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Problem Set 1

(Due at 11:59pm, Jan 10)

1. Suppose a firm’s tax rate is 25%.

a. What effect would a $10 million operating expense have on this year’s earnings? What effect would it have on next year’s earnings?

b. What effect would a $10 million capital expense have on this year’s earnings if the capital is depreciated at a rate of $2 million per year for five years? What effect would it have on next year’s earnings?

2. Consider two securities that pay risk-free cash flows over the next two years and that have the current market prices shown here:

a. What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $100 in two years?

b. What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $500 in two years?

c. Suppose a security with cash flows of $50 in one year and $100 in two years is trading for a price of $130. What arbitrage opportunity is available?

3. Suppose a security with a risk-free cash flow of $150 in one-year trades for $140 today. If there are no arbitrage opportunities, what is the current risk-free interest rate?

4. Marian Plunket owns her own business and is considering an investment. If she undertakes the investment, it will pay $4000 at the end of each of the next three years. The opportunity requires an initial investment of $1000 plus an additional investment at the end of the second year of $5000. What is the NPV of this opportunity if the interest rate is 2% per year? Should Marian take it?

5. You have just turned 30 years old, have just received your MBA, and have accepted your first job. Now you must decide how much money to put into your retirement plan. The plan works as follows: Every dollar in the plan earns 7% per year. You cannot make withdrawals until you retire on your sixty-fifth birthday. After that point, you can make withdrawals as you see fit. You decide that you will plan to live to 100 and work until you turn 65. You estimate that to live comfortably in retirement, you will need $100,000 per year starting at the end of the first year of retirement and ending on your 100th birthday. You will contribute the same amount to the plan at the end of every year that you work. How much do you need to contribute each year to fund your retirement?

6. Your friend tells you he has a very simple trick for shortening the time it takes to repay your mortgage by one-third: Use your holiday bonus to make an extra payment on January 1 of each year (that is, pay your monthly payment due on that day twice). Assume that the mortgage has an original term of 30 years and an APR of 12%.

a. If you take out your mortgage on January 1 (so that your first payment is due on February 1), and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage?

b. If you take out your mortgage on July 1 (so the first payment is on August 1), and you make the extra payment each January, in how many months will you pay off your mortgage?

c. How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan?

7. Suppose a 10-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading for a price of $1034.74.

a. What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?

b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?

8. The Isabelle Corporation rents prom dresses in its stores across the southern United States. It has just issued a five-year, zero-coupon corporate bond at a price of $74. You have purchased this bond and intend to hold it until maturity.

a. What is the yield to maturity of the bond?

b. What is the expected return on your investment (expressed as an EAR) if there is no chance of default?

c. What is the expected return (expressed as an EAR) if there is a 100% probability of default and you will recover 90% of the face value?

d. What is the expected return (expressed as an EAR) if the probability of default is 50%, the likelihood of default is higher in bad times than good times, and, in the case of default, you will recover 90% of the face value?

e. For parts (b–d), what can you say about the five-year, risk-free interest rate in each case?