Annuities:Futurevalue&PresentValue

ofanordinaryAnnuities

Department of Mathematical SciencesFaculty of Science

SSCM4863Room: C10 336/C22 441

Tel: 34321/34274/019-7747457

http://science.utm.my/norhaiza/

Annuities• Definition• Futurevalueofanordinaryannuity• Presentvalueofanordinaryannuity• Annuitiesdue• Perpetuities• Deferredannuities• Summaryofannuities

Annuities

• Anannuityisasequenceofperiodicpaymentso Oftenequalinamounto Madeatequalintervalsoftime

• Example:o Monthlyrentpaymentso Annualpremiumsforalifeinsurancepolicyo Monthlyhousingloanrepaymentso Regulardepositsinasavingsaccount

Paymentperiod=TimebetweensuccessivepaymentsTermofanannuity=Timefromthebeginningofthefirstpaymentperiodtotheendofthelastpaymentperiod

TypesofAnnuities

AnnuityCertain• HasaspecificstatednumberofPayments• Termofannuity isfixed• Datesofthe1st andlastpayments

arefixedEg.Housing loanrepayment

ContingentAnnuity• Hasnofixednumberofpayments• Termofannuitydependsonsome

uncertainevent

Eg.Lifeinsurancepayments(stopswiththedeathoftheinsured);Bond interestpayment.

OrdinaryAnnuity• Paymentsmadeattheendofeach

paymentperiodEg.Loanrepayment

AnnuityDue• Paymentsmadeatthebeginning of

eachpaymentperiodEg.Insurancepremium

Annuities• Definition• Futurevalueofanordinaryannuity• Presentvalueofanordinaryannuity• Annuitiesdue• Perpetuities• Deferredannuities• Summaryofannuities

FutureValueofanordinaryannuityDefinition• Amountdueattheendoftheterm

Exampleofordinaryannuityonatimediagram:

Time 0 n1

R

2

R

3

R

n-1

R

1period

term

Focaldate

• Interestperiodequalspaymentperiodasunitofmeasure

• Rastheregularpaymentateachperiod

FutureValueofanordinaryannuity

Wecancalculatethevalueoftheannuityatthefocaldatebyrepeatedapplicationofthecompoundinterestformula

Example1Findthefuturevalueofanordinaryannuityconsistingof4annualpaymentsofRM250eachat3%pa

Time 0 1

RM250

2

RM250

3

RM250

4

RM250

FV?

Valueof1aRM10KNow = 5167.19 1 +0.02 -./ = 𝑅𝑀4074.29

Valueof1st

paymentRM250(1)at4= 250 1 + 0.03 4

= 𝑹𝑴𝟐𝟕𝟑.𝟏𝟖

Valueof2nd

paymentRM250(2)at4= 250 1 + 0.03 /

= 𝑹𝑴𝟐𝟔𝟓.𝟑𝟐

Valueof3rd

paymentRM250(3)at4= 250 1 + 0.03 .

= 𝑹𝑴𝟐𝟓𝟕.𝟓𝟎

Valueof4th(final)paymentRM200(4)at4

= 𝑹𝑴𝟐𝟓𝟎

Thus,thefuturevalueoftheordinaryannuity (basedonendof termasfocaldate)=RM273.18+RM265.32+RM257.50+RM250=RM1045.91

FutureValueofanordinaryannuityAnalternativewaytocalculatethevalueofanordinaryannuityisusingthesumofgeometricprogression.

Consideranordinaryannuityofn paymentsofRM1eachasshownbelow

Focaldate

nn-2

RM1

Time 0 1

RM1

2

RM1

3

RM1

n-1

RM1 RM1

Here,tocalculatetheFVofthisannuity,weneedtoaccumulateeachpaymentofRM1totheendofthetermoftheannuity(ie.Focaldate)andaddthemtogether(similartoExample1)

FutureValueofanordinaryannuity(cont’d)

Focaldate

nn-2

RM1

Time 0 1

RM1

2

RM1

3

RM1

n-1

RM1 RM1

Thesumfromtheaccumulation foreachpaymentofRM1tothefocaldatecanbeexpressedas

𝐹𝑉 = 1 + 𝑖 (C-.)+ 1 + 𝑖 (C-/)+ 1 + 𝑖 (C-4)+… + 1 + 𝑖 4+ 1+ 𝑖 /+ 1 + 𝑖 .+1

wherethe1st paymentattheendoftheyearearnsinterestfor(n-1)years;the2nd paymentfor(n-2)yearsetc.Reordering theorder, wecanexpresstheequationaboveas:

𝐹𝑉 = 1 + 1 + 𝑖 . + 1 + 𝑖 / + 1 + 𝑖 4 + ⋯+ 1 + 𝑖 (C-4)+ 1 + 𝑖 (C-/)+ 1 + 𝑖 (C-.)

FutureValueofanordinaryannuity(cont’d)

Geometricprogression withntermssimilartothefollowingexpression.

𝐹𝑉 =𝑎 𝑟𝑛− 1(𝑟 − 1)

=1 (1 + 𝑖)𝑛− 1((1+ 𝑖) − 1)

=1 + 𝑖 𝑛− 1

𝑖

𝐹𝑉 = 1 + 1 + 𝑖 . + 1 + 𝑖 / + 1 + 𝑖 4 + ⋯+ 1 + 𝑖 (C-4)+ 1 + 𝑖 (C-/)+ 1 + 𝑖 (C-.)

𝑆𝑛 = 𝑎 + 𝑎𝑟. + 𝑎𝑟/ + 𝑎𝑟4 +⋯+ 𝑎𝑟(C-4)+𝑎𝑟(C-/)+𝑎𝑟(C-.)

Hence,wehave

𝐹𝑉 =1 + 𝑖 𝑛− 1

𝑖𝐹𝑉𝐼𝐹𝐴𝑖, 𝑛 =

1 + 𝑖 𝑛 − 1𝑖

𝐹𝑉𝐼𝐹𝐴𝑖, 𝑛 ie.Futurevalueinterestfactor

Similartermsused:

𝑛𝑖𝑠 =

1 + 𝑖 𝑛− 1𝑖

FutureValueofanordinaryannuity(cont’d)

è FVofanordinaryannuityofn paymentsofRM$Reach,

FVofanordinaryannuityofn paymentsofRM1each,

𝐹𝑉 =1 + 𝑖 𝑛− 1

𝑖

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

RevisitExample1Find thefuturevalueofanordinaryannuityconsistingof4annualpaymentsofRM250eachat3%pa

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

== 2501 + 0.03 4 − 1

0.0340.03𝑠250

Eq.10

Example2Findthefuturevalueattheendof15yearsofanannuityofRM100payableattheendofeachquarterif,𝑗R=12%

R=100;m=4;è i=jm/m=0.12/4=0.03t=15èn=mt=60

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

== 1001 + 0.03 60 − 1

0.03600.03𝑠100

= 𝑅𝑀16305.34

Example3AworkerissavingRM1000eachyearanddepositingitintoabank.Howmuchmoneywillshehaveattheendof40yearsforherretirementiftheinterestrateis9%pa?

4038

RM1K

Time 0 1

RM1K

2

RM1K

3

RM1K

39

RM1K RM1K

R=1000;n=40;i=0.09

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

== 10001 + 0.09 40 − 1

0.09400.09𝑠1000

= 𝑅𝑀337882.45

Theeffectofcompound interestearnedoveralongperiod isclearlyevident.

Exercise1.AcoupledepositsRM500every3monthsintoasavingaccountwhichpaysinterestat6%convertiblequarterly (i.e 1.5%perquarter).Howmuchmoneybeintheiraccounton1October1999immediatelyaftertheirdeposit, ifthefirstdepositwasmadeon1Jan1992? (𝑅𝑀20344.14)

2.AfrugalemployeeinvestsRM300fromhistaxreturneach31Aug.After10suchpayment,heincreaseshisdepositstoRM400p.a.Assuming thathehasbeenearning8%p.a.effective,whataccumulationwilltherebeafter15payments? (RM8732.29.Hint:all FVatfocaldate)

Annuities• Definition• Futurevalueofanordinaryannuity• Presentvalueofanordinaryannuity• Annuitiesdue• Perpetuities• Deferredannuities• Summaryofannuities

PresentValueofanordinaryannuityDefinition• Amountdueatthebeginning oftheterm(i.e oneperiodbeforethefirstpayment)

Exampleofordinaryannuityonatimediagram:

Time 0 n1

R

2

R

3

R

n-1

R

1period

Presentvalue

• Rastheregularpaymentateachperiod

PresentValueofanordinaryannuityPVvs.FV

Time 0 n1

R

2

R

3

R

n-1

R

1period

PRESENTVALUE

Time 0 n1

R

2

R

3

R

n-1

R

1period

FUTUREVALUE

NOTE:PVandFVarebothvaluesfromthesamesetofpaymentsbutoccurwhencalculatedatdifferentvaluationdates.

PresentValueofanordinaryannuityThus,therelationshipbetweenPVandFV:

PV=FVx(1+i)-n 𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

Recall

è 𝑃𝑉 =𝑛𝑖𝑠𝑅

= 𝑅1 + 𝑖 𝑛 − 1

𝑖 1 + 𝑖 − 𝑛

= 𝑅 .- .de-C

e

1 + 𝑖 −n

Wedefine

𝑛𝑖𝑎 =

1 − 1+ 𝑖 − 𝑛

𝑖

Thus,thePVoftheannuity:

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖

Othernotation:PresentValueInterestFactorforanAnnuity

𝑛𝑖𝑃𝑉𝐼𝐹𝐴𝑖, 𝑛 = 𝑎 =

1 − 1+ 𝑖 − 𝑛

𝑖

Eq.11

Example4HowmuchmoneyisneedednowtoprovideRM500attheendoftheyear(firstpayment1yearfromnow)for15yearsifthemoneyearnsinterestat12%p.a.effective?

R=500;n=15;i=0.12

== 5001 − 1+ 0.12 − 15

0.12150.12𝑎500

= 𝑅𝑀3405.43

Note:thefacevalueof15paymentsofRM500eachisRM7500.ButonlyRM3405.43isrequiredNOWtoprovidethesepayments.Thedifference isduetotheinterestearnedduring theterm

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖

Example5Astudentborrowedsomemoneytopurchaseacarwastorepaytheloanwithmonthly installmentsofRM150for3years.Calculatethevalueoftheserepaymentsatthebeginning oftheloaniftheinterestratewas(a)9%convertiblemonthly (b)12%convertiblemonthly

R=150;t=3;m=12è n=36;j12=0.09;è i=0.075

== 5001 − 1+ 0.075 − 36

0.075360.075𝑎150

= 𝑅𝑀4717.02

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖

(a) (b) 𝑅𝑀4516.12

Example6En.JosignedacontractthatcallsforadepositofRM1500andforthepaymentofRM2000ayearfor10years.Moneyisworth10%p.a.effective.(a) Whatisthecashvalueofthecontract?(b) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetobringhimselfuptodate?(c) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetodischargehisdebtcompletely?

Example6En.JosignedacontractthatcallsforadepositofRM1500andforthepaymentofRM2000ayearfor10years.Moneyisworth12%p.a.effective.(a) Whatisthecashvalueofthecontract?(b) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetobringhimselfuptodate?(c) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetodischargehisdebtcompletely?

R=2000;t=10yearsè n=10;i=0.12

101 2

RM2K

3

RM2K

4

RM2K RM2K

0

RM2KRM1.5K

?

=+1500 = 2000 .- .df../-.f

f../ +1500100.12𝑎2000

= 𝑅𝑀12800.45

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖+𝑅𝑀1500

Example6(cont’d)En.JosignedacontractthatcallsforadepositofRM1500andforthepaymentofRM2000ayearfor10years.Moneyisworth12%p.a.effective.(a) Whatisthecashvalueofthecontract?(b) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetobringhimselfuptodate?(c) IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd payment

isduetodischargehisdebtcompletely?(#exercise)

R=2000;t=10yearsè n=3;i=0.12

101 2

RM0

3

RM?

4

RM2K RM2K

0

RM0RM1.5K

FutureValueofRM2000annuityattime3

ThismeansEn Johastopaytheaccumulatedvalueofthe3paymentsatthetimeofthe3rd payment

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

== 20001 + 0.12 3 − 1

0.12100.12𝑠2000

= 𝑅𝑀6748.80

Example7AcompanyisconsideringthepossibilityofacquiringnewcomputerequipmentforRM600000cash.ThescrapvalueisestimatedtobeRM50000attheendofthe6-yearlifeoftheequipment.ThecompanycouldleasetheequipmentforRM150000peryear,payableattheendofeachyear.Ifthecompanycanearn16%p.a.onitscapital,advisethecompanywhethertobuyortolease NetPV=PVofCashinflows– PVofcashoutflow

èNetPV=PVofRM50000attime0– PVofcashoutflow

=RM20522.11– RM60000=-RM579477.89

IfBuy:

1 2 6RM50000

0RM60000

PVofRM50Kattime0= 𝑅𝑀50000(1+i)−n=𝑅𝑀50000(1+0.16)-6=RM20522.11

IfLease:

1 2 60

RM150K RM150K RM150KDebt

è NetPV=PVofcashinflow– PVofcashoutflow=0– RM552710.39=-RM552710.39

Should thecompanybuyorlease?WHY?

== 1500001− 1 +0.16 − 6

0.1660.16𝑎150000

= 𝑅𝑀552710.39

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖

Exercise1.En.JosignedacontractthatcallsforadepositofRM1500andforthepaymentofRM2000ayearfor10years.Moneyisworth12%p.a.effective.IfEn Jomissedthefirst2payments,whatmusthepayatthetimethe3rd paymentisduetodischargehisdebtcompletely? RM15876.31

2.AnannuitypaysRM500p.a.for5yearsandthenRM300p.a.for4years.Calculatethevalueofthisannuityoneyearbeforethefirstpaymentusinganannualinterestrateof11%. RM2400.30

3.Awomanhasaninsurancepolicywhosevalueatage65willprovidepaymentsofRM1500ayearfor15years,firstpaymentatage66.Iftheinsurancecompanypays9%paon itsfunds,whatisthepolicy’svalueatage65? RM12091.03