Annuities:Futurevalue&PresentValue

ofanordinaryAnnuities

Department of Mathematical SciencesFaculty of Science

SSCE 2193Room: C10 336/C22 441

Tel: 34321/34274/019-7747457

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

PVofAnnuities

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

FVofAnnuities

Annuities

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖

Eqn.11 Eqn.12

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

AnnuitiesdueDefinition• Anannuitydue isanannuitywhoseperiodicpaymentsarepaidatthe

beginningofeachpaymentperiod.• Thetermofanannuity:

Timeof1stpayment

Onepaymentperiodafterthedateoflastpayment

startterm endterm

AnnuitiesdueExampleofanannuitydueofnpayments:Paymentperiodandinterestperiodcoincide

0 n1

R

2

R

3

R

n-1

R

PresentValue

R

FutureValue

• Theseannuitiesarealsoknownasannuitiespayableinadvance.• Typicallyariseininsurancepremiums,annualsubscriptions

FutureValueofAnnuitiesdue

0 n1

R

2

R

3

R

n-1

R

PresentValue

R

FutureValue

RecallthattheFVofanannuitycanbedefinedastheequivalentvalueof thepaymentattheendoftheterm,è Bydefinition

FVofanannuitydueisanequivalentvaluedueoneperiodafterthelastpayment

FutureValueofAnnuitiesdue

ThusUsingEq.10,theFVof thepaymentsattheendofthe(n-1)th period is

è FVofanordinaryannuityofn paymentsofRM$Reach,

𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1

𝑖𝑛𝑖𝑠𝑅 Eq.10

Recall

𝑛𝑖𝑠𝑅

è TodeterminetheFVofanannuitydueattheendoftheterm,weaccumulatefor1interestperiod.i.e.𝑛

𝑖𝑠𝑅

𝐹𝑉 =𝑛𝑖𝑠𝑅 Eq.121 + 𝑖

𝐹𝑉 = 𝑅1 + 𝑖 𝑛− 1

𝑖1 + 𝑖

PresentValueofAnnuitiesdue

0 n1

R

2

R

3

R

n-1

R

PresentValue

R

FutureValue

RecallthatthePVofanannuitycanbedefined astheequivalentvalueof thepaymentatthebeginning oftheterm,è Bydefinition

PVofanannuitydue isanequivalentvaluedueoneperiodbeforethefirstpayment

PresentValueofAnnuitiesdue

ThusUsingEq.11,thePVofthepayments,1periodbeforethe1st paymentis

è PVofanordinaryannuityofn paymentsofRM$Reach,

Recall

𝑛𝑖𝑎𝑅

è TodeterminethePVofanannuitydue (i.e.onthedateofthe1st payment),weaccumulatefor1interestperiod.i.e.

Eq.13𝑃𝑉 =𝑛𝑖𝑎𝑅 1+ 𝑖

𝑃𝑉 = 𝑅1 − 1+ 𝑖 − 𝑛

𝑖1 + 𝑖

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖 Eq.11

𝑛𝑖𝑎𝑅

PVandFVforOrdinaryAnnuitiesvsAnnuitiesDue

PaymentifRnow+

Note:ThePVofannuitymayalsobeconsidered toequalthefollowing:

OrdinaryAnnuity AnnuityDue

PresentValue

FutureValue

𝑛𝑖𝑎𝑅

𝑛𝑖𝑠𝑅

𝑛𝑖𝑎𝑅 1+ 𝑖

𝑛𝑖𝑠𝑅 1 + 𝑖

Example1MariadepositsRM100atthebeginningofeachyearfor10yearsinanaccountpaying12%p.a.Howmuchisinheraccountattheendof10years?

Valueof1aRM10KNow = 5167.19 1 +0.02 789 = 𝑅𝑀4074.29

0 101

RM100

2

RM100

3

RM100

9

RM100RM100

FutureValueR=100;i=0.12;n=10

= 1 + 0.12 = 1001 + 0.12 10 − 1

0.121 + 0.12

100.12𝑠100

= 𝑅𝑀1956.46

𝐹𝑉 = 𝑅1 + 𝑖 𝑛− 1

𝑖1 + 𝑖

𝐹𝑉 =𝑛𝑖𝑠𝑅 1 + 𝑖

Example2Themonthly rentforaflatisRM520payableatthebeginning ofeachmonth. Ifmoneyisworth𝑗89=9%(a)whatistheequivalentyearlyrentalpayableinadvance?(b)whatisthecashequivalentof5yearsinrent?

Valueof1aRM10KNow = 5167.19 1 +0.02 789 = 𝑅𝑀4074.29

Example2Themonthly rentforaflatisRM520payableatthebeginning ofeachmonth. Ifmoneyisworth𝑗89=9%(a)whatistheequivalentyearlyrentalpayableinadvance? (b)whatisthecashequivalentof5yearsinrent?#exercise

R=520;m=12;i=jm/m=0.09/12=0.0075;t=1yearè n=12

ThisisthePresentvalueofanannuitydueof12paymentsofRM520eachat𝑗89=9%

𝑃𝑉 =𝑛𝑖𝑎𝑅 1+ 𝑖

𝑃𝑉 = 𝑅1 − 1+ 𝑖 − 𝑛

𝑖1 + 𝑖

= 1 + 0.0075 = 5201 − 1+ 0.0075 − 15

0.00751 + 0.0075

120.0075𝑎520

= 𝑅𝑀5990.75

Example3AdebtofRM10000withinterestat𝑗==11%istobepaidoffby8equalquarterlypayments,thefirstdue today.Find thequarterlypayment

R=?;m=4;i=jm/m=0.11/4=0.0275;n=8;PV=10000

ThisisthePresentvalueofanannuitydueof12paymentsofRM520eachat𝑗89=9%

𝑃𝑉 =𝑛𝑖𝑎𝑅 1+ 𝑖

𝑃𝑉 = 𝑅1 − 1+ 𝑖 − 𝑛

𝑖1 + 𝑖

10000 = 1 + 0.0275 = 𝑅1 − 1+ 0.0275 − 8

0.02751 + 0.0275

80.0275𝑎𝑅

𝑅 = 𝑅𝑀1371.85

85

R

1 2

R

3

R

4

R

6

RR

0 7

RR

10000

Exercise1.AdebtofRM1000withinteresttj12=18%istobepaidoffover18monthsbyequalmonthlypayments,thefirstduetoday.Find themonthlypayment.

(𝑅𝑀62.86)

2.AusedcarsellsforRM2550.Khairin wishestopayforitin18monthlyinstallments,thefirstdueonthedayofpurchase.If21%payablemonthly ischarged,find thesizeofthemonthlypayments. (𝑅𝑀163.51)

3.Afridge isbought forRM60depositandRM60amonth for15months. Ifinterestischargedatj12=18.5%,whatisthecashpriceofthefridge? (𝑅𝑀858.06)

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

PerpetuitiesDefinition• Perpetuityisanannuitywhosepaymentsbeginonafixeddateandcontinue

forever.

Examples:• seriesofinterestpaymentsfromasetofmoneyinvestedpermanentlyata

certaininterestrate• ascholarshippaidfromanendowmentonaperpetualbasis• Dividendspaidinrespectofpreferenceshares

Perpetuities:illustratedConsiderapersonwhoinvestsRM10000atrate10%p.a.,maintainstheoriginalinvestmentintactandcollectsRM1000interestattheendofeachyear.AslongastheinterestratedoesnotchangeandtheoriginalprincipalofRM10000iskeptintact,interestpaymentsofRM1000canbecollectedforever.

WesayinterestpaymentsofRM1000formaperpetuity.

0 1

RM1000

2

RM1000

3

RM1000 RM1000

RM10000

RM1000

ThepresentvalueofthisinfiniteseriesofpaymentsisRM10000,asshownbelow:.

PresentvalueofPerpetuitiesPVofaperpetuityisdefinedastheequivalentvalueofthesetofpaymentswhichbeginsatthebeginningoftheperpetuity’sterm.

0 1

R

2

R

3

R R

PresentValue

R

PVofaperpetuitymustbeequivalenttothesetofpaymentsRshownbelow:

TypesofPerpetuity:• Appliessimilarlytoannuities:eg.Ordinaryperpetuity,perpetuitydue,perpetuitydefered.Note:Ordinaryperpetuityisaseriesoflevelperiodicpaymentsmadeattheendofeachinterestperiod,whichcontinueforever.

PresentvalueofPerpetuitiesRecall:PVofanordinaryannuityiscanbedeterminedusingEqn.11below

𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅

1 − 1 + 𝑖 − 𝑛

𝑖 Eq.11

Tocalculatethepresentvalueofaperpetuity,weletn beinfinityinEqn.11.i.e.(1+i)-n becomeszero.Thuswehave,

𝑃𝑉 = 𝑅𝑖 Eq.14

𝑃𝑉 = 𝑅𝑎∞𝑖i.e.

Example4Howmuchmoneyisneededtoestablishascholarship fundpayingscholarshipsofRM1000eachhalfyeariftheendowmentcanbeinvestedatj2=10%andifthefirstscholarshipswillbeprovided:(a)Ahalfyearfromnow?(b) immediately?

Example4HowmuchmoneyisneededtoestablishascholarshipfundpayingscholarshipsofRM1000eachhalfyeariftheendowment canbeinvestedatj2=10%andifthefirstscholarshipswillbeprovided:(a)Ahalfyearfromnow?(b) immediately?

R=1000;m=2;i=jm/m=0.1/2=0.05

𝑃𝑉 = 𝑅𝑖

𝑃𝑉 =10000.05

= 𝑅𝑀20000

(a) (b)Sumofperpetuity+RM1K

0 1

RM1000

2

RM1000

3

RM1000

?

0 1

RM1000

2

RM1000

3

RM1000

?

RM1000

3

𝑅𝑀20000+𝑅𝑀1000= 𝑅𝑀21000

Exercise1.AcompanyisexpectedtopayRM3.50every3monthsonitspreferenceshares.Ifmoney isworthj4=16%,whatshould thesharebesellingfor?

RM87.50

2.ApreferencesharepaysaRM6dividend annually.Ifmoney isnowworth9%p.a,whatshouldaninvestorpayforthisstockifadividendhasjustbeenpaid?

RM66.67

3.ItcostsRM100attheendofeachmonth tomaintainarailwaylevelcrossing.Howmuchcantherailwayscontribute towardthecostofanunderpasswhichwilleliminatethelevelcrossingifmoneyisworth15%p.a.payablemonthly?

RM8000

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

DeferredAnnuitiesDefinition• Andeferredannuity isanannuitywhose1st paymentisduesometimelater

thantheendofthefirstinterestperiod.

è Anordinarydeferredannuityisanordinaryannuitywhosetermisdeferredforkperiods.

0 1 2 3

PresentValue

R

k k+1

R

k+2

R

k+n

FutureValuePeriod ofdeferment Ordinaryannuityofn

payments

Intheabovediagram:• theperiodofdefermentiskperiods• the1st paymentoftheordinaryannuityisattimek+1.Thisisbecausethetermofanordinaryannuitystarts1periodbeforeitsfirstpayment.Sowhenthedateof1st paymentgiven,weneedtodeterminetheperiodofdefermentbymovingbackoneperiod.

PresentValueofDeferredAnnuities

Tocalculatethepresentvalueofadeferredannuity,wefindthevalueofn paymentsoneperiodbeforethefirstpaymentandthendiscountthissumforkperiodstoobtain.

Eq.15𝑃𝑉 = 𝑅𝑎𝑛𝑖 (1 + 𝑖)-k

Example5Whatsumofmoneyshouldbesetasideatachild’sbirthtoprovideannualpaymentsofRM1500tocovertheexpensesfor4years’tertiaryeducationifthefirstpaymentistobemadeonthechild’s19th birthday?Thefundwillearninterestat12%p.aeffective

0 1 2 3

PresentValue

RM1.5K

18 19

RM1.5K RM1.5K

k+n=18+4=22

FutureValuePeriod ofdeferment

k=18Ordinaryannuityofn

paymentsn=4

R=1500;i=0.12;n=4;k=8 𝑃𝑉 = 𝑅𝑎𝑛𝑖 (1 + 𝑖)-k

= 1500𝑎40.12 (1 + 0.12)-8

= 15001 − 1 + 0.12 − 4

0.12(1 + 0.12)-8

=RM592.46

Exercise1.Findthepresentvalueofanordinaryannuitydeferred3years6monthswhichpaysRM500halfyearlyfor7yearsifinterestis7%paconvertiblehalf-yearly.

RM4291.72

2.Onhere55th birthday,Ms Smithdecidestosellherhouseandmoveintoanapartment.SheobtainsRM80Konthesaleofherhouseandinveststhismoney inafundpaying9%pa.Onher65th birthdayshemakesher1st withdrawalthatwillexhaustthefundover15years(ie 15withdrawals).Whatisthesizeofeachwithdrawal?

RM21555.41