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# Business Mathematics Jerome Chapter 07

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## Business Mathematics Jerome Chapter 07Presentation Transcript

• Applications 7-1 SI Applications of S imple I nterest Chapter 7 Chapter 7 McGraw-Hill Ryerson ©
• Applications 7-2 Learning Objectives Learning Objectives SI After completing this chapter, you will be able to: Calculate LO-1 LO-2 McGraw-Hill Ryerson © …Interest paid on Savings Accounts, Term Deposits, and Guaranteed Investment Certificates …the MARKET PRICE and rate of return for Treasury Bills and Commercial Paper also… also…
• Applications 7-3 Learning Objectives Learning Objectives SI LO-3 State …the Valuation Principle and apply it to the calculation of the Fair Market Value of an investment providing specified future cash flows Describe the… LO-4 …typical terms, conditions, and repayment arrangements for revolving (demand) loans, fixed-payment (demand) loans, and Canada Student Loans McGraw-Hill Ryerson ©
• Applications SI Notes about… LO-1 avings Accounts S hort -Term GICs McGraw-Hill Ryerson © 7-4
• Applications 7-5 SI About avings Accounts Interest often calculated on daily balance and posted monthly McGraw-Hill Ryerson ©
• Applications SI S hort -Term GICs About 7-6 G uaranteed I nvestment C ertificates “Guaranteed” refers to the unconditional guarantee of principal and interest by the parent financial institution. 30 – 364 day terms McGraw-Hill Ryerson © and… and…
• Applications S hort -Term GICs About SI 7-7 - Redeemable (or cashable GIC or term deposit) - Non- Redeemable Redeemable (or cashable GIC or term deposit) Can be “ cashed in ” before maturity date Non- Redeemable Cannot be “ cashed in ” before maturity Most date GICs are not redeemable before maturity McGraw-Hill Ryerson © also… also…
• Applications SI S hort -Term GICs About 7-8 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 McGraw-Hill Ryerson © also… also…
• Applications SI S hort -Term GICs About 7-9 …Mortgage rates are typically 1.5% to 2% higher than the interest rate paid are GIC on …Higher rates to paid investors. nonredeemable GIC’s …Higher rates are paid for longer terms (within the 30 to 364-day range) …Higher rates are paid on larger principal amounts McGraw-Hill Ryerson ©
• Applications SI McGraw-Hill Ryerson © S hort -Term GICs About 7 - 10
• Applications S hort -Term GICs About SI 7 - 11 e1 as C 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. McGraw-Hill Ryerson ©
• Applications SI S hort -Term GICs About 7 - 12 In order to retain the redemption privilege, how much Interest (in dollars) must be sacrificed on an investment of \$15,000 for 120 days? McGraw-Hill Ryerson ©
• Applications SI S hort -Term GICs Non-redeemable GIC Non-redeemable GIC 7 - 13 Redeemable GIC Redeemable GIC I = 15000 * .0315 * 120 / 365 I = 15000 * .028 * 120 / 365 = \$155.34 = \$155.34 Subtract = \$138.08 = \$138.08 Interest Sacrifice \$17.26 Interest Sacrifice \$17.26 McGraw-Hill Ryerson ©
• Applications SI e2 as C S hort -Term GICs 7 - 14 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 . McGraw-Hill Ryerson ©
• Applications SI S hort -Term GICs 7 - 15 more Interest will a 120-day \$22,000 GIC earn if it is held until maturity than if it is redeemed after 100 days ? How much McGraw-Hill Ryerson ©
• Applications SI S hort -Term GICs 7 - 16 120 Days 120 Days 100 Days 100 Days 4.5% 4.5% 3.0% for early 3.0% for early I = 22000 * .045 * 120 / 365 I = 22000 * .03 * 100 / 365 = \$325.48 = \$325.48 Subtract = \$180.85 = \$180.85 More Interest \$144.63 More Interest \$144.63 McGraw-Hill Ryerson ©
• Applications SI LO-2 McGraw-Hill Ryerson © 7 - 17
• Applications 7 - 18 SI Treasury Bills 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! McGraw-Hill Ryerson ©
• Applications 7 - 19 SI Treasury Bills 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 McGraw-Hill Ryerson ©
• Applications 7 - 20 SI Treasury Bills 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 What profit did the institution make on the transaction? on the transaction? McGraw-Hill Ryerson ©
• Applications 7 - 21 SI Treasury Bills What profit did the institution make What profit did the institution make on the transaction? on the transaction? Selling Price \$75000/[1 + .046(91/365)] = \$74,149.62 McGraw-Hill Ryerson © Profit Profit Selling Price Acquisitio n Price \$74,149.62 74,058.34 \$ 91.28
• Applications 7 - 22 SI Treasury Bills 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) (a) McGraw-Hill Ryerson © What price would she receive if the What price would she receive if the short-term interest rate for this maturity short-term interest rate for this maturity had fallen to 4.2% by the date of the sale? had fallen to 4.2% by the date of the sale?
• Applications 7 - 23 SI Treasury Bills (a) (a) What price would she receive if the What price would she receive if the short-term interest rate for this maturity short-term interest rate for this maturity had fallen to 4.2% by the date of the sale? 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% McGraw-Hill Ryerson ©
• Applications 7 - 24 SI Treasury Bills Determine the number of days remaining ……. 91-day … purchased (T-Bill)to maturity …sold (T-Bill) after ………. 30 days = 61 days Calculate the Selling Price based on the Discounted Rate of 4.2% PV of \$75,000 discounted at 4.2% for 61 days = \$75000/[1 + .042(61/365)] SP = \$74,477.23 SP = \$74,477.23 McGraw-Hill Ryerson ©
• Applications 7 - 25 SI Treasury Bills 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 (b) What Rate of Return (per annum) did the client realize while holding the T-bill? client realize while holding the T-bill? McGraw-Hill Ryerson ©
• Applications 7 - 26 SI Treasury Bills (b) What Rate of Return (per annum) did the (b) What Rate of Return (per annum) did the client realize while holding the T-bill? client realize while holding the T-bill? … Sold (T-Bill) after 30 days …Purchased (T-Bill) ……. Grew in 30 Days by… I Prt McGraw-Hill Ryerson © \$74,477.23 74,149.62 \$ 327.61 = Pt rr= II//Pt r = 327.61 / 74,149.62 (30/365) = .053755 = 5.38%
• Applications 7 - 27 SI LO-3 The Valuation Principle McGraw-Hill Ryerson ©
• Applications SI McGraw-Hill Ryerson © 7 - 28 The Valuation Principle …the Fair Market Value of an investment and is the Sum of the Present Value(PV) of the expected cash flows The Discount Rate used in the Present Value(PV) calculations should be the Marketdetermined Rate of Return required for
• Applications SI 7 - 29 The Valuation Principle 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%? McGraw-Hill Ryerson ©
• Applications 7 - 30 The Valuation Principle SI \$110 one year from now = Future Value(FV) \$110 represents 110% or 100% + 10% Return We are looking for the Present Value PV PV = FV /( 1 +rt) = 110/[1+0.10*(1)] = \$100 = \$100 McGraw-Hill Ryerson © \$100 is \$100 is the most you the most you should pay should pay
• Applications SI 7 - 31 The Valuation Principle The process of calculating a payment’s Present Value is often called “Discounting the payment” McGraw-Hill Ryerson ©
• Applications SI McGraw-Hill Ryerson © 7 - 32 The Valuation Principle
• Applications SI 7 - 33 The Valuation Principle An investment promises two payments of \$2000, on dates 1 and 4 months from now. What price will an investor pay today What price will an investor pay today if the required rate of return if the required rate of return is 8% is 8% ?? 1 month = 1/12 4 months = 4/12 Formula Formula PV PV = FV /(1+rt) (1) Using a Time Line display, and (1) Using a Time Line display, and (2) Both algebraically and by calculator (2) Both algebraically and by calculator McGraw-Hill Ryerson ©
• Applications 7 - 34 The Valuation Principle SI Months 2 3 1 (1) 4 \$2000 \$2000 Present Present Value Value (2) PV PV 2000/[1+0.08*(1/12)] = 1986.76 = 1986.76 + 2000/[1+0.08*(4/12)] = 1948.05 = 1948.05 PV= \$3934.81 PV= \$3934.81 McGraw-Hill Ryerson ©
• Applications 7 - 35 The Valuation Principle SI Today’s Value: 2000/[1+0.08*(1/12)] + 2000/[1+0.08*(4/12)] 2000/[1+0.08*(1/12)] + 2000/[1+0.08*(4/12)] .08 4 12 1 2000 McGraw-Hill Ryerson © 1 12 1 = 1948.05 1986.76 1948.05 = 1986.76 .08 2000 = 1986.76 = 1986.76 + = 1948.05 = 1948.05 PV= \$3934.81
• Applications SI LO-4 McGraw-Hill Ryerson © 7 - 36
• Applications About 7 - 37 SI … Loans where the lender has the right to demand full repayment of the loan at any time! …revolving loans … fixed payment loans McGraw-Hill Ryerson ©
• Applications About 7 - 38 SI ... 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 McGraw-Hill Ryerson © date
• Applications 7 - 39 SI 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. The debt was completely repaid on June 30 together with the accrued interest. What was (a) the total interest charge, and What was (a) the total interest charge, and (b) the (b) the total payment on June 30? McGraw-Hill Ryerson ©
• Applications 7 - 40 SI Record the information provided and known in a Table format Determine the number of days in each period Calculate the interest accrued to date of payment Calculate the balance after the payment Calculate the interest accrued and the final balance on June 30 McGraw-Hill Ryerson ©
• Applications 7 - 41 SI Record the information that is known! Interest Rate Date Prime Plus Total Days Interest Paid Balance \$ Accrued Mar 1 5 2 7.0 May18 4 2 6.0 May 20 4 2 6.0 June 30 4 2 6.0 McGraw-Hill Ryerson © 5,000.00
• Applications 7 - 42 SI Determine the number of days in each period Interest Rate Date Prime Plus Total Interest Days Paid Balance \$ Accrued Mar 1 5 2 7.0 60 May18 4 2 6.0 138 78 May 20 4 2 6.0 140 2 June 30 4 2 6.0 181 41 Look up Look up Days Days McGraw-Hill Ryerson © 5,000.00
• Applications 7 - 43 SI Calculate the interest accrued Interest Rate Date Prime Plus Total Interest Days Paid Balance \$ Accrued Mar 1 5 2 7.0 60 May18 4 2 6.0 138 78 May 20 4 2 6.0 140 2 June 30 4 2 6.0 181 41 5,000.00 …Calculation McGraw-Hill Ryerson ©
• Applications 7 - 44 SI Calculate the interest accrued Date IR Total Interest Days Accrued Balance \$5000 Mar 1 7.0 60 May18 6.0 138 78 = \$76.48 \$76.48 \$5000 * 0.07 * 78/365 = \$76.48 May 20 6.0 140 2 = \$ 1.64 \$ 1.64 \$5000 * 0.06 * 2/365 = \$ 1.64 June 30 6.0 181 41 Total = \$78.12 Total = \$78.12 …complete the Table McGraw-Hill Ryerson ©
• Applications 7 - 45 SI Calculate the balance after the payment Interest Rate Date Prime Plus Total Interest Days Paid Balance \$ Accrued Mar 1 5 2 7.0 60 May18 4 2 6.0 138 78 76.48 5,076.48 May 20 4 2 6.0 140 2 1.64 June 30 4 2 6.0 5,078.12 1,303.12 3,775.00 181 41 5,000.00 Payment \$1,225.00 + Interest 78.12 Payment \$1,225.00 + Interest 78.12 McGraw-Hill Ryerson ©
• Applications 7 - 46 SI Calculate the interest accrued and final balance on June 30 Interest Rate Date Prime Plus Total Interest Days Paid Balance \$ Accrued Mar 1 5 2 7.0 60 May18 4 2 6.0 138 78 76.48 May 20 4 2 6.0 140 2 1.64 June 30 4 2 6.0 181 41 5,000.00 5,076.48 5,078.12 1,303.12 3,775.00 25.44 3,800.44 Interest Accrued = \$3,775.00 * .06 * 41/365 = \$25.44 McGraw-Hill Ryerson ©
• Applications 7 - 47 SI 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%. McGraw-Hill Ryerson © to use…
• Applications 7 - 48 SI Record the information provided and known in a Table format Determine the number of days in the grace period Calculate the interest accrued during the grace period (a) Calculate the total interest paid in the first two regular payments (b) Calculate the balance after the second payment McGraw-Hill Ryerson ©
• Applications 7 - 49 SI Record the information that is known! Interest Rate Date Prime Plus Total 6 6.5 2.5 2.5 8.5 9.0 June 30 6.5 5.0 Days 11.5 Dec 31 Mar 29 End of Grace Period July 31 6.5 5.0 11.5 Aug 13 7.0 5.0 12.0 Aug 31 7.0 5.0 12.0 McGraw-Hill Ryerson © Interest Accrued Paid Balance \$ 5,800.00
• Applications 7 - 50 SI Determine the number of days in each period Interest Rate Date Prime Days Plus Total 6 6.5 2.5 2.5 8.5 9.0 88 88 June 30 6.5 5.0 11.5 181 93 Dec 31 Mar 29 July 1 Accrued End of Grace Period July 31 6.5 5.0 11.5 212 30 Aug 13 7.0 5.0 12.0 225 13 Aug 31 7.0 5.0 12.0 243 18 McGraw-Hill Ryerson © Look up Look up Days Days Interest Paid Balance \$ 5,800.00 Interest includes June 30 due to grace period
• Applications 7 - 51 SI Calculate the interest accrued during the grace period Interest Rate Date Prime Days Plus Total 6 6.5 2.5 2.5 8.5 9.0 88 5.0 11.5 181 Paid Balance \$ 88 June 30 6.5 Interest 93 Dec 31 Mar 29 Accrued 5,800.00 End of Grace Period July 31 6.5 5.0 11.5 212 30 Aug 13 7.0 5.0 12.0 225 13 Aug 31 7.0 5.0 12.0 243 18 McGraw-Hill Ryerson © …Calculation
• Applications 7 - 52 SI Calculate the interest accrued during the grace period IR Date Days Total Interest Balance \$5800 Accrued Dec 31 8.5 Mar 29 9.0 88 88 118.86 \$5800 * 0.085 * 88/365= \$118.86 \$118.86 June 30 11.5 181 93 131.56 \$5800 * 0.09 * 93/365= \$133.00 \$133.00 End of Grace Period July 31 11.5 212 30 Includes June 30 Includes June 30 Aug 13 12.0 225 13 Aug 31 12.0 243 18 Total = \$251.86 Total = \$251.86 McGraw-Hill Ryerson © …complete the Table
• Applications 7 - 53 SI Calculate the interest accrued during the grace period Interest Rate Date Prime Days Interest Plus Total 6 6.5 2.5 2.5 8.5 9.0 88 88 118.86 June 30 6.5 5.0 11.5 181 Paid 93 133.00 Dec 31 Mar 29 Accrued End of Grace Period July 31 6.5 5.0 11.5 212 30 Aug 13 7.0 5.0 12.0 225 13 Aug 31 7.0 5.0 12.0 243 18 McGraw-Hill Ryerson © Balance \$ 5,800.00 5,918.86 251.86 6,051.86 5,800.00
• Applications SI (a) 7 - 54 Calculate the total interest paid in the first two regular payments Interest Rate Date Prime Days Interest Plus Total 6 6.5 2.5 2.5 8.5 9.0 88 88 118.86 June 30 6.5 5.0 11.5 181 Paid 93 133.00 Dec 31 Mar 29 Accrued End of Grace Period July 31 6.5 5.0 11.5 212 30 Aug 13 7.0 5.0 12.0 225 7.0 5.0 12.0 243 18 5,800.00 5,918.86 251.86 13 Aug 31 McGraw-Hill Ryerson © Balance \$ 95.00 95.00 …Calculation 6,051.86 5,800.00
• Applications (a) 7 - 55 SI Calculate the total interest paid in the first two regular payments Balance \$5800.00 First Payment \$95.00 Add: Interest 54.82 July 1 - July 31 interest 5854.82 Less: Payment 95.00 \$5800 * .115 * 30/365 = \$54.82 Balance 5759.82 Second Payment \$95.00 Add: Interest 23.59 July 31 - August 13 interest Interest 34.09 \$5759.82 * .115 * 13/365 = \$23.59 5817.50 August 13 - August 31 interest Less: Payment 95.00 Balance 5722.50 \$5759.82 * .12 * 18/365 = \$34.09 McGraw-Hill Ryerson © …complete the Table
• Applications 7 - 56 (a) & (b) SI …the total interest paid and the balance after the second payment Interest Rate Date Prime Days Interest Plus Total 6 6.5 2.5 2.5 8.5 9.0 88 88 118.86 June 30 6.5 5.0 11.5 181 Paid 93 133.00 Dec 31 Mar 29 Accrued Balance \$ 5,800.00 5,918.86 July 31 6.5 5.0 11.5 212 30 54.82 95.00 6,051.86 5,800.00 5,759.82 Aug 13 7.0 5.0 12.0 225 13 23.59 5,783.41 Aug 31 7.0 5.0 12.0 243 18 End of Grace Period McGraw-Hill Ryerson © 251.86 34.09 95.00 Total Interest \$112.50 5,722.50
• Applications 7 - 57 SI This completes Chapter 7 McGraw-Hill Ryerson ©