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Final solution sketches

Final solution sketches. Note for multiple-choice questions: Choose the closest answer. PV of Perpetuity. Jil is due to receive $50,000 every 2 years, forever, starting one year from today. What is the PV of this perpetuity if her effective annual discount rate is 7%?

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Final solution sketches

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  1. Final solution sketches Note for multiple-choice questions: Choose the closest answer

  2. PV of Perpetuity • Jil is due to receive $50,000 every 2 years, forever, starting one year from today. What is the PV of this perpetuity if her effective annual discount rate is 7%? • Rate every 2 years = (1.07)2 – 1 = 14.49% • (In $1000’s): 50/.1449 * 1.07 = 369.22 • PV = $369,220

  3. Loan Payments • Remi is taking out a loan for a $20,000 car. He will make a 20% down payment and borrow the rest, and will pay off the loan with 40 monthly payments. Each month, he will pay off a constant amount of the principal. How much will the first payment be if Remi’s stated annual interest rate is 12%, compounded monthly? • Loan = 0.8 * $20,000 = $16,000 • 1st payment: • Principal = 16,000/40 = $400 • Interest = 16,000 * 0.01 = $160 • 1st payment = $400 + $160 = $560

  4. Bonds • William’s bond will mature in 10 years. The amount that he will receive in 10 years will be $8,000. Assume William’s effective annual discount rate is 5% over the next 3 years, and 9% thereafter. What is the PV of this bond? • $8,000 / (1.05)3(1.09)7 = $3780.39

  5. Discount Rates and IRRs • Liam is analyzing a project with cash flows at multiple times. He knows that if the discount rate is 14%, the NPV of the project is positive. If the discount rate is 18%, the NPV is negative. Which statement MUST be true? • NPV must hit zero somewhere between 14% and 18%, so an IRR must exist between 14% and 18%

  6. Leverage and Returns • Joel’s Textboks, Inc. is currently an unlevered company. The current expected return on equity is 12%. If the cost of debt is 7 an Joel’s Textbooks decides to move 60% of the value of the company to bonds, what will the new return on equity be? • 12% = 0.6 * 7% + 0.4 * X • 12% = 4.2% + 0.4 * X • 7.8% = 0.4 * X • X = 19.5%

  7. Value of Options • Tlinches stock sells for $75/share today. The value of the stock in one year will come from a uniform distribution, with lower bound $71 and upper bound $81. What is the PV of a European put option with exercise price $67 and expiration date one year from today? • The stock will never be below $67 on the expiration date, so PV = $0

  8. Decreasing Dividends • Silly Spring Stock plans on paying out a dividend of $1/share today. Each subsequent annual dividend will be 1 cent less than the previous, until the dividend paid is 1¢. After the 1¢ dividend is paid, the company will go out of business. The effective annual discount rate is 10%. What is the PV of this stock? • Upper bound on PV is a dividend that goes on forever and does not decrease • Upper bound: $1 / .1 = $10 • All other answer choices are greater than $10, so it is the closest answer

  9. Calculating Dividends • Slimy Mushroom Music (SMM) stock currently sells for $50/share. The beta value for SMM is 1. The risk-free rate of return is currently 4%, and the risk premium is currently 7%. SMM will pay a yearly dividend of $X every year forever, starting one year from today. What is X? • Discount rate for SMM is .04 + .07 = .11 • $50 = X / .11 • X = $5.50

  10. Peak Housing Prices • From lecture, in what year did median housing prices peak in the United States? • Answer: 2006 • From lecture slides

  11. Corporate Regulations • Which of the following was NOT listed as a corporate regulation in the textbook? • Answer: Prohibition of initial public offerings (IPOs) on American stock markets • We talked about recent IPOs in class

  12. Geometric Average Return • A stock is worth $500/share today. One year ago, the value was $450/share. Two years ago, the value was $400/share. Three years ago, the value was $350/share. What has been the geometric average rate of return over the last three years? • (500/350)1/3 – 1 = 12.625%

  13. Bonds with Changing Yields • A zero-coupon bond is purchased for $1,000 at 11 am today, with a face value of $1,250 to be paid two years from today. Later today, at 1:30 pm, the yield to maturity (compounded on an annual basis) changes to 10%. How much does the value of the bond go up between 11 am and 1:30 pm today? • Value at 11 am = $1,000 • Value at 1:30 pm = 1,250/(1.1)2 = $1,033.06 • Bond value went up by $33.06

  14. Portfolio Standard Deviation • Stocks M and N are uncorrelated with each other. Stock M has an expected return of 5% and a standard deviation of 6%. Stock N has an expected return of 8% and a standard deviation of 10%. What is the standard deviation of a portfolio that has 30% of stock M and 70% of stock N?

  15. Portfolio Standard Deviation • Variance = (.3)2(.06)2 + 0 + (.7)2(.1)2 = .000324 + 0 + .0049 = .005224 • Standard deviation = .072277 (Since M and N are uncorrelated)

  16. Option Value • U2B4 stock was valued at $60 three years ago, and has had a geometric average rate of return of 5% over the past three years. Savannah bought an American call option with an exercise price of $68 a few weeks ago. The option expires today. What is the current value of the option if Savannah’s effective annual discount rate is 6%? • Current value = 60 * (1.05)3 = $69.46 • Value of option = 69.46 – 68 = $1.46

  17. Project Discount Rates • Janie’s Noodles currently has a company-wide beta value of 2. A stock whose beta is 1 has an expected return of 5%. The risk-free rate is currently 3%. A new project that Janie’s Noodles is considering has a beta value of 1.5. What is the relevant discount rate for this new project? • Risk premium = 5% - 3% = 2% • X = 3% + 1.5 * (2%) = 6%

  18. Non-annual Discount Rates • If the effective annual discount rate is 10%, what is the effective discount rate for 9 months? • Rate every 3 months = (1.1)1/4 – 1 = 2.41137% • Rate every 9 months = (1.0241137)3 – 1= 7.40995%

  19. Probability of Positive Option Value • A stock is currently priced at $65. The stock’s price will change by $2 per year every year, starting 11 months from today. The stock’s price will go up with 50% probability and down with 50% probability. Each change in the stock’s price is independent of all other price changes.

  20. Probability of Positive Option Value • Sonia buys a European option with an exercise price of $70 and an expiration date in five years. What is the probability that the option will have a positive value on the expiration date? • Only times with positive value: UUUUU, DUUUU, UDUUU, UUDUU, UUUDU, UUUUD • 25 = 32 possible outcomes, equal chances • 6/32 = 3/16 probability of positive value

  21. Value of Bond Guarantees • Jenny’s Furniture Manufacturing (JFM) is considering moving away from California. In order to try to keep JFM in the state, the government is offering to guarantee any bond repayment for bonds issued by JFM this year. JFM’s normal cost of debt capital is 15%.

  22. Value of Bond Guarantees • If JFM is able to issue $1,000,000 in bonds at a coupon rate of 9%, what is the value of this implicit subsidy? • NPV = 1,000,000 – [90,000/1.15 + 90,000/1.152 + 90,000/1.153+ 90,000/1.154+ 90,000/1.155+ 1,090,000/1.156] • NPV = 1,000,000 – [78,260.87 + 68,052.93 + 59,176.46 + 51,457.79 + 44,745.91 + 471,237.08 • NPV = $227,068.96

  23. Standard Deviation of Stock Returns • Suppose the returns of a stock over a four-year period are 15%, -6%, 19%, and 22%. Find the standard deviation of this sample. • Arithmetic average = (15+(-6)+19+22) / 4 = 12.5% • Variance= 1/3 * [(.15-.125)2 + (-.06-.125)2 + (.19-.125)2 + (.22-.125)2] • Variance= 1/3 * [.000625 + .034225 + .004225 + .009025] = 1/3*[.0481] = .01603 • Standard deviation = (.01603)1/2 = .126623 • Standard deviation = 12.6623%

  24. Annuity Payments • An annuity will pay $5 one year from today, $(5+Y) two years from today, $(5+2Y) three years from today, and $(5+3Y) four years from today. The PV of this annuity is $15, and the effective annual discount rate is 20%. Find Y. • 5/1.2 + (5+Y)/1.22 + (5+2Y)/1.23 + (5+3Y)/1.24 = 15 • 12.9437 + 3.29861*Y = 15 • 3.29861*Y = 2.0563 • Y = 0.623384

  25. Equivalent Annual Cost • The Jack Crock Griller costs $6,000 to purchase today and lasts for 9 years. Annual maintenance costs are required. The first annual cost will be $100 one year from today. Each subsequent maintenance cost will be 6% higher than the previous year’s cost. The final cost will be 8 years from today.

  26. Equivalent Annual Cost • What is the equivalent annual cost if the stated annual discount rate is 12%, compounded every six months? • EAIR = (1 + .12/2)2 – 1 = 12.36% • PV of costs = 6000 + 100*[1/(.1236-.06) – 1/(.1236-.06) * (1.06/1.1236)8] = $6,585.30 • EAC: 6585.30 = X/.1236 * [1 – 1/(1.12369)] 6585.30 = 5.25612 * X • X = $1,252.88

  27. Funding Future Withdrawals • Wendy has two financial objectives. She would like to withdraw $500,000 every year for 30 years, starting 5 years from now. She would like to donate $3,000,000 to a charity 30 years from today. Wendy will make a single deposit in 3 years to exactly fund these goals. How much will she have to deposit if the EAIR = 4%?

  28. Funding Future Withdrawals • First objective:FVYear 4 = 500,000/.04 * [1 – 1/(1.0430)] = $8,646,016.65 • FVYear3= 8,646,016.65/1.04 = $8,313,477.55 • Second objective:FVYear3 = 3,000,000/(1.0427) =$1,040,449.71 • Total in year 3 = $9,353,927.26

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