1 Prentice Hall, 1998 Chapter 5 The Time Value of Money

2 Prentice Hall, 1998 Learning Objectives n Calculate present and future values of any set of expected future cash flows. n Explain how the present value and discount rate are inversely related. n Calculate payments on a debt contract. n Compute the APR and APY for a contract. n Value “special financing” offers.

3 Prentice Hall, 1998 Definitions and Assumptions n A point in time is denoted by the letter “t”. n Unless otherwise stated, t=0 represents today (the decision point). n Unless otherwise stated, cash flows occur at the end of a time interval. n Cash inflows are treated as positive amounts, while cash outflows are treated as negative amounts. n Compounding frequency is the same as the cash flow frequency.

4 Prentice Hall, 1998 The Time Line t=0 t=1t=2t=3t=4

5 Prentice Hall, 1998 The Time Line t=0 t=1t=2t=3t=4 Today

6 Prentice Hall, 1998 The Time Line t=0 t=1t=2t=3t=4 Today End of the third year

7 Prentice Hall, 1998 The Time Line t=0 t=1t=2t=3t=4 Today Beginning of the fourth year End of the third year

8 Prentice Hall, 1998 Future Value Formula Let PV= Present Value FV n = Future Value at time n r= interest rate (or discount rate) per period. r= interest rate (or discount rate) per period.

9 Prentice Hall, 1998 r = 0% r = 10% Future Value Factor Time FV Factor r = 5% r = 15%

10 Prentice Hall, 1998 Present Value Formula Let PV= Present Value FV n = Future Value at time n r= interest rate (or discount rate) per period. r= interest rate (or discount rate) per period.

11 Prentice Hall, 1998 Present Value Factors Time PV Factor r = 5% r = 10% r = 0% r = 15%

12 Prentice Hall, 1998 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, Assuming you hold the CD to maturity, what annual interest rate is the bank paying on this CD?

13 Prentice Hall, 1998 Solving for an Unknown Interest Rate (CD) PV = $3,477.87; FV 4 = $5,000; n = 4 years. SinceFVPVr) n n  (1

14 Prentice Hall, 1998 Solving for an Unknown Interest Rate (CD) PV = $3,477.87; FV 4 = $5,000; n = 4 years. SinceFVPVr) n n  (1

15 Prentice Hall, 1998 Annuities n An annuity is a series of identical cash flows that are expected to occur each period for a specified number of periods. n Thus, CF 1 = CF 2 = CF 3 = Cf 4 =... = CF n Examples of annuities: v Installment loans (car loans, mortgages). v Coupon payment on corporate bonds. v Rent payment on your apartment.

16 Prentice Hall, 1998 Types of Annuities n Ordinary Annuity: v An annuity with end-of-Period cash flows, beginning one period from today. n Annuity Due: v An annuity with beginning-of-period cash flows. n Deferred Annuity: v An annuity that begins more than one period from today.

17 Prentice Hall, 1998 Future Value of an Annuity FVA n = CF(1+r) 0 + CF(1+r) CF(1+r) n-1

18 Prentice Hall, 1998 Future Value of an Annuity FVA n = CF[summation {from 0 to n-1} of (1+r) t ] FVA n = CF(1+r) 0 + CF(1+r) CF(1+r) n-1 = CF[(1+r) 0 + (1+r) (1+r) n-1 ]

19 Prentice Hall, 1998 Future Value of an Annuity

20 Prentice Hall, 1998 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 Prentice Hall, 1998 Future Value of Your Savings  $,.

22 Prentice Hall, 1998 Present Value of an Annuity

23 Prentice Hall, 1998 Present Value of an Annuity

24 Prentice Hall, 1998 Present Value of an Annuity

25 Prentice Hall, 1998 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 Prentice Hall, 1998 Present Value of Your Bank Loan  $5,.

27 Prentice Hall, 1998 Payments of an Annuity (Given FVA n )

28 Prentice Hall, 1998 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 Prentice Hall, 1998 Saving for Retirement  33614$20,.

30 Prentice Hall, 1998 Payments of an Annuity (Given PVA n )

31 Prentice Hall, 1998 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 Prentice Hall, 1998 Installment Payments on a Loan  43205$3,.

33 Prentice Hall, 1998 Loan Amortization Schedule n It shows how a loan is paid off over time. n It breaks down each payment into the interest component and the principal component. n Let’s illustrate this using Rob’s 4-year $10,000 loan which calls for annual payments of $3, Recall that the interest rate on this loan is 14% per year.

34 Prentice Hall, 1998 Loan Amortization Schedule Period:1234 Start of Period of Period Interest for Period Balance Payment Principal Repaid End of Period of Period $ $1, $11, $3, $2, $7,967.95

35 Prentice Hall, 1998 Loan Amortization Schedule Start of Period of Period Interest for Period Balance Payment Principal Repaid End of Period of Period $ $1, $11, $3, $2, $7, $7, $1,115,51 $9, $3, $2, $5,651.42Period:1234

36 Prentice Hall, 1998 Loan Amortization Schedule Start of Period of Period Interest for Period Balance Payment Principal Repaid End of Period of Period $ $1, $11, $3, $2, $7, $7, $1,115,51 $9, $3, $2, $5, $5, $ $6, $3, $2, $3,010.57Period:1234

37 Prentice Hall, 1998 Loan Amortization Schedule Start of Period of Period Interest for Period Balance Payment Principal Repaid End of Period of Period $ $1, $11, $3, $2, $7, $7, $1,115,51 $9, $3, $2, $5, $5, $ $6, $3, $2, $3, $3, $ $3, $3, $3, $0.00Period:1234

38 Prentice Hall, 1998 Deferred Annuity n The first cash flow in a deferred annuity is expected to occur later than t=1. n The PV of the deferred annuity can be computed as the difference in the PVs of two annuities.

39 Prentice Hall, 1998 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 Prentice Hall, 1998 Deferred Annuity $500$500$500$500

41 Prentice Hall, 1998 Deferred Annuity $500$500$500$ $500$500$500$500$500$ $500$500 equals minus

42 Prentice Hall, 1998 Deferred Annuity PV of the deferred annuity = PV of 6 year ordinary annuity - PV of 2 year ordinary annuity.

43 Prentice Hall, 1998 Deferred Annuity   $2,.$867. $1,

44 Prentice Hall, 1998 Perpetuity n A perpetuity is an annuity with an infinite number of cash flows. n The present value of cash flows occurring in the distant future is very close to zero. v At 10% interest, the PV of $100 cash flow occurring 50 years from today is $0.85! v The PV of $100 cash flow occurring 100 years from today is less than one penny!

45 Prentice Hall, 1998 Present Value of a Perpetuity

46 Prentice Hall, 1998 Present Value of a Perpetuity

47 Prentice Hall, 1998 Present Value of a Perpetuity

48 Prentice Hall, 1998 Present Value of a Perpetuity n As n goes to infinity, 1/(1+r) n goes to 0 n and PVA perpetuity = CF/r

49 Prentice Hall, 1998 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? PV CF r perpetuity  $270. $2,

50 Prentice Hall, 1998 Multiple Cash Flows n PV of multiple cash flows = the sum of the present values of the individual cash flows. n 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.

51 Prentice Hall, 1998 Multiple Cash Flow Example Consider the following cash flows: TimeNotationCash Flow 0CF0- $ 2,000 1CF1+ $ 1,000 2CF2+ $ 1,500 3CF3+ $ 2,000 The interest rate (r) is 10% per period.

52 Prentice Hall, 1998 Time Line of Multiple Cash Flows t=0 t=1t=2 t=3 -$ 2,000 +$ 1,000 +$ 1,500 +$ 2,000

53 Prentice Hall, 1998 PV of Multiple Cash Flows PV      $2,. $1,. $1,. $2,. $2,$909.$1,.$1,. $1,. 000 (110) 000 (110) 500 (110) 000 (110)

54 Prentice Hall, 1998 FV (at t=3) of Multiple Cash Flows   $2,$1,$1650$2, $2,.

55 Prentice Hall, 1998 FV (at t=3) of Multiple Cash Flows Alternatively FVPVr), $1,..$2, ( (110)19800  

56 Prentice Hall, 1998 FV (at t=2) of Multiple Cash Flows FV (110)000 (1 (110) 500(110) 000 (110)     $2,.$1,. $1,. $2,. $2,.$1,.$1,$1,. $1,.

57 Prentice Hall, 1998 FV (at t=2) of Multiple Cash Flows Alternatively or FV FV r), $2,.. $1, ( (110)   

58 Prentice Hall, 1998 More Frequent Compounding n Interest may be compounded more than once per year. n The Annual Percentage Rate (APR) is the periodic rate times the number of periods per year. n The Annual Percentage Yield (APY) is the “true” annually compounded interest rate.

59 Prentice Hall, 1998 APR and APY for an Installment Loan Suppose you borrow $5,000 from the bank and promise to repay the loan in 12 equal monthly installments of $ each, with the first payment to be made one month from today. What is the APR? What is the APR? n What is the APY?

60 Prentice Hall, 1998 APR and APY for an Installment Loan n Solving this for r, we get r = 0.75% per month n APR = 0.75 x 12 = 9% per year

61 Prentice Hall, 1998 APR and APY for an Installment Loan n To calculate the APY, compound the periodic rate for the number of periods in one year n in this case, 12 months to make one year: or  %..

62 Prentice Hall, 1998 Effect of Compounding Frequency on Future Value Find the future value at the end of one year if the present value is $20,000 and the APR is 16%. Use the following compounding frequencies: n Annual Compounding n Semiannual Compounding n Quarterly Compounding n Monthly Compounding n Daily Compounding n Continuous Compounding

63 Prentice Hall, 1998 Annual Compounding Since m = 1, the periodic rate is 16%. FV (116)20000  $20,.$23,. APY = APR = 16%

64 Prentice Hall, 1998 Semi - Annual Compounding Since m = 2, the periodic rate is 8%. APR = 2 x 8 = 16% FV 2 = 20,000(1.08) 2 = $23,328.00

65 Prentice Hall, 1998 Quarterly Compounding Since m = 4, the periodic rate is 4%. APR = 4 x 4 = 16% FV (104)39717  $20,.$23,.

66 Prentice Hall, 1998 Effect of Compounding Frequency on Future Value CompoundingmFVAPY Annual Semi-Annual Quarterly Monthly Daily $23, $23, $23, $23, $23, % % % % % n APR = 16%

67 Prentice Hall, 1998 Continuous Compounding n With continuous compounding, m becomes very large. n As m approaches infinity, APY = e APR - 1, where e = n So APY = ( ) = or %. n FV = $20,000 ( ) 1 = $23,469.39

68 Prentice Hall, 1998 Partial Time Periods What is the future value 3.5 years from today of $5,000 at an APR of 12%?

69 Prentice Hall, 1998 Partial Time Periods What is the future value 3.5 years from today of $5,000 at an APR of 12%? FVPVr) (1000(112) $5,. $7,.  

70 Prentice Hall, 1998 Valuing Special Financing Offers Miller Motors is offering the following alternatives on a Dodge Intrepid, which has a stated price of $24,000. n $1500 “cash back,” or n “Special” 36-month 3.5% APR financing? If you can borrow from M&T Bank at 9% APR, which alternative is better?

71 Prentice Hall, 1998 Valuing Special Financing Offers The general procedure is to measure the opportunity cost of the special financing, and compare this cost to the net “cash-back” price.

72 Prentice Hall, 1998 Valuing Special Financing Offers First, compute the monthly payments for the special financing plan. 3.50% APR implies a monthly rate of 3.50%/12, or 0.292% per month.

73 Prentice Hall, 1998 Valuing Special Financing Offers First, compute the monthly payments for the special financing plan. 3.50% APR implies a monthly rate of 3.50%/12, or 0.292% per month. $  PMTpermonth

74 Prentice Hall, 1998 Valuing Special Financing Offers n Note: The monthly payments are “real,” even though the 3.5% APR is not “real.” Next, find the present value of these payments at the bank ’s lending rate. The 9% APR bank loan implies a monthly interest rate is 9%/12, or 0.75% per month. 0.75% per month.

75 Prentice Hall, 1998 Valuing Special Financing Offers Next, find the present value of these payments at the bank’s lending rate. The 9% APR bank loan implies a monthly interest rate is 9%/12, or 0.75% per month. 0.75% per month. The present value is:

76 Prentice Hall, 1998 Valuing Special Financing Offers Next, find the present value of these payments at the bank’s lending rate. The 9% APR bank loan implies a monthly interest rate is 9%/12, or 0.75% per month. 0.75% per month. The present value is: n This is a “real” price for buying the car.

77 Prentice Hall, 1998 Valuing Special Financing Offers Finally, compare this present value of $22, to the net price under the cash-back plan: Price net of $1,500 cash back = $24,000 - $1,500 = $22,500 This is also a “real” price for buying the car. Which price would you prefer?

78 Prentice Hall, 1998 Valuing Special Financing Offers The “special financing” at $22,114.96, or The “cash back” price at $22,500? The difference of $ between the two amounts is the Net Present Value (NPV) of the special financing plan.

79 Prentice Hall, 1998 Valuing Special Financing Offers Another way to look at this problem is to find the monthly payments if you borrow the “cash back” price ($22,500) from the bank at 9% APR.

80 Prentice Hall, 1998 Valuing Special Financing Offers Another way to look at this problem is to find the monthly payments if you borrow the ‘cash-back’ price ($22,500) from the bank at an APR of 9%. These payments are: $  PMTpermonth

81 Prentice Hall, 1998 Valuing Special Financing Offers n The monthly payments under the cash back plan would be $ n Under the special financing plan, the monthly payments are $ n Thus, you save $12.24 per month (for 36 months) under the special financing plan. n The PV of these savings at the bank’s rate of 0.75% per month is exactly the NPV of the special financing plan ( = $385.04)