Problems

Unless otherwise mentioned, all payments are made at the end of the period.

1. You have just won the lottery. At the end of each of the next 20 years you will receive a payment of $50,000. If the cost of capital is 10 percent per year, what is the present value of your lottery winnings?

2. A perpetuity is an annuity that is received forever. If I rent out my house and at the beginning of each year I receive $14,000, what is the value of this perpetuity? Assume an annual 10 percent cost of capital. (Hint: Use the PV function and let the number of periods be many!)

3. I now have $250,000 in the bank. At the end of each of the next 20 years I withdraw $15,000. If I earn 8 percent per year on my investments, how much money will I have in 20 years?

4. I deposit $2,000 per month (at the end of each month) over the next 10 years. My investments earn 0.8 percent per month. I would like to have $1 million in 10 years. How much money should I deposit now?

5. An NBA player is receiving $15 million at the end of each of the next seven years. He can earn 6 percent per year on his investments. What is the present value of his future revenues?

6. At the end of each of the next 20 years I will receive the following amounts:

   Open table as spreadsheet

   Years	Amounts
   1–5	$200
   6–10	$300
   11–20	$400

   Use the PV function to find the present value of these cash flows if the cost of capital is 10 percent. Hint: Begin by computing the value of receiving $400 a year for 20 years then subtract the value of receiving $100 a year for 10 years, etc.

7. We are borrowing $200,000 on a 30 year mortgage with an annual interest rate of 10 percent. Assuming end-of-month payments, determine the monthly payment, interest payment each month, and amount paid towards principal each month.

   Answer each question in Problem 7 assuming beginning-of-month payments.

8. Use the FV function to determine the value to which $100 accumulates in three years if you are earning 7 percent per year.

9. You have a liability of $1,000,000 due in 10 years. The cost of capital is 10 percent per year. What amount of money would you need to set aside at the end of each of the next 10 years to meet this liability?

10. You are going to buy a new car. The cost of the car is $50,000. You have been offered two payment plans:

    • A 10 percent discount on the sales price of the car, followed by 60 monthly payments financed at 9 percent per year.

    • No discount on the sales price of the car, followed by 60 monthly payments financed at 2 percent per year.

    If you believe your annual cost of capital is 9 percent, which payment plan is a better deal? Assume all payments occur at the end of the month.

11. I presently have $10,000 in the bank. At the beginning of each of the next 20 years I am going to invest $4,000 and I expect to earn 6 percent per year on my investments. How much money will I have in 20 years?

12. A balloon mortgage requires you to pay off part of a loan during a specified time period and then make a lump sum payment to pay off the remaining portion of the loan. Suppose you borrow $400,000 on a 20 year balloon mortgage and the interest rate is .5 percent per month. Your end-of-month payments during the first 20 years are required to pay off $300,000 of your loan and 20 years from now you will have to pay off the remaining $100,000. Determine your monthly payments for this loan.

13. An adjustable rate mortgage (ARM) ties monthly payments to a rate index (say, the US T-Bill rate). Suppose we borrow $60,000 on an ARM for 30 years (360 monthly payments). The first 12 payments are governed by the current T-Bill rate of 8 percent. In years 2–5, monthly payments are set at the year’s beginning monthly T-Bill rate + 2 percent. Suppose the T-Bill rates at the beginning of years 2–5 are as follows:

    Open table as spreadsheet

    Beginning of year	T-Bill rate
    2	10 percent
    3	13 percent
    4	15 percent
    5	10 percent

    Determine monthly payments during years 1–5 and each year’s ending balance.

14. Suppose we have borrowed money at a 14.4 percent annual rate and we make monthly payments. If we have missed four consecutive monthly payments, how much should next month’s payment be to catch up?

15. We want to replace a machine in 10 years and we estimate our cost will be $80,000. If we can earn 8 percent annually on our investments, how much money should we put aside at the end of each year to cover the cost of the machine?

16. We are buying a motorcycle. We pay $1,500 today and $182.50 a month for three years. If the annual rate of interest is 18 percent, what was the original cost of the motorcycle?

17. Suppose the annual rate of interest is 10 percent. We pay $200 a month for two years, $300 a month for a year, and $400 for two years. What is the present value of all our payments?

18. We can invest $500 at the end of each six-month period for five years. If we want to have $6,000 after five years, what is the annual rate of return we need on our investments?

19. I borrow $2,000 and make quarterly payments for two years. The annual rate of interest is 24 percent. What is the size of each payment?

20. I have borrowed $15,000. I am making 48 monthly payments and the annual rate of interest is 9 percent. What is the total interest paid over the course of the loan?

21. I am borrowing $5,000 and plan to pay back the loan with 36 monthly payments. The annual rate of interest is 16.5 percent. After one year, I pay back $500 extra and shorten the period of the loan to two years total. What will my monthly payment be during the second year of the loan?

22. With an Adjustable Rate Mortgage, you make monthly payments depending on the interest rates at the beginning of each year. We have borrowed $60,000 on a 30-year ARM. For the first year, monthly payments will be based on the current annual T-Bill rate of 9 percent. In years 2–5, monthly payments will be based on the following annual T-Bill rates +2 percent.

    • Year 2: 10 percent

    • Year 3: 13 percent

    • Year 4: 15 percent

    • Year 5: 10 percent

    The catch is that the ARM contains a clause which ensures that monthly payments can increase a maximum of 7.5 percent from one year to the next. To compensate the lender for this provision, the borrower adjusts the ending balance of the loan at the end of each year based on the difference between what the borrower actually paid and what he should have paid. Determine monthly payments during Years 1–5 of the loan.

23. You have a choice of receiving $8,000 each year beginning at age 62 and ending when you die, or receiving $10,000 each year beginning at age 65 and ending when you die. If you think you can earn an 8 percent annual return on your investments, which will net the largest amount?

24. You have just won the lottery and will receive $50,000 a year for 20 years. What rate of interest would make these payments the equivalent of receiving $500,000 today?

25. A bond pays a $50 coupon at the end of each of the next 30 years, and pays $1,000 face value in 30 years. If we discount cash flows at an annual rate of 6 percent, what would be a fair price for the bond?

26. You have borrowed $100,000 on a 40-year mortgage with monthly payments. The annual interest rate is 16 percent. How much will you pay over the course of the loan? With four years left on the loan, how much will you still owe?

27. I need to borrow $12,000. I can afford payments of $500 per month and the annual rate of interest is 4.5 percent. How many months will it take to pay off the loan?