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Time Value of Money: Future Value and Present Value Calculations

Learn about the time value of money and how to calculate future value and present value using different compounding periods. Understand the importance of cash flow timing and how it affects investment decisions.

josephnelson
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Time Value of Money: Future Value and Present Value Calculations

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  1. Chapter 4 Time Value of Money

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

  3. Time Value Basic Concepts Time lines Future value / Present value of lump sum FV / PV of annuity Perpetuities Uneven CF stream Compounding periods Nominal / Effective / Periodic rates Amortization Rates of return Amortization

  4. 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) Market interest rates Firm’s debt/equity mix Cost of debt Cost of equity Market risk aversion Firm’s business risk

  5. 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.

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

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

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

  9. Compounding $$ • Growing Money to accumulate value in future • Solve for Future Value (FV) • Mathematical process (multiply)

  10. 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.

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

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

  13. 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

  14. Four Ways to Find FVs Step-by-step approach using time line (as shown in Slides 7-10). Solve the equation with a regular calculator (formula approach). Use a financial calculator. Use a spreadsheet.

  15. Financial calculator: HP10BII Adjust display brightness: hold down ON and push + or –. Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3). To temporarily show all digits, hit Orange Shift key, then DISP, then =.

  16. HP10BII (Continued) To permanently show all digits, hit ORANGE shift, then DISP, then . (period key). Set decimal mode: Hit ORANGE shift, then ./, key. Note: many non-US countries reverse the US use of decimals and commas when writing a number.

  17. HP10BII: Set Time Value Parameters To set END (for cash flows occurring at the end of the year), hit ORANGE shift key, then BEG/END. To set 1 payment per period, hit 1, then ORANGE shift key, then P/YR.

  18. Financial Calculator Solution Financial calculators solve this equation: FVN + PV (1+I)N = 0. There are 4 variables. If 3 are known, the calculator will solve for the 4th.

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

  20. After 4 years • PV = $100 • N = 4 • i = 10% • FV = ? = $146.41

  21. After 4 years, but different compounding per year Semi-annual Quarterly PV = $100 N = 4 yrs x 4 = 16 periods i = 10% / 4 = 2.5% per period FV = ? = • PV = $100 • N = 4 yrs x 2 = 8 periods • i = 10% / 2 = 5% per period • FV = ? =

  22. Spreadsheet Solution Use the FV function: see spreadsheet in Ch04 Mini Case.xls = FV(I, N, PMT, PV) = FV(0.10, 3, 0, -100) = 133.10

  23. Discounting $$ • Money needed today to accumulate x$ value in future • Solve for Present Value (PV) • Mathematical process (divide)

  24. What’s the PV of $110 due in 1 year if I/YR = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% 110 PV = ?

  25. Solve FVN = PV(1 + I )N for PV N FVN 1 PV = = FVN (1+I)N 1 + I 1 110 PV = 1.10 PV= $110

  26. What’s the PV of $110 due in 1 year if I/YR = 10%? Semi-annually FV = $110 N = 1 yr x 2 = 2 periods i = 10% / 2 = 5.0% per period FV = ? = • FV = $110 • N = 1 yr • i = 10% • PV = ? =

  27. 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 = ?

  28. 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

  29. 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.

  30. Spreadsheet Solution Use the PV function: see spreadsheet in Ch04 Mini Case.xls = PV(I, N, PMT, FV) = PV(0.10, 3, 0, 100) = -75.13

  31. Cash Flow signs Investing $ today Borrowing $ today Take in (borrow) $ today in present to use now, then repay with interest in the future. Pay interest (expense), plus principal PV = + FV = <-> • Outlay (invest) $ today in present to earn greater return in the future. • Earn interest (revenue), plus principal • PV = <-> • FV = +

  32. Periods or Interest Rate unknown Solve for N Solve for i Deposit $100 today. You need $148.45 in 4 years. What’s the annual interest rate if the money is compounded quarterly? • Invest $100 today earning 10% & need $146.41. How long will it take

  33. Periods or Interest Rate unknown Solve for N Solve for i Deposit $100 today. You need $148.45 in 4 years. What’s the annual interest rate if the money is compounded quarterly? • Invest $100 today earning 10% & need $146.41. How long will it take

  34. Finding the Time to Double 0 1 2 ? 20% 2 -1 FV = PV(1 + I)N Continued on next slide

  35. 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

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

  37. Spreadsheet Solution Use the NPER function: see spreadsheet in Ch04 Mini Case.xls = NPER(I, PMT, PV, FV) = NPER(0.10, 0, -1, 2) = 3.8

  38. Solve for 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%

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

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

  41. Ordinary Annuity vs. Annuity Due • Series of equal payments made at fixed intervals or specified number of periods. • Ordinary Annuity @ end • Annuity Due @ beg

  42. Ordinary Annuity vs. Annuity Due Ordinary Annuity 0 1 2 3 I% $100=PMT $100 $100 Annuity Due 0 1 2 3 I% $100=PMT $100 $100

  43. 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

  44. 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: (1+I)N-1 = PMT I (1+0.10)3-1 = $331 = $100 0.10

  45. Financial Calculator Formula for Annuities Financial calculators solve this equation: (1+I)N-1 FVN + PV(1+I)N + PMT = 0 I There are 5 variables. If 4 are known, the calculator will solve for the 5th.

  46. 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.

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

  48. 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

  49. PV Annuity Formula The present value of an annuity with N periods and an interest rate of I can be found with the following formula: 1 1 = PMT − I I (1+I)N 1 1 = $100 − = $248.69 0.1 0.1(1+0.1)3

  50. 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.

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