes and i explicily shows a he changing risk involved in he exchange raes. ANALYSIS OF THE DISTRIBUTION OF CORPORATE VALUE Firs, we calculaed he corporae value by swiching all risks off by seing all sandard deviaions o zero. In his case here is no uncerainy, each and every parameer can be foreseen. In our heoreical example he value of he company is 1.497,62 million HUF, and i requires million HUF invesed capial. Therefore enering ino ha business has 137,62 million HUF ne presen value (Marke Value Added), meaning he projec is worh o finance. We applied he same real operaive (unlevered) cos of capial (r A =12%) in every scenario, which is a consrain of our analysis, as he cos of capial may vary according o he fac wheher he sales revenue derives from expor or domesic marke, or wheher i is ensured by fixed quaniy sold or i is random. On he oher hand he cos of capial is a funcion of numerous oher facors (like macroeconomic environmen, indusry or labor force needs), so he sysemaic risk is considered idenical in all cases. We assume ha he volailiy of he analyzed facors sem from individual and diversifiable risk, and consequenly do no enhance he risk premium of he projec. 16,0% 14,0% 12,0% 10,0% 8,0% 6,0% 4,0% 2,0% 0,0% Table 1: Saisics of he disribuion vol=5% vol=10% vol=15% Volailiy 5% 10% 15% Mean Sandard deviaion S. Dev/Mean 117% 203% 255% Maximum Minimum Median Quarile Quarile Quarile Probabiliy of defaul 20% 32% 37% A sligh volailiy (5%) of he producion resuls a negaive ne presen value in 20% of he cases, meaning ha he generaed cash flow is no enough o cover he required cos of capial. In he wors case (1 realizaion ou of 5000) he discouned value of he cash flow during he 20 years is only 68% of he invesed capial. The increasing volailiy causes rivially higher upside and downside deviaions. As he produced quaniy can flucuae in a +/-40% range around he mean, in he heoreical wors case he producion is he fixed minimum (600 housand pieces annually). In ha case he firm value is -667 million HUF. The erm probabiliy of defaul refers no o a real defaul in our analysis, bu he probabiliy ha he reurn fails o exceed he cos of capial, as we assume

4 ha financing is always available, he projec is o be coninued in case of negaive cash flow as well. Exchange rae volailiy The oher source of he dispersion of he corporae profi and so he corporae value derives from paymens in foreign currency. In he example shown in his paper only he sales revenue (expor) depends on foreign exchange rae, bu of course any oher iems of he cash flow could conain similar risk. The sensiiviy oward currency risk is deermined by he ne posiion in foreign currencies ha can be reduced or even perfecly eliminaed by naural hedge (maching foreign currency posiion of incoming and ougoing iems: e.g. expor revenue and foreign exchange loan) or financial derivaives. According o our assumpions he expeced change of he foreign exchange rae equals he difference of he inflaion he uncovered ineres rae pariy is held, and he sales price follows he change of he price level, herefore if he exchange rae volailiy is 0 percen, he denominaion of he sales revenue has no affec on he resuls. Table 2 shows he saisics of he disribuion of he corporae value by 50% and 100% expor raio (denominaed and seled in euro) and differen EUR/HUF exchange rae volailiy. The expeced value is he same almos in all cases, excep for he scenario of full expor and 15% currency rae volailiy. Here we can see a drasic drop (by 33%) of he mean of he corporae value, which is o be explained by he fac ha in conras o he process of he producion, we do no apply any consrain on he exchange rae flucuaion, so he appearance of exreme values has impressive effec. An exchange rae volailiy of 15% wihou any currency hedge causes for an exporer company he risk of failing he projec in half of he cases, and almos 150% of he invesed capial can be los (here we neglec he possibiliy of geing ou of he projec, so he owners mee heir paymen obligaion even if he ne presen value of he projec a he end of he year 10 is negaive). I is remarkable, ha he drif of he EUR/HUF exchange rae during he firs decade of he 2000s, was no only less han he ineres rae difference, bu i was in mos of he years negaive, resuling a forin-appreciaion. Table 2: Corporae value in case of FX-risk Expor raio 50% 100% Volailiy Volailiy 5% 10% 15% 5% 10% 15% Mean Sandard deviaion S. Dev/Mean 92% 185% 289% 188% 396% 916% Maximum Minimum Median Quarile Quarile Quarile Probabiliy of defaul 14% 31% 40% 30% 43% 50% Sochasic inflaion In our model exclusively he domesic inflaion is sochasic, we handle he foreign inflaion o be consan. The price level affecs boh he revenue and he cos side, so he change of he inflaion has a balanced influence on he profiabiliy of he company. The inflaion akes effec on he corporae value principally hrough he financing coss, as i is included in he nominal cos of capial. A a moderae inflaion level (3%), volailiy of inflaion can be negleced, as even an exreme (50%) volailiy causes jus a modes range of corporae value flucuaion (able 3). Volailiy of inflaion can cause negaive ne presen value if he inflaion exceeds 10%, wih a larger volailiy of 50%. In his exreme case he probabiliy of defaul is 5%, bu i is due o he higher cos of capial, and accordingly low expeced value, he affec of he volailiy is less significan.

5 Table 3: Affec of he price level and volailiy of inflaion on corporae value Inflaion 3% 10% Volailiy Volailiy 5% 10% 15% 50% 5% 10% 15% 50% Mean Sandard deviaion S. Dev/Mean 0% 1% 2% 5% 8% 15% 22% 64% Maximum Minimum Median Quarile Quarile Quarile Probabiliy of defaul 0% 0% 0% 0% 0% 0% 0% 5% Three sochasic facors Each of he simulaions above presen he effec of one sochasic facor on corporae value. Supposing all he hree parameers o be sochasic, we have o model he join disribuion of he hree risk facors. The facor volailiies used in he model are: 5% (producion); 10% (inflaion) 4 ; 10% (exchange rae) 5. We applied hree correlaion srucure: independen facors (correlaion coefficien: 0); perfec posiive correlaion meaning an increase in he producion is followed by increasing inflaion and exchange rae; and a real correlaion srucure, which we inended o consruc o be close o he real marke circumsances. Consequenly he correlaion beween inflaion and exchange rae is srong (δ=0.8), as weakening of he forin usually causes higher inflaion. We assume a weak, almos zero correlaion beween he producion and he inflaion (δ=0.1), because inflaion has a wosided effec on he produced quaniy. The correlaion of exchange rae and producion is assumed o be posiive, bu no oo srong (δ=0.4). Table 4 conains our resuls. The higher correlaion (economic predicabiliy) enhances he corporae value. The same resul was achieved by Csányi, Juhász and Megyik (1997) in heir simulaion of a corporae populaion. On he oher hand he deerminisic relaionship excludes he possibiliy ha he unfavorable marke movemens of he facors are balanced, he risks are less diversified, and so he confidence of he esimaion. 4 The sandard deviaion of he annual consumer price indices beween was 2,2%, according o he daa of he Hungarian Cenral Saisical Office. 5 The annual EUR/HUF exchange rae volailiy is 9,2% since 2000; he volailiy of he las 5 years is 12,2%, based on he daily ECB fixings. Table 4: Affec of correlaions of he sochasic facors Expor raio 50% correlaion real 0 1 Mean Sandard deviaion S. Dev/Mean 244% 229% 256% Maximum Minimum Median Quarile Quarile Quarile Probabiliy of defaul 37% 36% 38% According o ha he inervenion of he sae ino he economy (for example he home currency is under pressure because of an exreme foreign deb) is value desroying. However he bias caused by he sae can lead o less correlaed affecs of he economic facors. As in such cases he link beween differen risk facors may be shifed some negaive endencies in he real economy may be counerbalanced by he weakening of he correlaions. Bu as soon he disorions disappear (he increase of foreign deb slows) and he basic connecions come again o dominae and he effec of he real economic endencies (global and euro crisis) may influence he economy undimmed. The value of he projecs would change in line wih he basic endencies end heir risk will surely grow.

6 TERMINATION OF THE PRODUCTION I is an imporan assumpion of he above analysis, ha he company will operae for 20 years afer saring he projec (enering ino a conrac abou he annual delivery) and here is no chance o cease producion even if he discouned cash-flow of he furher years is negaive. The possibiliy of erminaion he projec earlier can be considered as a real opion. In case of arising financing need (negaive cash flow), i is always worh o examine, wheher he discouned cash flow of he furher years can cover he new invesmens and oherwise abandon i. We build his opion in our model allowing he projec owners o deny he addiional capial invesmen a he end of he 10 h year. In he absence of risky facors (all processes are deerminisic), his opion has of course no value. Using he volailiy figures and he real correlaion marix of he previous secion, he opion of early erminaion enhances he iniial corporae value by 11 million HUF, 7% of he oal projec value (able 5.). The real opion has no influence on he probabiliy of defaul, bu i moderaes he exen of he losses in he negaive oucomes. Table 5: Disribuion of he corporae value, wih and wihou he possibiliy of early erminaion Expor raio 50% no available Real opion available Mean Sandard deviaion S. Dev/Mean 244% 220% Maximum Minimum Median Quarile Quarile Quarile Probabiliy of defaul 37% 37% CONCLUSION In his paper we presened he consequences of he uncerainy deriving from sochasic producion and inflaion in addiion o he marke risk of currency raes on he corporae value. We consruced a simplified corporae model, which includes producion, invesmen, invenory, expor and impor. Our calculaions confirm he more significan role of he flucuaion of foreign exchange raes on he corporae value volailiy of an exporer company, han he volailiy of producion iself. The reason for his is he fac, ha he lower income of a decreasing producion is counerbalanced parly by he parallel reducion of he coss. The volailiy of inflaion proved o be less deermining, as we provided, ha inflaion affecs he similarly boh he incoming and ougoing iems of he profi and loss. Furhermore early erminaion of he projec as a real opion enhances he corporae value considerably. The exen of hese effecs can be criical in corporae operaion. We offered a framework ha expands he usual corporae valuaion models wih he mehods used in risk managemen and derivaives pricing. Mone Carlo simulaion is he mos adequae ool for ha. We plan o develop furher our research by releasing he assumpion of perfec marke liquidiy (financing sources being always available), in order o invesigae he effec of he capial srucure, and he differen ypes of loan. REFERENCES Bau III, D., Trefehen, L. N. (1997). Numerical Linear Algebra. Sociey for Indusrial and Applied Mahemaics, Philadelphia Brealey, R. A., Myers, S. C., Allen, F. (2008): Principles of Corporae Finance, ninh ediion, McGraw-Hill, New York Csányi, T., Juhász, P., Megyik, L. (1997): From Shorage Economics o he Shorage of Marke Economy, (in Hungarian: A hiánygazdaságól a gazdaság hiányáig), Éle és Irodalom, 28 November pp Damodaran, A. (2002): Invesmen Valuaion. Tools and Techniques for Deermining he Value of Any Asses. second ediion, Jonh Wiley & Sons, Inc., New York ECB Saisics: hp://sdw.ecb.europa.eu/browsetable.do?node= &cu RRENCY=HUF&FREQ=D&sfl1=4&sfl3=4&DATASET =0&SERIES_KEY=120.EXR.D.HUF.EUR.SP00.A downloaded: Hull, J. C. (2009): Opions, Fuures and oher Derivaives, sevenh ediion, Pearson/Prenice Hall Koller, T., Goedhar, M., Wessels, D. (2010): Valuaion, Measuring and Managing he Value of Companies, fifh ediion, John Wiley & Sons, New York KSH Saisics: hp://sainfo.ksh.hu/sainfo/haviewer.jsp; downloaded: Medvegyev, P., Száz, J. (2010): The Naure of Coninuous Surprices in he Financial Markes. (in Hungarian: A meglepeések jellege a pénzügyi piacokon), Je Se, Budapes Ross, S., A., Weserfiled, R., W., Jordan, B., D. (2008): Corporae Finance Fundamenals, eigh ediion, McGraw- Hill, New York Száz J. (2007): The Increase of he Corporae Capial and he Ho-Lee Model. (in Hungarian: A vállalai őkeállomány bővülése és a Ho-Lee model), In: Pénzügy-poliikai sraégiák a XXI. Század elején, Budapes, Akadémiai Kiadó

7 AUTHOR BIOGRAPHIES BARBARA DÖMÖTÖR is an Assisan Professor of he Deparmen of Finance a Corvinus Universiy of Budapes. Before saring her PhD sudies in 2008, she worked for several mulinaional banks reasury in field of srucuring currency and ineres rae risk hedging producs for corporae cliens. She works now on her docoral hesis abou corporae hedging. She lecures Corporae Finance, Financial Risk Managemen and Invesmen Analysis, her main research areas are financial markes, financial risk managemen and corporae hedging. Her address is: barbara.domoor@uni-corvinus.hu PÉTER JUHÁSZ, PhD, CFA is an Associae Professor of he Deparmen of Finance a Corvinus Universiy of Budapes. Besides eaching he also works as a rainer and managemen consulan mainly for SMEs and governmenal eniies. His field of research covers business valuaion and planning, valuaion of off-balance shee iems, corporae risk managemen and solving financial problems using VBA programming of MS Excel. His address is: peer.juhasz@unicorvinus.hu