Amercan Journal of Economcs and Busness Admnsraon 1 (): 194-05, 009 ISSN 1945-5488 009 Scence Publcaons A heory of he Rsks of Venure Capal Fnancng Edmund H. Manell Deparmen of Fnance, Lubn School of Busness, Pace Unversy, 1 Marne Avenue, Whe Plans, New York, USA, 10606 Absrac: Problem saemen: here s no general heory analyzng how he me-varyng cash flows of venure capals fnancng affec he lkelhood of success of a new venure. hs research addressed ha lacuna n he leraure. Approach: Research n he area of venure capal fnancng was needed because of he mporance of new venures as germnaors of echnologcal nnovaon. he research n hs sudy developed a general economc heory quanfyng he rsk of venure falure assocaed wh me-varyng cash flows of fnancng. Each occason when an enrepreneur made an overure o a venure capals o elc a fnancng commmen was defned o be a dsnc solcaon even. he seres of fnancal commmens elced from venure capalss were assumed o have he characerscs of ndependenly dsrbued random varables. I was assumed ha he enrepreneur mus secure a mnmal aggregae commmen n order o ensure developmen of he projec; falure o secure ha amoun caused he venure o be abored. he heory of sochasc processes was appled o derve he praccal mplcaons as regards he rsk of aboron. Resuls: I was shown ha he aggregae fnancng commmen secured by an enrepreneur n a fne me had sochasc properes correspondng o hose of a sascal renewal process. he research derved lmng condons on he probably ha enrepreneur s venure wll be abored because of hs falure o secure he mnmal aggregae commmen. he man resul was ha f he number of solcaons by he enrepreneur s large and he fnancal commmens were ndependenly dsrbued random varables wh fne means and varances, he probably dsrbuon governng venure survval s he Normal dsrbuon. Concluson: he sudy derved four analycal proposons quanfyng he radeoffs beween he rsk and he expeced reurn assocaed wh venure capal fnancng. he polcy mplcaons of he resuls mply he benefs of mgang nformaon asymmery. Some of he rsk faced by he enrepreneurs could be aenuaed f nformaon abou he rsk/reurn preferences of venure capalss were known o he enrepreneurs pror o solcaon. Some of he rsks faced by he venure capalss could be aenuaed f nformaon abou he rsk/reurn characerscs of he proposed nvesmen projec could be accuraely and ransparenly communcaed o he venure capals durng he solcaon even. If eher or boh of hese nformaon defcs were pallaed, he marke for venure capal would operae more effcenly. Key words: Enrepreneural solcaons, venure capal fnancng, sochasc processes, probably of enrepreneural nsolvency, renewal process INRODUCION he germnaon of corporae valy va venure capal s growng rapdly. By he end of year 001, companes fnanced wh venure capal snce he 1970s accouned for 5.9% of he jobs n he Uned Saes and 13.1% of US gross domesc produc n year 000 (Naonal Venure Capal Assocaon). hese sascs, as well as ohers ha can be adduced, evdence he remendous mpac ha venure capal has had on employmen and revenue generaon n he Uned Saes durng he pas hry years. Gven ha venure capal was less han one percen of US 194 nvesmen acvy durng mos of he perod suded, s mpac s remarkable [30]. In vew of he growng sgnfcance of he venure capal ndusry n he US, s odd ha so lle aenon has been gven o he developmen of a formal heory of enrepreneural behavor n he venure capal solcaon process. here s a large body of leraure descrbng he audes and he nvesmen behavor of he venure capalss [6-8,18,33]. here are emprcal sudes documenng he realzed raes-of-reurn o he venure capalss employng dfferen ex sraeges []. here s an abundan supply of how-o-do- books and arcles dspensng praccal gudance o enrepreneurs
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 seekng fnancng (An especally well known how-odo- book s Pra) []. here are books and arcles descrbng he fnancal conracng process (A general recen survey s Har [9]. A much more focused descrpon of fnancal conracng n he venure capal ndusry s Sahlman [7] ). I have no found any general heory purporng o analyze he neremporal properes of enrepreneural fundng. he lacuna s confrmed by a very recenly publshed sudy wheren he auhor commened...hough enrepreneural frms are a pvoal source of new employmen n Europe, he enrepreneur s perspecve on capal acquson s rarely dscussed n he leraure [6]. ha auhor conduced a case sudy, conssng manly f nervews wh successful enrepreneurs. However, whou a rgorous heory of he solcaon process, he sascs yelded by he daa colleced mgh yeld only a few banales [19]. hs sudy esablshes a heory of he venure capal fundng by focusng aenon on he acves of he enrepreneur a he ncpency of he nvesmen process: Namely he solcaon even. A solcaon even s a seres of acves wheren he enrepreneur (or he enrepreneural group) searches for a venure capals o solc, proposes an nvesmen o a venure capals and elcs a fnancal commmen or, more frequenly, a rejecon. A seres of such evens consues he solcaon process. hs way of descrbng a solcaon even makes possble o characerze as a bnary-valued varable;.e., eher he enrepreneur secures a fnancal commmen, or he does no. hs sudy descrbes he seres of solcaon evens as he realzaon of random varables. he successes and he falures of he enrepreneural group are shown o have characerscs conformng o hose of a sochasc process. I explo some of he properes of such processes o derve behavoral mplcaons as well as nferences peranng o he probably dsrbuons governng he success of he fundng solcaons. MAERIALS AND MEHODS nermedares, he marke for venure capal would end o dry up. hs s because relavely poorly nformed nvesors who were drawn no falng nvesmens would declne o provde venure capal fnance. he nvesmen allocaon problems assocaed wh rsk and uncerany are assgned o hose persons who are wllng and compeen o manage hem. Venure capalss are suffcenly specalzed and experenced n hgh-rsk nvesmens o cope wh problems of nformaon asymmery. hs managemen of nformaon asymmery ends o lead o he so-celled problem of adverse selecon. Venure capalss have been characerzed as nsuons ha resolve nformaon asymmeres [1,4]. Venure capal frms ofen fnance projecs whch, a he me of fundng, have neher revenues nor even a produc n exsence [14]. However, an enrepreneur (or an organzed group) wll ofen approach venure capalss (hereafer VCs ) for fundng a dfferen sages n he gesaon of he nascen venure. he mulple sages of he fnance solcaon acves have been adumbraed by Kozmesky e al. [15] : Seed fnancng-capal provded o an enrepreneur o prove a concep. I may nclude produc developmen bu does no nvolve nal markeng Sar-up fnancng-fnancng used n produc developmen and nal markeng Frs-sage fnancng-fnancng provded o companes ha have expended her nal capal (ofen n developng a prooype) and requre funds o nae commercal manufacurng and sales Second-sage fnancng-workng capal used for he nal expanson of a company ha s producng and shppng a produc and has growng accouns recevable and perhaps nvenores hrd-sage fnancng-funds provdng for major expanson of a company whose sales volume s ncreasng and ha s breakng even or sarng o show a prof Fourh-sage, mezzanne, or brdge fnancngcapal funds nvesed n a company expeced o go publc whn sx monhs o a year Synopss of he venure capal fundng process: From he perspecve of he mcroeconomc heory of When a VC makes a fundng commmen o an fnancal nsuons, he venure capals s a knd of enrepreneur almos always enals saged fnancng. fnancal nermedary beween enrepreneurs and he ha s, he VC makes a frm commmen o offer a nvesng publc [3]. he venure capals manages fracon of he funds needed, wh he undersandng funds for downsream nvesors (.e., he buyers of he ha fuure fundng s conngen on frm performance. sock when and f, he busness goes publc) who are hs s fnancally equvalen o akng a sequence of no neresed n drec nvesmen n hgh rsk/hgh call opons on he enrepreneur s projec. he VC hen reurn nvesmens. Whou such fnancal evaluaes wheher he projec has reached s 195
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 performance arges and decdes wheher o proceed wh addonal fundng. Consequenly, he nal fundng commmen s seldom for 100% of he enrepreneur s needs. o pu he fnancal sgnfcance of hese sages no praccal perspecve, n year 1999 roughly 30% apece was nvesed (by VCs) n frs and second rounds of fnancng and close o 40% was nvesed n hrd and laer rounds, wh he remander (apparenly close o zero) devoed o seed rounds [31]. he mul-sage characer of solcaon acves s emboded n he me seres of solcaon evens. Assumpons characerzng he solcaon process: he solcaon process ha s he subjec of hs sudy s characerzed by a se of smplfyng assumpons. hese are enumeraed below: 1. he enre me perod durng whch he solcaon acves are carred ou s called he fundng solcaon perod. he pon n me ha marks he ncepon of he solcaon acves s symbolzed by 0. he fundng solcaon perod s assumed o conss of nervals of equal lengh: = 1 for = 1,...,. In hs sudy s assumed ha s a large number. I s assumed ha n each nerval he enrepreneural group solcs a leas one source of venure capal fundng. A solcaon even consss of hree (or possbly four) dsnc phases: (a) denfy suable VCs o be solced, (b) make an overure o a suable VC o deermne wheher he wll eneran a full-blown proposal, (c) presen a busness plan o he VC who agrees o eneran he proposal and (d) carry ou negoaons for fundng and conrol. In order o smplfy he model s assumed ha solcaon acves ha are commenced n each me nerval are compleed n he same me nerval 3. I s assumed ha he fundng decsons of he VCs who are solced n each of he me nervals are made durng he same n he nerval n whch hey are solced 4. he lengh of he fundng solcaon perod s measured as = 1 5. I s assumed ha he ermnal pon of he solcaon perod for hs venure s mmuable (he consequences of an mmuable ermnal pon n he solcaon perod has been vvdly descrbed by Sahlman [7]. Afer pckng hmself up off he floor, he venure capals wll begn a process of 196 ryng o educae (he enrepreneur) abou he real world... he erms of he (venure capals s) couneroffer wll lkely call for saged nfusons of capal over me... Remember ha he venure s scheduled o run ou of capal perodcally; f canno rase capal a he second or hrd rounds, hen goes ou of busness and he enrepreneur s ou of a job ) If an affrmave fundng decson s made by a VC n he nerval, ha decson s manfesed as a moneary commmen n he same nerval. he commmen may consue a capal fundng appropraon for a sngle perod, or may consue seral commmens for fundng he venure n specfed amouns a predeermned pons n fuure me. Whaever form may ake, he commmen secured n nerval s symbolzed by I( ). I s assumed ha he enrepreneural group regards he sequence of commmens {I( ) = 1,,...} as a sochasc process whose elemens are ndependenly dsrbued on he non-negave real lne (here s some evdence of collaboraon among venure capal frms n he form of syndcaon of her nvesmens. For example, a 1994 sudy by Lerner [16] examnes hree raonales for he syndcaon of venure capal nvesmens. He used a sample of 71 prvae boechnology frms. Lerner found ha syndcaon s commonplace, even n he frs round nvesmens. Syndcaon ofen ensures ha he ownershp sake of he venure capals says consan n laer venure rounds. For he purposes of he analyses n hs paper a syndcae of VCs s defned as one VC acng ndependenly of oher VCs who are no members of he syndcae) Hsorcally, only one ou of 50-100 nvesmen opporunes revewed by any gven VC has acually been fnanced (ankersley [8] and Zder [33] Even hough he srucure of venure capal deals seems o pu he enrepreneurs a a seep dsadvanage, hey connue o subm far more plans han acually ge funded, ypcally by a rao of more han en o one. hs percenage has some emprcal valdy, as found by Maer and Walker [17]. Venure mpac, 007 page 10 saed: For every 100 busness plans ha come o a venure capal frm for fundng, usually only 10 or so ge a serous look and only one ends up beng funded ). o he exen ha hs hsorcal sasc s saonary, mos of he elemens of he seres {I( )} wll be equal o zero (One commenaor has suggesed ha f an
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 nvesmen opporuny s rejeced by a number of frms, may ge an overshopped repuaon because VCs freely rade nformaon and a urndown by one frm may nfluence ohers. See ankersley [8]. Assumpon (5) expressly precludes hs behavor). In addon, he enrepreneural group wll recognze ha hey mus expec o solc many VCs n order o secure adequae fundng. hs ranslaes o mean ha s lkely o be a large neger Anoher characersc of he venure capal ndusry ha affecs he valdy of Assumpon (5) s ha VC frms also proec hemselves from rsk by convesng wh oher VCs. I s he excepon, no he rule, for one VC o fnance an ndvdual company enrely. Raher, venure frms prefer o have wo or hree VCs nvolved n mos sages of fnancng [33]. hs s a form of syndcaon (supra). A reason for hs s ha agency coss of nvesng are reduced when oher repuable venure capalss are syndcang a deal and he nework allows an effcen ransfer of nformaon abou compeng frms and echnologes. o he exen ha hs preference s ubquous n he communy of VCs, wll end o nduce an ncrease n he number of VCs solced by an enrepreneural group. hs aude wll also end o cause o be large 6. I s assumed ha he members of he enrepreneural group formulae homogeneous expecaons abou he parameers of he dsrbuons governng he elemens n he sequence {I( )}. he expeced values of he aggregae commmens n each nerval are symbolzed by E[I( )] = µ and he varances of he aggregae commmens n each nerval are symbolzed by Varance of I( ) = var 7. I s assumed ha he enrepreneural group allocaes a budge of B dollars o underwre he expenses of carryng on solcaon acves durng he enre fundng solcaon perod. In each of he nervals he group wll spend a consan poron of he aggregae amoun budgeed. he budge allocaon n each solcaon nerval B s symbolzed by b 8. I s assumed ha he enrepreneural group has deermned he mnmum amoun of fnancng requred for a successful connuaon of her projec a he expraon of he fundng solcaon perod. ha mnmum s symbolzed by K dollars. Fundng a he mnmal level does no guaranee 197 success of he venure, bu s a sne qua non for success 9. I s assumed ha f he group does no secure adequae fundng on or before he expraon of he fundng solcaon perod he venure wll be abored 10. I s assumed ha f he enrepreneural group secures commmens equal o or exceedng K dollars pror o he expraon of he fundng solcaon perod, he solcaon acves wll connue o he end of he fundng solcaon perod. o proec her nvesmen or o compensae for he lack of avalable workng capal from radonal sources, VCs provde laer sage fnancng o her successful companes who oherwse would be unable o expand o a profable maury sage Properes of he commmen process: Consder he fnancal saus of he enrepreneural group when has compleed solcaons n of he nervals, ha s, here are - nervals remanng n whch he enrepreneural group wll connue s solcaon acves. Le S represen he aggregae capal fundng ha has been commed o he group a he end of nerval. hs s calculaed as: S = I( ) = 1 he dollar magnude of S can be descrbed n probablsc language; he enrepreneur has made a sequence of draws from ndependen probably dsrbuons. Each of hose draws resuls n a nonnegave random varable,.e., he dollar commmen. he sequence of dollar commmens sums o S. ha sum consss of he realzaons of random varables drawn from he process{i( ) = 1,,...}. Cox [4] descrbes hs sysem as an (ordnary) renewal process (he analycal approach o renewal heory adoped by Cox cass n erms of a populaon of componens wh falure mes governed by a connuous random varable. He saes: I s agan o be sressed ha he erms componen and falure-me can be gven many dfferen nerpreaons. In hs paper he analogue of he n h falure-me (.e., he n h renewal) consss of he nvesmen commmen n he n h solcaon nerval). Perodcally (A recurrng even ξ s called perodc f here exss an neger λ f1 such ha ξ can occur only a rals number λ,λ,3λ, he greaes λ wh hs propery s called he perod of ξ. In hs paper he
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 perodcy of he process {S } s a consequence of Assumpon (1)), he sum S s renewed by he random varable I( ) such ha n nerval he frs dfference of he sum s S S 1 = I( ). A bref descrpon of he applcaon of renewal processes o fnance can be found n Zemba and Vckson [3]. A general reamen can be found n he monograph by Cox ced above. Consdered ex ane a he end of nerval, he ne aggregae nvesmen capal ha wll be avalable a he end of he fundng solcaon perod s a random varable. I can be expressed as Eq. 1: S = I( ) + S + B b (1) + 1 For any fxed value of <, ex pos he paral sum S s a known value. hus we can condense he hree consan erms on he rgh hand sde of Eq. 1 and defne: K = (K S ) (B b) () he quany K defned n Eq. measures he margnal requred commmen; ha s, he aggregae commmen ha mus be secured by he enrepreneural group n he fuure - nervals. he frs parenhecal erm on he rgh sde of Eq. represens he defc remanng beween (a) he mnmum capalzaon requred for a launch of he venure and (b) he gross commmens o me. he second parenhecal erm represens he dfference beween he aggregae amoun budgeed o underwre he solcaon acves and he poron of ha wll be expended by dae. Expresson (1) can now be adaped o he express he probably ha he aggregae nvesmen capal commed a he end of he fundng solcaon me horzon wll be nadequae o launch he venure, gven he aggregae commmens o me. ha s o say, he condonal probably of undercapalzaon calculaed ex ane a me wh ex pos knowledge of commmens o ha me. he probably s expressed as Proposon 1. Proposon 1: If he number of solcaons s large and he capal commmens by he VCs are ndependenly dsrbued random varables wh fne means and varances, he probably ha he venure wll be abored s a normally dsrbued funcon of he mnmal aggregae capal commmen. Proof: Pursuan o Assumpon (9) above, he probably ha he venure wll be abored s equal o he probably ha he venure wll be undercapalzed a he end of he solcaon perod. he probably dsrbuon s symbolzed as P( ). he probably of undercapalzaon can be wren as Eq. 3: P(undercapalzaon) = P(S p K) = P I( ) p K (3) = + 1 Pursuan o Assumpons (1) and (5), he number of ndependenly dsrbued random varables summed s -. If ha number s large, hen he sum sasfes a condon suffcen o nvoke he Cenral Lm heorem (.e., CL). he applcaon of he CL n hs conex saes ha he probably dsrbuon governng he renewal equaon I( ) s approxmaely = + 1 Normal. he formal expresson of he CL enals some addonal noaon. Le I and le = + 1 = µ σ = var. = + 1 he former s he sum of he commmens expeced n he remanng - solcaon evens and he laer s he varance of ha sum. Le he varable Z be defned as: = + 1 Z = I( ) I σ he CL saes: P(Z X) F(X) lm where, F(.) symbolzes he cdf of he Normal dsrbuon wh a mean of 0 and a varance equal o 1. hs heorem allows us o derve an expresson ha s an approxmae measure of he probably denfed n Eq. 3. hs s accomplshed usng smple algebra: P(undercapalzon) = P I( ) p K = + 1 = P I + σz p K K = P Z p σ K I F σ I hs complees he proof of Proposon 1. 198
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 I s a propery of all cumulave probably dsrbuons ha hey are monoone non-decreasng funcons of her argumen (s). hus, n hs conex s obvous ha any ncrease n he argumen X = (K I ) σ wll lessen he lkelhood ha he venure wll secure s capalzaon requremens. Perhaps less obvous s he effec on he probably of undercapalzaon of a change n he varance of he randomly dsrbued commmens n he remander of he solcaon perod, ceers parbus. hs ssue bears on he behavor of he enrepreneural group n carryng ou s solcaon acves subsequen o me. he mplcaons of he CL can be summarzed n Proposon. Proposon : a. If he expeced aggregae commmen exceeds he margnal requred commmen, an ncrease n he varance of he aggregae commmen wll ncrease he probably of undercapalzaon b. If he expeced aggregae commmen s less han he margnal requred commmen, an ncrease n he varance of he aggregae commmens wll lessen he probably of undercapalzaon c. If he expeced aggregae commmen equals he margnal requred commmen, a change n he varance of he aggregae he fuure commmen wll have no effec on he probably of undercapalzaon Proof: he proof of Proposon proceeds by analyzng he paral dervave of he argumen of he Normal cdf wh respec o σ, ha s, akng he paral dervave of X w.r.. σ, we have: X I K σ = σ I s obvous ha sgn [ X σ ] = sgn[ I K ]. Recall ha I represens he expeced value of he sum of he randomly dsrbued commmens n he fuure - perods. Also recall ha K represens he margnal requred commmen. he followng nferences are mmedae: a. If he aggregae expeced commmen a me exceeds he margnal requred commmen, hen X σ f 0 b. If he aggregae expeced commmen a me s less han he margnal requred commmen, hen X σ p 0 199 c. If he aggregae expeced commmen a me s equal o he margnal requred commmen, hen X σ = 0 Pursuan o he srcly monoonc propery of he Normal cdf, an ncrease n X wll ncrease he probably of undercapalzaon. Lkewse, a decrease n X wll lessen he probably of undercapalzaon. he drec mplcaon of monooncy s ha f he varance of he aggregae fuure commmens ncreases, ceers parbus, he probably of undercapalzaon wll be ncreased when he sum of her expeced values exceeds he arge fundng. he probably of undercapalzaon wll decrease when he varance of fuure commmens ncreases, ceers parbus, f arge fundng exceeds he sum of he expecaons. hs complees he proof of Proposon. Behavoral mplcaons of proposon : he behavoral mplcaons of proposon reflec he collecve aude owards rsk manfesed by he enrepreneural group. I s well beyond he scope of hs sudy o underake a comprehensve dscusson of he defnon and he feasbly of esablshng a collecve aude owards rsk. hs sudy assumes ha he members of he group agree o ac as f hey shared a common aude. A he hreshold of he behavoral analyss wll be useful o prescnd he dfferen manfesaons of rsk faced by he enrepreneural group. One apparen rsk s he rsk ha he group wll fal o secure aggregae commmens ha are requred o launch he venure; ha s, he rsk ha S p K. ha rsk can be called he ermnal rsk. he oher knd of rsk s he uncerany manfesed n he varaon of he nerm cash flow commmens anedang he ermnal pon of he fundng solcaon perod. ha rsk s measured by he elemens of he seres { σ = 1,,...}. he reason for dsngushng beween hese wo manfesaons of rsk s o esablsh a group ulyorderng relaon beween hem. As beween hese wo manfesaons of rsk, s assumed ha he enrepreneural group assgns prmacy o he ermnal rsk. he collecve group aude regardng he ermnal rsk can be expressed n erms of he behavor amed a aanng he objecve of he group: he group adops a polcy of managng and carryng ou he solcaon process n order o mnmze he probably ha he venure wll fal o mee s capalzaon requremens a he ermnal pon of he fundng solcaon perod. All oher objecves of he group are
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 subordnaed o ha prmary objecve. he group may manfes rsk-aversve or rsk-seekng audes only o he exen ha hose audes do no movae behavor ha s nconssen wh aanmen of he group s prmary objecve. Behavoral mplcaons of a group rsk-aversve aude: Assume ha he collecve group aude s one of rsk averson. hs s consrued o mean ha he group wll avod any purposeful acon ha causes an ncrease n he rskness of he cash flow of he fuure commmens, ceers parbus. hs presumed aude reflecs recen research n he area of manageral audes and pracces for managng rsk. I embodes he psychologcal approach developed by, ner ala [1,5,10,9]. All hese papers are case sudes adducng emprcal evdence respecng manageral audes owards defnng, measurng and managng rsk). he fundamenal noon s one of loss averson ---- noably he avodance of he maxmum concevable loss. he maxmum loss, n he conex of hs sudy, would be he falure by he enrepreneur o elc aggregae commmens suffcen o mee s capalzaon requremens a he ermnal pon of he fundng solcaon perod. A falure of ha magnude resuls n he deah of he nvesmen projec. From he perspecve of he enrepreneur, nohng could be worse. ha aude has been emprcally suded n several recen publcaons. Gven hs consrucon, Proposon 1 mples he followng: A rsk-aversve group wll manage s fuure solcaon acves n such a way as o cause an ncrease n he rskness of he fuure commmens f he expeced value of hose aggregae commmens exceeds he margnal requred commmen. hs s an nsance n whch a rsk-aversve group wll seek ou addonal rsk n s nerm cash flow n order o lessen he ermnal rsk of falng o mee s argeed objecve. hs counernuve resul s a drec mplcaon of par (b) of Proposon 1. he rskness of he aggregae fuure commmens s measured by he varance of he sum of hose randomly dsrbued commmens. Proposon 1(b) esablshes ha an ncrease n ha varance wll lessen he probably of undercapalzaon, gven he nequaly specfed n ha par of he proposon. Behavoral mplcaons of a group rsk-seekng aude: A group rsk-seekng aude s consrued o mean ha he group wll conduc s solcaon acves n such as way as o cause an ncrease n he rskness of he cash flow of he fuure commmens, ceers parbus. Gven hs consrucon, Proposon 1 mples he followng behavor: If he expeced aggregae commmen exceeds he margnal requred commmen, a rsk-seekng group wll carry on s solcaon acves n such a way as o lessen he rskness of he cash flow of fuure commmens. hs counernuve resul s a drec mplcaon of par (a) of Proposon 1. Behavoral mplcaons of group rsk-neuraly: A collecve aude of rsk-neuraly means ha he group regards he changes n he varably of he fuure commmens as rrelevan o s solcaon acves, ceers parbus. Proposon 1(c) mples ha f he expeced aggregae commmen s equal o he margnal requred commmen, he group wll no aler s solcaon behavor n response o changes n he rskness of he fuure commmens unless hose changes produce an nequaly of he wo capal amouns. Mean-varance analyss when he solcaon nervals are ndependen random varables: Assumpon 1 defned he solcaon nervals as fxed and of equal lengh. hs sudy relaxes ha assumpon o enhance he versmlude of he heory. he Synopss of hs sudy suggesed ha here are a leas hree and perhaps four dsnc phases of he solcaon process n each nerval. he amoun of me allocaed o phase (a) s enrely conrolled by he enrepreneural group (one well-known enrepreneur wh a rack record of successful experences n rasng venure capal delvered a speech n whch he revealed o ncpen enrepreneurs (and ohers) some of he fund-rasng pracces ha worked well for hm.. Frs on hs ls of ps was an exhoraon o... carefully arge four or fve venure frms ha have nvesmen sraeges conssen wh your company s needs. Hoffsen [11] ); he amoun of me allocaed o phases (b), (c) and (d) s only parally conrolled by he group. he search acvy n phase (a), o he exen ha s carred ou carefully and wh due aenon o he knd of nformaon o be dscovered, wll ofen represen he mos me consumng poron of he oal solcaon acves (An example of he due aenon o he nformaon dscovered by he search s he avodance of approachng a laer sage nvesor f he enrepreneural group s lookng for seed money). 00
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 {I( )} are..d. where E[I( )] = µ and Varance[I( )] = σ. he counerpar elemens of he processes { = 1,,...} and {I( ) = 1,,...} are no, n general, ndependen. he correlaon beween correspondng elemens s symbolzed by: Cov[,I( )] ρ = σ σ Fg. 1: Dsrbuon of he sources of enrepreneural fundng n he US n 003 [30] he sysemac componen of he solcaon nerval reflecs he delberaons and he search acves carred ou by he enrepreneural group o denfy VCs o approach. As a general maer, here are a leas sx dsnc sources of fundng, each wh s own specal neress and audes owards rsk. Fgure 1 dsplays he relave dsrbuon of he sources of enrepreneural fundng n year 003. he enrepreneur mus allocae suffcen me o denfy he opmal VCs o approach. o he exen ha he VC parcpans n he process exercse conrol over he empo of he solcaon acves n phases (b-d), he me requred o complee he solcaon n each nerval canno be known by he enrepreneurs wh cerany ex ane. he enrepreneural group regards each solcaon nerval as conssng of he sum of a sysemac par and a randomly dsrbued par. Hereafer he solcaon nervals { = 1,,...} are assumed o be..d. random varables wh E[ ] = and Varance [ ] = σ. I assume he enrepreneural group knows (or can esmae) he numercal values of hese parameers. Defne he random nerval n as he cumulave me elapsed beween he ncepon of solcaon acvy (a me 0 ) and he concluson of he solcaon even n he nerval n. ha cumulave elapsed me nerval s calculaed as: n n = = 1 I assume Cov[,I( j)] = 0 for j. Defne he sochasc process {N( ) = 1,,...} as a counng process such ha N( ) represens he oal number of solcaon evens compleed up o me. Formally, N( ) has he followng properes: () N( ) > 0 for all () N( ) s neger valued () If j <, hen N( j ) < N( ) (v) For j <, N( ) - N( j ) equals he number of solcaon evens ha have occurred n he nerval - j hese properes suffce o esablsh N( ) = 1,,..} as a renewal process [5]. Now we can defne a sochasc process ha measures he aggregae capal commmens ha have been secured from he oal number of VCs n all he solcaon evens up o me n. he noaon can be smplfed by omng he subscrp appended o n. Hereafer, N() wll sgnfy he oal number of VCs who have been solced n he arbrary nerval [ 0, ]. he aggregae commmens n ha nerval s: S = I( 1) + I( ) +... + I( N() ) (4) he sochasc process {S } s called a renewal reward process by Ross [5] and a cumulave renewal process by Cox [4] and Karln [13]. All hese researchers analyze paramerc and oher characerscs of hs process. Cox dsplays he lmng properes of he mean and varance of {N()} for large as follows: Assumpon 5 s also amended. Hereafer he E[N()] elemens of he process of he capal commmens 01 and Var[N()] σ 3 ( ) (5)
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 he counerpar parameers of {S } for are gven by Cox [4] as follows: µ = (6) E[S ] σ µ µ Var[S ] = + σ σσ 3 ρ ( ) ( ) (7) Equaon 6 can be subsued no Eq. 7 o express he varance of {S } as a funcon of s expeced value. he expresson s wren n Eq. 8: 1 σ Var[S ] [E(S )] E(S ) = σσ ρ + σ (8) Fg. : Level curves dsplayng he rsk/reurn radeoff for dfferen correlaons beween he randomly dsrbued solcaon nerval and he randomly dsrbued fnance commmen n he nerval Equaon 8 esablshes ha he varance of {S } s a quadrac funcon of s expeced value, ner ala. For arbrary numercal values of he parameers, σ and σ, he shape of he locus of Eq. 9 s deermned by he algebrac sgn of he correlaon beween he solcaon nervals and he commmens secured n hose nervals. Fgure 3 llusraes he relaonshp beween rsk and expeced reurn. Fgure dsplays hree members of a famly of loc ha dffer from each oher only wh respec o a sngle parameer; namely he algebrac sgn of he correlaon beween he randomly dsrbued solcaon nerval [.e., ] and he randomly dsrbued capal commmen secured n ha nerval [.e., I( )]. he Fg. 3 shows how he varance of he aggregae commmens a me s funconally relaed o he expeced value of ha aggregae a he same me, assumng dfferen algebrac sgns of he correlaons. Generally, he loc are shaped as one would expec; namely slopng upward o he rgh. hs s merely a graphc confrmaon of he proposon ha as he expeced value of he aggregae commmens ncreases, s rskness lkewse ncreases. here s nohng remarkable n ha resul. However, he orderng of he curves mples a less obvous resul; namely ha as he correlaon decreases, ceers parbus, he rskness ncreases. Compare, for example, a case where he capal commmen s sascally ndependen of he solcaon nerval wh he case where he wo are posvely correlaed. hen for any arbrary expeced aggregae, he rskness wll be larger f he correlaon s ndependen han f he correlaon s posve. he loc dsplayed n Fg. sugges nferences summarzed n Proposon 3. 0 Fg. 3: Level curves of he varance showng he cerany equvalen varyng wh he correlaon Proposon 3: Gven arbrary values for hree parameers: he mean solcaon nerval, he varance of he solcaon nerval and he varance of he randomly dsrbued capal commmens: a. If he randomly dsrbued nvesmen commmens are no posvely correlaed wh me allocaed by he enrepreneur o solcaon of hose commmens, hen he varance of he aggregae capal commmens s a monoone ncreasng funcon expeced value of he aggregae capal commmen b. If he randomly dsrbued nvesmen commmens are posvely correlaed wh me allocaed by he enrepreneur o solcaon of hose commmens, hen varance of he aggregae capal commmens s nally a decreasng funcon and hen an ncreasng funcon of he expeced value of he aggregae capal commmens
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 As a praccal maer, one would expec ha f an enrepreneural group allocaes more me and effor o he solcaon process, ceers parbus, he ndvdual capal commmens ha he group can expec o secure wll lkewse ncrease,.e., ρ>0. he counernuve mplcaon of par (b) of Proposon 3 s ha for some range of ncreases n he expeced aggregae commmens, he rskness of he aggregae capal commmen decreases. Solcaon behavor for consan absolue rsk averson: Defne a collecve uly funcon for he enrepreneural group as U(X) where X s a moneary amoun. Le U be a hree-mes connuously dfferenable funcon ha assgns o s argumen a collecve uly for he enrepreneural group. I assume ha he funcon U has he properes U >0 and U >0. hese properes of he aggregae uly funcon suffce o defne he group s collecve aude owards rsk. If X s a random varable wh an expeced value symbolzed by X, hen he so-called coeffcen of absolue rsk averson s defned as θ (X) = U(X) / U (X). he usefulness of he coeffcen of absolue rsk averson n hs conex s ha can be appled o denfy he cerany equvalen of he unceran oucome of he solcaon process. Mlgrom and Robers [0] explan ha he cerany equvalen s he amoun of ncome, payable for ceran, ha he (decson maker) regards as equvalen n value o he orgnal, random ncome. Le S symbolze he cerany equvalen of S. ha cerany equvalen can be calculaed as (Mlgrom and Robers [0] for a dervaon of Eq. 9: S = E[S 1 ] θ (E[S ])Var(S ) Where: U (E[S ]) θ (E[S ]) = (9) U (E[S ]) he cerany equvalen can be used o denfy he mean and he varance of he aggregae capal commmen n he solcaon process mpled by he correlaon beween he randomly dsrbued solcaon nervals and he random commmens secured n hose nervals. In order o acheve hs s necessary o assume ha he coeffcen of absolue rsk averson s a non-negave consan hereafer symbolzed by θ (consdered from a behavoral perspecve, consancy of he coeffcen of absolue rsk averson s consrued o mean ha he rsk premum ha an enrepreneur would pay o elmnae a gven amoun of varance does no depend on he expeced level of he aggregae commmen). If he enrepreneur assgns a fxed value K as a cerany equvalen, hen Eq. 9 graphs as a sragh lne ha can be supermposed on he loc dsplayed on Fg. 1. he resul of ha superposon appears n Fg. 3. On Fg. 3 he dashed sragh lne s he graphc represenaon of he enrepreneur s ceranly equvalen symbolzed by Eq. 9. For any fxed value of capal, say K, he dashed lne shows he combnaon of he expeced aggregae commmen/rsk of commmen ha he enrepreneur denfes as he cerany equvalen of K. ha s o say, a any pon on he dashed lne he enrepreneur would be ndfferen beween he fxed capal commmen K and a randomly deermned aggregae commmen wh an expeced value/sandard devaon gven by he coordnaes of he pon on he lne. he cerany equvalen locus nersecs each of he level curves assocaed wh dfferen correlaons. For a fxed value of he cerany equvalen K, can be shown ha he slope of he cerany equvalen locus s equal o. hs mples ha he slope of he θ cerany equvalen locus wll decrease as he coeffcen of absolue rsk averson ncreases, ceers parbus. he behavoral mplcaons of he geomery can be summarzed n he followng Proposon 4. Proposon 4: For a fxed cerany equvalen of he aggregae capal commmen, he mean and he varance of he aggregae capal commmen dsplay he followng properes: a. If he correlaon beween he solcaon nervals and he capal commmens secured n hose nervals s no posve, an ncrease n he enrepreneur s coeffcen of absolue rsk averson wll be assocaed wh a larger mean and varance b. If he correlaon beween he solcaon nervals and he capal commmens secured n hose nervals s posve, an ncrease n he enrepreneur s coeffcen of absolue rsk averson wll be assocaed wh a larger mean and a varance ha frs decreases and hen ncreases Par (a) of Proposon 4 has a counernuve sense; suggess ha as he enrepreneur s averson o 03
Am. J. of Economcs and Busness Admnsraon, 1 (): 194-05, 009 rsk ncreases, ceers parbus, he wll carry ou he solcaon acves n such a way as o resul n an ncrease n he expecaon and he uncerany of he aggregae capal commmens. Bu hs nference s srcly rue only n he (arguably) unrealsc cases where he correlaons beween he me allocaed o he solcaon evens s no posvely correlaed wh he capal commmens secured by hose evens. In he mos plausble scenaro where he correlaon s posve, namely Par (b) of he Proposon, Fg. 3 demonsraes ha an ncrease n absolue rsk averson s assocaed wh a larger expeced value and a smaller varance for a range of dfferen cerany equvalens. RESULS AND DISCUSSION nformaon abou he rsk/reurn characerscs of he proposed nvesmen projec could be accuraely and ransparenly communcaed o he VC durng he solcaon even. If eher or boh of hese nformaon defcs were pallaed, he marke for venure capal would operae more effcenly. How mgh he pallaon be effeced? o he exen ha he venure capal ndusry has he characerscs of a publc good, governmenal nervenon mgh be jusfed as a means of mgang some of he problems posed by he nformaon asymmery. he esablshmen of an nformaon deposory, operaed by a dsneresed governmenal auhory, could help o dssemnae nformaon among nvesors as well as enrepreneurs. hs sudy examnes he rsk-reurn relaonshps n a scenaro wheren an enrepreneur solcs fnancal commmens from a seres of venure capalss. he four proposons derved n hs sudy are based on he sochasc properes of he sequence of solcaon evens as well as on he rsk-audnal characerscs of he enrepreneur. he sudy apples ergodc heorems o derve sascal and behavoral nferences bearng on he parameers of he sochasc processes characerzng he solcaon evens. A man resul of hs sudy s Proposon 1; derves a lmng probably dsrbuon governng he lkelhood ha an enrepreneur wll succeed n solcng from venure capalss he fnancal commmens ha he needs o launch hs venure. Proposons and 3 explo he properes of he sochasc process of solcaon evens o derve nferences bearng on he relaonshp beween he varaon n he seres of capal commmens and he enrepreneur s expecaon of aggregae fnancng. Proposon 4 consrues he ways n whch he sochasc properes of he solcaon evens nfluence he solcaon behavor of he enrepreneur. ha proposon apples he heory of cerany equvalens and he coeffcen of absolue rsk averson o show ha changes n eher of hese audnal parameers wll affec he radeoff beween he rsk and he expeced reurn n he solcaon process. Implcaons for polcy: he polcy mplcaon of he conclusons perans o he problemac consequences of nformaon asymmery. On he one hand, some of he uncerany faced by he enrepreneurs could be aenuaed f nformaon abou he rsk/reurn preferences of VCs were known o he enrepreneurs pror o solcaon. On he oher hand, some of he rsks faced by he VCs could be aenuaed f 04 CONCLUSION he sudy derved four analycal proposons quanfyng he rade-offs beween he rsk and he expeced reurn assocaed wh venure capal fnancng. he polcy mplcaons of he resuls mply he benefs of mgang nformaon asymmery. Some of he rsk faced by he enrepreneurs could be aenuaed f nformaon abou he rsk/reurn preferences of venure capalss were known o he enrepreneurs pror o solcaon. Some of he rsks faced by he venure capalss could be aenuaed f nformaon abou he rsk/reurn characerscs of he proposed nvesmen projec could be accuraely and ransparenly communcaed o he venure capals durng he solcaon even. If eher or boh of hese nformaon defcs were pallaed, he marke for venure capal would operae more effcenly. REFERENCES 1. Adma, A.R. and P. Pflederer, 1994. Robus fnancal conracng and he role of venure capalss. J. Fnance, 49: 371-403. DOI: 10.307/39157. Barry C.B., C.J. Muscarella, W.I.I.I. Peavy and M.R. Vesuypens, 1999. he role of venure capal n he creaon of publc companes. J. Fn. Econ., 7: 447-47. DOI: 101016/0304-405X(90)90064-7 3. Chan, Y.S., 1983. On he posve role of fnancal nermedaon n allocaon of venure capal n a marke wh mperfec nformaon. J. Fnance, 38: 1543-1569. DOI: 10.307/37586 4. Cox, D.R., 196. Renewal heory. 1s Edn., John Wley and Sons, Inc., New York., ISBN: 0410570X, pp: 14.
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