Business Mathematics Jerome Chapter 08

McGraw-Hill Ryerson©

8 - 1Compound

Interest

Compound

Interest 88 88


CompoundCompound

Chapter 8


8 - 2Compound

Interest

Compound

Interest 88 88

Calculate the…

Learning ObjectivesLearning 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-1LO-1


8 - 3Compound

Interest

Compound

Interest 88 88

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 ObjectivesLearning Objectives

LO-2LO-2


8 - 4Compound

Interest

Compound

Interest 88 88

Compound

Interest

Compound

Interest 88 88

LO-1LO-1


8 - 5Compound

Interest

Compound

Interest 88 88

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

Simple Interest!

Formula Formula I = Prt

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 8 8 8 8


8 - 6Compound

Interest

Compound

Interest 88 88

…the interest on the principal plus the

interest of prior periods

e.g. Principal + prior period interest = $1100.00

Interest for the next period is calculated on $1100.00.This method will continue over the life of the

loan or investment. (See later example)

$1000.00 $100.00

Compound Interest- Future Value


8 - 7Compound

Interest

Compound

Interest 88 88

…is the compounded 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!



8 - 8Compound

Interest

Compound

Interest 88 88

…the calculation of interest over the life of the loan or investment

Example: Principal + prior period interest = $1100.00

Interest is now calculated on $1100.00

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

Principal(Compounded) * 0.10 = $110.00

New P $1210.00 to start next period

Graphically…



8 - 9Compound

Interest

Compound

Interest 88 88

100110

121

1000

1210

1331

1100100 100

110

Time(Years)0 1 2 3 4

Amount $1000Amount $1000

110

InterestInterestInterestInterest

100

InterestInterest

133.1

Compounding Period

Compounding Period

Compounding Period

Compounding Period

InterestInterest

121



8 - 10Compound

Interest

Compound

Interest 88 88

What happens if the interest rate changes during the life of

an investment?

Example…Example…



8 - 11Compound

Interest

Compound

Interest 88 88

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?



8 - 12Compound

Interest

Compound

Interest 88 88

$1000

Year 1 Year 2 Year 3 Year 4

$1040 $1123.20 $1010.88

$1000 * (1 + .04)

= $1040

$1040 * (1 + .08)

= $1123.20 = $1010.88 = $1101.86

$1123.20 *(1 - .10)

$1010.88 *(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?



8 - 13Compound

Interest

Compound

Interest 88 88

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) = $1101.86

1 -10%

Solving Alternative

Solving Alternative

Solve for all 4 years at

once!

Solve for all 4 years at

once!



8 - 14Compound

Interest

Compound

Interest 88 88

Compounding Frequencies and Periods

FrequencyFrequency No. per YearNo. per Year Period Period

Annually 1 1 year

Semiannually 2 6 months

Quarterly 4 3 months

Monthly 12 1 month

Daily 365 1 day


8 - 15Compound

Interest

Compound

Interest 88 88 Formula Formula Development of a

n Total Number of PeriodsPeriods

Determining values for n and i

Nominal or Annual Rate ( j )

Periodic Rate per period ( i )Number of compoundings per year m


8 - 16Compound

Interest

Compound

Interest 88 88 Formulae Formulae

To Determine nTo Determine n

To Determine iTo Determine i

# of Compounding Frequencies p.a.(m)Time(Years)

Annual Interest Rate(j)

# of Compounding Frequencies p.a. (m)

*


8 - 17Compound

Interest

Compound

Interest 88 88

3 *3 *3 *

Annually Semiannually

Quarterly

= 3 = 6 = 12

1

2

4

nn

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) *Formula


8 - 18Compound

Interest

Compound

Interest 88 88

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 - iRate - i

6% /6% /6% /

1

2

4

= 6%

= 3%

= 1.5%

Annually Semiannually

Quarterly

Annual Interest Rate (j)# of Compounding Frequencies per

year(m)

Formula Formula

No.No.


8 - 19Compound

Interest

Compound

Interest 88 88

Formula Formula Development of a for Future Value

PV= Present Value(Principal)

i = rate per period n = number of periods

FV = PV(1 + i)n

Where…


8 - 20Compound

Interest

Compound

Interest 88 88

FV = PV(1 + i)n

Formula 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…


4 X 2 = 8$1000

.08/2 = .04


8 - 21Compound

Interest

Compound

Interest 88 88

FV = PV(1 + i)n

Formula Formula

Solve…

FV = $1000(1 + .04)8

= $1000(1.368569) = $1,368.57

Principal $1,000.00 + Interest 368.57Compounded $1,368.57

Using PV = $1000 n = 8 i= .04



8 - 22Compound

Interest

Compound

Interest 88 88

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

calculate compound interest:


8 - 23Compound

Interest

Compound

Interest 88 88

Solve… $1000(1 + .04)8

.04

1

8

1000

$1,368.57$1,368.57

Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing

1368.57


8 - 24Compound

Interest

Compound

Interest 88 88

Find the following KEYS: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.


Changes the sign of the data value of the number

being displayed.


8 - 25Compound

Interest

Compound

Interest 88 88

Some calculators have the yx 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



NegativeNegative (1.04)-8 is…1.04 8


8 - 26Compound

Interest

Compound

Interest 88 88

This calculator can store up to 10 values.



Used to Store or save displayed values.

Used to Recall the saved values.

Let’s PractiseLet’s Practise

Therefore, the calculator must be informed as to

where the values are to be stored.


8 - 27Compound

Interest

Compound

Interest 88 88 Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing

Using the key

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:

…where you stored the value!


8 - 28Compound

Interest

Compound

Interest 88 88

Some key Keys!Some key Keys!


8 - 29Compound

Interest

Compound

Interest 88 88

The nominal interest rate (Interest/Year)

1. Number of compoundings (for lump payments)

2. Number of payments (for annuities)

Represents the Periodic Annuity Payment

(used in chapter 10)

Tells the calculator to compute (CPT)

Present Value or initial(first) lump sum


Future Value or terminal(last) lump sum


8 - 30Compound

Interest

Compound

Interest 88 88

However, we can now input the number of compoundings per year into the financial calculator. This can be performed by using

the symbol


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

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

Previously, it was noted that

To access this symbol use:

…and you will see


8 - 31Compound

Interest

Compound

Interest 88 88 The 12

is a default setting

The 12 is a

default setting

This display is referred to as “the worksheet”.

… represents the number of Payments per Year

… represents the number of Compoundings per Year

To access use:

Note: You can override these values by entering new ones!

…more…more

Appearsautomatically

Appearsautomatically


8 - 32Compound

Interest

Compound

Interest 88 88

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

IllustrationIllustration


8 - 33Compound

Interest

Compound

Interest 88 88

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

IllustrationIllustration

… represents the number of Compoundings per Year

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 scrollWe must key in this sequence to close any worksheet

you have opened.

We must key in this sequence to close any worksheet

you have opened.


8 - 34Compound

Interest

Compound

Interest 88 88

There are two methods that can be used to

calculate compound interest:

Using the TI BAII Plus financial calculator!


8 - 35Compound

Interest

Compound

Interest 88 88

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 TI BAII Plus financial calculator 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


financial keys

Step 2Step 2

Using


8 - 36Compound

Interest

Compound

Interest 88 88

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


keys

Step 2Step 2

$1,368.57$1,368.57


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

amount?


8 - 37Compound

Interest

Compound

Interest 88 88

…there is no need to keep inputting each time! 0

You only need to input the values that have changed!

The calculator remembers this step!


8 - 38Compound

Interest

Compound

Interest 88 88

Cash FlowsCash Flows

… payments received e.g. receipts

Treated as:Treated as:

Positives +Positives + Negatives -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


8 - 39Compound

Interest

Compound

Interest 88 88

What is the effect on the Future Value of

different Compounding Periods of

Interest?


8 - 40Compound

Interest

Compound

Interest 88 88

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 ?


AnnualAnnual FVA = 100(1.06)3 $119.10$119.10

Semi-Semi- FVS = 100(1.03)6 $119.41$119.41

Semi = 6%/2

QuarterlyQuarterly FVQ = 100(1.015)12 $119.56$119.56

Quarterly = 6%/4


8 - 41Compound

Interest

Compound

Interest 88 88

Fu

ture

Val

ue

S o

r F

VF

utu

re V

alu

e

S o

r F

V

1 2 3 4 5 6 7 8 9 10 110

50

100

150

200

250

FV=PV(1+i)n

Time(Years)

S=P(1+rt)

Original PrincipalOriginal Principal

Interest onOriginal Principal

Interest on

Interest


The Components of the Future Value of $100


8 - 42Compound

Interest

Compound

Interest 88 88

ComparisonsComparisons


8 - 43Compound

Interest

Compound

Interest 88 88

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

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

Annual Simple Interest Rate of 10%

Annual Simple Interest Rate of 10%

Annual Compound Rate of 10%

Annual Compound Rate of 10%

Simple Vs Compound Interest

What is Al’s

Simple Interest and

Maturity Value?

What is Al’s

Simple Interest and

Maturity Value?

What is Al’s

Interest and

Compounded Value?

What is Al’s

Interest and

Compounded Value?


8 - 44Compound

Interest

Compound

Interest 88 88

Simple Vs Compound Interest

FV = PV(1 + i)nFormulae Formulae I = Prt

I = $1,000 * .10 * 5

= $500

FV = $1,000 + $500

= $1,500

FV = $1000(1.1)5

= $1,000 *1.6105

= $1,610.51

n = 5 * 1 = 5

I = FV – PV = $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

Compare


8 - 45Compound

Interest

Compound

Interest 88 88

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 at

Various Rates of Interest Compounded Annually

Future Values of $100 at

Various Rates of Interest Compounded Annually

100

6%

10%

8%

12%

Fu

ture

Val

ue

FV

Fu

ture

Val

ue

FV

Compound

Interest

Compound

Interest 88 88


8 - 46Compound

Interest

Compound

Interest 88 88

Ending Balance

Ending Balance

Compounding Period

Compounding Period

$1,000$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%+ 6%

Annual

Semiannual

Quarterly

Daily


8 - 47Compound

Interest

Compound

Interest 88 88

0

500

1000

1500

2000

2500

5 10 15 20 25

Fu

ture

Val

ue

FV

Fu

ture

Val

ue

FV

Time (years)

12% Compounded

monthly

Future Values of $100 at the same Nominal Rate but

Different Compounding Frequencies

12% Compounded

Annually

Compound

Interest

Compound

Interest 88 88

100


8 - 48Compound

Interest

Compound

Interest 88 88

Calculate the Future Value of $2,000

compounded daily for 4 years

at 4.5%.

n = i =

= $2,000 * 1.1972 = $2,394.41= $2,000 * 1.1972 = $2,394.41FV = $2000(1+ .045/365)1460

FV = PV(1 + i)n

Formula Formula

Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest InterestCompound

Interest

Compound

Interest 88 88

4 * 365 = 1460 .045 /365 = 0.0001232


8 - 49Compound

Interest

Compound

Interest 88 88

2394.41.045365

1

1460

2000

Solve FV = $2000(1+ .045/365)1460

= $2,394.41= $2,394.41

Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest Interest


8 - 50Compound

Interest

Compound

Interest 88 88

FV= - 2394.414.5

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


financial keys


financial keys

Step 2Step 2

Compounding Compounding DailyDaily Interest InterestCompounding Compounding DailyDaily Interest Interest

365


compounded

daily for 4 years at 4.5%.


compounded

daily for 4 years at 4.5%.


8 - 51Compound

Interest

Compound

Interest 88 88

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?

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


8 - 52Compound

Interest

Compound

Interest 88 88

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 = .045/4FV1 = 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) = 8n = 2.5*12 = 30


8 - 53Compound

Interest

Compound

Interest 88 88

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


keys


keys

Step 2Step 2

$6,561.75$6,561.75


FV1 = PV2FV1 = PV2

FV2

4

6,571.75


8 - 54Compound

Interest

Compound

Interest 88 88

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


keys


keys

Step 2Step 2

$7,470.61$7,470.61


FV2FV2

12


8 - 55Compound

Interest

Compound

Interest 88 88

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?


8 - 56Compound

Interest

Compound

Interest 88 88

0 1 year 2 years

$5000i = .07/12

FV1 - $2500 = PV2

FV1 = 5000(1+.07/12)12

= 5000(1.072290) = 5361.45PV2 = 5361.45 – 2500.00 = 2861.45

FV2

i = .07/12FV2 = = 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 Balance New Balance

= $3068.30 – 2500.00

= $568.30


8 - 57Compound

Interest

Compound

Interest 88 88

5000

Step 1Step 1

12

$2,861.45$2,861.45


FV1 – 2500 = PV2FV1 – 2500 = PV2

FV2

You borrowed $5000 at 7% compounded

monthly. On the 1st. and 2nd anniversaries of

the loan, you made payments of $2500.

What is the balance

outstanding immediately after the 2nd payment?

2500

7.0

12

FV= -5361.45-2861.45


8 - 58Compound

Interest

Compound

Interest 88 88

-2861.45

$568.30$568.30


FV2FV2

You borrowed $5000 at 7% compounded

monthly. On the 1st. and 2nd anniversaries of

the loan, you made payments of $2500.

What is the balance

outstanding immediately after the 2nd payment?

2500

Step 2Step 2

- 2,861.45 3068.30 568.30


8 - 59Compound

Interest

Compound

Interest 88 88


8 - 60Compound

Interest

Compound

Interest 88 88 Formula for Present Value

PV = FV(1 + i)-nFormula Formula

Keys

i1

$PV$PV

This is the only change to the

usual sequence!


8 - 61Compound

Interest

Compound

Interest 88 88

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 Present Value Calculating Present Value

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



8 - 62Compound

Interest

Compound

Interest 88 88


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


1.02

6

1500

0.887971,331.96

= $1500 * .8880

= $1,331.96

(a)


8 - 63Compound

Interest

Compound

Interest 88 88

3 * 2

4

2

1500

0PV= -1,331.96(b)


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?

$1331.96$1331.96


8 - 64Compound

Interest

Compound

Interest 88 88


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

j =

m =

FV =

i =

n =


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

= $6000 * .9512

= $5,707.40

.05

365

6000

365

1

0.00011.0010.95125,707.405%

365

.05/365

1*365 = 365

$6000


8 - 65Compound

Interest

Compound

Interest 88 88

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


$5707.40$5707.40


8 - 66Compound

Interest

Compound

Interest 88 88

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-lineStep 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



Equivalent Payments Equivalent Payments


8 - 67Compound

Interest

Compound

Interest 88 88

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-lineStep 1Step 1

$2200 $2200

0 1 year 2 years 3 years 4 years

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

Step 2Step 2Find the FV of the payment that is moved

from Year 1 to Year 3

Find the FV of the payment that is moved


FV1

= 2200(1+.05/12)24

= 2200(1.1049) = 2430.87

(a)

FV1PV1

PV2

FV2

Now 0

2430.87



8 - 68Compound

Interest

Compound

Interest 88 88 (b)

2*12

5

2200

0

Step 2Step 2Find the FV of the payment that is moved


Find the FV of the payment that is moved


$2200 $2200


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

FV1PV1

PV2

FV2

12

Now 0

2430.87


8 - 69Compound

Interest

Compound

Interest 88 88

Find the PV of the payment that is moved


Find the PV of the payment that is moved


Step 3Step 3 i = .05/12 n =1*12=12

PV2

PV2 = 2200(1+.06/12)-12

= 2200(0.9513) = 2092.92

(a)

$2200 $2200


FV1PV1

$4523.79

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

0

2430.87



8 - 70Compound

Interest

Compound

Interest 88 88

n =1*12=12

$2200 $2200


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

2,092.924,523.79


8 - 71Compound

Interest

Compound

Interest 88 88

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

Making a choice!…Making a choice!…


8 - 72Compound

Interest

Compound

Interest 88 88

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!…Making a choice!…

Which rate should an investor choose for an investment of $1,000?


8 - 73Compound

Interest

Compound

Interest 88 88

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%

compoundedannually

5 * 2

6.5

2

1376.89

j = 6.5%

compoundedsemi-annually

j = 6.4%

compoundedmonthly5 * 12

6.4

12

1375.96ComparisonsComparisons


8 - 74Compound

Interest

Compound

Interest 88 88 ComparisonsComparisons

the 6.5% compounded semi-annuallyprovides for the best

rate of return on investment!

the 6.5% compounded semi-annuallyprovides for the best

rate of return on investment!

ResultsResultsj = 6.6%

compounded

annuallyj = 6.5%

compoundedsemi-annuallyj = 6.4%

compounded

monthly

1376.531376.53

1376.891376.89

1375.961375.96


8 - 75Compound

Interest

Compound

Interest 88 88

of Interest Rates


8 - 76Compound

Interest

Compound

Interest 88 88

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

cannot drop


8 - 77Compound

Interest

Compound

Interest 88 88

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 Interest


8 - 78Compound

Interest

Compound

Interest 88 88

Canadian

Savings

Bonds


8 - 79Compound

Interest

Compound

Interest 88 88

- 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

Canadian

Savings

Bonds

To view current rates of interest and redemption values

Go to http://www.cis-pec.gc.ca/


8 - 80Compound

Interest

Compound

Interest 88 88


8 - 81Compound

Interest

Compound

Interest 88 88

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.

Canadian

Savings

Bonds


8 - 82Compound

Interest

Compound

Interest 88 88


8 - 83Compound

Interest

Compound

Interest 88 88

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.

Concepts


8 - 84Compound

Interest

Compound

Interest 88 88


8 - 85Compound

Interest

Compound

Interest 88 88

… 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!


8 - 86Compound

Interest

Compound

Interest 88 88

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)

= $3604.50


8 - 87Compound

Interest

Compound

Interest 88 88 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

$3604.50$3604.50


8 - 88Compound

Interest

Compound

Interest 88 88

This completes Chapter 8This completes Chapter 8

Business Mathematics Jerome Chapter 08

Education

Transcript of Business Mathematics Jerome Chapter 08