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A Simple Discounting Rule

Author(s): Fischer BlackReviewed work(s):Source: Financial Management, Vol. 17, No. 2 (Summer, 1988), pp. 7-11Published by: Blackwell Publishing on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665521 .

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A S i m p l e Discountingu l e

FischerBlack

FischerBlack s apartner t Goldman,Sachs&Company, ewYork,NY.

0 How can we discount uturecash flowsfrom either

an actual or proposedinvestment?When the futurecash flows arecertain,we discountthemat a risklessratewith anappropriatematurity.nprinciple,we canobservethe rate we wantby lookingat the priceof a

purediscountbond.In practice,however, uch bonds

maybeunavailable, ndpricingacouponbondmaybe

complicatedbytaxfactorsand call features.Evenso,we can probably ome up with a reasonableapprox-imateprice.

When he futurecash lowsareuncertain, iscount-

ingbecomes morecomplex. ngeneral, he right pro-cedurewill be one that dependson the covariance

between the cash flow and aggregateconsumption,which can be highlystate-dependent. In a simplermodel,the discountingprocedurewill dependon the

covariancesbetween the change n the expectedcash

flowandthe market eturnsbetweennowandthe timeof the cashflow.2

In certainspecialcases, the correct rule is to dis-count the expectedcash flow at a constantrate that

dependson the beta of the cash flow'svalueand themarket'sexpectedreturn.This rule requires hat weknowthree itemsbesidespurediscountbondprices:the expectedcashflow,thecashflowbeta,and the ex-

pectedreturnon themarket.All are difficult o estim-ate.3If we startwith thejoint distribution f the cashflowandpriormarket eturns,wewillusethe betaandthe expectedmarketreturn n estimatingheexpected

cashflow; henwewillreusethem to discount t.Undercertain onditions hese two usesforthe betaand the expectedmarketreturnwill canceleachotherout. Thepresentvalue of the cashflowwill thenbe in-

I amgratefulto RicharadRuback for many helpful discussions of this

topic, and to CarlissBaldwin,Eugene Fama, Michael Jensen, PhillipJones, MarkLatham,Robert Merton, StewartMyers,Stephen Ross,and William Schwert for comments on an earlier draft.

IForexamplesof models in which this is true, see [2,8,9,11,14]. Cor-nell [5] points out some practicaldifficulties in using this procedure.

2For example, see Brennan [4], Treynor and Black [15], and Fama

[7].

3These difficulties are discussed by Myers and Turnbull [12] and byFama [7].

7

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8 FINANCIAL ANAGEMENT/SUMMER988

dependentof both the beta andthe expectedmarketreturn.We cansolve for the presentvaluebyestimat-

ing the cashflow-assuming thatall priormarketre-turnswereequalto the interestrate.Thenwediscountthe cashflow at the same nterestrate.

Inpractice, hismaybe aneasy ruleto use. We donot estimatea joint distribution f cash flows with

priormarket eturns.Wejust estimate he cashflowsfor one possiblesequenceof market eturns. We dothis assumingthat subnormaleconomic conditionscause themarket eturno be only equalto interest,attherisklessrate, n eachperiod).Thenwe discountatthe interest ate.We do not even haveto estimate he

expectedcash flows. A simplerway may be to esti-

mate the cashflow assumingan arbitraryeta and anarbitraryxpectedmarket eturn.Thenwe discountata rate thatuses the same beta andthe sameexpectedmarket eturn.

It is importanto start with conditionalexpectedcashflows, rather hanunconditional nes. It is easierto formconditional xpectations nd it certainlyakesless effort o discount onditional xpectedcashflowsthanunconditional nes.

I. AnExampleAn examplewill helpus to explainhow to use the

rule.Assume hat he short-termiskless nterest ate sconstantat 10%peryear,and that heexpectedreturnon themarkets somehowknown o beconstant t20%

per year. Now supposethata firmis consideringaninvestmentwitha cashpayoffat the endof eachyearthatdependson themarket eturn or thatyear.Whenthe marketreturn s 10%, the payofffor the year is

$150,000. When hemarket eturn s 20%,thepayoffis $250,000. The payoff increases or decreasesby$10,000for eachpercentage oint ncrease rdecreaseinthe market eturn or theyear.Eachyear,thepayoffwill dependon thatyear'smarketreturn,but not on

returns orprioryearsor onanything lse. Thepayoffswill follow thispattern earafteryearinto the indefi-nite future.

To estimate he presentvalue of the payoffsfromthis nvestment,we use thesimplediscountinguleandignore he20%expectedreturn n themarket,makingno attempto estimate he betaof theproject,noranyof its cashflows. We assume hateachyear's payoff s$150,000, because that is what it will be when thereturnon the market s equalto the constant nterestrate, 10%. Then we discount a perpetual treamof$150,000payoffsat 10%,whichgives a present alueof $1,500,000.

We can also value thesepayoffsstarting romthe

expectedpayoffsandusing the payoffbetas. In this

example, hebeta oreachpayoffdepends n time: t ishighintheyearbefore hepayoffdate,butzero before

that,since the payoffdependsonly on the market e-turn n theyearbeforethepayoff.Thus,thediscount-

ing procedurehatusesbetas s complex so complexthat here s noeasyway to show how it applies o this

example.

II.Excess ReturnsLet's assumeaworldwhere hesimplecapitalasset

pricingmodelholds neach nfinitesimal eriod.Thereare no taxesor transaction osts.Supposeyouareof-

fereda cashflow at the end of a shortperiodstartingat timet,wherethe dollaramountof the cash lowwillbe equal o thepercentage xcessreturnon themarketon a dollar investmentover that period.The excessreturn s the market eturnminus nterestattheshort-termriskless nterestrate for the period.Whatwould

youpayforthis cashflow?You wouldpaynothing.You canget the samecash

flow withoutputtingup anymoney.You simplybor-row to investin the marketat the startof the period.Sinceyou put up no money,the presentvalue of the

payofffrom this investmentmust be zero. Thus the

presentvalueof the excess returnon the marketoversome futureperiodmustbe zero.4Similarly, he pres-ent valueof any multipleof the excessreturnon themarketmust be zero. You just borrowa differentamounto invest nthemarket. nfact,theamount ouborrow andepend n anyway onpastmarket eturns.Thepresent alueof a multipleof themarket's xcessreturn,where the multiplierdepends n any way on

pastmarket eturns,will be zero. Thesame is trueofexcessreturns nsecurities ther han he market.Andthemultiplierandependon anyvariables hatwill beknownat timet, even though heymaybe unknown

today.

Ill. JointNormality nd OtherDistributions

In thesimplest asethe cash lowandmarket eturnsin prior periodsfollow a joint normaldistribution.Thusthe cashflow is a constant,plusconstants imesthe variousmarket eturns, lusa random ash low n-

4This is a special case of the arbitrageresult obtained by Ross [13].Margrabe [10] has a very similar result.

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BLACK/A IMPLEDISCOUNTING ULE 9

dependentof all the marketreturns.The expectedvalue of the random ash flow is zero.

Thevalueofasum s thesumof thevalues.Thevalueof a constant s obtainedbydiscounting he constant

usingthepriceof a discountbond.The value of a con-stantmultipleof the marketreturn s the value of thesame multipleof the marketexcessreturnplus thevalue of the samemultipleof interestfor the period.Since the excessreturnportionhasavalueof zero,we

just discount he interestportionusingthe priceof adiscountbond. The value of the randomcashflow iszero.

In otherwords,we startwith a cash flow writtenasa linear functionof marketreturns.We replaceeach

market eturnbyinterestat the risklessrate or theap-propriateperiod,and discountby multiplyingby the

priceof a purediscountbond that maturesat the timeof the market eturn.Weignore hepurely andompartof the cash low.Since heperiodsare nfinitesimal,wedo notworry bout hedifference etween hestartandtheend of the period.

Nowsupposethat the cash flowis a constant imesone plusthereturnonthemarket ortheentireperiodbetweennowandthe time of the cash low.5 fweknowin advance he varianceand expectedreturnson the

market, he resultingdistributionwill be lognormal.

Wewill not be able to describe t asjoint normalwiththe marketreturns n priorperiods.But the presentvalue of a constanttimes one plus the marketreturnoveranyperiodmustbe thatconstant,becausewecan

duplicate he payoffby investing hat constant n themarket.

We can also solveforthat constantby assuminghemarketreturnequalsthe interest n eachperiod,andthen discountusingthe sameinterest rates.Thus the

simpleruleworks or the lognormal aseas well as the

joint normalcase.Infact, tworkswhether he marketreturn s lognormalor not. It worksfor any process

governing he marketreturn.The presentvalue of aconstant imesone plusthe marketreturn s that con-stant.

Byasimilarargument,hesimplerulealsoworks orsecurities other than the market.It works when the

cash low is a constant imesoneplusthe returnon anysecurityor the entireperiodbetweennowand he timeofthecash low.It evenworks or anarbitrary ortfolio.The cash flow can be a constant times one plus the

returnon anyportfoliofor the entireperiodbetweennowand the timeof thecashflow.Thecompositionof

the portfoliocanbe changing onstantly.

IV.Multiplicative evisionsAssumethatwe revise ourexpectationabouta fu-

turecashflowaccordingo thefollowingrule.6Theex-

pectedcash flow at the startof each periodtimes arandom evision actor s theexpectedcash flow at theendoftheperiod.The revision actoralwayshasamean

of one,and ollowsajointnormaldistributionwiththemarketreturn or the sameperiod.At the end of thelast period,the expectedcash flow equalsthe actualcash flow. Successive evisionsare ndependent.

The one-periodversion of this case has been dis-cussedabove. It is ajointnormalcase.)Note that the

expectedcash flow conditionalon the marketreturn

being equalto interest or theperiod s lower hanthe

expectedcashflow. Wedo not need the expectedcashflow at all. We simplydiscount the conditional ex-

pectedcashflow at theriskless nterestrate.We repeat heprocess or the firstperiodof a two-

periodexample.The valueof thecashflow atthe endof thefirstperiod sjointnormalwiththe firstperiod'srevision.To find hevalueat the startof thefirstperiod,wefigure heexpected alueconditionalon the market

returnbeing equalto riskless nterestfor the period,and discountat the interest rate. We can repeatthis

process oranynumber f steps.Thesimpleruleworks

again.We find theexpectedvalueof the cashflow,as-

suming hat he actualmarket eturn quals nterest orthe periodforeveryperiodbetweennow and the timeof the cash flow. We then discount this value usingthosesame nterestrates.

V.OptionsWhen won't the simple discountingrule work?

When the cash flow is a non-linear unction of themarketreturn n anygiven period,holdingfixed the

marketreturns n otherperiods.This means the rulewill not workfor an option.7An optionon one plus

5This is a special case of the resultobtained by Margrabe[10] and byRoss

[13].

6This is a rule discussed by Fama [7].

7This is pointed out by Myersand Turnbull [12]. A more general dis-

counting procedure that does applyto options is given in [4,6,14,15].A still more general procedure is derivedby Banzand Miller [1].Theyuse the notion of estimatingcash flowscontingent on what the market

does in a more general way than I. Thus the simple discounting rule

can be regarded as a special case of their analysis. It can also be

regardedas anapplicationof Theorem 2in Breeden and Litzenberger

[3].

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10 FINANCIAL ANAGEMENT/SUMMER988

the marketreturn, or example,hasa payoffequaltothe truncateddistributionof possibleend-of-period

valuesof the marketportfolio.The untruncateddis-tributioncanbe discountedwith the simplerule,butnot the truncateddistribution.The option alwayshasa positivevalue,but applying he simplediscountingrule to the truncated istributionwillgivea zerovalueiftheoption'sexerciseprice ssufficiently igh.So,the

simplerulewill notwork.

VI.Estimationand ExtensionsMostthinkthatit is easierto form conditional x-

pectationshanunconditionalnes. Why?If we formourconditional xpectationsby doingregressionson

pastdataandplugging ntheconditionswe want,whycan'twejusttakethesamplemean rom hesamepastdata,anduse it for anunconditionalxpectation?t isofteneasier oestimateheslopeof arelationnvolvingsecurityreturnshanto estimate ts intercept,buttheconditional xpectationdependson boththeslopeandtheintercept.Wheyshouldthe interceptbe estimated

anymoreaccuratelyhanthe samplemean frompastdata?

Perhapsmost do not thinkof formingexpectationsfrompastdata n thisway. Maybe heystartwith ideasaboutwhat hecashflow willbeunder arious tatesof

theworld,multiplyby theprobabilities f the states,andsum.Thiswill give theunconditionalxpectation.If this is whatpeopledo, thenforminga conditional

expectationwill be easierthanformingan uncondi-tionalone; theywill not need as many probabilities,norneed to do as muchsumming.Or,perhapsmanag-ersknow the linearrelationbetween he cash flow and

priormarket eturns. n thatcase, theymustknowthe

expectedmarketreturns o know the unconditional

expectedcashflow. Itis easier o form he conditional

expectedcashflow sinceyou do not needto estimatethe expectedmarketreturn or eachperiod.

It turnsout, though,

that thepresent

value of thecashflowis independent f whatwe assumeaboutthe

expectedmarket eturnn this case.Thehigher he ex-

pectedmarketreturn, he higherthe expected uturecash low,butthehigher he discount ate.Thuswecan,if wewish,follow the modifiedprocedure. magineaneconomic cenario, stimate ts cash low,andcalculatethe marketreturn or it. Thendiscount he cashflow

usinga ratebasedon anexpectedmarketreturn qualto that market eturn.

This scenariomaybe easierto imagine ince t neednot be as drearyas one wherethe marketreturnonlyequals nterest n eachperiod.It maybe easier to es-

timate the cash flow conditionalon good economicconditionsand then to discount as if the expected

marketreturnwereequalto the actualmarketreturnfor those conditions.Using this modifiedprocedure,the betawe use will not matteras long as we use thesamebetaforprojectinghe cash lowand or discount-

ing.This meanswewill not have oworryabouthavingjustthe rightbeta.

Outlinedbelow are somepossibleextensionsof thebasicrule.

(i) In a simplemodel with taxes,wherepersonalandcorporate ax ratesare the same,but wherefirmspaynodividends nd ndividuals aynocap-italgains axes, herulewoulduse theafter-taxn-

terest rate. We would first estimate cash flowsconditionalon marketreturns hat arealwayse-

qual o interestat the after-tax ate.Thenwe woulddiscountat the after-tax nterestrate.

(ii) In morecomplexmodelswhere here s anop-timal capitalstructure,and wheredividend and

capital gains taxesmatter,the rule will be more

complex.If we areworriedabouta farmerwhosecash lowsdependoncommodity ricesandcorre-

sponding uturesprices,andnot inanyotherwaysonthestockmarket eturn,henwe canusea mod-ifiedrule.Weestimate he farmer's ash flowsas-

sumingall futurespricesremainunchanged.Thenwe discountas before.

(iii) Finally,we can use the consumptionversionof thecapitalassetpricingmodel.8 n this caseweestimatethe cash flowsassuming hat a portfoliowhosereturnsperfectly orrelatedwithtotalcon-

sumptionhasa return n eachperiodequalto in-terestatthe rateappropriateorthatperiod.Thenwe discountas before.

References1. R.W. Banz andM.H. Miller, "Prices orState-ContingentClaims:

Some EstimatesandApplications,"Journal of Business (October

1978), pp. 653-672.

2. D. Breeden, "AnIntertemporalCapitalAsset PricingModel with

Stochastic Consumption and Investment Opportunities,"Jour-

nal of Financial Economics (September 1979), pp. 265-296.

3. D.T. Breeden and R.H. Litzenberger, "Pricesof State-Contin-

gent Claims Implicit in Option Prices," ournalof Business (Oc-

tober 1978), pp. 621-651.

4. M.J. Brennan, "AnApproach to the Valuation of Uncertain In-

come Streams,"Journal fBusiness.(October 1978),pp.653-672.

8See, for example, Breeden [2].

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BLACK/A IMPLEDISCOUNTING ULE 11

5. B. Cornell, "The Consumption Based Asset Pricing Model: A

Note on Potential Tests and Applications,"Journal of Financial

Economics (March 1981), pp. 103-108.

6. U. Dothan and J.Williams, "Term-RiskStructures and the Valu-

ation of Projects,"Journalof Financial and QuantitativeAnalysis

(November 1980), pp. 875-905.

7. E.F. Fama, "Risk-AdjustedDiscount Rates and Capital Budget-

ingUnder Uncertainty,"JournalofFinancial Economics (August

1977), pp. 3-24.

8. J.B. Long, "Consumption-InvestmentDecisions and Equilibri-um in the Securities Market," n Studies in the Theoryof Capital

Markets,M. C. Jensen (ed.), New York, Praeger, 1972, pp. 146-

222.

9. R.E. Lucas, Jr., "Asset Prices in an Exchange Economy," E-

conometrica (November 1978), pp. 1429-1445.

10. W. Margrabe, "Alternative Investment Performance Fee Ar-

rangements and Implications for SEC Regulatory Policy,"Bell

Journal of Economics (Autumn 1976), pp. 716-718.

11. R.C. Merton, "AnIntertemporal Capital Asset Pricing Model,"

Econometrica (September 1973), pp. 867-887.

12. S.C. Myers and S.M. Turnbull, "CapitalBudgeting and the Capi-tal Asset Pricing Model: Good News and Bad News,"Journal ofFinance (May 1977), pp. 321-332.

13. S.A. Ross, "A Simple Approach to the Valuation of Risky

Streams,"Journal of Business (July 1978), pp. 453-475.

14. M. Rubinstein,"The Valuation of UncertainIncome Streamsand

The Pricing of Options," Bell Journal of Economics (Autumn

1976), pp. 407-425.

15. J.L. Treynorand F. Black, "Corporate nvestmentDecisions," in

Modern Developments in Financial Management, S.C. Myers

(ed.), New York, Praeger, 1976, pp. 310-327.

ASSISTANTR ASSOCIATEPROFESSORF R E A L ESTATE

A N D FINANCE

The Universityof Connecticuts seekinga doctorallyqualifiedcandidaten real estate orrelated ield(mustanticipate efenseprior o9/1/89). ForAssociateProfessor onsideration,musthave active researchprogramwith high qualityscholarlypublications.Must havedemonstratednterestandcompetence n teachingat all levels including he Ph.D. Primaryteachingareais real estate (decisionmaking/valuation/finance);econdary inance fieldsdesired:eithercorporate, nstitutions, nternational, isk and insurance,or investments.Service to and interactionwith real estate

community/professionequired.Tenure

track;start9/1/89. Submitresumeby 12/23/88forpreferentialcreening,oruntilposition s filledto: ProfessorKeith B. Johnson,Head,Department f Finance,SBA U-41F, UniversityofConnecticut,368 FairfieldRoad,Storrs,CT06268. Womenandminoritiesareencouragedto apply:an EEO/AA institution. Search #8A373).

TI I -1

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