1 / 83

Future value Present value Annuities

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.

warner
Download Presentation

Future value Present value Annuities

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 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.

More Related