1 / 116

CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics. Future value Present value Rates of return Amortization Annuities, AND Many Examples. MINICASE 2 SIMPLE?. p. 88. Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS.

elan
Download Presentation

CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CHAPTER 2Discounted Cash Flow AnalysisTime Value of MoneyFinancial Mathematics • Future value • Present value • Rates of return • Amortization • Annuities, AND • Many Examples

  2. MINICASE 2SIMPLE? p. 88 Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS.

  3. MINICASE 2 • Why is financial mathematics (time value of money) so important in financial analysis?

  4. a.Time lines show timing of cash flows. ALWAYS A GOOD IDEA TO DRAW A TIME LINE. 0 1 2 3 i% CF0 CF1 CF2 CF3 Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1, or the beginning of Period 2; and so on.

  5. Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Years i% 100

  6. Time line for an ordinary annuity of $100 for 3 years. 0 1 2 3 i% 100 100 100

  7. Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 0 1 2 3 i% -50 100 75 50

  8. b(1) What’s the FV of an initial$100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? Finding FVs is compounding.

  9. b(1) What’s the FV of an initial$100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? 110 ? Finding FVs is compounding.

  10. After 1 year FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years FV2 = FV1(1 + i) = PV(1 + i)2 = $100(1.10)2 = $121.00.

  11. After 3 years FV3 = PV(1 + i)3 = 100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n.

  12. Four Ways to Find FVs • Solve the equation with a regular calculator • Use tables • Use a financial calculator • Use a spreadsheet

  13. USING TABLES: See handout • 3 PERIODS • 10 % • = 1.3310 • times 100 = $133.10 • SAY GOOD-BYE TO USING TABLES!

  14. Financial Calculator Solution Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th.

  15. Here’s the setup to find FV: 3 10 -100 0 N I/YR PV PMT FV INPUTS OUTPUT 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END

  16. b(2) What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% PV = ? 100

  17. Solve FVn = PV(1 + i )n for PV: PV = FVn (1 + i)n n PV = $100/() = = $100(0.7513) = $75.13. 3 1.10

  18. Financial Calculator Solution 3 10 0 100 N I/YR PV PMT FV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

  19. EXCEL SOLUTION • LOOK AT FUNCTION’S PAGE FOR EXCEL/LOTUS.

  20. Spreadsheet Solution • Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. • = FV(Rate, Nper, Pmt, PV) • = FV(0.10, 3, 0, -100) = 133.10

  21. Spreadsheet Solution • Use the PV function: see spreadsheet. • = PV(Rate, Nper, Pmt, FV) • = PV(0.10, 3, 0, 100) = -75.13

  22. c. If sales grow at 20% per year, how long before sales double? Solve for n: Time line ? FVn = PV(1 + i)n 2 = 1(1.20)n (1.20)n = 2 n ln(1.20) = ln 2 n(0.1823) = 0.6931 n = 0.6931/0.1823 = 3.8 years.

  23. 20 -1 0 2 N I/YR PV PMT FV 3.8 Beware:Some Calculators round up. INPUTS OUTPUT Graphical Illustration: FV 2 3.8 1 Years 0 1 2 3 4

  24. Spreadsheet Solution • Use the NPER function: see spreadsheet. • = NPER(Rate, Pmt, PV, FV) • = NPER(0.20, 0, -1, 2) = 3.8 Correction

  25. ADDITIONAL QUESTION • A FARMER CAN SPEND $60/ACRE TO PLANT PINE TREES ON SOME MARGINAL LAND. THE EXPECTED REAL RATE OF RETURN IS 4%, AND THE EXPECTED INFLATION RATE IS 6%. WHAT IS THE EXPECTED VALUE OF THE TIMBER AFTER 20 YEARS?

  26. ADDITIONAL QUESTION • Bill Veeck once bought the Chicago White Sox for $10 million and then sold it five years later for $20 million. In short, he doubled his money in five years. What compound rate of return did Veeck earn on his investment?

  27. RULE OF 72 • A good approximation of the interest rate--or number of years--required to double your money. • n * krequired to double = 72 • In this case, • 5 * krequired to double = 72 • k = 14.4 • Correct answer was 14.87, so for ball-park approximation, use Rule of 72.

  28. ADDITIONAL QUESTION • John Jacob Astor bought an acre of land in Eastside Manhattan in 1790 for $58. If average interest rate is 5%, did he make a good deal?

  29. d. What’s the difference between an ordinary annuity and an annuity due?

  30. d. What’s the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT 36

  31. HINT • ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.

  32. e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 10% 100 100 100 FV =

  33. e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331

  34. FV Annuity Formula • The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

  35. Financial Calculator Formula for Annuities Financial calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Correct but confusing!

  36. Financial Calculator Solution INPUTS 3 10 0 -100 331.00 N I/YR PV PMT FV OUTPUT Have payments but no lump sum PV, so enter 0 for present value.

  37. Spreadsheet Solution • Use the FV function: see spreadsheet. • = FV(Rate, Nper, Pmt, Pv) • = FV(0.10, 3, -100, 0) = 331.00

  38. e(2). What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 _____ = PV

  39. What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.69 = PV

  40. 3 10 100 0 -248.69 INPUTS N PV PMT FV I/YR OUTPUT Have payments but no lump sum FV, so enter 0 for future value.

  41. Spreadsheet Solution • Use the PV function: see spreadsheet. • = PV(Rate, Nper, Pmt, Fv) • = PV(0.10, 3, 100, 0) = -248.69

  42. e(3). Find the FV and PV if theannuity were an annuity due. 0 1 2 3 10% 100 100 100

  43. Could, on the 12C, switch from “End” to “Begin”; i.e.f Begin. Then enter variables to find PVA3 = $273.55. 3 10 100 0 -273.55 INPUTS N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.

  44. Another HINT • FV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE FV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k) (slide 30) • PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k)

  45. HINT, illlustrated • The PV of this regular annuity was 248.69. • Multiply this by (1 + .10), and you get: 273.55, the PV of the annuity due. • This avoids the necessity of having to switch from end to begin.

  46. PV and FV of Annuity Due vs. Ordinary Annuity • PV of annuity due: • = (PV of ordinary annuity) (1+i) • = (248.69) (1+ 0.10) = 273.56 • FV of annuity due: • = (FV of ordinary annuity) (1+i) • = (331.00) (1+ 0.10) = 364.1

  47. Switch from “End” to “Begin”. Then enter variables to find PVA3 = $273.55. INPUTS 3 10 100 0 -273.55 N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.

  48. Excel Function for Annuities Due Change the formula to: =PV(10%,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(10%,3,-100,0,1)

  49. EXCEL SOLUTION

  50. (f) What is the PV of this uneven cashflow stream? 0 1 2 3 4 10% 100 300 300 -50 ______ = PV

More Related