Constant in Ides (1978) - Market Risk Adjustment in Project Valuation

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American Finance Association

Market Risk Adjustment in Project ValuationAuthor(s): George M. ConstantinidesReviewed work(s):Source: The Journal of Finance, Vol. 33, No. 2 (May, 1978), pp. 603-616Published by: Blackwell Publishing for the American Finance AssociationStable URL: http://www.jstor.org/stable/2326572 .

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THE JOURNALOF FINANCE VOL. XXXIII,NO. 2 MAY 1978

MARKET RISK ADJUSTMENT IN PROJECT VALUATION

GEORGE M. CONSTANTINIDES*

I. INTRODUCTION

THIS PAPER DEVELOPS A RULE which reduces the problem of valuation in the

presence of market risk to the problem of valuation in a world where the market

price of risk is zero. Given a valuation problem in an intertemporal, continuous

time framework, the rule isapplied

in twosteps as follows: We first replace one

or

more of the model parameters by their "effective values" in a specified way. Then

we discount all expected cash flows at the riskless rate of return as if the market

price of risk were zero. The broad- applicability of the valuation rule is illustrated

through the diverse examples of asset valuation, option pricing, determination of

the optimal capital structure of a firm, and cash management.

We motivate the discussion of this paper by briefly examining a forerunner to

our valuation rule, the certainty equivalence approach to evaluating a stream of

cash flows. In a single period model, the risk-adjusted net present value

RANPV(X) of the cash flow X, realized at the end of the period, is given by the

single period Sharpe-Lintnercapital asset pricing model (CAPM) as

XX-(RM-RF )cov(RM X)/aMRANPV(X )==+RF

Essentiallythis formula states that the expected cash flow X is adjustedto itscertainty equivalent X- (RM RF)cov(RM,X)/a, which is subsequently dis-

counted at the risklessrateof return,RF. The certaintyequivalenceapproachmayalso be viewed as follows: The expected cash flow X is first adjusted to its"effective value" X- (RM RF)cov(RM,X)/ am and is then discounted at the

risklessrateof returnRF, as if the marketpriceof risk were zero. Viewedin thisway,thecertainty quivalenceapproach esemblesour valuationrule.Indeed t willbe apparent aterin the paperthat the certaintyequivalenceapproach s a specialcase of our valuationrule.

The paperis organizedas follows: SectionII states the assumptionswhichleadto Merton's 1973) intertemporalCAPM.In sectionIII we define the problemandderive the valuationrule in the simplestcase. Three examplesare consideredinsectionIV wherethe rule is appliedto the valuationof an asset, the pricingof a

europeancall option, and the determination f the optimal capitalstructureof afirm. In sectionV we generalize he rule to controlledprocessesand applyit to acashmanagementproblem.In section VI we provideconcludingremarks.

*Carnegie-Mellon University Graduate School of Industrial Administration, Pittsburgh, Pa. 15213.

603



604 The Journal of Finance

II. THE CAPITAL ASSET PRICING MODEL

The assumptionswhich lead to the intertemporalCAPM, as developedby Merton

(1973),are statedbelow.

A. Perfect MarketsThe securitiesof firms are traded n a perfectcapitalmarket.In particularhere

are no transactionscosts in trading securities; there are no taxes; securitiesareinfinitely divisible; nvestorscan borrow and lend at the same interestrate;shortsales of all securities,with full use of the proceeds,are allowed;and tradinginsecurities akesplacecontinuously n time.

B. Securities

The prices of securities are lognormally distributed.For each security, the

expectedrateof returnper unit time,a,, and the varianceof returnper unit time,a,2,exist and are finite with ai2>0. The opportunityset is non-stochastic n thesense that ai, a,2,the covarianceof returnsper unit time auj,and the risklessborrowing-lendingater, are all nonstochastic unctionsof time.

C. Investor Preferences

Eachinvestormaximizeshis strictlyconcaveand time-additiveutilityfunctionofconsumptionover his lifespan.Investorshavehomogeneousexpectationsregardingthe investmentopportunity et.

Under these assumptions,Merton(1973) proved that the equilibrium ecurity

returns atisfy the CAPMrelationship'

a,- r = aiM/M (1)

where the subscriptM refersto the marketportfolioand A-(aM - r)/ aM.

We shall be studyingprojectswhich are tradedby firmsin a perfectmarket,butwhich are not necessarilytradedin the capital market. We now prove that theequilibrium eturnof any project,whichis heldby a firm,also satisfies the CAPMrelationship 1). Hamada (1969) showed that a firm undertakesa project only if

ap- r> (aM -

r)aPM/am,where the

subscriptP

designatesthe

project.The firm

realizes a windfallgain if the inequalityis strict. Under the assumptionof free

1. This result follows from Merton's 1973) theorem1. Some minor differencesbetween Merton'sassumptions nd theassumptionsmade hereare noted.Thesedifferencesdo not affect the derivation fthe theorem:

a) UnlikeMerton,we makea distinctionbetween he terms"project"nd"security".Dependingonthe interpretation f the project, n specificapplicationsnvestors n the capitalmarketmay ormay not be allowedto invest n the projectdirectly.

b) Mertonassumed hatfirms ssueonly equity.We assume hatfirmsmayalso issuedebt.

c) Mertonassumed hat firmspayno dividendsandaccomplishhe transfer f cash to shareholdersthrough harerepurchase.We allow dividendsas wellas sharerepurchase.In the next sectionwe shallbe considering rojectswith state dependent eturns nd which, herefore,

violate assumptionB, that the opportunityet is non-stochastic. t has been proven n Constantinides(1977) that the assumption hat the opportunity et is non-stochasticmay be replacedby the weakerassumptionhat theefficient rontier andnot necessarilyhe entire nvestment pportunity et) is stateindependent.Projectswith statedependent eturnsarein agreementwiththe latterassumption.



Market Risk Adjustment n Project Valuation 605

entry by new firms,the equilibriumpriceof the project s sufficientlyhigh so thatno firm realizes a windfall gain throughundertaking he project.Then the equi-libriumreturnof thisprojectsatisfies(1).

AssumptionA may be relaxed to allow for the case where a risklessasset does

not exist. It is a simple matterto repeat Merton's 1973) derivation n the absenceof a risklessasset, andobtain the continuous-time ersionof Black's 1972) CAPM.Equation 1) stillobtains,where now r is interepreted s the expectedreturnof thezero-betaportfolio.For the sake 'ofconcreteness,n the discussionof the followingsectionswe shall assumethat a risklessasset exists and we shallbe referringo r astherisklessrate.However,ourentire discussionappliesalso to the case where theredoes not exista risklessasset. One would simply have to replaceour references othe risklessrate by the expectedreturnon the zero-betaportfolio.

III. THE GENERAL MODEL

We considera projectof marketvalue V(x, t) which is completelyspecifiedby thestate variablex, and time t. In specific applicationswhichappear n latersections,the projectwillbe interpreted s beingan investment,an option,a claimon a firm,or a subsystemof the firm. The project generatesa streamof cash flows and themarketvalue of the project represents he time- and risk-adjusted alue of thesecash flows.

We assumethat the state variablex changes n the time interval t, t + dt) by

dx =judt+ a dw (2)

where t= p(x, t), a = a(x, t) and dw is the incrementof a Wienerprocessw(t).2 Wealso assume that the cash returngenerated by the projectin the time interval(t, t + dt) is cdt, where c = c(x, t).

The return on the projectin the time interval (t, t + dt) is the sum of capitalappreciation V(x, t) andcash returnc dt. Assume that the function V(x, t) is twicecontinuouslydifferentiablew.r.t.x and once continuouslydifferentiablew.r.t. t.Wemay thenwritedV(x, t) through to's Lemma4 s

/V(x 2)=( + d+adV(x, t) =V, + pLVx+

2- Vxx)dt+ aVxdw (3)

2. For a heuristicdiscussionof the Wiener process see Kushner 1971), chap. 10. For a rigorousdiscussion, ee Breiman 1968).

A Wienerprocess s definedas follows:Forany partitiono < tI < t2 < . . . of the time interval to,oo),the randomvariablesw(t1) - w(t0), w(t2) - w(t1), w3) -w(t2),..., are independentand normallydis-tributedwithmeanzero and variance I - tO,2- tI, t3 - t2, . ., respectively.Therefore, or any infinites-

simaltimeinterval, herandomdisturbance f the state is normallydistributedwithmean u(x,t)dtandvarianceu2(x, ) dt.Weassume hat thefunctionsu(x, ) anda(x, t) satisfy henecessary egularityonditions, o that the

stochasticprocessdefinedby (2) exists.

3. The differentiability ssumptionmust be verified n every specificapplication,once the functionV(x, t) hasbeenexplicitly valuated.

4. For a discussionof Ito's Lemmasee Kushner 1971)Chap. 10.



606 TheJournal of Finance

whereV=-aV(x, t)/ at etc. The rateof returnon the project s

dV(x,t)+cdt 1 / aV

V(x' t) =vtCc

+l+ 2 Vxx)t Vd (4)

withexpectedvalue per unit timeap, and covariancewiththe marketperunit timecPM givenby

ap= c+V x+2 Vxx/V(5)

and

GPM= PuMa Vx/ V (6)

where p= p(x, t) is the instantaneouscorrelationcoefficient between dw and themarketreturn.

Under the assumptionsmade in section II, the project return satisfies theintertemporalCAPM of equation(1). We substitute(5) and (6) in (1) and uponsimplification btain5

c-rV+V+ +(t,-Xpa)Vx+ Vxx= 0. (7)

The solution to this partial differentialequation which satisfies the relevant

boundaryconditionsgivesthe marketvalue of the project.Forreasonsthat will beapparentater,we rewrite 7) as

c-rV+ +V,*+ 2x+ Vx =0 (8)2

where

,u* (x, t) _ (x, t) - Ap(x, t)a(x, t). (9)

Consider, or the moment,a capitalmarketwhichpays

nopremium

ormarketrisk, i.e. aM - r= 0. Denote by V(x, t) the value of the above project in this market.

We set aM r = 0 in (7) and obtain the following differential equation for V(x, t)

c-rV+ V,+ttvx+a

V^ =0? (8')

Since the boundaryconditions are independentof the marketrisk premium,6heboundary onditions mposedon Vareidentical o thoseimposedon V.

Comparison f equations 8) and(8') indicatesthat V(x, t) maybe consideredas

the marketvalue of the projectin a capitalmarketwhich pays no premiumformarketrisk, providedthe drift,u(x,t) of the state variable s replacedby ,u*(x,t).

5. This procedurewas first used by Merton (1970) in derivingthe differentialequationwhich issatisfiedby thepriceof a european all option.

6. This point willbe clarified n the next section whereapplications re discussedand theboundaryconditionsareexplicitly tated.




This observation uggests he following rule for determining he marketvalue of aproject.

Rule

Stepone: Replacethe drift L(x, ) by,L*(x, t) = ,u(x, t) - Xp(x, t)a (x, t).

Step two: Evaluate he streamof cash flows as if the marketprice of risk werezero, i.e., discountexpectedcash flows at the risklessrate.

The rule is easily generalized o the case where the state variablex is a vectorx = [x,x2,... ,x] rather han a scalar.Changes n the elementsof x satisfya jointdiffusion process such that E(dxi) = ,idt and cov(dxi, dxj)= ijdt, where pi and aijare functionsof x and t. Proceedingas before we derive the following differentialequationwhichis analogous o equation(7)

n In

c-rV+ VI+ (Li-ApiMai) Vi+ 2 EZ = O.

The rule still holds, that is to say, we first replace the drift parameterspi by

p.*= i - Apimaj, i = 1,., n, and ignore market risk thereafter.

The rule is also easily generalized o the case where the cash flow generatedbythe projectin the time interval (t, t + dt) is stochastic and is given by cdt+ sdz

instead of cdt, where c = c(x, t), s = s(x, t) and dz is the increment of a Wienerprocessz(t) whichin general s correlatedwith dw of equation(2). Equation 4) isreplacedby

dV(x, t) + cdt sdz [ + v + a

V(x, t) c~k+ V+L 2 Vxxjdt

+ aVxdw+ sdz] (4')

and equation(7) is replacedby

(c-APCS)-rV + V + (tL-Xpa) Vx+ 2g Vxx=? (7')

wherep,= p (x, t) is the instantaneouscorrelationcoefficient between dz and themarketreturn.The latter equation suggests a slightly generalizedversionof therule: The drift parameterp,(x,t) is replacedby ,u*(x,t) of equation (9) as before.Furthermorehe expectedcash flow rate c(x,t) is replacedby c*(x,t) where

c*(x, t) = c (x, t) - Ap, (x, t)s (x, t). (9')

Note that the adjustmentor marketriskof the drift parameter (x,t) is formallyidentical to the correspondingadjustmentof the drift parameterp(x, t). Uponadjustmentof the driftparametersp, c we evaluate the streamof cash flows as ifthe market price of risk were zero. It is worthwhile o note that the correlation




between the Wiener processes w(t) and z(t) does not influence the valuationprocedure.

Exampleswhich illustrate he usefulnessand generalapplicability f the rule arepresented n the next two sections.

IV. APPLICATIONS OF THE RULE

A. Valuationof an Asset

Consideran asset with lifetimet. In any time interval t, t + dt) the asset earnsdxwhere dx = ,udt + a dw and jt, a, p, are constant (as before, p is the correlation

coefficientof dwand marketreturn).All earningsare payable n cash to the ownerof the assetat time T. No interestaccrueson the earnings rom the time that theyareearneduntilthe timethat they are paid.The latterassumptions made in order

to simplifythe example.We denote the cumulative arningsof the asset at time t by x(t) and the valueof

the assetby V(x, t). Theboundarycondition s V(x, T)= x, whichsaysthat at timeT, all cumulativeearningsare paid in cash and the value of the asset at T equalsthe cumulative arnings.Notice that this boundarycondition does not involve themarketprice of risk. Also c(t) = 0. We now apply the rule developedin the lastsection as follows:

Step one: Replace the expected earnings rate ,t by ji* = - Xpc.

Step two: Evaluate the asset as if the market price of risk were zero. The

expectedcumulativeearningsat time T areji* T, and the net presentvalue of ,* T at time zero is tt*Te rT Thus the value of the asset at

timezero is (tp-Xpa)Te-rT.

Moregenerally,f jt, p,a are functionsof time, the valueof the assetat timezerois e rTfO( t -Xp)dt.

The procedurewhich we have followed may now be related to the certaintyequivalenceapproach: t- Xpurepresents he certaintyequivalentof the expectedearningsrate jt, which is subsequentlydiscountedat the risklessrate.Howeverin

other applications,where the drift parameter t may not be interpretedas anearnings rate, the analogywith the certainty equivalenceapproach s no longervalid.

B. Valuationof an Option

Cofisidera europeancall optionwith marketvalue V(x, t) at time t, wherex isthe marketpriceof the securityon which the optionis written.The securityprice7

7. Wedo not assume hat the securityon which the option is written s necessarilypricedaccording

to the CAPM 1).A feasiblesenario s as follows:An option s written n the U.S. on a foreignsecurity.Institutionalonstraints,uch as taxes,transactionsosts,andspecialrestrictions rohibit radingof the

foreignsecurity n the U.S. capitalmarket. n general, hissecurity s not pricedaccording o the U.S.CAPM.Uponexerciseof theoption,theholderof theoptionreceives hedollarvaluewhichtheforeignsecuritycommands n the foreigncapitalmarket n lieu of the certificateof the foreign security.For

simplicitywe disregardluctuations n the exchangerate.Moregenerally he option may be writtenon some indexx, such as a consumerprice index,where

againthe dynamicsof x aregiven by dx/x = adt+ G'dw,witha,a' constants.




is lognormallydistributed, .e., dx/xx= adt+u'dw, wherea, a' are constants.In

terms of our earlier notation, dx=1pdt+adw, where /t =ax and a=a'x. The

maturitydateof the optionis T and the exerciseprice s E. No dividendsarepaid.

Since no cash flows are received before time T, we set c(t) = 0. The boundary

condition is V(x, T) = max[O,x- E] and is independent of the market price of risk.Followingourrule, we proceedas follows:

Step one: Replace ,t by j*'p - Xpa= (a - Xpa')x. Price changes are now

described by dx/x = a* dt + o'dw, where a* = a - Xpa'.

Step two: Ignoremarketrisk and discount expectedcash flows at the risklessrate. Thus

V(x0,O)= erT (XT-E) dF(xT IxO)

whereF(xT I O) is the probabilitydistribution f x(T) conditionalon

x(O)= xo.Thesolutionof thisintegralgives8

V(xO, 0) xoe(a* -r)TN ln(xo/E) + (a* + a'2/2) T)

- TN ln(xo/ E) +(ca*- /'2/2)T) (10)

(J'~~~~~~(0

whereN(-) is the standarderror unction.

In the specialcase whenthe securityon which the optionis written, s traded n

the capital market without marketimperfections,by the CAPM (1) we obtain

a -r=Xpa' which implies a*=r. Then (10) becomes the familiar Black-Scholes

(1973)formula or the pricingof an option

V(x, O)= xoN ( ln(xo/ E) + (r + u'2/2) T)

-Ee-T ln (x?/ E ) +(r - a2/2) T)(1

C. Theoryof the Firm and Optimal Capital Structure

We consider a firm with equity Ve and a discountbond Vb at time t =0. The

bond maturesat time t= 1 and the firm liquidatesat the same time. We assume

that the firmmay take no action at times within the time interval (0,1) such as

issuingor buyingback equityand debt. Whereasthe latterassumption s restric-

tive, it is typicalof one-periodmodelsof the firm where the firm makesdecisions

only at the points in time t= 0, 1, and is made here solely for the purpose of

illustratinghe applicationof our rule to a particular lass of modelsof the firm.

8. This integral is explicitly evaluated in Sprenkle (1964), the appendix.




We denote by x(1) the value of the firmassets at t= 1, before taxes and beforethe repayment f the bond. If the firm s able to meetits obligation o bondholders,the bondholders re paidin full and the equityholderseceivethe remainingwealthafter taxes are paid. If thefirm s unableto meetits obligation o bondholders, hen

the firm is declaredbankrupt.Dependingon the specificmodel assumptions hefirm is liquidatedor reorganized, he equityholders eceive nothingand the bond-holders receive the residualwealth x(l) net of bankruptcycosts and possible taxpayments.Without discussingthe details of these modelswe may say that, givensome x(l), the stockholdersand bondholdersreceive at time one Ye(x(l)) andYb(x(l)) respectively, s functionsof x(l).9 The valuesof equityand debt at timezero are then the expected values of Ye(x(l)) and Yb(x(l)), respectively, discounted

for time and risk. Whereas earlier models had ignored the adjustmentof thediscount rate for marketrisk, more recently Rubinstein(1972) and Kim have

adjustedthe expected value of Yeand Yb for both time and marketriskthroughapplicationof the one-periodCAPM. With costly bankruptcy his adjustment s

computationallycomplex and it typically involves the determinationof partialmoments.On theoreticalgroundsthis procedures objectionable or the followingreason:Evenif the distribution f x(l) is normal, he distributions f Yeand yb arenot normal (at best they are truncated normal), and a one-periodCAPM isapplicableonly underthe assumptionthat investorshave quadraticutility func-tions.Gonzalez,Litzenberger nd Rolfo (1977)have recentlyanalyzedthe absurd-ity of the application of the one-periodCAPM to cash flows which are not

normally distributed.Specifically they showed that the implicit assumptionofquadraticutility function leads to an optimal capital structureof the firm, forwhich the firm returnsare stochasticallydominatedby the returnsof a firm with anon-optimal apitalstructure.

We shall demonstrate hat the adjustmentof Yeand Yb or marketriskthroughapplicationof our rule is straightforwardnd free from the fallaciesdiscussedbyGonzalezet al. (1977).We make two assumptions.Firstwe assume that the equityand debt are tradedcontinuouslyover the time interval[0,1]. Second we assumethat the random variablex(l) is distributedas follows: Let the random variable

z(t) be.defined bydz(t) = ,(z, t)dt + a(z, t)dw

z (0) =

wheredwis the incrementof a Weinerprocess.We definex(l) by

x(1)= z(1).

This definitionof x(l) coversa wideclassof distributions. n particular,f jt, a are

constant, x(l) is normallydistributedwith mean jt and variance a2;

ift =

zZC= z, whereii,a are constant, nx(1) is normallydistributedwith meanla a /2

and variance a2We applyour rule to thisproblemas follows.

9. For detailed specificationsof Ye(x(l)) and Yb(x(l)) see for exampleKraus and Litzenberger(1973)and Baron 1975)for one periodmodelsand Scott (1976) for a multiperiodmodel.




Step 1: We replace ,t by [* = y - Xpu.

Step 2: We discountYeand Yb as if the marketpriceof risk werezero, i.e.

Ye= e-rEYe(x(1))

and

Yb= e rEYb(X(l)).

Thisprocedure s free fromthe criticismof Gonzalezet al. becausenowhere s itassumed hatinvestorshavequadraticutilities.The procedures also computation-ally simplerthan the procedure nvolved in the applicationof the single-periodCAPM. Furthermore,our rule provides the following insight to problems incorporation finance: A t least in those cases where the cash flows are normally or

lognormallydistributed, considerationsof market risk may shift some of the modelparameters, but leave the structure of the model and the qualitative implicationsregarding optimalcapital structureunchanged.

V. VALUATION OF CONTROLLED PROJECTS

In general he managerof a projectmayinfluencethe cashflows of the project.Asan illustration,we assumethat the project s a machinewhichis characterized y asingle state variablex, wherex is some measure of efficiency.Assume that themachine deteriorates n a random fashion. Let u(t) be the expenditurerate for

maintenanceof the machine. The managerof the machinemay choose the timepathof u(t) and in this sense he controls the project.Maintenancenfluences thedeterioration of the machine as dx = ,(x, u, t)dt + a(x, u, t)dw. Maintenance also

influences the cash flow c(x, u, t)dt generated over (t, t + dt). The problem then is to

determine he maintenancepolicy u(t) which maximizes he valueof the streamofcash flows, V(x, t).

Underthe assumption hat IcI< so, Iul< oo, it is shownin the appendixthat Vsatisfies

max[c-rV+ V,+(II-Xpa)Vx+ a Vxx]=O (12)

which is an extensionof equation(7). The argumentwhichfollowsequation(7) isapplicable o equation(12) also. Thereforeour valuationruleappliesto controlledprojecfsas well.

As anotherapplicationto a controlledproject,we consider V(x, t) to be thevalueof a firmwhich is characterized y a singlestatevariable, heworkingcapitalof the firm,x. The controlvariable s the rate at whichdividendsarepaid,if u>0;it is the rate at whichthe firmraisescapital,if u<0. Then c= u, that is to say, u

standsfor the cash flows of the equityholders f this firm.Assume that the changeof the working capital level, dx=[t(x,u,t)dt+a(x,u,t)dw is the only stochastic

elementof this model. Then our rulespecifiesthat we may replacey by tt*= -

Xpaand ignoremarketriskthereafter,husgreatlysimplifying he discussion.We considernext a different ype of controlprocessand the applicabilityof our

rule. For purposes of illustration we assume that the "project"is the cash




management ystemof a firm: it generatesa stream of cash flows which are theholdingcost of cash, penaltycost of negativecash balances and transactions ostsin transferring ash in and out of interest bearingbonds. The firm controls thisproject, n the sense that it follows a cash managementpolicy with the goal to

minimize the present value of the stream of these costs. This model has beenextensively tudied n theliterature'0 ndertheassumption hatthe objectiveof thefirm is to minimizethe time-adjustedcosts of the system. The latter objectiveignores the adjustment f expected costs for marketrisk.We show that a straight-forwardapplicationof our rule bridgesthediscrepancybetweenthe two objectives.

Let x be the level of cash reservesat time t and V(x, t) be themarketvalue of allfuturecosts incurred n the operationof the system.The demandfor cash in time(t, t + dt) is dx = y dt+ adw. A fixed cost is incurred each time that bonds are

liquidated to increase the level of cash reserves and each time that cash is

convertedinto bonds. Assume that the cash policy is describedby three timedependentparameters (t) > xo(t)> x(t) such that the cash level is adjusted o xo(t)whenever x > (t) or x< x(t); and the cash level is not adjusted as long asx(t)< x < 5?(t). It can be shown" that the boundaryconditionson the functionV(x, t) do not involve the marketprice of risk: At those well defined instances(x > x(t) or x < x(t)) whenit is optimalto adjust he cashlevel,theadjustment ostis deterministic nd nothingin this processinvolves the marketprice of risk. Wenow applythe rule as follows:

Step one: Replace the drift parametert(x,t) of the demand for cash by

It*(x, t) = 4(x,t)-

Xp(x, t)a(x, t).Steptwo: Evaluate he streamof costs as if the marketpriceof riskwere zero.

Then themarketvalue costscoincidewith the time-adjusted xpectedcosts.

In otherwordsthe effect of marketriskon the marketvalue costs and on theparametersx(t), xo(t), x(t) of the cash policy is fully capturedby replacingtheactualdrift jt by the effective drift ,u*,and ignoringmarketrisk thereafter.Forexample,high positivecorrelationbetween the demandfor cash and marketreturn

has the same effect on the parametersof the optimal policy as low expecteddemandfor cash does, otherthings equal.

VI. CONCLUDING REMARKS

We have limited our discussion to diffusion processes (2) and have ignoreddiscontinuous tochasticprocesses n a continuous-timeramework, uch as Pois-son or Pareto-Levyprocesses. The reason for this omission is that Merton's

10. SeeConstantinides1976)for a reviewof cashmanagementmodels.

11. Theboundary onditionson the functionsV(x,t) areV(x(t), t) = K1+ V(xo(t), t)

V(x-(t), t) = K2+ V(xO(t), t)

and

vSe C st), t)= Vt (xo(t), t) = Vf(x (t), t)= O.

See Constantinides and Richard (1978) for the derivation of these conditions.




intertemporalCAPM (1) is derivedunder the assumption hat the rate of return sgeneratedby a diffusionprocess.We do not, as yet, have an intertemporalCAPMfor discontinuous tochasticprocesses.It must be stressed, hough,that the familyof diffusionprocesses 2) is quite rich,since yt and a are in generalfunctions of x

and t.We have also limitedour discussion to a continuoustime frameworkand have

excluded a discrete time framework.After all, in a discretetime frameworkaCAPMis applicable o everytimeperiod,and the marketvalueof a streamof cashflows may, in principle,be evaluated by the sequentialapplicationof the one-periodCAPM.Indeed thisprocedurehas been outlinedby Bogueand Roll (1974).Theoretical bjections o theapplicationof the discrete-timeCAPMwere discussedin our section IV-C. Additional difficultiesin the derivationof our rule areillustratedwhen we attemptto duplicate he developmentof sectionIII in discrete

time. Assume that time periods are of length 8 and that the state equation isxZ(t 8)- x(t) = wherethe distributionof z is known. The rate of returnon the

.~~~~z

project is

( V(x(t)+ z,t+ 8)- V(x(t),t)+ c}/ V(x(t),t)8. (13)

In section III we arejustifiedin expanding(13) in terms of a Taylorseries andderiving (3), because 8 is small (d->Oin continuous time) and the stochasticincrementsz follow the stochasticprocess (2). In discretetime6 is not necessarilysmalland the derivationof sectionIII is not valid.

It is helpfulto place our rulein perspectivewith relateddevelopmentsn finance.The Modigliani-Millerarbitrageand the CAPM are two powerful financialmethodologies,each howeverwith limitations: The arbitrageapproach requiresthat the securitiesunder considerationbelong to the same risk class whereas theCAPM approach requiresthat the assumptions,which lead to the CAPM, aresatisfied. Table 1 summarizessome developmentsin finance through the twomethodologies.

TABLE 1

THE ARBITRAGEAPPROACHAND THECAPMAPPROACH N SOMEDEVELOPMENTSN FINANCE

Arbitrage Approach CAPM approach

(Requires the exist- (Conditional on the

tence of a risk assumptions which lead

class) to the CAPM)

Modigliani-Miller Modigliani and Miller Hamada (1969), Mossin (1966)

propositions and others

(no tax)

Option pricing Black and Scholes (1973) Black and Scholes (1973)

Merton (1970)

Generalized project Cox and Ross (1976) This paper

valuation: Reduc-

tion to a riskless

world.




Blackand Scholes(1973)observed that over an infinitessimal ime interval anoptionand its underlying ecurity i.e. the securityon whichthe option is written)belong to the same riskclass. They applied an arbitrageargumentand evaluatedtheoptionin termsof theunderlying ecurity.Cox and Ross (1976)generalized his

approachand noted that, if a financial claim is spannedby securitieswhich aretraded in the market, he financial claim may be evaluatedthroughthe arbitrageapproach,whereinvestors'preferencesonly play the indirectrole of determiningsome equilibrium arameter alues. In particular he claimmay be evaluated n ariskneutralworld,thus considerably implifying he valuation.

A paralleldevelopmenthas taken place through he CAPM and withoutassum-ing the existenceof a risk class. Hamada(1969),Mossin (1966) and othersderivedthe Modigliani-Millerropositions hroughthe CAPM. Black and Scholes (1973)and Merton(1970) derived the option pricingdifferentialequation through the

CAPM. However,if the underlying security is traded in the market,then thesecurityand the option belong to the same risk class, and the option pricingdifferential quation s valid, evenwhen the assumptions, eading to the CAPM, donot hold. For this reasonthe derivationof the option pricingequationthrough hearbitrage rgumentdominates he derivation hrough he CAPM, provided hat theunderlying ecurity s traded n the market.

The rule developed n this paperbears some similarity o the Cox-Ross rule inthe sense that both rules reduce a valuation problem under market risk to avaluationproblem n a riskneutralworld. Neither of the two rules dominatesthe

other: Our rule is limited to situations where the CAPM is valid, while theCox-Rossprocedure s limitedto situations where a risk class exists. Thus ourexamplein section IV-B on option pricingmay not be solved by the Cox-Rossprocedure,as long as the underlying ecurity s not traded n the market;and ourexamplein section V on cash managementmay not be solved by the Cox-Rossprocedureas long as a securityor portfolio does not exist which perfectlyhedgesagainstthe variabilityn the demandfor cash.

APPENDIX

Let V(x, t) be the market value of the project, if the project is optimally controlled.Assume that the project is controlled by u dt in time (t, t + dt) and is optimally

controlled thereafter. Since the control u is not necessarily optimal, the marketvalue of the project at t is J(x, t), where J(x, t) < V(x, t). The value of the project at

t + dt is J(x + dx, t + dt), where J(x + dx, t + dt)= V(x + dx, t + dt). The rate of

return on the project is

V(x + dt,t + dt) + c dt - J(x, t) V(x, t)-J(x, t)

J(x, t) J(x, t)

J(X,t) (c+ V+ 2VX+2 VXX)dt+ dw (Al)

withexpectedvalue perunit time'ap,and covariancewith the marketper unit time




aPMgivenby

ap J(x,t))() + ( +t) c V++ 2Vxx) (A2)

and

aPM PaMaVX/ J(x, t). (A3)

If the project s in equilibrium n the capital market, t satisfies the intertemporalCAPMof equation(1). We substitute A2) and (A3) in (1) and obtain

V(X, t) -J(X, t) +V - =Xpc;Vx

J(x t)dt + J( l t) (c+ V,Kx 2 Vxx)-r= J t) . (M)

But V(x, t) - J(x, t) > 0 and the above simplifies nto

J(x t)kc+ + 2vx+2Vxx) -rA J(x t) or

c - rJ(x, t) + V,+ (u - Xpc)Vx+ 2 V< 0, or

cr V+ V,+ (u -Xpa) Vx+-

vx< ?(5

2 xO . (A5)

If u is optimal hen V(x, t) - J(x, t) = 0 and(A5)is satisfiedas an equality.Thus the

optimalu is givenby

max c-rV+V,K+(u-Xpa)Vx+ a Vxx]=. (A6)

REFERENCES

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Constant in Ides (1978) - Market Risk Adjustment in Project Valuation

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