ࡱ> )+({ ,bjbjzz 4>~ff8 DPL$r"@@@#######$%B(#vvv#@@#v@@#v#:]!,!@ƆVd! ##0$!R(.(!(!hJ< m##B$vvvv(f : Can you afford your dream car? Compound interest and amortization schedules When you borrow money to buy a house or a car, your lender calculates your monthly payment by a method called amortization. This is the process of paying off a debt by making a certain number of equal payments at specific intervals (generally monthly). These payments include compound interest. With each payment, the amount of interest declines (as the unpaid balance on the loan declines), while the amount paid toward principal increases. If you make equal payments monthly, then the payment amount is calculated by the formula payment = (loan amount) X  EMBED Equation.3 X  EMBED Equation.3 , where t is the number of years to repay the loan. We can simplify the above formula by letting P represent the amount borrowed (the principal), and m represent the monthly interest rate (i.e. m = interest rate/12). Then the monthly payment is given by payment =  EMBED Equation.3  Lets look at an example. Suppose your rich aunt agrees to lend you $3000 at an interest rate of 3%, amortized over 2 years. Your monthly interest rate is  EMBED Equation.3  = 0.0025 and your monthly payment is payment =  EMBED Equation.3  = 128.94 Now lets figure out how much of each payment goes to interest and how much to the principal? Each month you must calculate interest on the current loan balance. If the monthly interest rate is 0.0025 and the initial balance is $3000, then the first months interest is $7.50. So $121.44 is applied toward principal, leaving a new balance of $2878.56. Continuing in this manner, you can construct an amortization schedule for your loan. An amortization schedule gives the amount of each payment that goes to interest, the amount that goes to principal , and the new balance after the payment is made. The following table gives the first four months of the amortization schedule for this loan. Payment No.Interest on unpaid balanceAmount paid toward principalNew Balance$3000.001$7.50$121.44$2878.562$7.20$121.74$2756.823$6.89$122.05$2634.774$6.59$122.35$2512.42When you finish all your payments, you might be curious about the amount of interest you actually paid. If you make 24 payments of $128.94, you repay your aunt  EMBED Equation.DSMT4  $3094.56, so you have paid  EMBED Equation.DSMT4  $94.56 interest on the loan. Search the Internet and find the car of your dreams new or used. State the price of the car and assume that you will borrow all but the $10,000 down payment. a) Find the sales tax rate in your county and determine the total price of your car (retail cost + tax). (1) b) Indicate how much you plan to borrow. (1) You decide to put the loan on your credit card, which charges an annual percentage rate of 14.24 %. Assuming you will repay the loan in 5 years (t = 5), find your monthly payment using the formula on the previous page. (You may also find the explanations on pp. 393 -396 in your text helpful.) (3) Construct an amortization table like the one above, either by hand or using EXCEL, for the first 12 payments on the loan, assuming that the loan is amortized over 5 years. Please use 4 DECIMAL PLACE accuracy in your calculations (i.e. round the monthly interest rate to 4 decimal places), but list your dollar amounts rounded to two decimal places. (5) a) Over the entire length of the loan, how much interest will you pay? (2) b) What is your total cost for the car? (retail price, tax, and loan costs). (1) After more research, you find two different lenders offering the following options: 5.5 % amortized over 4 years 6.0% amortized over 6 years. a) If you can afford a monthly car payment of no more than $500, can you afford to borrow the money from either of these lenders? Explain. Please show all work. (2) b) Of the three scenarios above, which is the better deal? Explain please be specific and detailed! (3) Creativity and appearance: (2) BONUS: Earn 5 extra points: Over three years ago, you leased your car for 39 months at $400 a month, and are deciding whether to trade it in, or buy the car with a 6.5% interest rate. Find the blue book value of your car now, and set up a loan for the current value of the car at the 6.5% rate for 5 years. When the car loan has been completely repaid, how much will you have paid for this car (including the amount paid on the lease)? (5)     PAGE  PAGE 1 Adapted from Davis, M. & Edwards, C, (2001). Elementary Mathematical Modeling. Upper Saddle River, NJ: Prentice Hall, pp. 204-205. 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