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# Future value Present value Annuities

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1. Chapter 2 Time Value of Money • Future value • Present value • Annuities • TVM is one of the most important concepts in finance: A dollar today is worth more than a dollar in the future. Why is this true??How does this affect us?? • HW: 2-1 through 2-5, pg 84 –B&E

2. Time lines show timing of cash flows. 0 1 2 3 i% CF0 CF1 CF2 CF3 Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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

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

5. 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

6. What’s the FV of an initial \$100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? Finding FVs (moving to the right on a time line) is called compounding.

7. 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)(1+i) = PV(1+i)2 = \$100(1.10)2 = \$121.00.

8. 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.

9. Future Value Relationships

10. Multi-Period Compounding Examples You put \$400 into an account that pays 8 % interest compounded annually, quarterly. How much will be in your account in 6 years? Set: P/YR = 1, END, Format ->Dec=4 ,CLR TVM N=6, I/Y=8, PV=-400 -> FV= 634.75 Interest is compounded 4 times per year, so: 8 % / 4 = 2 % interest rate per period 6 yrs x 4 = 24 periods FV = \$643.37 What do you get if you compound daily instead? I//Y=8/365 (not .08!!!!), N=6*365 FV = \$646.40

11. Three Ways to Find FVs • Solve the equation with a regular calculator. • Use a financial calculator. • Use a spreadsheet.

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

13. 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.

14. 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

15. 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% 100 PV = ?

16. Solve FVn = PV(1 + i )n for PV: 3 1    PV = \$100    1.10   = \$100 0.7513 = \$75.13.

17. 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.

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

19. Finding the Time to Double 0 1 2 ? 20% 2 -1 FV = PV(1 + i)n \$2 = \$1(1 + 0.20)n (1.2)n = \$2/\$1 = 2 nLN(1.2) = LN(2) n = LN(2)/LN(1.2) n = 0.693/0.182 = 3.8. e=2.7183 , ln(e)=1 10^2=100, LOG(100)=2 , Rule of 72 >> 72/periods = IPER

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

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

22. 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%.

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

24. Spreadsheet Solution • Use the RATE function: • = RATE(Nper, Pmt, PV, FV) • = RATE(3, 0, -1, 2) = 0.2599

25. What’s the difference between an ordinaryannuity and an annuitydue? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT PV FV

26. 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

27. Penny :Super TVM Question Suppose you can take a penny and double your money every day for 30 days. What will you be worth? Wait!! Guess a value before you calculate. Iper=100%, n=30, pv=-.01, pmt=0, Fv = \$10,737,418.24 1cent,2,4,8,16,32,64,128,256,512,1024,2048,4096 cents…

28. 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:

29. 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.

30. 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.

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

32. 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, FV=0

33. 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:

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

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

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

37. 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

38. 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 FVA3 = \$364.10.

39. 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)

40. What is the PV of this uneven cashflow stream? 4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV

41. Input in “CFLO” register: CF0 = 0 (Typically initial investment so –ve) CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 • Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

42. Spreadsheet Solution A B C D E 1 0 1 2 3 4 2 100 300 300 -50 3 530.09 Excel Formula in cell A3: =NPV(10%,B2:E2)

43. Distinguishing Between Different Interest Rates kSIMPLE = Simple (Quoted) Rateused to compute the interest paid per period APR = Annual Percentage Rate= kSIMPLEperiodic rate X the number of periods per year EAR = Effective Annual Ratethe annual rate of interest actually being earned

44. Nominal rate (iNom) • Stated in contracts, and quoted by banks and brokers. • Not used in calculations or shown on time lines • Periods per year (m) must be given. • Examples: • 8%; Quarterly • 8%, Daily interest (365 days)

45. Periodic rate (iPer ) • iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. • Used in calculations, shown on time lines. • Examples: • 8% quarterly: iPer = 8%/4 = 2%. • 8% daily (365): iPer = 8%/365 = 0.021918%.

46. Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

47. FV Formula with Different Compounding Periods (e.g., \$100 at a 12% nominal rate with semiannual compounding for 5 years) mn i   Nom FV = PV 1 . +     n m 2x5 0.12   FV = \$100 1 +     5S 2 = \$100(1.06)10 = \$179.08. With annual cmpndg the A=\$176.23

48. FV of \$100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= \$100(1.12)5 = \$176.23. FV(Semiannual)= \$100(1.06)10=\$179.08. FV(Quarterly)= \$100(1.03)20 = \$180.61. FV(Monthly)= \$100(1.01)60 = \$181.67. FV(Daily) = \$100(1+(0.12/365))(5x365) = \$182.19.

49. Effective Annual Rate (EAR = EFF%) • The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding Example: Invest \$1 for one year at 12%, semiannual: FV = PV(1 + iNom/m)m FV = \$1 (1.06)2 = 1.1236. • EFF% = 12.36%, because \$1 invested for one year at 12% semiannual compounding would grow to the same value as \$1 invested for one year at 12.36% annual compounding.

50. An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. • Banks say “interest paid daily.” Same as compounded daily.