1 / 81

Chapter 5

Chapter 5. The Time Value of Money. Learning Objectives . Calculate present and future values of any set of expected future cash flows. Explain how the present value and discount rate are inversely related. Calculate payments on a debt contract. Compute the APR and APY for a contract.

Download Presentation

Chapter 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 5 The Time Value of Money

  2. Learning Objectives • Calculate present and future values of any set of expected future cash flows. • Explain how the present value and discount rate are inversely related. • Calculate payments on a debt contract. • Compute the APR and APY for a contract. • Value “special financing” offers.

  3. Definitions and Assumptions • A point in time is denoted by the letter “t”. • Unless otherwise stated, t=0 represents today (the decision point). • Unless otherwise stated, cash flows occur at the end of a time interval. • Cash inflows are treated as positive amounts, while cash outflows are treated as negative amounts. • Compounding frequency is the same as the cash flow frequency.

  4. The Time Line t=1 t=2 t=3 t=4 t=0

  5. The Time Line Today t=1 t=2 t=3 t=4 t=0

  6. The Time Line End of the third year Today t=1 t=2 t=3 t=4 t=0

  7. The Time Line End of the third year Beginning of the fourth year Today t=1 t=2 t=3 t=4 t=0

  8. Future Value Formula Let PV = Present Value FVn = Future Value at time n r = interest rate (or discount rate) per period.

  9. Future Value Factor 10.00 r = 15% 8.00 6.00 r = 10% 4.00 FV Factor r = 5% 2.00 r = 0% 0.00 0 5 10 15 Time

  10. Present Value Formula Let PV = Present Value FVn = Future Value at time n r = interest rate (or discount rate) per period.

  11. Present Value Factors 1.00 r = 0% 0.80 r = 5% 0.60 r = 10% 0.40 PV Factor 0.20 r = 15% 0.00 0 5 10 15 Time

  12. Solving for an Unknown Interest Rate (CD) The First Commerce Bank offers a Certificate of Deposit (CD) that pays you $5,000 in four years. The CD can be purchased today for $3,477.87. Assuming you hold the CD to maturity, what annual interest rate is the bank paying on this CD?

  13. Solving for an Unknown Interest Rate (CD) PV = $3,477.87; FV4 = $5,000; n = 4 years. n = + Since FV PV (1 r) n

  14. Solving for an Unknown Interest Rate (CD) PV = $3,477.87; FV4 = $5,000; n = 4 years. n = + Since FV PV (1 r) n

  15. Annuities • An annuity is a series of identical cash flows that are expected to occur each period for a specified number of periods. • Thus, CF1 = CF2 = CF3 = Cf4 = ... = CF • Examples of annuities: • Installment loans (car loans, mortgages). • Coupon payment on corporate bonds. • Rent payment on your apartment.

  16. Types of Annuities • Ordinary Annuity: • An annuity with end-of-Period cash flows, beginning one period from today. • Annuity Due: • An annuity with beginning-of-period cash flows. • Deferred Annuity: • An annuity that begins more than one period from today.

  17. Future Value of an Annuity FVAn = CF(1+r)0 + CF(1+r)1 + . . . + CF(1+r)n-1

  18. Future Value of an Annuity FVAn = CF(1+r)0 + CF(1+r)1 + . . . + CF(1+r)n-1 = CF[(1+r)0 + (1+r)1 + . . . + (1+r)n-1 ] FVAn = CF[summation {from 0 to n-1} of (1+r)t ]

  19. Future Value of an Annuity

  20. Future Value of Your Savings Suppose you save $1,500 per year for 15 years, beginning one year from today. The savings bank pays you 8% interest per year. How much will you have at the end of 15 years?

  21. Future Value of Your Savings = $ 40 , 728 . 17

  22. Present Value of an Annuity

  23. Present Value of an Annuity

  24. Present Value of an Annuity

  25. Present Value of Your Bank Loan Cindy agrees to repay a loan in 24 monthly installments of $250 each. If the interest rate on the loan is 0.75% per month, what is the present value of the loan payments?

  26. Present Value of Your Bank Loan = $5 , 472 . 28

  27. Payments of an Annuity (Given FVAn)

  28. Saving for Retirement You wish to retire 25 years from today with $2,000,000 in the bank. If the bank pays 10% interest per year, how much should you save each year to reach your goal?

  29. Saving for Retirement = $20 , 336 . 14

  30. Payments of an Annuity (Given PVAn)

  31. Installment Payments on a Loan Rob borrows $10,000 to be repaid in four equal annual installments, beginning one year from today. What is Rob’s annual payment on this loan if the bank charges him 14% interest per year?

  32. Installment Payments on a Loan = $3 , 432 . 05

  33. Loan Amortization Schedule • It shows how a loan is paid off over time. • It breaks down each payment into the interest component and the principal component. • Let’s illustrate this using Rob’s 4-year $10,000 loan which calls for annual payments of $3,432.05. Recall that the interest rate on this loan is 14% per year.

  34. Loan Amortization Schedule Period: 1 2 3 4 Principal @ Start of Period $10000.00 Interest for Period $1,400.00 Balance $11,400.00 Payment $3,432.05 Principal Repaid $2,032.05 Principal @ End of Period $7,967.95

  35. Period: 1 2 3 4 Loan Amortization Schedule Principal @ Start of Period $10000.00 $7,967.95 Interest for Period $1,400.00 $1,115,51 Balance $11,400.00 $9,083.47 Payment $3,432.05 $3,432.05 Principal Repaid $2,032.05 $2,316.53 Principal @ End of Period $7,967.95 $5,651.42

  36. Period: 1 2 3 4 Loan Amortization Schedule Principal @ Start of Period $10000.00 $7,967.95 $5,651.42 Interest for Period $1,400.00 $1,115,51 $791.20 Balance $11,400.00 $9,083.47 $6,442.62 Payment $3,432.05 $3,432.05 $3,432.05 Principal Repaid $2,032.05 $2,316.53 $2,640.85 Principal @ End of Period $7,967.95 $5,651.42 $3,010.57

  37. Period: 1 2 3 4 Loan Amortization Schedule Principal @ Start of Period $10000.00 $7,967.95 $5,651.42 $3,010.57 Interest for Period $1,400.00 $1,115,51 $791.20 $421.48 Balance $11,400.00 $9,083.47 $6,442.62 $3,432.05 Payment $3,432.05 $3,432.05 $3,432.05 $3,432.05 Principal Repaid $2,032.05 $2,316.53 $2,640.85 $3,010.57 Principal @ End of Period $7,967.95 $5,651.42 $3,010.57 $0.00

  38. Deferred Annuity • The first cash flow in a deferred annuity is expected to occur later than t=1. • The PV of the deferred annuity can be computed as the difference in the PVs of two annuities.

  39. Deferred Annuity An annuity’s first cash flow is expected to occur 3 years from today. There are 4 cash flows in this annuity, with each cash flow being $500. At an interest rate of 10% per year, find the annuity’s present value.

  40. 0 1 2 3 4 5 6 Deferred Annuity $500 $500 $500 $500

  41. 0 1 2 3 4 5 6 $500 $500 $500 $500 $500 $500 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Deferred Annuity $500 $500 $500 $500 equals minus $500 $500

  42. Deferred Annuity PV of the deferred annuity = PV of 6 year ordinary annuity - PV of 2 year ordinary annuity.

  43. Deferred Annuity = - $2 , 177 . 63 $867 . 77 = $1 , 309 . 86

  44. Perpetuity • A perpetuity is an annuity with an infinite number of cash flows. • The present value of cash flows occurring in the distant future is very close to zero. • At 10% interest, the PV of $100 cash flow occurring 50 years from today is $0.85! • The PV of $100 cash flow occurring 100 years from today is less than one penny!

  45. Present Value of a Perpetuity

  46. Present Value of a Perpetuity

  47. Present Value of a Perpetuity

  48. Present Value of a Perpetuity • As n goes to infinity, 1/(1+r)n goes to 0 • and PVAperpetuity = CF/r

  49. Present Value of a Perpetuity What is the present value of a perpetuity of $270 per year if the interest rate is 12% per year? CF $270 = = = PV $2 , 250 perpetuity r 0 . 12

  50. Multiple Cash Flows • PV of multiple cash flows = the sum of the present values of the individual cash flows. • FV of multiple cash flows at a common point in time = the sum of the future values of the individual cash flows at that point in time.

More Related