Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 8.5, Slide 1 Consumer Mathematics The Mathematics of Everyday Life 8
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 8.5, Slide 2 Amortization 8.5 Calculate the payment to pay off an amortized loan. Construct an amortization schedule. Find the present value of an annuity. Calculate the unpaid balance on a loan.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 3 Amortization The process of paying off a loan (plus interest) by making a series of regular, equal payments is called amortization, and such a loan is called an amortized loan.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 4 Example: An amortized loan of $10,000 is made to pay off a car in 4 years. If the yearly interest rate is 18%, what is your monthly payment? Amortization Solution: We know the following values. (continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 5 Amortization We must solve for R in the equation
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 6 Example: $5,000 is borrowed at a 12% annual interest rate, and will be paid back in three equal monthly installments of $1, Construct an amortization schedule for this loan. Amortization Schedule (continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 7 Amortization Schedule Solution: First month’s interest: (continued on next slide) Money applied to principal: $1, – $50 = $1,
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 8 Amortization Schedule Second month’s balance: $5,000 – $1, = $3, The table shows the rest of the computations for this problem.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 9 Example: You wish to borrow $120,000 to buy a house. A bank offers a 30-year mortgage at an annual rate of 7%. The monthly payment is $ Construct an amortization schedule for the first three payments on this loan. Amortization Schedule (continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 10 Amortization Schedule Solution:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 11 Finding the Present Value of an Annuity
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 12 Example: You can afford to spend $200 each month on car payments. A bank offers you a 4- year car loan with an annual rate of 12%. what is the present value of this annuity? Solution: We can use the formula for finding payments on an amortized loan: (continued on next slide) Finding the Present Value of an Annuity
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 13 Finding the Present Value of an Annuity We know and.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 14 Example: Suppose you have a 30-year mortgage for $100,000 at an annual interest rate of 9%. After 10 years, you refinance. How much remains to be paid on your mortgage? The remaining 20 years is financed at an annual interest rate of 7.2%. What are the monthly payments? How much will you save in interest in 20 years by paying the lower rate? (continued on next slide) Finding the Unpaid Balance of a Loan
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 15 Solution: We can find that the monthly payment is $ based on the fact that your loan was a 30-year loan. The unpaid balance U on the loan is (continued on next slide) Finding the Unpaid Balance of a Loan
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 16 We know and. (continued on next slide) Finding the Unpaid Balance of a Loan Therefore, you still owe $89, on this mortgage.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 17 (continued on next slide) Finding the Unpaid Balance of a Loan In essence, you are taking out a new loan with and
Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 8.5, Slide 18 Finding the Unpaid Balance of a Loan Payment reduction: $ – $ = $ per month. Total amount paid over 20 years at the old interest rate: 240 × $ = $193, Total amount paid over 20 years at the new interest rate: 240 × $ = $168, Amount saved: $193, – $168, = $24,