lecture notes for chapter 13(1)

8/15/2019 Lecture Notes for Chapter 13(1)

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13-2

1. Diferentiate between contingent annuities an annuitiescertain.

!. Ca"cu"ate the #uture $a"ue o# an orinar% annuit% an an

annuit%ue &anua""% an b% tab"e "oo'up.

LU13-1: Annuities: Ordinary Annuity and Annuity Due(Find Future Value)

(EA)NING UNIT *+,ECTI-ES

LU 13-2: Present Value of an Ordinary Annuity (FindPresent Value)1. Ca"cu"ate the present $a"ue o# an orinar% annuit% b%

tab"e "oo'up an &anua""% chec' the ca"cu"ation.

!. Co&pare the ca"cu"ation o# the present $a"ue o# one"u&p su& $ersus the present $a"ue o# an orinar%annuit%.

LU 13-3: Sinkin Funds (Find Periodi! Pay"ents)

1. Ca"cu"ate the pa%&ent &ae at the en o# eachperio b% tab"e "oo'up.

!. Chec' tab"e "oo'up b% using orinar% annuit%tab"e.


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13-3

ANNUITIES

Annuities have many uses in addition to lottery payoffs. Some of these uses are insurance companies'

pension installments, Social Security payments, home mortgages, businesses paying off notes, bond

interest, and savings for a vacation trip or college education.

Eample 1

!hat happens "hen you have the "inning lottery tic#et$ %ou ta#e it to the lottery head&uarters. !hen

you turn in the tic#et, do you immediately receive a chec# for 1 million$ (o. )ottery payoffs are not

usually made in lump sums. )ottery "inners receive a series of payments over a period of time* usually years. +his stream of payments is an annuity. y paying the "inners an annuity, lotteries do

not actually spend 1 million. +he lottery deposits a sum of money in a financial institution. +he

continual gro"th of this sum through compound interest provides the lottery "inner "ith a series of

payments.

Eample 2

any parents of small children are concerned about being able to afford to pay for their children's

college educations. Some parents start an annuity by depositing a series of payments in a financialinstitution usually of equal amounts over a period of time) from the time "hen the child is in diapers.

+he interest on these deposits is compounded until the child is 1/, "hen the parents "ithdra" the sum

of all deposits plus the interest that accumulates for college epenses.

!e begin the chapter by eplaining the difference bet"een calculating the future value of an ordinary

annuity and an annuity due. +hen you learn ho" to find the present value of an ordinary annuity. +hechapter ends "ith a discussion of sin#ing funds.


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13-0

#O$POU%D&%' &%*S (FUU*VALU)

er" of t+e annuity ,

The ti&e #ro& the beginningo# the rst pa%&ent perio tothe en o# the "ast pa%&ent

perio

Future alue of annuity ,

The #uture o""ar a&ount o# aseries o# pa%&ents p"us

interest

Present alue of anannuity , he a&ount o#

&one% neee to in$est toa%in orer to recei$e a strea& o#pa%&ents #or a gi$en nu&ber

o# %ears in the #uture

Annuity ,

A series o# pa%&ents


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13-

FUTU)E -A(UE *F AN ANNUIT/

• Annuity is stream of e&ual payments made at periodic times.• +he future value of an annuity is the future dollar amount of a series of payments plus interest. +he term of

the annuity is the time from the beginning of the first payment period to the end of the last payment period.• +he concept of the future value of an annuity is illustrated in the follo"ing figure• At end of period 1 1 is invested.• At end of period 2: An additional 1 is invested. +he 1 from period 1 earns interest and is no" "orth 1./.

+he 1 invested at the end of period 2, does not earn any interest, because it "as invested at the en# of the

period. +he 2. is no" "orth 2./• At end of period 3 An additional 1 is invested. +he 3. is no" "orth 3.2. 4emember that the last dollar

invested earns no interest.


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13-5

Step 1. For perio 10 no interest ca"cu"ation is necessar%0 since&one% is in$este at the en o# the perio.

Step 2. For perio !0 ca"cu"ate interest on the ba"ance an athe interest to the pre$ious ba"ance.

Step 3. A the aitiona" in$est&ent at the en o# perio !to the new ba"ance.

CA(CU(ATING FUTU)E -A(UE *F AN*)DINA)/ ANNUIT/ ANUA((/

Step 4. )epeat Steps ! an 2 unti" the en o# the esireperio is reache.


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CA(CU(ATING FUTU)E -A(UE *FAN *)DINA)/ ANNUIT/ ANUA((/

Fin the $a"ue o# an in$est&ent a#ter 2 %ears #or a320444 orinar% annuit% at 56.

CA(CU(ATING FUTU)E -A(UE *F


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Step 1. Ca"cu"ate the nu&ber o# perios an rate perperio.

Step 2. (oo' up the perios an rate in an orinar% annuit%

tab"e. The intersection gi$es the tab"e #actor #or the#uture $a"ue o# 31.

Step 3. u"tip"% the pa%&ent each perio b% the tab"e#actor. This gi$es the #uture $a"ue o# the annuit%.

CA(CU(ATING FUTU)E -A(UE *FAN *)DINA)/ ANNUIT/ +/ TA+(E(**KU7

Future value of = Annuity payment xOrdinary annuity

ordinary annuity each period tablefactor


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*)DINA)/ ANNUIT/ TA+(E8C*7*UND SU *F AN ANNUIT/ *F31 9TA+(E 12.1:


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7erios 9N: ; 2 < 1 ;2

FUTU)E -A(UE *FAN *)DINA)/ ANNUIT/

Fin the $a"ue o# an in$est&ent a#ter 2 %ears #or a 320444orinar% annuit% at 56.

)ate 9): ; 56=1 ;562.!>?> 9tab"e #actor: < 320444 ;3@02@.!4


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C(ASSIFICATI*N *F ANNUITIES

#ontinent annuities ,

ha$e no


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C(ASSIFICATI*N *F ANNUITIES

Ordinary annuity ,regu"ar eposits=pa%&ents

&ae at the end o# theperio

Annuity due ,

regu"ar eposits=pa%&ents&ae at the beginning o# the

perio

Jan. 31 Monthly Jan. 1

June 30 Quarterly April 1

Dec. 31 Semiannually July 1

Dec. 31 Annually Jan. 1

"e can divide each of the ma8or annuity groups 9ontingent annuities and Annuities

certain: into Ordinary annuity an Annuity due:


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CA(CU(ATING FUTU)E -A(UE *FAN

ANNUIT/ DUE ANUA((/Step 1. Ca"cu"ate the interest on the ba"ance #or the perio an

a it to the pre$ious ba"ance.

Step 2. A aitiona" in$est&ent at the beginning o# theperio to the new ba"ance.

Step 3. )epeat Steps 1 an ! unti" the en o# the esire

perio is reache.


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CA(CU(ATING FUTU)E -A(UE *FAN ANNUIT/ DUE ANUA((/

Fin the $a"ue o# an in$est&ent a#ter 2 %ears #or a320444 annuit% ue at 56.

CA(CU(ATING FUTU)E -A(UE *F


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CA(CU(ATING FUTU)E -A(UE *FANANNUIT/ DUE +/ TA+(E (**KU7

Step 1. Ca"cu"ate the nu&ber o# perios an rate perperio. A one e


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FUTU)E -A(UE *F AN ANNUIT/DUE

Fin the $a"ue o# an in$est&ent a#ter 2 %ears #or a320444 annuit% ue at 56.

7erios 9N: ; 2 < 1 ; 2 B 1

; >

>.4?1 9tab"e #actor: < 320444 ;312015.24

)ate 9): ; 56=1 ; 56

314015.24312015.24 320444 ;


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P*A# .U&/

1-Using Table 13.1, (a) find the value of an investment after 4 years on an ordinary annuity of $4,000made semiannually at 10% and (b) re!al!ulate, assuming an annuity due."-#ally eaver on a lottery and ill re!eive a !he!& for $4,000 at the beginning of ea!h ' months forthe ne(t ) years. *f #ally de+osits ea!h !he!& into an a!!ount that +ays '%, ho mu!h ill he have atthe end of the ) years

;or step by step solution "atch the video for )< 13-1 =o to c=ra"->ill?s 9onnect@ Assignment @ Buestion

1@ 9lic# the eoo# C resources options drop do"n menu@ scroll do"n to )


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13-1/

• )et's assume that "e "ant to #no" ho" much money "e need to invest today to receive a stream of payments for a given number of years in the future. +his is called the present value of an ordinary annuity.

• *n the folloing figure you !an see that if you anted to ithdra $1at the end of one +eriod, you ouldhave to invest 3 !ents today. *f at the end of ea!h +eriod for three +eriods you anted to ithdra $1, you#ould have to +ut $".) in the ban& today at % interest. /ote that e go from the future ba!& to the+resent.

Nu&ber o# perios

3.@!@

31.522

3!.1

7)ESENT -A(UE *F AN ANNUIT/

CA(CU(ATING 7)ESENT -A(UE *F


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CA(CU(ATING 7)ESENT -A(UE *FAN *)DINA)/ ANNUIT/ +/ TA+(E(**KU7

Step 1. Ca"cu"ate the nu&ber o# perios an rate perperio.

Step !. (oo' up the perios an rate in the present$a"ue o# an annuit% tab"e. The intersection gi$es

the tab"e #actor #or the present $a"ue o# 31.

Step 2. u"tip"% the withrawa" #or each perio b% thetab"e #actor. This gi$es the present $a"ue o# anorinar% annuit% .

Preent value of Annuity Preent value of

ordinary annuity payment payment ordinary annuitytable

= x

7)ESENT -A(UE *F AN ANNUIT/ *F


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7)ESENT -A(UE *F AN ANNUIT/ *F319TA+(E 12.!:


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7)ESENT -A(UE *F AN ANNUIT/

,ohn Fitch wants to recei$e a 350444annuit% in 2 %ears. Interest on theannuit% is 56 se&iannua""%. ,ohn wi""&a'e withrawa"s at the en o# each

%ear. ow &uch &ust ,ohn in$esttoa% to recei$e a strea& o# pa%&ents#or 2 %ears.

N ; 2 < 1 ; 2perios

Dnterest F

Gayment F

End of %ear 3 F

Dnterest F

Dnterest F

Gayment F

Gayment F

) ; 56=1 ; 56

!.1 9tab"e #actor: <350444 ;

3!40?1?.54


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(U7 SUS -E)SUS ANNUITIES

,ohn Sans &ae eposits o# 3!44 to F"oor +an'0 which pa%s 56interest co&poune se&i annua""%. A#ter %ears0 ,ohn &a'es no &oreeposits. hat wi"" be the ba"ance in the account ? %ears a#ter the "asteposit

N ; < ! ; 14perios

N ; ? < ! ; 1!perios

Step 1!

Step 2!

) ; 56=! ; >6

1!.44?1 9tab"e #actor: <3!44 ;

3!0>41.!!

Future value of anannuity

Future value of a lumpum

) ; 56=! ; >6

1.?414 9tab"e #actor: <3!0>41.!! ;

3205>>.2


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(U7 SUS -E)SUS ANNUITIES

e" )ich ecie to retire in 5 %ears to New e40444 atthe en o# each %ear ! %ears a#ter he retires Assu&e e" canin$est &one% at 6 interest co&poune annua""%.

N ; ! < 1 ; ! perios) ; 6=1 ; 6

Step 1! Preent value of an annuity Step 2! Preent value of a lumpum

) ; 6=1 ; 6

1>.4@2@ < 3>40444 ;

3?20?

N ; 5 < 1 ; 5 perios

.??5 < 3?20? ;325104.4?


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13-20

P*A# .U&/

1-!hat must you invest today to receive an 1/, annuity for years semiannually at a 1H annual rate$ All

"ithdra"als "ill be made at the end of each period.

2-4ase >igh School "ants to set up a scholarship fund to provide five 2, scholarships for the net 1 years. Df

money can be invested at an annual rate of 7H, ho" much should the scholarship committee invest today$

3-Ioe !ood decided to retire in years in AriJona. !hat amount should Ioe invest today so he can "ithdra" 5,

at the end of each year for 3 years after he retires$ Assume Ioe can invest money at 5H compounded annually.

;or step by step solution "atch the video for )< 13-2 =o to c=ra"->ill?s 9onnect@

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SINKING FUNDS9FIND 7E)I*DIC 7A/ENTS:

Sin'ing #un ; Future < Sin'ing

#unpa%&ent $a"ue tab"e

• A sinin! fund is a financial arrangement that sets aside regular periodic payments of a particular amount of money. 9ompound interest

accumulates on these payments to a specific sum at a predetermined

future date.

• 9orporations use sin#ing funds to discharge bonded indebtedness, toreplace "orn-out e&uipment, to purchase plant epansion, and so on.

• A sin#ing fund is a different type of an annuity. Dn a sin#ing fund, youdetermine the amount of periodic payments you need to achieve a given

financial goal. Dn the annuity, you #no" the amount of each payment and

must determine its future value.

)et's "or# "ith the follo"ing formula


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SINKING FUND TA+(E +ASED *N 31

9Tab"e 1!.2:


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SINKING FUND

To retire a bon issue0 oore Co&pan% nees 3?40444 in 15 %ears#ro& toa%. The interest rate is 146 co&poune annua""%. hatpa%&ent &ust oore &a'e at the en o# each %ear Use Tab"e 12.2.

N ; 15 < 1 ; 15 perios

#+e!k

Future -a"ue o# an annuit%tab"e

N ; 150 ); 146

H *f ue to rouning

) ; 146=1 ; 146

4.4!1@ < 3?40444 ;31021>

31021> < >.@@! ;3@[email protected]


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13-2/

P*A# .U&/

Today, 2rro om+any issued bonds that ill mature to a value of $0,000 in10 years. 2rros !ontroller is +lanning to set u+ a sin&ing fund. *nterest ratesare 1"% !om+ounded semiannually. #hat ill 2rro om+any have to setaside to meet its obligation in 10 years he!& your anser. 5our anser illbe off due to the rounding of Table 13.3.

;or step by step solution "atch the video for )< 13-3 =o to c=ra"->ill?s

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P*O0L$ 13-13

Solution"

7erios ; ! %ears < 1 ; !perios

To he"p %ou reach nancia" securit% upon retire&ent0 %ou shou"in$est !46 o# %our inco&e annua""%. I# %ou auto&atica""% trans#erre320444 at the en o# each %ear to a retire&ent account earning >6interest co&poune annua""%0 how &uch wou" %ou ha$e a#ter !%ears 24 %ears #$ 13%1&2'

Interest rate per perio ; >6=1 ;>6320444 < >1.?>@ ; 31!>[email protected] a#ter ! %ears

7erios ; 24 %ears < 1 ; 24 perios

Interest rate per perio ; >6=1 ; >6

320444 < ?.45>@ ; 31?50!>.4 a#ter 24%ears


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13-3

7)*+(E 121

Solution"

!4 perios0 1!6 9Tab"e 12.1:

31!044 !.4!> ; 3@440?

,ose# Co&pan% borrowe &one% that &ust be repai in !4 %ears. Theco&pan% wants to &a'e sure the "oan wi"" be repai at the en o#%ear !40 so it in$ests 31!044 at the en o# each %ear at 1!6 interestco&poune annua""%. hat was the a&ount o# the origina" "oan #$13%1&2'


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13-31

P*O0L$ 13-1

Solution"

7erios ; >4 %ears 1 ; >4 perios

3!40444 .4452 ; 3!04 each %ear

+an'rate.co& reporte on a shoc'ing statistic8 on"% >6 o# wor'ersparticipate in their co&pan%Js retire&ent p"an. This &eans that >?6 onot. ith such an uncertain #uture #or Socia" Securit%0 this can "ea$ea"&ost 1 in ! ini$iua"s without proper inco&e uring retire&ent. ,i""Co""ins0 !40 ecie she nees to ha$e 3!40444 in her retire&entaccount upon retiring at ?4. ow &uch oes she nee to in$est each%ear at 6 co&poune annua""% to &eet her goa" (ip" She is settingup a sin'ing #un. #$ 13%3&1'

Interest rate per perio ; 6=1 ; 6


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7)*+(E 12!2

Solution"

5 perios 56= > ; !6

*n ,oe artinJs grauation #ro& co""ege0 ,oeJs unc"e pro&ise hi& agi#t o# 31!0444 in cash or 3@44 e$er% uarter #or the ne %earsa#ter grauation. I# &one% cou" be in$este at 56 co&pouneuarter"%0 which ofer is better #or ,oe #$ 13%1&2'0 #$ 13%2&1'

or 3@44 < 15.?2@! ; 31?0.!5 9Tab"e 12.1: < .!5> 9Tab"e1!.2:

[email protected]!60 1? perios

3@44 < 12. ;31!0!1@.@29Tab"e

12.!:


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7)*+(E 12!

A "oca" Dun'inJ Donuts #ranchise &ust bu% a new piece o# euip&entin %ears that wi"" cost 3550444. The co&pan% is setting up a sin'ing#un to nance the purchase. hat wi"" the uarter"% eposit be i# the#un earns 56 interest#$ 13%3&1'

Solution"

!4 perios0 !6 9Tab"e 12.2:

.4>1! 3550444 ; 320?!.?4 uarter"% pa%&ent


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7)*+(E 12!?

i'e acaro is se""ing a piece o# "an. Two ofers are on the tab"e.orton Co&pan% ofere a 3>40444 own pa%&ent an 320444 a%ear #or the ne

lecture notes for chapter 13(1)

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