equivalent

n

EquivalentAnnualAnnuity

Mutuallyexclusivealternativesforperformingsomervicetypicallyhaveunequallives.

Forexample,suppoanexpressmailrviceproviderhastoaddtoitsfleetofplanesto

plicity,assumeithastwochoices,a

nity,

thenet

prentvalueanalysisofthetwoalternativeswillmissthepointthattheinvestment

projectis"providingexpandedairfreightcapacityfortheindefinitefuture."The

equivalentannualannuityistheappropriatetoolforthisproblem.

Theequivalentannualannuityforaninvestmentisthelevelannuityoverthe

investment'slifethathasaprentvalueequaltotheinvestment'

analternative

alt

thathaslifelengthofnyears,realrequiredrateofreturn

real

randnet

prentvalueofNPValt

,theequivalentannualannuityinrealtermsis

real

EA=,

real

NPValtPVAFrn

(1)

where

real

EA

standsforequivalentannuity(inrealterms)and,

real

PVAFrn

isthe

prentvalueannuityfactorforthegivenrate

real

r

nuityfactor

isgivenintheconddisplayedexpressionandinfootnote5onpage41in(BMA).

Usingthixpression,(1)becomes

real

EA

=





1

11

n

realreal

n

real

NPValtrr

r





.(2)

Goingbacktotheoriginalproblem,foreachalternativeforsupplyingthervice

wecalculateits

real

EA

andwechoothealternativethathasthelargest

real

EA

.Itisthat

n

alternativethatmaximizesthenetprentvalueofsupplyingtherviceindefinitely.

Suppointheexpressmailrviceexamplethattheappropriaterealrateofreturnis5%

andthenetprentvalueoftheudairplaneis$150,000,butbecauofitslongerlife

thenetprentvalueofthenewairplaneis$275,haveNPVud=

150,000andNPVnew=275,enetprentvalueanalysiswouldindicate

thatthenewplanewouldmaximizenetprentvalue.

Butremember,theexpressmailrvicefirmneedsthixpandedairfreight

ouldconsiderusingtheudairplanefor8

yearsandreplacingitwithanotherudairplaneforanother8years,rly,we

shouldconsiderusingthenewairplanefor20yearsandreplacingitwithanothernew

airplaneforanother20years,evalueaddedofeachalternativeisthenthe

lueaddedisthesameas

theprentvalueoftheperpetuityoftheequivalentannualannuitygivenby(2).

Thiscomputationfortheudairplaneis

real

EA

=





8

8

150,000.051.05

1.051

=

150,000

6.4632

=$23,208.

Thetruevalueaddedofthisalternativeistheprentvalueofaperpetuityof$23,208or

NPV

=23,208.05=$464,165.

Thesamecomputationforthenewairplaneis

real

EA

=





20

20

275,000.051.05

1.051

=

275,000

12.4622

=$22,067.

Therefore,thetruevalueaddedofthenewairplanealternativeistheprentvalueofa

perpetuityof$22,067or

n

NPV=

22,067.05

=$441,334.

Usingtheudairplanetosupplytheexpandedairfreightcapacityfortheindefinite

ntvalueannuityfactors

ethefactors

in(BMA)AppendixTable3,page970.

InSection6.3in(BMA)theconceptofequivalentannualcostisudtorank

alternativesforsupplyingsomervicewherethealternativesforsupplyingthervice

differintermsoflifelengthandcostsbutdonotdifferintermsofrevenuestheygenerate.

Forsuchproblems,NPValtcanbejusttheprentvalueofcostsofeachalternative

alt

andinthisca,(2)ision

thatmaximizesvalueaddedisthenthealternativethathasthesmallestequivalentannual

cost.

Onthebottomofpage127andthetopofpage128,(BMA)cautionabout

calculatingequivalentannualcostusingnominalcashflowsandthiscautionis

visioncalculatinganominalannuityusingonelifelengthasin(2)

aofnominalcashflowsyoucannolongeruthe

NPV

foronelife

lengthlikewedidin(1)and(2)tallow

NPV

togrowattherateof

inflationratebedenotedbyi,let

nom

r

denotethenominalrequiredrateofreturn,and

recalltherelationshipbetweenrealandnominalratesofreturn

1

nom

r

=11

real

ri

.(3)

Seepage50andfootnote10,page128,in(BMA).

n

Again,letNPValtdenotethenetprentvalueofonealternativealtfor

ndofthefirstn

years,ifthealternativeisrepeatedallcostsandbenefitswillhaverinbytheinflation

rateandthenetprentvalueofthereplacementwillbe1nNPValti.Attheendof

anothernyears,ifthealternativeisreplaced,itsreplacementwillhavenetprentvalue

of21nNPValti,prentvalue,innominalterms,oftheinfinite

quenceofreplacementsis

NPV

=













2

2

111

111

nnkn

nnkn

nomnomnom

NPValtiNPValtiNPValti

NPValt

rrr









=











2111nnkn

realrealreal

NPValtNPValtNPValt

NPValt

rrr





(4)

=

2

111

1

111nnkn

realrealreal

NPValt

rrr















=

1

1

1

1

n

real

NPValt

r

























(5)

=





1

11

n

real

n

real

NPValtr

r





.(6)

Toget(4)weudtherelationshipin(3)togetthat





1

1

n

n

nom

i

r





=



1

1n

real

r

.

Toget(5)weudtheformulaforthesumofapowerries

n

21xx

=

1

1x

forany

,01xx

(efootnotes3,4page40in(BMA)),whereherex=11n

real

r.

Thatisalotofworkbuttheresult(6)iswhatwewant,thenetprentvalueinnominal

termsoftheentirequenceofreplacements.

withthe

onvertthis

enominalamountthat

hasprentvalueinperpetuitythatisthesamenetprentvalue,wherethevalueofthe

(6),

letting

nom

EAdenotethiquivalentannualnominalannuity,wehavethatbydefinition

nom

nom

EA

r

=

NPV

=





1

11

n

real

n

real

NPValtr

r





.(7)

Solving(7)for

nom

EAgives

nom

EA

=





1

11

n

nomreal

n

real

NPValtrr

r





.(8)

Sochoosingthe

alt

thathasthelargest

NPV

isthesameaschoosing

alt

thathasthe

largest

nom

EA

,solongasall

alt

shavethesamerequiredrateofreturn.

Itisalsothecathatchoosingthe

alt

withthelargest

nom

EA

isthesameas

choosingthe

alt

withthelargest

real

EA

.Toethis,lookatequation(2)(2)

and(8)givesusthat

nom

EA

=





1

11

n

nomreal

n

real

NPValtrr

r





=





1

11

n

realreal

n

real

NPVrr

r





nom

real

r

r







n

=

real

EAnom

real

r

r







.(9)

Solongastheratio

nomreal

rrispositive,rankingaltsonthebasisofequivalentannual

annuity

ratiomaybegreaterthanoneiftheinflationrateispositiveorlessthanoneifthe

inflationrateisnegative.

Noticethatby(7)thevalueofthenominalperpetuityisthe

NPV

ofthe

alt

.

Dividingbothsidesof(9)by

nom

r

givesthesatisfyingresultthat

nom

nom

EA

r

=

NPV

=real

real

EA

r

.(10)

Thevalueadded,thenetprentvalue,ofthe

alt

isthesame,theprentvalueofthe

perpetuityofequivalentannualannuity,whetherthatannuityisinrealornominalterms.

Lookbackattheexpressmailrviceexampleandassumethattheinflationrate

is3%.Using(3)abovegivesthat1

nom

r=(1.05)(1.03)=1.0815sothat

nom

ris8.15%.In

thiscawecanuourpreviouscalculationsof

real

EAforeachalternativetocalculate

nom

EA

udairplanethiscalculationis

nom

EA

=

real

EAnom

real

r

r







=23,208

.0815

.05







=23,208(1.63)=$37,829.

Forthenewairplane,thecalculationis

nom

EA

=

real

EAnom

real

r

r







=22,067(1.63)=$35,969.

n

Again,theudairplanealternativehasthelargestequivalentannualannuityinnominal

termsbecauthiquivalentannualannuityis1.63timestheequivalentannualannuity

inrealterms.

References:

(BMA)y,,,PrinciplesofCorporateFinance,8

th

ed.,

McGraw-Hill,Inc.,icular,ection6.3foraprentationofthe

equivalentannualcostconceptwithalltheappropriatecautionsaboutitsuextendingto

thebroaderconceptofequivalentannualannuity.

Thematerialhereontheconceptofequivalentannualannuityistakenfrom

n,CapitalBudgetingandLong-TermFinancingDecisions,3rded.,

DrydenPress,1999,Chapters7and10.