1. © Prentice Hall, 2000 1 Chapter 4 Foundations of Valuation: Time Value Shapiro and Balbirer: Modern Corporate Finance: A Multidisciplinary Approach to Value Creation Graphics by Peeradej Supmonchai

2. © Prentice Hall, 2000 2 Learning Objectives è Explain why money has time value and the importance of the interest rate in the valuation process. è Use the concepts of compound interest to determine the future value of both individual amounts as well as streams of payments. è Use discounting to determine the present value of both individual amounts as well as streams of payments.

3. © Prentice Hall, 2000 3 Learning Objectives (Cont.) è Explain how the concept of present value can be used to value assets ranging from plant and equipment to marketable securities. è Understand the difference between the stated and annual percentage rate (APR), and how this difference influences the present and future values of a stream of payments. è Understand the concept of an investment’s net present value (NPV) and how it relates to the building of shareholder value.

4. © Prentice Hall, 2000 4 Time Value of Money The time value of money is based on the simple idea that a dollar today is worth more than a dollar tomorrow. How much more depends on time preferences of individuals for consumption of goods and services, the rates of return that can be earned on available investments, and the expected rate of inflation.

5. © Prentice Hall, 2000 5 Future Value Formula FV = PV [(1+ k) n ] Where: k = the periodic interest rate n = the number of periods

6. © Prentice Hall, 2000 6 Future Value of $1,000 Investment ACCOUNT BALANCES FOR $1,000 INVESTMENT FIVE YEARS AT 6 PERCENT INTEREST BEGINNING INTEREST EARNED ENDING YEAR BALANCE DURING YEAR BALANCE 1 $1,000.00 $60.00 $1,060.00 2 1,060.00 63.60 1,123.60 3 1,123.60 67.42 1,191.02 4 1,191.02 71.46 1,262.48 5 1,262.48 75.75 1,338.23

7. © Prentice Hall, 2000 7 Future Value Interest Factor Future Value Interest Factor= [(1+ k) n ] Where: k = the periodic interest rate n = the number of period

8. © Prentice Hall, 2000 8 Determinates of Future Value è Amount Invested è Interest Rate è Number of Compounding Periods

9. © Prentice Hall, 2000 9 Future Value of $1 r=10% r=5% r=3% r=1% Period Future Value

10. © Prentice Hall, 2000 10 Frequency of Compounding The future value in n years, when interest is paid m times a year is: F n,m = PV [ (1+k/m) nxm ] Where: k = the annual interest rate

11. © Prentice Hall, 2000 11 Frequency of Compounding - An Example Suppose you invested $1,000 for five years at a six percent interest rate. If interest were compounded semi- annually, the future value would be: F n,m = $1,000[1+(0.06/2)] 5x2 = $1,000[1.3439] = $1,343.90

12. © Prentice Hall, 2000 12 Annual Percentage Rate (APR) APR=FVIF k/m,m - 1 =[1+(k/m) m - 1]

13. © Prentice Hall, 2000 13 Annual Percentage Rate - An Example Suppose a U.S. corporate bond paying interest semiannually has a quoted rate of 9 percent. Its APR is: APR = [ (1.045) 2 ] -1 = 9.2 percent

14. © Prentice Hall, 2000 14 Financial Calculator Keystrokes N or n = the number of periods interest is compounded I or I/Y = the periodic interest rate FV = the future value of a current or present amount PV = the current or present value of a future amount PMT = the periodic payment or receipt. Used when dealing with a stream payments which are the same in each period. CPT = the “compute” button. Some calculators require that you hit this key prior to running a calculation.

15. © Prentice Hall, 2000 15 Financial Calculator Solutions - An Example Future value of $1,000 earning 6 percent for 5 years NIPVPMTFV Inputs 5 6 1,000 Answer:1,338.23

16. © Prentice Hall, 2000 16 Present Value Formula FV PV=  (1+ k) n Where: k = the discount rate n = the number of years

17. © Prentice Hall, 2000 17 Present Value - An Example Suppose you have the opportunity to buy a piece of land for $10,000 today, and sell it in eight years for $20,000. Is this a “good deal” if you can put your money in a risk-equivalent that is expected to earn 10 percent a year compounded annually?

18. © Prentice Hall, 2000 18 Present Value - An Example The present value of the $20,000 you expect to receive at the end of eight years is: PV = $20,000 [ 1/(1.10) 8 ] = $9330.15 This is a “bad deal” since the present value of return in eight years is less than the cost of the land.

19. © Prentice Hall, 2000 19 Calculator Solution NIFVPMTPV Inputs 8 10 20,000 Answer:9,330.15

20. © Prentice Hall, 2000 20 Present Value Interest Factor (PVIF) 1 PVIF =  (1+ k) n Where: k = the discount rate n = the number of years

21. © Prentice Hall, 2000 21 Valuing a Zero-Coupon Bond Suppose that a zero-coupon bond matures in 20 years at a face value of $10,000. If an investor’s opportunity cost of money is 8 percent, the value of the bond would be: PV = FV(PVIF 8,20 ) = $10,000(0.2145) = $2,145.00

22. © Prentice Hall, 2000 22 Valuing a Zero-Coupon Bond - Calculator Solution NIFVPMTPV Inputs 20 8 10,000 Answer:2,145.48

23. © Prentice Hall, 2000 23 Present Value of $1 Present Value Period

24. © Prentice Hall, 2000 24 The Discounting Period When interest is compounded more than once a year, the present value is: 1 PV=FV  (1+ k/m) nxm Where: k = the discount rate n = the number of years m = the number of times that interest is paid a year

25. © Prentice Hall, 2000 25 The Discount Period - An Example If you can earn 8 percent, compounded semiannually, the value of a zero- coupon bond maturing in 20 years at a face amount of $10,000 would be PV = FV(PVIF 4,40 ) = $10,000(0.2083) = $2,083.00

26. © Prentice Hall, 2000 26 Present Value of a Constant Perpetuity CF PV=  k Where: CF = Cash Flow per Period k = Opportunity Cost of Money

27. © Prentice Hall, 2000 27 Present Value of a Constant Perpetuity - An Example Suppose a console pays £50 a year, and the investor’s opportunity cost of money is 10 percent. The price of the console is: £50 Price=  0.10 = £ 500

28. © Prentice Hall, 2000 28 Present Value of a Growing Perpetuity CF PV=  (k - g) Where: CF = Cash Flow per Period k = Opportunity Cost of Money g=Growth Rate per Period

29. © Prentice Hall, 2000 29 Present Value of a Growing Perpetuity - An Example A firm’s cash flows are estimated to be $200,000 next year and are expected to grow at a five percent annual rate of return indefinitely. If the appropriate discount rate is 10 percent, the value of the firm is: $200,000 Value=  (0.10 - 0.05) =$4,000,000

30. © Prentice Hall, 2000 30 Annuities An annuity is a series of equal cash flows per period for a specified number of periods. There are two basic kinds of annuities: è Annuity Due è Deferred Annuity

31. © Prentice Hall, 2000 31 Present Value of Annuity (PVA) Present Value Present Value Present Value PVA n = of Payment + of Payment +   + of Payment in Period 1 in Period 2 in Period n = PMT(PVIF k,1 ) + PMT(PVIF k,2 )    PMT(PVIF k,n )

32. © Prentice Hall, 2000 32 Present Value Interest Factor of an Annuity (PVIFA) (1+ k) n - 1 PVIFA n,m =  k(1+ k) n Where: k = the discount rate n = the number of years

33. © Prentice Hall, 2000 33 Present Value of an Annuity - An Example Suppose you are negotiating with a supplier to buy a piece of equipment that will reduce production costs. The after-tax savings are expected to be $50,000 a year for the next six years. How much is the equipment worth if your company’s opportunity cost of capital is 10 percent?

34. © Prentice Hall, 2000 34 Present Value of an Annuity - Solution PV = PMT (PVIFA 6,10 ) =$50,000 (4.35526) =$217,763

35. © Prentice Hall, 2000 35 Installment Payments on a Loan Suppose a small business borrows $200,000 from a bank at an interest rate of 12 percent compounded annually. The loan, including interest, is to be repaid in equal installments starting next year. The annual payments would be: $200,000 $200,000 PMT =  =  PVIFA 12,3 2.4018 =$83,269.80

36. © Prentice Hall, 2000 36 LOAN AMORTIZATION SCHEDULE $200,000 LOAN @ 12 PERCENT INTEREST Interest Principal Year-End Year Payment Portion Repayment Balance 1 $83, 269.80 $24,000.00 $59,269.80 $140,730.20 2 83,269.80 16,887.62 66,383.20 74,348.00 3 83,269.80 8,921.76 74,348.04 ( 0.04 )

37. © Prentice Hall, 2000 37 Future Value of an Annuity (FVA) Future Value Future Value Future Value Future Value FVA n = of Payment + of Payment +  + of Payment of Payment in Period 1 in Period 2 in Period n - 1 in Period n FVA n = PMT(1+k) n–1 + PMT(1+k) n–2 +   + PMT(1+k) 1 + PMT

38. © Prentice Hall, 2000 38 Future Value of an Annuity - An Example Suppose you were to receive $1,000 a year for three years, and then deposit each receipt in an account paying 8 percent interest, compounded annually. How much would you have at the end of three years?

39. © Prentice Hall, 2000 39 Future Value of an Annuity - Solution CALCULATING THE FUTURE VALUE OF A 3-YEAR ANNUITY Period Cash Flow Future Value 1 $1,000 x (1.08) 2 = $1,166.40 2 1,000 x (1.08) 1 = 1,080.00 3 1,000 x (1.08) = 1,000.00 $3,246.40

40. © Prentice Hall, 2000 40 Future Value Interest Factor for an Annuity (FVIFA) (1+k) n  1 FVIFA k,n =  k Where: k = the discount rate n = the number of years

41. © Prentice Hall, 2000 41 The Annuity Period FVA nm = PMT [FVIFA k/m, nm ] PVA nm = PMT [PVIFA k/m, nm ]

42. © Prentice Hall, 2000 42 Valuing Social Security Suppose you’re 25 years old and have just graduated with an engineering degree. You begin work for a company under a lifetime contract where your salary would remain unchanged at $30,000 a year until retirement in 40 years. Suppose that Social Security has been privatized, so that your 6.2 percent payment, plus the employers’ matching contribution can be put into a personal retirement account. With a salary of $30,000 a year, this means that $310 a month for 480 months will be put in an account earning 6 percent. You can also continue with the existing Social Security program, in which case $310/month would be sent to the government and credited to your account.

43. © Prentice Hall, 2000 43 Value of the Private Retirement Account FVA=PMT[FVIFA 0.50,480 ] = $310 [1,991.49] = $617,362.13

44. © Prentice Hall, 2000 44 VALUE OF $1,232 A MONTH SOCIAL SECURITY PAYMENT Life Expectancy Present Value of Beyond Age 65 Social Security Benefits Years (Months) Discounted @0.5 Percent 5 (60) $ 63,725.89 10 (120) 110,970.50 15 (180) 145,996.33 20 (240) 171,963.51 25 (300) 191,214.85 30 (360) 205,487.27 40 (480) 223,913.02

45. © Prentice Hall, 2000 45 Present Value of Uneven Cash Flow Stream - Equipment Problem Revisited After-Tax Year Cash Flow X PVIF@10% = Present Value 1 $50,000 0.9091 $45,455.00 2 48,000 0.8264 39,667.20 3 45,000 0.7513 33,808.50 4 40,000 0.6830 27,320.00 5 35,000 0.6209 21,731.50 6 40,000* 0.5645 22,580.00 Total Present Value = $190,562.20 * Includes an estimated $10,000 salvage value

46. © Prentice Hall, 2000 46 Present Value of Uneven Cash Flow Streams - Valuing John Smoltz’s Contract Year Payment X PVIF@8% = Present Value 1997 $7,000,000 0.9259 $6,481,481 1998 7,750,000 0.8573 6,644,376 1999 7,750,000 0.7938 6,152,200 2000 8,500,000 0.7350 6,247,754 Total Contract Value = $25,525,811

47. © Prentice Hall, 2000 47 Net Present Value (NPV) è The difference between the present value of an investment’s cash flows and its cost. è Measures how much better off we’ d be by taking on the investment. If the discount rate used in calculating present values represents the stockholders opportunity cost of money, taking on positive NPV projects will create shareholder value.

48. © Prentice Hall, 2000 48 Determinants of the Opportunity Cost of Money è Risk è Inflation è Taxes è Maturity

49. © Prentice Hall, 2000 49 Risk è Default Risk è Price, or Variability Risk è Type of Claim

50. © Prentice Hall, 2000 50 Expected Inflation - The Fisher Effect r = a + i + ai Where: r = the nominal interest rate a = the real or inflation-adjusted interest rate i = the expected rate of inflation

51. © Prentice Hall, 2000 51 Treasury Bill Rates versus Inflation Real Rate Inflation Interest rate % Year

52. © Prentice Hall, 2000 52 Generalized Fisher Effect 1 + r h 1 + i h  =  1 + r f 1 + i f Where: r h = the home country interest rates r f = the foreign currency interest rates i h = the home country inflation rates i f = the foreign country inflation rates