anuitas ii
TRANSCRIPT
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Chapter 11
OrdinaryAnnuities
McGraw-Hill Ryerson©
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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
Or"inary +imple Annuity
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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"
.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 muchwoul" you have to save
each month, if you coul"in'est with a return of
123 compoun"e"monthly4
inding !ayment Size of an
Or"inary +imple Annuity
inding ! ayment Size of an
Or"inary +imple Annuity
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[ ]
9hich )ormula49hich )ormula4
Algebraic Metho" of +ol'ing for *MAlgebraic Metho" of +ol'ing for *M
%%
77
Applying Metho"!Applying Metho"!
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inding !ayment Size of an
Or"inary +imple Annuity
inding ! ayment Size of an
Or"inary +imple Annuity
9hich )ormula49hich )ormula4
.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!
@+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
.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!
@+a'e up for 7years an" then buy the
car for cashHow
much woul" you have tosave each month, ifyou coul" in'est with a
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"$$
Btract necessary "ata
)5 6 $22 n 6 7D1$ 6 7
i ( 12E1$ c ( 1 *M 6 4*5 6 2
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1$12
$22
71
1
.our life partnersomehow con'ince" youthat you can/t affor" the
car of your "reams,price" at 0$22 "#t least
not right now$. .ou area"'ise" to!
@+a'e up for 7
years an" then buy thecar for cashHow
much woul" you have tosave each month, ifyou coul" in'est with a
return of 123
.our life partnersomehow con'ince" youthat you can/t affor" the
car of your "reams,price" at 0$22 "#t leastnot right now$. .ou are
a"'ise" to!@+a'e up for 7
years an" then buy thecar for cash
Howmuch woul" you have tosave each month, ifyou coul" in'est with a
return of 123 compoun"e" monthly4
222%122% 17872787222%F$$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$
$1 payments 6 1 yrs $months$1 payments 6 1 yrs $months
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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-$2FF$
$2 payments 6 1 yrs $months$2 payments 6 1 yrs $months
$2FF$
)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
inding !ayment Size in a
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
has a promotionon a be"room set selling
for 0$$F2 Juyers will pay
“ no money down and no payments for /0 months .” he first of $7
e?ual monthly payments is"ue 1$ months from the
purchase "ate 8hat
should the monthly payments be if .or=)urniture earns 123
compoun"e" monthly onits account recei'able
during both the deferral period an"
the repayment period 4
inding !ayment Size in a
Keferre" Annuity
inding !ayment Size in a
Keferre" Annuity
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$7F2
$7
)5 6 $,7F2
Iow fin" the *M of the annuity !
2
$7 monthly payments of 011%Fwill repay the loan
$7 monthly payments of 011%F will repay the loan
*5 6 - $,7F2*M 6 11%F
.or= )urniture
has a promotionon a be"room set selling
for 0$$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
has a promotionon a be"room set selling
for 0$$F2 Juyers will pay
“ no money down and no payments for /0 months .” he first of $7
e?ual monthly payments is"ue 1$ months from the
purchase "ate 8hat
should the monthly payments be if .or=)urniture earns 123
compoun"e" monthly onits account recei'able
during both the deferral period an"
the repayment period 4 )ormula solution)ormula solution
inding !ayment Size in a
Keferre" Annuity
inding !ayment Size in a
Keferre" Annuity
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)5 6 *5:1 N i ;n)ormula)ormula
)5 6 $$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
has a promotionon a be"room set selling
for 0$$F2 Juyers will pay“no money down and no
payments for /0 months.” he first of $7
e?ual monthly payments is"ue 1$ months from the
purchase "ate 8hat
should the monthly payments be if .or=)urniture earns 123
compoun"e" monthly onits account recei'able
during both the deferral
period an"the repayment period 4
.or= )urniture
has a promotionon a be"room set selling
for 0$$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
$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
inding !ayment Size in a
Keferre" Annuity
inding !ayment Size in a
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
inding umber 9f !ayments in a
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
of 01222 will beta=en from thefun" five years
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%$$
inding umber 9f !ayments in a
Keferre"Annuity
inding umber 9f !ayments in a
Keferre"
Annuity
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1222
Iow fin" the *M of the annuity !
2
771 ?uarterly payments will "eplete thefun":77 full payments an" 1 partial;
771 ?uarterly payments will "eplete thefun":77 full payments an" 1 partial;
$81%$$
)5 6 $8,1%$$*5 6 - $81%$$I 6 771
)ormula solution)ormula solution
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
of 01222 will beta=en from thefun" five years
from now . How
many with"rawals
will it
inding umber 9f !ayments in a
Keferre"Annuity
inding umber 9f !ayments in a
Keferre"
Annuity
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)5 6 *5:1 N i ;n)ormula)ormula
)5 6 $2222:1 N 22E7;18
6 0$8,1%$$
)in" the )5 of0$2,222 in 7F years
*M
*5 Di
)i 1ln
1ln
n L
ln:12$;
6 771 paymentsor 11 years
0$2,222 isin'este"
in a fun"earning
3 compoun"e"%uarterly
he first?uarterly
with"rawal
of 01222 will beta=en from thefun" five years
from now. How
many with"rawalswill it
0$2,222 isin'este"
in a fun"earning
3 compoun"e"
%uarterlyhe first
?uarterlywith"rawal
of 01222 will beta=en from thefun" five years
from now . How
many with"rawals
will it
inding umber 9f !ayments in a
Keferre"Annuity
inding umber 9f !ayments in a
Keferre"
Annuity
Lln 1 - $81%$$ 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$*M 6 -%F2I 6 7177
)ormula solution)ormula solution
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+ince you get
pai" e'ery secon"
hurs"ay you
"eci"e to pay 1=-
every two wee&s to ma=e your
bu"geting easier
ind the new term
of your mortgage ifthe interest charges
remain at F3
compounded
monthly
Ketermine c Step 1Step 1
C6 1$ 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 $*M
=
)5−
i +
ni ;:1 1L
:a; Multiply both si"es by i% Continue to isolate n !
*M
)5=
−
i +
ni ;:1 1L Di Di
L *M)5
−+
ni ;:1 1 Di
:b; Reorgani(e e?uation =*M
)5+
ni ;:1 1 Di
:c; ow a=e the natural logarithm
:ln or ln 5 ; of both si"es n ln:1N i ; = lnL *M D i )5
1 N
:"; Solving for n…
divide both sides by
ln:1Ni ;
ln:1Ni ; ln:1Ni ;
n ln:1N i ; = lnL *M D i )51 N*M
)5 D i
)
i 1ln
1 Nln
nL
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.ou alrea"y ha'e
1/ saved
for your "ownpayment ?
compounded monthly
on all of your sa'ings
1$
$
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
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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
$$
%%
)in" the )5 of the 012 222 in $ years
)in" the )5 of the 0%F2 per pay
A"" totals together
he 012 222 continues to earn
interest during the new savings period:
he 012 222 continues to earn
interest during the new savings period:
.ou alrea"y ha'e
1/ saved
for your "own
payment
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)ormula +olution)ormula +olution
)5 6 *5:1 N i ;n)ormula)ormula
6 12222:1 N 22E1$; $7
6 011,$
11
$$
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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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
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his completes Chapter 11