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Chapter 4

Chapter 4. Time Value of Money. Time Value Topics. Future value Present value Rates of return Amortization. Determinants of Intrinsic Value: The Present Value Equation. Net operating profit after taxes. Required investments in operating capital. −. Free cash flow (FCF). =. FCF 1.

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Chapter 4

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  1. Chapter 4 Time Value of Money

  2. Time Value Topics Future value Present value Rates of return Amortization

  3. Determinants of Intrinsic Value: The Present Value Equation Net operating profit after taxes Required investments in operating capital − Free cash flow (FCF) = FCF1 FCF2 FCF∞ ... Value = + + + (1 + WACC)1 (1 + WACC)2 (1 + WACC)∞ Weighted average cost of capital (WACC) Cost of debt Cost of equity

  4. Why is timing important? You are asked to choose from the following options: 1. Receive $1 million today 2. Receive $1 million 10 years from now Would you choose 1 or 2?

  5. Money has time value • Most people prefer to receive it sooner rather than later because they place a higher value on the cash received earlier.

  6. Time value of money: Practical relevance Examples • Retirement plan • Mortgage payment • Pricing a financial securities • Helping your company to decide which project to undertake

  7. Time lines show timing of cash flows. 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.

  8. Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Year I% 100

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

  10. Time line for uneven CFs 0 1 2 3 I% -50 100 75 50

  11. Preparing BAII Plus for use • Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’. • Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’. • Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

  12. FV of an initial $100 after3 years (I = 10%) 0 1 2 3 10% 100 FV = ? Finding FVs (moving to the right on a time line) is called compounding.

  13. After 1 year FV1 = PV + INT1 = PV + PV (I) = PV(1 + I) = $100(1.10) = $110.00

  14. After 2 years FV2 = FV1(1+I) = PV(1 + I)(1+I) = PV(1+I)2 = $100(1.10)2 = $121.00

  15. After 3 years FV3 = FV2(1+I)=PV(1 + I)2(1+I) = PV(1+I)3 = $100(1.10)3 = $133.10 In general, FVN = PV(1 + I)N

  16. Four Ways to Find FVs Step-by-step approach using time line (as shown in Slides 12-15). Solve the equation with a regular calculator (formula approach). Use a financial calculator. Use a spreadsheet.

  17. Financial Calculator Solution Financial calculators solve this equation: FVN= PV (1+I)N There are 5 variables. If 4 are known, the calculator will solve for the 4th.

  18. Here’s the setup to find FV INPUTS 3 10 -100 0 N I/YR PV PMT FV 133.10 OUTPUT

  19. Spreadsheet Solution Use the FV function = FV(I, N, PMT, PV) = FV(0.10, 3, 0, -100) = 133.10

  20. What’s the PV of $100 due in 3 years if I/YR = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% 100 PV = ?

  21. Solve FVN = PV(1 + I )N for PV N FVN 1 PV = = FVN (1+I)N 1 + I 3 1 PV = $100 1.10 = $100(0.7513) = $75.13

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

  23. Spreadsheet Solution Use the PV function: = PV(I, N, PMT, FV) = PV(0.10, 3, 0, 100) = -75.13

  24. Finding the Time to Double 0 1 2 ? 20% 2 -1 Q: if deposit $1 today, and i=20%, when will it double?

  25. Time to Double (Continued) $2 = $1(1 + 0.20)N (1.2)N = $2/$1 = 2 N LN(1.2) = LN(2) N = LN(2)/LN(1.2) N = 0.693/0.182 = 3.8

  26. Financial Calculator Solution INPUTS 20 -1 0 2 N I/YR PV PMT FV 3.8 OUTPUT

  27. Spreadsheet Solution Use the NPER function: = NPER(I, PMT, PV, FV) = NPER(0.10, 0, -1, 2) = 3.8

  28. Finding the interest rate 0 1 2 3 ?% 2 -1 FV = PV(1 + I)N $2 = $1(1 + I)3 (2)(1/3) = (1 + I) 1.2599 = (1 + I) I = 0.2599 = 25.99%

  29. Financial Calculator INPUTS 3 -1 0 2 N I/YR PV PMT FV 25.99 OUTPUT

  30. Spreadsheet Solution Use the RATE function: = RATE(N, PMT, PV, FV) = RATE(3, 0, -1, 2) = 0.2599

  31. Exercises Suppose you deposit $150 in an account today and the interest rate is 6 percent p.a.. How much will you have in the account at the end of 33 years? You deposited $15,000 in an account 22 years ago and now the account has $50,000 in it. What was the annual rate of return that you received on this investment? You currently have $38,000 in an account that has been paying 5.75 percent p.a.. You remember that you had opened this account quite some years ago with an initial deposit of $19,000. You forget when the initial deposit was made. How many years (in fractions) ago did you make the initial deposit?

  32. Perpetuity 1 Perpetuity: a stream of equal cash flows ( C ) that occur at the end of each period and go on forever. PV of perpetuity =

  33. Perpetuity 2 • We use the idea of a perpetuity to determine the value of • A preferred stock • A perpetual debt

  34. Perpetuity questions • Suppose the value of a perpetuity is $38,900 and the discount rate is 12 percent p.a.. What must be the annual cash flow from this perpetuity? Verify that C = $4,668. • An asset that generates $890 per year forever is priced at $6,000. What is the required rate of return? Verify that r = 14.833 %

  35. Ordinary Annuity • Ordinary annuity: a cash flow stream where a fixed amount is received at the end of every period for a fixed number of periods.

  36. 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

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

  38. Spreadsheet Solution Use the FV function: = FV(I, N, PMT, PV) = FV(0.10, 3, -100, 0) = 331.00

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

  41. Spreadsheet Solution Use the PV function: = PV(I, N, PMT, FV) = PV(0.10, 3, 100, 0) = -248.69

  42. Annuity, find FV You open an account today with $20,000 and at the end of each of the next 15 years, you deposit $2,500 in it. At the end of 15 years, what will be the balance in the account if the interest rate is 7 percent p.a.? PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?

  43. Annuity, find I/Y You lend your friend $100,000. He will pay you $12,000 per year for the ten years and a balloon payment at t = 10 of $50,000. What is the interest rate that you are charging your friend? PV=-100,000, FV=50,000, PMT=12,000, N = 10, I/Y=?

  44. Annuity, find PMT Next year, you will start to make 35 deposits of $3,000 per year in your Individual Retirement Account (so you will contribute from t=1 to t=35). With the money accumulated at t=35, you will then buy a retirement annuity of 20 years with equal yearly payments from a life insurance company (payments from t=36 to t=55). If the annual rate of return over the entire period is 8%, what will be the annual payment of the annuity?

  45. Annuity Due • Annuity due: a cash flow stream where a fixed amount is received at the beginning of every period for a fixed number of periods.

  46. Ordinary Annuity vs. Annuity Due Ordinary Annuity 0 1 2 3 I% PMT PMT PMT Annuity Due 0 1 2 3 I% PMT PMT PMT

  47. Ordinary Annuity vs. Annuity Due $300 $300 $300 T = 1 T = 3 T = 0 T = 2 $300 $300 $300 T = 3 T = 0 T = 1 T = 2

  48. a relationship between ordinary annuity and annuity due? PV of annuity due = (PV of ordinary annuity) x (1 + r) FV of annuity due = (FV of ordinary annuity) x (1 + r)

  49. Find the FV and PV if theannuity were an annuity due. 0 1 2 3 10% 100 100 100

  50. 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) = $364.10

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