Business Mathematics Jerome Chapter 07

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Applications ofApplications of

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Chapter 7Chapter 7

IISS imple nterest

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Calculate

Learning ObjectivesLearning

ObjectivesAfter completing this chapter, you will be able to:

…the MARKET PRICE and rate of return for Treasury Bills and

Commercial Paper

…Interest paid on Savings Accounts, Term Deposits, and

Guaranteed Investment Certificates

also…also…

LO-1LO-1

LO-2LO-2

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State…the Valuation Principle and apply it to

the calculation of the Fair Market Value of an investment providing

specified future cash flows

…typical terms, conditions, and repayment arrangements for revolving (demand)

loans, fixed-payment (demand) loans,

and Canada Student Loans

Describe the…

Learning ObjectivesLearning

ObjectivesLO-3LO-3

LO-4LO-4

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avings Accounts

Short-Term GICs

LO-1LO-1

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Interest

About

often calculated on daily balanceand posted monthly

avings Accounts

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Guaranteed Investment Certificates

30 – 364 day terms

“Guaranteed” refers to the unconditional guarantee of

principal and interest by the parent financial institution.

Short-Term GICsAbout

and…and…

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- Non-Redeemable

- Redeemable (or cashable GIC or term

deposit)

Non-Redeemable

Redeemable (or cashable GIC or term

deposit)Can be “cashed in” before maturity date

Most GICs are not redeemable before maturity

Cannot be “cashed in” before maturity date

Short-Term GICsAbout

also…also…

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Can be purchased from banks, credit unions, trust companies, and caisses

populaires (in Quebec).

(you are in effect lending money to the financial

institution)The financial institution uses the funds raised from selling GICs to make loans …

most commonly, mortgage loans

Short-Term GICsAbout

also…also…

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…Higher rates are paid for longer terms (within the 30 to 364-day

range)

…Higher rates are paid on non-redeemable GIC’s

…Higher rates are paid on larger

principal amounts

…Mortgage rates are typically 1.5% to 2% higher

than the interest rate paid to GIC

investors.

Short-Term GICsAbout

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Case 1

For amounts of $5000 to $99,999 and terms of 90 to 179 days,

the Royal Bank pays an interest rate of 2.80% pa on redeemable GICs and 3.15% on non-redeemable GICs.

Short-Term GICsAbout

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In order to retain the redemption privilege,

how much Interest (in dollars) must be sacrificed on an

investment of $15,000 for 120 days?

Short-Term GICsAbout

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Non-redeemable GIC Non-redeemable GIC Redeemable GICRedeemable GIC

I = 15000 * I = 15000 *

= $155.34= $155.34 = $138.08= $138.08

.0315 * 120 / 365 .028 * 120 / 365

Interest Sacrifice $17.26Interest Sacrifice $17.26

Subtract

Short-Term GICs

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For amounts between $10,000 and

$24,999, a bank pays a rate of 4.5% on

GICs with maturities in the 91 to 120-

day range. However, early redemption will result

in a rate of 3% being applied.

Case 2Short-Term GICs

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How much more Interest will a

120-day $22,000 GIC earn

if it is held until maturity

than if it is redeemed after 100

days?

Short-Term GICs

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120 Days 120 Days 100 Days100 Days

I = 22000 * I = 22000 *

= $325.48= $325.48 = $180.85= $180.85

.045 * 120 / 365 .03 * 100 / 365

More Interest $144.63More Interest $144.63

Subtract

Short-Term GICs

4.5%4.5% 3.0% for early3.0% for early

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LO-2LO-2

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Paper contracts issued to lenders by the

federal government and several provincial governments when they

borrow money for terms of less than one year.

…the FACE VALUE (Maturity Value) of a T-Bill includes the Interest

that the T-bill will earn during its life!

Treasury Bills

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Find the price of a 91-day $75,000 T-bill on its date of issue if the

going rate of interest is 5.1%.

PV = $75,000/[1+(0.051*91/365)]

= $74,058.34

Treasury Bills

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Suppose that the institutional purchaser of the T-bill in the previous example sells it to a client on the same day at a higher price

that represents a lower yield to the

client of 4.6%.

What profit did the institution make on the transaction?

What profit did the institution make on the transaction?

Treasury Bills

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What profit did the institution make on the transaction?

What profit did the institution make on the transaction?

Selling Price

$75000/[1 + .046(91/365)]

ProfitProfit

= $74,149.62$74,149.62

Selling Selling PricePrice

74,058.34AcquisitioAcquisitio

n Pricen Price

$ 91.28

Treasury Bills

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Suppose the client who purchased the 91-day, $75,000 T-bill in the previous example for

$74,149.62, sold the T-bill after 30 days in order to invest the proceeds elsewhere!

(a) What price would she receive if the short-term interest rate for this maturity

had fallen to 4.2% by the date of the sale?

(a) What price would she receive if the short-term interest rate for this maturity

had fallen to 4.2% by the date of the sale?

Treasury Bills

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(a) What price would she receive if the short-term interest rate for this maturity

had fallen to 4.2% by the date of the sale?

(a) What price would she receive if the short-term interest rate for this maturity

had fallen to 4.2% by the date of the sale?

Determine the number of days remaining to maturity

Calculate the Selling Price based on the Discounted Rate of 4.2%

Treasury Bills

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Determine the number of days

remaining to maturity…purchased (T-Bill) ……. 91-day

…sold (T-Bill) after ………. 30 days= 61 days

PV of $75,000 discounted at 4.2% for 61 days

Calculate the Selling Price based on the Discounted Rate of 4.2%

= $75000/[1 + .042(61/365)] SP = $74,477.23 SP = $74,477.23

Treasury Bills

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Suppose the client who purchased the 91-day, $75,000 T-bill in the previous example for

$74,149.62, sold the T-bill after 30 days in order to invest the proceeds elsewhere!

(b) What Rate of Return (per annum) did the client realize while holding the T-bill?

(b) What Rate of Return (per annum) did the client realize while holding the T-bill?

Treasury Bills

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74,149.62$74,477.23

(b) What Rate of Return (per annum) did the client realize while holding the T-bill?

(b) What Rate of Return (per annum) did the client realize while holding the T-bill?

…Purchased (T-Bill) …….

…Sold (T-Bill) after 30 days

Grew in 30 Days by… $ 327.61

IPrt

r = I / Ptr = I / Pt

r = 327.61 /

= .053755 = 5.38%

Treasury Bills

74,149.62 (30/365)

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The Valuation PrincipleThe Valuation Principle

LO-3LO-3

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…the Fair Market Value of an investment and is the Sum of the Present Value(PV) of the expected cash

flowsThe Discount Rate used in the Present Value(PV)

calculations should be the

Market-determined Rate of Return required

for this type of investment

The Valuation PrincipleThe Valuation Principle

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Consider an investment that will deliver a single payment of $110 one year from

now.

What is the most you should pay to buy the investment if you

require a minimum return of 10%?

The Valuation PrincipleThe Valuation Principle

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$110 one year from now = Future Value(FV)

$110 represents 110% or 100% + 10% Return

We are looking for the Present Value

$100 is the most you

should pay

$100 is the most you

should pay = $100= $100

=

The Valuation PrincipleThe Valuation Principle

110/[1+0.10*(1)]PVPV = FV /(1+rt)

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The process of calculating a payment’s Present Value is often called

“Discounting the payment”

The Valuation PrincipleThe Valuation Principle

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An investment promises two payments of $2000, on dates 1 and 4 months from now.

What price will an investor pay today

if the required rate of return

is 8% ?

What price will an investor pay today

if the required rate of return

is 8% ?

1 month = 1/124 months = 4/12

(1) Using a Time Line display, and

(2) Both algebraically and by calculator

(1) Using a Time Line display, and

(2) Both algebraically and by calculator

The Valuation PrincipleThe Valuation Principle

Formula Formula = FV /(1+rt)PVPV

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PV= $3934.81 PV= $3934.81

= 1948.05= 1948.05

2000/[1+0.08*(1/12)] 2000/[1+0.08*(4/12)]

= 1986.76= 1986.76

(1)$2000$2000

Months 1 2 3 4 Months 1 2 3 4

Present Value

Present Value

PVPV(2)

+

The Valuation PrincipleThe Valuation Principle

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.08

1

121

2000

.08

4

12

PV= $3934.81PV= $3934.81= 1948.05= 1948.05= 1986.76= 1986.76 +

2000/[1+0.08*(1/12)] + 2000/[1+0.08*(4/12)]2000/[1+0.08*(1/12)] + 2000/[1+0.08*(4/12)]Today’s Value:

= 1986.76= 1986.76 1948.05 1948.05

The Valuation PrincipleThe Valuation Principle

1

2000

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LO-4LO-4

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About

… Loans where the lender has the right to demand full repayment of the loan

at any time!

…revolving loans

… fixed payment loans

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About

... Borrower may repay any portion of the loan at

any time without penalty

… Interest rate charged is usually “floating”

… Interest is calculated on the same

‘statement date’ each month

… Interest is calculated up to but

not including the statement date

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The debt was completely repaid on June 30 together with the accrued interest.

What was (a) the total interest charge, and (b) the

total payment on June 30?

What was (a) the total interest charge, and (b) the

total payment on June 30?

You negotiated a $5000 demand loan on March 1 at prime plus 2%.

The prime rate was 5% on March 1 and decreased to 4% on May 18.

On May 20 you paid $1225 towards principal plus the accrued interest.

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Determine the number of days in each period

Calculate the interest accrued to date of payment

Record the information provided and known in a Table format

Calculate the balance after the payment

Calculate the interest accrued and the final balance on June 30

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Date Interest

Accrued

Paid

5,000.00

Balance $

Mar 1

May18

May 20

June 30

5 2 7.0

4 2 6.0

4 2 6.0

4 2 6.0

Record the information that is known!

Days

Interest Rate

TotalPrime Plus

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Determine the number of days in each period

Look upDays Look up

Days

60

138

140

181

78

2

41

Date Prime Plus Interest

Accrued

Paid

5,000.00

Balance $

Mar 1

May18

May 20

June 30

5 2 7.0

4 2 6.0

4 2 6.0

Total

Interest Rate

4 2 6.0

Days

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Calculate the interest accrued

…Calculation

60

138

140

181

78

2

41

Date Prime Plus Interest

Accrued

Paid

5,000.00

Balance $

Mar 1

May18

May 20

June 30

5 2 7.0

4 2 6.0

4 2 6.0

Total

Interest Rate

4 2 6.0

Days

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Calculate the interest accrued

$5000 * 0.07 * 78/365

$5000 * 0.06 * 2/365

= $76.48= $76.48

= $ 1.64= $ 1.64

Total = $78.12Total = $78.12

60

138

140

181

78

2

41

Interest

AccruedDays

I R

7.0

6.0

6.0

Total

6.0

Date

Mar 1

May18

May 20

June 30

$76.48

$ 1.64

Balance $5000

…complete the Table

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60

138

140

181

78

2

41

Date Prime Plus Interest

Accrued

Paid

5,000.00

Balance $

Mar 1

May18

May 20

June 30

5 2 7.0

4 2 6.0

4 2 6.0

Total

Interest Rate

4 2 6.0

Days

Payment $1,225.00 + Interest 78.12Payment $1,225.00 + Interest 78.12

5,076.48

3,775.001,303.12

76.48

1.64 5,078.12

Calculate the balance after the payment

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Interest Accrued = $3,775.00 * .06 * 41/365 = $25.44

60

138

140

181

78

2

41

Date Prime Plus Interest

Accrued

Paid

5,000.00

Balance $

Mar 1

May18

May 20

June 30

5 2 7.0

4 2 6.0

4 2 6.0

Total

Interest Rate

4 2 6.0

Days

5,076.48

3,775.001,303.12

76.48

1.64 5,078.12

Calculate the interest accrued and final balance on June 30

25.44 3,800.44

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You completed your college programme in December. On June 30, you paid all of the interest that had accrued

(at prime +2.5%) on your $5800 Canada Student Loan during the six-month grace period.

You selected the fixed rate option (prime+5%) and agreed to make end-of-month payments of $95

beginning July31.

The prime rate began the grace period at 6% and rose by 0.5% effective March 29.

On August 13, the prime rate rose another 0.5%. to use…

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Determine the number of days in the grace period

Calculate the interest accrued during the grace period

Record the information provided and known in a Table format

(a) Calculate the total interest paid in the first two regular payments

(b) Calculate the balance after the second payment

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Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Record the information that is known!

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

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Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

88

93

30

13

88

181

18

212

225

243

Determine the number of days in each period

Look upDays Look up

Days

Interest includes June 30 due to grace period

July 1

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Calculate the interest accrued during the grace period

…Calculation

88

93

30

13

88

181

18

212

225

243

Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

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88

93

30

13

88

181

18

212

225

243

DaysDate

Dec 31

Mar 29

June 30

July 31

Aug 13

Aug 31

IR

8.5

9.0

11.5

11.5

12.0

12.0

Total

End of Grace Period

Calculate the interest accrued during the grace period

Balance $5800

$5800 * 0.085 * 88/365=

$5800 * 0.09 * 93/365=

$118.86$118.86

$133.00$133.00

Total = $251.86Total = $251.86

Interest

Accrued

118.86

131.56

…complete the Table

Includes June 30Includes June 30

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Calculate the interest accrued during the grace period

118.86

133.00

5,918.86

6,051.86

251.86 5,800.00

88

93

30

13

88

181

18

212

225

243

Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

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95.00

…Calculation95.00

(a)

118.86

133.00

5,918.86

6,051.86

251.86 5,800.00

88

93

30

13

88

181

18

212

225

243

Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

Calculate the total interest paid in the first two regular payments

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First Payment $95.00

July 1 - July 31 interest

$5800 * .115 * 30/365 = $54.82

Second Payment $95.00

July 31 - August 13 interest

$5759.82 * .115 * 13/365 = $23.59August 13 - August 31 interest

Balance $5800.00Add: Interest 54.82

Less: Payment 95.00 5854.82

Balance 5759.82Add: Interest 23.59 Interest 34.09

5817.50Less: Payment 95.00Balance 5722.50

$5759.82 * .12 * 18/365 = $34.09

…complete the Table

Calculate the total interest paid in the first two regular payments

(a)

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118.86

133.00

5,918.86

6,051.86

251.86 5,800.00

88

93

30

13

88

181

18

212

225

243

Date Interest

Accrued

Paid

5,800.00

Balance $

Dec 31

Mar 29

June 30

July 31

6 2.5 8.5

6.5 2.5 9.0

6.5 5.0 11.5

6.5 5.0 11.5

Days

Aug 13

Aug 31

7.0 5.0 12.0

7.0 5.0 12.0

Interest Rate

TotalPrime Plus

End of Grace Period

…the total interest paid and the balance after the second payment

(a) & (b)

95.00

95.00

54.82

23.59

34.09

5,759.82

5,783.41

5,722.50

Total Interest $112.50

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This completes Chapter 7This completes Chapter 7