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![Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family purchased a home 10 years ago for
$80,000. The home was financed by paying 20% down and
signing a 30-year mortgage at 9% on the unpaid balance. The
net market value of the house (amount recieved after
subtracting all costs involved in selling the house) is now
$120,000, and the family wishes to sell the house. How much
equity (to the nearest dollar) does the family have in the house
now after making 120 monthly payments?
[Equity = (current net market value) - (unpaid loan balance)]
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Math 1300: Section 3-4 Present Value of an Ordinary Annuity; Amortization
The document discusses present value calculations for annuities and amortization schedules. It provides the formula to calculate the present value of an ordinary annuity. It then works through two examples, one calculating the present value needed to fund annual withdrawals over 20 years, and the other calculating the monthly payment for a $50,000, 20-year mortgage at 7.2% interest compounded monthly.
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Math 1300: Section 3-4 Present Value of an Ordinary Annuity; Amortization
- 1. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Math 1300 Finite Mathematics Section 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Present 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Present 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
- 5. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i where PV = present value of all payments university-logo Jason Aubrey Math 1300 Finite Mathematics
- 6. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i where PV = present value of all payments PMT = periodic payment university-logo Jason Aubrey Math 1300 Finite Mathematics
- 7. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i where PV = present value of all payments PMT = periodic payment i = rate per period university-logo Jason Aubrey Math 1300 Finite Mathematics
- 8. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i where 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Theorem (The present value of an ordinary annuity) 1 − (1 + i)−n PV = PMT i where PV = present value of all payments PMT = periodic payment i = rate per period n = number of periods Note: Payments are made at the end of each period. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 10. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: American General offers a 10-year ordinary annuity with a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $5,000 annually over the 10-year period? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 11. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: American General offers a 10-year ordinary annuity with a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $5,000 annually over the 10-year period? r 0.0665 Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000. So, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 12. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: American General offers a 10-year ordinary annuity with a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $5,000 annually over the 10-year period? r 0.0665 Here 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: American General offers a 10-year ordinary annuity with a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $5,000 annually over the 10-year period? r 0.0665 Here 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: Recently, Lincoln Benefit Life offered an ordinary annuity that earned 6.5% compounded annually. A person plans to make equal annual deposits into this account for 25 years in order to then make 20 equal annual withdrawals of $25,000, reducing the balance in the account to zero. How much must be deposited annually to accumlate sufficient funds to provide for these payments? How much total interest is earned during this entire 45-year process? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 15. Present Value ofan Ordinary Annuity Amortization Amortization Schedules We first find the present value necessary to provide for the withdrawals. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 16. Present Value ofan Ordinary Annuity Amortization Amortization Schedules We first find the present value necessary to provide for the withdrawals. In this calculation, PMT = $25,000, i = 0.065 and n = 20. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 17. Present Value ofan Ordinary Annuity Amortization Amortization Schedules We first find the present value necessary to provide for the withdrawals. 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules We first find the present value necessary to provide for the withdrawals. 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Now 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Now 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Now 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Now 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Now 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 − 1 Thus, depositing $4,677.76 annually for 25 years will provide for 20 annual withdrawals of $25,000. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 24. Present Value ofan Ordinary Annuity Amortization Amortization Schedules The interest earned during the entire 45-year process is university-logo Jason Aubrey Math 1300 Finite Mathematics
- 25. Present Value ofan Ordinary Annuity Amortization Amortization Schedules The interest earned during the entire 45-year process is interest = (total withdrawals) − (total deposits) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 26. Present Value ofan Ordinary Annuity Amortization Amortization Schedules The 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules The 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A family has a $50,000, 20-year mortgage at 7.2% compounded monthly. Find the monthly payment. 0.072 We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240; PV = $50, 000 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 34. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A family has a $50,000, 20-year mortgage at 7.2% compounded monthly. Find the monthly payment. 0.072 We 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A family has a $50,000, 20-year mortgage at 7.2% compounded monthly. Find the monthly payment. 0.072 We 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A family has a $50,000, 20-year mortgage at 7.2% compounded monthly. Find the monthly payment. 0.072 We 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For the same mortgage, find the unpaid balance after 5 years. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 38. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For the same mortgage, compute the unpaid balance after 10 years. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 42. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For the same mortgage, compute the unpaid balance after 10 years. Here n = 240 − 120 = 120 and so, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 43. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For the same mortgage, compute the unpaid balance after 10 years. Here n = 240 − 120 = 120 and so, 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
- 44. Present Value ofan Ordinary Annuity Amortization Amortization Schedules For the same mortgage, compute the unpaid balance after 10 years. 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Amortization 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A family purchased a home 10 years ago for $80,000. The home was financed by paying 20% down and signing a 30-year mortgage at 9% on the unpaid balance. The net market value of the house (amount recieved after subtracting all costs involved in selling the house) is now $120,000, and the family wishes to sell the house. How much equity (to the nearest dollar) does the family have in the house now after making 120 monthly payments? [Equity = (current net market value) - (unpaid loan balance)] university-logo Jason Aubrey Math 1300 Finite Mathematics
- 78. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1. Find the monthly payment: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 79. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1. Find the monthly payment: r 0.09 Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075 and n = 360. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 80. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1. Find the monthly payment: r 0.09 Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075 and n = 360. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
- 81. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1. Find the monthly payment: r 0.09 Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075 and 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1. Find the monthly payment: r 0.09 Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075 and 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2. Find unpaid balance after 10 years (the PV of a $514.96 per month, 20-year annuity): 0.09 Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 85. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2. Find unpaid balance after 10 years (the PV of a $514.96 per month, 20-year annuity): 0.09 Here 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2. Find unpaid balance after 10 years (the PV of a $514.96 per month, 20-year annuity): 0.09 Here 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3. Find the equity: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 88. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3. Find the equity: Equity = (current net market value) − (unpaid loan balance) = $120, 000 − $57, 235 = $62, 765 Thus, if the family sells the house for $120,000 net, the family will have $62,765 after paying off the unpaid loan balance of $57,235. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 91. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: A person purchased a house 10 years ago for $120,000 by paying 20% down and signing a 30-year mortgage at 10.2% compounded monthly. Interest rates have dropped and the owner wants to refinance the unpaid balance by signing 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 1: Find monthly payments. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 93. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2: Find amount owed after 10 years (at the time of refinancing). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 98. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2: Find amount owed after 10 years (at the time of refinancing). Here we apply the formula with i = 0.0085 and n = 240. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 99. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2: Find amount owed after 10 years (at the time of refinancing). 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 2: Find amount owed after 10 years (at the time of refinancing). 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3: We now calculate the owner’s monthly payment after refinancing. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 102. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3: We now calculate the owner’s monthly payment after refinancing. 0.075 Here we apply the formula with i = 12 = 0.00625 and n = 240. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 103. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3: We now calculate the owner’s monthly payment after refinancing. 0.075 Here we apply the formula with i = 12 = 0.00625 and n = 240. 1 − (1 + i)−n PV = PMT i university-logo Jason Aubrey Math 1300 Finite Mathematics
- 104. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3: We now calculate the owner’s monthly payment after refinancing. 0.075 Here we apply the formula with i = 12 = 0.00625 and n = 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 3: We now calculate the owner’s monthly payment after refinancing. 0.075 Here we apply the formula with i = 12 = 0.00625 and n = 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 4: We now compare the amount he would have spent without refinancing to the amount he spends after refinancing. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 107. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 4: We now compare the amount he would have spent without 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Step 4: We now compare the amount he would have spent without 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules After 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules After 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules After 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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: You want to purchase a new car for $27,300. The dealer offers you 0% financing for 60 months or a $5,000 rebate. You can obtain 6.3% financing for 60 months at the local bank. Which option should you choose? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 113. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: You want to purchase a new car for $27,300. The dealer offers you 0% financing for 60 months or a $5,000 rebate. You can obtain 6.3% financing for 60 months at the local bank. Which option should you choose? To answer this question, we determine which option gives the lowest monthly payment. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 114. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Example: You want to purchase a new car for $27,300. The dealer offers you 0% financing for 60 months or a $5,000 rebate. You can obtain 6.3% financing for 60 months at the local bank. Which option should you choose? To answer this question, we determine which option gives the lowest monthly payment. Option 1: If you choose 0% financing, your monthly payment will be $27, 300 PMT1 = = $455 60 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 115. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $22,300 for 60 months at 6.3% compounded monthly. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 116. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $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. Present Value ofan Ordinary Annuity Amortization Amortization Schedules Option 2: Suppose that you choose the $5,000 rebate and borrow $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 You should choose the rebate. You will save $455 - $434.24 = $20.76 monthly, or ($20.76)(60) = $1,245.60 over the life of the loan. university-logo Jason Aubrey Math 1300 Finite Mathematics