Revisiting Uncertain Investment and Financing Decisions using Entry Probability

Transcription

1 Journal of Applied Finance & Baning, vol. 3, no. 3, 03, ISSN: (prin version), (online) Scienpress Ld, 03 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy Po-Yuan Chen Absrac This paper exends he wor of Chen and Chang (00) and aemps o presen a model for he opimal invesmen hreshold and he real opion value under price uncerainy from a differen aspec of enry probabiliy. I measure a financing policy by he deb raio, a weigh for he proporion of funding by deb relaive o funding by equiy. The weigh is exogenously embedded in he sochasic opimizaion of an invesmen opporuniy under price uncerainy. An enry probabiliy, indicaing he lielihood of an invesmen acion opimally aen a a fuure ime insan, is derived. Sensiiviy analyses are performed o invesigae he effecs of parameers such as price volailiy and price drif. The resuls demonsrae ha price drif and price volailiy migh have differen effecs, depending on he ime elapsed and on he deb raio. In general, higher deb raio conribues o higher invesmen hreshold and higher real opion value, confirming ha financial leverage migh serve as a value driver. However, as he deb raio increases, he invesmen hreshold migh become oo high o reach, indicaing ha waiing is he bes choice. Such a circumsance is also verified by he enry probabiliy approaching zero, which implies he far-reaching invesmen hreshold. JEL classificaion numbers: C6, G, G3 Keywords: Price uncerainy, Financing policy, Invesmen hreshold, Real opion, Enry probabiliy Dr. Po-yuan Chen serves as associae professor a Deparmen of Financial and Tax Planning, Jinwen Universiy of Science and Technology, Taiwan, R.O.C. miecpy@ms7.hine.ne Aricle Info: Received : February 9, 03. Revised : March 6, 03. Published online : May, 03

2 80 Po-Yuan Chen Inroducion As he produc life cycle in a highly compeiive mare of consumer elecronics becomes shorer and shorer, he launch of a new produc as an invesmen opporuniy faces more and more riss han before. The overall ris in an invesmen opporuniy can be measured by boh operaional and financial leverage. The operaional leverage relaes o he mare uncerainy such as price volailiy in a produc mare, whereas he financial leverage relaes o financing uncerainy such as variable ineres paymens in a capial mare. An invesmen, financed parly from he soc mare and parly from he deb mare, is he one wih financial leverage. Even hough he ineres paymens in funding by deb increase he expendiures, hey also decrease he ax payables. Such a radeoff is referred o as he effec of ax shield. In oher words, he ineres paymens have wo opposing effecs on projec value: increasing expendiures and saving he cash ouflows of axes. Besides, ax exempion or ax incenive may be provided by governmens in many counries o boos he developmen of an indusry, such as he semiconducor indusry in Taiwan. Modigliani and Miller [] advocaed he irrelevance of invesmen and financing decisions in he absence of ax and banrupcy cos. However, he furher developmen of heir heory releases he ax-free and banrupcy-cos-free assumpions and hen swiches from independence o inerdependence beween invesmen and financing decisions (Brennan and Schwarz [], Mellon and Parsons [3], Mauer and Trianis [4], Mauer and O [5], Hennessy and Whied [6], and Timan and Tsyplaov [7]). The rac advocaing relevance beween invesmen and financing decisions becomes more and more popular. For example, Srebulaev [8] furher disinguished funding policy beween exernal deb and inernal equiy sources. Among hese scholars who focus on inerdependence beween invesmen and financial decisions, Timan and Tsyplaov lays emphasis on firms invesmen flexibiliy, allowing a firm o underae any invesmen projec a he opimal enry ime. The sudy of Azofra and Miguel [9] empirically verified a direc relaionship beween invesmen and deb. The empirical sudy of Su and Vo [0] found ha combined effec of corporae sraegy and capial srucure explains firms performance very well. In conras, I inend o propose an opimizaion model o deal wih he relaionship beween invesmen iming and funding policy from a differen aspec of enry probabiliy. Regarding he applicaions of enry probabiliy, Kamien and Schwarz [] used enry probabiliy o invesigae he effec of pricing policy on he enry probabiliy of compeing suppliers. Besides, Michell [] examined he enry probabiliy and enry iming for indusrial incumbens preparing o ener he corresponding echnical subfields. In conras, I employ he enry probabiliy o revisi he relevance beween he invesmen iming and he financing policy of capial srucure. Unlie mos applicaions of opimizaion inending o solve for radeoffs beween wo opposing forces, his wor adops sochasic opimizaion in ime domain. For example, Chen [3] considers wo opposing forces: qualiy loss and invenory cos in he opimizaion of he expeced oal cos o obain he opimal producion run lengh and process mean. Conrarily, I consider a price process wih sochasic naure o evolve. The sochasic price comprises sochasic profi flow, which in urn comprises he value process of an invesmen. As ime passes, he value process migh evolve upwards or downwards in he fuure. There mus exis a cerain ime insan in he fuure when maximum expeced presen value occurs. To obain he opimal value in he sochasic conex, sochasic opimizaion is herefore employed by his wor.

3 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 8 In his wor, I follow he same derivaion procedures as Chen and Chang [4] proposed o invesigae he impac of financing policy on invesmen sraegy under uncerainy. The invesmen projec value is derived by discouning all he fuure profis, which evolve as a sochasic process due o he assumpion of randomness in price. Based on he hisorical price quoes for he LED componens from Sepember 999 o June 009 acquired from he daabase of Taiwan Economic Journal (TEJ), an example is illusraed in figure o demonsrae he sochasic price process. The figure also demonsraes ha he quaniy demanded is negaively correlaed wih he corresponding price. Such phenomenon is modeled by he inverse demand funcion in his model. Figure : The monhly price process denominaed in he US dollar for LED componens. This paper is organized as follows: Secion inroduces he previous researches on capial srucure in corporae finance. Secion presens he formulaion of projec value in he conex of sochasic price under an inverse demand funcion. The analyical soluions for invesmen hreshold, real opion value and corresponding enry probabiliy are hen derived. In Secion 3, I presen he analyical resuls in figures o analyze he sensiiviy. Secion 4 concludes his paper. The Framewor. Noaion and Modeling In his model, following he framewor of Chen and Chang [4], I calculae he expeced presen value of fuure cash flows generaed from an invesmen projec facing he price uncerainy. Because of he sochasic effecs in he price, he uncerainy will be propagaed ino he fuure cash flows, evolving sochasically on he coninuous ime basis, and hen ino he discouned value, whose expecaion represens he projec value in our model. I is assumed ha he price is a sochasic process and denoed by p, which evolves as a geomeric Brownian moion as shown in equaion (). dp p d p dz, ()

4 8 Po-Yuan Chen where is a consan price drif, is a consan price volailiy, and Z is a sandard Brownian moion wih a filraion p ( ), 0 ; hus dz is he normally disribued incremen wih he expeced value 0 and he variance d. I assume ha he mare demand is price sensiive and can be defined by an inverse demand funcion of Carruh e al. [5] as follows. Q P. () The firm s marginal cos is assumed o be proporional o is revenue as he assumpion of Schwarz and Moon [6], who divided he cos srucure ino wo componens: a fixed erm and a variable erm which is proporional o he revenue. I also assume ha he firm could conrol is cos rae a a fixed level relaive o he revenues hrough is managerial aciviies such as is bargaining power o negoiae wih he supplier. I denoe he marginal cos incurred a ime as v, which is defined as follows. v P, (3) where is a consan cos rae. I is assumed ha he mare share of he firm is consan and denoed by m. I denoe he revenue a ime as R and he oal cos asc, which can be expressed as follows. R PmQ, (4) and C vmq. (5) I assume ha oal capial demand for he firm is proporional o is revenues, and denoed by K. Then I have K br, (6) whereb 0. I assume ha he capial demand of he projec is financed hrough issuing deb and equiy wih weighs w and w, respecively. The weigh w is wha Kim e al. [7] used o measure he capial srucure, and referred o as he deb raio, which is defined as he raio of deb o oal capial demand. According o business pracice, I se0w. When w 0, he invesmen projec is all-equiy financed wihou financial leverage. In conras, when w, he invesmen projec is an all-deb one wihou any soc issued. I denoe he deb and equiy amouns as D and E. The following wo equaions describe he relaionships beween revenue and deb amoun, as well as equiy amoun. D wk, (7) and E ( w) K. (8) I denoe he funding raes from boh crediors and shareholders as r D and r E. The ax rae is denoed by. As proposed by Lin [8], he weighed average of he cos of capial (WACC) for he firm is employed as he discoun rae, which is denoed by and defined as follows: ( r ( ) r ) w r. (9) D E E

5 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 83 The ineres paymen o he crediors is Dr D. I hen denoe he profi flow afer ax as, which can be formulaed as follows: ( R C D rd )( ). (0) The funding rae from crediors has wo opposing effecs on he projec value: a negaive effec on he profi flow and a posiive effec on he discoun rae. In pracice, shareholders require a higher rae of reurn for heir invesmens because of heir assuming higher ris han crediors. I have r D r E. For he ineres paymen, higher w leads o higher deb amoun and hus higher ineres paymen, as well as lower profi flow. In conras, according o equaion (9), higher w leads o lower discoun rae and hus higher presen value of any fuure profi flow. I assume ha he firm obains susainable growh in is invesmen projec hrough coninuous innovaion. Consequenly, he fuure profi flows las indefiniely. The invesmen projec value under price uncerainy can be obained by he same procedures as Chen and Chang [0] derived: ( s) p ( wbrd )( ) m V E s e ds () P () ( )( ) where ( )( ) >0.. Derivaion of Analyical Soluion By Io s lemma, I can expand he incremen of he invesmen projec value a ime from equaion () as follows. dv ( )( d dw ) ( )( ) d. () V Rewrie equaion (), I can derive he dynamics of he invesmen projec value as follows. dv ud vdz V, where u ( )( ) and v ( ). (3) Subsequenly, he firm could deermine he opimal enry ime for he invesmen. I should pay he irreversible sun cos ( I ) for he insallaion of he operaion faciliies and hen acquire he projec value a ime s ( V s ) in reurn. The invesmen profi a ime s isv s I.Therefore, he opimizaion for he invesmen could be described as equaion (4). The objecive funcion, referred o as he real opion value, is denoed by FV as follows: s p F V max E Vs I e ( ). (4) s 0 The opimal enry ime for he invesmen is he corresponding ime when he opimizaion is achieved. To opimize equaion (4), he corresponding Bellman equaion should be employed.

6 84 Po-Yuan Chen Fd E( df p ( )). (5) The expeced capial appreciaion can be expressed as follows: p E( df ( )) uv FV v V FVV d. (6) Subsiuing equaion (6) ino (5), I can obain he differenial equaion as follows: uv FV v V FVV F 0. (7) Solving for Equaion (7), I can finally obain he opion value FV ( ). Assume ha he opion value is in he following form: ( )= ( ) d d F V c V c ( V ) (8) where c, c are wo consan coefficiens. Equaion (8) is reduced o he following equaion: F( V )= c ( V ) d. I denoe he value hreshold asv.the oher wo boundary condiions are as follows: F( V )V I, (0) and F ( V ). () V According o he wor of Dixi and Pindyc [9], I can derive coefficiens d, c and V from Equaions (9), (0) and () as follows: d I d d d hresholds c () and d V I (3) d where u u d ( ) ( ), v v v u ( ), v ( )( ), and ( rd ( ) re ) w re. Subsiuing equaion () ino (9), I can finally obain he real opion value as follows. (9)

7 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 85 d ( wbrd )( ) m F( V) = c P (4) ( )( ) From equaions (3) and (4), i can be shown ha he price drif and volailiy, he deb raio, cos and ax raes, and he price elasiciy of demand joinly deermine no only he invesmen hreshold bu also he real opion value. Consequenly, he firm should inensively observe hose facors for is opimal enry decision. In order o undersand he lielihood of enry for he invesmen opporuniy, I derive he enry probabiliy shown as equaion (5), where N ( ) is a sandard normal disribuion funcion. prob( V V ) s ( u0.5 v )( s) v s prob( V e V ) V ln ( u 0.5 v )( s ) (5) V prob( ) v s N ( ) V ln ( u 0.5 v )( s ) V where. v s In order o have a more complee undersanding of he effec of deb raio on he enry probabiliy, I derive equaion (6) (proved in Appendix) o demonsrae he sensiiviy of enry probabiliy o deb raio. prob( Vs V ) w V I e v s V m( ) P 0.5 d h v d v d ( brd ) v where u h ( ), v v (6)

8 86 Po-Yuan Chen V V v s wbrd, and r ( ) r. ln ( u 0.5 v )( s ) D E, Similarly, equaion (7) can be derived o demonsrae he sensiiviy of enry probabiliy o ime index s ( s ). prob( Vs V ) u 0.5v e s v s (7) 3 Sensiiviy Analysis In his secion, I shall discuss he effecs of parameers on he hreshold and he real opion value. Based on equaions (3) and (4), sensiiviy analyses are performed and illusraed in figures 3 hrough 4 by he same parameer se as Chen and Chang [4] adoped: 0., 0.3, m 0.8, rd 4%, re 5%, 5%, 0.4 and 0.9. In figure, as Chen and Chang [4] discussed, he invesmen hreshold slighly increases wih increasing price drif, bu he curves wih differen price drif are almos overlapped, indicaing ha price drif does no affec he enry hreshold. In figure 3, i is shown ha he higher he price volailiy, he higher he hreshold. The deb raio srenghens he increasing effec of price volailiy on hreshold, causing he invesmen hreshold o rise sharply. The sharp rise of enry hreshold may cause he enry less possible. This phenomenon could be explained laer in erms of enry probabiliy. Figure : The sensiiviy of invesmen hreshold o deb raio under varying price drif

9 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 87 Figure 3: The sensiiviy of invesmen hreshold o deb raio under varying price volailiy As Chen and Chang [4] discussed, figure 4 shows ha he higher he price drif, he higher he real opion value, while higher deb raio raises he real opion value more sharply. On he conrary, in figure 5, he higher he price volailiy, he lower he real opion value. However, he impac of deb raio is no as significan as in figure 9. Figure 4: The sensiiviy of real opion value o deb raio under varying price drif

10 88 Po-Yuan Chen Figure 5: The sensiiviy of real opion value o deb raio under varying price volailiy The invesmen aciviy is acivaed when he sochasic invesmen projec value goes up hrough he hreshold. Therefore, he higher he hreshold, he less liely he invesmen is underaen. Based on equaion (5), one example is illusraed in figure 6 o demonsrae he effecs of deb raio and ime duraion on he enry probabiliy. For example, when deb raio is 0., he opimal enry ime may be laer han year 6 wih enry probabiliy.0. To have a more complee undersanding of he effecs of deb raio and ime duraion on he enry probabiliy, equaion (6) and (7) are derived o demonsrae he sensiiviy of enry probabiliy o deb raio, and o ime duraion. I is shown in figure 7 ha he change rae of enry probabiliy wih respec o deb raio is equal o or less han zero, indicaing ha he enry probabiliy is a decreasing funcion of deb raio. Conrarily, i can be shown in figure 8 ha he change rae of enry probabiliy wih respec o ime index (ime lengh) is equal o or greaer han zero, indicaing ha he enry probabiliy is an increasing funcion of ime. Figures 7 and 8 confirm he sensiiviy in figure 6. Figures 9- illusrae he impacs of price drif and volailiy on he enry probabiliy. A ime, he enry probabiliy could be a decreasing funcion of price volailiy. However, as ime elapses owards year 0, he enry probabiliy gradually becomes an increasing funcion of price volailiy as shown in figures 0 and. In conras, he enry probabiliy more consisenly demonsraes an increasing funcion of price drif as shown in figures 9-. Figure 6: The sensiiviy of enry probabiliy o deb raio and o ime lengh (s) when =0

11 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 89 Figure 7: The sensiiviy for rae of change in enry probabiliy o deb raio Figure 8: The sensiiviy for rae of change in enry probabiliy o ime index (s) Figure 9: The sensiiviy of enry probabiliy o price drif and price volailiy a ime (s=) Figure 0: The sensiiviy of enry probabiliy o price drif and price volailiy a ime 5 (s=5)

12 90 Po-Yuan Chen Figure : The sensiiviy of enry probabiliy o price drif and price volailiy a ime 0 (s=0) 4 Conclusion This paper exends he wor of Chen and Chang [4] by inegraing he sochasic price in invesmen valuaion o invesigae he impacs of price uncerainy, ax, deb raio, demand elasiciy, and mare share on invesmen hreshold and on real opion value. In his framewor, he uncerainy of fuure profis comes from he sochasic price. To consruc he value dynamics for he invesmen opporuniy in quesion, I employ a sochasic value process driven by he sochasic price as Chen and Chang [4] proposed. The profi flows are discouned and summed up as expeced invesmen projec value. Analyical soluions for invesmen hreshold, he real opion value and he corresponding enry probabiliy are obained. Based on he analyical resuls, sensiiviy of invesmen hreshold, real opion value, and enry probabiliy are analyzed. The analyical resul demonsraes ha he higher he deb raio, he higher he invesmen hreshold is. In oher words, i is less liely o underae such an invesmen projec when he deb raio is higher. However, once he invesmen is underaen, he higher he deb raio, he higher he real opion value is. ACKNOWLEDGEMENTS: I deeply appreciae he gran wih reference no. NSC0-40-H-8-00 from Naional Science Commiee (NSC) of Taiwan, financially sponsoring he research aciviies relaed o he developmen of he framewor in his paper. References [] Modigliani, F., and Miller M. H., The Cos of Capial, Corporaion Finance, and he Theory of Invesmen, The American Economic Review, 48, (958), [] Brennan, M., and Schwarz, E., Opimal Financial Policy and Firm Valuaion, Journal of Finance, 39, (984), [3] Mello, A., and Parsons, J., Measuring he Agency Cos of Deb, Journal of Finance, 47, (99), [4] Mauer, D., and Trianis, A., Ineracions of Corporae Financing and invesmen decisions: A Dynamic Framewor, Journal of Finance, 49, (994),

13 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 9 [5] Mauer, D., and O, S.H., Agency Coss, Invesmen Policy and Opimal Capial Srucure: The Effec of Growh Opions, In Projec Flexibiliy, Agency, and Compeiion: New developmens in he Theory and Applicaion of Real Opions, Brennan M.J. and Trigeorgis L.(eds), Oxford Universiy Press, (000), [6] Hennessy,C., and Whied,T., Deb Dynamics, Journal of Finance, 60, (005), [7] Timan, S., and Tsyplaov, S., A Dynamic Model of Opimal Capial Srucure, Review of Finance,, (007), [8] Srebulaev, I., Do Tess of Capial Srucure Theory Mean Wha They Say?, Journal of Finance, 6, (007), [9] Azofra, V., and Miguel, A., La inerrelacion de las decisiones financieras en la gran empresa indusrial espanola, Invesigaciones Economicas, (990), [0] Su,G.S., and Vo,H.T., The Relaionship Beween Corporae Sraegy, Capial Srucure, and Firm Performance: An Empirical Sudy of he Lised Companies in Vienam, Inernaional Research Journal of Finance and Economics, 50, (00), 6-7. [] Kamien, M.I. and Schwarz, N.L., Limi Pricing and Uncerain Enry, Economerica, 39 (3), (97), [] Michell, W., Wheher and When: Probabiliy and Timing of Incumbens Enry ino Emerging Indusrial Subfields, Adminisraive Science Quarerly, 34 (), (989), [3] Chen, C.H., The Modified Economic Manufacuring Quaniy Model for Produc wih Qualiy Loss Funcion, Tamang Journal of Science and Engineering,, (009), 09-. [4] Chen, P.Y. and Chang, H.J., The Effecs of Financing Policy on Invesmen Sraegy under Price Uncerainy, Inernaional Research Journal of Finance and Economics, 59, (00), [5] Carruh, A., Dicerson, A. and Henley, A., Wha Do We Know Abou Invesmen Under Uncerainy?, Journal of Economic Surveys, 4, (000), [6] Schwarz, E.S., and Moon, M., Raional Pricing of Inerne Companies Revisied, The Financial Review, 36, (00), 7-6. [7] Kim, C.S., Mauer, D.C., and Sherman, A.E., The Deerminans of Corporae Liabiliy: Theory and Evidence, Journal of Financial and Quaniaive Analysis, 33, (998), [8] Lin, T.T., The Deerminan of Producion Enry and Exi Model on Financing Behavior, European Journal of Operaional Research, 96, (009), [9] Dixi, A.K., and Pindyc, R.S., Invesmen under Uncerainy, The Princeon Universiy Press, N.J., USA, (994).

14 9 Po-Yuan Chen Appendix Based on equaion (5), he enry probabiliy is as follows. prob( V V ) N ( ) s e x dx Taing he firs derivaive wih respec o deb raio w, we can derive he analyical soluion as follows. prob( Vs V ) w e w e x x dx dx w e w V ln ( u 0.5 v )( s ) V e v s w V ln V e v s w V ln V e v s w V V V e v s V w

15 Revisiing Uncerain Invesmen and Financing Decisions using Enry Probabiliy 93 d I d ( wbrd )( ) m P c ( )( ) E V e v s V w V e v s V u u (( rd ( ) re ) w r ) E ( ) ( ) v v v I u u (( rd ( ) re ) w re ) ( ) ( ) v v v ( wbrd )( ) m P ( rd ( ) re ) w re ( )( ) w V e v s V m( ) P I ( wbr )( r ( ) r ) ( )( ) ( br ) D D E D 0.5 d ( ) ( ( ) ) wbr h r r D D E d ( )( ) v d ( )( ) ( wbr D ) ( )( )

16 94 Po-Yuan Chen V e I v s V m( ) P 0.5 d ( wbrd )( rd ( ) re ) ( wbr ) D h ( rd ( ) r ) E ( brd ) d ( )( ) v d wbrd ( )( ) V e I v s V m( ) P d ( wbrd )( rd ( ) r ) E 0.5 d ( ) ( ( ) ) wbrd h rd r E ( )( ) ( br ) v d D wbrd wbrd wbr D ( )( ) ( )( ) ( )( ) V I e v s V m( ) P 0.5 d h ( r ( ) ) ( )( ) D r E d ( ) ( )( ) v d brd ( wbrd ) wbrd 0.5 d h v V d v d ( br ) I D v e v s V m( ) P u where h ( ), v v V ln ( u 0.5 v )( s ) V, v s wbr, D and r ( ) r. D E