Math 2UU3 * Problem set 11

Transcription

1 Math 2UU3 * Problem set You have two options to invest $1500: (a) collect $150 at the end of each year (b) collect 6% interest at the end of each year. Determine which option is better in the short run, i.e., for the first few years. Which option is better in the long run, i.e., yields better return after many years? Figure out when (answer in the form after... years ) option (b) becomes better, i.e., yields more money, than option (a). Make a rough sketch to illustrate how both options behave over time. 2. You bought a car for $32,000. The moment you drive off a dealer s lot your car loses somewhere between 9% and 11% of its value (you can confirm this easily, Google search by car depreciation). After that, your car loses between 15% and 25% of its value per year. Taking midpoints of the ranges given, find out how much your car is worth after 1, 2, 3, 4 and 10 years. 3. You bought a car for $50,000. Assume that it depreciates in value as follows: 11% as you drive off a dealer s lot, and then 22% per year. You gave a downpayment of $5,000, and the dealer gave you a 0% interest loan to pay off the remaining balance. Your monthly loan payments are $400. How much is your car worth at the end of 5 years, and how much do you still owe on it? This is an example of negative equity. This term is usually applied to mortgages, and it means that the market value of your car is lower than the outstanding amount of your loan. 4. (a) Calculate the EAR (effective annual rate) corresponding to 7% APR (annual percentage rate) when it is compounded annually, semi-annually, monthly, weekly, daily, hourly, and once a second. (b) What would be the effective rate if (theoretically) the interest were computed once every infinitely short instant of time? 5. Assume that the annual percentage rate is 30%. (a) Compute the effective rate when interest is computed semi-annually. Using a calculator, approximate the doubling time for the effective rate, and compare with what the rule of 70 gives you. (b) Compute the effective rate when interest is computed daily. Using a calculator, approximate the doubling time, and compare with what the rule of 70 gives you. 6. According to CreditCards.com site there were 68.5 million Visa and Mastercard credit cards in circulation in Canada (October 2015). As well, we find out that the average Canadian holds $3,954 in credit card debt. (a) On average, how many Visa and Mastercard credit cards does each Canadian have? 1

2 (b) The same site claims that about 40% of Canadians carry their balance forward, i.e., do not pay their monthly balance in full. How many Visa and Mastercard credit cards carry forward a (non-zero) balance? (c) Assume that on each Visa and Mastercard from (b) there is a carry forward balance of $500, and assume that in a given month there are no extra charges on these credit cards. As well, asume that the APR is 19.99%. Compute the total interest by using the formula that the banks use (i.e., based on average daily balance and daily interest rate) and compare with the total interest if the monthly rate were used instead. 7. The APR on a credit card is 19.99%. In April 2018, the ending daily balances were: $1200 on 1 April, $1450 on 4 April, $550 on 10 April, $800 on 14 April, $1100 on 22 April, and $1200 on 30 April. As the credit card carried over unpaid balance, find out what was the interest charged for April Assume, as is common practice, that the interest in computed based on a daily interest rate. 8. For this question, use the credit card payment calculator (the link is on our web page as well). Assume that you owe $3500 on a credit card whose APR is 18%, and you decide to pay the minimum payment of $70 each month. Also assume that you will not be using this card any more, so there will be no new purchases on it. You pay the minimum payment at the start of each month. (a) Start a calculation by hand: compute the balance on your card at the end of the first three months. Assume that all months have 30 days, and all years have 365 days. (b) Using the calculator above, find out how long it will take to pay off the full balance on your credit card. (c) How much in interest would you pay? What percent of your total payments is the interest? (d) Enter the minimum payment of $45 each month into the calculator. Interpret the answer that you get. 9. In calculating daily periodic rate to apply to credit card interest some banks divide the APR by 360, rather than by 365. Why do they do that? 10. Use the mortgage calculator (the link is also on our home page), or any other suitable mortgage calculator, to answer the following questions. You plan to borrow $450,000 to buy a house. In each case, record: monthly payments (so payment frequency is once per month), principal and interest paid after 5 years (this is called term), and the total interest over the life of the mortgage. (a) Interest rate is 3.25%. Amortization (life of the mortgage, i.e., how long it will take to pay it off) is 25 years. (b) As in (a), except that the interest rate is 3.45%. How to the two options, (a) and (b), compare? 2

3 (c) Interest rate is 2.25%. Amortization is 15 years. How does it compare to (a)? (d) You start as in (a), but after one year the bank increases the interest rate to 5.15% (for instance, they can do this because you signed up for a variable interest rate). How does it compare to (a), after you ve been paying your mortgage for 5 years? 11. Read the article, at least the first five paragraphs, of Subprime mortgage crisis on Wikipedia, as background for events that lead to a huge banking crisis and recession in in the U.S. (which did affect Canada, but to a lesser extent). In particular, we read When U.S. home prices declined steeply after peaking in mid- 2006, it became more difficult for borrowers to refinance their loans. As adjustable-rate mortgages began to reset at higher interest rates (causing higher monthly payments), mortgage delinquencies soared. We will use the mortgage calculator from the previous exercise to understand this sentence. Assume that someone borrowed $150,000 at 2% interest rate (which was realistic at the time), with amortization of 25 years. (a) What were the monthly payments? (b) How much of the mortgage is still left to be paid off after two years? (c) Assume that, after two years the bank increased the rate to 9% (since it s adjustablerate mortgage, the bank is allowed to do that; and that s exactly what happened). What is the monthly payment now? People were caught by surprise, and many could no longer afford to pay monthly mortgage payments. Banks foreclosed on the loans, and they ended losing lots of money. Many people did not read/ were not aware of the small print about adjustable-rate mortgages in their mortgage contracts. Promise to yourself that you will never sign a money-borrowing agreement/ contract with a bank or any other entity without reading and understanding every word of the small print on every page of the contract. 12. To answer these questions consult the table of historic values of the CPI on the last page of this problem set. (Source: Statistics Canada, Consumer Price Index, historical summary, (a) What was the inflation rate in 2011 (i.e., percent change from 2010 to 2011)? (b) In 1997, you paid $67.00 for a pair of sneakers. How much would you pay for the same pair of sneakers in 2017? (c) Express $350 (in 2017 dollars) in terms of 2004 dollars. (d) Express $350 (in 2004 dollars) in terms of 2017 dollars. (e) What was the inflation rate between 2000 and 2011? (f) Use the inflation calculator to check your answers to questions (b)-(e). 13. Suppose that your salary in 2013 was $ For 2014, you negotiate an increase of 5%. What is the actual increase in salary if you include inflation? Use at the table from 3

4 the previous question. Express the increase in both absolute and relative terms. 14. If a student s GPA (on the scale from 0 to 12) is 7.45 based on 90 units, what grades must she earn for the remaining 30 units to end up with a GPA of 8.00? 15. The final course grade formula is given by: 30% for tests, 25% for assignments, and 45% for exam. A student earned 70% on tests and 50% on assignments. Can he end up with a grade of 80% or above in the course? What is the maximum grade he can earn? 16. The CPI for food is the weighted average of the cost of food at home and the cost of eating out. For the purpose of this exercise assume that the cost of food at home is $600, and the cost of eating out is $400. (a) Assume that the increase in the cost of food at home is 5%, and the increase in the cost of eating out is 8%. If the respective weights (food at home and eating out) are 75% and 25%, what is the increase in the CPI for food? (b) Assume that the increase in the cost of food at home is 5%, and the increase in the cost of eating out is 8% (same as in (a)). If the weight for food at home is 89%, what is theincreaseinthecpiforfood? 17. A car sells for $25000 and an SUV for $ (a) What is the average car price if the weights of 50% for both a car and an SUV are used? (b) A car dealer sells 12 cars and 4 SUVs. What is the weighted average of the price of a car sold? (The weights are based on the number of cars sold.) (c) A car dealer sells 3 cars and 13 SUVs. What is the weighted average of the price of a car sold? (The weights are based on the number of cars sold.) (d) What are the smallest and the largest possible values for a weighted average of cars and SUVs? 18. Suppose that A is a weighted average of X and Y. Is it possible for A to decrease even though both X and Y are increasing? 4

5 CONSUMER PRICE INDEX IN CANADA, HISTORIC VALUES THE END 5