3 MateriTerbuka Compound Interest

(1)

8 - 1

Compound

Interest

Compound Interest

8 C

ompound

(2)

8 - 2

Compound

Interest

Compound Interest

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 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 …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 …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 …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

Compound

Interest

Compound Interest

8

100 11 0 121

1000 1210 1331

1100 ₁₀₀ ₁₀₀11₀

Time(Years)

0 1 2 3 4

Amount $1000Amount $1000

110 InterestInterest InterestInterest

100

InterestInterest 133.1

Compoundin g Period Compoundin

g Period

Compoundin g Period

InterestInterest

121

Compound Interest

(10)

8 - 10

Compound

Interest

Compound Interest

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 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 $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

Compound

Interest

Compound Interest

8

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

It is rare for interest to be compounded only once per year! 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?

1000

(1.

04 )(1.

08 )(.

90 )(1.

09 ) =

$

1

101.86 1 -10%

Solving

Alternative

Solving

Alternative

Solve for all

4 years at

once!

Solve for all

4 years at

once!

Compound Interest

(14)

8 - 14

Compound

Interest

Compound Interest

8 Compounding Frequencies and Periods

FrequencyFrequency No. per YearNo. per Year Period Period

Annually

1 1 year

Semi

annually

2 6 months

Quarterly

4 3 months

Monthly

12 1 month

Daily 365

1 day

(15)

8 - 15

Compound

Interest

Compound Interest

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 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

Compound

Interest

Compound Interest

8

3 *

Annually

Semiannually

Quarterly

= 3

= 6

= 12

1

2

4 n

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

Compound

Interest

Compound Interest

8

If you compounded

$100

for 3 years at

6%

annually, semiannually, or quarterly, what are the values for n and i ?

Determining values for

i

Rate -

Rate - i

i

/

1 2 4

=

6%

=

3%

=

1.5%

Annually

Semiannually

Quarterly

Annual Interest Rate (j)

# of Compounding Frequencies per

year(m)

Formula

(19)

8 - 19

Compound

Interest

Compound Interest

8 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

)

(20)

8 - 20

Compound

Interest

Compound Interest

8 FV

=

PV

(1 +

i

)

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 FV

=

PV

(1 +

i

)

Formula

Solve…

FV = $1000(1 +

.04

)

=

_{$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 Solve… $1000(1 +

.04

)

.04

1

8 1000

$1,368.57

$

1 ,

368.57 Use

a

calculator and algebraic sequencing

calculator

and algebraic sequencing

(24)

8 - 24

Compound

Interest

Compound Interest

8 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 )

_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

where the values are to be stored.

(27)

8 - 27

Compound

Interest

Compound Interest

8 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:

..choose from 0 - 9

…’clear’ display The key stroke sequence ‘to recall’ is:

…where you stored the value!

(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

m

t

(used in chapter 10)

Tells the calculator to compute (CPT)

Present Value or initial(first) lump sum

Find the following

KEYS

:

Future Value or terminal(last) lump sum

(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

… represents the number of

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 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

Compound

Interest

Compound Interest

8

Setting a new value for

P/Y

will

automatically change the entry for

C/Y

to the same value

as the default, i.e.

P/Y

Setting a new value for P/Y will

automatically change the entry for C/Y to the same value

as the default, i.e. P/Y

Illustration

… represents the number of

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

.

…to scroll We must key in this

sequence to close any worksheet

you have opened.

We must key in this sequence to close any worksheet

(34)

8 - 34

Compound

Interest

Compound Interest

8

There are

two methods

that can be used to

calculate compound interest:

Using

the

TI BAII Plus

financial

calculator

!

(35)

8 - 35

Compound

Interest

Compound Interest

8

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?

Using the

Using

the

TI BAII Plus

financial calculator

Set the

frequency

of interest compounding

Set the

frequency

of interest compounding

Step 1Step 1

Input values

into the financial keys

Input values

into the financial keys

Step 2Step 2

(36)

8 - 36 Compound Interest Compound Interest

8

FV= 1368.57

8.0

2

1000 Set the frequency of interest compounding Set the frequency of interest compounding

Step 1Step 1

4 * 2

0 Input values into the financial keys Input values into the financial keys

Step 2Step 2

$1,368.57

$1,

368.57 Using the

Using

the

TI BAII Plus

financial calculator

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? 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

(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 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 …

… payments made e.g. cheques

(39)

8 - 39

Compound

Interest

Compound Interest

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

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

=

100 (1.

06 )

3 $119.10

$

1

19 .

10 Semi-

_FV

₌

₁₀₀

_(1.

₀₃

₎

6 $119.41

$

1

19 .

41 Semi

=

6%

/2

Quarterly

_FV

=

100 (1.

015 )

12 $119.56

$

1

19 .

56

(41)

8 - 41 Compound Interest Compound Interest

8

F u tu re V al u e

S

o

r

F

V

F u tu re V al u e

S

o

r

F

V

0 50 100 150 200 250

FV

=

PV

(1+

i

)

Time(Years)

S=

P

(1+

r

t

)

Original Principal

Interest on Original Principal Interest on Interest

Compound Interest

- Future Value

(42)

8 - 42

Compound

Interest

Compound Interest

8

(43)

8 - 43

Compound

Interest

Compound Interest

8

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

What is Al’s

S

imple

I

nterest and

M

aturity

V

alue?

What is Al’s

S

imple

I

nterest and

M

aturity

V

alue?

What is Al’s

I

nterest and

C

ompounded

V

alue? What is Al’s

I

nterest and

(44)

McGraw-Hill Ryerson©

8 - 44

Compound

Interest

Compound Interest

8 Simple

Vs

Compound

Interest

FV

=

PV

(1 +

i

)

Formulae Formulae

I

=

P

r

t

I

= $1,000 *

.10

* 5

= $

500

FV = $1,000 + $500

=

$1,

500 FV

= $1000(1.

1 )

**= $1,000 *1.6105**

=

$1,

610.51 n

= 5

*

1

5 I =

–

= $1610.51 - $1000

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)

8 - 45 Compound Interest Compound Interest

8

0 200 400 600 800 1000 1200 1400 1600 1800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Years to Maturity,

n

Future Values of

$100

Various Rates of Interest

Compounded

Annually

Future Values of

$100

Various Rates of Interest

Compounded

Annually

100 6% 10% 8% 12% F u tu re V al u e

F

V

F u tu re V al u e

F

V

Compound Interest Compound Interest

8

(46)

8 - 46 Compound Interest Compound Interest

8

Ending Balance Ending Balance Compounding Period Compounding Period

$1,000

Nominal Rates of Interest

Compared

$1,

060.00 $1,

060.90 $1,

061.36 $1,

061.83

Beginning Balance Beginning Balance Nominal Rate Nominal Rate

+ 6%

Annual Semiannual

Quarterly Daily

(47)

8 - 47 Compound Interest Compound Interest

8

0 500 1000 1500 2000 2500

5 10 15 20 25

F u tu re V al u e

F

V

F u tu re V al u e

F

V

Time (years)

12% Compounded

monthly

Future Values of

$100

at the same Nominal

Rate

b

Different Compounding Frequencies

12%

Co

mp

oun

ded

An

nua

lly

Compound Interest Compound Interest

8

(48)

8 - 48

Compound

Interest

Compound Interest

8

Calculate the

Future Value

$2,000

compounded

daily

for

4 years

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)

Formula

Compounding

Daily

Interest

Compound

Interest

Compound Interest

8

(49)

8 - 49

Compound

Interest

Compound Interest

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 Compound Interest Compound Interest

8 FV

- 2394.41

4.5

2000

$2,394.41

$

2 ,

394 .

41

Set the frequency of interest compounding Set the frequency of interest compounding

Step 1Step 1

4 * 365

0 Input values into the financial keys Input values into the financial keys

Step 2Step 2

Compounding

Daily

Interest

365

Calculate the Future Value of $2,000 compounded

daily for 4 years at 4.5%.

Calculate the

Future Value of $2,000

compounded

daily for 4 years at 4.5%.

(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 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? 6000 Set the frequency of interest compounding Set the frequency of interest compounding

Step 1Step 1

4 * 2

4.5 Input values into the financial keys Input values into the financial keys

Step 2Step 2

$6,561.75

$6,

561.75 Using the

Using

the

TI BAII Plus

financial calculator

FVFV₁1 = PV = PV₂2

4

(54)

8 - 54 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? 7470.61 Set the frequency of interest compounding Set the frequency of interest compounding

Step 1Step 1

2.5*12

5.2 Input values into the financial keys Input values into the financial keys

Step 2Step 2

$7,470.61

Using the

Using

the

TI BAII Plus

financial calculator

FVFV₂2

(55)

8 - 55

Compound

Interest

Compound Interest

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.

What is the balance outstanding immediately following the second payment?

(56)

8 - 56

Compound

Interest

Compound Interest

8

0 1 year 2 years

$5000

i = .07/12

FV₁ - $2500 = PV₂

FV₁ = 5000(1+.07/12)12

= 5000(1.072290) = 5361.45

PV₂ = 5361.45 – 2500.00 = 2861.45

FV₂

i = .07/12

FV₂ =

= 2861.45(1.072290) = $3068.30

2861.45 (1+.07/12)12

n = 12 _{n = 12}

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

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

Compound

Interest

Compound Interest

8

5000

Step 1Step 1

$2,861.45

$2,

861.45 Using the

Using

the

TI BAII Plus

financial calculator

FVFV₁1 – 2500 = PV – 2500 = PV₂2

FV₂

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

7.0

(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

8

(60)

8 - 60

Compound

Interest

Compound Interest

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

usual sequence!

(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

Compound

Interest

Compound Interest

8 PV = FV(1 +

i

)

-n

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

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

PV = $1500(1+.02)-6

n = 3 * 2 = 6

Calculating

P

resent

V

alue

1.02 6

1500

0.88797 1,331.96 = $1500 * .8880

= $1,331.96

(a)

(63)

8 - 63

Compound

Interest

Compound Interest

8

3 * 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

Compound

Interest

Compound Interest

8 PV = FV(1 +

i

)

-n

Formula

What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year?

j = m =

FV =

i = n =

Calculating

P

resent

V

alue

PV = $6000(1+.05/365)-365

= $6000 * .9512 = $5,707.40

.05

365

6000

365

0.00011.001 0.9512 5,707.40

365

.05/365

1*365 = 365 $6000

(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

365

PV= - 5,707.40

6000

Calculating

P

resent

V

alue

(66)

8 - 66

Compound

Interest

Compound Interest

8

Two payments of $2200 each must be made 1 and 4 years

from now. If money can earn 5% compounded monthly,

what single payment 3 years from now would be

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

Find the PV of the payment that is moved from Year 4 to Year 3 Find the PV of the payment that

is moved from Year 4 to Year 3

Step 3Step 3

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

(67)

8 - 67 Compound Interest Compound Interest

8

Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded

monthly, what single payment 3 years from now

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

Step 1Step 1

$2200 $2200

0 1 year 2 years 3 years 4 years

i = .05/12 n = 2*12 = 24

Step 2Step 2

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

FV₁

= 2200(1+.05/12)24

= 2200(1.1049) = 2430.87

(a)

FV₁ PV₁

PV₂

FV₂

Now

0

2430.87

(68)

8 - 68

Compound

Interest

Compound Interest

8 (b)

2*12

2200

Step 2Step 2

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

$2200 $2200

0 1 year 2 years 3 years 4 years

i = .05/12 n = 2*12 = 24

FV₁ PV₁

PV₂

FV₂

Now

0

2430.87

(69)

8 - 69

Compound

Interest

Compound Interest

8

Find the PV of the payment that is moved from Year 4 to

Year 3

Find the PV of the payment that is moved from Year 4 to

Year 3

Step 3Step 3 _i₌_.05_/₁₂ _{n =1*}PV2₁₂₌₁₂

PV₂ = 2200(1+.06/12)-12

= 2200(0.9513) = 2092.92

(a)

$2200 $2200

0 1 year 2 years 3 years 4 years

FV₁ PV₁

$4523.79

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

0

2430.87

(70)

8 - 70 Compound Interest Compound Interest

8

n =1*12=12 $2200 $2200

0 1 year 2 years 3 years 4 years

PV₂ FV₁

PV₁

(b)

1*12

2200 Some of the values

have not changed so

there is no need to enter them

again!

Some of the values

have not changed so

there is no need to enter them

again!

$4523.79

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

0

2430.87 2,092.92_4,523.79

(71)

8 - 71

Compound

Interest

Compound Interest

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 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!…

Which rate should an investor choose

for an investment of $1,000?

(73)

8 - 73

Compound

Interest

Compound Interest

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. Which

rate should an investor choose for

an investment of $1,000?

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. Which

rate should an investor choose for

an investment of $1,000?

5*1 6.6 1000 0 1 1376.53

j = 6.6%

compounded

annually ₅_{* 2}

6.5

1376.89

j = 6.5%

compounded

semi-annually

j = 6.4%

compounded monthly₅_{* 12}

6.4

1375.96

(74)

8 - 74

Compound

Interest

Compound Interest

8 Comparisons

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!

Results

j = 6.6%

compounded annually

j = 6.5%

compounded

semi-annually

j = 6.4%

compounded monthly

1376.531376.53 1376.891376.89 1375.961375.96

(75)

8 - 75

Compound

Interest

Compound Interest

8

(76)

8 - 76 Compound Interest Compound Interest

8 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

Compound

Interest

Compound Interest

8 Regular Interest

version

Regular Interest

version

Compound Interest

_version

Compound Interest

version

Interest is

paid

to the investor

every year or every 6

months

Interest

is

paid

to the investor

every year or every 6

months

Interest is periodically

converted to principal

and

paid at maturity

Interest

is periodically

converted to principal

and

paid at maturity

Payment of

Interest

Payment of

(78)

8 - 78

Compound

Interest

Compound Interest

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

8

(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

8

(83)

8 - 83

Compound

Interest

Compound Interest

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

8

(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.

No interest will be received in the interim!

(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

/

n

= 18 *

2

= 36

PV = 10000(1+.0575

/

2)-36

= 10000(0.3605) = $3604.50

(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

(88)

8 - 88

Compound

Interest

Compound Interest

8 This completes Chapter 8

(1)

8 - 83

Compound

Interest

Compound Interest

8 The fair market value of

an investment

is the

sum

of the

P

resent

V

alues of the

expected cash flows.

The

discount rate

used

should be

the

prevailing market

determined rate

of return

required

on this type of

investment.

(2)

8 - 84

Compound

Interest

Compound Interest

(3)

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.

No interest will be received in the interim!

(4)

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

/

n

= 18 *

2

= 36

PV = 10000(1+.0575

/

2)-36

= 10000(0.3605) = $3604.50

(5)

8 - 87

Compound

Interest

Compound Interest

8

Suppose a $10,000 face value strip bond

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

(6)

8 - 88

Compound

Interest

Compound Interest