1. Introduction to Valuation- The Time Value of Money

2. 2 Time Value Topics Future value Present value Rates of return Amortization

3. The Time Value of Money Compounding and Discounting Single Sums

4. We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. Today Future

5. If we can measure this opportunity cost, we can: Translate $1 today into its equivalent in the future (compounding). Translate $1 in the future into its equivalent today (discounting ? Today Future Today ? Future

6. Time Lines of The Cash Flows

7. 7 Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 R% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

8. 8 Ordinary Annuity PMT 0123 R% PMT 0123 R% PMT Annuity Due Ordinary Annuity vs. Annuity Due

9. Future Value

10. 10 Time line for a $100 lump sum due at the end of Year 2. 100 012 Year R%

11. 11 Time line for an ordinary annuity of $100 for 3 years 100 0123 R%

12. 12 Time line for uneven CFs 100 50 75 0123 R% -50

13. 13 Compound Interest Interest is earned on previously earned interest $100 invested at 10% with annual compounding 1 st yearinterest is $10.00Principal is $110 2 nd yearinterest is $21.0Principal is $121.0 3 rd yearinterest is $33.10Principal is $133.10 Total interest earned: $33.10

14. 14 FV of an initial $100 after 3 years (R = 10%) FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding. 100

15. 15 After 1 year FV 1 = PV + RNT 1 = PV + PV (R) = PV(1 + R) = $100(1.10) = $110.00.

16. 16 After 2 years FV 2 = FV 1 (1+R) = PV(1 + R)(1+R) = PV(1+R) 2 = $100(1.10) 2 = $121.00.

17. 17 After 3 years FV 3 = FV2(1+R)=PV(1 + R) 2 (1+R) = PV(1+R) 3 = $100(1.10) 3 = $133.10 In general, FV N = PV(1 + R) N.

18. 18 Financial calculators solve this equation: FV N + PV (1+R) N = 0. There are 4 variables. If 3 are known, the calculator will solve for the 4th. Financial Calculator Solution

19. 19 Calculator Keys Texas Instruments BA-II Plus FV = future value PV = present value I/Y = period interest rate P/Y must equal 1 for the I/Y to be the period rate Interest is entered as a percent, not a decimal N = number of periods Remember to clear the registers (CLR TVM) after each problem Other calculators are similar in format

20. 20 310-100 0 NI/YR PV PMTFV 133.10 INPUTS OUTPUT The setup to find FV

21. 21 PMT INPUTS OUTPUT FV PV I/YR N N If the $100 is entered as a positive Number, then -133.10 3 10 100 0

22. 22 Quick Quiz – Part I What is the difference between simple interest and compound interest? Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. How much would you have at the end of 15 years using compound interest? How much would you have using simple interest?

23. Present Value

24. 24 10% What’s the PV of $100 due in 3 years if R = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 PV = ?

25. 25 Solve FV N = PV(1 + R ) N for PV PV = FV N (1+R) N = FV n () 1 1 +R N  PV= $100 1 1.10 = $1000.7513 = $75.13.       3

26. 26 3 10 0100 N I/YR PV PMTFV -75.13 Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT Financial Calculator Solution

27. 27 Present Values – Example 2 You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? PV = 150,000 / (1.08) 17 = 40,540.34

28. 28 Present Values – Example 3 Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? PV = 19,671.51 / (1.07) 10 = 10,000

29. 29 Present Value – Important Relationship I For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% 5 years: PV = 500 / (1.1) 5 = 310.46 10 years: PV = 500 / (1.1) 10 = 192.77

30. 30 Present Value – Important Relationship II For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? Rate = 10%: PV = 500 / (1.1) 5 = 310.46 Rate = 15%; PV = 500 / (1.15) 5 = 248.59

31. 31 Quick Quiz – Part II What is the relationship between present value and future value? Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?

32. 32 20% 2 012? FV= PV(1 + R) N Continued on next slide Finding the Time to Double

33. 33 Time to Double (Continued) $2= $1(1 + 0.20) N (1.2) N = $2/$1 = 2 N LN(1.2)= LN(2) N= LN(2)/LN(1.2) N= 0.693/0.182 = 3.8.

34. 34 Time to Double– Example 2 Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? r = (20,000 / 10,000) 1/6 – 1 =.122462 = 12.25%

35. 35 Discount Rate – Example 3 Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it? r = (75,000 / 5,000) 1/17 – 1 =.172688 = 17.27%

36. 36 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT Financial Calculator Solution

37. 37 Finding the Interest Rate or Discount Rate Often we will want to know what the implied interest rate is in an investment Rearrange the basic PV equation and solve for r FV = PV(1 + R) N r = (FV / PV) 1/N – 1

38. 38 ?% 2 0123 FV= PV(1 + R) N $2= $1(1 + R) 3 (2) (1/3) = (1 + R) 1.2599= (1 + R) I= 0.2599 = 25.99%. Finding the Interest Rate

39. 39 3 -1 0 2 NI/YR PV PMTFV 25.99 INPUTS OUTPUT Financial Calculator

40. 40 Finding the Interest Rate- Example 2 Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it? r = (75,000 / 5,000) 1/17 – 1 =.172688 = 17.27%

41. 41 Finding the Number of Periods Start with basic equation and solve for N (remember your logs) FV = PV(1 + R) N N = ln(FV / PV) / ln(1 + R) You can use the financial keys on the calculator as well; just remember the sign convention.

42. 42 Number of Periods – Example 1 You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? N= ln(20,000 / 15,000) / ln(1.1) = 3.02 years

43. 43 Number of Periods – Example 2 Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

44. 44 Number of Periods – Example 2 Continued How much do you need to have in the future? Down payment =.1(150,000) = 15,000 Closing costs =.05(150,000 – 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750 Compute the number of periods Using the formula N= ln(21,750 / 15,000) / ln(1.075) = 5.14 years Per a financial calculator: PV = -15,000, FV = 21,750, I/Y = 7.5, CPT N = 5.14 years

45. 45 Quick Quiz – Part III What are some situations in which you might want to know the implied interest rate? You are offered the following investments: You can invest $500 today and receive $600 in 5 years. The investment is considered low risk. You can invest the $500 in a bank account paying 4%. What is the implied interest rate for the first choice and which investment should you choose?

46. 46 Spreadsheet Example Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv) The formula icon is very useful when you can’t remember the exact formula Click on the Excel icon to open a spreadsheet containing four different examples.

47. The Time Value of Money Compounding and Discounting Cash Flow Streams 01 234

48. Annuity: a sequence of equal cash flows, occurring at the end of each period. 01 234 Annuities

49. 49 Ordinary Annuity PMT 0123 R% PMT 0123 R% PMT Annuity Due Ordinary Annuity vs. Annuity Due

50. 50 What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

51. 51 FV Annuity Formula The future value of an annuity with n periods and an interest rate of R can be found with the following formula:

52. 52 3 10 0 -100 331.00 NI/YRPVPMTFV Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT Financial Calculator Solution

53. 53 What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

54. 54 PV Annuity Formula The present value of an annuity with n periods and an interest rate of i can be found with the following formula:

55. 55 Have payments but no lump sum FV, so enter 0 for future value. 3 10 100 0 NI/YRPVPMTFV -248.69 INPUTS OUTPUT Financial Calculator Solution

56. 56 Annuity – Sweepstakes Example Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

57. 57 Buying a House You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

58. 58 Buying a House - Continued Bank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment =.28(3,000) = 840 30*12 = 360 N.5 I/Y -840 PMT CPT PV = 140,105 Total Price Closing costs =.04(140,105) = 5,604 Down payment = 20,000 – 5,604 = 14,396 Total Price = 140,105 + 14,396 = 154,501

59. 59 Annuities on the Spreadsheet - Example The present value and future value formulas in a spreadsheet include a place for annuity payments Click on the Excel icon to see an example

60. 60 Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

61. 61 PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: = (PV of ordinary annuity) (1+R) = (248.69) (1+ 0.10) = 273.56 FV of annuity due: = (FV of ordinary annuity) (1+R) = (331.00) (1+ 0.10) = 364.1

62. 62 310 100 0 -273.55 NI/YRPVPMTFV INPUTS OUTPUT PV of Annuity Due: Switch from “End” to “Begin

63. 63 310 0 100 -364.1 NI/YRPVPMTFV INPUTS OUTPUT FV of Annuity Due: Switch from “End” to “Begin

64. Other Cash Flow Patterns 0123 The Time Value of Money

65. 65 What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

66. Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. You can think of a perpetuity as an annuity that goes on forever.

67. PMT R PV = So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity

68. What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 PMT $10,000 R.08 R.08 = $125,000 PV = =

69. 69 Nominal rate (R NOM ) Stated in contracts, and quoted by banks and brokers. Not used in calculations or shown on time lines Periods per year (M) must be given. Examples: 8%; Quarterly 8%, Daily interest (365 days)

70. 70 Periodic rate (I PER ) R PER = R NOM /M, where M is number of compounding periods per year. M = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Used in calculations, shown on time lines. Examples: 8% quarterly: R PER = 8%/4 = 2%. 8% daily (365): R PER = 8%/365 = 0.021918%.

71. 71 The Impact of Compounding Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated R% constant? Why?

72. 72 The Impact of Compounding (Answer) LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

73. 73 FV Formula with Different Compounding Periods FV = PV1.+ R NOM M N M N     

74. 74 $100 at a 12% nominal rate with semiannual compounding for 5 years = $100(1.06) 10 = $179.08 FV = PV1+ R NOM M N M N      FV = $1001+ 0.12 2 5S 2x5      

75. 75 FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= $100(1.12) 5 = $176.23. FV(Semiannual)= $100(1.06) 10 =$179.08. FV(Quarterly)= $100(1.03) 20 = $180.61. FV(Monthly)= $100(1.01) 60 = $181.67. FV(Daily)= $100(1+(0.12/365)) (5x365) = $182.19.

76. 76 Effective Annual Rate (EAR = EAR%) The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding.

77. 77 Effective Annual Rate Example Example: Invest $1 for one year at 12%, semiannual: FV = PV(1 + R NOM /M) M FV = $1 (1.06)2 = 1.1236. EAR% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

78. 78 Comparing Rates An investment with monthly payments is different from one with quarterly payments. Must put on EAR% basis to compare rates of return. Use EAR% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

79. 79 Decisions, Decisions You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = (1 +.0525/365) 365 – 1 = 5.39% Second account: EAR = (1 +.053/2) 2 – 1 = 5.37% Which account should you choose and why?

80. 80 EAR (or EAR%) for a Nominal Rate of of 12% EAR Annual = 12%. EAR Q =(1 + 0.12/4) 4 - 1= 12.55%. EAR M =(1 + 0.12/12) 12 - 1= 12.68%. EAR D(365) =(1 + 0.12/365) 365 - 1= 12.75%.

81. 81 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if M = 1. If M > 1, EAR% will always be greater than the nominal rate.

82. 82 When is each rate used? R NOM : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

83. 83 R PER : Used in calculations, shown on time lines. If R NOM has annual compounding, then R PER = R NOM /1 = R NOM. When is each rate used? (Continued)

84. 84 When is each rate used? (Continued) EAR (or EAR%): Used to compare returns on investments with different payments per year. Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

85. 85 Amortization Loans Amortized Loan: a loan that is repaid in equal payments over its life. Each periodic payment includes not only interest but also a portion of principal.

86. 86 Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

87. 87 Step 1: Find the required payments. PMT 0123 10% -1,000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

88. 88 Step 2: Find interest charge for Year 1. INT t = Beg bal t (R) INT 1 = $1,000(0.10) = $100.

89. 89 Repmt = PMT - INT = $402.11 - $100 = $302.11. Step 3: Find repayment of principal in Year 1.

90. 90 Step 4: Find ending balance after Year 1. End bal= Beg bal - Repmt = $1,000 - $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

91. 91 Amortization Table YEAR BEG BALPMTINT PRIN PMT END BAL 1$1,000$402$100$302$698 269840270332366 3 402373660 TOT1,206.34206.341,000

92. 92 Interest declines because outstanding balance declines.

93. 93 Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables. Amortization

94. 94 Amortized Loan with Fixed Principal Payment - Example Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year. Click on the Excel icon to see the amortization table

95. 95 Amortized Loan with Fixed Payment - Example Each payment covers the interest expense plus reduces principal Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000. What is the annual payment? 4 N 8 I/Y 5,000 PV CPT PMT = -1,509.60 Click on the Excel icon to see the amortization table

96. 96 Fractional Time Periods On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

97. 97 R PER = 11.33463%/365 = 0.031054% per day. FV=? 012273 0.031054% -100 Convert interest to daily rate

98. 98   FV = $1001.00031054 = $1001.08846= $108.85. 273 Find FV

99. 99 273-100 0 108.85 INPUTS OUTPUT N I/YRPVFV PMT R PER =R NOM /M =11.33463/365 =0.031054% per day. Calculator Solution

100. 100 Non-matching rates and periods What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

101. 101 Time line for non-matching rates and periods 01 100 23 5% 45 6 6-mos. periods 100

102. 102 Non-matching rates and periods Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

103. 103 1st Method: Compound Each CF 01 100 23 5% 456 100100.00 110.25 121.55 331.80 FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $331.80.

104. 104 2nd Method: Treat as an annuity, use financial calculator Find the EAR for the quoted rate: EAR = ( 1 + ) - 1 = 10.25%. 0.10 2 2

105. 105 3 10.25 0 -100 INPUTS OUTPUT N I/YR PVFV PMT 331.80 Use EAR = 10.25% as the annual rate in calculator.

106. 106 What’s the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59

107. 107 Comparing Investments You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

108. 108 R PER =0.018538% per day. 1,000 0365456 days -850 Daily time line

109. 109 Three solution methods 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EAR%

110. 110 1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FV Bank =$850(1.00018538) 456 =$924.97 in bank. Buy the note: $1,000 > $924.97.

111. 111 456-850 0 924.97 INPUTS OUTPUT NI/YRPVFV PMT R PER =R NOM /M =6.76649%/365 =0.018538% per day. Calculator Solution to FV

112. 112 Find PV of note, and compare with its $850 cost: PV=$1,000/(1.00018538) 456 =$918.95. 2. Greatest Present Wealth

113. 113 456.018538 0 1000 -918.95 INPUTS OUTPUT NI/YRPVFV PMT 6.76649/365 = PV of note is greater than its $850 cost, so buy the note. Raises your wealth. Financial Calculator Solution

114. 114 Find the EAR% on note and compare with 7.0% bank pays, which is your opportunity cost of capital: FV N = PV(1 + R) N $1,000= $850(1 + R) 456 Now we must solve for R. 3. Rate of Return

115. 115 456-850 0 1000 0.035646% per day INPUTS OUTPUT NI/YRPV FV PMT Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. EAR = EAR%= (1.00035646) 365 - 1 = 13.89%. Calculator Solution

116. 116 P/YR=365 NOM%=0.035646(365)= 13.01 EAR%=13.89 Since 13.89% > 7.0% opportunity cost, buy the note. Using interest conversion

117. 117 Growing Perpetuities The cash flows of a growing perpetuity grow at a constant rate forever. The present value of a growing perpetuity is: PV = PMT R - g

118. 118 Present Value of Growing Annuity Cash flows in business are very likely to grow over time, due either to real growth or inflation. The present value of a growing annuity is for a finite number of growing cash flows:

119. 119 Present Value of Growing Annuity- Example Ruben Ramirez, a first year MBA student at CSUF, is going to be offered a job at $100,000 a year after his graduation. He anticipates his salary increasing by 6% a year until his retirement in 30 years. Given an interest rate of 10 percent, what is the present value of his lifetime salary?