From Maura:
Ethan was out today so I stepped in for him. I started by asking the class what they had been working on last time. It took a bit of pushing but they finally told me that they had looked at credit cards and interest rates. They had the main philosophical ideas: don’t carry a balance, think carefully before getting a credit card, etc. They also told me that linear growth and exponential growth are different and that exponential growth can go faster. Good. I told them that we’d look at this in a different context and asked them to start problem 10.6.4. Essentially this problem has them choose a house or condo that they are interested in buying, find an interest rate and time to payment, then use the spreadsheet to calculate monthly payments and total interest paid. It took awhile for them to get moving on it but once they got the idea, they were creative in terms of where they looked for houses (Allston, Dorchester, Hawaii) and what types of houses they were interested in. After about 10 minutes they shared the different housing prices, interest rates and monthly payments. We talked about the advantages of buying a house (tax deduction, beating inflation, your own place to live, equity) and advantages of renting (Massachusetts tax deduction, flexibility, may be cheaper, less responsibility). One student had asked about APR so we moved on to talk about what that meant (I think we need to be clearer in the book about this by the way).
I gave them the basic idea: to get the APR using an interest rate for a fixed period of time (day, week, month), just multiply. We looked at some examples from the credit card problems: a 1.9% monthly interest rate ends up being 22.8% annually. A 0.5% weekly rate ends up being 26% annually. The point is that you can compare rates easily when you have an annual basis for comparison. We looked at two examples. One was the monthly vs. weekly rate comparison. The second was an example based on buying a car. I made it up on the spot which is always risky, but it worked out ok. The question is to figure out which is the better deal, in the long run, for taking a 5-year loan to buy a $20,000 car. The first option is to pay $5,000 down and finance the remainder at 0.8% monthly interest. The second option is to finance the entire purchase of $20,000 at 0.25% monthly interest. We can find the annual rates: in the first option, the annual rate is 0.8*12%=7.2%. In the second option, it’s 0.25*12%=6%. So the second option has a lower rate, but on the other hand we are borrowing a larger amount. We put the two examples into the spreadsheets and worked it out. In the first example, you would pay $20,000 (the purchase price) + 2,906.13 (interest) = 22,906.13. In the second example, you would pay $20,000 (the purchase price) + 3,199.36 (interest) = 23,199.36. The first deal is the better one provided you have the $5,000 downpayment. They asked if this really happens. Absolutely it does! I told them to check some car websites to see examples and that when I bought a car this past April and did the calculation myself to see what the better deal was.
We talked again after this about APR and how it’s a useful estimate but that it doesn’t tell the whole story. The method of calculating APR ignores the effects of compounding; that is, of interest being calculated on interest. I gave a quick example: suppose you put $1000 in a savings account and earn $200 interest at the end of the year. If you keep all of that in your account then at the end of the next year you’ll earn interest on $1,200 and it will be more than an additional $200 interest. That’s because you are earning interest on the original $1000 AND on the $200 of interest. I don’t think they quite got it so I had them start problem 10.6.10 on payday loans. In this problem we look at a real example of borrowing $100 for two weeks and paying it back with $20 interest. We were running short on time so we focused on the question of checking that this is a daily interest rate of 1.43% and that it translates into an annual rate of 522%. Then I asked how we would calculate the effective annual rate, the rate that takes compounding into account. I reminded them that they’ve done this already and that it’s an exponential approach. There are a lot of piece to pull together here: use the 1+ technique, use 0.0143 instead of 1.43, use the exponent. We put it together and calculated (1+0.0143)^365=178.13. They remembered that we have to multiply by 100 to get the percentage. So in fact the effective annual interest rate is 17,813%. Wow. I brought them back to the original question: if you borrowed $100 and never paid it back, and if the interest kept accumulating, then a year later you would owe $17,813.34. I hope they will think about this more as they do that problem for homework.
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