INVESTMENT UNDER UNCERTAINTY: STATE PRICES IN INCOMPLETE MARKETS

Transcription

1 INVESTMENT UNDER UNCERTAINTY: STATE PRICES IN INCOMPLETE MARKETS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ENGINEERING-ECONOMIC SYSTEMS AND OPERATIONS RESEARCH AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Thomas Edward Hoff Augus, 997

2 Copyrigh by Thomas E. Hoff, 997 All Righs Reserved ii

3 I cerify ha I have read his disseraion and ha in my opinion i is fully adequae, in scope and in qualiy, as a disseraion for he degree of Docor of Philosophy. John P. Weyan (Principal Advisor) I cerify ha I have read his disseraion and ha in my opinion i is fully adequae, in scope and in qualiy, as a disseraion for he degree of Docor of Philosophy. James L. Sweeney I cerify ha I have read his disseraion and ha in my opinion i is fully adequae, in scope and in qualiy, as a disseraion for he degree of Docor of Philosophy. David G. Luenberger Approved for he Universiy Commiee on Graduae Sudies: iii

4 Absrac An invesmen s price is he sae-price weighed sum of is fuure payoffs when markes are complee. This is a well esablished fac in he financial economics field. The invesmen valuaion problem in incomplee markes, however, has araced he ineres of several addiional fields, including decision analysis and real opions, and has led o a variey of valuaion approaches. The approaches differ in heir reamen of he invesor s exising porfolio, marke opporuniies o hedge risks, he abiliy o re-opimize afer adding a new invesmen o one s porfolio, and he invesmen s divisibiliy. This research develops a single valuaion approach ha produces resuls consisen wih financial economics when markes are complee bu is also applicable when he invesmen is no divisible and he decision maker can only borrow and lend a he risk-free discoun rae (markes are incomplee). Resuls sugges ha, given a ime- and sae-separable uiliy funcion, a decision maker s buying price for an invesmen is approximaely equal o he sae-price weighed sum of is fuure payoffs (see Theorems. and 3.); resuls are exac when markes are complee or he uiliy funcion is exponenial. Sae prices in incomplee markes have a similar definiion as sae prices in complee markes in ha hey are approximaely marginal uiliy based prices. The approach may have compuaional advanages because he prices can be esimaed wihou solving a full uiliy maximizaion problem for each new invesmen ha is evaluaed. The mos imporan applicaion of his work will be in he evaluaion of projecs ha are indivisible and have managerial flexibiliy in markes ha are incomplee (i.e., an imporan caegory of real opion problems). A comprehensive example of an elecrical uiliy s use of disribued generaion o provide sysem capaciy (raher han he ypical approach of upgrading ransmission and disribuion faciliies) illusraes how o apply he mehod. The example shows ha he correlaion beween a new projec s payoffs and he exising porfolio has a large effec on he invesmen decision. iv

5 Acknowledgemens Many people have helped me during my sudies a Sanford. The mos imporan is my advisor John Weyan. John has been a caring, commied, and knowledgeable advisor and a good friend hroughou my ime a Sanford. My commiee members, Jim Sweeney and David Luenberger, have also been helpful. Jim Sweeney se a good example of o wed academic heories wih real life problems, encouraged me o produce my bes research, and provided good feedback and direcion hroughou my research. I have benefied from David Luenberger s leadership in he invesmen science area as well as from his oher books. I was helped by discussions a various sages of my research wih people from Sanford including Darrell Duffie, Ron Howard, Blake Johnson, Karl Knapp, Gil Masers, Ross Shacher, and Tom Varner, and hose ouside he universiy including Shimon Awerbuch (Independen Economis), Seve Chapel (Elecric Power Research Insiue), Chuck Feinsein (Sana Clara Universiy), Spiros Marzoukas (George Washingon Universiy), Pee Morris (Applied Decision Analysis), Ren Orans (Energy and Environmenal Economics), Jamie Reed (Incenives Research), Tom Ruherford (Universiy of Colorado a Boulder), Jim Smih (Duke Universiy), Elizabeh Teisberg (Universiy of Virginia), and Howard Wenger (Pacific Energy Group). Saff members Maggie Barsow Taylor, Susan Clemen, Nancy Florence, Danielle Herrmann, Pam McCrosky, and Roz Morf have been supporive hroughou he process. Above all, I am hankful o God for he opporuniy o have come o Sanford wih he suppor of friends and family. A special hanks o my wife, Elaine, my children, Rachel, Jonahan, David, and Samuel, and my parens, Sam and Jo Hoff. v

6 Table of Conens. Inroducion.... Background.... Financial Economics..... Valuaion by Arbirage Equilibrium Valuaion Valuaion by Single-Agen Opimaliy Real Opions Decision Analysis Mehod Comparison Expeced Uiliy Resuls can be Inconsisen wih a Marke Valuaion Expeced Uiliy Theory Lacks a Temporal Elemen Expeced Uiliy Theory Does No Allow for Prior Cons. Changes....6 Objecive...3. Single-Period Model...4. Seing...4. Indifference Condiion Opimaliy Condiion Buying Price Muli-Period Model Seing Necessary Condiions Buying Price Buying Price Wihin a Pariion Applicaion: Invesmen in Disribued Elecriciy Resources Expeced Disribued Generaion Cos Incorporaion of Risk Aiiude Demand Uncerainy Demand Uncerainy and Firm Value Uncerainy (Incomplee Markes) Demand Uncerainy and Firm Value Uncerainy (Parially Complee Markes) Comparison of Resuls Conclusions and Fuure Research Appendix: Proofs Appendix: Pariions and Time- and Sae-Separable Uiliy Funcions Bibliography...58 vi

7 Lis of Figures Figure -. Focus of various fields when evaluaing invesmens... Figure -. Iniial condiion and opimal cons./wealh wih and wihou ne payoffs....6 Figure -. Opimal consumpion/wealh wih and wihou ne payoffs (Example.)...8 Figure 3-. Uncerain saes (numbers) and pariions (ovals) of S....4 Figure 3-. Two ways of delineaing he ne payoffs...6 Figure 3-3. Opimal consumpion wih and wihou ne payoffs....7 Figure 3-4. Illusraion of Assumpion Figure 3-5. Esimaed and exac buying price for logarihmic uiliy funcion Figure 4-. Simple Example...38 Figure 4-. Firm s profis and disribued generaion coss...43 Figure 7-. Projec payoffs and resoluion of uncerainy wih expeced uiliy Figure 7-. Consumpion and resoluion of uncerainy wih expeced uiliy...56 vii

8 . Inroducion. Background Idenifying beer ways o evaluae invesmens is an aciviy in which here is widespread ineres. I is of ineres o individuals wih very limied resources as well as o governmens wih vas financial resources. Individuals face invesmen problems when purchasing a washing machine, when deciding wheher o inves in a college educaion, when invesing in he sock marke, and when buying a home. Firms face invesmen problems when designing heir research and developmen programs, when inroducing new producs, and when making be-he-company ype decisions. Governmens face invesmen problems when designing welfare programs, when supporing he developmen of new echnologies, and when deciding wheher or no o ener ino an inernaional confronaion. This ineres in invesmen problems has araced he aenion of several fields of research and has led o a variey of valuaion heories and approaches. Two of he mos imporan fields are financial economics and decision analysis/decision sciences. A hird ha is growing in imporance is he field of real opions. One issue ha ends o disinguish hese hree fields is heir focus. As illusraed in Figure., financial economics focuses on he marke. I provides a marke-based valuaion of an invesmen by examining all invesmen opporuniies ha are available in he marke. Decision analysis focuses on he decision maker. Lile hough is ypically given o he marke and how i affecs he decision. Real opions focuses on he invesmen. Some applicaions of real opions are concerned wih how he invesmen ineracs wih he marke while ohers are only concerned abou he decision maker. In some ways, i can be viewed as a field ha is in beween financial economics and decision analysis.

9 Financial Economics Focus on Marke Real Opions Focus on Invesmen Decision Analysis Focus on Decision Maker Figure -. Focus of various fields when evaluaing invesmens. This chaper provides a brief discussion of hese fields wih a focus on some of he issues ha will apply o he res of his work.. Financial Economics The exisence of sae prices is he unifying principle in asse pricing in financial markes. If sae prices exis, here is one price for each sae of he world a each dae. Sae prices ake ino accoun uncerainy and discouning over ime. A sae price is he curren price of a securiy ha pays off $ if a paricular sae occurs in he fuure and pays off $ in all oher saes. The price of any securiy is deermined according o hese sae prices and he securiy s cash flows. Securiy price is he sae-price weighed sum of is fuure cash flows. Tha is, one muliplies he sae price imes he securiy s cash flow for each sae and for each dae and hen sums he resuls. Deerminaion of sae prices is based on he consrains placed on asse prices. The hree basic consrains on asse prices are he absence of arbirage (i.e., here is no way o lock in a risk-free profi by simulaneously enering ino wo or more marke ransacions), marke equilibrium, and single-agen opimaliy. Sae prices exis when any of hese hree consrains is saisfied. Tha is, he absence of arbirage, marke equilibrium, and singleagen opimaliy each imply he exisence of a se of sae prices and complee markes (Duffie 99). The following subsecions describe how o deermine sae prices using hese hree consrains.

10 .. Valuaion by Arbirage Valuaion by arbirage deermines he prices of derivaive securiies based on he prices of oher securiies ha are raded in he marke. (A derivaive securiy is a securiy whose value depends on he values of oher more basic underlying variables). The arbirage approach akes he price processes of a se of marke-raded securiies as given, demonsraes ha hey are free from arbirage, and hen uses his arbirage-free se of prices o price he derivaive securiy. The bes known opion pricing formulas based on arbirage are hose developed by Black and Scholes (973) and he binomial opion pricing mehod developed by Cox, Ross, and Rubinsein (979). The Black-Scholes approach prices a sock opion by creaing a dynamically hedged porfolio ha consiss of he underlying sock and a riskfree asse. The porfolio is seleced so ha is cash flow is idenical o he opion s cash flow in every sae. Securiies wih idenical cash flows have he same price o avoid arbirage. Thus, he opion s price is idenical o he porfolio s price. The binomial opion pricing approach () generaes he disribuion of he fuure sock price based only on he risk-free rae and he sock s volailiy (he risk premium of he sock is irrelevan); () calculaes he expeced value of he sock price minus he exercise price in he range where his value is posiive using a se of risk-neural probabiliies; and (3) discouns he resul a he risk-free rae. The sae prices are idenical o he riskneural probabiliies imes he risk-free discoun rae. Harrison and Kreps (979) demonsrae ha maringales provide a direc way o price derivaive asses based on hese prices. Y is a maringale if he expeced value of Y a any fuure ime given he curren available informaion equals he curren value of Y. More < 6 A ; if EY6 s Y6 precisely, a process Y = Y ; =,,..., T is a maringale adaped o an informaion srucure F = = T F = for all s where E F is he expecaion condiional on F. The probabiliies in he expecaion ha make Y a maringale are he sae prices normalized by he risk-free discoun rae. The necessary and sufficien condiion for price processes no o admi arbirage opporuniies is ha hey are relaed o maringales hrough a normalizaion and change of 3

11 probabiliy (Duffie 99 and Huang and Lizenberger 988). Wha is he normalizaion ha makes a coningen claim (e.g., an opion on a sock) ino a maringale? Several auhors have shown ha any coningen claim on an asse can be priced in a world wih sysemaic risk by replacing is acual growh rae wih a cerainy-equivalen rae (i.e., is risk-free rae) and hen behaving as if he world were risk neural (Cox and Ross 976, Consaninides 978, and Cox, Ingersoll, and Ross 985). Tha is, he normalizaion in a marke conex is ha a non-dividend paying sock earns a he risk-free rae (regardless of is risk), he expecaion of he normalized sock price minus he exercise price is aken as if he agen is risk-neural, and he resul is discouned a he risk-free discoun rae. In his case, he sae prices are idenical o he risk-neural probabiliies imes he risk-free discoun rae. Sae prices in a wo-period world can be deermined using an arbirage approach as follows. Suppose ha he number of linearly independen securiies (N) equals he number S of saes (S). The period sae-price vecor is ψ = ψ ψ... ψ T ; he period securiy-price vecor is q = q q q N T... ; and he period securiy cash flow marix D is an N by S marix where elemen D ij is he cash flow of securiy i in sae j. A sysem ha is free from arbirage requires ha q = Dψ. D is inverible because he securiies are linearly independen and markes are complee (i.e., S = N). Thus, he sae price vecor equals he inverse of he cash flow marix imes he period securiy prices, or ψ = D q... Equilibrium Valuaion While valuaion by arbirage is he mos widely used approach o pricing derivaive securiies in financial markes, i is no he only approach. Anoher approach is equilibrium valuaion. This approach was pioneered by Arrow (964) and Debreu (959). Rubinsein (976) demonsraed how equilibrium valuaion is linked o valuaion by arbirage. Equilibrium valuaion in a pure rade economy wih complee markes begins wih a group of agens. Each agen has an iniial endowmen and a sricly increasing, sricly 4

12 concave, differeniable uiliy funcion, and opimizes consumpion using a rading sraegy such ha he marke is in equilibrium. The associaed equilibrium allocaion is Pareo opimal. This means ha here is no oher feasible allocaion ha makes all agens a leas as well off and a leas one agen beer off. Since here is an equilibrium ha is Pareo opimal, here is no arbirage, and herefore here is a se of sae prices (Duffie 99). The equilibrium approach differs from he arbirage approach in ha no prices are given iniially. Raher, all prices in he economy are deermined hrough an economic opimizaion. The resul of he opimizaion is a se of sae prices based on marginal uiliies. Huang and Lizenberger (988) show ha when he allocaion of sae coningen claims is efficien and individuals have ime-addiive, sae-independen, sricly increasing, sricly concave, and differeniable uiliy funcions, sae prices are deermined as if here were a single individual in he economy endowed wih he aggregae endowmen. Sae prices are he se of prices ha make he represenaive agen s iniial endowmen he opimal consumpion choice. This problem is solved by maximizing he represenaive agen s uiliy maximizaion problem subjec o is endowmen, = e e R R S, e; +, e + B. The firs order necessary condiions of he maximizaion problem Max u z 6+ p uz6 subjec o he consrain ha ψ z ψ ψ z,z + ψ z ψ e + ψ e along wih marke clearing condiions require ha i i i 38 6 pu' e = u ' e for i S (p is he probabiliy vecor p = p p p S where p i is he probabiliy ha sae i will occur). Summing he firs order necessary condiions over all i and reaing ψ as he numeraire resuls in he raio of he period expeced marginal uiliy divided by he period marginal uiliy being equal o he sum of sae 6 6 p u' e prices. Tha is, ψ = u ' e. The sum of sae prices equals he price of a risk-free asse (i.e., ha asse has a cash flow of $ in all saes in period ), so ha ψ =ψ, where ψ is he risk-free discoun rae. This and he previous equaion are subsiued 5

13 ino he firs order necessary condiions o obain he se of sae prices, namely ha i ψ 38 6 pu' e i i = ψ p u' e 38 6 for i S. The se of pu i e i ' p u ' e he risk-neural probabiliies because S i= i i pu' e p u ' e 38 6 risk-neural probabiliies imes he risk-free discoun rae...3 Valuaion by Single-Agen Opimaliy for i S are inerpreed as =. Thus, he sae prices equal he The leas common approach o deermine sae prices is hrough he use of single agen opimaliy. This approach deermines sae prices by examining individual consumer behavior. I urns ou ha his may be he mos imporan approach from he perspecive of his research. Given ha an agen wih a sricly increasing uiliy funcion is consuming opimally and markes are complee, he raio of wo sae prices equals he raio of he respecive marginal uiliies. The mechanics and resuls of his approach are very similar o he equilibrium valuaion approach. The primary difference in he resul is ha he individual agen s uiliy funcion replaces he represenaive agen s uiliy funcion and individual agen consumpion replaces marke endowmen in he sae prices. The consumer s uiliy maximizaion problem can be formulaed by allowing he consumer o purchase some porfolio of marke-raded securiies ha have saeconingen payoffs ha resul in he opimal consumpion over ime. Alernaively, he problem can be formulaed by aking he se of sae prices as given and hen allowing he consumer o opimize sae-by-sae consumpion; ha is he approach aken here. The firs order condiions for opimaliy for he problem Max U c, c c 6 subjec o he, c consrain ha c + ψ c e + ψ e are U λ c 6 = and U i ψ λ c i 38 = for i S. Luenberger (997) shows ha a similar resul can be obained when markes are incomplee as long as he invesmens are infiniely divisible. The prices could be calculaed using valuaion by arbirage. 6

14 Summing over all i, U i resul ha ψ = ψ previous secion (i.e., U c form as before (i.e., ψ c 6 ψ = λ = ψ λ. The binding budge consrain leads o he U i 38 c. If he uiliy funcion is an addiive expeced uiliy as in he U c 6,c u c p u c i i pu ' i 38 c = ψ p u' c6 = + ) hen he sae prices have he same ). Once again, he sae prices can be inerpreed as he risk-neural probabiliies imes he risk-free discoun rae..3 Real Opions Financial opion evaluaion mehods have more recenly been applied o evaluae he flexibiliy associaed wih physical invesmens. Some have labeled his exension real opions. Real opion evaluaions accoun for he value of flexibiliy embedded wihin projecs. Like he field of financial opion valuaion, his is a large and growing field. 3 The field of real opions is an imporan one from he perspecive of his research because i represens a field ha is beween financial economics and decision analysis. The field of real opions focuses on he invesmen and how o capure he value of he flexibiliy of he invesmen. This focus is bes illusraed by Dixi and Pindyck (994). The auhors ake wo approaches o evaluaing invesmens. One is a coningen claims analysis and he oher is a dynamic programming analysis. The coningen claims analysis, which is essenially he same as a financial economics opions valuaion, consrucs a risk neural porfolio and applies he principle of no arbirage o value he invesmen. The criical assumpion implici in he coningen claims analysis is ha sochasic changes in he invesmen s value are spanned by exising asses in he economy. This requires ha capial markes are sufficienly complee so ha a dynamic porfolio of asses could be consruced whose price is perfecly correlaed wih he value of he invesmen. This is a crucial poin, 7

15 because his assumpion is widely made when evaluaing real opions. For example, his is a fundamenal assumpion in he book Real Opions by Trigeorgis (996). Dixi and Pindyck value an invesmen using a dynamic programming approach when spanning condiions do no exis. The applicaion of his approach saes ha, over a shor inerval of ime, he oal expeced reurn of he invesmen opporuniy is equal o is expeced rae of capial appreciaion. According o Dixi and Pindyck (994, p. 47), a difficuly of his approach is ha i is based on an arbirary and consan discoun rae. I is no clear where his discoun rae should come from, or even ha i should be consan over ime. Dixi and Pindyck (994, p.5) elaborae on his criical poin (wih he emphasis being mine): Hence, he coningen claims soluion o our invesmen problem is equivalen o a dynamic programming soluion, under he assumpion of risk neuraliy (ha is he discoun rae [of he dynamic programming approach] is equal o he risk-free rae). Thus, wheher or no spanning holds, we can obain a soluion o he invesmen problem, bu wihou spanning, he soluion will be subjec o an assumed discoun rae. In eiher case, he soluion will have he same form, and he effecs of changes [in cerain key variables] will likewise be he same. One poin is worh noing, however. Wihou spanning, here is no heory for deermining he correc value for he discoun rae (unless we make resricive assumpions abou invesors or managers uiliy funcions). The CAPM, for example, would no hold, and so i could no be used o calculae a riskadjused discoun rae in he usual way. The implicaion of his is ha, while you can use he dynamic programming approach when markes are incomplee, here is no heoreically correc way o selec he correc discoun rae..4 Decision Analysis The field of decision analysis (and decision sciences) focuses on he decision maker. A wide range of heories are capured under his ile. Some of hese heories include expeced uiliy, subjecive expeced uiliy, prospec heory, rank-dependen uiliy, saedependen subjecive expeced uiliy, ec. I would be difficul o give an adequae reamen o all of hese heories in his secion. 3 See he bibliography in a recen book on he subjec (Dixi and Pindyck, 994) or Trigeorgis (996). 8

16 The major poin of his secion is ha all of hese heories are unied by heir focus on he decision maker. Wheher he heory is normaive in ha i is inended o sugges how decisions should be made or wheher i is descripive in ha i is inended o describe how decisions are acually made, he cenral focus of all of hese heories is on he decision maker. Two heories are briefly discussed here: expeced uiliy (and subjecive expeced uiliy) and prospec heory. The expeced uiliy heories have been around he longes (Bernoulli 738, and von Neumann and Morgensern 944) and are probably sill he mos widely used. Fishburn (98) presens an excellen summary of he various axiomaic approaches o arrive a he expeced uiliy heories. Howard (99, pp ) akes anoher approach in his ariculaion of he foundaions of decision analysis in five rules of hough. While here are a variey of ways he parameers in he expeced uiliy funcion can be expressed, a ypical one is ha he uiliy of a projec wih payoffs of x equals S i i pu3w + x8, where p i is he probabiliy of he payoff x i occurring and w is iniial i= wealh. One wans o deermine he cerain equivalen CE ha makes he decision maker s uiliy wih he payoffs he same as he uiliy wih he cerain equivalen (i.e., S 6 3 i8). i= i uw + CE = puw + x A second heory ha is of paricular ineres o his research is prospec heory (Kahneman and Tversky 979). Prospec heory was developed in a response o observed descripive violaions of expeced uiliy heory. Prospec heory suggesed ha he payoffs x be evaluaed using a funcion of he form S i= i i ϕ p u x, where ϕ3p 8 i is a decision weigh; fuure work exended he definiion of he decision weigh o depend on oher variables as well as wheher here was a loss or a gain (Hogarh and Einhorn 99, Tversky and Wakker 995). While proponens of prospec heory sae ha he decision weighs are no probabiliies, he improved descripive capabiliy of he inclusion of decision weighs will be seen o be of value laer in his research. 9

17 .5 Mehod Comparison To se he sage for he conribuion ha his research makes, i is imporan o idenify some of he weaknesses associaed wih decision analysis as i is ypically applied. As saed above, i would be difficul o do his for all of hese heories. Forunaely, he hree primary weaknesses idenified in he following secions are applicable o essenially all of he heories. For his reason, he basis of comparison is expeced uiliy heory..5. Expeced Uiliy Resuls can be Inconsisen wih a Marke Valuaion Firs, he valuaion resuls from an expeced uiliy approach are no necessarily consisen wih a financial economics approach. Consider he following illusraion. Suppose ha a risk-averse decision maker is evaluaing hree projecs. Projec X pays x, which is greaer han, wih probabiliy p else i pays xp, ;, p6. Projec Y pays y, which is greaer han, wih probabiliy -p else i pays, py ;, p6. Projec X+Y is he sum of projec X and projec Y so ha i pays x wih probabiliy p and y wih probabiliy -p xpy, ;, p6. Le V X be he value for he financial economics approach and he cerain equivalen for he expeced uiliy approach of projec X. The value of wo projecs combined equals he values of he individual projecs summed ogeher when using a financial economics approach, i.e., VX+ Y= VX+ VY. The cerain equivalen of wo projecs combined, however, does no necessarily equal he cerain equivalens of he individual projecs summed ogeher when using an expeced uiliy approach. The financial economics approach sars wih he assumpion of complee markes. This means ha a sae-price vecor exiss. Thus, he sae-price weighed sum off he payoffs for projec X is VX = x ψ ψ = ψ x. Likewise, V = Y y = ψ ψ ψ y and VX+ Y= x y ψ ψ = ψ x+ ψ y. This means ha VX+ Y= VX+ VY. The expeced uiliy approach requires ha uw + VX6= puw + x6+ puw 6 6 and Y for projecs X and Y. This means ha uw + V = puw + puw + y

18 6 6 6 and VY = u pu w + p u w + y w VX = u pu w + x + p u w w Suppose ha VX+ Y= VX+ VY. Subsiuing for VX + VY, adding w o boh sides, and X+ Y6 = B. The expeced uiliy approach requires ha X+ Y for he projec X+Y. The previous equaion aking he uiliy resuls in uw + V = uu puw + x + puw + u pu w + p u w + y w uw + V = puw + x + puw + y simplifies o his only when ua+ B = ua+ ub This is he case when u is linear and he decision maker is risk-neural. This violaes he iniial assumpion ha he decision maker is risk-averse. Thus, an expeced uiliy approach for a risk-averse decision maker canno obain a value for all hree of hese projecs ha is consisen wih a financial economics approach..5. Expeced Uiliy Theory Lacks a Temporal Elemen Second, an expeced uiliy formulaion lacks a emporal elemen alhough he parameers used in he analysis are ofen from differen ime periods. Specifically, he expeced uiliy analysis evaluaes he expeced uiliy of wealh (a he ime when he decision is made) plus he projec s payoff (a he ime when he uncerainy is resolved). A difficuly wih his formulaion is ha he exisence of uncerainy in any decision problem requires ha here is some period of ime beween when he decision is made and he uncerainy is resolved (Pope 985). For example, if he decision is made in period and he uncerainy is resolved in period, wealh is aken a period bu he projec s payoff is aken a period. One soluion o his problem is o ake wealh and payoff from he same period wih he appropriae period being he ime when he payoff occurs. Tha is, wealh is aken from period raher han period. An implicaion of his is ha wealh in he expeced uiliy formulaion is no longer cerain bu can vary. The payoff iself is no longer he key facor. Raher, i is he covariance beween wealh and payoff ha is crucial. 4 I is assumed ha he decision maker s uiliy funcion is inverible.

19 Kasanen and Trigeorgis (995) formulae a decision analyic problem using his approach of having wealh and he payoffs occur a he same ime. They assume ha here exiss a uiliy funcion for he whole economy. They hen ake a firs order Taylor series expansion of heir formulaion around marke wealh. The resul hey derive is he same as one would obain by aking he sae prices defined using an equilibrium approach (secion..3), replacing endowmen wih wealh, calculaing he sae-price weighed sum of he payoffs, and applying he definiion of covariance. Tha is, hey show ha an expeced uiliy framework can be linked o finance heory by allowing wealh o vary..5.3 Expeced Uiliy Theory Does No Allow for Prior Consumpion Changes Third, expeced uiliy heory does no allow for changes in prior consumpion ha can occur when a cerain equivalen is given raher han a projec. Tha is, offering a cerain equivalen for he projec s payoffs in isolaion from oher decisions can aler decisions ha mus be made before he uncerainy is resolved. Maheson and Howard (989, p. 44) recognize his when hey sae ha he approach [of calculaing he cerain equivalen of he projec] is appropriae when here is no opporuniy o uilize he informaion abou he oucomes as i is revealed. Likewise, Keeney and Raiffa (976, p. 5) poin ou ha he ime resoluion of uncerainy affecs earlier acs, and Becker and Sarin (989) and LaValle (989, 99) sae ha decision rees canno be simplified by cerain equivalen subsiuions wihou poenially affecing preferences for iniial acs. Viewed from an economics perspecive, Mossin (969), Spence and Zeckhauser (97), and Dreze and Modigliani (97) observe ha induced preference for income will no in general saisfy he von Neumann-Morgensern axioms even if preference for consumpion has an expeced uiliy represenaion. In response o some of hese observaions, Kreps and Poreus (978) propose a generalizaion of von Neumann-Morgensern uiliy called emporal von Neumann- Morgensern preference. Kreps and Poreus (979a) derive he necessary and sufficien condiions for induced preference o saisfy he von Neumann-Morgensern axioms in a wo-period model and show ha hey are quie sringen. In a consumpion-savings problem, he condiions ranslae ino a uiliy funcion of he form

20 Uc, c = fc + gc hc + c, a special case of which is a uiliy funcion ha is exponenial in period : Uc, c = u c exp λ c. Smih and Nau (995) exend his resul o parially complee markes wih more han wo ime periods wih similar resuls (i.e., here is a ime-separable uiliy funcion ha is exponenial in every ime period excep period )..6 Objecive The objecive of his research is o develop a valuaion approach ha is heoreically consisen wheher markes are complee or incomplee and invesmens are no infiniely divisible. The only hings ha will change depending upon marke condiions are he parameer inpus ino he valuaion framework. Tha is, he goal is o provide an evaluaion framework ha produces resuls ha are idenical o a financial economics approach when markes are complee bu is applicable when markes are incomplee. The ouline of he repor is as follows. Chaper develops he approach in a discree ime, wo-period world. Chaper 3 exends he resuls o muliple ime periods. Chaper 4 illusraes how o apply he mehod o a real world problem. Conclusions and recommendaions for furher research are presened in Chaper 5. Proofs are presened in Chaper 6 and a discussion abou ime- and sae-separable uiliy funcions is given in Chaper 7. 3

21 . Single-Period Model This chaper develops a single valuaion approach applicable in complee and incomplee markes in a discree ime, single-period seing. Assumpions and definiions are presened in Secion.. A decision maker s buying price for an invesmen wih uncerain payoffs saisfies he condiion ha uiliy wih he ne payoffs (see Definiion.) equals uiliy wihou he ne payoffs (Secion.) and he condiion ha uiliy canno be improved by changing consumpion or marke ransacions (Secion.3). An example is included in Secion. abou how his approach eliminaes he need o separaely define a buying price and a selling price. Secion.4 inroduces he assumpion of a ime- and sae-separable uiliy funcion and demonsraes ha he decision maker s buying price for he invesmen is approximaely equal o he sae-price weighed sum of is payoffs; resuls are exac when markes are complee or he uiliy funcion is addiive exponenial. Exponenial and logarihmic examples are included.. Seing This secion assumes a discree ime, single-period seing. Period has no uncerainy and period has a finie se, ;,,..., S@, saes of uncerainy, one of which will be revealed o be rue in period. Complee markes means ha a unique sae-price vecor exiss wih a price for all saes (Duffie 99). Incomplee markes means ha he only asse available is one ha allows risk-free borrowing and lending. Boh ypes of markes allow for he purchase of infiniely divisible asses. All vecors in his secion have S elemens unless specifically noed and are prined in bold ype. z R S means ha z has S elemens so ha z = z z... z S and R S means ha here are no sign resricions on he elemens of z; z R S + means ha every elemen of z is non-negaive; and z R S ++ means ha every elemen of z is sricly posiive. Subscrips refer o imes and superscrips refer o saes. Definiion.: S R is he period price of an invesmen ha has period payoffs of x R S. Uncerainy is resolved in period and here are no sign resricions on he invesmen price or is payoffs. S may be known or unknown. The objecive of his work 4

22 is o eiher find S if i is unknown or o deermine if he invesmen should be made if i is known. Definiion.: B R is he decision maker s period buying price for he invesmen s period payoffs x. I is he amoun ha he decision maker is willing o pay in period in order o receive he payoff vecor x. This definiion is made more precise in Theorem.. The ne payoffs are he payoffs minus he period buying price; i.e., x B R S. The period buying price equals he period buying price discouned a he risk-free rae since he period buying price is a cerain cash flow; i.e., B =ψ B, where ψ is he risk-free discoun facor ( superscrip) beween period and period ( subscrip). The decision maker is beer off buying he invesmen if he price is less han he buying price; i.e., S incomplee markes. Definiion.3: c < B. The price does no have o equal he buying price in R and w R S are he changes in period consumpion and period wealh due o re-opimizaion afer he ne payoffs are added. For example, suppose ha an iniial opimizaion saes ha opimal wealh in period sae i is w i. The addiion of he invesmen's payoff in sae i as well as he buying price for he i i invesmen changes his o w + x B. The decision maker hen re-opimizes and he i i i new wealh is w + x B + w. Assumpion.: The decision maker has a sricly increasing uiliy funcion ha maps period consumpion and period, sae-dependen wealh o a real number. S UR : + R+ R where U = Uz,z 6. z R + is period consumpion and z R S + is period, sae-dependen wealh. A soluion exiss o he uiliy maximizaion problem, max Uz, z6, where z, z 6 is budge-feasible. A soluion also exiss when any finie z, z uncerain payoff is offered o he decision maker. Assumpion.: As illusraed in Panel of Figure -, he decision maker begins wih an iniial wealh w and possibly a se of oher pre-exising uncerain payoffs summarized by X R S ; noe ha X is no he same as x. The decision maker 5

23 maximizes uiliy given hese iniial condiions prior o adding he ne payoffs. The resul is ha he opimal period consumpion/period wealh pair is c, w 6 before he ne payoffs are added. The decision maker hen adds in he ne payoffs and re-opimizes wih he resul ha he opimal period consumpion/period wealh pair is c, w 6, where c = c + c and w = w + w + x B. Period Period X Iniial Wealh/ Pre-exising Uncerain Payoffs w X X S w Opimize Consumpion/ Wealh c w w S w + w + x -B Add Ne Payoffs and Re-opimize c + c w + w + x -B w S + w S + x S -B Figure -. Iniial condiion and opimal cons./wealh wih and wihou ne payoffs.. Indifference Condiion Two condiions mus be saisfied in order for B o be he buying price of x. Firs, he decision maker s uiliy wih he ne payoffs mus be he same as he decision maker s uiliy wihou he ne payoffs. Second, he decision maker canno improve uiliy by changing consumpion or marke ransacions. The firs condiion is developed in his secion and he implicaions of he second condiion are developed in Secion.3. 6

24 Theorem.: If B is he period buying price for he payoffs x hen he uiliy of he original opimal consumpion/wealh pair plus he ne payoffs plus changes in consumpion/wealh due o re-opimizaion (Panel of Figure -) equals he uiliy of he original consumpion/wealh pair (Panel of Figure -). Uc, w = Uc, w 6 6 where c = c + c and w = w + w + x B. 7 (.) Theorem. differs from a ypical expeced uiliy approach in four ways. Firs, here are no consrains on he uiliy funcion s form or he separabiliy of he argumens. Second, wealh occurs in period and can be sae-dependen. Third, he decision maker can re-opimize afer he ne payoffs are added. Fourh, all changes o he uiliy funcion s argumens occur on one side of he equaion by adding he payoffs, subracing he buying price, and allowing he decision maker o re-opimize. This fourh poin is aracive because here is no need o formulae separae problems depending upon wheher he decision maker is buying or selling he invesmen; all problems are formulaed as if he invesmen has been purchased. Consider he following example. Example.: Buy Invesmen. A decision maker is deciding wheher or no o pay $6K in period for an invesmen ha has period payoffs of $K or $4K. Wihou he invesmen, opimal period consumpion is $K and period wealh will be $K in all saes. Thus, w = and x = 4. As shown in he lef side of 3 8 = Figure -, Theorem. requires ha U + c, + w B, 4 + w B U(,,). The decision maker will be beer off o buy he invesmen if $6K < ψ B, where ψ is he risk-free discoun facor beween period and period. Sell Invesmen. Conversely, assume a decision maker owns an invesmen ha will pay off eiher $K or $4K in period in addiion o oher period wealh. The decision maker has been offered $6K in period for he invesmen and is deciding wheher or no o accep he offer. Afer opimizing (bu before calculaing he buying price for he payoffs), he decision maker decides ha opimal period consumpion is $K and period wealh will be $K or $4K. Thus, w = 4 and x = 4. As shown in he righ side of Figure -, Theorem. requires ha

25 The decision maker will be U + c, + w B, + w B = U,, 4 beer off o sell he invesmen if $6K < ψ B (i.e., if $6K > -ψ B ). Buy Invesmen Period Period Sell Invesmen (i.e., buy negaive invesmen) Period Period Original c /w 4 Original c /w + Ne Payoffs + Re-opimizaion + c + w -B 4+ w -B + c + w -B + w -B Figure -. Opimal consumpion/wealh wih and wihou ne payoffs (Example.)..3 Opimaliy Condiion The second condiion ha mus be saisfied when here is an iner-emporal componen o he uiliy funcion (i.e., here exiss uiliy associaed wih period consumpion) or when markes are complee is ha he decision maker canno improve uiliy by changing consumpion or marke ransacions. This condiion is rue by assumpion and has several implicaions as summarized in he following corollaries; proofs for he corollaries are in he firs Appendix. Corollary.: The opimaliy condiion implies ha period marginal uiliy discouned a he risk-free rae beween period and period minus he sum of period marginal uiliies equals zero. ψ U U = w c where U U w U w w =! period and period.... U w S " $# and ψ (.) is he risk-free discoun facor beween 8

26 Corollary.: When markes are incomplee, he change in period wealh due o reopimizaion is consan across all saes and equals he opposie of he escalaed change in period consumpion. w = c /ψ. (.3) Corollary.3: When markes are complee, here is no change in period consumpion (so ha c i i w = w for i S)..4 Buying Price = c ) or oal period wealh wih he ne payoff (so ha Assumpion.3: The uiliy funcion is assumed o be ime- and sae-separable for he remainder of his secion =u3z8 u3z8 u3z8 S S Uz,z = u z + u z (.4) where u z.... This assumpion is more general han assuming ha he form of he uiliy funcion is expeced uiliy (i.e., u z 6= pu3z8 pu3z 8 pu3z 8 S S form of he uiliy funcion o be sae-dependen. ) because i allows he Theorem.: For ime- and sae-separable uiliy funcions, he period buying price B for period payoffs of x is approximaely equal o he sae-price weighed sum of he payoffs. Resuls are exac when markes are complee or he period uiliy funcion is linear or exponenial. B ψ x. (.5) ψ The sae prices equal ψ = u u i u % & K ' K i i i i u w u w i i = for w + x B i i w+ x B i i i i = u ' w for w + x B = 3 8 R S, where u = u u... u S and ( ) K * K Proof: According o Theorem. for a ime- and sae-separable uiliy funcion, B is he period buying price for payoffs x if uc6+ u w6 = uc6+ uw6. This can. 9

27 be rewrien as u c u c + u w u w = and hen as u c + u w + x B = u % &K 'K u c = 6 wih u defined above and 6 uc6 ( c c )K ' for 6 for *K = u c c =. Dividing by u (i is sricly posiive because every elemen of u is sricly posiive) and adding B resuls in " " u u B = c! $# +! $# + u x w u u u. B is discouned o period a he riskfree rae since i occurs a period wih cerainy wih he resul ha he period buying price is B = x +! ψ " ψ u u c $# + ψ w. u u (.6) Theorem. is proven if he second and hird erms of (.6) are approximaely equal o " ψ u ψ zero; i.e., when w! u $# + u c. Consider wo cases. u Case : Complee Markes. Corollary.3 implies ha c = and saes ha i i w = w for i S when markes are complee. The hird erm of (.6) equals because c =. The second erm equals zero because ψ ψ u! $# = u w ', which are he u! u' w6 $ sae prices from a financial economics approach and he sae price weighed sum of he " 6" # change in period wealh mus equal zero o avoid arbirage opporuniies. Case : Incomplee Markes. The change in period wealh associaed wih he reopimizaion is he same across all saes when markes are incomplee and, according o Corollary., i equals w! (.6) resuls in ψ u u u c =. Subsiuing his ino he second and hird erms of ψ " $# c. This erm is approximaely equal o zero since he numeraor in he square brackes is approximaely equal o he opimaliy condiion from Corollary..

28 Linear and exponenial uiliy funcions represen a special case when markes are incomplee so ha Equaion (.6) is saisfied exacly. This is because here is no change in period consumpion. Corollary.4: When markes are incomplee, he change in period consumpion mus be zero if and only if he period uiliy funcion is linear or exponenial (see Appendix for proof). The following wo examples illusrae how Theorem. can be applied. Example.: Invesmen Opporuniy Using an Exponenial Uiliy Funcion. A firm is deciding wheher or no o inves $.6M in period in a projec ha is equally likely o pay off eiher nohing or $.M in period. The risk-free discoun rae is percen, he z 6 6 / z / z firm s uiliy funcion is Uz, = u z pe pe dollars, and markes are incomplee., where z is in millions of Consider he cases when here is posiive and negaive correlaion beween payoffs and exising wealh (i.e., x =.. and w =.. or w =.. ) as well as he case when wealh is cerain ( w =.. ). Sae prices are calculaed using Theorem. wih he resuls ha: he buying price when here is posiive correlaion is $. M = $.M $.M ; he buying price when wealh is cerain is $. 5M = $.M $.M ; and he buying price when here is negaive correlaion is $. 3M = $.M $.M. 5 This suggess ha he firm should only inves when here is negaive correlaion because he projec acs as a hedge agains uncerainy (i.e., his is he only case when he invesmen price is less han he buying 5 The definiion of sae prices in (.5) for a period exponenial uiliy funcion resuls in a sae-price vecor of ψ = ψ x B x B ( w x ) ( w x ) pe + / ρ pe + / ρ +, where B = ρln x x x x w w pe ρ + pe. / / ρ Noice ha he period buying price equals he buying price wih he payoffs minus he buying price wihou he payoffs. This leads o he observaion ha an alernaive approach o his problem is o calculae he buying price for wo porfolios (one wih and one wihou he payoffs) and he difference beween he wo is he buying price for he payoffs. For example, ake he case when here is negaive correlaion. The buying price for he porfolio of exising wealh is $9.6M and he buying price for he

29 price $. 6M< $. 3M). I is imporan o realize ha he large difference in buying prices is due o he correlaion, no o he size of he payoffs; e.g., i can be shown ha he relaive difference in he buying prices is sill very dramaic when he projec is reduced o a fracion of he size of he original projec. Example.3: Insurance Opporuniy Using a Logarihmic Uiliy Funcion. A decision maker currenly has $, and owns a home and wans o find his or her buying price for he purchase of fire insurance. The home can be sold for $, in period bu has a 5 percen chance of burning down before period (i.e., p = ). The decision maker s uiliy funcion is Uz, z = ln z + pln z + pln z, where z is in housands of dollars. The homeowner can borrow and lend a percen per period and markes are incomplee. Wihou insurance, i is opimal for he homeowner o spend $6,538 in period and o inves he res so ha he or she will have $9,88 in period. Thus, including he value of he house, w = Raher han performing a full opimizaion, assume ha here is no change in period consumpion or period wealh so ha he decision maker s oal wealh wih he ne payoff equals w = 988. B 988. B. A fixed-poin approach is used o calculae B, where ψ B6 x = ψ B. The sae-price vecor is ψ = and he period buying price is B = $6, 3; his resul is almos idenical o a full opimizaion. porfolio of exising wealh plus he payoffs is $.9M. Thus, he buying price for he payoffs is $.3M.

30 3. Muli-Period Model Chaper presened a single valuaion approach applicable in complee and incomplee markes in a discree ime, single-period seing. This chaper exends he resuls o a muli-period seing. While his secion follows he forma from Chaper, he way in which uncerainy is revealed over ime needs o be deermined in a muli-period seing. There are several ways o make his deerminaion. One approach is o use a probabiliy space (Duffie 99). When he evens of he probabiliy space are all aken ogeher, he resul is a filraion ha represens how informaion is revealed hrough ime. Anoher approach is o pariion he sae space. While he filraion approach has is advanages in coninuous ime, he pariions approach is aken here because of is inuiive appeal. This secion develops a single valuaion approach applicable in complee and incomplee markes in a discree ime, muli-period seing. Assumpions and definiions are presened in Secion 3., wih a paricular focus on pariions. Secion 3. shows ha he buying price for any invesmen saisfies he condiion ha uiliy wih he ne payoffs equals uiliy wihou he ne payoffs and he condiion ha uiliy canno be improved by changing consumpion or marke ransacions. Secion 3.3 inroduces he assumpion of a ime- and sae-separable uiliy funcion and demonsraes ha he decision maker s buying price for an invesmen is approximaely equal o he sae-price weighed sum of is payoffs; resuls are exac when markes are complee or he uiliy funcion is addiive exponenial. Secion 3.4 ieraively calculaes he buying price using a dynamic programming approach. Exponenial and logarihmic examples are included. 3. Seing Definiion 3.: S is he se of all possible saes ha can occur a imes equal o,,, T. 6 Uncerainy is resolved as ime progresses so ha a ime, saes ha have occurred prior o ime are known. This se of saes s, s,, s up o ime is denoed as s ;@ is he se of all possible saes ha can occur over imes +, ;@. Thus, S s S, T given ha he paricular saes s,..., s occurred a imes,,. 3

31 Definiion 3.: A pariion of S ;@ s is a collecion of disjoin non-empy subses of S ;@ s whose union is S ;@ s. One way o pariion S s is in erms of he curren sae and fuure pariions. Le s s ;@ ;@ refer o sae s a ime given ha saes ;@ s,..., s have occurred a imes,, -; s = when equals. A pariion a ime of S ;@ s for T is he se ha conains he curren sae and he se of all pariions a he nex period given ha he curren sae has occurred. Tha is, he ;@ > ;@ = ;@ BC pariion P s = s P s, s + for T. Figure 3- illusraes wha pariions look like when he sae space is S = 7 The period pariion is he larges oval, he period pariions are he medium sized ovals, and he period pariions are he smalles ovals. The period pariion conains all subsequen pariions and hus he enire sae space since here is no uncerainy in period. Period Period Period Figure 3-. Uncerain saes (numbers) and pariions (ovals) of S. 6 7 As in Secion, all variables ha have more han one elemen are presened in bold ype. Sae is no explicily included in periods and for noaional simpliciy because i is he same in all saes. 4

32 Definiion 3.3: S R is he period price of an invesmen ha has ime- and sae- dependen payoffs x s P ; P s s P R in pariion P ;@; s i.e., x = x x,, T. P S ; J L Definiion 3.4: B P ; s@ τ R is he decision maker s period τ buying price for he P invesmen s payoffs of x s in period τ in he pariion P s. I is he amoun ha he decision maker is willing o pay P ;@ in order o receive he payoff vecor x s I is s τ assumed ha τ T. Tha is, B P canno occur before he pariion is defined or afer he end of he analysis. The definiion of B P τ ; s@ is made more precise in Theorem P 3.. The ne payoffs associaed wih he payoffs x s J P s P s P s P s τ τ T L and he buying price B P x,, x B,,x where he vecor corresponds o he number of saes a ime τ in pariion P JsL. Definiion 3.5: c R S are he changes in consumpion due o re-opimizaion afer he ne payoffs are added. Example 3.: In order o illusrae he definiion of ne payoffs, consider he sae space from Figure 3- wih an invesmen ha has period payoffs given ha sae occurred in period. Since all payoffs occur wihin he pariion P, he payoffs are = x, x B and he ne payoffs can be wrien as eiher B x =,, x B or = x B, x B B as illusraed in Figure 3-; he ime subscrips on he saes have been dropped for noaional simpliciy. τ ; s@ are 5

33 Period Period Period Period Period Period -B A x x B x -B x -B Figure 3-. Two ways of delineaing he ne payoffs. Assumpion 3.: The decision maker has a sricly increasing uiliy funcion ha maps consumpion z over all imes and saes o a real number. UR : S + R where U = U 6 z and S z R +. Uiliy of consumpion in he final period can be viewed as uiliy of wealh. A soluion exiss o he uiliy maximizaion problem, max Uz6, where z is budge-feasible. z Assumpion 3.: c is he soluion o he uiliy maximizaion problem before he ne payoffs are added and c is he soluion afer he ne payoffs are added, where c is he sum of he original opimal consumpion (c), he ne payoffs J P ; s@ P ; s@ P ; s@ P ; s@ τ τ T x,, x B,,x, and he changes in consumpion due o reopimizaion ( c for and he pariion is P s 3. Necessary Condiions L T ). Tha is, c = c+ x+ c excep for he case when =τ ;@ in which case c P c x c τ = P τ + P P P τ B + τ ; s@ s s s s τ. As in he single-period model from Secion, wo condiions mus be saisfied in order for B P τ ; s@ P o be he period τ buying price for he payoffs x s Firs, he decision maker s uiliy wih he ne payoffs mus be he same as he decision maker s uiliy wihou he ne payoffs. Second, he decision maker canno improve uiliy by changing consumpion or marke ransacions. The firs condiion is developed in Theorem 3.. 6