DOUMNTO D DISUSIÓN DD/0/6 Discoun aes or Seed apial Invesmens *Samuel Mongru Monalván (Universidad del Pacíico) mongru_sa@up.edu.pe *GAD, ITSM, ampus Querearo, Mexico Marzo, 206
Absrac So ar, he esimaion o discoun raes required by enrepreneurs has remained a mysery. Mongru and amirez (2006) made a conribuion o his area by deriving he lower bound discoun rae or a non-diversiied enrepreneur in an emerging mare. However, hey used a quadraic uiliy uncion, which does no have desirable assumpions. In his research one exends he previous wor by deriving expressions o discoun raes using a Hyperbolic Absolue is Aversion (HAA) uiliy uncion ha includes he quadraic and he logarihmic orms as special cases. Furhermore, one also assumes he enrepreneur wih he lowes ris-aversion ha invess almos all his capial in his projec or irm and whose level o wealh approaches o zero. One inds ha boh expressions depend upon enrepreneur s risaversion and a measure o he projec oal ris. Mainaining consan he ris-ree rae, we simulae he expressions o discoun rae or he quadraic orm and he logarihmic orm. As expeced, he enrepreneur s required reurns (discoun raes) are highly sensiive in boh speciicaions and all values were lower han 50% and mos o hem were lower han 25%, bu higher han he assumed ris-ree rae. Keywords: Seed apial, discoun raes, enrepreneurship JL codes: L26 and M3 2
. Inroducion In a pioneering wor, McInish and Kudla (98) argued ha he so called wo-und separaion heorem, originally developed by Tobin (958), does no hold in he case o closelyheld irms and small irms. Hence, he appropriae discoun rae or valuaing hese irms would be required raes o reurn rom he owners o he business insead o a mare discoun rae. The wo-und separaion heorem saes ha he opimal porolio o risy securiies is exacly he same regardless o he invesors ris preerences. This propery is exremely imporan or asse pricing because i allows us o esimae he cos o equiy capial under equilibrium condiions, such as in he case o he apial Asse Pricing Model (APM). ecenly, Breuer and Gürler (2007) have shown ha in order o mee he wo-und separaion heorem wo resriced condiions mus be me: he uiliy uncion mus be always deined and marginal uiliy mus be posiive. Unorunaely, hese well-deined problems resric seriously he applicaion o he wo-und separaion heorem. This problem is even worse given he ac ha no all preerences can be represened wih uiliy uncions. This is he main argumen ha Meyer (2007) advocaes o avor marginal uiliy uncion insead o he uiliy uncion isel because he ormer encompasses a bigger possible se o ris preerences. Anoher avenue ha he lieraure has pursued is o impose cerain resricions o he disribuion o soc reurns in order o aain he wo-und separaion heorem. oss (978) has shown he condiions ha mus be ulilled o apply he heorem independenly o he invesor uiliy uncion. ei e.al (999) have shown ha, alhough general ellipical disribuions can guaranee he wo-und separaion propery, he ideniicaion o he rue reurn disribuion is raher diicul, no o say impossible. I is clear by now ha using disribuion properies o soc reurns, is no a good avenue o guaranee he wo-und separaion heorem. Furhermore, o ollow he speciicaion o uiliy uncions ha belong o he Hyperbolic Absolue is Aversion (HAA) amily, only guaranee his separaion propery under resriced condiions. The previous discussion is imporan i one is ineresed in obaining he mare value o an invesmen projec. I his is no he case, he argumen relaed o he disribuion o soc reurns becomes irrelevan as he capial mare is no going o be a benchmar and he argumen relaed o ris preerence (uiliy uncion and marginal uiliy) becomes more relevan because i is he only way o speciy he ris preerences and discoun raes or seed capial projecs and irms. Unorunaely, here is a cos o using subjecive discoun raes because he esimaed Ne Presen Value (NPV) urns ou o be jus one proiabiliy indicaor insead o being a normaive invesmen rule (Zuria, 2005). In oher words, wo enrepreneurs could assess dierenly he same invesmen projec being boh esimaions valid. This is no possible under mare-derived condiions such as he ones imposed by he apial Asse Pricing Model (APM). as and Sigliz (970) managed o show he condiions under which he wo-und separaion heorem can be applied under hyperbolic preerences. 3
Anoher problem relaed wih he esimaion o subjecive discoun raes is he possible bias in he esimaion o he parameers. These dieren biases have been explained by Fuenzalida e al. (2007), so one mus use a heurisic procedure in order o avoid as much as possible he poenial biases. Despie hese disadvanages, i is convenien o derive ormulas or he projec s discoun raes under cerain assumpions because hese will provide magniudes o how much we could gain or we could lose wih he projec in dieren scenarios hrough a prospecive analysis or valuaing he projec. In his prospecive analysis wha maers are he oreseen coningen sraegies or increasing he lielihood o success o he projec raher han using jus one indicaor (i.e. NPV) o accep or rejec he invesmen proposal. In ac, Mongru and amirez (2006) esimaed a ormula or projec discoun rae or invesors wih quadraic preerences in an incomplee mare. They concluded ha, or enrepreneurs wih he lower ris aversion, he discoun rae was no unique in an incomplee mare. ven more, he discoun rae is bounded rom below and we can only esimae he maximum value or he invesmen projec, bu no is minimum value. In his siuaion he discoun rae will depend upon hree parameers: he ris-ree rae, he reward-o-variabiliy raio (TV) o he projec, and he projec oal ris. Mongru (205) expanded he ris analysis process proposed by Fuenzalida e al. (2007) o esimae he projec s oal ris. This expanded process includes a previous prospecive analysis, so he projec s NPV will be esimaed or dieren scenarios and where he enrepreneurs' ris-aversion coeicien could change. In his way, one may esimae projec s NPV using he esimaed oal ris or each year o he ime-horizon wih dieren discoun raes and wihin each scenario. Alhough he discoun rae expression provided by Mongru and amirez (2006) is simple, conains unrealisic assumpions because hey used a quadraic uiliy uncion ha implies a consan absolue ris aversion (AA) and i is well-nown ha his uiliy uncion is no increasing everywhere 2. The main goal o his research is o derive discoun raes using ris-preerences according o HAA uiliy uncions. I urns ou o be ha discoun rae expressions depend upon a measure o he projec s oal ris and he enrepreneurs' ris aversion coeicien. The paper is organized as ollows, he nex secion deines he concep o a discoun rae according o he eaures o seed capial invesmens, hen in he hird secion one explains briely all he new se o uiliy uncions available and jusiies he use o he HAA amily. In he ourh secion one derives an expression or he discoun rae using he HAA amily and assuming incomplee mares. In he ih secion, wih he aid o simulaion, one explores he properies o he discoun rae expressions. In he las secion one concludes he wor. 2 The radiional assumpions relaed o uiliy uncions are: he uiliy uncion is an increasing uncion everywhere o consumpion or wealh (meaning ha more is beer), he uiliy uncion is concave and wice diereniable, where he irs derivaive is posiive (marginal uiliy) and he second is negaive (Gerber and Paumi, 999). 4
2. Deiniion o a discoun rae in he conex o seed capial invesmens In he conex o seed capial invesmens, he discoun rae is he reurn ha enrepreneur would require rom his invesmen projec. In his secion one no only deines he discoun rae, bu also, one discusses he dieren eaures ha a uiliy uncion mus have in order o be applied in he conex o seed capial invesmens. Time preerences, ime value o money and ris There are wo parallel srands o lieraure relaed o he esimaion o he discoun rae: he social choice lieraure and privae choice lieraure. Alhough, in boh cases, he discoun rae is being used o discoun uure ne beneis, wha i represens is quie dieren. In erms o he social choice lieraure wha maers is he ime preerence problem ha is he rade-o beween presen consumpion and uure consumpion. Given ha his rade-o is done on an individual basis, hen one needs o aggregae hese individual beneis ino a measure o social benei (ameron and Gerdes, 2003). Hence, much o he lieraure has been devoed o esimae individual discoun raes using exponenial or hyperbolic uiliy uncions, and surveys and simulaions o validae hem. The main conclusion so ar is ha individuals opinions abou social discoun raes vary subsanially across samples depending on he conex or maing a choice and he echniques used o elici opinions. The main problem is ha mos individuals do no now he meaning o a discoun rae and i hey now i, nohing guaranees ha hey are being able o exeriorize he magniude o heir individual discoun rae. Hence, according o he social choice lieraure, here is an urgen need o gaher only expers opinions and o ranslae he language o a discoun rae ino is consiuens ha, hopeully, are more amiliar o expers raher han nonexpers. In he case o he privae choice lieraure, one is aced wih only one choice: o inves or no o inves in a cerain invesmen projec. However, here is also he problem o social aggregaion ha needs o be solved. The discoun rae in his conex is undersood as an opporuniy cos ha considers no only he ime value o money, bu ha i also includes he ris o which he invesor is acing. In oher words, i is an opporuniy cos commensurae o he projec s ris level. Here, here are wo common inerpreaions ha lead o mare discoun raes or o subjecive discoun raes. In he ormer case, i is being assumed ha each invesor holds a well-diversiied porolio o invesmens; hence he only needs o worry abou he projec s mare ris, which is he projec s conribuion o he ris o his well-diversiied porolio. In he laer case, one assumes ha invesors do no hold a porolio o invesmen projecs and in his way hey are compleely exposed o he projec s oal ris. Naurally, hese are wo exremes assumpions wihin a range o possible degrees o diversiicaion, bu hey are quie convenien because hese assumpions le us o wor wih more racable models. The aggregaion problem in he case o mare discoun raes is solved by using he wo-und separaion heorem. Speciically, one posulaes he exisence o an invesor who hold a well-diversiied invesmen porolio (mare porolio) ha represens all individual invesors in he mare, so each one o he asses is priced according o is mare ris. Finally, 5
by using he mare discoun rae one is esimaing he value o he invesmen projec rom he mare s poin o view (he represenaive invesor). In he case o individually-based discoun raes he aggregaion problem is o a dieren naure. Here we need a consensus orecas o he expers opinion relaed o he criical variables o a single invesmen projec. However, nohing guaranees ha his consensus orecas is valid or he mare as a whole. In ac, i is only helpul o a single invesmen projec. Hence, according o hese wo exreme poins o view, he mare ris is he relevan one or well-diversiied invesors and he oal ris is he relevan one or non-diversiied invesors ha are usually he owners o seed capial projecs and irms (rison, 2005). Hence, i is o paramoun imporance he proper esimaion o boh riss. The uiliy uncion or seed capial invesmens In a seminal wor, McMahon and Sanger (995) discussed dieren acors ha aec small irms objecive uncion. They argued ha he owner-manager s uiliy uncion depends upon pecuniary reurns rom he business (P) (i.e. reurn on invesmen - OI); non-pecuniary saisacions rom he business ha are in he inancial domain (N) (i.e. a good inancial healh, operaive lexibiliy, and so on); non-pecuniary saisacions rom he business ha are ouside he inancial domain (Nn) (i.e. preerred liesyle, sel-eseem, and so on); and oal ris, which is he sum o sysemaic or mare ris and he unsysemaic or speciic ris ( ). Hence, he uiliy uncion or a small irm s owner-manager will be as ollows: U w UP,N,N, () n In his speciicaion, he owner-manager s wealh is aeced by all he acors already described. Furhermore, hese auhors wen urher explaining why he irms reurn, irm s ris, irm s liquidiy, he degree o diversiicaion o he owners-managers, he ranserabiliy o inancial and human capial, he inancial lexibiliy and he desire o conrol he irm would impac his uiliy uncion hrough heir dieren variables. Alhough his uiliy uncion seems plausible as a generalizaion or small irms, here are wo remars: here are many ypes o small irms and hence some o hem are going o be less aeced by some acors, and mos o hese acors inluence also he uiliy uncion o a corporae irm. The poin here is no he dierence beween irms' sizes, bu he dieren relaive impac o each o hese acors in he enrepreneur's uiliy uncion. For insance, he possibiliy o ranser inancial and human capial rom one generaion o he oher is imporan or cerain ypes o small irms such as amily irms. In his case, he ounder derives par o his saisacion by looing a how his heirs ae he business properly. However, or he owner-manager o a new venure i is more imporan o have inancial lexibiliy because he needs unds, which are usually no provided by venure capialiss in he early sage o he business. In oher words, he mus be able o show o poenial crediors or shareholders ha he is able o cope wih dieren scenarios using coningen sraegies. Oher acors, such he irm s reurn, liquidiy, desire o conrol and accounabiliy are also imporan or corporae irms. In ac, here is no way in which a company can survive in he long-erm i i does no have liquidiy and i i does no care or he irm s saeholders. O 6
course, here one mus recognize ha liquidiy could be a maer o lie and deah or a small irm, bu no necessarily or a corporae irm. The degree o diversiicaion o he owner-manager and he irm s ris are crucial because hey change he valuaion ools or small irms and in paricular or seed capial invesmens. A seed capial projec is usually run by a single non-diversiied enrepreneur ha aces he projec s oal ris. A mare valuaion does no apply here because here is no a represenaive invesor, only maer he enrepreneur and he expers opinions concerning a paricular invesmen proposal. Besides, given ha he enrepreneur pus all his money in he projec or irm hen he will ace he projec s oal ris. Furhermore, in his case he radiional value-addiiviy principle breas up, because he enrepreneur could derive some posiive synergies by invesing in dieren projecs 3. Anoher issue worh o discuss is wheher o include or no in he uiliy uncion he non-pecuniary saisacions ha do no lie wihin he inancial domain (Nn). Alhough i is imporan o have a preerred liesyle, securiy by being sel-employed, employmen o amily members, and sel-eseem; one mus recognize ha mos o hese saisacions do no apply or all small irms and some o hem are beyond he irm s ae, so i is debaable wheher o include hem or no in he enrepreneur s uiliy uncion. For insance, having a good sel-eseem is a personal aribue ha a person mus culivae regardless wheher he becomes an enrepreneur or no. O course, i he is ineresed o become an enrepreneur, as a preerred liesyle, he will ge an imporan saisacion by being an enrepreneur independenly i his business becomes successul or no. In oher words, i sel-eseem and become an enrepreneur by convicion are wo aribues beyond he irm s ae, is i imporan o value he marginal saisacion ha he enrepreneur may obain hrough his enrepreneurial advenure? Due o he ac ha he enrepreneur will obain saisacion anyway by running his business, one may hin abou hese non-pecuniary saisacions as belonging o a superior indierence curve, bu all wih he same slope. In oher words, i is lie having a consan ha moves he uiliy uncion upward, bu hese non-pecuniary saisacions do no resric he shape and he opimum soluion. Hence one may decide no o include his consan in he enrepreneur s uiliy uncion 4. oncerning he non-pecuniary saisacions relaed o he inancial domain (N), hey could be included hrough dieren scenarios as coningen sraegies ha improve he projec inancial healh and is lielihood o success. O course one mus also include hem in he enrepreneur s uiliy uncion and obain a discoun rae accordingly, bu one may also include hem hrough a prospecive analysis o he projec 5. 3 The value-addiiviy principle saes ha he value o a irm, or a well-diversiied invesor, is equal o he value o he irm wihou he projec plus he NPV o he projec. This is possible because only maers he irm s conribuion o he well-diversiied invesor s porolio, so any poenial synergy beween he projec and he irm does no have a mare value because he invesor could diversiy beer and quicer han he irm. In he case o a non-diversiied enrepreneur, his principle breas-up. 4 This decision amouns o picure a sel-coniden enrepreneur who decided o be an enrepreneur by convicion and who runs his business in a sricly proessional way.. 5 For example, James (999) has included he ranserabiliy issue ino he owner-manager s uiliy uncion o a amily irm. 7
In brie, one believes ha he uiliy uncion or enrepreneurs who are acing a seed capial invesmen may be a subse o he uiliy uncion described by McMahon and Sanger (995): w UP, U (2) Noe ha P is relaed o pecuniary reurns rom he business, especially he reurn on invesmen (OI) ha helps o provide a good salary or owner-manager. Given ha mos enrepreneurs wih seed capial invesmen opporuniies are no diversiied, he second parameer ( ) is relaed o he projec's oal ris. The non-pecuniary saisacion relaed o he inancial domain (N) are addressed direcly in he prospecive analysis and he pecuniary saisacions ouside o he inancial domain (Nn) are no considered in he enrepreneur or owner-manager s uiliy uncion. 3. The choice o he uiliy uncion or discoun raes The ield o ris preerences (uiliy uncions) was osered by he wor o Arrow (965, 97) and Pra (964). In paricular, hey boh proposed he ollowing speciicaion o uiliy uncions 6 : U e (3) This uncion is well-nown as he negaive exponenial and i has a consan absolue A ). They also proposed he ollowing speciicaion 7 : ris-aversion coeicien ( c U (4) This uiliy uncion is nown as he power uiliy uncion where he parameer ( ) represens he level o relaive ris-aversion. Unorunaely, boh speciicaions only le us vary he magniude o he ris-aversion (absolue or relaive), bu hey do no allow us o change he slope o hese ris-aversion measures (Meyer, 2007). The slope represen he behaviour o he individual s ris- aversion because i may depic decreasing (or increase) o he absolue ris-aversion (DAA or IAA).In one hand, a DAA behaviour implies ha richer people are less ris adverse han poorer people, so hey require a smaller paymen. In he oher hand IAA implies ha richer people require a larger paymen han poorer people in order o ener in a loery game. Naurally, he DAA case is more realisic han he IAA case. 6 One can also express he uiliy uncion in erms o wealh () insead o consumpion () wihou loss o generaliy. 7 The coeicien o relaive ris-aversion (A) is equal o c c Ac and i implies ha risaversion also depends upon he individual s level o consumpion, which in urn depends on his wealh. 8
In 97, Meron proposed he Hyperbolic uiliy uncion wih Absolue is-aversion (HAA) ha has wo big advanages: i includes he previous speciicaions as paricular cases and i includes a wider range o ris preerence speciicaions. Furhermore a paricular choice o uncion parameers can change he slope o he ris aversion coeiciens (AA and A): U (5) here: 0 ; 0 ; and 0 For insance, i 0 hen absolue ris-aversion is decreasing, while i 0 hen absolue ris-aversion is increasing. I hen he coeicien o relaive ris-aversion will be equal o. I hen and he so called generalized power uiliy uncion appears: U (6) From his speciicaion, one could obain he negaive exponenial uncion, he logarihmic uiliy uncion and he quadraic uiliy uncion: I 0 and 0, hen he generalized power uiliy uncion (expression 6) converges o he negaive exponenial uncion given by expression 3 and i has an absolue ris-aversion coeicien equal o. I hen an aine ransormaion o expression 6 converges o he logarihmic uiliy uncion: Ln U here: 0 This speciicaion has a decreasing absolue ris-aversion coeicien equal o A. I hen he relaive ris-aversion coeicien is equal o. I hen he aine ransormaion o expression 6 will converge o he quadraic uiliy uncion: 9
U 2 2 here: 0. This speciicaion has an increasing absolue ris-aversion coeicien equal o: ecenly, some new uiliy uncions have been pu orward in he lieraure. For insance Saha (993) inroduced he expo-power (P) uiliy uncion. The uncional orm is as ollows: U e ih he ollowing resricions: and 0 Given hese resricions, he absolue ris-aversion coeicien is equal o he ollowing expression: A. As Meyer (2007) poined ou, his is a wo parameer model because he consan does no ener he ris-aversion coeicien. The main advanage o he P uncional orm over he HAA is ha he P orm reduces o he AA orm wih inie parameer values. emember ha in he HAA orm (expression 6) one needs o assume ha 0 in order o have he AA orm. Xie (2000) ried o improve also a uncional orm over he HAA uncion, so he proposed he Power is Aversion (PA) uncional orm. The advanage o his new uncional orm over he HAA is ha i remains well-deined over a longer range or parameer values (Meyer, 2007): c U exp ih he ollowing resricions: 0 and 0 In he previous expression he exponenial operaor (exp) reers o he ranscendenal number e. The absolue ris-aversion coeicien is given by he ollowing expression: A Finally, onnie (2006) has provided he Flexible Three Parameer (FTP) uncional orm ha encompasses he P and PA orms. The uncional orm is given by he ollowing expression: 0
U A hen =0 i reduces o he PA (Meyer, 2007) The absolue ris aversion coeicien is given by he ollowing expression: Despie he ac ha using an HAA uiliy uncion jus capures a more resricive range o invesors preerences han he alernaives uiliy unciuons (P, PA and FTP orms), i is he mos used closed-orm speciicaion in inancial economics. For insance, he HAA uiliy uncions have been applied or esimaing discoun raes in he case o banrupcy (Barhelemy e. al., 2006), or porolio perormance (Breuer and Gürler, 2006), or porolio allocaion wih hedge unds (Popova, 2006), and or expanding condiions or pricing in incomplee mares (Menoncin (998), Luenberger (2002), and Guu and ang (2008)). I one aims o capure oher ypes o ris preerences, i would be beer o ollow he approach o Meyer (2007). The approach o Meyer is based upon deining a closed-orm uncion or he marginal uiliy uncion insead o he uiliy uncion isel. This is because, he marginal uiliy also represens a ris preerence, however no all persons have a closedorm marginal uiliy. In his wor, one believes ha i is possible o use a broad (closed-orm) uiliy uncion o esimae he lower bound o a discoun rae and hen, hrough a prospecive ris-analysis, one may esimae he resuling expression s parameers. In ac, Mongru and amirez (2006) have shown ha wih incomplee mares here could be many discoun raes so he possibiliies o speciy dieren values or he parameers are enormous. 4. Discoun raes or seed capial invesmens under incomplee mares In his secion, one uses an aine ransormaion o expression 6 o solve he problem saed by Mongru and amirez (2006): how much o consume oday and omorrow provided ha each individual has an iniial wealh and each one can inves in one invesmen projec or mae a deposi in a savings accoun. However, in order o pu he problem more realisic, one analyses he case when he enrepreneur s iniial wealh level approaches o zero, when he is he enrepreneur wih he lowes ris-aversion and invess all his money in he projec. I is imporan o sae beorehand ha he soluion his problem can be considered a generalized soluion only he in he sense ha i accouns or a broader se o ris preerences raher han he quadraic uiliy uncion used by Mongru and amirez (2006). However, i should be clear by now ha using expression 6 is only a subse o all possible ris preerences.
2 All he remaining assumpions made by Mongru and amirez (2006) are also made in his derivaion namely, i is an individual opimizaion meaning ha each individual mus deine he parameers o he resuling discoun rae expression according o his ris aversion and he speciics o he projec. However, one can also deine he lower bound o he discoun rae wih he enrepreneur wih he lowes ris-aversion, which implies an absolue ris aversion coeicien equal o. One sars by opimizing he ollowing wo-period objecive uncion (one also assumes ha he uiliy uncion is ime separable):, U U Subjec o he ollowing consrain: here: : epresens he projec expeced reurn in period + : The enrepreneur iniial wealh level. : The enrepreneur s consumpion o oday : The enrepreneur s consumpion o omorrow The sochasic discoun acor (SDF) o he previous problem is equal o: m (7) In order o esimae he parameers o he SDF or his individual, one may sae he ollowing sysem o equaions (Mongru and amirez, 2006): m m Using he SDF rom equaion 7, his sysem o equaion ranslaes ino he ollowing sysem:
3 (8) (9) From equaions 8 and 9, i ollows he ollowing condiion: 0 (0) Binomial 0, 0 () Simpliying even more expression i yields: n Binomial, (2) xpression 2 is very imporan because i allow us o derive he expressions o he discoun raes or wo ris preerences: quadraic and logarihmic. I is no possible o obain he discoun rae corresponding o he case o a negaive exponenial uiliy uncion because 0 and. One mus remember ha he coeicien o absolue ris-aversion is equal o. The case o a quadraic uiliy uncion appears when he ollowing condiions are me:, = - and 0. By replacing hese condiions in equaion 2 yields:, Binomial (3) Simpliying he previous expression yields: (4) Now, one mus consider he ollowing wo condiions. The irs condiion is relaed o he enrepreneur wih he lowes ris-aversion coeicien and he second condiion reers o he low iniial level o wealh o he enrepreneur
4 0 A Given hese wo condiions, equaion 4 provides he lower bound or he discoun rae or enrepreneurs wih quadraic preerences: V TV (5) here: TV is he reward-o-variabiliy raio and V is he coeicien o variaion However, or he enrepreneur wih he lowes ris-aversion coeicien, one may assume ha TV, hence equaion 5 boils down o: V (6) Noe ha in his case, he discoun rae has hree parameers: he ris-ree rae (), he coeicien ha measure he magniude o he enrepreneur ris-aversion ( ) and he coeicien o variaion ha comes rom he seed capial projec. Besides, he absolue coeicien o ris-aversion is increasing because 0 and. Alernaively, one may sudy he case o he logarihmic ris preerence. This siuaion appears when he ollowing condiions are me:, = and 0. By replacing hese condiions in equaion 2 yields:, Binomial (7) Simpliying his expression aing ino accoun he ollowing propery yields expression 8: 2 2 2
V V (8) Again in expression 8 one needs o consider he aoremenioned wo addiional resricions: A 0 eplacing hese wo resricions in equaion 8, yields he lower bound discoun rae or an enrepreneur wih a logarihmic ris preerence: V 2 V (9) Noe ha in his case he discoun rae has also hree parameers: he ris-ree rae (), he ris-aversion, and he projec s oal ris. However, he projec oal ris is being represened wih wo esimaors: he sandard deviaion (and variance) o he projec reurns and he coeicien o variaion o he projec reurns. This speciicaion implies a decreasing ris-aversion coeicien and0. 5. Properies o he discoun rae expressions In his secion one ideniies he criical parameers in boh expressions or he lower bound o he discoun rae (expressions 6 and 9) and hen one perorms a simulaion analysis 5. Analysis o he discoun rae wih quadraic preerences Figure No shows he resuls o he sensiiviy analysis using he expression 6. As one may observe, he criical variable is he enrepreneur's ris-aversion coeicien and i has an asymmeric impac over he esimaed discoun rae because a decrease o 0% in he alpha parameer increases he projec s discoun rae in more han 30%.. 5
Figure No : riical parameers or he esimaion o discoun raes wih quadraic preerences 3 Tornado Graph or Discoun rae/6-6.29% is-aversion magniude (alpha) / is-aversion magniude (alpha)/4 30.252% 2 oeicien o variaion / oeicien o variaion/5-7.059% 7.059% is ree rae/3-2.94% 2.94% Perc% hg in oupu a end o bars 0-30.69-25.58-20.46-5.35-0.23-5.2.00 5.2 0.23 5.35 20.46 25.58 30.69 Percen% hange in Discoun rae/6 Source: Own elaboraion Figure No 2 shows he resuls o simulaing discoun raes assuming a consan risree rae o 5%, a consan coeicien o variaion equal o 0.5, and an alpha ha is normally disribued wih a mean o 2 and a sandard deviaion o 20%. The resuling discoun raes end o cluser around 20%. Alhough his is he lower bound, one may assume a dieren scenario and by geing dieren values or he parameers (especially or he coeicien o variaion) Figure No 2: Simulaed values or he discoun rae wih quadraic preerences 0.332 Disribuion or Discoun rae/d6 0.265 POBABILITY 0.99 0.33 0.066 0.000 3.52 2.79 30.06 38.34 46.6 54.88 63.5 Values in 0^-2 6
Source: Own elaboraion 5.2 Analysis o he discoun rae wih logarihmic preerences Figure No 3 shows he criical variables or his ype o discoun rae. The main conclusion is ha he alpha and he projec oal ris are crucial parameers o esimae he projec s discoun raes. An increase o 0% in he ris aversion parameer alpha will decrease he projec discoun rae in %. I is ineresing o noe ha alhough he alpha parameer is sill imporan, is relaive impac has been diminished. 4 Figure No 3: riical parameers or he esimaion o discoun raes wih logarihmic preerences Tornado Graph or Discoun rae/4 -.07% is-aversion magniude (alpha) / is-aversion magniude (alpha)/ 2.02% 3 -.07% Projec oal ris / Projec oal ris/3 2.02% 2 oeicien o variaion / oeicien o variaion/2-3.04% 4.054% -.892% is ree rae/0.892% Perc% hg in oupu a end o bars 0-2.99-0.82-8.66-6.49-4.33-2.6.00 2.6 4.33 6.49 8.66 0.82 2.99 Percen% hange in Discoun rae/4 Source: Own elaboraion Figure No 4 shows he simulaion resuls or he discoun raes wih logarihmic preerences. Here, one uses a ris ree rae o 5%, a coeicien o variaion o 0.5, an alpha parameer ha is normally disribued wih a mean o 0.5 and a sandard deviaion o 0%. Besides he projec oal ris is disribued according o a uniorm disribuion wih a lower value o 0.2 and an upper value o 0.4. From his analysis one may conclude ha he join eec o boh criical variables end o increase he simulaed values o he discoun raes. However, he ne eec is sill asymmeric and i is sewed o he righ as i should be because hey represen lower values. Figure No 4: Simulaes values or he discoun rae wih logarihmic preerences 7
Disribuion or Discoun rae/g4 0.78 0.42 POBABILITY 0.07 0.07 0.036 0.000 8.9 8.33 27.76 37.9 46.62 56.05 65.48 Values in 0^-2 Source: Own elaboraion 6. onclusion In his wor one has obained wo expressions or esimaing subjecive discoun raes using he HAA uiliy uncion. In his way, one has obained more generalized resuls han hose o Mongru and amirez (2006). Furhermore, hese wo expressions have been obained assuming an enrepreneur wih he lowes ris-aversion coeicien, who almos lacs iniial wealh, and who pus all his money in he invesmen opporuniy (non-diversiied enrepreneur). One believes ha his enrepreneur s proile is very common in emerging mares. The uiliy speciicaion applied in his wor only includes pecuniary reurns (such as he projec reurn on invesmen - OI and oal ris). Depending on he speciicaion, oal ris is measured using he sandard deviaion or he coeicien o variaion. From he resuls, i seems ha depending on he enrepreneur s ris preerences, he ris-aversion parameer could be more or equally imporan as he measure o he projec s oal ris. However, given he assumpions, i seems more realisic o use he logarihmic speciicaion insead o he quadraic one because he ormer assumes DAA. Given he imporance o he enrepreneur s ris aversion, i is imporan ha uure wor is direced owards improve he measuremen o i. A he end, wha maers in he valuaion o seed capial invesmens is consisency, no accuracy in he values. 8
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