# Chapter 4

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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. 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. Simple vs Compound Interest • Simple Interest: • I = P x R x T • Interest \$ amount = Principal x Interest Rate x Time • NO accumulation of interest earning interest

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

11. 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.

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

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

14. 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

15. Four Ways to Find FVs Step-by-step approach using time line (as shown in Slides 11-14). Solve the equation with a regular calculator (formula approach). Use a financial calculator. Use a spreadsheet.

16. 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 =.

17. 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.

18. 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.

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

20. Here’s the setup to find FV INPUTS N I/YR PV PMT FV OUTPUT Clearing automatically (shift Clear-All) sets everything to 0, but for safety enter PMT = 0.

21. After 4 years • PV = • N = • i = • FV = ? =

22. After 4 years, but different compounding per year Semi-annual Quarterly PV = \$100 N = periods i = per period FV = ? = • PV = \$100 • N = 4 yrs • i = 10% • FV = ? =

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

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

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

26. 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

27. What’s the PV of \$110 due in 1 year if I/YR = 10%? Annual Compounding Semi-annually FV = \$ N = periods i = per period PV = ? = • FV = \$ • N = • i = • PV = ? =

28. 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 = ?

29. 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

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

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

32. 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 = +

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. 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

35. 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? RATE(Nper,Pmt,pv,fv,type) =RATE(4*4,0,-100,148.45)*4 =(2.5%)*4 = 10% • Invest \$100 today earning 10% & need \$146.41. How long will it take? • NPER(rate,pmt,pv,fv,type) • =NPER(0.1,0,-100,146.41)=4

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

37. 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

38. Financial Calculator Solution INPUTS N I/YR PV PMT FV OUTPUT

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

40. 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%

41. Financial Calculator INPUTS N I/YR PV PMT FV OUTPUT

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

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

44. 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

45. What’s the FV of a 3-year ordinary annuity of \$100 at 10%? 0 1 2 3 10% 100 100 100 ??? ??? FV = ???

46. 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

47. 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 (PV, PMT, N, I, FV. If 4 are known, calculator solves for 5th. Pay attention to inflows & outflows (signs).

48. Financial Calculator Solution INPUTS PMT N PV FV I/YR OUTPUT Have payments but no lump sum PV, so enter 0 for present value.

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

50. What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 ??? ??? ??? ????? = PV