Note for multiple-choice questions: Choose the closest answer

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

 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

 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

 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

 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

 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…

  50,000 (1 + IRR) 3 IRR = 0.0507
= 58,000

 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 _____.

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

 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

 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 _____.

 PV, set equal to 0    0 = -500 + 200/1.07 + 250/1.07
2 0 = -94.72 + X/1.07
4 X = $124.16
+ X/1.07
4

 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.

 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

 October 28, 2012, 5.5% interest rate, compounded continuously   1.5 years of compounding FV = $1,000 * e 1.5(0.055) = $1,085.99

 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

   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%

 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.)

 What is the NPV of all of the costs of Machine A?
  1000 + 200/1.06
2 $1,504.34 + 200/1.06
3 + 200/1.06
4

 What is the NPV of all of the costs of Machine B?
  1500 + 100/1.06 + 100/1.06
2 + 100/1.06
4 + 100/1.06
5 $1,921.24
+ 100/1.06
3

 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

 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

 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

 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 41,000/1.071
T = 500/.071 (1 – 1/1.071
T ) 6.822 = 1.071
T T = log 6.822 / log 1.071 = 27.99
Round to the nearest # of years: T = 28

 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.)

 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

 What is the NPV of delaying the introduction of Super Water by two years?
 NPV = -0.3 + 0.6*5/1.05
2 + 0.4*2/1.05
2  NPV = 3.1467

 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