Chapter 5

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.