Compound Interest

(1)

8 - 1 8 - 1 Compound Compound Interest Interest Compound Compound Interest Interest

8

C

ompound

Chapter 8

(2)

(3)

8

To better understand howTo better understand how C_{is calculated, let’s reiew how we calculate}_{is calculated, let’s reiew how we calculate}Compound Iompound Interestnterest

!

!impleimple IInterest"nterest"

#ormula

I $

%

r

t

The &ormula on which we base our The &ormula on which we base our

calculation is' calculation is'

Here we hae an amount, the

Here we hae an amount, the %rincipal%rincipal, , which which isis multiplied by the

multiplied by the Interest RateInterest Rate and the and the Time Time oer oer which the Interest is earned"

which the Interest is earned"

(s we will now see,

(s we will now see, Compound InterestCompound Interest uses uses the

the S S um ofum of P P & & I I as a base on as a base on which to calculatewhich to calculate

new

Interest Interest !!

Compound Compound Interest Interest Compound Compound Interest Interest

8

(4)

(5)

8

8 'the

'the interest

interest

on the

on the principal

principal

plus

the

interest

o& prior

o&

prior

periods

e.g.

%rincipal

)

prior

period

interest

$

$ *

*+

+

+



Interest

&or the

&or the next

period

is

is calculated on *

calculated on *+

+

+



This method will continue over the life of the

loan

loan or

or investment.

investment. (See

(See later

later example)

example)

$1000.00

$100.00

Compound Interest

(6)

8 - 4 Compound Interest Compound Interest

8

8 'is the compounded amount and

is the #I.(/ amount o& the loan

or inestment at the

end of the last period!

Contrast this with…

is the alue o& a loan or

inestment TO"!

Compound Interest

(7)

8

8 'the calculation o& interest oer

the li&e o& the loan or inestment

Example:

%rincipal ) prior period interest $ *++

Interest is now calculated on *++

Let’s assume that

the

interest rate is 10% pa.

%rincipal0Compounded1

2 + $ *++

New P $110.00 to start next period

Graphically'

Compound Interest

(8)

8

100 11 0 11 1000 110 1!!1 1100 100 100 11 0

"ime#ears

0

1 

!

&

(mount *+(mount *+ 110 InterestInterest InterestInterest 100 InterestInterest 1!!.1 Compoundin g Period Compoundin g Period Compoundin g Period Compoundin g Period InterestInterest 11

Compound Interest

- Future Value

(9)

(10)

8

8 3hat happens i& the

interest

rate chan#es

durin4 the li&e o&

an inestment5

'(ample…

Compound Interest

(11)

8

8 6ou hold an inestment &or a period o& 7 years

Rates o& return &or each year are 7, 8,

-+ and 9 respectiely

I& you inested *+

at the be4innin4 o& the term, how much will you

hae at the

end o& the last

year5

Compound Interest

(12)

8

8 *+

6ear +

6ear :

6ear ;

6ear 7

*+7

*++:;:

*++88

*+ 2

0+ ) 71

$ *+7

*+7 2

0+ ) 81

$ *++:;:

$ *++88

$ *+++8<

*++:;: 2

0+

-

+1

*++88 2

0+ )91

'(lternatie'(lternatie 6ou hold an inestment &or a period o& 7 years Rates o& return &or each year are 7, 8, -+ and 9 respectiely I& you inested *+ at the be4innin4 o& the term, how much will you hae at the end o& the last year5

Compound Interest

(13)

8

It is rare &or interest to be compounded onl$ once per $ear!

It is rare &or interest to be compounded onl$ once per $ear! 6ou hold an inestment &or a period o& 7 years

Rates o& return &or each year are 7, 8, -+ and 9 respectiely I& you inested *+ at

the be4innin4 o& the term, how much will you

hae at the end

o& the last year5

+0+710+810910+91 $ *+++8<

+ -+

Solvin#

(lternatie

Solvin#

(lternatie

Solve for all

% $ears at

once!

Solve for all

% $ears at

once!

Compound Interest

(14)

8

8 Compoundin4 #re=uencies and %eriods

)re*uen+,

o. per ear

Period

(nnually

+

+ year

Semi

annually

:

< months

>uarterly

7 ; months

Monthly

+:

+ month

(15)

8 - 12

Compound Interest Compound

Interest

8

8 ?eelopment o& a

_#ormula

#ormula

n

Total umber

o&

_Periods

Periods

?eterminin4 alues &or

n

and

i

ominal or nnual

Rate

0 A 1

Periodic

Rate per period

0 i

1 umber of compoundin#s per $ear

m

(16)

8

8 #ormulae

#ormulae

"o etermine

n

"o etermine

i

B o& Compoundin4 #re=uencies pa0m1

Time06ears1

nnual

Interest Rate

0A1

B o& Compoundin4 #re=uencies pa

0m1

(17)

8

8 ;

2 ;

2 (nnually

!emiannually

>uarterly

$ ;

$ <

$ +:

+

:

7 n

n

?eterminin4 alues &or

n

I& you compounded

*+

&or

; years at

<

annually, semiannually, or =uarterly

, what are the alues &or n and i 5

o.

B o& Compoundin4

#re=uencies per year 0m1

Time06ears1

2 #ormula

(18)

8

If you compounded

*+ for

; years at

<

annually, semiannually, or =uarterly ' what are the alues &or n and i 5

?eterminin4 alues &or

i

Rate - i

<



<



<



+ : 7

$ <

$ ;

$ +@

(nnually

!emiannually

>uarterly

nnual Interest Rate 0A1

/ o Compounding )re*uen+ies per

,ear#m

#ormula

(19)

8

8 #ormula

#ormula

?eelopment o& a

&or #uture Dalue

%D$

%resent Dalue0 Principal 1

i $

rate per period

n $ number o& periods

#D $ %D0+ ) i1

n

(20)

8

8 #D $ %D0+ ) i1

n

#ormula

!tee !mith deposited *+, in a sain4s account &or 7 years at a rate o& 8

compounded semiannually

3hat is !tee’s interest and compounded amount5

EFtract necessar$ data

%D $

n $

i $

ol2e…

Compound Interest

- Future Value

7  : $ 8

*+

8: $ 7

(21)

8

8 #D $ %D0+ ) i1

n

#ormula

ol2e…

#D $ *+0+ ) 71

8

$ *+0+;<8@<91

$ *+,;<8@

%rincipal *+, ) Interest ;<8@ Compounded *+,;<8@ sin4 %D $ *+ n $ 8

i

$ 7

Compound Interest

- Future Value

(22)

8

8 3hat is the e&&ect on the



uture



alue

o&

different

Compoundin4 %eriods

o&

I

nterest5

(23)

8

I& you compounded

*+

&or

; years

at

<

annuall$' semi annuall$' or *uarterl$ '

what are the &inal amounts that you would hae at the end o& the three 0;1 years 5

Compound Interest

- Future Value

(nnual

_#D

(

$ +0+<1

;

_*++9+

*++9+

!emi-

_#D

!

$ +0+;1

<

_*++97+

*++97+

Semi

$ <:

>uarterly

_#D

>

$ +0++@1

+:

_*++9@<

*++9@<

+uarterl$ $ <7

(24)

8

(l Jones deposited *+, in a sain4s account &or @ years at + pa (l Jones deposited *+, in a sain4s account

&or @ years at + pa

(nnual

!

imple

I

nterest

Rate o& +

(nnual

!

imple

I

nterest

Rate o& +

(nnual

_{Rate o& +}

C

ompound

(nnual

C

ompound

Rate o& +

imple Vs Compound

Interest

3hat is (l’s

!

imple

I

nterest and

M

aturity

D

alue5 3hat is (l’s

!

imple

I

nterest and

M

aturity

D

I

nterest and

C

ompounded

D

I

nterest and

C

ompounded

D

alue5

(25)

(26)

8 - 22

Compound Interest Compound

Interest

8

8 imple Vs Compound

_Interest

#D $ %D0+ ) i1

n #ormulae#ormulae

I $ %rt

I

$ *+, 2 + 2 @

$ *@

#D $ *+, ) *@

$

*+,@

#D $ *+0++1

@

$ *+, 2+<+@

n

$ @

2 +

$

@

I

$ #D

K

%D $ *+<+@+ - *+ !imple!imple CompoundCompound

(l Jones deposited *+, in a sain4s account &or @ years at +(l Jones deposited *+, in a sain4s account &or @ years at +

$ *<+@+

i $ +

Compare

(27)

8

Endin4 Lalance Endin4 Lalance Compoundin4 %eriod Compoundin4 %eriod

*+,

ominal 3ates o Interest

Compared

*+,<

*+,<9

*+,<+;<

*+,<+8;

Le4innin4 Lalance Le4innin4 Lalance .ominal Rate .ominal Rate

) <

(nnual !emiannual >uarterly ?aily

(28)

8

Calculate the

#uture Dalue

o&

*:,

compounded

dail$

&or

7 years

at

7@

n

$

i

$

*:,

2

2 ++9: $

++9: $

*:,;977+

#D $ *:0+) 7@;<@1

+7<

#D $ %D0+ ) i1

n

#ormula

Compoundin4

Compoundin4 ail$

Compoundin4

ail$ Interest

Interest

Compound Interest Compound Interest

8

7 2 ;<@ $

+7<

7@



;<@ $ +:;:

(29)

8

6ou inested *< at 7@ compounded *uarterl$ (&ter : years, the rate chan4ed to @:

compounded monthl$

3hat amount will you hae 7+_{: years a&ter the initial}

inestment5

(30)

8

6ou inested *< at 7@ compounded *uarterl$ (&ter : years, the rate chan4ed to @:

compounded monthl$

3hat amount will you hae 7+_: years a&ter the

initial inestment5



:

years

7@

years

*< i $ 7@7 #D₊ $ %D_: #D₊ $ <0+)7@718 $ <0+9;<1 $ <@<+@ #D_: i $ @:+: #D_: $ $ <@<+@0++;8@1 $ *7<+ <@<+@0+)@:+:1; n $ 0:271 $ 8 n $ :@2+: $ ;

(31)

8

8 %repare a time-line’ as part o& the solution

6ou borrowed *@ at  compounded monthl$ Nn the &irst and second anniersaries o& the loan, you made payments o& *:@

3hat is the balance outstandin4

(32)

8

 + year : years *@ i $ +: #D₊ - *:@ $ %D_: #D₊ $ @0+)+:1+: $ @0+::91 $ @;<+7@ %D_: $ @;<+7@ K :@ $ :8<+7@ #D_: i $ +: #D_: $ $ :8<+7@0+::91 $ *;<8; :8<+7@ 0+)+:1+: n $ +: n $ +:

6ou borrowed *@ at  compounded monthl$

Nn the &irst and second anniersaries o& the loan, you made payments o& *:@ 3hat is the balance outstandin4 immediatel$ followin# the second pa$ment,

.ew Lalance.ew Lalance

$ *;<8; K :@ $ *@<8;

(33)

8

(34)

(35)

8

6ou eFpect to need *+,@ in ; years 6our banO o&&ers 7 interest compounded semiannuall$

How much money must you put in the banO toda$ 0%D1 to reach your 4oal in

; years5

Calculatin4 P resent  alue

%repare the solution'0a1 al4ebraically, and 0b1 by &inancial calculator

(36)

8

8 %D $ #D0+ ) i1

-n

#ormula

i $ 7: $ :

6ou eFpect to need *+,@ in ; years

6our banO o&&ers 7 interest compounded semiannuall$ How much money must you put in the

banO toda$ 0%D1 to reach your 4oal in ; years5

P $ *+@0+):1-< n $ ; 2 : $ <

Calculatin4 P resent  alue

+: < +@ 889 +,;;+9< $ *+@ 2 888 $ *+,;;+9<

0a1

(37)

8

8 %D $ #D0+ ) i1

-n

#ormula

3hat amount must you inest now at @ compounded dail$ to accumulate to *< a&ter + year5

A $ m $

#D $ i $ n $

Calculatin4 P resent  alue

P $ *<0+)@;<@1-;<@ $ *< 2 9@+: $ *@,7 @ ;<@ < ;<@ + +++ 9@+: @,7 @ ;<@ @;<@ +2;<@ $ ;<@ *<

(38)

8

Two pa$ments o& *:: each must be made + and 7 years from now I& money can earn @ compounded monthl$,

what sin4le payment ; years &rom now would be

e*uivalent to the two scheduled pa$ments, ?raw a Time-line?raw a Time-line

Step 1

#ind the #D o& the payment that is moed &rom 6ear + to 6ear ; #ind the #D o& the payment that

is moed &rom 6ear + to 6ear ;

Step 2

#ind the %D o& the payment that is moed &rom 6ear 7 to 6ear ; #ind the %D o& the payment that

is moed &rom 6ear 7 to 6ear ;

Step 3

%repare the solution'0a1 al4ebraically, and 0b1 by &inancial calculator

(39)

8

Two pa$ments o& *:: each must be made + and 7 years from now I& money can earn @ compounded monthl$,

what sin4le payment ; years &rom now

would be e*uivalent to the two scheduled pa$ments, ?raw a Time-line?raw a Time-line

Step 1

*:: *::

 + year : years ; years 7 years

i $ @+: n $ :2+: $ :7

Step 2

#ind the #D o&

the payment that is moed &rom 6ear + to

6ear ;

#ind the #D o& the payment that is moed &rom 6ear + to

6ear ; #D₊ $ ::0+)@+:1:7 $ ::0++791 $ :7;8 0a1 #D₊ %D₊ %D_: #D_: ow