Anuitas II

8/17/2019 Anuitas II

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McGraw-Hill Ryerson©

Chapter 11

OrdinaryAnnuities



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Calculate the…

Learning Objectives

After completing this chapter, you will be able to

! number of payments in or"inary

an" "eferre" annuities

! payment size in or"inary an"

"eferre" annuities

… interest rate in or"inary annuities

#O-1

#O-1

#O-$

#O-$

#O-%

#O-%


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)in"ing the *ayment +i(e!

*M


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.our life partner somehow con'ince" you that you can/taffor" the car of your "reams, price" at 0$22 .ou are

a"'ise" to! “Save up for 4 years and then buy the car forcash.” How much woul" you have to save each month, if

you coul" in'est with a return of123 compoun"e" monthly4

.ou nee" to "eci"e if this situation in'ol'es! a *5 or a )5 an" then use the appropriate formula

*M

As you ha'e to save up the 0$,22, ie in the future,

)5 6 0$,22

Assume you ha'e no sa'ings ! *5 6 2

inding !ayment Size of an

Or"inary +imple Annuity

inding ! ayment Size of an



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.our life partnersomehow con'ince" youthat you can/t affor" the

car of your "reams,price" at 0$22 "#t least

not right now$. .ou area"'ise" to!

“Save up for 4

years and then buy the car for cash.” How much

woul" you have to saveeach month, if you coul"in'est with a return of

123 compoun"e"




“Save up for 4 years and then buy the car for cash.”

How muchwoul" you have to save

each month, if you coul"in'est with a return of

123 compoun"e"monthly4





7

1$

*M6

-

78277

12 2

$22

)ormula solution

)ormula solution


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[)5 6 *M (1+ i )n - 1i ]i *5 6 *M

1-(1+ i )-n[ ]

9hich )ormula49hich )ormula4

Algebraic Metho" of +ol'ing for *MAlgebraic Metho" of +ol'ing for *M

:a;


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Calculate the ?uantity within the s%uare brac&ets

'earrange the e%uation to sol'e for *M .

i [)5 6 *M (1+ i )n - 1]

i *5 6 *M 1-(1+ i )

-n[ ]


Algebraic Metho" of +ol'ing for *MAlgebraic Metho" of +ol'ing for *M

%%

77

Applying Metho"!Applying Metho"!


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@+a'e up for 7

years an" then buy thecar for cashHow

much woul" you have tosave each month, ifyou coul" in'est with a

return of 123




@+a'e up for 7years an" then buy the

car for cashHow


return of 123 compoun"e" monthly4

[)5 6 *M (1+ i )n - 1i ]

As the annuity/s )5 is =nown,

therefore, the )5 formula is use"$$

Btract necessary "ata

)5 6 $22 n 6 7D1$ 6 7

i ( 12E1$ c ( 1 *M 6 4*5 6 2


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1$12

$22

71

1




@+a'e up for 7

years an" then buy thecar for cashHow


return of 123


car of your "reams,price" at 0$22 "#t leastnot right now$. .ou are

a"'ise" to!@+a'e up for 7

years an" then buy thecar for cash

Howmuch woul" you have tosave each month, ifyou coul" in'est with a

return of 123 compoun"e" monthly4

222%122% 17872787222%F$$F78277

[)5 6 *M (1+ i )n - 1i ])ormula

)ormula

)5 6 $22 n 6 7D1$ 6 7

i ( 12E1$ c ( 1 *M 6 4

*5 6 2

!another eample!another eample


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)he


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.our parents are "iscussing the terms of the0122 222 mortgage that they ha'e offere" to

hol" in the purchase of your first homehey are consi"ering aninterest rate of F3 compoun"e" monthly


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.our parents are

"iscussing the terms of

the 122 222 mortgagethat they ha'e offere" tohol" in the purchase ofyour first home hey

are consi"ering an

interest rate of F3compoun"e" monthly


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Choose appropriate formula an" +ol'e

As the annuity/s *5 is =nown, the *5 formula is use"$$

i 6 2FE1$n 61$

D$2 6$72 *5 6 0122222

i *5 6 *M 1-(1+ i )

-n[ ])ormula)ormula

1$2F 1

122 222

$72 1

2227$1227$ 2%-2%171F1F%2221FF88+i(e of monthly

mortgagepayment

+i(e of monthlymortgage

payment


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Amount

0

How much interest will you pay your

parents o'er the $2 year perio"4

Monthly *ayment Iumber of *ayments

F88 $72 1F,%8272

Amount Jorrowe" 122,22222

otal


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*M 6 -22

As this amountof interest

shoc=s you,you "iscuss the

possibility ofma=ing paymentsof 022Emonth,

to sa'e some time

an" interestcosts

Ketermine the time

it will ta&e you tore a our

22

I 6 $1F$

)ormula solution)ormula solution

$1 payments 6 1 yrs $months$1 payments 6 1 yrs $months


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)ormula)ormula )i

!*)

!+ i

+

∗

1ln

1lnL n

2227$

i ( .- /0 *M ( 12*5 ( 1/, C 6 1n 2

1$2F 1

22

122 222

1

1227$ 2227$2F8F$-2727-2827F-$1F$$1F$



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1 Jase formula +

=

i

3ni ;*M*5

:11

L $ o isolate n, "i'i"e bothsi"es by *M

*M*M

!Continue!!Continue!

Ke'eloping the )ormula)ormula

*M=

*5+

i

3ni ;:11L

+

=

i

3ni ;*M*5

:11 L

)ormula)ormula )i

*M

+

*5 i D

1ln

1lnL n


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:a; Multiply both si"es by i

% Continue to isolate n

*M=

*5+

i

3ni ;:11L

*M=

*5 + 3ni ;:11

i L Di Di

:b; Reorgani(e e?uation

:c; ow a=e the natural logarithm

:ln or ln 5 ; of both si"es3n6 ln

*M

Di =

*5−

3n1 +i ;:1

3n+i ;:1

+i ;:1 = ln

:"; Solving for n…

divide both sides by

ln:17i ; ln"/7i $ ln"/7i $

!from $

=

*M

Di *51−

L *MDi *51

−

3n6 ln + i ;:1 = lnL *MDi *51 −*M*5Di )i 1ln

1ln

nL


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2222 $1F$ 1F$,$722

otal


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i ( .- /0 *M ( 120-*5 ( 1/, C 6 1n 2

2227$

1$2F 1

$F

122 222

1

1227$ 2227$2F7-27$F%-2FF2-$2FF$


$2FF$

)ormula)ormula )i

*M

+

*5 i D

1ln

1lnL n


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:b;otal


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.or= )urniture has a promotion on a be"room setselling for 0$$F2 Juyers will pay

“ no money down and no payments for /0 months .”

he first of $7 e?ual monthly payments is "ue

1$ months from the purchase date8hat should the monthly payments be if .or= )urniture earns 123 compoun"e" monthly on itsaccount recei'able during both the deferral period

an" the repayment period 4

+ince you want the furniture now, this in'ol'es a *5

*M

*5 6 0$$F2 9nce you repay the loan, )5 6 2

!ayments are deferred for 11 months

KB)BRRA#

inding !ayment Size in a

Keferre" Annuity


Keferre" Annuity


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.or= )urniture has a promotion on abe"room set selling for 0$$F2 Juyers will pay

“no money down and

no payments for /0 months.” he first of$7 e?ual monthly payments is "ue 1$ months from the purchase date

8hat should the monthly payments be if.or= )urniture earns 123 compoun"e" monthly

on its account recei'able during both the deferral period an" the repayment period 4

.or= )urniture has a promotion on abe"room set selling for 0$$F2 Juyers will pay

“ no money down and

no payments for /0 months .” he first of$7 e?ual monthly payments is "ue 1$ months from the purchase date

8hat should the monthly payments be if.or= )urniture earns 123 compoun"e" monthly

on its account recei'able during both

the deferral period an" the repayment period 4


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d 6 11i 6 212E1$

n 6 $7

0$$F2 *M *M *M *ayments

0$$F2

*5Annuity

)5

*5 of the payments at

the en" of month 11

*5 of the payments at

the en" of month 11

)5 of the 0$,$F2 loan

at the

en" of month 11

)5 of the 0$,$F2 loan

at the

en" of month 11

6

Months2 11 1$ 1% %F %


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11

1$

)5 6 $,7F2

12 2

)in" the amount owe"after 11 months

$$F2

0$,7F2 is the *5 of the annuity0$,7F2 is the *5 of the annuity

.or= )urniture

has a promotionon a be"room set selling

for 0$$F2 Juyers will pay“no money down and no

payments for /0 months.” he first of $7

e?ual monthly payments is"ue 1$ months from the

purchase "ate 8hat

should the monthly payments be if .or=)urniture earns 123

compoun"e" monthly onits account recei'able

during both the deferral

period an"the repayment period 4

.or= )urniture


for 0$$F2 Juyers will pay

“ no money down and no payments for /0 months .” he first of $7


purchase "ate 8hat



during both the deferral period an"

the repayment period 4


Keferre" Annuity


Keferre" Annuity


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$7F2

$7

)5 6 $,7F2

Iow fin" the *M of the annuity !

2

$7 monthly payments of 011%Fwill repay the loan

$7 monthly payments of 011%F will repay the loan

*5 6 - $,7F2*M 6 11%F

.or= )urniture





purchase "ate 8hat





.or= )urniture





purchase "ate 8hat




the repayment period 4 )ormula solution)ormula solution


Keferre" Annuity


Keferre" Annuity


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)5 6 *5:1 N i ;n)ormula)ormula

)5 6 $$F2:1 N 212E1$;11

6 0$,7F2

$7F2

+

=

i

3ni ;*M*5

:11L 6 *M L1-:1N12E1$;-$7

12E1$

6 011%F*M

.or= )urniture





purchase "ate 8hat





.or= )urniture





purchase "ate 8hat




the repayment period 4

$7 monthly payments of 011%F

will repay the loan

$7 monthly payments of 011%F

will repay the loan

)in" the amount owe"after 11 months


Keferre" Annuity


Keferre" Annuity


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i.e....umber 9f !ayments


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0$2,222 is in'este" in a fun"earning 3 compoun"e" %uarterly

he first ?uarterly with"rawal

of 01,222will be ta=en from the fun" five

years from now . Howmany with"rawals will it ta=e to "eplete

the fun"4

I

*ayments are "eferre" for 18 ?uarters

K ;;''#<

inding umber 9f !ayments in a

Keferre"Annuity


Keferre"

Annuity

he )5 of 0$2,222 after the "eferral, becomes the *5 of the annuity


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.ears

2 7F F

d 6 18

0$2,222

*51

n 6 4

his )51 then becomes the !+ of the

annuity of 01222E?uarter

he 0$2222

earnsinterest for 7

years 8months

*ayments of 01222E?uarter

)51

i 6 2 2E7 6 2$*M 6 01222


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0$2,222 isin'este"

in a fun"earning

3 compoun"e"%uarterly

he first?uarterly

with"rawal

of 01222 will beta=en from thefun" five years

from now. How

many with"rawalswill it

0$2,222 isin'este"

in a fun"earning

3 compoun"e"

%uarterlyhe first

?uarterlywith"rawal


from now . How

many with"rawals

will it

18

7 2

)in" the )5 of0$2,222 in 7F years

$2222

0$8,1%$$ is the *5 of the annuity0$8,1%$$ is the *5 of the annuity

)5 6 $8,1%$$


Keferre"Annuity


Keferre"

Annuity


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1222

Iow fin" the *M of the annuity !

2

771 ?uarterly payments will "eplete thefun":77 full payments an" 1 partial;

771 ?uarterly payments will "eplete thefun":77 full payments an" 1 partial;

$81%$$

)5 6 $8,1%$$*5 6 - $81%$$I 6 771


0$2,222 isin'este"

in a fun"earning


he first?uarterly

with"rawal


from now. How


0$2,222 isin'este"

in a fun"earning

3 compoun"e"

%uarterlyhe first

?uarterlywith"rawal


from now . How

many with"rawals

will it


Keferre"Annuity


Keferre"

Annuity


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)5 6 $2222:1 N 22E7;18

6 0$8,1%$$

)in" the )5 of0$2,222 in 7F years

*M

*5 Di

)i 1ln

1ln

n L

ln:12$;

6 771 paymentsor 11 years

0$2,222 isin'este"

in a fun"earning


he first?uarterly

with"rawal


from now. How


0$2,222 isin'este"

in a fun"earning

3 compoun"e"

%uarterlyhe first

?uarterlywith"rawal


from now . How

many with"rawals

will it


Keferre"Annuity


Keferre"

Annuity

Lln 1 - $81%$$ D2$1222


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9hen!number of compoundings per year

number of payments per year

≠


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+ince you get pai"

e'ery secon"

hurs"ay you

"eci"e to pay 1=-

every two wee&s to ma=e your

bu"geting easier

ind the new term

of your mortgage ifthe interest charges

remain at F3

compounded

monthly

1$

$%F2

*E. 6 $

71F bi-wee=ly

payments or

1F yrs 117 months

71F bi-wee=ly

payments or1F yrs 117 months

CE.6 1$*M 6 -%F2I 6 7177



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+ince you get

pai" e'ery secon"

hurs"ay you

"eci"e to pay 1=-

every two wee&s to ma=e your

bu"geting easier

ind the new term

of your mortgage ifthe interest charges

remain at F3

compounded

monthly

Ketermine c Step 1Step 1

C6 1$ E $ 6 71F

i $ 6 :1Ni ;c - 1

i $ 6 :1N 2FE1$; 71F-1

i $ 6 22218

&se c to "etermine i $ Step 2Step 2

C 6number of compoundings per year number of payments per year

Step 3Step 3


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as the 'alue for

@i in the appropriate annuity formula

Step 3Step 3 &se this rate i $ 6 22218

)ormula)ormula )i

*M

+

*5 i D

1ln

1lnL n

12218 222182F7$-27F1-2$

1

%F2122 222

1

22218-7177

71F payments or 1F yrs 117 months71F payments or 1F yrs 117 months

0122 222


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of*ayments

of*ayments

*aymentAmount

*aymentAmount

otal Costotal Cost+cenario+cenario

1

$

0122,222 wenty-year

Mortgage

P


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%F2

.ou are nowconsi"ering delayingthe purchase

of your first house toallow for a largerdown payment


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Ketermine c Step 1Step 1

C6 1$ E $ 6 71F

i $ 6 :1Ni ;c - 1

i $ 6 :1N 2E1$;71F

-1

i $ 6 222%1

&se c to "etermine i $ Step 2Step 2

C 6number of compoundings per year

number of payments per year

Step 3Step 3

.ou are now

consi"ering delayingthe purchase of your first house to

allow for a largerdown payment


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.ou are now

consi"ering delayingthe purchase of your first house to

allow for a largerdown payment


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1 Jase formula−

=

i

*M)5 +

ni ;:1 1

L $ o isolate n, "i'i"e bothsi"es by *M *M*M

!continue"!!continue"!

Ke'eloping the )ormula)ormula

*M=

)5

)ormula)ormula )i

*M

+

)5 i D

1ln

1lnL n

N

−

=

i *M)5

+

ni ;:1 1L

−

i + ni ;:1 1L

11 46


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! from $*M

=

)5−

i +

ni ;:1 1L

:a; Multiply both si"es by i% Continue to isolate n !

*M

)5=

−

i +

ni ;:1 1L Di Di

L *M)5

−+

ni ;:1 1 Di

:b; Reorgani(e e?uation =*M

)5+

ni ;:1 1 Di

:c; ow a=e the natural logarithm

:ln or ln 5 ; of both si"es n ln:1N i ; = lnL *M D i )5

1 N

:"; Solving for n…

divide both sides by

ln:1Ni ;

ln:1Ni ; ln:1Ni ;

n ln:1N i ; = lnL *M D i )51 N*M

)5 D i

)

i 1ln

1 Nln

nL

11 47


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.ou alrea"y ha'e

1/ saved

for your "ownpayment ?

compounded monthly

on all of your sa'ings

1$

$

Alrea"yentere"Alrea"yentere" I 6 %$F

%F2

12222

$F222

%F bi-wee=ly payments

6 appro 1 yr Fmonths

%F bi-wee=ly payments

6 appro 1 yr Fmonths

11 48


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.ou alrea"y ha'e

1/ saved

for your "own

payment


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% +teps

)ormula +olution)ormula +olution

his is more complicate"to sol'e whenusing

algebraic e?uationsQ

11

$$

%%

)in" the )5 of the 012 222 in $ years

)in" the )5 of the 0%F2 per pay

A"" totals together

he 012 222 continues to earn

interest during the new savings period:

he 012 222 continues to earn

interest during the new savings period:

.ou alrea"y ha'e

1/ saved

for your "own

payment


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)ormula +olution)ormula +olution


6 12222:1 N 22E1$; $7

6 011,$

11

$$

i $ 6 :1Ni ;c - 1

6 :1N 2E1$;71F-1

6 222%1

%%

0

11,$

6 %F2 L:1N22%1;F$ P1

22%1

6 01821$7 18,21$7

%1,7%21$otal

=

*M)5

−

i

+

ni ;:1 1

L

.ou alrea"y ha'e

1/ saved

for your "own

payment


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11 52


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A life insurance company a"'ertises that0F2,222 will purchase a $2-year annuity

paying 0%711% at the end of eachmonth9hat nominal rate of return "oes the

annuity in'estment earn4

A life insurance company a"'ertises that0F2,222 will purchase a $2-year annuity

paying 0%711% at the end of each

month9hat nominal rate of return "oes the

annuity in'estment earn4

1

1$

%711%$72

2F2222

CE. 6 1


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!to sol'e for i without

a financial calculator

11 54


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his completes Chapter 11

Documents

Anuitas II