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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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(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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• 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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• )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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• 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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• 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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• Example 2continueduarterly

monthly

&".7"5&!3&"""!&".39"5&!

&".39"5&!""633.123&"""!12

&/&1,633.

&/

1&#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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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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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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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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Compounding – Early

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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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• 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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• 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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• +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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• 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!

&#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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• *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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• + loan is made $or !35"" 4ith an interest rate o$

9# and payments made annually $or / years.

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• + loan is made $or !/6"" 4ith an +(* o$ 1

and payments made monthly $or &/ months.

math 1100 - chapter 13-15[1]

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