FIN2010 Problem Set 1
Due 2025-2-24 at 5PM
1. Suppose you took a $100,000 15 year fixed-rate mortgage at 4.5% (APR) 3 years ago. Now the interest rate has dropped to 4% in the market, and you are considering refinance your mortgage. Hint: refinance means that you take out a new loan and pay off the old loan.
(1) What was the original monthly payment?
(2) Suppose you just made the 36th monthly payments of the original mortgage today. What is the remaining mortgage balance?
(3) If you refinance the mortgage with another bank which offers 4% (APR) and keep the remaining term (that is, 12 years until the mortgage is paid off), what would the new monthly payment be?
2. A foundation announces that it will be offering CUHK(SZ) scholarship to one student. The first
scholarship is to be offered exactly one year from now. When the scholarship is offered, the student will receive ¥100,000 annually for a period of four years, beginning from the date the scholarship is offered.
This student is then expected to repay the principal amount received (¥400,000) in 10 equal annual installments, interest-free, starting two years after the last payment of the scholarship. This implies that the foundation is really giving an interest-free loan under the guise of a scholarship. The discount rate is 6% and is expected to remain unchanged.
(1) What is the PV of the CUHK(SZ) scholarship from the perspective of the student (the scholarship
includes both the money provided from the foundation and the repayments made to the foundation)?
(2) The foundation plans to provide this scholarship to one student every single year forever. (The first
recipient of the scholarship receives the first payment of ¥100,000 one year from today, The second recipient of the scholarship receives the first payment of ¥100,000 two years from today, etc) In order to fund all these scholarships, how much money does the foundation need to have today?
3. The annual membership fee at your health club is $750 a year and is expected to increase at 5% per year. A life membership is $7,500 and the discount rate is 12%. You either pay the annual membership every year (starting right now), or pay the life membership right now. In order to justify taking out the life membership, what would be your minimum life expectancy?
4. You are considering buying a car worth $30,000. The dealer, who is anxious to sell the car, offers you an attractive financing package. You have to make a down-payment of $3,500, and pay the rest over 3 years with monthly payments. The dealer will charge you interest at a constant APR of 2%, which is lower than the market interest rate other dealers offer.
(1) What is the monthly payment to the dealer?
(2) The dealer offers you a second option: you pay cash, but get a $2,500 discount. Should you go for the loan or should you pay cash? Assume that the market annual interest rate other deals offer is at 5%
(APR).
5. Your cousin is entering medical school next fall and asks you for financial help. He needs $65,000 each year at the beginning of the school year for the first two years. After that, he is in residency for two years and will be able to pay you back $10,000 each year (paid after working for 1 year and 2 years, respectively). Then he graduates and becomes a fully qualified doctor, and will be able to pay you $40,000 each year (paid at the end of each year). He promises to pay you $40,000 for 5 years after he graduates. Are you taking a financial loss or gain by helping him out? Assume that the interest rate is 5% and that there is no risk of him breaking his promise.
6. Currently a 20-year Treasury bond with 4% semiannual coupon is traded at a yield of 5% (APR).
(1) Is the price above or below 100% of par value?
(2) Calculate the current price of the bond as percentage of the par value.
(3) If the yield suddenly increased by 0.1%, how much would the price change? Would the price increase or decrease?