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Math 1300: Section 3-4 Present Value of an Ordinary Annuity; Amortization

Lecture on Section 3-4 of Barnett's "Finite Mathemaitcs."

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Math 1300: Section 3-4 Present Value of an Ordinary Annuity; Amortization

  1. 1. Present Value of an Ordinary Annuity Amortization Amortization Schedules Math 1300 Finite MathematicsSection 3.4 Present Value of an Annuity; Amortization Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
  2. 2. Present Value of an Ordinary Annuity Amortization Amortization SchedulesPresent Value Present value is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk. university-logo Jason Aubrey Math 1300 Finite Mathematics
  3. 3. Present Value of an Ordinary Annuity Amortization Amortization SchedulesPresent Value Present value is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis. university-logo Jason Aubrey Math 1300 Finite Mathematics
  4. 4. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  5. 5. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT iwhere PV = present value of all payments university-logo Jason Aubrey Math 1300 Finite Mathematics
  6. 6. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT iwhere PV = present value of all payments PMT = periodic payment university-logo Jason Aubrey Math 1300 Finite Mathematics
  7. 7. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT iwhere PV = present value of all payments PMT = periodic payment i = rate per period university-logo Jason Aubrey Math 1300 Finite Mathematics
  8. 8. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT iwhere PV = present value of all payments PMT = periodic payment i = rate per period n = number of periods university-logo Jason Aubrey Math 1300 Finite Mathematics
  9. 9. Present Value of an Ordinary Annuity Amortization Amortization SchedulesTheorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT iwhere PV = present value of all payments PMT = periodic payment i = rate per period n = number of periodsNote: Payments are made at the end of each period. university-logo Jason Aubrey Math 1300 Finite Mathematics
  10. 10. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period? university-logo Jason Aubrey Math 1300 Finite Mathematics
  11. 11. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period? r 0.0665Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.So, university-logo Jason Aubrey Math 1300 Finite Mathematics
  12. 12. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period? r 0.0665Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.So, 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  13. 13. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period? r 0.0665Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.So, 1 − (1 + i)−n PV = PMT i 1 − (1.0665)−10 PV = ($5, 000) = $35, 693.18 .0665 university-logo Jason Aubrey Math 1300 Finite Mathematics
  14. 14. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: Recently, Lincoln Benefit Life offered an ordinaryannuity that earned 6.5% compounded annually. A personplans to make equal annual deposits into this account for 25years in order to then make 20 equal annual withdrawals of$25,000, reducing the balance in the account to zero. Howmuch must be deposited annually to accumlate sufficient fundsto provide for these payments? How much total interest isearned during this entire 45-year process? university-logo Jason Aubrey Math 1300 Finite Mathematics
  15. 15. Present Value of an Ordinary Annuity Amortization Amortization SchedulesWe first find the present value necessary to provide for thewithdrawals. university-logo Jason Aubrey Math 1300 Finite Mathematics
  16. 16. Present Value of an Ordinary Annuity Amortization Amortization SchedulesWe first find the present value necessary to provide for thewithdrawals.In this calculation, PMT = $25,000, i = 0.065 and n = 20. university-logo Jason Aubrey Math 1300 Finite Mathematics
  17. 17. Present Value of an Ordinary Annuity Amortization Amortization SchedulesWe first find the present value necessary to provide for thewithdrawals.In this calculation, PMT = $25,000, i = 0.065 and n = 20. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  18. 18. Present Value of an Ordinary Annuity Amortization Amortization SchedulesWe first find the present value necessary to provide for thewithdrawals.In this calculation, PMT = $25,000, i = 0.065 and n = 20. 1 − (1 + i)−n PV = PMT i 1 − (1.065)−20 PV = ($25, 000) = $275, 462.68 .065 university-logo Jason Aubrey Math 1300 Finite Mathematics
  19. 19. Present Value of an Ordinary Annuity Amortization Amortization SchedulesNow we find the deposits that will produce a future value of$275,462.68 in 25 years. university-logo Jason Aubrey Math 1300 Finite Mathematics
  20. 20. Present Value of an Ordinary Annuity Amortization Amortization SchedulesNow we find the deposits that will produce a future value of$275,462.68 in 25 years.Here we use FV = $275,462.68, i = 0.065 and n = 25. university-logo Jason Aubrey Math 1300 Finite Mathematics
  21. 21. Present Value of an Ordinary Annuity Amortization Amortization SchedulesNow we find the deposits that will produce a future value of$275,462.68 in 25 years.Here we use FV = $275,462.68, i = 0.065 and n = 25. (1 + i)n − 1 FV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  22. 22. Present Value of an Ordinary Annuity Amortization Amortization SchedulesNow we find the deposits that will produce a future value of$275,462.68 in 25 years.Here we use FV = $275,462.68, i = 0.065 and n = 25. (1 + i)n − 1 FV = PMT i (1.065)25 − 1 $275, 462.68 = PMT .065 university-logo Jason Aubrey Math 1300 Finite Mathematics
  23. 23. Present Value of an Ordinary Annuity Amortization Amortization SchedulesNow we find the deposits that will produce a future value of$275,462.68 in 25 years.Here we use FV = $275,462.68, i = 0.065 and n = 25. (1 + i)n − 1 FV = PMT i (1.065)25 − 1 $275, 462.68 = PMT .065 .065 PMT = ($275, 462.68) = $4, 677.76 (1.065)25 − 1Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000. university-logo Jason Aubrey Math 1300 Finite Mathematics
  24. 24. Present Value of an Ordinary Annuity Amortization Amortization SchedulesThe interest earned during the entire 45-year process is university-logo Jason Aubrey Math 1300 Finite Mathematics
  25. 25. Present Value of an Ordinary Annuity Amortization Amortization SchedulesThe interest earned during the entire 45-year process is interest = (total withdrawals) − (total deposits) university-logo Jason Aubrey Math 1300 Finite Mathematics
  26. 26. Present Value of an Ordinary Annuity Amortization Amortization SchedulesThe interest earned during the entire 45-year process is interest = (total withdrawals) − (total deposits) = 20($25, 000) − 25($4, 677.76) university-logo Jason Aubrey Math 1300 Finite Mathematics
  27. 27. Present Value of an Ordinary Annuity Amortization Amortization SchedulesThe interest earned during the entire 45-year process is interest = (total withdrawals) − (total deposits) = 20($25, 000) − 25($4, 677.76) = $383, 056 university-logo Jason Aubrey Math 1300 Finite Mathematics
  28. 28. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization In business, amortization is the distribution of a single lump-sum cash flow into many smaller cash flow installments, as determined by an amortization schedule. university-logo Jason Aubrey Math 1300 Finite Mathematics
  29. 29. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization In business, amortization is the distribution of a single lump-sum cash flow into many smaller cash flow installments, as determined by an amortization schedule. Unlike other repayment models, each repayment installment consists of both principal and interest. Amortization is chiefly used in loan repayments (a common example being a mortgage loan) and in sinking funds. university-logo Jason Aubrey Math 1300 Finite Mathematics
  30. 30. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Payments are divided into equal amounts for the duration of the loan, making it the simplest repayment model. university-logo Jason Aubrey Math 1300 Finite Mathematics
  31. 31. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Payments are divided into equal amounts for the duration of the loan, making it the simplest repayment model. A greater amount of the payment is applied to interest at the beginning of the amortization schedule, while more money is applied to principal at the end. university-logo Jason Aubrey Math 1300 Finite Mathematics
  32. 32. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment. university-logo Jason Aubrey Math 1300 Finite Mathematics
  33. 33. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment. 0.072We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;PV = $50, 000 university-logo Jason Aubrey Math 1300 Finite Mathematics
  34. 34. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment. 0.072We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;PV = $50, 000 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  35. 35. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment. 0.072We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;PV = $50, 000 1 − (1 + i)−n PV = PMT i 1 − (1.006)−240 $50, 000 = PMT .006 university-logo Jason Aubrey Math 1300 Finite Mathematics
  36. 36. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment. 0.072We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;PV = $50, 000 1 − (1 + i)−n PV = PMT i 1 − (1.006)−240 $50, 000 = PMT .006 .006 PMT = ($50, 000) = $393.67 1 − (1.006)−240 university-logo Jason Aubrey Math 1300 Finite Mathematics
  37. 37. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, find the unpaid balance after 5 years. university-logo Jason Aubrey Math 1300 Finite Mathematics
  38. 38. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, find the unpaid balance after 5 years. We use the value of PMT=$393.67 to find the unpaid balance after 5 years. In these "unpaid balance after" problems, n represents the number of interest periods remaining.Here n = 240 − 60 = 180. Therefore, university-logo Jason Aubrey Math 1300 Finite Mathematics
  39. 39. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, find the unpaid balance after 5 years. We use the value of PMT=$393.67 to find the unpaid balance after 5 years. In these "unpaid balance after" problems, n represents the number of interest periods remaining.Here n = 240 − 60 = 180. Therefore, 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  40. 40. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, find the unpaid balance after 5 years. We use the value of PMT=$393.67 to find the unpaid balance after 5 years. In these "unpaid balance after" problems, n represents the number of interest periods remaining.Here n = 240 − 60 = 180. Therefore, 1 − (1 + i)−n PV = PMT i 1 − (1.006)−180 PV = (393.67) = $43, 258.22 .006 university-logo Jason Aubrey Math 1300 Finite Mathematics
  41. 41. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, compute the unpaid balance after 10years. university-logo Jason Aubrey Math 1300 Finite Mathematics
  42. 42. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, compute the unpaid balance after 10years.Here n = 240 − 120 = 120 and so, university-logo Jason Aubrey Math 1300 Finite Mathematics
  43. 43. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, compute the unpaid balance after 10years.Here n = 240 − 120 = 120 and so, 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  44. 44. Present Value of an Ordinary Annuity Amortization Amortization SchedulesFor the same mortgage, compute the unpaid balance after 10years.Here n = 240 − 120 = 120 and so, 1 − (1 + i)−n PV = PMT i 1 − (1.006)−120 PV = ($393.67) = $33, 606.26 .006 university-logo Jason Aubrey Math 1300 Finite Mathematics
  45. 45. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Example: Construct the amortization schedule for a $5,000 debt that is to be amortized in eight equal quarterly payments at 2.8% compounded quarterly. university-logo Jason Aubrey Math 1300 Finite Mathematics
  46. 46. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Example: Construct the amortization schedule for a $5,000 debt that is to be amortized in eight equal quarterly payments at 2.8% compounded quarterly. First we calculate the quarterly payment. Here we have m = 4; n = 8; i = m = 0.028 = 0.007. Then r 4 university-logo Jason Aubrey Math 1300 Finite Mathematics
  47. 47. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Example: Construct the amortization schedule for a $5,000 debt that is to be amortized in eight equal quarterly payments at 2.8% compounded quarterly. First we calculate the quarterly payment. Here we have m = 4; n = 8; i = m = 0.028 = 0.007. Then r 4 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  48. 48. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Example: Construct the amortization schedule for a $5,000 debt that is to be amortized in eight equal quarterly payments at 2.8% compounded quarterly. First we calculate the quarterly payment. Here we have m = 4; n = 8; i = m = 0.028 = 0.007. Then r 4 1 − (1 + i)−n PV = PMT i 1 − (1.007)−8 $5, 000 = PMT .007 university-logo Jason Aubrey Math 1300 Finite Mathematics
  49. 49. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Example: Construct the amortization schedule for a $5,000 debt that is to be amortized in eight equal quarterly payments at 2.8% compounded quarterly. First we calculate the quarterly payment. Here we have m = 4; n = 8; i = m = 0.028 = 0.007. Then r 4 1 − (1 + i)−n PV = PMT i 1 − (1.007)−8 $5, 000 = PMT .007 .007 PMT = ($5, 000) = $644.85 1 − (1.007)−8 university-logo Jason Aubrey Math 1300 Finite Mathematics
  50. 50. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 1 2 3 4 5 6 7 8 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  51. 51. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 2 3 4 5 6 7 8 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  52. 52. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 2 $644.85 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  53. 53. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 2 $644.85 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  54. 54. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 2 $644.85 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  55. 55. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  56. 56. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  57. 57. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  58. 58. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  59. 59. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  60. 60. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  61. 61. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  62. 62. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  63. 63. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  64. 64. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  65. 65. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  66. 66. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  67. 67. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  68. 68. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  69. 69. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  70. 70. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  71. 71. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  72. 72. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 $635.50 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  73. 73. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 $635.50 $640.35 8 $644.85 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  74. 74. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 $635.50 $640.35 8 $644.85 $4.48 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  75. 75. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 $635.50 $640.35 8 $644.85 $4.48 $640.37 Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  76. 76. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAmortization Schedules Period Payment Payment Int. Bal. Reduction Unpaid Bal. 0 $5,000 1 $644.85 $35 $609.85 $4,390.15 2 $644.85 $30.73 $614.12 $3,776.03 3 $644.85 $26.43 $618.42 $3,157.61 4 $644.85 $22.10 $622.75 $2,534.87 5 $644.85 $17.74 $627.11 $1,907.76 6 $644.85 $13.35 $631.50 $1,276.26 7 $644.85 $8.93 $635.50 $640.35 8 $644.85 $4.48 $640.37 $0.00* Interest owed during a period = university-logo (Balance during period)(Interest rate per period) Jason Aubrey Math 1300 Finite Mathematics
  77. 77. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A family purchased a home 10 years ago for$80,000. The home was financed by paying 20% down andsigning a 30-year mortgage at 9% on the unpaid balance. Thenet market value of the house (amount recieved aftersubtracting all costs involved in selling the house) is now$120,000, and the family wishes to sell the house. How muchequity (to the nearest dollar) does the family have in the housenow after making 120 monthly payments?[Equity = (current net market value) - (unpaid loan balance)] university-logo Jason Aubrey Math 1300 Finite Mathematics
  78. 78. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1. Find the monthly payment: university-logo Jason Aubrey Math 1300 Finite Mathematics
  79. 79. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1. Find the monthly payment: r 0.09Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075and n = 360. university-logo Jason Aubrey Math 1300 Finite Mathematics
  80. 80. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1. Find the monthly payment: r 0.09Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075and n = 360. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  81. 81. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1. Find the monthly payment: r 0.09Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075and n = 360. 1 − (1 + i)−n PV = PMT i 1 − (1.0075)−360 $64, 000 = PMT .0075 university-logo Jason Aubrey Math 1300 Finite Mathematics
  82. 82. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1. Find the monthly payment: r 0.09Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075and n = 360. 1 − (1 + i)−n PV = PMT i 1 − (1.0075)−360 $64, 000 = PMT .0075 .0075 PMT = ($64, 000) = $514.96 1 − (1.0075)−360 university-logo Jason Aubrey Math 1300 Finite Mathematics
  83. 83. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity): university-logo Jason Aubrey Math 1300 Finite Mathematics
  84. 84. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity): 0.09Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075. university-logo Jason Aubrey Math 1300 Finite Mathematics
  85. 85. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity): 0.09Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  86. 86. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity): 0.09Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075. 1 − (1 + i)−n PV = PMT i 1 − (1.0075)−240 PV = ($514.96) = $57, 235 .0075 university-logo Jason Aubrey Math 1300 Finite Mathematics
  87. 87. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3. Find the equity: university-logo Jason Aubrey Math 1300 Finite Mathematics
  88. 88. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3. Find the equity: Equity = (current net market value) − (unpaid loan balance) = $120, 000 − $57, 235 university-logo Jason Aubrey Math 1300 Finite Mathematics
  89. 89. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3. Find the equity: Equity = (current net market value) − (unpaid loan balance) = $120, 000 − $57, 235 = $62, 765 university-logo Jason Aubrey Math 1300 Finite Mathematics
  90. 90. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3. Find the equity: Equity = (current net market value) − (unpaid loan balance) = $120, 000 − $57, 235 = $62, 765Thus, if the family sells the house for $120,000 net, the familywill have $62,765 after paying off the unpaid loan balance of$57,235. university-logo Jason Aubrey Math 1300 Finite Mathematics
  91. 91. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: A person purchased a house 10 years ago for$120,000 by paying 20% down and signing a 30-year mortgageat 10.2% compounded monthly. Interest rates have droppedand the owner wants to refinance the unpaid balance bysigning a new 20-year mortgage at 7.5% compounded monthly.How much interest will the refinancing save? university-logo Jason Aubrey Math 1300 Finite Mathematics
  92. 92. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1: Find monthly payments. university-logo Jason Aubrey Math 1300 Finite Mathematics
  93. 93. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1: Find monthly payments. The owner put 20% down at the time of purchase. Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000. university-logo Jason Aubrey Math 1300 Finite Mathematics
  94. 94. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1: Find monthly payments. The owner put 20% down at the time of purchase. Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000. We also have that r 0.102 m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085. university-logo Jason Aubrey Math 1300 Finite Mathematics
  95. 95. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1: Find monthly payments. The owner put 20% down at the time of purchase. Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000. We also have that r 0.102 m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085. 1 − (1.0085)−360 $96, 000 = PMT 0.0085 university-logo Jason Aubrey Math 1300 Finite Mathematics
  96. 96. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 1: Find monthly payments. The owner put 20% down at the time of purchase. Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000. We also have that r 0.102 m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085. 1 − (1.0085)−360 $96, 000 = PMT 0.0085 PMT = $856.69 university-logo Jason Aubrey Math 1300 Finite Mathematics
  97. 97. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2: Find amount owed after 10 years (at the time ofrefinancing). university-logo Jason Aubrey Math 1300 Finite Mathematics
  98. 98. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2: Find amount owed after 10 years (at the time ofrefinancing).Here we apply the formula with i = 0.0085 and n = 240. university-logo Jason Aubrey Math 1300 Finite Mathematics
  99. 99. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2: Find amount owed after 10 years (at the time ofrefinancing).Here we apply the formula with i = 0.0085 and n = 240. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  100. 100. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 2: Find amount owed after 10 years (at the time ofrefinancing).Here we apply the formula with i = 0.0085 and n = 240. 1 − (1 + i)−n PV = PMT i 1 − (1.0085)−240 PV = ($856.69) = $87, 568.38 .0085 university-logo Jason Aubrey Math 1300 Finite Mathematics
  101. 101. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3: We now calculate the owner’s monthly payment afterrefinancing. university-logo Jason Aubrey Math 1300 Finite Mathematics
  102. 102. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3: We now calculate the owner’s monthly payment afterrefinancing. 0.075Here we apply the formula with i = 12 = 0.00625 andn = 240. university-logo Jason Aubrey Math 1300 Finite Mathematics
  103. 103. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3: We now calculate the owner’s monthly payment afterrefinancing. 0.075Here we apply the formula with i = 12 = 0.00625 andn = 240. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  104. 104. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3: We now calculate the owner’s monthly payment afterrefinancing. 0.075Here we apply the formula with i = 12 = 0.00625 andn = 240. 1 − (1 + i)−n PV = PMT i 1 − (1.00625)−240 $87, 568.38 = PMT .00625 university-logo Jason Aubrey Math 1300 Finite Mathematics
  105. 105. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 3: We now calculate the owner’s monthly payment afterrefinancing. 0.075Here we apply the formula with i = 12 = 0.00625 andn = 240. 1 − (1 + i)−n PV = PMT i 1 − (1.00625)−240 $87, 568.38 = PMT .00625 .00625 PMT = ($87, 568.38) = $705.44 1 − (1.00625)−240 university-logo Jason Aubrey Math 1300 Finite Mathematics
  106. 106. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing. university-logo Jason Aubrey Math 1300 Finite Mathematics
  107. 107. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing. If the owner did not refinance, he would pay a total of 856.69 × 240 = $205, 605.60 in principal and interest during the last 20 years of the loan. university-logo Jason Aubrey Math 1300 Finite Mathematics
  108. 108. Present Value of an Ordinary Annuity Amortization Amortization SchedulesStep 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing. If the owner did not refinance, he would pay a total of 856.69 × 240 = $205, 605.60 in principal and interest during the last 20 years of the loan. This would amount to a total of $205, 605.60 − $87, 568.38 = $118, 037.22 in interest. university-logo Jason Aubrey Math 1300 Finite Mathematics
  109. 109. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAfter refinancing, the owner pays a total of$705.44x240 = $169, 305.60 in principal and interest. university-logo Jason Aubrey Math 1300 Finite Mathematics
  110. 110. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAfter refinancing, the owner pays a total of$705.44x240 = $169, 305.60 in principal and interest.This would amount to a total of$169, 305.60 − $87, 568.38 = $81, 737.22 in interest. university-logo Jason Aubrey Math 1300 Finite Mathematics
  111. 111. Present Value of an Ordinary Annuity Amortization Amortization SchedulesAfter refinancing, the owner pays a total of$705.44x240 = $169, 305.60 in principal and interest.This would amount to a total of$169, 305.60 − $87, 568.38 = $81, 737.22 in interest.Therefore refinancing results in a total interest savings of $118, 037.22 − $81, 737.22 = $36, 299.84. university-logo Jason Aubrey Math 1300 Finite Mathematics
  112. 112. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose? university-logo Jason Aubrey Math 1300 Finite Mathematics
  113. 113. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose?To answer this question, we determine which option gives thelowest monthly payment. university-logo Jason Aubrey Math 1300 Finite Mathematics
  114. 114. Present Value of an Ordinary Annuity Amortization Amortization SchedulesExample: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose?To answer this question, we determine which option gives thelowest monthly payment.Option 1: If you choose 0% financing, your monthly paymentwill be $27, 300 PMT1 = = $455 60 university-logo Jason Aubrey Math 1300 Finite Mathematics
  115. 115. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly. university-logo Jason Aubrey Math 1300 Finite Mathematics
  116. 116. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.We compute the PMT for a loan with PV = $22,300,i = 0.063 = 0.00525 and n = 60. 12 university-logo Jason Aubrey Math 1300 Finite Mathematics
  117. 117. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.We compute the PMT for a loan with PV = $22,300,i = 0.063 = 0.00525 and n = 60. 12 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
  118. 118. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.We compute the PMT for a loan with PV = $22,300,i = 0.063 = 0.00525 and n = 60. 12 1 − (1 + i)−n PV = PMT i 1 − (1.00525)−60 $22, 300 = PMT 0.00525 university-logo Jason Aubrey Math 1300 Finite Mathematics
  119. 119. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.We compute the PMT for a loan with PV = $22,300,i = 0.063 = 0.00525 and n = 60. 12 1 − (1 + i)−n PV = PMT i 1 − (1.00525)−60 $22, 300 = PMT 0.00525 PMT = $434.24 university-logo Jason Aubrey Math 1300 Finite Mathematics
  120. 120. Present Value of an Ordinary Annuity Amortization Amortization SchedulesOption 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.We compute the PMT for a loan with PV = $22,300,i = 0.063 = 0.00525 and n = 60. 12 1 − (1 + i)−n PV = PMT i 1 − (1.00525)−60 $22, 300 = PMT 0.00525 PMT = $434.24You should choose the rebate. You will save $455 - $434.24 =$20.76 monthly, or ($20.76)(60) = $1,245.60 over the life of theloan. university-logo Jason Aubrey Math 1300 Finite Mathematics

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