Systems Analysis Laboratory Research Reports E16, June 2005 PROJECT VALUATION IN MIXED ASSET PORTFOLIO SELECTION

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1 Helsii Uiversity of Techology Systes Aalysis Laboratory Research Reports E6, Jue 25 PROJECT VALUATION IN MIXED ASSET PORTFOLIO SELECTION Jae Gustafsso Bert De Reyc Zeger Degraeve Ahti Salo ABTEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI 53

2 Title: Authors: Proect Valuatio i Mixed Asset Portfolio Selectio Jae Gustafsso Cheye Capital Maageet Liited Storoway House, 3 Clevelad Row, Lodo SWA DH, U.K. ae.gustafsso@cheyecapital.co Date: Jue, 25 Bert De Reyc Lodo Busiess School Reget's Par, Lodo NW 4SA, U.K. bdereyc@lodo.edu Zeger Degraeve Lodo Busiess School Reget's Par, Lodo NW 4SA, U.K. zdegraeve@lodo.edu Ahti Salo Systes Aalysis Laboratory Helsii Uiversity of Techology P.O. Box, 25 HUT, FINLAND ahti.salo@t.fi Status: Systes Aalysis Laboratory Research Reports E6 Jue 25 Abstract: We exaie the valuatio of proects i a settig where a ivestor, such as a private fir, ca ivest i a portfolio of proects as well as i securities i fiacial arets. I proect valuatio, it is iportat to cosider all alterative ivestets opportuities, such as other proects ad fiacial istruets, because each iposes a opportuity cost o the proect uder cosideratio. Therefore, covetioal ethods based o decisio aalysis ay lead to biased estiates of proect values, because they typically cosider proects i isolatio fro other ivestet opportuities. O the other had, optios pricig aalysis, which does cosider fiacial ivestet opportuities, assues that a proect s cash flows ca be replicated usig fiacial assets, which ay be difficult i practice. The ai cotributio of this paper is the developet of a procedure for proect valuatio i a settig where the fir ca ivest i a portfolio of proects ad securities, but where exact replicatio of proect cash flows is ot ecessarily possible. We cosider both sigle-period ad ulti-period odels, ad show that the valuatio procedure exhibits several iportat aalytical properties. We also ivestigate the pricig behavior of ea-variace ivestors through a set of uerical experiets. Keywords: fiace, capital ratioig, portfolio selectio, decisio aalysis, decisio aalysis, ixed iteger prograig. 54

3 Itroductio I the literature o corporate fiace (e.g. Brealey ad Myers 2, Chapter 9), it is proposed that a proect s cash flows should be discouted at the rate of retur of a security that is equivalet i ris to the proect. Uder the Capital Asset Pricig Model (CAPM; Sharpe 964, Liter 965), two assets are equivalet i ris if they have the sae beta. Therefore, a appropriate discout rate for a proect is r = r + β ( E r r ), f [ M] f with r f the ris-free iterest rate, cov[ r, r ]/ var[ r ] β = the proect s beta, r M the rado p M M rate of retur of the aret portfolio, ad r p the rado rate of retur of the proect. The use of the CAPM relies o the assuptio that the fir is a public copay axiizig its share price. Whe the arets abide by the CAPM assuptios, the share price is axiized whe all proects with a rate of retur higher tha what is give by the CAPM are started (Rubistei 973). Yet, ay copaies ae decisios about acceptig ad reectig proects before their iitial public offerig (IPO). At this stage, the CAPM assuptios are ot typically satisfied, because shareholders are ofte etrepreeurs ad veture capitalists. Also, sice there is o share price to axiize, such firs are typically assued to axiize the value of their proect portfolio istead. I such a settig, the fir is regarded as a idividual ivestor with welldefied prefereces over risy assets. Several ethods have bee proposed to value proects i a settig where the fir is a idividual ivestor, icludig decisio aalysis (Frech 986, Clee 996) ad optios pricig aalysis, also referred to as cotiget clais aalysis ad real optios aalysis (Dixit ad Pidyc 994, Trigeorgis 996). However, covetioal ethods based o decisio aalysis ay lead to biased estiates of proect values, because they typically cosider proects i isolatio fro other ivestet opportuities, eglectig the opportuity costs that these ivestet opportuities ipose. Optios pricig aalysis does accout for the effect that fiacial istruets have o proect values, but its applicability is liited, because replicatig the proect s cash flows usig fiacial istruets ay be difficult i practice. Sith ad Nau (995) have exteded decisio aalytic ethods to iclude the possibility of tradig securities. They do ot, however, cosider other proects i the portfolio. 55

4 The ai cotributio of this paper is the developet of a procedure for proect valuatio i a settig, where the fir ca ivest i a portfolio of proects ad securities, but where replicatio of proect cash flows with securities is ot ecessarily possible. We call this settig a ixed asset portfolio selectio (MAPS) settig. The valuatio procedure is based o the cocepts of breaeve sellig ad buyig prices (Lueberger 998, Sith ad Nau 995, Raiffa 968), which are the ivestor s ow sellig ad buyig prices for a proect. We forulate the procedure for both expected utility axiizers ad ea-ris optiizers. Usig the Cotiget Portfolio Prograig (CPP) fraewor (Gustafsso ad Salo 25), we develop a geeral ulti-period MAPS odel, which allows us to study a broad variety of ris-averse preferece odels i a ulti-period settig. This odel accoodates Marowitz s (952) ea-variace (MV) odel, Koo ad Yaazai s (99) ea-absolute deviatio (MAD) odel, the ea-lower seiabsolute deviatio (MLSAD) odel (Ogrycza ad Ruszczysi 999), ad Fishbur s (977) ea-ris odels where ris is associated with deviatios below a fixed target value. We also prove several aalytical properties for breaeve sellig ad buyig prices. MAPS-based proect valuatio is iportat i practice as well as i theory. O the oe had, MAPS-based proect valuatio is required whe a corporatio or a idividual aes ivestets both i securities ad proects, or other lupy ivestet opportuities. For exaple, i Jue 24, Microsoft had ivested early $44.6 billio i short-ter ivestets i fiacial arets i additio to carryig out a large uber of proects at the sae tie (Microsoft 24). Also, ay ivestet bas ivest i a portfolio of publicly traded securities ad udertae ucertai oe-tie edeavors, such as veture capital ivestets ad orgaizatio of IPOs. O the other had, oe of the fudaetal probles i the literature o corporate fiace is the valuatio of a sigle proect while taig ito accout the opportuity costs iposed by securities (see e.g. Brealey ad Myers 2). This is a MAPS settig that icludes oe proect ad several securities. This paper is structured as follows. Sectio 2 itroduces sigle-period MAPS odels ad discusses differet forulatios of the portfolio selectio proble. Sectio 3 presets the ultiperiod MAPS odel. I Sectio 4, we itroduce the valuatio cocepts, ad produce theoretical results o the valuatio properties of differet types of ivestors. Sectio 5 deostrates the pricig properties of ea-variace ivestors through a series of uerical experiets. I 2 56

5 Sectio 6, we suarize our fidigs ad discuss the aagerial iplicatios of our results. 2 Mixed Asset Portfolio Selectio I a MAPS proble, available ivestet opportuities are divided ito two categories: () securities, which ca be bought ad sold i ay quatities, ad (2) proects, lupy all-or-othig type ivestets. Fro a techical poit of view, the ai differece betwee these two types of ivestets is that the proects decisio variables are biary, while those of the securities are cotiuous. Aother differece is that the cost, or price, of securities is deteried by a aret equilibriu odel, such as the CAPM, while a proect s ivestet cost is a edogeous property of the proect. Portfolio selectio odels ca be forulated either i ters of rates of retur ad portfolio weights, lie i Marowitz-type forulatios, or by usig a budget costrait, expressig the iitial wealth level, ad axiizig the ivestor s terial wealth level. Whe properly applied, both approaches yield idetical results. We use the secod approach with MAPS, because it is ore suitable to proect portfolio selectio. We first forulate sigle-period MAPS odels, where the ivestets are ade at tie ad the obective at tie is optiized. These odels will allow us to geerate several isights ad show how MAPS is related to Marowitz (952) ad the CAPM. We the develop the ulti-period MAPS odel based o Cotiget Portfolio Prograig (CPP, Gustafsso ad Salo 25). Early portfolio selectio forulatios (Marowitz 952) were bi-criteria decisio probles iiizig ris while settig a target for expectatio. Later, the ea-variace odel was forulated i ters of expected utility theory (EUT) usig a quadratic utility fuctio. However, there are o siilar utility fuctios for ost other ris easures, icludig the widely used absolute deviatio (Koo ad Yaazai 99). Therefore, we distiguish betwee two classes of portfolio selectio odels: () preferece fuctioal odels, such as the expected utility odel, ad (2) bi-criteria optiizatio odels or ea-ris odels. A sigle-period MAPS odel usig a preferece fuctioal ca be forulated as follows. Let there be risy securities, a ris-free asset (labeled as the th security), ad proects. Let the 3 57

6 price of asset i at tie be S i ad the correspodig (rado) price at tie is S i. The price of the ris-free asset at tie is ad + at period, where r f is the ris-free iterest rate. The rf aouts of securities i the portfolio are deoted by x, i =,...,. The ivestet cost of proect i tie is C ad the (rado) cash flow at tie is i C. The biary variable z idicates whether proect is started or ot. The ivestor s budget is b. We ca the forulate a MAPS odel usig a preferece fuctioal U as follows: (i) axiize utility at tie : axu S ixi + C z, xz i= = subect to (ii) budget costrait at tie : i i i= = (iii) biary variables for proects: { } Sx+ Cz = b z, =,..., (iv) cotiuous variables for securities: x free i =,...,. The budget costrait is forulated as equality, because i the presece of a ris-free asset all of the budget will be expeded at the optiu. I this odel ad throughout the paper, it is assued that there are o trasactio costs or capital gais tax, ad that the ivestor is able to borrow ad led at the ris-free iterest rate without liit. These assuptios ca be relaxed without itroducig prohibitive coplexities. For expected utility theory, the preferece U X = E u( X), where u is the ivestor s vo Neua-Morgester utility fuctioal is fuctio. Whe the ivestor is able to deterie a certaity equivalet for ay rado variable X, U ca be expressed as a strictly icreasig trasforatio of the ivestor s certaity equivalet operator CE. Hece, the obective ca also be writte as ax CE[ X ], which gives the total value of the ivestor s portfolio. i I additio to preferece fuctioal odels, ea-ris odels have bee widely used i the literature. We cocetrate o these odels, because uch of the oder portfolio theory, icludig the CAPM, is based o a ea-ris odel, aely the Marowitz (952) eavariace odel. Table describes three possible forulatios for ea-ris odels: ris iiizatio, where ris is iiized for a give level of expectatio (Lueberger 998), expected value axiizatio, where expectatio is axiized for a give level of ris (Eppe et al. 989), ad the additive forulatio, where the weighted su of ea ad ris is axiized (Yu 985). I Table, ρ is the ivestor s ris easure, µ is the iiu level for expectatio, 4 58

7 ad R is the axiu level for ris. The paraeters λ are tradeoff coefficiets. The Marowitz (952) odel ca be uderstood as a special case of a ea-variace MAPS odel where the uber of proects is zero. Table. Forulatios of the ea-ris optiizatio proble. Ris iiizatio Expected value axiizatio i ρ xz, Sx i i + Cz i= = Obective ax E Sx i i + Cz, xz i= = Costraits E S ixi + C z µ i= = i i i= = ρ Sx+ Cz = b Sx+ Cz R i i i= = i i i= = Sx+ Cz = b Geeral additive Sharpe (97) Ogrycza ad Ruszczysi (999) ax λ E Sx i i + Cz λ2 ρ Sx i i + Cz, Sx i i + Cz = b xz i= = i= = i= = ax λ E Sx i i + Cz ρ Sx i i + Cz, xz i= = i= = ax E Sx i i + Cz λ ρ Sx i i + Cz, xz i= = i= = i i i= = Sx+ Cz = b i i i= = Sx+ Cz = b The geeral additive for ca be tured ito the odel eployed by Sharpe (97) by dividig the additive for by λ 2, provided it is ozero, ad ito the for used by Ogrycza ad Ruszczysi (999) by dividig it by λ. Apart fro expectatio ad ris costraits, the Karush- Kuh-Tucer (KKT) coditios of all of the forulatios are idetical ad therefore will yield the sae efficiet frotiers, as log as optial solutios i the additive forulatio reai bouded. However, if liitless borrowig ad shortig are allowed, the additive forulatio ca give ubouded solutios uless a ris costrait is itroduced. Yet, i this case the forulatio essetially coicides with expected value axiizatio i ters of KKT coditios. Both ris iiizatio ad expected value axiizatio odels have advatages ad 5 59

8 disadvatages. Ris iiizatio requires the ivestor to set a iiu level for expectatio, a readily uderstadable quatity, while the iterpretatio of a axiu ris level i expected value axiizatio ay ot always be clear. However, expected value axiizatio allows us to iclude ultiple ris costraits, e.g. oe for variace, oe for expected dowside ris, ad oe for the probability of gettig a outcoe below a particular level, so that the ris profile of the portfolio ca be atched to the ris prefereces of the ivestor as closely as possible. Due to this flexibility we focus o this forulatio i the followig. 3 A Multi-Period MAPS Model 3. Fraewor We develop a ulti-period MAPS odel usig the Cotiget Portfolio Prograig (CPP) fraewor developed by Gustafsso ad Salo (25). I this fraewor, ucertaities are odeled usig a state tree, represetig the structure of future states of ature, as depicted i the leftost chart i Figure. The state tree eed ot be bioial or syetric; it ay also tae the for of a ultioial tree with differet probability distributios i its braches. I each oterial state, securities ca be bought ad sold i ay, possibly fractioal quatities. Cotiue? 3% ω 5% ω Yes ω States ω 5% 5% ω ω 2 7% 4% 6% ω 2 ω 2 ω 22 35% 2% 3% Start? Yes No Cotiue? Yes No Yes Start? No ω 2 No Yes No ω 2 ω ω 2 ω 2 ω 22 ω 2 ω 22 2 Tie 2 Tie Tie Figure. A state tree, decisio sequece, ad a decisio tree for a proect. 2 Proects are odeled usig decisio trees that spa over the state tree. The two right-ost charts i Figure describe how proect decisios, whe cobied with the state tree, lead to proectspecific decisio trees. The specific feature of these decisio trees is that the chace odes are shared by all proects, sice they are geerated usig the coo state tree. Security tradig is ipleeted through state-specific tradig variables, which are siilar to the oes used i 6 6

9 fiacial odels of stochastic prograig (e.g. Mulvey et al. 2) ad i Sith ad Nau s (995) ethod. Siilar to a sigle-period MAPS odel, the ivestor sees either to axiize the utility of the terial wealth level, or the expectatio of the terial wealth level subect to a ris costrait. 3.2 Model Copoets The two ai copoets of the odel are (i) states ad (ii) the ivestor s ivestet decisios, which iply the cash flow structure of the odel States Let the plaig horizo be {,...,T }. The set of states i period t is deoted by T Ω t, ad the set of all states is Ω= Ωt. The state tree starts with base state ω i period. Each o-terial t= state is followed by at least oe state. This relatioship is odeled by the fuctio B : Ω Ω which returs the iediate predecessor of each state, except for the base state, for which the fuctio gives B( ω) = ω. The probability of state ω, whe B( ω ) has occurred, is give by p ( ) B( ω ) ω. Ucoditioal probabilities for each state, except for the base state, ca be coputed recursively fro the equatio p( ω) = p ( )( ω) p( B( ω)). The probability of the base state is p( ω ) = Ivestet Decisios B ω Let there be securities available i fiacial arets. The aout of security i bought i state ω is idicated by tradig variable xi, ω, i =,...,, ω Ω, ad the price of security i i state ω is deoted by Si ( ω ). Uder the assuptio that all securities are sold i the ext period, the cash flow iplied by security i i state ω ω is Si( ω) ( xi, B( ω) xi, ω ). I base state ω, the cash flow is S ( ω ). i x ω i, The ivestor ca ivest privately i proects. The decisio opportuities for each proect, =,...,, are structured as a decisio tree, where we have a set of decisio poits D ad fuctio ap( d ) that gives the actio leadig to decisio poit { d D\ d}, where d is the first decisio poit of proect. Let A d be the set of actios that ca be tae i decisio poit d D. For each actio a i A d, a biary actio variable z a idicates whether the actio is selected or ot. Actio variables at each decisio poit d are boud by the restrictio that oly 7 6

10 oe z a, a A, ca be equal to oe. The actio i decisio poit d is chose i state ω ( d). d For a proect, the vector of all actio variables z a relatig to the proect, deoted by z, is called the aageet strategy of proect. The vector of all actio variables of all proects, deoted by z, is the proect portfolio aageet strategy. We call the pair ( xz, ), coposed of all tradig ad actio variables, the ixed asset portfolio aageet strategy Cash Flows ad Cash Surpluses p Let CF ( z, ω) be the cash flow of proect i state ω with proect aageet strategy z. Whe C ( ω ) is the cash flow i state ω iplied by actio a, this cash flow is give by a CF ( z, ω) = C ( ω) z p a a d D: a Ad B ω( d ) Ω ( ω) where the restrictio i the suatio of the decisio poits guaratees that actios yield cash flows oly i the prevailig state ad i the future states that ca be reached fro the prevailig state. The set B ( ω) B Ω ( ω) = ω Ω such that B ( ω) = ω, where B ( ω) B( B ( ω)) Ω is defied as { } = is the th predecessor of ω ( B ( ω) = ω ). The cash flows fro security i i state ω Ω are give by ( ) s Si ω xi, ω if ω = ω CFi ( x i, ω) = Si ( ω) ( xi, B( ω ) xi, ω ) if ω ω Thus, the aggregate cash flow CF( xz,, ω) i state ω Ω, obtaied by suig up the cash flows for all proects ad securities, is s p CF( xz,, ω) = CF ( x, ω) + CF ( z, ω) i i i= = Si ( ω) xi, ω + Ca( ω) za, if ω = ω i= d D: a A d B ω( d ) Ω ( ω) = Si ( ω) ( xi, B( ω) xi, ω) + Ca( ω) za, if ω ω i= d D: a Ad B ω( d ) Ω ( ω) Together with the iitial budget of each state, cash flows defie cash surpluses that would result i state ω Ω if the ivestor chose portfolio aageet strategy ( xz, ). Assuig that excess cash is ivested i the ris-free asset, the cash surplus i state ω Ω is give by 8 62

11 b( ω) + CF( xz,, ω) if ω = ω CSω =, b( ω) + CF( xz,, ω) + ( + rb( ω) ω) CSB( ω) if ω ω where b( ω ) is the iitial budget i state ω Ω ad rb ( ω ) ω is the short rate at which cash accrues iterest fro state B( ω ) to ω. The cash surplus i a terial state is the ivestor s terial wealth level i that state. 3.3 Optiizatio Model Whe usig a preferece fuctioal U, the obective fuctio for the MAPS odel ca be writte as a fuctio of cash surplus variables i the last tie period, i.e. axu CS xzcs,, ( ) T where CS T deotes the vector of cash surplus variables related to period T. Uder the riscostraied ea-ris odel, the obective is to axiize the expectatio of the ivestor s terial wealth level: ax p( ω) CS ω xzcs,, ω ΩT Table 2. Multi-period MAPS odels. Obective fuctio Budget costraits Decisio cosistecy costraits Ris costraits Variables Preferece fuctioal odel axu xzcs,, ( CS ) T CF( xz,, ω ) CS = b( ω ) ω Mea-ris odel ax p( ω) CS ω xzcs,, ω ΩT CF( xz,, ω) + ( + r ) CS CS = b( ω) ω Ω\{ ω } a A d a Ad z = =,..., a B( ω) ω B( ω) ω { } z = z ( ) d D \ d =,..., a ap d { } za, a Ad d D =,..., xi, ω free ω Ω i =,..., free CS ω ω Ω + ( ) ρ, R + CS ω τ( CS T) ω + ω = ω ΩT za {,} a Ad d D =,..., xi, ω free ω Ω i =,..., CS ω free ω Ω ω Ω ω Ω ω + ω T T 9 63

12 Three types of costraits are iposed o the odel: (i) budget costraits, (ii) decisio cosistecy costraits, ad (iii) ris costraits, which apply to ris-costraied odels oly. The ulti-period MAPS odel uder both a preferece fuctioal ad a ea-ris odel is give i Table 2. The costraits are explaied i ore detail i the followig sectios Budget Costraits Budget costraits esure that there is a oegative aout of cash i each state. They ca be ipleeted usig cotiuous cash surplus variables CS ω, which easure the aout of cash i state ω. These variables lead to the budget costraits CF( xz,, ω ) CS = b( ω ) ω CF( xz,, ω) + ( + r ) CS CS = b( ω) ω Ω\{ ω } B( ω) ω B( ω) ω Note that if CS ω is egative, the ivestor borrows oey at the ris-free iterest rate to cover a fudig shortage. Thus, CS ω ca also be regarded as a tradig variable for the ris-free asset Decisio Cosistecy Costraits Decisio cosistecy costraits ipleet the proects decisio trees. They require that (i) at each decisio poit reached, oly oe actio is selected, ad that (ii) at each decisio poit that is ot reached, o actio is tae. Decisio cosistecy costraits ca be writte as a A d z = =,..., a { } za = zap( d) d D\ d =,...,, a Ad where the first costrait esures that oe actio is selected i the first decisio poit, ad the secod ipleets the above requireets for other decisio poits Ris Costraits A ris-costraied odel icludes oe or ore ris costraits. We focus o the sigle costrait case. Whe ρ deotes the ris easure ad R the ris tolerace, a ris costrait ca be expressed as ρ( CS ) R. T I additio to variace (V), several other ris easures have bee proposed i the literature o portfolio selectio. These iclude seivariace (Marowitz 959), absolute deviatio (Koo ad Yaazai 99), lower sei-absolute deviatio (Ogrycza ad Ruszczysi 999), ad their 64

13 fixed target value couterparts (Fishbur 977). Seivariace (SV), absolute deviatio (AD) ad lower sei-absolute deviatio (LSAD) are defied as µ X, SV: σ = ( x µ ) 2 df ( x) X X X AD: δ = x µ df ( x), ad X X X µ X µ X, LSAD: δ = x µ df ( x) = ( µ x) df ( x) X X X X X where µ X is the ea of rado variable X ad F X is the cuulative desity fuctio of X. The fixed target value statistics are obtaied by replacig µ X by soe costat target value τ. All of these easures ca be forulated i a MAPS progra through deviatio costraits. I geeral, deviatio costraits are expressed as + CS ω τ( CS T) ω + ω = ω ΩT, where τ ( CS ) is a fuctio defiig the target value fro which the deviatios are calculated, ad + ω ad T ω are oegative deviatio variables which easure how uch the cash surplus i state ω Ω T differs fro the target value. For exaple, whe the target value is the ea of the terial wealth level, the deviatio costraits are writte as + CS p( ω ) CS + = ω Ω, ω ω ω ω ω ΩT Usig these deviatio variables, soe coo dispersio statistics ca be writte as follows: AD: + p( ω) ( ω + ω ). ω ΩT LSAD: p( ω) ω. V: SV: ω ΩT + 2 p( ω) ( ω + ω) ω ΩT 2 p( ω) ( ω ). ω ΩT The respective fixed-target value statistics ca be obtaied with the deviatio costraits + τ + = ω Ω, CS ω ω ω T where τ is the fixed target level. EDR, for exaple, ca the be obtaied fro the su p( ω) ω. ω ΩT It is also possible to set a liit for the (critical) probability of gettig a outcoe below soe T 65

14 target level. Matheatically, the critical probability is defied as crit P ( τ ) = F ( X < τ ), X X where F X deotes the cuulative distributio fuctio of rado variable X ad τ is the desired target level. This ris easure ca be ipleeted by usig the followig liear costraits: τ CS Mξ ω Ω, ad ω { } ω ξω, ω Ω T, T where τ is the target value fro which critical probability is calculated ad M is soe very large uber. Critical probability ca the be costraied fro above usig the costrait pωξω R. ω ΩT Other Costraits Other types of costraits ca also be icluded i the odel, such as o-egativity restraits that prevet short sellig, upper bouds for the uber of shares bought, ad credit liit costraits (Marowitz 987). I the esuig sectios, we assue, for the sae of siplicity, that o such additioal costraits have bee iposed. 4 Proect Valuatio 4. Breaeve Buyig ad Sellig Prices Because we cosider proects as o-tradable ivestet opportuities, there is o aret price that ca be used to value the proect. I such a situatio, it is reasoable to defie the value of the proect as the aout of oey at preset, tie, that is equally desirable to the proect. I a portfolio cotext, this ca be iterpreted so that the ivestor is idifferet betwee the followig two portfolios: (A) a portfolio with the proect ad (B) a portfolio without the proect ad cash equal to the value of the proect. However, we ay alteratively defie the value of a proect as the idifferece betwee the followig two portfolios: (A2) a portfolio without the proect ad (B2) a portfolio with the proect ad a debt equal to the value of the proect. The proect values obtaied i these two ways will ot, i geeral, be the sae. Aalogously to Lueberger (998), Sith ad Nau (995), ad Raiffa (968), we call the first value the breaeve sellig price (BSP), as the portfolio copariso ca be uderstood as a sellig process, ad the secod type of value the breaeve buyig price (BBP). 2 66

15 A crucial eleet i BSP ad BBP is the defiitio of equal desirability of two differet portfolios. I preferece fuctioal odels, the ivestor is, by defiitio, idifferet betwee two portfolios wheever their utility scores are equal. I a ea-ris settig, a ivestor is idifferet betwee two portfolios if the eas ad riss of the two portfolios are idetical. More geerally, whe the riss are cosidered as costraits, the ivestor is idifferet betwee two portfolios if their expectatios are equal ad they both satisfy the ris costraits. Hece, we obtai the pairs of optiizatio probles show i Table 3. Here, the variable z a* is the actio variable associated with ot startig proect ; thus, z a* = idicates that ivestor ivests i the proect, whereas z * = eas that the proect is ot started. I the case that the proect startig decisio a is a biary go / o go decisio, we ca alteratively ipose the restrictios o the variable idicatig proect startig istead. Fidig a BSP ad BBP is a iverse optiizatio proble: oe has to fid the values for the paraeters v ad v so that the optial value of the secod optiizatio proble atches the optial value of the first proble. 4.2 Iverse Optiizatio Procedure s Iverse optiizatio probles have recetly attracted iterest aog researchers, such as Ahua ad Orli (2). I a iverse optiizatio proble, the challege is to fid the values for a set of paraeters, typically a subset of all odel paraeters, that yields the desired optial solutio. Iverse optiizatio probles ca broadly be classified ito two groups: (i) fidig a optial value for the obective fuctio, ad (ii) fidig a solutio vector. Ahua ad Orli (2) discuss probles of the secod id, whereas the proble of fidig a BSP or BBP falls withi the first class. b I priciple, the tas of fidig a BSP is equivalet to fidig a root to the fuctio s ( s ) ( s + f v = Ws v) Ws, where W + s is the optial value of the portfolio optiizatio proble i s the status quo ad Ws ( v ) is the correspodig optial value i the secod settig as the s fuctio of paraeter v. Siilarly, the BBP ca be obtaied by fidig the root to the fuctio b b + b f ( v ) = W W ( v ). Note that these fuctios are icreasig with respect to their b b paraeters. To solve such root-fidig probles, we ca use ay of the usual root-fidig algoriths (see e.g. Belegudu ad Chadrupatla 999) such as the bisectio ethod, the secat ethod, ad the false positio ethod, which have the advatage that they do ot require the 3 67

16 owledge of the fuctios derivatives, which are ot typically ow. If the first derivatives are ow, or whe approxiated uerically, we ca use the Newto-Raphso ethod to obtai quicer covergece. 4.3 Geeral Aalytical Properties 4.3. Sequetial Cosistecy Breaeve sellig ad buyig prices are ot, i geeral, equal to each other. While this discrepacy is accepted as a geeral property of ris prefereces i expected utility theory (Raiffa 968), it ay also see to cotradict the ratioality of these valuatio cocepts. It ca be argued that if the ivestor were willig to sell a proect at a lower price tha at which he/she would be prepared to buy it, the ivestor would create a arbitrage opportuity ad lose a ifiite aout of oey whe aother ivestor repeatedly bought the proect at its sellig price ad sold it bac at the buyig price. I a reverse situatio where the ivestor s sellig price for a proect is greater tha the respective buyig price, the ivestor would be irratioal i the sese that he/she would ot tae advatage of a arbitrage opportuity if such a opportuity existed where it would be possible to repeatedly buy the proect at the ivestor s buyig price ad sell it at a slightly higher price which is below the ivestor s breaeve sellig price. However, these arguets eglect the fact that the breaeve prices are affected by the budget ad that therefore these prices ay chage after obtaiig the proect s sellig price ad after payig its buyig price. Ideed, it ca be show that i a sequetial settig where the ivestor first sells the proect, addig the sellig price to the budget, ad the buys the proect bac, the ivestor s sellig price ad the respective (sequetial) buyig price are always equal to each other. This observatio is foralized as the followig propositio ad it holds for ay preferece odel accoodated by the odel i Sectio 3. The proof is give i the Appedix. PROPOSITION. A proect s breaeve sellig (buyig) price ad its sequetial breaeve buyig (sellig) price are equal to each other Cosistecy with Cotiget Clais Aalysis Optio pricig aalysis, or cotiget clais aalysis (CCA; Lueberger 998, Brealey ad Myers 2, Hull 999), ca be applied to value proects wheever the cash flows of a proect 4 68

17 ca be replicated usig fiacial istruets. Accordig to CCA, the value of proect is give by the aret price of the replicatig portfolio (or tradig strategy) less the ivestet cost of the proect: CCA i i i= v C S x = +, where x i is the aout of security i i the replicatig portfolio ad the proect i at tie. C is the ivestet cost of It is straightforward to show that, whe CCA is applicable, i.e. if there exists a replicatig portfolio, the the breaeve buyig ad sellig prices are equal to each other ad yield the sae result as CCA (Sith ad Nau 995). For exaple, whe v CCA is positive, we ow that ay ivestor will ivest i the proect, sice it is possible to ae oey for sure by ivestig i the proect ad shortig the replicatig portfolio. Furtherore, ay ivestor will start the proect eve b CCA whe he/she is forced to pay a su v less tha v to gai a licese to ivest i the proect, because it is ow possible to gai v CCA b v for sure. O the other had, if v b is greater tha CCA v, the ivestor will be better off by ivestig the replicatig portfolio istead ad hece he/she will ot start the proect. A siilar reasoig applies to breaeve sellig prices. These observatios are foralized i Propositio 2. The proof is obvious ad hece oitted. Due to the cosistecy with CCA, the breaeve prices ca be regarded as a geeralizatio of CCA to icoplete arets. PROPOSITION 2. If there is a replicatig portfolio for a proect, the breaeve sellig price ad breaeve buyig price are equal to each other ad yield the sae result as CCA Sequetial Additivity The BBP ad BSP for a proect deped o what other assets are i the portfolio. The value obtaied fro breaeve prices is, i geeral, a added value, which is deteried relative to the situatio without the proect. Whe there are o other proects i the portfolio, or whe we reove the fro the odel before deteriig the value of the proect, we spea of the proect s isolated value. We defie the respective values for a set of proects as the oit added value ad oit value. Figure 2 illustrates the relatioship betwee these cocepts. 5 69

18 Nuber of proects beig valued Sigle proect Portfolio of proects Isolated Value Added Value Joit Value Joit Added Value No other proects Additioal proects Nuber of other proects i the portfolio Figure 2. Differet types of valuatios for proects. Isolated proect values are, i geeral, o-additive; they do ot su up to the value of the proect portfolio coposed of the sae proects. However, i a sequetial settig where the ivestor buys the proects oe after the other usig the prevailig buyig price at each tie, the obtaied proect values do add up to the oit value of the proect portfolio. These prices are the proects added values i a sequetial buyig process, where the budget is reduced by the buyig price after each step. We refer to these values as sequetial added values. This sequetial additivity property holds regardless of the order i which the proects are bought. Idividual proects ca, however, acquire differet added values depedig o the sequece i which they are bought. These observatios are prove by the followig propositio. The proof is i the Appedix. PROPOSITION 3. The breaeve buyig (sellig) prices of sequetially bought (sold) proects add up to the breaeve buyig (sellig) price of the portfolio of the proects regardless of the order i which the proects are bought (sold). 4.4 Aalytical Properties of Mea-Ris Preferece Models 4.4. Equality of Prices ad Solutio through a Pair of Optiizatio Probles Whe the ivestor is a ea-ris optiizer ad the ris easure is idepedet of a additio of a costat to the portfolio, lie variace, breaeve sellig ad buyig prices are idetical, provided that uliited borrowig ad ledig are allowed. I cotrast, they are typically differet uder expected utility theory ad uder ost o-expected utility odels. Also, for ea-ris optiizers the breaeve prices ca be coputed directly by solvig the expectatios of terial wealth levels whe the ivestor ivests ad does ot ivest i the proect ad discoutig their differece bac to its preset value at the ris-free iterest rate. Therefore, with 6 7

19 ea-ris odels, there is o eed to resort to possibly laborious iverse optiizatio. We foralize these clais i Propositio 4. The proof is i the Appedix. PROPOSITION 4. Let the ivestor be a ea-ris optiizer with ris easure ρ that satisfies ρ X + b = ρ X for all rado variables X ad costats b. Whe liitless borrowig ad ledig are allowed, the breaeve sellig price ad the breaeve buyig price of ay give proect are idetical. Moreover, the prices are equal to v = + W W T, + r ( t) ( f ) t= where W + is the expectatio of the terial wealth level whe the ivestor ivests i the proect, W is the expectatio of the terial wealth level whe the ivestor does ot ivest i the proect, ad r f, (t) is the ris-free iterest rate fro tie t to t+. Propositio 4 is striig both i its geerality ad siplicity, sice either the give valuatio forula, or the equality of breaeve prices, geerally holds uder expected utility theory. The results are also ituitively appealig: the ivestor places oly a sigle price o a give proect, ad this price is related to the ivestor s terial wealth levels with ad without the proect. This idicates that ea-ris odels exhibit a very reasoable type of pricig behavior Relatioship to the Capital Asset Pricig Model Accordig to the CAPM, the aret price of ay asset is give by the certaity equivalet forula (see, e.g., Lueberger 998): E C cov C, r E CAPM [ r ] r v C = + + r var r + r where M M f f M f C is the ivestet cost of the asset,, C is the rado value of the asset at tie, ad r M is the rado retur of the aret portfolio. For o-aret-traded assets lie proects, the outcoe of the forula ca be iterpreted as the price that the arets would give to the asset if it were traded. I geeral, breaeve sellig ad breaeve buyig prices are icosistet with CAPM valuatios, because soe of the CAPM assuptios do ot hold i a MAPS settig. For exaple, private proects are ot icluded i the derivatio of the CAPM aret equilibriu, yet they ay 7 7

20 be strogly correlated with the securities. Therefore, results that hold for the CAPM do ot ecessarily hold here. I particular, the optial fiacial portfolio for the ivestor is ot ecessarily a cobiatio of the ris-free asset ad the aret fud cosistig of all securities i proportios accordig to their aret capitalizatio. However, there is a special case whe the ivestor s optial fiacial portfolio is what the CAPM predicts, aely whe the proect portfolio is ucorrelated with the aret securities ad the ivestor is a ea-variace optiizer. This is prove i Propositio 5. Nevertheless, eve though the optial fiacial portfolio falls o the capital aret lie with ad without the proect beig valued, the CAPM valuatio is icorrect eve i this case, as a extra ris preiu ter appears i the breaeve sellig ad buyig prices, as show i Propositios 6 ad 7. Proofs for the propositios ca be foud i the Appedix. Note that this is a particularly iterestig special case, because proect outcoes do ot usually deped o the fluctuatios of fiacial arets. Propositio 5 is to be uderstood as a ixed asset portfolio extesio of the usual Separatio Theore (Tobi 958), applicable whe proects are ucorrelated with securities. PROPOSITION 5. If the ivestor is a ea-variace optiizer ad proects are ucorrelated with securities, the optial fiacial portfolio is a cobiatio of the aret fud ad the ris-free asset. PROPOSITION 6. If the coditios of Propositio 5 hold ad the copositio of the optial proect portfolio is the sae with ad without proect, the breaeve sellig price of proect is var Cz var Cz E C = = s E[ rm] rf v C = + + b M, + rf + rf var Sx i i i= where r is the rado rate of retur of the aret portfolio ad b M M is the aout of oey spet o securities i the status quo. That is, b b x C z =. M = PROPOSITION 7. If the coditios of Propositio 5 hold ad the copositio of the optial 8 72

21 proect portfolio is the sae with ad without proect, the breaeve buyig price of proect is var Cz var Cz E C = = b E[ rm] rf v C = + b M, + rf + rf var Sx i i i= where b b x C z =. M = Note that b M is coputed here slightly differetly tha i Propositio 6 due to a differet situatio i the status quo. Despite the differeces i the forulas i Propositios 6 ad 7, we ow fro Propositio 4 that the forulas will give the sae result. The followig propositio geeralizes Propositios 6 ad 7 to a case where the optial proect portfolio chages with ad without the exaied proect. PROPOSITION 8. If the coditios of Propositio 5 hold, the differece i budget required to ae two portfolios with differet proects (the decisio variables of the first are deoted by z, =,...,, ad those of the secod by z, =,..., ) equally desirable is E C ( ) var var z z Cz Cz E [ r ] M r = f C ( z z = = ) b M = + r f + rf var Sx i i i= = + +, where b b x C z =. M = As the CAPM predicts that the value of a ucorrelated proect is equal to its NPV discouted at the ris-free iterest rate, this suggests that breaeve buyig ad sellig prices are, i geeral, differet fro CAPM values, eve whe the optial fiacial portfolio falls o the capital aret lie. However, the prices approach CAPM values whe the aout of securities i the portfolio icreases (i.e. whe the ivestor s ris tolerace icreases). This ca be verified by ultiplyig b M iside the paretheses. The last ter i Propositio 8 ow becoes 9 73

22 var Cz var Cz E 2 [ rm ] r = = b M b + M var r + r M f which goes to as b M goes to ifiity. f, Covergece of the proect values to CAPM prices as the ivestor s ris tolerace goes to ifiity applies i geeral to all proects, regardless of their correlatio with each other or with the aret. As log as the optial proect portfolio is the sae with ad without the proect beig valued, every proect s BSP ad BBP will coverge towards the CAPM price of the proect, as show i Propositio 9; otherwise, they will coverge towards the CAPM value of a differece portfolio z-z specified i Propositio. The proofs of the propositios are i the Appedix. Note that these liit results are valid regardless of whether the arets actually abide by the CAPM. If there is a discrepacy betwee securities expected returs ad the capitalizatio weights that a ea-variace ivestor would fid appropriate, or vice versa, the aret portfolio i Propositios 5 is to be uderstood as the portfolio that would be the aret portfolio if all ivestors i the aret were ea-variace optiizers. PROPOSITION 9. If the ivestor is a ea-variace optiizer ad the copositio of the optial proect portfolio is the sae with ad without proect, the proect s breaeve buyig ad sellig prices coverge towards the proect s CAPM price E C cov C, r E b s CAPM [ r ] r v v v C = = = + + r var r + r as the ivestor s ris tolerace goes to ifiity. M M f f M f PROPOSITION. If the ivestor is a ea-variace optiizer, a proect s breaeve buyig ad sellig prices coverge towards E C ( ) cov ( ), z z C z z r M E b s CAPM [ r ] M r = = v v v C ( z z = = = ) + + r var r + r = f M f f as the ivestor s ris tolerace goes to ifiity. Here, z, =,...,, deote the decisio variables for the optial proect portfolio with the proect ad z, =,..., those without the 2 74

23 proect. Apart fro this liit behavior, aother case where breaeve buyig ad sellig prices coicide with CAPM recoedatios is whe a replicatig portfolio exists for the proect. This is a direct result of the fact that each of the three valuatio ethods is cosistet with CCA. 4.5 Valuatio of Opportuities ad Real Optios Whe valuig a proect, we ca either value a already started proect or a opportuity to start a proect. The differece is that, although the value of a started proect ca be egative, that of a opportuity to start a proect is always o-egative, because a ratioal ivestor does ot start a proect with a egative value. While BSP ad BBP are appropriate for valuig started proects, ew valuatio cocepts are eeded for valuig opportuities. Sice a opportuity etails the right but ot the obligatio to tae a actio, we eed sellig ad buyig prices that rely o coparig the situatios where the ivestor ca ad caot ivest i the proect, istead of does ad does ot. The lowest price at which the ivestor would be willig to sell a opportuity to start a proect ca be obtaied fro the defiitio of the breaeve sellig price by reovig the requireet to start the proect i the status quo, i.e. by reovig the costrait z a* = i the top-left quadrat of Table 3. We defie this price as the opportuity sellig price (OSP) of the proect. Liewise, the opportuity buyig price (OBP) of a proect ca be obtaied be reovig the startig requireet i the secod settig, i.e. the equatio z a* = i the botto-right quadrat of Table 3. It is the highest price that the ivestor is willig to pay for a licese to start the proect. Opportuity sellig ad buyig prices have a lower boud of zero; it is also straightforward to show that the opportuity prices ca be coputed by taig a axiu of ad the respective breaeve price. Table 4 gives a suary of breaeve ad opportuity sellig ad buyig prices. Opportuity buyig ad sellig prices ca also be used to value real optios (Trigeorgis 996) that ay be cotaied withi the proect portfolio. These optios result fro aageet s flexibility to adapt later decisios to uexpected future developets. Typical exaples iclude possibilities to expad productio whe arets are up, to abado a proect uder bad aret coditios, ad to switch operatios to alterative productio facilities. Real optios ca be valued uch i the sae way as opportuities to start proects. However, istead of coparig 2 75

24 portfolio selectio probles with ad without a possibility to start a proect, we will copare portfolio selectio probles with ad without the real optio. This ca typically be ipleeted by disallowig the ivestor fro taig a particular actio (e.g. expadig productio) i the settig where the real optio is ot preset. Sice breaeve prices are cosistet with CCA, also opportuity prices have this property, ad hece they ca be regarded as a geeralizatio of the stadard CCA real optio valuatio procedure to icoplete arets. Table 4. A suary of a ivestor s buyig ad sellig prices. Valuatio cocept Idea Differece fro Properties breaeve prices Breaeve buyig ad sellig prices Copariso of settigs where the ivestor does ad does ot - Breaeve buyig price ad sellig price are i geeral Opportuity buyig ad sellig prices ivest i the proect. Copariso of settigs where the ivestor ca ad caot ivest i the proect. 5 Nuerical Experiets The ivestor is ot obliged to ivest i the proect. differet fro each other. Lower boud of. Equal to axiu of ad the respective breaeve price. I this sectio, we deostrate the use of a MAPS odel i proect valuatio through a series of uerical experiets. We eploy the ulti-period MAPS odel but costruct oly a sigleperiod odel, because this allows creatig several isights, is easier to follow ad replicate, ad because we ca cotrast the results with the CAPM, also a sigle-period odel. We have also coducted further experiets featurig real optios ad a ulti-period MAPS odel i two worig papers (De Reyc et al. 24, ad Gustafsso ad Salo 24). The preset experiets cofir soe of the theoretical results obtaied i the previous sectios ad also cast light o the followig issues: How are proect values affected by the presece of other proects i the portfolio? 2 How are proect values affected by the opportuity to ivest i securities? 3 How are proect values affected by the ivestor s ris tolerace (axiu level for ris)? 4 How are proect values related to the CAPM? 5 How does the presece of twi securities affect the value of a proect? 5. Experietal Set-up The experietal set-up icludes 8 equally liely states of ature, four proects, A, B, C, ad D (Table 5), ad two securities, ad 2, which together costitute the aret portfolio (Table 6)

25 The settig ca be exteded to iclude ore securities, but for the sae of siplicity we liit our aret portfolio to two assets oly. This does ot ifluece the geerality of our results. Note that proects C ad D ad security 2 are the sae that Sith ad Nau (995) use i their exaples. Table 7 shows the correlatio betwee the assets cash flows. Nubers i italic i Tables 5 ad 6 are coputed values. The ris-free iterest rate is 8%. Table 5. Proects. Proect A B C D Ivestet cost $8 $ $4 $. State $5 $4 $8 $67.68 State 2 $5 $4 $8 $67.68 State 3 $5 $5 $6 $. State 4 $5 $ $6 $. State 5 $5 $7 $8 $67.68 State 6 $5 $ $8 $67.68 State 7 $5 $9 $6 $. State 8 $5 $9 $6 $. Expected outcoe $ $23.75 $2 $33.84 St. dev. of outcoe $5 $28.26 $6 $33.84 Beta Maret price (NPV) $2.59 $.33 -$4. $25.7 The aret prices i Tables 5 ad 6 are obtaied by usig the CAPM, where the expected rate of retur of the aret portfolio has bee chose so that the price of security 2 is $2, the price used by Sith ad Nau (995). Techically, this ca be accoplished by icludig all of the assets ito the aret portfolio, with proects havig zero issued shares, ad by fidig the aret prices, excludig proects ivestet costs, that iiize the su of squared errors (SSE) betwee the rate of retur give by the CAPM forula ad the real expected rate of retur, as coputed fro the aret price. The prices coverge, resultig i SSE equal to zero. The desired expected rate of retur is 5.33%. The stadard deviatio of the aret portfolio is the 35.32%. I Table 6, the security capitalizatio weights represet the ratio betwee the aret capitalizatio of the security ad that of the etire aret. These weights are of iterest, because the CAPM predicts that a MV ivestor will always ivest i a cobiatio of the ris-free asset ad the aret fud, where securities are preset accordig to their capitalizatio weights

26 Table 6. Securities. Security 2 Maret price $39.56 $2. Shares issued 5,,,, Capitalizatio weight 74.79% 25.2% State $6 $36 State 2 $5 $36 State 3 $4 $2 State 4 $3 $2 State 5 $6 $36 State 6 $5 $36 State 7 $4 $2 State 8 $3 $2 Beta Expected retur 3.76% 2.% St. dev. of retur 28.26% 6.% Table 7. Correlatio atrix. A B C D 2 A.398 B C D The experiet coprises several steps. We start with a settig where the ivestor ca ivest oly i the proect beig valued, which is the exteded to cover a portfolio of all proects. We the add securities ad 2. Apart fro the very first case, we assue that liitless borrowig at the ris-free rate ad shortig of securities are allowed. As predicted by Propositio 4, the breaeve sellig ad buyig prices are idetical i this case ad hece we display both of the prices i oe etry. Uless otherwise oted, i each of the steps we use a budget of $5. The axiu ris level is defied i ters of the rate of retur of the portfolio. For exaple, a 5% ris level iplies that the axiu stadard deviatio for the terial wealth level is $ Proect Values without Securities I geeral, the value of a proect depeds o what would happe to the ivested fuds if the proect were ot started. I the absece of other proects ad securities, three cases described i Table 8 are possible. These cases both provide a bechar for proect valuatios obtaied later, 24 78

27 ad show that eve apparetly sall differeces i available ivestet opportuities ca ae a sigificat differece i the values of proects. I the first case, where uused fuds are lost, the et preset value of the proect is udefied, ad hece we give the future value (FV) of the proect istead. Table 8. Proects values whe oly the proect beig valued is available. Proect What happes to uused fuds: A B C D Fuds are lost (FV) $. $23.75 $2. $33.84 No iterest gaied $2. $23.75 $6. $33.84 Deposited at ris-free iterest rate $2.59 $4.58 $7. $3.33 Let us ext assue that the ivestor is able to ivest i all of the proects ad i the ris-free asset. The values of the proects are described i Table 9 as a fuctio of the ivestor s ris tolerace. I the optial policy, the ivestor starts the proects with positive value. Table 9. Proects values whe all of the proects are available. Mea-Variace Ris level A B C D 5% -$.99 $.99 -$38.8 $8.74 2% $2.59 $4.58 -$2.6 $ % $5.48 $7.47 -$5.48 $ % ad up $2.59 $4.58 $7. $3.33 $4. $2. Proect value $. -$2. 2% 5% 8% 2% 24% 27% 3% A B C D -$4. -$6. Ris level Figure 2. Proect values for MV ivestor. Table 9 ad Figure 2 show that the proect values behave rather erratically whe the ris tolerace 25 79

28 is varied. For exaple, at a 2% ris level, the MV values for proects A ad B are idetical to the values they have i isolatio ($2.59 ad $4.58), but the they drop to $5.48 ad $7.47 at 25% ad rise bac to the values i isolatio for higher ris tolerace levels. Ituitively, oe ight expect proect values to rise as the ris costrait is relaxed, because ore proects ca be icluded ito the portfolio so that fewer proects ipose a opportuity cost. At the liit, this is correct: whe the ris costrait is relaxed eough to allow the iclusio of all profitable proects, the proect values coicide with the prices o the last row of Table 8. For iterediate ris values, however, aother effect cofouds this result: the value of a proect depeds o the proects that fit ito the portfolio whe the proect is started ad whe it is ot. For exaple, the decrease of the price of proects A ad B whe the ris costrait is icreased fro 2% to 25% is due to the fact that, whe the ris costrait is 25%, proect C ca be started if either of the proects is ot icluded ito the portfolio, but this is ot the case at 2%. So at a 25% ris costrait, proect C iposes a opportuity cost o these proects, but ot at 2%, sice the proect does ot fit ito the portfolio regardless the decisio o proects A ad B. This couterituive result is typical for proect values i a MAPS settig. 5.3 Proect Values with Securities We will ow ivestigate what happes to the proect values whe securities are also available. I this case, the ris costrait will always be bidig as log as the rate of retur of the optial security portfolio for the ivestor is higher tha the ris-free iterest rate: expectatio of the whole portfolio will be axiized by purchasig as uch of the optial security portfolio as possible ad by borrowig the ecessary fuds at the ris-free iterest rate. Whe the ivestor does ot ivest i the proects, i.e. i a pure CAPM settig, the optial security portfolio for a MV ivestor is to buy 75% of security ad 25% of security 2 at all ris levels. Note that these weights are the capitalizatio weights of the securities i Table 6. Colus 2 5 of Table describe the proect values assuig that the ivestor ca oly ivest i the proect beig valued ad i securities. The presece of securities chages the proect values for two reasos. O the oe had, securities ipose a additioal opportuity cost, which lowers proect values. O the other had, it is possible to hedge agaist proect riss by buyig egatively correlated or shortig positively correlated securities, which icreases the proect values. 26 8