MGT 143 CHAP 4 TIME VALUE OF MONEY

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MGT 143 CHAP 4 TIME VALUE OF MONEY

  1. 1. Chapter 5 - The TimeChapter 5 - The Time Value of MoneyValue of Money © 2005, Pearson Prentice Hall
  2. 2. The Time Value of MoneyThe Time Value of Money Compounding andCompounding and Discounting Single SumsDiscounting Single Sums
  3. 3. We know that receiving P1 today is worthWe know that receiving P1 today is worth moremore than P1 in the future. This is duethan P1 in the future. This is due toto opportunity costsopportunity costs.. The opportunity cost of receiving P1 inThe opportunity cost of receiving P1 in the future is thethe future is the interestinterest we could havewe could have earned if we had received the P1earned if we had received the P1 sooner.sooner. Today Future
  4. 4. If we can measure this opportunityIf we can measure this opportunity cost, we can:cost, we can:
  5. 5. If we can measure this opportunityIf we can measure this opportunity cost, we can:cost, we can:  Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future (compounding)(compounding)..
  6. 6. If we can measure this opportunityIf we can measure this opportunity cost, we can:cost, we can:  Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future (compounding)(compounding).. Today ? Future
  7. 7. If we can measure this opportunityIf we can measure this opportunity cost, we can:cost, we can:  Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future (compounding)(compounding)..  Translate P1 in the future into its equivalent todayTranslate P1 in the future into its equivalent today (discounting)(discounting).. Today ? Future
  8. 8. If we can measure this opportunityIf we can measure this opportunity cost, we can:cost, we can:  Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future (compounding)(compounding)..  Translate P1 in the future into its equivalent todayTranslate P1 in the future into its equivalent today (discounting)(discounting).. ? Today Future Today ? Future
  9. 9. Compound InterestCompound Interest and Future Valueand Future Value
  10. 10. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year?
  11. 11. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year? 0 1 PV =PV = FV =FV =
  12. 12. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year? Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 1N = 1 PV = -100PV = -100 FV =FV = P106P106 00 11 PV = -100PV = -100 FV =FV =
  13. 13. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year? Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 1N = 1 PV = -100PV = -100 FV =FV = P106P106 00 11 PV = -100PV = -100 FV =FV = 106106
  14. 14. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year? Mathematical Solution:Mathematical Solution: FV = PV (FVIFFV = PV (FVIF i,ni,n )) FV = 100 (FVIFFV = 100 (FVIF .06,1.06,1 ) (use FVIF table, or)) (use FVIF table, or) FV = PV (1 + i)FV = PV (1 + i)nn FV = 100 (1.06)FV = 100 (1.06)11 == P106P106 00 11 PV = -100PV = -100 FV =FV = 106106
  15. 15. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?
  16. 16. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years? 00 55 PV =PV = FV =FV =
  17. 17. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years? Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 5N = 5 PV = -100PV = -100 FV =FV = P133.82P133.82 00 55 PV = -100PV = -100 FV =FV =
  18. 18. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years? Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 5N = 5 PV = -100PV = -100 FV =FV = P133.82P133.82 00 55 PV = -100PV = -100 FV =FV = 133.133.8282
  19. 19. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years? Mathematical Solution:Mathematical Solution: FV = PV (FVIFFV = PV (FVIF i,ni,n )) FV = 100 (FVIFFV = 100 (FVIF .06,5.06,5 ) (use FVIF table, or)) (use FVIF table, or) FV = PV (1 + i)FV = PV (1 + i)nn FV = 100 (1.06)FV = 100 (1.06)55 == PP133.82133.82 00 55 PV = -100PV = -100 FV =FV = 133.133.8282
  20. 20. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  21. 21. 0 ? PV =PV = FV =FV = Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  22. 22. Calculator Solution:Calculator Solution: P/Y = 4P/Y = 4 I = 6I = 6 N = 20N = 20 PV =PV = -100-100 FV =FV = P134.68P134.68 00 2020 PV = -100PV = -100 FV =FV = Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  23. 23. Calculator Solution:Calculator Solution: P/Y = 4P/Y = 4 I = 6I = 6 N = 20N = 20 PV =PV = -100-100 FV =FV = P134.68P134.68 00 2020 PV = -100PV = -100 FV =FV = 134.134.6868 Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  24. 24. Mathematical Solution:Mathematical Solution: FV = PV (FVIFFV = PV (FVIF i,ni,n )) FV = 100 (FVIFFV = 100 (FVIF .015,20.015,20 )) (can’t use FVIF table)(can’t use FVIF table) FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn FV = 100 (1.015)FV = 100 (1.015)2020 == P134.68P134.68 00 2020 PV = -100PV = -100 FV =FV = 134.134.6868 Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  25. 25. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  26. 26. Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years? 0 ? PV =PV = FV =FV =
  27. 27. Calculator Solution:Calculator Solution: P/Y = 12P/Y = 12 I = 6I = 6 N = 60N = 60 PV =PV = -100-100 FV =FV = P134.89P134.89 00 6060 PV = -100PV = -100 FV =FV = Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  28. 28. Calculator Solution:Calculator Solution: P/Y = 12P/Y = 12 I = 6I = 6 N = 60N = 60 PV =PV = -100-100 FV =FV = P134.89P134.89 00 6060 PV = -100PV = -100 FV =FV = 134.134.8989 Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  29. 29. Mathematical Solution:Mathematical Solution: FV = PV (FVIFFV = PV (FVIF i,ni,n )) FV = 100 (FVIFFV = 100 (FVIF .005,60.005,60 )) (can’t use FVIF table)(can’t use FVIF table) FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn FV = 100 (1.005)FV = 100 (1.005)6060 == P134.89P134.89 00 6060 PV = -100PV = -100 FV =FV = 134.134.8989 Future Value - single sumsFuture Value - single sums If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have, how much would you have in the account after 5 years?in the account after 5 years?
  30. 30. Future Value - continuous compoundingFuture Value - continuous compounding What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
  31. 31. Future Value - continuous compoundingFuture Value - continuous compounding What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with continuous compoundingcontinuous compounding, after 100 years?, after 100 years? 0 ? PV =PV = FV =FV =
  32. 32. Mathematical Solution:Mathematical Solution: FV = PV (eFV = PV (e inin )) FV = 1000 (eFV = 1000 (e .08x100.08x100 ) = 1000 (e) = 1000 (e 88 )) FV =FV = P2,980,957.P2,980,957.9999 00 100100 PV = -1000PV = -1000 FV =FV = Future Value - continuous compoundingFuture Value - continuous compounding What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
  33. 33. 00 100100 PV = -1000PV = -1000 FV =FV = P2.98mP2.98m Future Value - continuous compoundingFuture Value - continuous compounding What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with continuous compoundingcontinuous compounding, after 100 years?, after 100 years? Mathematical Solution:Mathematical Solution: FV = PV (eFV = PV (e inin )) FV = 1000 (eFV = 1000 (e .08x100.08x100 ) = 1000 (e) = 1000 (e 88 )) FV =FV = P2,980,957.P2,980,957.9999
  34. 34. Present ValuePresent Value
  35. 35. Present Value - single sumsPresent Value - single sums If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
  36. 36. 0 ? PV =PV = FV =FV = Present Value - single sumsPresent Value - single sums If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
  37. 37. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 1N = 1 FV =FV = 100100 PV =PV = -94.34-94.34 00 11 PV =PV = FV = 100FV = 100 Present Value - single sumsPresent Value - single sums If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
  38. 38. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 1N = 1 FV =FV = 100100 PV =PV = -94.34-94.34 PV =PV = -94.-94.3434 FV = 100FV = 100 00 11 Present Value - single sumsPresent Value - single sums If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
  39. 39. Mathematical Solution:Mathematical Solution: PV = FV (PVIFPV = FV (PVIF i,ni,n )) PV = 100 (PVIFPV = 100 (PVIF .06,1.06,1 ) (use PVIF table, or)) (use PVIF table, or) PV = FV / (1 + i)PV = FV / (1 + i)nn PV = 100 / (1.06)PV = 100 / (1.06)11 == P94.34P94.34 PV =PV = -94.-94.3434 FV = 100FV = 100 00 11 Present Value - single sumsPresent Value - single sums If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
  40. 40. Present Value - single sumsPresent Value - single sums If you receive P100 five years from now, what isIf you receive P100 five years from now, what is the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
  41. 41. 0 ? PV =PV = FV =FV = Present Value - single sumsPresent Value - single sums If you receive P100 five years from now, what isIf you receive P100 five years from now, what is the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
  42. 42. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 5N = 5 FV =FV = 100100 PV =PV = -74.73-74.73 00 55 PV =PV = FV = 100FV = 100 Present Value - single sumsPresent Value - single sums If you receive P100 five years from now, what isIf you receive P100 five years from now, what is the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
  43. 43. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 6I = 6 N = 5N = 5 FV =FV = 100100 PV =PV = -74.73-74.73 Present Value - single sumsPresent Value - single sums If you receive P100 five years from now, what isIf you receive P100 five years from now, what is the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%? 00 55 PV =PV = -74.-74.7373 FV = 100FV = 100
  44. 44. Mathematical Solution:Mathematical Solution: PV = FV (PVIFPV = FV (PVIF i,ni,n )) PV = 100 (PVIFPV = 100 (PVIF .06,5.06,5 ) (use PVIF table, or)) (use PVIF table, or) PV = FV / (1 + i)PV = FV / (1 + i)nn PV = 100 / (1.06)PV = 100 / (1.06)55 == P74.73P74.73 Present Value - single sumsPresent Value - single sums If you receive P100 five years from now, what isIf you receive P100 five years from now, what is the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%? 00 55 PV =PV = -74.-74.7373 FV = 100FV = 100
  45. 45. Present Value - single sumsPresent Value - single sums What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
  46. 46. 00 1515 PV =PV = FV =FV = Present Value - single sumsPresent Value - single sums What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
  47. 47. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 7I = 7 N = 15N = 15 FV =FV = 1,0001,000 PV =PV = -362.45-362.45 Present Value - single sumsPresent Value - single sums What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years from now if your opportunity cost is 7%?from now if your opportunity cost is 7%? 00 1515 PV =PV = FV = 1000FV = 1000
  48. 48. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 7I = 7 N = 15N = 15 FV =FV = 1,0001,000 PV =PV = -362.45-362.45 Present Value - single sumsPresent Value - single sums What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years from now if your opportunity cost is 7%?from now if your opportunity cost is 7%? 00 1515 PV =PV = -362.-362.4545 FV = 1000FV = 1000
  49. 49. Mathematical Solution:Mathematical Solution: PV = FV (PVIFPV = FV (PVIF i,ni,n )) PV = 100 (PVIFPV = 100 (PVIF .07,15.07,15 ) (use PVIF table, or)) (use PVIF table, or) PV = FV / (1 + i)PV = FV / (1 + i)nn PV = 100 / (1.07)PV = 100 / (1.07)1515 == P362.45P362.45 Present Value - single sumsPresent Value - single sums What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years from now if your opportunity cost is 7%?from now if your opportunity cost is 7%? 00 1515 PV =PV = -362.-362.4545 FV = 1000FV = 1000
  50. 50. Present Value - single sumsPresent Value - single sums If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5 years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of return?return?
  51. 51. 00 55 PV =PV = FV =FV = Present Value - single sumsPresent Value - single sums If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5 years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of return?return?
  52. 52. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 N = 5N = 5 PV = -5,000PV = -5,000 FV = 11,933FV = 11,933 I =I = 19%19% 00 55 PV = -5000PV = -5000 FV = 11,933FV = 11,933 Present Value - single sumsPresent Value - single sums If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5 years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of return?return?
  53. 53. Mathematical Solution:Mathematical Solution: PV = FV (PVIFPV = FV (PVIF i,ni,n )) 5,000 = 11,933 (PVIF5,000 = 11,933 (PVIF ?,5?,5 )) PV = FV / (1 + i)PV = FV / (1 + i)nn 5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)55 .419 = ((1/ (1+i).419 = ((1/ (1+i)55 )) 2.3866 = (1+i)2.3866 = (1+i)55 (2.3866)(2.3866)1/51/5 = (1+i)= (1+i) i =i = .19.19 Present Value - single sumsPresent Value - single sums If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5 years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of return?return?
  54. 54. Present Value - single sumsPresent Value - single sums Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays 9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long will it take for your account to grow to $500?will it take for your account to grow to $500? 00 PV =PV = FV =FV =
  55. 55. Calculator Solution:Calculator Solution:  P/Y = 12P/Y = 12 FV = 500FV = 500  I = 9.6I = 9.6 PV = -100PV = -100  N =N = 202 months202 months Present Value - single sumsPresent Value - single sums Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays 9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long will it take for your account to grow to P500?will it take for your account to grow to P500? 00 ?? PV = -100PV = -100 FV = 500FV = 500
  56. 56. Present Value - single sumsPresent Value - single sums Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays 9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long will it take for your account to grow to P500?will it take for your account to grow to P500? Mathematical Solution:Mathematical Solution: PV = FV / (1 + i)PV = FV / (1 + i)nn 100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN 5 = (1.008)5 = (1.008)NN ln 5 = ln (1.008)ln 5 = ln (1.008)NN ln 5 = N ln (1.008)ln 5 = N ln (1.008) 1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months
  57. 57. Hint for single sum problems:Hint for single sum problems:  In every single sum present value andIn every single sum present value and future value problem, there are fourfuture value problem, there are four variables:variables: FVFV,, PVPV,, ii andand nn..  When doing problems, you will be givenWhen doing problems, you will be given three variables and you will solve for thethree variables and you will solve for the fourth variable.fourth variable.  Keeping this in mind makes solving timeKeeping this in mind makes solving time value problems much easier!value problems much easier!
  58. 58. The Time Value of MoneyThe Time Value of Money Compounding and DiscountingCompounding and Discounting Cash Flow StreamsCash Flow Streams 0 1 2 3 4
  59. 59. AnnuitiesAnnuities  Annuity:Annuity: a sequence ofa sequence of equalequal cashcash flowsflows, occurring at the, occurring at the endend of eachof each period.period.
  60. 60.  Annuity:Annuity: a sequence ofa sequence of equalequal cashcash flows, occurring at the end of eachflows, occurring at the end of each period.period. 0 1 2 3 4 AnnuitiesAnnuities
  61. 61. Examples of Annuities:Examples of Annuities:  If you buy a bond, you willIf you buy a bond, you will receive equal semi-annual couponreceive equal semi-annual coupon interest payments over the life ofinterest payments over the life of the bond.the bond.  If you borrow money to buy aIf you borrow money to buy a house or a car, you will pay ahouse or a car, you will pay a stream of equal payments.stream of equal payments.
  62. 62.  If you buy a bond, you willIf you buy a bond, you will receive equal semi-annual couponreceive equal semi-annual coupon interest payments over the life ofinterest payments over the life of the bond.the bond.  If you borrow money to buy aIf you borrow money to buy a house or a car, you will pay ahouse or a car, you will pay a stream of equal payments.stream of equal payments. Examples of Annuities:Examples of Annuities:
  63. 63. Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  64. 64. 0 1 2 3 Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  65. 65. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 FV =FV = P3,246.40P3,246.40 Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years? 0 1 2 3 10001000 10001000 10001000
  66. 66. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 FV =FV = P3,246.40P3,246.40 Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years? 0 1 2 3 10001000 10001000 10001000
  67. 67. Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  68. 68. Mathematical Solution:Mathematical Solution: Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  69. 69. Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  70. 70. Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or) Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  71. 71. Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  72. 72. Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 =- 1 = P3246.40P3246.40 .08.08 Future Value - annuityFuture Value - annuity If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much would you have after 3 years?would you have after 3 years?
  73. 73. Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  74. 74. 0 1 2 3 Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  75. 75. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 PV =PV = P2,577.10P2,577.10 0 1 2 3 10001000 10001000 10001000 Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  76. 76. Calculator Solution:Calculator Solution: P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 PV =PV = P2,577.10P2,577.10 0 1 2 3 10001000 10001000 10001000 Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  77. 77. Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  78. 78. Mathematical Solution:Mathematical Solution: Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  79. 79. Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n )) Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  80. 80. Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n )) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or) Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  81. 81. Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n )) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii Present Value - annuityPresent Value - annuity What is the PV of $1,000 at the end of each of theWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  82. 82. Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n )) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii 11 PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 == P2,577.10P2,577.10 .08.08 Present Value - annuityPresent Value - annuity What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
  83. 83. Other Cash Flow PatternsOther Cash Flow Patterns 0 1 2 3 The Time Value of Money
  84. 84. PerpetuitiesPerpetuities  Suppose you will receive a fixedSuppose you will receive a fixed payment every period (month, year,payment every period (month, year, etc.) forever. This is an example ofetc.) forever. This is an example of a perpetuity.a perpetuity.  You can think of a perpetuity as anYou can think of a perpetuity as an annuityannuity that goes onthat goes on foreverforever..
  85. 85. Present Value of aPresent Value of a PerpetuityPerpetuity  When we find the PV of anWhen we find the PV of an annuityannuity,, we think of the followingwe think of the following relationship:relationship:
  86. 86. Present Value of aPresent Value of a PerpetuityPerpetuity  When we find the PV of anWhen we find the PV of an annuityannuity,, we think of the followingwe think of the following relationship:relationship: PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
  87. 87. Mathematically,Mathematically,
  88. 88. Mathematically,Mathematically, (PVIFA i, n ) =(PVIFA i, n ) =
  89. 89. Mathematically,Mathematically, (PVIFA i, n ) =(PVIFA i, n ) = 1 -1 - 11 (1 + i)(1 + i)nn ii
  90. 90. Mathematically,Mathematically, (PVIFA i, n ) =(PVIFA i, n ) = We said that a perpetuity is anWe said that a perpetuity is an annuity where n = infinity. Whatannuity where n = infinity. What happens to this formula whenhappens to this formula when nn gets very, very large?gets very, very large? 1 -1 - 11 (1 + i)(1 + i)nn ii
  91. 91. When n gets very large,When n gets very large,
  92. 92. When n gets very large,When n gets very large, 1 - 1 (1 + i)n i
  93. 93. When n gets very large,When n gets very large, this becomes zero.this becomes zero. 1 - 1 (1 + i)n i
  94. 94. When n gets very large,When n gets very large, this becomes zero.this becomes zero. So we’re left with PVIFA =So we’re left with PVIFA = 1 i 1 - 1 (1 + i)n i
  95. 95.  So, the PV of a perpetuity is verySo, the PV of a perpetuity is very simple to find:simple to find: Present Value of a Perpetuity
  96. 96. PMT i PV =  So, the PV of a perpetuity is verySo, the PV of a perpetuity is very simple to find:simple to find: Present Value of a Perpetuity
  97. 97. What should you be willing to pay inWhat should you be willing to pay in order to receiveorder to receive P10,000P10,000 annuallyannually forever, if you requireforever, if you require 8%8% per yearper year on the investment?on the investment?
  98. 98. What should you be willing to pay inWhat should you be willing to pay in order to receiveorder to receive P10,000P10,000 annuallyannually forever, if you requireforever, if you require 8%8% per yearper year on the investment?on the investment? PMT P10,000PMT P10,000 i .08i .08 PV = =PV = =
  99. 99. What should you be willing to pay inWhat should you be willing to pay in order to receiveorder to receive P10,000P10,000 annuallyannually forever, if you requireforever, if you require 8%8% per yearper year on the investment?on the investment? PMT P10,000PMT P10,000 i .08i .08 = P125,000= P125,000 PV = =PV = =
  100. 100. Ordinary AnnuityOrdinary Annuity vs.vs. Annuity DueAnnuity Due P1000 P1000 P1000P1000 P1000 P1000 4 5 6 7 8
  101. 101. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8
  102. 102. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 5 6 7
  103. 103. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 5 6 7 PVPV inin ENDEND ModeMode
  104. 104. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 5 6 7 PVPV inin ENDEND ModeMode FVFV inin ENDEND ModeMode
  105. 105. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 6 7 8
  106. 106. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 6 7 8 PVPV inin BEGINBEGIN ModeMode
  107. 107. Begin Mode vs. End ModeBegin Mode vs. End Mode 1000 1000 10001000 1000 1000 4 5 6 7 84 5 6 7 8 year year year 6 7 8 PVPV inin BEGINBEGIN ModeMode FVFV inin BEGINBEGIN ModeMode
  108. 108. Earlier, we examined thisEarlier, we examined this “ordinary” annuity:“ordinary” annuity:
  109. 109. Earlier, we examined thisEarlier, we examined this “ordinary” annuity:“ordinary” annuity: 0 1 2 3 10001000 10001000 10001000
  110. 110. Earlier, we examined thisEarlier, we examined this “ordinary” annuity:“ordinary” annuity: Using an interest rate of 8%, weUsing an interest rate of 8%, we find that:find that: 0 1 2 3 10001000 10001000 10001000
  111. 111. Earlier, we examined thisEarlier, we examined this “ordinary” annuity:“ordinary” annuity: Using an interest rate of 8%, weUsing an interest rate of 8%, we find that:find that:  TheThe Future ValueFuture Value (at 3) is(at 3) is P3,246.40P3,246.40.. 0 1 2 3 10001000 10001000 10001000
  112. 112. Earlier, we examined thisEarlier, we examined this “ordinary” annuity:“ordinary” annuity: Using an interest rate of 8%, weUsing an interest rate of 8%, we find that:find that:  TheThe Future ValueFuture Value (at 3) is(at 3) is P3,246.40P3,246.40..  TheThe Present ValuePresent Value (at 0) is(at 0) is P2,577.10P2,577.10.. 0 1 2 3 10001000 10001000 10001000
  113. 113. What about this annuity?What about this annuity?  Same 3-year time line,Same 3-year time line,  Same 3 P1000 cash flows, butSame 3 P1000 cash flows, but  The cash flows occur at theThe cash flows occur at the beginningbeginning of each year, ratherof each year, rather than at thethan at the endend of each year.of each year.  This is anThis is an “annuity due.”“annuity due.” 0 1 2 3 10001000 10001000 10001000
  114. 114. 0 1 2 3 Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3?
  115. 115. Calculator Solution:Calculator Solution: Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 FV =FV = P3,506.11P3,506.11 0 1 2 3 -1000-1000 -1000-1000 -1000-1000 Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3?
  116. 116. 0 1 2 3 -1000-1000 -1000-1000 -1000-1000 Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Calculator Solution:Calculator Solution: Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = -1,000PMT = -1,000 FV =FV = P3,506.11P3,506.11
  117. 117. Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
  118. 118. Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
  119. 119. Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
  120. 120. Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii (1 + i)(1 + i)
  121. 121. Future Value - annuity dueFuture Value - annuity due If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at the end of year 3?the end of year 3? Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i) FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 =- 1 = P3,506.11P3,506.11 (1 + i)(1 + i) (1.08)(1.08)
  122. 122. Present Value - annuity duePresent Value - annuity due What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%? 0 1 2 3
  123. 123. Calculator Solution:Calculator Solution: Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = 1,000PMT = 1,000 PV =PV = P2,783.26P2,783.26 0 1 2 3 10001000 10001000 10001000 Present Value - annuity duePresent Value - annuity due What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
  124. 124. Calculator Solution:Calculator Solution: Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8 N = 3N = 3 PMT = 1,000PMT = 1,000 PV =PV = P2,783.26P2,783.26 0 1 2 3 10001000 10001000 10001000 Present Value - annuity duePresent Value - annuity due What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
  125. 125. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution:
  126. 126. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
  127. 127. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
  128. 128. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or)
  129. 129. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii (1 + i)(1 + i)
  130. 130. Present Value - annuity duePresent Value - annuity due Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the ordinary annuity one more period:ordinary annuity one more period: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i) PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii 11 PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 == P2,783.26P2,783.26 .08.08 (1 + i)(1 + i) (1.08)(1.08)
  131. 131.  Is this anIs this an annuityannuity??  How do we find the PV of a cash flowHow do we find the PV of a cash flow stream when all of the cash flows arestream when all of the cash flows are different? (Use a 10% discount rate.)different? (Use a 10% discount rate.) Uneven Cash FlowsUneven Cash Flows 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
  132. 132.  Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one. We have to discount each cash flowWe have to discount each cash flow back separately.back separately. 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000 Uneven Cash FlowsUneven Cash Flows
  133. 133.  Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one. We have to discount each cash flowWe have to discount each cash flow back separately.back separately. 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000 Uneven Cash FlowsUneven Cash Flows
  134. 134.  Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one. We have to discount each cash flowWe have to discount each cash flow back separately.back separately. 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000 Uneven Cash FlowsUneven Cash Flows
  135. 135.  Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one. We have to discount each cash flowWe have to discount each cash flow back separately.back separately. 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000 Uneven Cash FlowsUneven Cash Flows
  136. 136.  Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one. We have to discount each cash flowWe have to discount each cash flow back separately.back separately. 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000 Uneven Cash FlowsUneven Cash Flows
  137. 137. periodperiod CFCF PV (CF)PV (CF) 00 -10,000-10,000 -10,000.00-10,000.00 11 2,0002,000 1,818.181,818.18 22 4,0004,000 3,305.793,305.79 33 6,0006,000 4,507.894,507.89 44 7,0007,000 4,781.094,781.09 PV of Cash Flow Stream: P4,412.95PV of Cash Flow Stream: P4,412.95 00 11 22 33 44 -10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
  138. 138. Annual Percentage Yield (APY)Annual Percentage Yield (APY) Which is the better loan:Which is the better loan:  8%8% compoundedcompounded annuallyannually, or, or  7.85%7.85% compoundedcompounded quarterlyquarterly??  We can’t compare these nominal (quoted)We can’t compare these nominal (quoted) interest rates, because they don’t include theinterest rates, because they don’t include the same number of compounding periods persame number of compounding periods per year!year! We need to calculate the APY.We need to calculate the APY.
  139. 139. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
  140. 140. Annual Percentage Yield (APY)Annual Percentage Yield (APY) APY =APY = (( 1 +1 + )) mm - 1- 1quoted ratequoted rate mm
  141. 141. Annual Percentage Yield (APY)Annual Percentage Yield (APY)  Find the APY for the quarterly loan:Find the APY for the quarterly loan: APY =APY = (( 1 +1 + )) mm - 1- 1quoted ratequoted rate mm
  142. 142. Annual Percentage Yield (APY)Annual Percentage Yield (APY)  Find the APY for the quarterly loan:Find the APY for the quarterly loan: APY =APY = (( 1 +1 + )) mm - 1- 1quoted ratequoted rate mm APY =APY = (( 1 +1 + )) 44 - 1- 1 .0785.0785 44
  143. 143. Annual Percentage Yield (APY)Annual Percentage Yield (APY)  Find the APY for the quarterly loan:Find the APY for the quarterly loan: APY =APY = (( 1 +1 + )) mm - 1- 1quoted ratequoted rate mm APY =APY = (( 1 +1 + )) 44 - 1- 1 APY = .0808, or 8.08%APY = .0808, or 8.08% .0785.0785 44
  144. 144. Annual Percentage Yield (APY)Annual Percentage Yield (APY)  Find the APY for the quarterly loan:Find the APY for the quarterly loan:  The quarterly loan is more expensive thanThe quarterly loan is more expensive than the 8% loan with annual compounding!the 8% loan with annual compounding! APY =APY = (( 1 +1 + )) mm - 1- 1quoted ratequoted rate mm APY =APY = (( 1 +1 + )) 44 - 1- 1 APY = .0808, or 8.08%APY = .0808, or 8.08% .0785.0785 44
  145. 145. Practice ProblemsPractice Problems
  146. 146. ExampleExample  Cash flows from an investment areCash flows from an investment are expected to beexpected to be P40,000P40,000 per year at theper year at the end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you require arequire a 20%20% rate of return, what israte of return, what is the PV of these cash flows?the PV of these cash flows?
  147. 147. ExampleExample 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040  Cash flows from an investment areCash flows from an investment are expected to beexpected to be P40,000P40,000 per year at theper year at the end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you require arequire a 20%20% rate of return, what israte of return, what is the PV of these cash flows?the PV of these cash flows?
  148. 148.  This type of cash flow sequence isThis type of cash flow sequence is often called aoften called a ““deferred annuitydeferred annuity.”.” 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  149. 149. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  150. 150. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  151. 151. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  152. 152. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  153. 153. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  154. 154. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  155. 155. How to solve:How to solve: 1)1) Discount each cash flow back toDiscount each cash flow back to time 0 separately.time 0 separately. Or,Or, 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  156. 156. 2)2) Find the PV of the annuity:Find the PV of the annuity: PVPV:: End mode; P/YR = 1; I = 20;End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5PMT = 40,000; N = 5 PV =PV = P119,624P119,624 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  157. 157. 2)2) Find the PV of the annuity:Find the PV of the annuity: PVPV3:3: End mode; P/YR = 1; I = 20;End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5PMT = 40,000; N = 5 PVPV33== P119,624P119,624 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  158. 158. 119,624119,624 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  159. 159. Then discount this single sum back toThen discount this single sum back to time 0.time 0. PV: End mode; P/YR = 1; I = 20;PV: End mode; P/YR = 1; I = 20; N = 3; FV = 119,624;N = 3; FV = 119,624; Solve: PV =Solve: PV = P69,226P69,226 119,624119,624 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040
  160. 160. 69,22669,226 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040 119,624119,624
  161. 161.  The PV of the cash flowThe PV of the cash flow stream isstream is P69,226P69,226.. 69,22669,226 00 11 22 33 44 55 66 77 88 P0P0 00 00 00 4040 4040 4040 4040 4040 119,624119,624
  162. 162. Retirement ExampleRetirement Example  After graduation, you plan to investAfter graduation, you plan to invest P400P400 per monthper month in the stock market.in the stock market. If you earnIf you earn 12%12% per yearper year on youron your stocks, how much will you havestocks, how much will you have accumulated when you retire inaccumulated when you retire in 3030 yearsyears??
  163. 163. Retirement ExampleRetirement Example  After graduation, you plan to investAfter graduation, you plan to invest P400P400 per month in the stock market.per month in the stock market. If you earnIf you earn 12%12% per year on yourper year on your stocks, how much will you havestocks, how much will you have accumulated when you retire in 30accumulated when you retire in 30 years?years? 00 11 22 33 . . . 360. . . 360 400 400 400 400400 400 400 400
  164. 164. 00 11 22 33 . . . 360. . . 360 400 400 400 400400 400 400 400
  165. 165.  Using your calculator,Using your calculator, P/YR = 12P/YR = 12 N = 360N = 360 PMT = -400PMT = -400 I%YR = 12I%YR = 12 FV =FV = P1,397,985.65P1,397,985.65 00 11 22 33 . . . 360. . . 360 400 400 400 400400 400 400 400
  166. 166. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30?
  167. 167. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30? Mathematical Solution:Mathematical Solution:
  168. 168. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30? Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
  169. 169. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30? Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
  170. 170. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30? Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii
  171. 171. Retirement ExampleRetirement Example If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at the end of year 30?the end of year 30? Mathematical Solution:Mathematical Solution: FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n )) FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table) FV = PMT (1 + i)FV = PMT (1 + i)nn - 1- 1 ii FV = 400 (1.01)FV = 400 (1.01)360360 - 1 =- 1 = P1,397,985.65P1,397,985.65
  172. 172. If you borrowIf you borrow P100,000P100,000 atat 7%7% fixedfixed interest forinterest for 3030 yearsyears in order toin order to buy a house, what will be yourbuy a house, what will be your monthly house paymentmonthly house payment?? House Payment ExampleHouse Payment Example
  173. 173. House Payment ExampleHouse Payment Example If you borrowIf you borrow P100,000P100,000 atat 7%7% fixedfixed interest forinterest for 3030 years in order toyears in order to buy a house, what will be yourbuy a house, what will be your monthly house payment?monthly house payment?
  174. 174. 0 1 2 3 . . . 360 ? ? ? ?
  175. 175.  Using your calculator,Using your calculator, P/YR = 12P/YR = 12 N = 360N = 360 I%YR = 7I%YR = 7 PV = P100,000PV = P100,000 PMT =PMT = -P665.30-P665.30 00 11 22 33 . . . 360. . . 360 ? ? ? ?? ? ? ?
  176. 176. House Payment ExampleHouse Payment Example Mathematical Solution:Mathematical Solution:
  177. 177. House Payment ExampleHouse Payment Example Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ))
  178. 178. House Payment ExampleHouse Payment Example Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n )) 100,000 = PMT (PVIFA100,000 = PMT (PVIFA .07,360.07,360 )) (can’t use PVIFA table)(can’t use PVIFA table)
  179. 179. House Payment ExampleHouse Payment Example Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n )) 100,000 = PMT (PVIFA100,000 = PMT (PVIFA .07,360.07,360 )) (can’t use PVIFA table)(can’t use PVIFA table) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii
  180. 180. House Payment ExampleHouse Payment Example Mathematical Solution:Mathematical Solution: PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n )) 100,000 = PMT (PVIFA100,000 = PMT (PVIFA .07,360.07,360 )) (can’t use PVIFA table)(can’t use PVIFA table) 11 PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn ii 11 100,000 = PMT 1 - (1.005833 )100,000 = PMT 1 - (1.005833 )360360 PMT=P665.30PMT=P665.30 .005833.005833
  181. 181. Team AssignmentTeam Assignment Upon retirement, your goal is to spendUpon retirement, your goal is to spend 55 years traveling around the world. Toyears traveling around the world. To travel in style will requiretravel in style will require P250,000P250,000 perper year at theyear at the beginningbeginning of each year.of each year. If you plan to retire inIf you plan to retire in 3030 yearsyears, what are, what are the equalthe equal monthlymonthly payments necessarypayments necessary to achieve this goal? The funds in yourto achieve this goal? The funds in your retirement account will compound atretirement account will compound at 10%10% annually.annually.
  182. 182.  How much do we need to have byHow much do we need to have by the end of year 30 to finance thethe end of year 30 to finance the trip?trip?  PVPV3030 = PMT (PVIFA= PMT (PVIFA .10, 5.10, 5) (1.10) =) (1.10) = = 250,000 (3.7908) (1.10) == 250,000 (3.7908) (1.10) = == P1,042,470P1,042,470 2727 2828 2929 3030 3131 3232 3333 3434 3535 250 250 250 250 250250 250 250 250 250
  183. 183. Using your calculator,Using your calculator, Mode = BEGINMode = BEGIN PMT = -P250,000PMT = -P250,000 N = 5N = 5 I%YR = 10I%YR = 10 P/YR = 1P/YR = 1 PV =PV = P1,042,466P1,042,466 2727 2828 2929 3030 3131 3232 3333 3434 3535 250 250 250 250 250250 250 250 250 250
  184. 184.  Now, assuming 10% annualNow, assuming 10% annual compounding, what monthlycompounding, what monthly payments will be required for youpayments will be required for you to haveto have P1,042,466P1,042,466 at the end ofat the end of year 30?year 30? 2727 2828 2929 3030 3131 3232 3333 3434 3535 250 250 250 250 250250 250 250 250 250 1,042,4661,042,466
  185. 185. • Using your calculator,Using your calculator, Mode = ENDMode = END N = 360N = 360 I%YR = 10I%YR = 10 P/YR = 12P/YR = 12 FV = P1,042,466FV = P1,042,466 PMT =PMT = -P461.17-P461.17 2727 2828 2929 3030 3131 3232 3333 3434 3535 250 250 250 250 250250 250 250 250 250 1,042,4661,042,466
  186. 186.  So, you would have to placeSo, you would have to place P461.17P461.17 inin your retirement account, which earnsyour retirement account, which earns 10% annually, at the end of each of the10% annually, at the end of each of the next 360 months to finance the 5-yearnext 360 months to finance the 5-year world tour.world tour.

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