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Test 1 solution sketches

Test 1 solution sketches. Note for multiple-choice questions: Choose the closest answer. Present value calculation. Assume that you receive $600 annually forever. Assume that the effective annual discount rate is 8%. Determine the present value given the following assumptions.

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Test 1 solution sketches

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

  2. Present value calculation • Assume that you receive $600 annually forever. Assume that the effective annual discount rate is 8%. Determine the present value given the following assumptions.

  3. Present value calculation • You receive the first payment today • Perpetuity formula assumes first payment is made one year from today • We must add in an additional payment today • Solution: PV = 600 + 600 / .08 = 8100

  4. Present value calculation(This was a hard question) • You receive the first payment one year from today, but you receive a payment of $300 every six months • This is trickier to do: Two ways to do it • Perpetuity every six months (and subtract first payment) • Discount the second payment each year before calculating the PV of the annuity

  5. You receive the first payment one year from today, but you receive a payment of $300 every six months • Method 1: Perpetuity every six months (and subtract first payment) • Discount rate every six months is sqrt(1.08) – 1 = .03923 • PV = 300 / .03923 – 300 / 1.03923 = $7358.44

  6. You receive the first payment one year from today, but you receive a payment of $300 every six months • Method 2: Discount the second payment each year • Discounting the second payment by six months: 300 / 1.03923 = 288.68 • PV = (300 + 288.68) / .08 = $7358.44

  7. IRR • Suppose Joanne Green invests in a new technology that makes tubeless toilet paper. Her annual discount rate is 8%. Her investment today is $50,000. The only positive cash flow she receives is three years from now, for $58,000. Her annual internal rate of return for this project is…

  8. IRR • 50,000 (1 + IRR)3 = 58,000 • IRR = 0.0507

  9. Real payments • Billy Bo Bob is set to receive a nominal payment of $100,000 six years from now. There is no inflation over the next year, followed by 5% annual inflation each of the following five years. The real payment six years from now is _____.

  10. Real payments • No factoring of inflation the first year, 5% each of the following five years • Real payment is 100,000 / (1.05)5 = $78,352

  11. Real interest rate • If Treasury bonds have an effective annual inflation rate of 7% and the annual nominal interest rate is 8.9%, then the annual real rate of interest is _____. • 1 + nominal = (1 + real) (1 + inflation) • 1.089 = (1.07) (1 + h) • h = .01776

  12. Final payment • Liv invests $500 in a project today, and receives $200 one year from today and $250 two years from today. Her NPV from the project is $0, and she receives one final payment four years from today. If her annual discount rate is 7%, then the final payment is _____.

  13. Final payment • PV, set equal to 0 • 0 = -500 + 200/1.07 + 250/1.072 + X/1.074 • 0 = -94.72 + X/1.074 • X = $124.16

  14. Future value • Today is April 28, 2011. You invest $1,000 today. Find the future values on the following dates, given the stated nominal annual interest rates and frequency of compounding.

  15. Future value • April 28, 2021, 3% interest rate, compounded monthly • Monthly stated interest rate of 3% annually is equal to 0.25% monthly • 10 years of compounding = 120 months • $1,000 today has a future value of… • $1,000 (1.0025)120 = $1,349.35

  16. Future value • October 28, 2012, 5.5% interest rate, compounded continuously • 1.5 years of compounding • FV = $1,000 * e1.5(0.055) = $1,085.99

  17. Twice-a-year compounding • You invest $50,000 today and the future value 12 years from now is $100,000. Interest is compounded twice per year. The stated annual interest rate is _____. • Assume r is stated annual interest rate • Then r/2 is stated interest rate every 6 months • 50,000 (1 + (r/2))24= 100,000 • (1 + (r/2))24= 2 • R = 0.0586

  18. Loss on an investment • Suppose that I purchase a rare 1909-S VDB one-cent coin today for $2,000. In five years, I sell it for $1,500. If I compound my rate of return annually, what is my annual rate of return on this investment? • 2,000 (1 + r)5 = 1,500 • r = -0.0559 = -5.59%

  19. PROBLEM: Machines • You have been asked to analyze the costs of two different machines. If you buy Machine A, you have to pay $1,000 today (year 0), and maintenance costs of $200 in each of years 2, 3, and 4. If you buy Machine B, you have to pay $1,500 today and $100 maintenance costs in each of years 1, 2, 3, 4, and 5. Machine A lasts 4 years, and Machine B lasts 5 years. The effective annual discount rate is 6%. (Note: All dollar amounts in this problem are in real terms.)

  20. Part (a) • What is the NPV of all of the costs of Machine A? • 1000 + 200/1.062 + 200/1.063 + 200/1.064 • $1,504.34

  21. Part (b) • What is the NPV of all of the costs of Machine B? • 1500 + 100/1.06 + 100/1.062 + 100/1.063 + 100/1.064 + 100/1.065 • $1,921.24

  22. Part (c) • What is the equivalent annual cost of Machine A? • $1,504.34 = C * annuity factor • To calculate annuity factor, you plug in r = .06, T = 4 • Annuity factor is 3.4651 • C = $434.14

  23. Part (d) • What is the equivalent annual cost of Machine B? • $1,921.24 = C * annuity factor • To calculate annuity factor, you plug in r = .06, T = 5 • Annuity factor is 4.2124 • C = $456.10

  24. Part (e) • If both machines can be easily replaced in the future, which machine will you buy? Explain in 15 words or less. • Buy Machine A, because the EAC is lower

  25. Bank account:How many years to $41,000? • PV = C * annuity factor • PV is $41,000 discounted by T years • C = 500 • To calculate annuity factor, you plug in r = .071 with unknown T • 41,000/1.071T = 500/.071 (1 – 1/1.071T) • 6.822 = 1.071T • T = log 6.822 / log 1.071 = 27.99 • Round to the nearest # of years: T = 28

  26. Aucks, Inc.:Super Water • Suppose that Aucks, Inc. has formulated a new drink, Super Water. If Super Water is a success, the present value of the payoff is $5 million (when the product is brought to market). If the product fails, the present value of the payoff is $2 million. If the product goes directly to market, there is a 30% chance of success and a 70% chance of failure. Alternatively, Aucks can delay the introduction of Super Water by 2 years and spend $300,000 today to test market the product. Test marketing will improve the product and increase the chance of success to 60%. The appropriate annual discount rate is 5%. (Note: Ignore previous costs in the calculations below.)

  27. Part (a) • What is the NPV of going directly to market? • All numbers below are in millions of dollars • NPV = 0.3(5) + 0.7(2) = 2.9

  28. Part (b) • What is the NPV of delaying the introduction of Super Water by two years? • NPV = -0.3 + 0.6*5/1.052 + 0.4*2/1.052 • NPV = 3.1467

  29. Part (c) • If you were advising Aucks what to do in 20 words or less, what would you say? Include any justification for your advise. • Wait, because the NPV is higher by test marketing

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