# Paying off a debt. Ethan D. Bolker Maura B. Mast. December 4, 2007

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1 Paying off a debt Ethan D. Bolker Maura B. Mast December 4, 2007 Plan Lecture notes Can you afford a mortgage? There s a \$250,000 condominium you want to buy. You ve managed to scrape together \$50,000 for the down payment (savings, your parents,...) but will have to take out a mortgage for the \$200,000 balance. Your credit is good, so you ve convinced the bank to let you borrow that money at 7% interest each year. Suppose you are now paying \$1,600 per month in rent, which comes to \$19,200 per year. You think you can afford to use \$17,000 of that for mortgage payments; you ll need the rest for condo fees. How long will it take you to pay off your loan? How much total interest will you pay? At the end of the first year you owe the bank \$200, = \$214, 000, the \$200,000 principal and \$14,000 in interest. You pay them \$17,000, so your new balance is \$200, \$17, 000 = \$197, 000 That s less than what you owed at the start of the year, but not nearly as much less as you d like. You still owe most of the \$200,000 in spite of your \$17,000 payment, since most of that went for interest. 1

3 The condo may well be worth more after 25 years than the total you paid for it even including the interest on the mortgage. Inflation is pretty nearly inevitable over the years. We ve computed the total interest in 2007 dollars, but the actual value of that money when you pay it to the bank each year will be less, in then current dollars. Words of warning. The purpose of this exercise is to show you how, in principle, you pay off a loan by paying some interest and some principal periodically. That s just one of the financial things you ll need to understand when you think about buying a house or condo. Just asking the bank for an interest rate and using it in this spreadsheet isn t sufficient. There are many books and web pages that may help here s just one we found with a simple search: Credit card interest In the previous section we saw why borrowing is sometimes wise. Now lets look at a time when it s not. We ll read this sample credit card bill, from practicalmoneyskills.com/english/resources/tutor/statements/credit_ state.php# If you have a credit card you get a bill like this once a month. John Doe (the owner of this card) charged \$ in merchandise during this billing period. He s decided not to use this card any longer, and will settle his debt by paying as 3

4 little as possible each month. When will he be debt free, and how much interest will he have paid? The arithmetic for this problem is exactly like the arithmetic for mortgage payments. After he makes the first payment he will owe \$ \$20 = \$ since \$ = \$2.07 of his \$20 payment is the interest charge. The rest is subtracted from \$ to reduce the balance owed. Here is the story for the first seven payments Month Balance Interest His last payment is for \$13.10 along with the \$13.10 \$ = \$0.22 interest. His total interest payments add up to \$8.08. That doesn t seem too terrible. It s \$8.08/\$ = = 6.5%. less than the mortgage rate for that condo. But don t be fooled. This is a monthly bill. The law requires the credit card company to tell you the annual rate somewhere on the monthly bill. In this case it s it s 19.80%. And you can check their arithmetic: 1.65% 12 = But in fact there s a subtle computation here that hides the true interest rate. Suppose you carry a balance of \$100 for the whole year. Each month the credit card company adds the interest for the month, and then charges you interest on the interest. At the end of the year this will have happened twelve times and you will owe them \$ = \$ which corresponds to an effective annual rate of 21.7% almost 2 full percentage points higher than the advertised rate of 19.8%. Applying interest to interest is called compounding. It s a good thing for your investments, since they grow exponentially. It s a bad thing for your credit card balance (but good for the credit card company). It s why the credit card company 4

7 The actual figures show that in 25 years you will have accumulated \$118,016. Of that amount, you contibuted just \$ = \$30, 000; \$88,016 is interest. If in fact you saved \$100 each month rather than \$1200 each year you d have even more. How much more we leave as an exercise for the reader. 7

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