Math 1100 - Chapter 13-15[1]
Transcript of Math 1100 - Chapter 13-15[1]
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13.1 Compound Interest
• Simple interest – interest is paid only on the principal
• Compound interest – interest is paid on both principal and interest, compounded at regularintervals
• Example a !1""" principal paying 1"# simpleinterest a$ter 3 years pays .1 × 3 × !1""" % !3""
I$ interest is compounded annually, it pays .1 × !1""" % !1"" the $irst year, .1 × !11"" % !11"the second year and .1 × !1&1" % !1&1 the thirdyear totaling !1"" ' !11" ' !1&1 % !331 interest
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13.1 Compound Interest
(eriod InterestCredited
)imesCredited
per year
*ate per compounding
period
+nnual year 1 * Semiannual months &
-uarterly uarter /
0onthly month 1&
& R
/ R
1& R
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13.1 Compound Interest
• Compound interest $ormula
0 % the compound amount or $uture value( % principal
i % interest rate per period o$ compounding
n % number o$ periodsI % interest earned
P M I and i P M n −=+= 12
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13.1 Compound Interest
• )ime alue o$ 0oney – 4ith interest o$ 5#
compounded annually.
&""" &"1" &"&"
1"
"5.121"""!
121"""!
=
+ ni
1"
"5.12
1"""!
12
1"""!=
+
n
i
1"""!
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13.1 Compound Interest
• Example !6"" is invested at 7# $or years. 8indthe simple interest and the interest compounded
annually
Simple interest
Compound interest
33!"7.6""! =××== PRT I
56./""!6""!56.1&""!56.1&""!"7.126""!12
=−=−===+=
P M I i P M
n
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13.1 Compound Interest
• Example !3&""" is invested at 1"# $or &years. 8ind the interest compounded yearly,semiannually, uarterly, and monthlyyearly
semiannually
&".69!3&"""!&".3669!
&".3669!"5.123&"""!12 /
=−=−===+=
P M I
i P M n
7&"!3&"""!367&"!
367&"!1".123&"""!12 &
=−=−===+=
P M I
i P M n
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13.1 Compound Interest
• Example 2continueduarterly
monthly
&".7"5&!3&"""!&".39"5&!
&".39"5&!""633.123&"""!12
&/&1,633.
&/
1"
=−=−===+=
=×===
P M I
i P M
ni
n
69.966!3&"""!69.36966!
69.36966!"&5.123&"""!12 6
=−=−=
==+=
P M I
i P M n
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13.& :aily and Continuous
Compounding• :aily compound interest $ormula divide i by 35and multiply n by 35
• Continuous compound interest $ormula
P M I and P M ni −=+= 353512
year per rater years y Pe M yr
=== ;
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13.& :aily and Continuous
Compounding• )ime alue o$ 0oney – 4ith 5# interest
compounded continuously.
&""" &"1" &"&"
"52.1"
1"""!
1"""!
e
e yr
="52.1"
1"""!1"""!
ee yr
= 1"""!
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13.& :aily and Continuous
Compounding• Example 8ind the compound amount i$ !&9"" is
deposited at 5# interest $or 1" years i$ interest is
compounded daily.
13./761!
12&9""!
12
35"
35
#5
35
35
=
+=
+= ni P M
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13.& :aily and Continuous
Compounding• Example 8ind the compound amount i$ !1&"" is
deposited at 6# interest $or 11 years i$ interest is
compounded continuously.
"6.193!1&""!"6.&693!
"6.&693!
1&""! "6."11
=−=−=
=== ×
P M I
e Pe M yr
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13.& :aily and Continuous
Compounding – Early
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13.& :aily and Continuous
Compounding – Early ob ?ashir deposited !""" in a /@year
certi$icate o$ deposit paying 5# compounded daily.Ae 4ithdre4 the money 15 months later. )he
passboo= rate at his ban= is 3B # compounded daily.8ind his amount o$ interest.
>ob receives 15@3 % 1& months o$ 3.5 # interest
compounded daily
73.&1"!
12"""!
12
35
35#5.3
35
35
=
+=
+= ni P M
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13.3 8inding )ime and *ate
• iven a principal o$ !1&,""" 4ith a compoundamount o$ !17,31.9/ and interest rate o$ 6#compounded annually, 4hat is the time period inyearsD
8rom +ppendix : table pg 6"52 i % 6# 4e $ind thatn % 5 years
n
n
ni P M
"6.12/93&6.11&"""
9/.1731
#612""",1&!9/.31,17!
12
==
+=
+=
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13.3 8inding )ime and *ate
• Example8ind the time to double your investmentat #.
I$ you try di$$erent values o$ n on your calculator,the value that comes closest to & is 1&. )here$orethe investment doubles in about 1& years.
n
n
ni P M
".12&
#121&
12
=
+=
+=
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13.3 8inding )ime and *ate
• Exampleiven an investment o$ !13&"",compound amount o$ !&&6"." invested $or 6years, 4hat is the interest rate i$ interest iscompounded annuallyD
8rom +ppendix : table pg 6"32 i % 7# 4e $indthat $or n%6, column + % 1.71616 so i % 7#.
6
6
1271616.113&""
".&&6"
1213&""".&&6"
12
i
i
i P M n
+==
+=+=
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13./ (resent alue at Compound
Interest• Exampleiven an amount needed 2$uture
value o$ !33"" in / years at an interest rate o$
11# compounded annually, $ind the presentvalue and the amount o$ interest earned.
19.11&!61.&17333""
61.&173!11.12
33""#111233""
12
/
/
=−
==+=
+=
P
P
i P M n
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13./ (resent alue at Compound
Interest• Example +ssume that money can be invested at
6# compounded uarterly.
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1/.1 +mount 28uture alue o$ an +nnuity
• +nnuity – a seuence o$ eual payments
• (ayment period – time bet4een payments
• FGrdinary annuityH – payments at the end o$ the pay period
• F+nnuity dueH @ payments at the beginning o$ the
pay period• FSimple annuityH – payment dates match the
compounding period 2all our annuities are simple
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1/.1 +mount 28uture alue o$ an +nnuity
• +mount o$ an annuity @ S 2$uture value o$ n
payments o$ R dollars $or n periods at a rate o$ i
per period
• se you calculator instead o$ using appendix :.
( )in
n
s Ri
i RS ⋅=
−+= 11
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1/.1 +mount 28uture alue o$ an +nnuity
• Example 8or S % !&1,""", payments 2 R) o$ !15""at the end o$ each @month period i % 1"#compounded semi@annually. 8ind the minimumnumber o$ payments to accumulate &1,""".
)rying di$$erent values $or n, the expression goes over 1/4hen n % 11 2Exact value % /.&"7671215""%!&131".16
( ) ( )
( )
−==
−=
−+=
"5."
1"5.11/
15""
""",&1
"5."1"5.115""1115""!""",&1!
"
"
n
nn
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1/.& (resent alue o$ an +nnuity
• Example
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1/.3 Sin=ing 8unds
• Sin=ing $und – a $und set up to receive periodic
payments.
)he purpose o$ this $und is to raise an amount o$
money at a $uture time.
• >ond – promise to pay an amount o$ money at a
$uture time.
2Sin=ing $unds can be set up to cover the $acevalue o$ bonds.
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1/.3 Sin=ing 8unds
• +mount o$ a sin=ing $und payment
• Same $ormula as in section 1/.1, except solved $or
the variable *.
( )in
n s
S iiS R 1
11 ⋅=
−+=
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1/.3 Sin=ing 8unds
• Example 15 semiannual payments are made into asin=ing $und at 7# compounded semiannually so
that !/65" 4ill be present. 8ind the amount o$
each payment rounded to the nearest cent.
( )35.&51!
1"35.12
"35./65"!
11
"35.&
"7.
&
#7
15
=−
×=
−+=
===
ni
iS R
i
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1/.3 Sin=ing 8unds
• Example + retirement bene$it o$ !1&,""" is to be paidevery months $or &5 years at interest rate o$ 7#compounded semi@annually. 8ind 2a the present value to$und the end@o$@period retirement bene$it. 2b the end@o$@
period semi@annual payment needed to accumulate the
value in part 2a assuming regular investments $or 3" yearsin an account yielding 6# compounded semi@annually.
( )( )
"./6,&61!
195/7&."
56/9&7./""",1&
"35.1"35.
1"35.1""",1&
5"&5&,"35.&
"7.
&
#7
5"
5"
=
=
=
−=
=×====
A
ni
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1/.3 Sin=ing 8unds
• Example2part b – amount to save every months$or 3" years $or this annuity
( ) ( )
9.116&!519&7.9
"/.&61,/6."
1"/.1
"/. &61,/6."
11
"&3","/.&
"6.
&
#6
"
=
=
−⋅=
−+⋅=
=×====
ni
iS R
ni
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15.1 Gpen@End Credit
• Gpen@end credit – the customer =eeps ma=ing payments until no outstanding balance is o4ed2e.g. charge cards such as 0asterCard and isa
• *evolving charge account – a minimum amountmust be paid account might never be paid o$$
• 8inance charges – charges beyond the cash price,
also re$erred to as interest payment• Gver@the@limit $ee – charged i$ you exceed your
credit limit
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15.1 Gpen@End Credit
• Example 8ind the $inance charge $or an
average daily balance o$ !6/31.1" 4ith
monthly interest rate o$ 1./#
$inance charge
"/.116!
1".6/31!"1/."
=
×=
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15.1 Gpen@End Credit
• Example 8ind the interest $or the $ollo4ing
account 4ith monthly interest rate o$ 1.5#
(revious balance !/1&./6
Lovember 5 >illing date
Lovember 16 (ayment !15"
Lovember 3" :inner and play !6/.5"
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15.1 Gpen@End Credit
• Example2continued
• +verage balance % 1"&/.9÷3" % !3/1.5• 8inance charge % ."15 × 3/1.5 % !5.1&• >alance at end % 3/.96 ' 5.1& % !35&.1"
:ate ; days until chg balance 2&×23 Lovember 5 13 !/1&./6 53&.&/
Lovember 16 1& !&&./6 31/9.7 Lovember 3" 5 !3/.96 173/.9
:ecember 5 3" 2total days 1"&/.9
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15.& Installment Moans
• + loan is FamortiNedH i$ both principal and interestare paid o$$ by a seuence o$ periodic payments.8or a house this is re$erred to as mortgage
payments.
• Menders are reuired to report $inance charge2interest and their annual percentage rate 2+(*
• +(* is the true e$$ective annual interest rate $or aloan
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15.& Installment Moans
• In order to $ind the +(* $or a loan paid ininstallments, the total installment cost, $inancecharge, and the amount $inanced are needed
1. )otal installment cost % :o4n payment ' 2amount o$each payment × number o$ payments
&. 8inance charge % total installment cost – cash price
3. +mount $inanced % cash price – do4n payment
/. et
5. se table 15.& to get the +(*
1""!$inanced+mount
charge8inance ×
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15.& Installment Moans
• Example iven the $ollo4ing data, $ind the
$inance charge and the total installment cost
)otal installment cost
8inance charge
+mount8inanced
:o4n(ayment
Cash(rice
; o$ payments
+mount o$ payment
!5" !1&5 !775 &/ !3&
116!775!693!
693!3&!&/1&5!
=−==×+=
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15.& Installment Moans
• Example iven the $ollo4ing data, $ind theannual percentage rate using table 15.&
$rom table 15.& ; payments % 1&, +(* isapproximately 13#
+mount8inanced
8inanceCharge
; o$ payments
!3/5 !&/.& 1&( )
1/.71""
3/5
&.&/
1""!1
&2
≈×
=×
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15.3 Early (ayo$$s o$ Moans
• nited States rule $or early payo$$ o$ loans
1. 8ind the simple interest due $rom the date the
loan 4as made until the date the partial paymentis made.
&. Subtract this interest $rom the amount o$ the payment.
3. +ny di$$erence is used to reduce the principal/. )reat additional partial payments the same 4ay,
$inding interest on the unpaid balance
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15.3 Early (ayo$$s o$ Moans
• Example iven the $ollo4ing note, $ind the balance dueon maturity and the total interest paid on the note.
1. 8ind the simple interest $or " days and subtract it
$rom the payment.
&. Subtract it $rom the payment
3. *educe the principal by the amount $rom 2&
(rincipal Interest )ime in days (artial payments
!56"" 1"# 1&" !&5"" on day "
7.9!3"
"1"."56""! =××== PRT I
33.&/"3!7.9&5""! =−
7.339!33.&/"3!56""! =−
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15.3 Early (ayo$$s o$ Moans
• Example2continuedInterest due at maturity
>alance due on maturity 2add reduced principal to interest
)otal interest paid on the note 2add interest paid to interest due at
maturity &6.3/53!1.57.339 =+
1.5!3"
"1"."7.339! =××== PRT I
&6.153!1.57.9 =+
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15.3 Early (ayo$$s o$ Moans
• *ule o$ 76 2sum@o$@the@balances method
Lote 21'&'3''1& – sum o$ the month numbers adds
up to 76 used to derive the $ormula.
% unearned interest, 8 % $inance charge, L % number
o$ payments remaining, and ( % total number o$
payments
++
= P N
P
N F U 1
1
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15./ (ersonal (roperty Moans
• 8rom section 1/.&, the present value o$ an
annuity 2 A made up o$ payments o$ R dollars
$or n periods at a rate o$ i per period( )
( )
( )( )
−++=
+−+
=
11
1
1
11
n
n
n
n
i
ii A R so
ii
i R A
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15./ (ersonal (roperty Moans
• + loan is made $or !35"" 4ith an interest rate o$
9# and payments made annually $or / years.
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15./ (ersonal (roperty Moans
• + loan is made $or !/6"" 4ith an +(* o$ 1
and payments made monthly $or &/ months.