Incorporaing Risk Preferences ino Real Opions Models Mura Isik Assisan Professor Agriculural Economics and Rural Sociology Universiy of Idaho 8B Ag Science Building Moscow, ID 83844 Phone: 08-885-714 E-mail: misik@uidaho.edu May 17, 004 Paper prepared for presenaion a he American Agriculural Economics Associaion Annual Meeing, Denver, Colorado, Augus 1-4, 004 Copyrigh 004 by Mura Isik. All righs reserved. Readers may make verbaim copies of his documen for non-commercial purposes by any means, provided ha his copyrigh noice appears on all such copies.
Incorporaing Risk Preferences ino Real Opions Models Absrac This paper develops a framework o link he expeced uiliy analysis o real opions models in order o capure he join effecs of risk aversion and irreversibiliy associaed wih real invesmens. I aims a modifying he heory of invesmen under uncerainy by incorporaing decision makers risk preferences and allows explicily analyzing he impacs of risk aversion, uncerainy and irreversibiliy on decisions such as invesmen and resource allocaions. I addresses he shorcomings of he commonly used expeced uiliy and invesmen under uncerainy models by generalizing he heory of irreversible invesmen under uncerainy by allowing for risk-averse invesors. We found ha uncerainy, irreversibiliy and risk aversion are imporan deerminans of he opimal iming of irreversible decisions. Ignoring risk preferences in real opions models would lead o over or underesimaion of magniude of invesmens. Key Word: expeced uiliy, invesmen under uncerainy, irreversibiliy, real opions, risk aversion. JEL Classificaion: D81, G1 1
1. Inroducion The assumpion ha a decision maker maximizes expeced uiliy has been a frequenly employed model specificaion. The conribuions of Arrow and Pra led o a large number of mainly heoreical papers concerning economic decisions involving risk and uncerainy. The expeced uiliy framework has been widely used o examine various economic and social problems in economics and in agriculural economics. There now exis a subsanial se of definiions, heorems, and empirical procedures available o hose applying his paradigm (see Meyer (00)). The expeced uiliy framework incorporaes uncerainy and risk preferences of decision makers and has been he foundaion of many economic analyses. A special case of he expeced uiliy framework derived wih he exponenial uiliy funcion is he mean-variance framework, which was popularized by Tobin (1958) for porfolio allocaion decisions. The mean-variance framework has been much used as a basis of various empirical problems such as invesmen in new echnologies and resource managemen decisions in analyzing he impacs of risk aversion and uncerainy. Recen sudies in economics and finance have inroduced he use of real opions models. These heoreical models o capial invesmen decisions of firms depend on he financial opions and have been recenly popularized by Dixi and Pindyck (1994) in modeling irreversible invesmen decisions. These models sress he irreversibiliy of mos invesmen decisions and he ongoing uncerainy of he economic environmen in which hose decisions are made. This new approach recognizes he opion value of waiing for beer informaion. The heory of irreversible invesmen has been much used in boh he empirical research and he heoreical lieraure in analyzing risk-neural decision makers invesmen decisions (Brennan and Schwarz, 1985; Dixi, 1989; McDonald and Siegel, 1985; Majd and Pindyck, 1987; Myers and
Majd, 1990; Pindyck, 1988; Dixi and Pindyck, 1994; Trigeorgis, 1996). One characerisic of he heory of irreversible invesmen under uncerainy is ha i explicily incorporaes he value of waiing or cos of commimen in analyzing decisions ha are a leas parially irreversible. This heory gives decision makers an opporuniy o delay irreversible decisions such as invesmen in new echnologies o learn more abou marke and economic condiions before making irreversible decisions. However, unil now his heory has only been developed in he cases of risk neuraliy using dynamic programming and risk aversion using coningen claims analysis. Coningen claims analysis incorporaes risk (i.e., marke price risk) hrough using risk adjused rae of reurn insead of discoun rae in he analysis. Alhough his procedure adjuss he rae of reurn required according o he variabiliy of reurn in he marke 1, i does no explicily ake ino accoun he decision makers subjecive degree of risk aversion. Individuals are ofen faced wih a variey of flexibiliies (for insance healh coverage opions) which can no be valued properly wihou aking ino accoun subjecive degree of risk aversion. The marke price of risk does no capure his effec. Many invesmen decisions such as wheher o expand he capaciy of heir curren operaions or exi he indusry involve sunk coss of invesmen and uncerainy abou prices, demand, or cos. Mos firms have he opporuniy o delay he invesmen decisions o learn more abou prices, coss, and oher marke condiions before making decisions ha are a leas parially irreversible. Firms currenly operaing also have he opion o choose risk-free invesmen alernaives such as long-erm saving accouns or governmen bonds o diversify heir porfolio. Alhough such invesmen alernaives are considered o be risk free and reduce he variabiliy of 1 The capial asses pricing model is usually used o deermine he risk-adjused rae of reurn as: µ = r + φσδ pm, where r is he risk free ineres rae, or risk and σ is he variabiliy of reurns of he asse. δ pm is he correlaion beween he asse and he marke, φ is he marke price 3
income, firms decisions o enroll in such programs are a leas parially irreversible. An example of such decisions is he farmer paricipaion in he conservaion reserve program in he U.S. Insead of operaing heir risky farming operaions, farmers have he opion o volunarily reire heir land for 10-15 years o receive annual renal paymens. However, hey face a decision wheher o coninue o operae heir risky operaions or paricipae in an irreversible program ha provides risk-free reurns over a fixed period of ime. If farmers paricipae in he program, hey can reduce he variabiliy of heir income. Alhough he paricipaion is likely o reduce he variabiliy of reurns, i involves an irreversible decision. I is herefore imporan o consider boh risk aversion and irreversibiliy of he decision in modeling such decisions. An imporan issue is o deermine which of he heoreical frameworks reviewed above is appropriae in modeling decisions such as invesmen decisions of firms ha are facing a decision o inves in an irreversible program or coninue o operae heir risky operaions. Real opions models and expeced uiliy framework are useful in modeling decision-making under uncerainy for a wide variey of problems. The heory of irreversible invesmen under uncerainy has been criicized because i does no allow decision makers subjecive degree of risk aversion o be explicily incorporaed in decision-making process. Specifically, his heory does no ake ino accoun he impacs of reducions in variabiliy of he firm s porfolio on invesmen decisions. The expeced uiliy framework, on he oher hand, allows decision makers risk preferences and herefore he risk premium o be explicily incorporaed ino he firms decision making. However, he expeced uiliy heory does no consider he imporance of irreversibiliy of invesmen decisions and abiliy o delay irreversible decisions. Anoher example is o decide wheher o inves in he risk free long-erm governmen cerificae of deposi or inves in he sock marke. 4
The purpose of his paper is o develop a framework ha allows he incorporaion of decision makers risk preferences ino real opions (invesmen under uncerainy) models. The framework developed in his paper aims a modifying he heory of irreversible invesmen under uncerainy by incorporaing decision makers risk preferences ino real opions models. Addiionally, i allows explicily analyzing he impacs of risk aversion, uncerainy and irreversibiliy on decisions such as invesmen and resource allocaions. I, herefore, addresses he shorcomings of he commonly used expeced uiliy and invesmen under uncerainy models. This paper conribues o he lieraure on invesmen under uncerainy by generalizing he heory of irreversible invesmen under uncerainy by allowing for risk-averse invesors. The resuls indicae ha uncerainy, irreversibiliy and risk aversion all play an imporan role in deermining he opimal iming of irreversible decisions. Ignoring risk preferences in real opions models would lead o over or underesimaion of magniude of invesmens.. Theoreical Model To illusrae he impac of risk aversion in real opions model, we develop a simple model of decision-making under uncerainy and irreversibiliy. We consider a decision maker who mus decide wheher o coninue o operae he curren operaion of a firm or inves in a riskless porfolio. The decision maker operaing he firm faces wih various sources of uncerainy such as demand, price, or weaher. Therefore, operaing he firm is a risky business. We denoe he expeced presen value of he reurns from he curren operaion a ime as E R ). Le he variance of he reurns be represened by Var R ). The decision maker has he opion of invesing in a risk- free porfolio wih he presen value of reurns V a ime. I is for simpliciy ( ( 5
assumed ha he invesmen in he risk-free alernaive is compleely irreversible 3. Le he discoun rae be represened by ρ. Noe ha he hreshold reurns required from he risk free porfolio o shu down he curren operaion of he firm under risk neuraliy is equal o he expeced reurns from he curren operaion, E R ). We now derive he firm s opimal decision ( rule wih an expeced uiliy model and wih a real opions model. Then, we inroduce an alernaive model ha combines hese wo models o address he shorcomings of hese wo models. Decision wih Expeced Uiliy Framework We firs examine he firm s opimal invesmen sraegy under uncerainy and risk aversion using a uiliy funcion. A risk-averse decision maker maximizes he expeced uiliy of wealh, EU ( R). To deermine he opimal invesmen sraegy, we derive he cerainy equivalen wealh from he expeced uiliy. We firs define he risk premium as he amoun of money ha an individual is willingly o pay o avoid uncerainy of income and ge he expeced value of he income for sure. The risk premium (P) can be derived from U( E( R) P) = EU ( R). Using a second-order Taylor series approximaion, he risk premium as: Var( R) UWW( E( R)) P =, U ( E( R)) W where UWW( E( R)) λ = is called he coefficien of absolue risk aversion evaluaed a he mean U ( E( R)) W wealh W. The cerainy equivalen level of he wealh hen can be wrien as: R CE λ = E( R ) Var( R ). (1) 3 Real world invesmen decisions are much more complex han he case considered here. Many invesmen decisions are a leas parially irreversible and firms could have several alernaives o inves. To focus on he impac of risk preferences on irreversible invesmen decisions, we make several simplifying assumpions in he model. 6
The hreshold reurns in which he decision maker would be indifferen beween choosing o operae he risky operaion of he firm and invesing in he riskless porfolio can be obained * λ from (1) as: V E( R ) Var( R ). Thus, he decision maker would inves in he risk-free porfolio a ime if he reurn of he risk-free porfolio were greaer han he hreshold reurn, λ i.e., V E( R ) Var( R ). This indicaes ha here is a radeoff beween he expeced reurn and reducions in he variabiliy of he reurn. The decision maker is willingly o reduce he expeced income by invesing in he risk-free porfolio in order o reduce he variabiliy of he reurns. The amoun ha he decision maker would be willingly o pay o receive he expeced income for sure is equal o he risk premium. Decision under Uncerainy and Irreversibiliy The firm s invesmen sraegy under uncerainy and irreversibiliy is modeled using wo alernaive approaches: dynamic programming echniques and coningen claims analysis, as in Dixi and Pindyck (1994). We assume ha R is sochasic and evolves according o he following geomeric Brownian moion processes represened by: dr = α Rd + σrdz. () The variance of R, R ) α σ Var (, can be obained from () as: Var( R ) = R0 e ( e 1) The decision problem a each ime is o maximize he ne reurns from he invesmen by choosing an opimal ime as: ρ ( R ) e F( R) = max E V. (3) We can obain wo alernaive soluions o he firm s invesmen decision. Firs, dynamic opimizaion echniques are used o derive he opimal invesmen rule. The Bellman equaion is [ F( )] ρ F ( R) d = E R. Using Io s Lemma o expand he righ-hand side of his expression, F (R). 7
can be shown o saisfy he following differenial equaion 0.5( R F ) + αrf ρf = 0 σ, RR R where F R and F are he derivaives of F (R). We solve his differenial equaion wih respec RR o he following boundary condiions: F ( 0) = 0, F( R) = V R, and ( R) = 1. Solving he differenial equaion subjec o he boundary condiions reveals ha he hreshold reurn o be received a which i is opimal o inves in he risk-free porfolio is given by 4 : F R where < 0 * β 1 V = ( β E R ) β is he smaller roo 0.5 β ( β 1) αβ ρ = 0 β 1 σ. Noe ha > 1 β (4) since β < 0. This decision rule requires he decision maker o inves in he riskless porfolio if he expeced presen value of he reurn from he riskless porfolio ( V ) is greaer han he expeced β 1 reurn from he risky operaion ( E ( R ) ) by a facor > 1. This is because he model of β he invesmen under uncerainy incorporaes he value of waiing or cos of he commimen in he invesmen decisions, requiring he firm o demand a premium o accoun he value of waiing. The exen o which uncerainy and irreversibiliy of he invesmen affec he decisionmaking depends on he magniude of he muliple β β 1. This facor increases wih an increase in σ and/or a decrease in α. Second, we use coningen claims analysis o incorporae risk using risk-adjused rae of reurn (µ ) insead of he exogenously given discoun rae in he analysis (as in Dixi and 4 The model developed here can be generalized o he case where boh V and R are sochasic and follow as geomeric Brownian moion. In ha case, β 0 is he smaller roo of 0.5 ( σ γσ σ + σ ) β ( β 1) + ( α α ) β ( ρ α ) V V R covariance beween changes in R and V. R V R R < =0, where he parameer γ represens he 8
Pindyck (1994)). The soluion o he above problem is similar o he dynamic programming, wih β being he smaller roo 0.5σ β ( β 1) ( r µ + α) β r = 0 < 0, where r is he risk free rae of reurn. If he discoun rae is equal o he risk free rae of reurn, he coningen claims soluion o he invesmen decision is equal o he dynamic programming soluion. Thus, opion price heory incorporaes he marke risk ino he model by using risk adjused rae of reurn insead of he discoun rae. However, his model does no incorporae subjecive degree of risk aversion. This decision rule derived using he real opions model explicily incorporaes he value of waiing ino he firm s invesmen decisions. However, i does no explicily ake ino accoun he decision maker s subjecive degree of risk aversion in deermining wheher o inves as well as he rade-off beween he expeced reurns and he variabiliy of reurns because i assumes risk neuraliy. The decision rule under uncerainy and irreversibiliy presened above is he opposie of ha wih he expeced uiliy framework. The hreshold reurn required o inves in he risk-free alernaive under he opion value framework is higher han ha under uncerainy and risk aversion of individuals. Decision under Uncerainy, Irreversibiliy, and Risk Aversion We now explicily incorporae he degree of risk aversion, uncerainy abou he reurns and irreversibiliy of he invesmen ino he firm s decision making by combining he wo alernaive models presened above. The decision problem of he firm is o maximize he cerainy equivalen reurns from he invesmen in he irreversible risk-free porfolio by choosing an opimal ime as: λ ρ Var( R e F( R) = max E V R ). (5) Alhough his model is similar o (3), i allows incorporaing he subjecive degree of risk aversion and exends ha model beyond risk neuraliy using uiliy funcions. Unil now, 9
dynamic programming has only been applied o he problem of irreversibiliy under he assumpion of risk neuraliy or under he marke price risk. In his paper we consider he economically relevan problem faced by risk-averse invesors who conemplae an irreversible invesmen in an asse. Use of he dynamic programming mehods described above reveals ha he hreshold reurns o be received a which i is opimal o inves is given by: V * β 1 λ = E( R ) Var( R ). (6) β The hreshold reurn in (6) incorporaes he impacs of boh risk aversion and he value of waiing on he invesmen decisions. An increase in risk aversion and/or variabiliy of he reurn increases he risk premium and herefore decreases he hreshold reurn required o inves in he risk-free porfolio given in (6). This herefore encourages he invesmen in he risk-free porfolio β 1 by risk-averse firms. On he oher hand, an increase in he value of waiing (i.e., ) β increases he hreshold reurn and herefore discourages he invesmen in he risk-free porfolio. Thus, here is a rade-off beween he value of waiing and reducions in he variabiliy of reurn in he porfolio. Equaion (6) akes ino accoun boh hese effecs in evaluaing he firm s invesmen decisions. The hreshold reurn under boh risk aversion and irreversibiliy of he invesmen given in (6) is greaer han ha under only risk aversion. The hreshold in (6) is also greaer han ha β 1 λ under cerainy and risk-neuraliy ( R ), if E( R ) β Var( R ) β >. These resuls indicae ha he invesmen rule under boh risk aversion and irreversibiliy of he decision is differen han ha under only risk aversion or under only cerainy and risk-neuraliy. These resuls also imply ha ignoring risk preferences in developing real opion opions models can 10
lead o over or underesimaion of magniude of invesmens. These resuls indicae how incorporaing subjecive degree of risk aversion changes he naure of he opimal invesmen rule. I is herefore imporan o incorporae he subjecive degree of risk aversion in modeling irreversible invesmen decisions. 3. Numerical Example We now examine he exen o which risk aversion, uncerainy and irreversibiliy affec invesmen decisions of acive firms by providing a numerical example. In his example, we consider a firm ha is currenly operaing a widge facory and decides wheher o coninue o operae he widge facory or inves in a risk-free porfolio. To keep maers as simple as possible, we assume ha he facory produces one widge per year forever wih zero operaing cos. Currenly, he expeced ne presen value of invesmen over he cos of invesmen is assumed o be $00, bu nex year he price will change hus he ne prese value of he invesmen could change. I is assumed ha he firm has an opion o inves in a risk-free alernaive ha pays $185 (he ne presen value of invesmen). However, his invesmen decision is assumed o be irreversible 5. We assume ha he risk-free rae of ineres is 10%. I is simpliciy assumed ha he uiliy funcion is represened by a negaive exponenial funcion: U λr = e, where λ is he risk aversion coefficien. Given ha R is normally disribued, λ cerainy equivalen level of income can be wrien as: RCE = E( R ) Var( R ). The risk aversion coefficien λ is assumed o be 0.015. The parameers used in he numerical simulaion are presened in Table 1. Table presens he firm s alernaive invesmen sraegies under various models 5 I is, however, possible o consider a more realisic scenario in which he invesmen decision is parially irreversible. In ha case, he impac of uncerainy and irreversibiliy on invesmen would be less han he case of complee irreversibiliy. 11
presened above. Under cerainy and risk neuraliy, he firm does no inves in he risk-free porfolio because he expeced ne prese value of operaing he widge facory ($00) is greaer han he expeced ne presen value of he earnings from he invesmen in he risk-free asse ($185). Thus, he firm s opimal decision is o coninue o operae he widge facory under he assumpion of cerainy and risk neuraliy. We hen calculae he criical values a which is opimal o inves in he risk-free asse under uncerainy and risk aversion for various values of Var (R). The criical values a which i is opimal o inves presened in Table indicae ha a risk-averse firm would inves in he riskfree asse in mos of he cases examined here. The reason is ha he cerainy equivalen level of he reurns akes ino accoun he risk premium and herefore he required hreshold reurns are much lower han hose under cerainy and risk-neuraliy. This is because a risk-averse firm would accep less reurn in order o reduce he variabiliy of he reurns and herefore is likely o inves more in he risk-free asse. Under he real opion model, he criical values a which i is opimal o inves in he riskfree porfolio are higher han he expeced ne prese value from he invesmen in he risk-free porfolio ($185). Thus, he firm would delay he invesmen decision in he risk-free porfolio. Insead, he firm coninues o operae he widge facory because i akes ino accoun he irreversibiliy of he invesmen decision and he value of waiing. On he oher hand, under boh he risk aversion and irreversibiliy of he invesmen, we ake ino accoun he risk preferences of he decision maker as well as he radeoff beween he expeced reurn and he variance of reurn and he impac of value of waiing in he decision-making. In his case, as he variance of he reurn of he widge facory increases, he criical value a which i is opimal o inves in he riskless porfolio decreases significanly and herefore he firm decides o inves in he riskless 1
porfolio. This framework allows he risk-averse firm o ake ino accoun he radeoff beween he reducions in he variabiliy of he firm s reurns and he irreversibiliy of he alernaive invesmen opion. Thus, he framework developed in his paper incorporaes he wo imporan characerisics of he commonly employed expeced uiliy and invesmen under uncerainy models. These resuls emphasize he imporance of incorporaing risk preferences in real opions models. 4. Conclusions Two imporan economic models, he expeced uiliy framework and invesmen under uncerainy, have been widely used o examine various economic and social problems in economics and in agriculural economics involving uncerainy and irreversibiliy. In his paper, we consider he appropriaeness of hese models in modeling acive firms irreversible decisions such as invesmen and porfolio allocaions. We develop an alernaive model ha addresses he shorcomings of he expeced uiliy and real opions models. The model developed in his paper combines he wo imporan characerisics of hese heoreical models, risk preferences of decision makers and he irreversibiliy of invesmen decisions. The paper makes conribuions o he lieraure by generalizing he heory of irreversible invesmen under uncerainy by allowing for risk-averse invesors and by showing how incorporaing subjecive degree of risk aversion changes he naure of he opimal invesmen rule. The resuls indicae ha uncerainy, irreversibiliy and risk aversion all play significan role in deermining he opimal iming of irreversible decisions such as invesmen and resource allocaions. Under he expeced uiliy framework, a risk-averse firm would inves in he risk-free asse in mos of he cases examined in he numerical examples because he cerainy equivalen level of he reurns akes ino accoun he risk premium. Under he real opion framework, he 13
criical values a which i is opimal o inves in he risk-free porfolio are higher han he expeced ne prese value of he alernaive invesmen because his framework akes ino accoun he value of waiing. The model developed here akes ino accoun he radeoff beween he reurn and variance of reurn and he impac of value of waiing in he decision-making. As he variance of he reurn of he widge facory increases he firm decides o inves more in he riskless porfolio o reduce he variabiliy of income. These resuls underline he imporance of incorporaing he degree of risk aversion ino real opions models. Our resuls show ha ignoring risk preferences in real opions models can lead o significan over or underesimaion of magniude of invesmens. The model developed in his paper can be helpful in analyzing risk-averse decision makers irreversible decisions in economics and finance. Furher research in his area is needed o incorporae many imporan feaures of he real world invesmen decisions under uncerainy. Empirical applicaions of he model developed in his paper are also needed o deermine he exen o which risk aversion and irreversibiliy of he invesmen impac he invesmen decisions of firms in various indusries. 14
Table 1. Parameers Used in he Numerical Example Example # a s Var (R) b b 1 1 0.05 0.15 1005.9 1.19 0.05 0.30 4163.1 1.65 3 0.05 0.45 99.7.9 4 0.05 0.60 19156.1 3.14 15
Table. Criical Values a Which I is Opimal o Inves in he Riskless Porfolio Example # Cerainy and Risk Neuraliy Uncerainy and Risk Aversion Real Opion Real Opion Wih Risk Aversion (Dynamic Programming) Dynamic Programming Coningen Claims Analysis* 1 00.0 19.5 38.7 3.8 9.7 00.0 168.8 39.3 309.3 77.9 3 00.0 15.6 458.9 487.1 88. 4 00.0 56.5 68. 771.8 176.9 * µ = 0. 10, r = 0. 05. 16
References Brennan, M.J., and E.S. Schwarz, 1985, Evaluaing Naural Resource Invesmen, Journal of Business, 58, 153-157. Dixi, A., 1989, Enry and Exi Decisions under Uncerainy, Journal of Poliical Economy, 97, 60-638. Dixi, A. K., and R.S. Pindyck, 1994, Invesmen under Uncerainy, Princeon Universiy Press, Princeon, NJ. Majd, S., and R. Pindyck, 1987, Time o Build, Opion Value, and Invesmen Decisions, Journal of Financial Economics, 18, 7-7. McDonald, R., and D. Siegel, 1985, Invesmen and Valuaion of Firms When There is an Opion o Shudown, Inernaional Economic Review, 6, 331-349. Meyer, J., 00, Expeced Uiliy as a Paradigm for Decision Making in Agriculure, in A Comprehensive Assessmen of he Role of Risk in U.S. Agriculure, edied by Richard E. Jus and Rulon D. Pope, pp -19, Kluwer Academic Publishers. Myers, S.C., and S. Majd, 1990, Abandonmen Value and Projec Life, Advances in Fuures and Opions Research, 4, 1-. Pindyck, R.S., 1988, Irreversible Invesmen, Capaciy Choice and he Value of he Firm, American Economic Review, 79, 969-985. Tobin, J., 1958, Liquidiy Presence as Behavior Toward Risk, Review of Economic Sudies, 5:65-86. Trigeorgis, L., 1996. Real Opions, Cambridge, MA, MIT Press. 17