5-1 What is an opportunity cost? How is this concept used in TVM analysis, and where is it shown on a time line? Is a single number used in all situations? Explain. 5-2 Explain whether the following statement is true or false: $100 a year for 10 years is an annuity, but S100 in Year 1, $200 in Year 2, and $400 in Years 3 through 10 does not consti- tute an annuity. However, the second series contains an annuity. 5-3 If a firm's earnings per share grew from $1 to $2 over a 10-year period, the total growth would be 100%, but the annual growth rate would be less than 10%. True or false? Explain. (Hitt: If you aren't sure, plug in some numbers and check it out.) 5-4 Would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% interest compounded daily? Explain. 5-5 To find the present value of an uneven series of cash flows, you must find the PVs of the individual cash flows and then sum them. Annuity procedures can never be of use, even when some of the cash flows constitute an annuity, because the entire series is not an annu- ity. True or false? Explain. 5-6 The present value of a perpetuity is equal to the payment on the annuity, PMT, divided by the interest rate, 1: PV = PMT/1. What is the future value of a perpetuity of PMT dollars per year? (Hint: The answer is infinity, but explain why.) 5-7 Banks and other lenders are required to disclose a rate called the APR. What is this rate? Why did Congress require that it be disclosed? Is it the same as the effective annual rate? If you were comparing the costs of loans from different lenders, could you use their APRs to determine the loan with the lowest effective interest rate? Explain. 5-8 What is a loan amortization schedule, and what are some ways these schedules are used? is 5-1 FUTURE VALUE If you deposit $2,000 in a bank account that pays 6% interest annually, how much will be in your account after 5 years? 5-2 PRESENT VALUE What is the present value of a security that will pay $29,000 in 20 years if securities of equal risk pay 5% annually? 5-3 FINDING THE REQUIRED INTEREST RATE Your parents will retire in 19 years. They cur- rently have $350,000 saved, and they think they will need $800,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any addi- tional funds? 5-4 TIME FOR A LUMP SUM TO DOUBLE If you deposit money today in an account that pays 4% annual interest, how long will it take to double your money? 5-5 TIME TO REACH A FINANCIAL GOAL You have $33,556.25 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $220,000. You expect to earn 12% annually on the account. How many years will it take to reach your goal? 5-6 FUTURE VALUE: ANNUITY VERSUS ANNUITY DUE What's the future value of a 5%, 5-year ordinary annuity that pays $800 each year? If this was an annuity due, what would its future value be? 5-7 PRESENT AND FUTURE VALUES OF A CASH FLOW STREAM An investment will pay $150 at the end of each of the next 3 years, $250 at the end of Year 4, $300 at the end of Year 5, and $500 at the end of Year 6. If other investments of equal risk earn 11% annually, what is its present value? Its future value? 5-8 LOAN AMORTIZATION AND EAR You want to buy a car, and a local bank will lend you $40,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest Chapter 5 Time Value of Money 185 rate will be 8% with interest paid monthly. What will be the monthly loan payment? What will be the loan's EAR? 5-9 PRESENT AND FUTURE VALUES FOR DIFFERENT PERIODS Find the following values using the equations and then a financial calculator. Compounding/discounting occurs annually a. An initial $600 compounded for 1 year at 6% b. An initial S600 compounded for 2 years at 6% c. The present value of $600 due in 1 year at a discount rate of 6% d. The present value of $600 due in 2 years at a discount rate of 6% 5-10 PRESENT AND FUTURE VALUES FOR DIFFERENT INTEREST RATES Find the following val- ues. Compounding/discounting occurs annually. a. An initial $200 compounded for 10 years at 4% b. An initial $200 compounded for 10 years at 8% c The present value of $200 due in 10 years at 4% d. The present value of $1,870 due in 10 years at 8% and at 4% e Define present value and illustrate it using a time line with data from part d. How are present values affected by interest rates