3 MateriTerbuka Compound Interest

(1)

8 - 1

Compound

Interest

Compound

Interest

8

8 C

C

ompound

(2)

8 - 2

Compound

Interest

Compound

Interest

8

8 Calculate the…

Learning Objectives

After completing this chapter, you will be able to:

…Maturity Value of compound interest for

Guaranteed Investment Certificates (GICs)

…Maturity Value(MV), Future Value (FV), and Present

Value(PV) in

compound interest applications

, by both the

algebraic method

and the

pre-programmed financial calculator

method

…Price of "strip" bonds

LO-1

(3)

8 - 3

Compound

Interest

Compound

Interest

8

8 Calculate the…

…

Redemption Value

of a compound interest bearing Canada Savings Bond

…Payment on any date

that is equivalent to one or

more payments on other dates

…Economic Value

of a payment stream

And be able to…

…Adapt the concepts and equations

of compound

interest to cases of compound growth

Learning Objectives

LO-2

(4)

8 - 4

Compound

Interest

Compound

Interest

8

Compound

Interest

Compound

Interest

8

LO-1

(5)

8 - 5

Compound

Interest

Compound

Interest

8

To better understand how Compound Interest is calculated, let’s review how we calculate

Simple Interest!

Formula

I =

P

r

t

The formula on which we base our calculation is…

Here we have an amount, the Principal, which is multiplied by the Interest Rate and the Time over

which the Interest is earned!

As we will now see, Compound Interest uses the Sum of P & I as a base on which to calculate

new

Interest!

Compound

Interest

Compound

Interest

(6)

8 - 6

Compound

Interest

Compound

Interest

8

8 …the

interest

on the principal

plus

the

interest

of

prior

periods

e.g.

Principal

+

prior

period

interest

= $1

100 .00

Interest

for the

period

is

calculated on $1

100 .00.

This method will continue over the life of the

loan or investment. (See later example)

$1000.00 $100.00

Compound Interest

(7)

8 - 7

Compound

Interest

Compound

Interest

8

8 …is the

compo

unded

amount

and

is the FINAL amount of the loan

or investment at the

end of the last period!

Contrast this with…

...is the value of a loan or

investment

TODAY!

Compound Interest

(8)

8 - 8

Compound

Interest

Compound

Interest

8

8 …the calculation of

interest over

the

life

of the

loan or investment

Example:

Principal

+

prior

period

interest

= $1

100 .00

Interest

is

now calculated on $1

100 .00

Let’s assume that the interest rate is 10% pa.

Principal(Compo

unded

)

*

0.10 = $

110.

00 New

P

$

1

210 .00 to start next period

Graphically…

Compound Interest

(9)

8 - 9

Amount $1000Amount $1000

(10)

8 - 10

Compound

Interest

Compound

Interest

8

8 What happens if the

interest

rate changes

during the life of

an

investment

?

Example…Example…

Compound Interest

(11)

8 - 11

Compound

Interest

Compound

Interest

8

8 You hold an investment for a period of 4 years.

Rates of return

for each year are

4%, 8%,

-10%

and

9% respectively

.

If you invested $1000

at the beginning of the term, how much will you

have at the end of the last

year?

Compound Interest

(12)

8 - 12

Compound

Interest

Compound

Interest

8

8 $1000

Year

1 Year

2 Year

3 Year

4 $10

40 $1

123.20 $1

010.88 $1000 *

(1

+

.04

)

= $10

40 $10

40 *

(1 +

.08

)

= $1

123.20 = $1

010.88 = $1

101.86 $1

123.20 *

(1

-

.10

)

$1

010.88 *

(1 +

.09

)

…Alternative…Alternative You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

Compound Interest

(13)

8 - 13 compounded only once per year!

(14)

8 - 14

Compound

Interest

Compound

Interest

8

8 Compounding Frequencies and Periods

FrequencyFrequency No. per YearNo. per Year Period Period

Annually

1 1 year

(15)

8 - 15

Compound

Interest

Compound

Interest

8

8 Development of a

Formula

n

Total Number

of

_PeriodsPeriods

Determining values for

n

and

i

Nominal

or

Annual

Rate

( j )

Periodic

Rate per period

(

i

)

(16)

8 - 16

Compound

Interest

Compound

Interest

8

8 Formulae

Formulae

To Determine To Determine

n

To Determine To Determine

i

# of Compounding Frequencies p.a.

(m)

Time(Years)

Annual

Interest

Rate

(j)

# of Compounding Frequencies p.a.

(m)

(17)

8 - 17

Determining values for

n

If you compounded

$100

for

3 years at

6%

annually

,

semiannually

,

or quarterly

, what are the values for n and i ?

No.No.

# of Compounding Frequencies per year (m)

Time(Years)

*

(18)

8 - 18

Determining values for

i

Rate -

Rate - i

i

Annual Interest Rate (j)

# of Compounding Frequencies per

year(m)

Formula

(19)

8 - 19

Compound

Interest

Compound

Interest

8

8 Formula

Formula

Development of a for

F

uture

V

alue

PV

=

Present Value(

Principal

)

i

=

rate

per period

n =

number

of periods

FV

=

PV

(1 +

i

)

n

(20)

8 - 20

Compound

Interest

Compound

Interest

8

8 FV

=

PV

(1 +

i

)

n

Formula

Steve Smith deposited $1,000 in a savings account for

4 years at a rate of 8%

compounded semiannually. What is Steve’s interest and compounded amount?

Extract

necessary

data...

PV

=

n

=

i

=

Solve…

Compound Interest

- Future Value

4 X

2 =

8 $1000

(21)

8 - 21

Compound

Interest

Compound

Interest

8

8 FV

=

PV

(1 +

i

)

n

Formula

Solve…

FV = $1000(1 +

.04

)

8

=

_{$1000(1.368569)}

= $1,

368.57

Principal $1,000.00

+ Interest 368.57

Compounded $1,368.57

Using PV = $1000 n = 8

i

= .04

Compound Interest

(22)

8 - 22

Compound

Interest

Compound

Interest

8

BOTH ways will

be shown!

BOTH

ways will be

shown!

Use

a

calculator

and

algebraic

sequencing

Use

the

TI BAII Plus

financial

calculator

!

There are

two methods

that can be used to

(23)

8 - 23

Compound

Interest

Compound

Interest

8

8 Solve… $1000(1 +

.04

)

8

.04

1

8 1000

$1,368.57

$

1 ,

368.57 Use

Use

a

calculator and algebraic sequencing

calculator

and algebraic sequencing

(24)

8 - 24

Compound

Interest

Compound

Interest

8

8 Find the following

Find the following

KEYS

:

The Power function Key. Used to calculate the

value of exponents.

Used to access symbols located “above”

another key, i.e. its acts like the SHIFT key on a computer

keyboard.

Use

a

calculator and algebraic sequencing

calculator

and algebraic sequencing

Changes the sign of the data value of the number

(25)

8 - 25

Compound

Interest

Compound

Interest

8

Some calculators have the

y

x symbol above the calculator key.

(

1.04 )

8

_is…

The key stroke sequence to evaluate an EXPONENT that is…

1.04

8 1.368569

0.73069

PositivePositive

Find the following

KEYS

:

Use

a

calculator and algebraic sequencing

calculator

and algebraic sequencing

(26)

8 - 26

Compound

Interest

Compound

Interest

8

This calculator can store up to

10 values.

Find the following

KEYS

:

Use

a

calculator and algebraic sequencing

calculator

and algebraic sequencing

Used to

Sto

re or save displayed values.

Used to

R

e

c

a

l

the saved values.

Let’s PractiseLet’s Practise Therefore, the calculator must be informed as to

(27)

8 - 27

Compound

Interest

Compound

Interest

8

8 Use

Use

a

_{calculator and algebraic sequencing}

calculator

and algebraic sequencing

Using the key

e.g. you want to store the value ’45’. The key stroke sequence ‘to store’ is:

45

..choose from 0 - 9

…’clear’ display

The key stroke sequence ‘to recall’ is:

(28)

8 - 28

Compound

Interest

Compound

Interest

8

(29)

8 - 29

Compound

Interest

Compound

Interest

8

The nominal interest rate (

I

nterest

/

Y

ear)

1. Number of compoundings (for lump payments)

2. Number of payments (for annuities)

Represents the Periodic Annuity

P

ay

m

en

t

(used in chapter 10)

Tells the calculator to compute (CPT)

Present Value or initial(first) lump sum

Find the following

KEYS

:

(30)

8 - 30

Compound

Interest

Compound

Interest

8

However, we can now input the number of compoundings per year into the financial calculator.

This can be performed by using the symbol

Find the following

KEYS

:

…it is rare for interest to be compounded only once per year!

Previously, it was noted that

To access this symbol use:

(31)

8 - 31

Compound

Interest

Compound

Interest

8

The 12 is a default

setting The 12

is a default

setting This display is referred to as “the worksheet”.

… represents the number of

P

ayments per

Y

ear

C

ompoundings per

Y

ear To access use:

Note

:

You can override these values by entering new ones!

…more

Appears automatically

(32)

8 - 32

Compound

Interest

Compound

Interest

8

8 must be given

the same value as

If

the calculation

does not

involve

more than one payment

If

the calculation

does not

involve

more than one payment

(33)

8 - 33

as the default, i.e. P/Y

Illustration

C

ompoundings per

Y

ear

In Compound Interest,

P/Y

must be given the same value as

C/Y.

In Compound Interest,

P/Y

must be given the same value as C/Y

.

(34)

8 - 34

Compound

Interest

Compound

Interest

8

There are

two methods

that can be used to

calculate compound interest:

(35)

8 - 35

compounded amount?

Using the

Using

the

TI BAII Plus

financial calculator

(36)

8 - 36

Using the

Using

the

TI BAII Plus

financial calculator

Steve Smith What is Steve’s

interest and What is Steve’s

interest and

compounded

(37)

8 - 37

Compound

Interest

Compound

Interest

8

…there is no need to keep

inputting each time!

0 You only need

to input the values that have changed!

(38)

8 - 38

Compound

Interest

Compound

Interest

8

8 Cash Flows

Cash Flows

… payments received e.g. receipts

Treated as:

Positives

+

Negatives

-

-..a term that refers to

payments

that can be either …

..a term that refers to

payments

that can be either …

(39)

8 - 39

Compound

Interest

Compound

Interest

8

8 What is the effect on the

F

uture

V

alue

of

different

Compounding Periods

of

(40)

8 - 40

Compound

Interest

Compound

Interest

8

If you compounded

$100

for

3 years

at

6%

annually, semiannually, or quarterly, what are the final amounts that you would have at

the end of the three (3) years ?

Compound Interest

- Future Value

Annual

_FV

A

=

100 (1.

06 )

3 $119.10

$

1

19 .

10 Semi-

Semi-

_FV

_S

₌

₁₀₀

_(1.

₀₃

₎

6 $119.41

$

1

19 .

41 Semi

=

6%

/2

Quarterly

_FV

Q

=

100 (1.

015 )

12 $119.56

$

1

19 .

56

(41)

8 - 41

(42)

8 - 42

Compound

Interest

Compound

Interest

8

(43)

8 - 43

Al Jones deposited $1,000 in a savings account for 5 years at 10% p.a..

Annual

S

imple

I

nterest

Rate

of 10%

Annual

S

imple

I

nterest

Rate

of 10%

Annual

_Rate

C

_{of 10%}

C

_{of 10%}

ompound

Simple

Vs

Compound

Interest

C

ompounded

V

alue? What is Al’s

I

nterest and

(44)

McGraw-Hill Ryerson©

Simple

Vs

Compound

Interest

FV

=

PV

(1 +

i

)

n

Formulae Formulae

I

=

P

r

t

SimpleSimple CompoundCompound

Al Jones deposited $1,000 in a savings account for 5 years at 10%Al Jones deposited $1,000 in a savings account for 5 years at 10%

= $

610.51 i

=

.10

Compare

(45)

(46)

8 - 46

(47)

(48)

8 - 48

Compound

Interest

Compound

Interest

8

Calculate the

Future Value

of

$2,000

compounded

daily

for

4 years

at

4.5%.

n

=

i

=

$2,000

$

2,000

*

1.1972 =

$2,394.41

$

2 ,

394.41 FV

= $

2000

(1+

.045

/365)

1460

FV

=

PV

(1 + i)

n

Formula

Compounding

Daily

Interest

Compound

Interest

Compound

Interest

8

(49)

8 - 49

Compound

Interest

Compound

Interest

8

8 2394.41

.045

₃₆₅

1 1460

2000

Solve FV =

$2000(1+

.045

/

365 )

1460

= $2,394.41

= $

2 ,

394 .

41 Compounding

(50)

8 - 50

Compounding

Daily

Interest

(51)

8 - 51

Compound

Interest

Compound

Interest

8

You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2%

compounded monthly.

What amount will you have 41/2 years after the initial

investment?

(52)

8 - 52

Compound

Interest

Compound

Interest

8

You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2%

compounded monthly.

What amount will you have 41_/2 years after the

initial investment?

0 2 years

4.5 years

$6000

i = .045/4

FV₁ = PV₂

FV₁ = 6000(1+.045/4)8

= 6000(1.0936) = 6561.75

FV₂

i = .052/12

FV₂ =

= 6561.75(1.1385) = $7470.61

6561.75(1+.052/12)30

n = (2*4) = 8

(53)

8 - 53

Using the

Using

the

TI BAII Plus

financial calculator

FVFV₁1 = PV = PV₂2

FV

4

(54)

8 - 54

Using the

Using

the

TI BAII Plus

financial calculator

FVFV₂2

(55)

8 - 55

Compound

Interest

Compound

Interest

8

8 Prepare a ‘time-line’ as part of the solution

You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of $2500.

(56)

8 - 56

you made payments of $2500. What is the balance outstanding immediately following the second payment?

New BalanceNew Balance

= $3068.30 – 2500.00

(57)

8 - 57

Using the

Using

the

TI BAII Plus

financial calculator

FVFV₁1 – 2500 = PV – 2500 = PV₂2

anniversaries of

the loan, you made payments of $2500.

What is the balance outstanding

immediately after the 2nd_payment?

2500

7.0

12

(58)

8 - 58

Compound

Interest

Compound

Interest

8

-2861.45

$568.30

Using the

Using

the

TI BAII Plus

financial calculator

FVFV₂2

You borrowed $5000 at 7%

compounded monthly.

On the 1st_._and₂nd

anniversaries of

the loan, you made payments of $2500.

What is the balance outstanding

immediately after the 2nd_payment?

2500

Step 2Step 2

(59)

8 - 59

Compound

Interest

Compound

Interest

(60)

8 - 60

Compound

Interest

Compound

Interest

8

8 Formula for

P

resent

V

alue

PV = FV(1 +

i

)

-n

Formula

Keys

i

1 $PV

$

PV

This is the only change to the

(61)

8 - 61

Compound

Interest

Compound

Interest

8

You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually.

How much money must you put in the bank today (PV) to reach your goal in

3 years?

Calculating

P

resent

V

alue

Prepare the solution…(a) algebraically, and (b) by financial calculator

(62)

8 - 62

Formula

i = .04/2 = .02

You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the

(63)

8 - 63

Compound

Interest

Compound

Interest

8

3 * 2

4

2

1500

0 PV= -1,331.96

(b)

Calculating

P

resent

V

alue

You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the

bank today (PV) to reach your goal in 3 years?

(64)

8 - 64

Formula

(65)

8 - 65

Compound

Interest

Compound

Interest

8

What amount must you invest now at 5% compounded

daily to accumulate to $6000 after 1 year?

1 * 365

5

365

PV= - 5,707.40

6000

0

Calculating

P

resent

V

alue

(66)

8 - 66

equivalent to the two scheduled payments? Draw a Time-lineDraw a Time-line

Step 1Step 1

Find the FV of the payment that is moved from Year 1 to Year 3 Find the FV of the payment that

is moved from Year 1 to Year 3

Step 2Step 2

(67)

8 - 67

would be equivalent to the two scheduled payments? Draw a Time-lineDraw a Time-line

(68)

(69)

8 - 69

Finally, this PV amount can be added to that put into memory…

0

2430.87

(70)

8 - 70 to that put into memory…

0

(71)

8 - 71

Compound

Interest

Compound

Interest

8

8 What

regular payment

will an investor receive

from a $10,000,

3 year

,

monthly payment

GIC

earning a

nominal rate of 4.8%

compounded monthly

?

Interest

rate per payment interval is:

i

=

j

/

m

= .

048/

12 = 0.0040

…the monthly

payment will be:

…the monthly

payment will be:

PV *

I

**= $10000 * 0.0040**

= $

40.00

(72)

8 - 72

Compound

Interest

Compound

Interest

8

8 Suppose a bank quotes

nominal annual interest rates

of

6.6%

compounded annually

,

6.5%

compounded

semi

-annually

,

and

6.4%

compounded monthly

on

five-year

GICs.

Making a choice!…

(73)

Suppose a bank quotes nominal 6.4% compounded monthly

on five-year GICs. Which

rate should an investor choose for

an investment of $1,000?

Suppose a bank quotes nominal

annual interest

6.5% compounded semi-annually, and

6.4% compounded monthly

on five-year GICs. Which

rate should an investor choose for

(74)

8 - 74

the 6.5% compounded

semi

-annually

provides for the best

rate of return on investment!

the

6.5%

compounded

semi

-annually

provides for the best

rate of return on investment!

(75)

8 - 75

Compound

Interest

Compound

Interest

8

(76)

8 - 76

of

Interest Rates

Fixed Rate

…the interest rate does not change over the term of the GIC.

Fixed Rate

…the interest rate does not change over the term of the GIC.

An investment in a GIC might have a…

Step-up Rate

…the interest rate is increased every 6 months or every year

according to a pre-determined schedule.

Step-up Rate

…the interest rate is

increased every 6 months or every year

according to a pre-determined schedule.

Variable Rate

... is adjusted every year or every 6 months to reflect market rates… may be a minimum

“floor” below which rates

cannot drop

Variable Rate

... is adjusted every year or every 6 months to reflect market rates… may be a minimum

“floor” below which rates

(77)

8 - 77

version

Compound Interest

_version

Compound Interest

version

Interest is

paid

to the investor

Interest is periodically

converted to principal

and

paid at maturity

Interest

is periodically

(78)

8 - 78

Compound

Interest

Compound

Interest

8

8 C

anadian

S

avings

(79)

8 - 79

Compound

Interest

Compound

Interest

8

- Can be purchased from financial institutions but funds go to federal government to help finance its debt

- usual term is 10 or 12 years

- variable interest rates

- interest rate is changed on each anniversary, with minimum rates for subsequent 2 years

C

anadian

S

avings

B

onds

To view current rates of interest and redemption values

(80)

8 - 80

Compound

Interest

Compound

Interest

(81)

8 - 81

Compound

Interest

Compound

Interest

8

All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest.

The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates.

All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest.

The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates.

C

anadian

S

avings

(82)

8 - 82

Compound

Interest

Compound

Interest

(83)

8 - 83

Compound

Interest

Compound

Interest

8

8 The

fair market value of

an investment

is the

sum

of the Present

Values of the

expected cash flows

.

The

discount rate

used

should be

the

prevailing market

determined rate

of return

required

on this type of

investment.

(84)

8 - 84

Compound

Interest

Compound

Interest

(85)

8 - 85

Compound

Interest

Compound

Interest

8

… owner will receive a single payment (called the face value of the bond) on the

bond’s maturity date

… owner will receive a single payment (called the face value of the bond) on the

bond’s maturity date

… the maturity date could be as much as 30

years in the future.

No interest will be received in the interim!

… the maturity date could be as much as 30

years in the future.

(86)

8 - 86

Compound

Interest

Compound

Interest

8

Suppose a $10,000 face value strip bond matures 18 years from now.

The owner of this bond will receive a payment of $10,000 in 18 years.

What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%,

compounded semi-annually?

FV = $10000

i

= .0575

/

2

n

= 18 *

2

= 36

PV = 10000(1+.0575

/

2)-36

= 10000(0.3605)

(87)

8 - 87

Compound

Interest

Compound

Interest

8

Suppose a $10,000 face value strip bond

matures 18 years from now. The owner of this bond will receive a payment of $10,000 in 18 years.What is the appropriate price to pay for the bond today if the prevailing rate of return is

5.75%, compounded semi-annually?

j =

5.75%

m =

2 FV = $10000

n =

18 *

2 = 36

18 * 2

5.75

2

PV = -3,604.50

10000

0

(88)

8 - 88

Compound

Interest

Compound

Interest