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# Business Mathematics Jerome Chapter 14

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## Business Mathematics Jerome Chapter 14Presentation Transcript

• Amortization Amortization 3 of Loans 14 - 1 of Loans Chapter 14 McGraw-Hill Ryerson© of
• Amortization Amortization 3 of Loans Learning Objectives of Loans 14 - 2 After completing this chapter, you will be able to: Calculate LO 1. LO 1. …the principal balance after any payment using both the Prospective Method and the Retrospective Method LO 2. LO 2. … the final loan payment when it differs from the others LO 3. LO 3. … the principal and interest components of any payment McGraw-Hill Ryerson© And… And…
• Amortization Amortization 3 of Loans of Loans Learning Objectives 14 - 3 Calculate LO 4. LO 4. LO 5. LO 5. McGraw-Hill Ryerson© … mortgage payments for the initial loan and its renewals … mortgage loan balances and amortization periods to reflect prepayments of principal
• Amortization Amortization 3 of Loans of Loans LO 1. LO 1. 14 - 4 A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period. (1) Calculate the monthly payment. (2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from \$20,000? PV = \$20000 FV = 0 n =12* 20 = 240 1. 1. PMT = 12 9 0 240 -179.95 20 000 2. & 3. 2. & 3. McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans 2. 2. 14 - 5 (2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from \$20,000? PV = ? FV = 0 n =12*20 = 240 PMT = 179.95 179.95 PV = 20,000.5345 179.95 3. 3. McGraw-Hill Ryerson© The difference of \$0.5345 is due to rounding the The difference of \$0.5345 is due to rounding the monthly payment to the nearest cent! monthly payment to the nearest cent!
• Amortization Amortization 3 of Loans of Loans 14 - 6 A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period. Calculate the exact balance after 5 years Calculate the exact balance after 5 years assuming the final payment will be adjusted for assuming the final payment will be adjusted for the effect of rounding the regular payment. the effect of rounding the regular payment. Calculate the exact n for monthly payments of \$179.95 to repay a \$20,000 loan... 20 000 N= McGraw-Hill Ryerson© 239.982
• Amortization Amortization 3 of Loans of Loans 14 - 7 A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period. Calculate the exact balance after 5 years Calculate the exact balance after 5 years assuming the final payment will be adjusted for assuming the final payment will be adjusted for the effect of rounding the regular payment. the effect of rounding the regular payment. P/V N = = 17,741.05 179.9821 239.982 After 5 years, 239.982 – 60 = 179.982 payments remain. Therefore, balance (after 5 years) = PV of 179.982 payments of \$179.95 60 McGraw-Hill Ryerson©
• 14 - 8 Amortization Amortization 3 of Loans of Loans Consider that… Consider that… An Original Loan = The PV of ALL of the Payments (discounted at the contractual rate of interest on the loan) Also, that… Also, that… A Balance = The PV of the remaining Payments (discounted at the contractual rate of interest on the loan) Then… Then… McGraw-Hill Ryerson©
• 14 - 9 Amortization Amortization 3 of Loans of Loans …this can be expressed as …the Statement of Economic Equivalence …this can be expressed as …the Statement of Economic Equivalence For a focal date of the original date of the loan, (Original Loan) McGraw-Hill Ryerson© PV of first x Payments PV of the Balance just after the xth Payment Focal Date… Focal Date…
• Amortization Amortization 14 - 10 Retrospective Retrospective 3 of Loans Retrospective Method for Loan Balances of Loans Suppose we locate the Focal Date… of the xth payment, the Statement of Economic Equivalence becomes… FV of the Original Loan FV of the Payments already made Balance This is now rearranged to isolate the “Balance” This is now rearranged to isolate the “Balance” Balance McGraw-Hill Ryerson© FV of the Original Loan FV of the Payments already made
• 14 - 11 Amortization Amortization 3 of Loans of Loans Prospective Method for Loan Balances … is based on PAYMENTS YET to be MADE!` Retrospective Retrospective Retrospective Method for Loan Balances … is based on PAYMENTS ALREADY MADE!` McGraw-Hill Ryerson© Application Application
• 14 - 12 Amortization Amortization 3 of Loans of Loans A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments of \$179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. Calculate the exact balance after 5 years. Solve using… Retrospective Method Prospective Method Then compare… McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans 14 - 13 Retrospective Method for Loan Balances A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments of \$179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. Calculate the exact balance after 5 years. 12 * 5 Years 12 * 5 Years Balance = FV of \$20,000 – FV of first 60 payments FV= 17,741.05 12 179.95 20,000 McGraw-Hill Ryerson© 60 9
• Amortization Amortization 3 of Loans of Loans 14 - 14 Prospective Method for Loan Balances A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments of \$179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. Calculate the exact balance after 5 years. Total payments = 12* 20 Years = 240 - 60 made = 180 remaining 12* 20 Years = 240 Balance = PV of remaining 180 payments PV= 17,741.88 12 179.95 0 McGraw-Hill Ryerson© 180 9
• Amortization Amortization 3 of Loans of Loans 14 - 15 Comparison of Methods Retrospective Method Retrospective Method for Loan Balances for Loan Balances FV= 17,741.05 Prospective Method Prospective Method for Loan Balances for Loan Balances PV= 17,741.88 Difference (\$0.83) is because the Prospective Method assumes that the final payment is the same as all the others. The Retrospective Method is based on payments already made. McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans LO 2. LO 2. 14 - 16 A \$20,000 mortgage loan at 9% compounded monthly requires monthly payments of \$179.95 during its 20-year amortization period. Calculate the size of the final payment. Calculate the size of the final payment. Final Payment = (1+i) * (Balance after 2nd to last payment) Balance after 239 payments = FV of \$20,000 after 239 months – FV of 239 payments 179.95 12 239 9 FV= 20,000 Final Payment = (1+0.09/12) * 175.42 = \$176.74 = \$176.74 McGraw-Hill Ryerson© - 175.42
• Amortization Amortization 3 of Loans of Loans A. 14 - 17 Meditech Laboratories borrowed \$28,000 at Meditech Laboratories borrowed \$28,000 at 10%, compounded quarterly, 10%, compounded quarterly, to purchase new testing equipment. to purchase new testing equipment. Payments of \$1,500 are made every 3 months. Payments of \$1,500 are made every 3 months. A. Calculate the balance after the 10th payment. A. Calculate the balance after the 10th payment. B. Calculate the final payment. B. Calculate the final payment. Balance after 10 payments = FV of \$28,000 after 10 quarters – FV of 10 payments 4 FV= - 19,037.29 1500 28,000 B. McGraw-Hill Ryerson© 10 1. 2. 3. 1. 2. 3. Balance after Balance after 10 payments 10 payments 10 Needed
• Amortization Amortization 3 of Loans of Loans 14 - 18 Meditech Laboratories borrowed \$28,000 at Meditech Laboratories borrowed \$28,000 at 10%, compounded quarterly, 10%, compounded quarterly, to purchase new testing equipment. to purchase new testing equipment. Payments of \$1,500 are made every 3 months. Payments of \$1,500 are made every 3 months. A. Calculate the balance after the 10th payment. A. Calculate the balance after the 10th payment. B. Calculate the final payment. FV N == -673.79 25.457 1. 1. Calculate …the number of payments 0 2. 2. Calculate …the balance after the 2nd to last payment 25 McGraw-Hill Ryerson© 3. 3.
• Amortization Amortization 3 of Loans of Loans 14 - 19 Meditech Laboratories borrowed \$28,000 at Meditech Laboratories borrowed \$28,000 at 10%, compounded quarterly, 10%, compounded quarterly, to purchase new testing equipment. to purchase new testing equipment. Payments of \$1,500 are made every 3 months. Payments of \$1,500 are made every 3 months. A. Calculate the balance after the 10th payment. A. Calculate the balance after the 10th payment. B. Calculate the final payment. B. Calculate the final payment. 3. 3. Calculate …the final payment Final Payment = (1+0.10/4) * 673.79 = \$690.63 = \$690.63 McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans 14 - 20 of Loans LO 3. LO 3. A \$9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term by equal monthly payments. A. Calculate the interest and principal components of the 29th payment. B. How much interest will be paid in the second year of the loan? McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans 14 - 21 A \$9,500 personal loan at 10.5% compounded A \$9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term monthly is to be repaid over a 4-year term by equal monthly payments. by equal monthly payments. A. Calculate the interest and principal components A. Calculate the interest and principal components of the 29th payment. B. How much interest will of the 29th payment. B. How much interest will be paid in the second year of the loan? be paid in the second year of the loan? First: … find the size of the monthly payment PV = 9500 n = 12(4) = 48 i = .105/12 PMT = - 243.23 12 48 9500 McGraw-Hill Ryerson© 10.5 0
• Amortization Amortization 3 of Loans of Loans 14 - 22 A \$9,500 personal loan at 10.5% compounded A \$9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term monthly is to be repaid over a 4-year term by equal monthly payments. by equal monthly payments. A. Calculate the interest and principal components A. Calculate the interest and principal components of the 29th payment. of the 29th payment. First: … find the balance after the 28 payments A. 243.23 FV = - 243.23 PMT = -4445.06 28 Interest Component of Payment 29 = i * Balance after 28th payment = 0.105/12* 4445.06 = \$38.89 Principal Component = PMT – Interest Component = \$243.23 - \$38.89 = \$204.34 = \$204.34 McGraw-Hill Ryerson©
• 14 - 23 A \$9,500 personal loan at 10.5% compounded A \$9,500 personal loan at 10.5% compounded 3 of Loans monthly is to be repaid over a 4-year term of Loans monthly is to be repaid over a 4-year term by equal monthly payments. by equal monthly payments. B. How much interest will be paid in the B. How much interest will be paid in the second year of the loan? second year of the loan? First:… find the balance after 1 Year, and the balance after 2 Years Amortization Amortization B. 12 FV = -7483.53 -5244.84 Balance Balance after 2 years after 1 year year after 21years 24 Total Principal paid in year 2 = \$7,483.53 - \$5,244.84 = \$2,238.69 = \$2,238.69 Total Interest paid in year 2 = 12(\$243.23) - \$2,238.69 = \$680.07 = \$680.07 McGraw-Hill Ryerson©
• 14 - 24 Amortization Amortization 3 of Loans of Loans … is a loan secured by some physical property McGraw-Hill Ryerson©
• 14 - 25 Amortization Amortization 3 of Loans Mortgage Loans of Loans …Basic Concepts and Definitions M O RT Borrower …the borrower is called the mortgagor McGraw-Hill Ryerson© GAGE AP PLI CATIO N Lender …the lender is called the mortgagee
• 14 - 26 Amortization Amortization 3 of Loans of Loans Mortgage Loans …Basic Concepts and Definitions M O RT AGE A Face Value of mortgage =GoriginalLprincipal amount PP ICATI ON Term … From … date on which loan advanced To … date on which the remaining Principal Balance is due and payable common periods are 20 and 25 years. …most …most common periods are 20 and 25 years. Interest Rate McGraw-Hill Ryerson© …usually a lender will commit to a fixed interest rate for only a shorter period or term (6 months to 7 years)
• 14 - 27 Amortization Amortization 3 of Loans of Loans M O RT GAGE AP PLI CATIO N A Mortgage Loan at 8.5% A Mortgage Loan at 8.5% compounded semiannually compounded semiannually with aa25-year amortization period with 25-year amortization period Graphic Illustrations Graphic Illustrations McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans 14 - 28 The Composition of Mortgage Payments during a The Composition of Mortgage Payments during a 25-year 25-year 100 Amortization Amortization 90 Principal Component 80 Approximately Approximately 40% 40% Interest % 70 60 Approximately Approximately 60% 60% 50 40 Interest Component 30 20 Yea r 14 10 0 McGraw-Hill Ryerson© 0 5 10 Years 15 20 25
• 14 - 29 Amortization Amortization 3 of Loans Mortgages Declining Balance during a Mortgages Declining Balance during a of Loans 25-year 25-year Amortization Amortization 100,000 90,000 Principal Balance \$ 80,000 70,000 60,000 Principal declines slower in earlier years 50,000 40,000 30,000 20,000 10,000 0 McGraw-Hill Ryerson© 0 5 10 Years 15 20 25
• Amortization Amortization 3 of Loans of Loans McGraw-Hill Ryerson© 14 - 30
• 14 - 31 Amortization Amortization 3 of Loans of Loans M O RT GAGE AP PLI CATIO N …need to satisfy all 3 of the following Ratios… Loan-to-Value Ratio (LVR) Loan-to-Value Ratio (LVR) Gross Debt Service Ratio (GDS) Gross Debt Service Ratio (GDS) Total Debt Service Ratio (TDS) Total Debt Service Ratio (TDS) McGraw-Hill Ryerson©
• 14 - 32 Amortization Amortization 3 of Loans of Loans M Loan-to-Value Ratio (LVR) Loan-to-Value Ratio (LVR)ORTGAGE AP PLI CATIO Principal Amount of Loan 100% N Lending Value of Property ≤ x 75% 75% Gross Debt Service Ratio (GDS) Gross Debt Service Ratio (GDS) Total monthly payments for Mortgage, Property taxes, and Heat Gross Monthly Income x 100% ≤ 32% 32% Total Debt Service Ratio (TDS) Total Debt Service Ratio (TDS) Total monthly payments for Mortgage, Property taxes, Heat and Other Debts Gross Monthly Income x 100% ≤ 40% 40% McGraw-Hill Ryerson©
• 14 - 33 3 You have saved \$35,000 for the down payment of Loans of Loans on a home. You want to know the maximum conventional mortgage loan for which you can qualify in order to determine the …highest price you can pay \$3,200 grossmonthly income is for a home. Personal … gross monthly income is \$3,200 Personal … 18 payments of \$300 per month remaining on … 18 payments of \$300 per month remaining on Data Data a car loan a car loan … property taxes of \$150 per month and … property taxes of \$150 per month and heating costs of \$100 per month heating costs of \$100 per month … the bank has upper limits of 32% for the … the bank has upper limits of 32% for the GDS Ratio and 40% for the TDS Ratio GDS Ratio and 40% for the TDS Ratio Amortization Amortization What maximum monthly mortgage payment do the GDS What maximum monthly mortgage payment do the GDS and TDS ratios permit? and TDS ratios permit? McGraw-Hill Ryerson©
• 14 - 34 Amortization Amortization 3 of Loans of Loans Gross Debt Service Ratio (GDS) Gross Debt Service Ratio (GDS) Total monthly payments for Mortgage, Property taxes, and Heat Gross Monthly Income x 100% Maximum Mortgage payment + 150 + 100 \$3,200 Maximum Mortgage payment = \$774 = \$774 McGraw-Hill Ryerson© = ≤ 32% 32% 32% 32% = .32(3200) - 250
• 14 - 35 Amortization Amortization 3 of Loans of Loans Total Debt Service Ratio (TDS) Total Debt Service Ratio (TDS) Total monthly payments for Mortgage, Property taxes, Heat and Other Debts Gross Monthly Income x 100% Maximum mortgage payment + 150 + 100+ 300 \$3,200 Maximum Mortgage payment = 40% 40% 40% 40% = .40(3200) - 550 = \$730 = \$730 McGraw-Hill Ryerson© ≤
• 14 - 36 Amortization Amortization 3 of Loans What is the maximum mortgage for which you qualify? Use a 25-year amortization and an interest rate of 8% compounded semiannually for a five-year term. of Loans 12 2 P/Y = 95,648.21 12 C/Y = 2 P/V= 0 8 730 300 McGraw-Hill Ryerson© 0 Maximum Maximum Mortgage Mortgage
• Amortization Amortization 3 of Loans of Loans 14 - 37 Based on a \$35,00 down payment and the maximum loan possible, what is the highest price you can pay for a home? Loan-to-Value Ratio (LVR) Loan-to-Value Ratio (LVR) Principal Amount of Loan Lending Value of Property x \$95, 600 Minimum house value Minimum house value McGraw-Hill Ryerson© = 100% = \$95, 648 75% 75% ≤ 75% 75% 75% 75% = \$127,530.67
• Amortization Amortization 3 of Loans of Loans 14 - 38 Based on a \$35,00 down payment and the maximum loan possible, what is the highest price you can pay for a home? At this price, the minimum down payment is: \$127,531– \$95,648= \$ 31,883 … the maximum price you can afford to pay for a home is… \$35,000 – 31,882 \$35,000 – 31,882 (Minimum down (Minimum down payment payment )) = over DP = over DP McGraw-Hill Ryerson© \$3,118 + \$127, 531 = \$130,649
• 14 - 39 Amortization Amortization 3 of Loans of Loans MORT G A GE A PPL I C ATION Common Prepayment Common Prepayment Privileges & Penalties McGraw-Hill Ryerson©
• Amortization Amortization 3 of Loans of Loans Common Common Fully Open No restrictions or penalties on extra payments by the borrower! McGraw-Hill Ryerson© Privileges 14 - 40 Prepayment &Prepayment Penalties Partially Open Limited penalty-free prepayment Lump or Balloon Payments 10% or 15% of the original amount Increasing the Regular Payment…permanently Once a year by 10% or 15% “Double-Up” Pay twice the amount for any monthly payment Closed No prepayment without a penalty
• Amortization Amortization 3 of Loans of Loans Privileges Common Common 14 - 41 Prepayment &Prepayment Penalties Financial Penalties i d es p ro v l tract ancia Con a fin any for n lty o t not pena men epay ically pr pecif tted s ermi p McGraw-Hill Ryerson© The most common prepayment penalty is the greater of: Three months’ interest on the Three months’ interest on the amount prepaid, amount prepaid, or or The lender’s reduction in interest The lender’s reduction in interest revenue from the prepaid amount revenue from the prepaid amount (over the remainder of the (over the remainder of the mortgage’s term) mortgage’s term) Example Example
• Amortization Amortization 3 of Loans of Loans LO 4. LO 4. LO 5. LO 5. 14 - 42 The interest rate for the first 5-year term of a \$100,000 mortgage loan is 7.5% compounded semiannually. The mortgage requires monthly payments over a 25 year amortization period. The mortgage contract gives the borrower the right to prepay up to 10% of the original mortgage loan, once a year, without interest penalty. Suppose that, at the end of the second year of the mortgage, the borrower makes a prepayment of \$10,000. a) How much will the amortization period be shortened? a) How much will the amortization period be shortened? b) What will be the principal balance b) What will be the principal balance at the end of the five-year term? at the end of the five-year term? McGraw-Hill Ryerson© Solving Steps Solving Steps
• Amortization Amortization 3 of Loans 14 - 43 of Loans 1. 1. Calculate …the payments based on a 25-year amortization 2. 2. Calculate …the balance after 24 payments 3. 3. - Reduce this balance by \$10,000 4. 4. Calculate …the number of monthly payments needed to pay off this new balance 5. 5. Calculate …the reduction in the original 25-year amortization period McGraw-Hill Ryerson©
• 14 - 44 Amortization Amortization 3 of Loans of Loans 1. 1. Calculate …the payments based on a 25-year amortization PV = 100,000 n = 25*12 = 300 i = .075/2 12 2 PMT= -731.55 c = 2/12 monthly monthly payment payment 100,000 300 7.5 0 McGraw-Hill Ryerson© 2. 3. 2. 3.
• 14 - 45 Amortization Amortization 3 of Loans of Loans 2. 2. Calculate …the balance after 24 payments 87,007.25 FV= -97,007.25 24 balance after the balance after balance after the balance after 24 payments prepayment 24 payments prepayment 731.55 3. 3. - Reduce this balance by \$10,000 10,000 McGraw-Hill Ryerson© 4. 5. 4. 5.
• 14 - 46 Amortization Amortization 3 of Loans of Loans 4. 4. Calculate …the number of monthly payments needed to pay off this new balance N= 214.60 215 more payments to 215 more payments to pay off the mortgage pay off the mortgage 87,007.25 0 5. 5. Calculate …the reduction in the original 25-year amortization period …with the prepayment: 24 + 215 = 239 Total payments Therefore, 300-239 = 61 months saved... i.e. 5 yrs 1 month McGraw-Hill Ryerson©
• 14 - 47 Amortization Amortization 3 of Loans of Loans Interactive Mortgage Payoff Chart…online Interactive Mortgage Payoff Chart…online www.mcgrawhill.ca/college/jerome/ www.mcgrawhill.ca/college/jerome/ Click On: Select: Select: Click On: Click On: Click On: Click On: Select: Select: -or-orMcGraw-Hill Ryerson© 4th Edition Student Centre
• Amortization Amortization 3 of Loans 14 - 48 of Loans This completes Chapter 14 McGraw-Hill Ryerson©