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2 - 1 CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future value Present value Rates of return Amortization Annuities,

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Presentation on theme: "2 - 1 CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future value Present value Rates of return Amortization Annuities,"— Presentation transcript:

1 2 - 1 CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future value Present value Rates of return Amortization Annuities, AND Many Examples

2 2 - 2 MINICASE 2 SIMPLE? p. 88 Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS.

3 2 - 3 MINICASE 2 Why is financial mathematics (time value of money) so important in financial analysis?

4 2 - 4 a.Time lines show timing of cash flows. ALWAYS A GOOD IDEA TO DRAW A TIME LINE. CF 0 CF 1 CF 3 CF i% 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; and so on.

5 2 - 5 Time line for a $100 lump sum due at the end of Year Years i%

6 2 - 6 Time line for an ordinary annuity of $100 for 3 years i%

7 2 - 7 Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through i% -50

8 2 - 8 b(1) Whats the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding.

9 2 - 9 b(1) Whats the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding. 110 ?

10 After 1 year FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years FV 2 = FV 1 (1 + i) = PV(1 + i) 2 = $100(1.10) 2 = $

11 FV 3 = PV(1 + i) 3 = 100(1.10) 3 = $ In general, FV n = PV(1 + i) n. After 3 years

12 Solve the equation with a regular calculator Use tables Use a financial calculator Use a spreadsheet Four Ways to Find FVs

13 USING TABLES: See handout 3 PERIODS 10 % = times 100 = $ SAY GOOD-BYE TO USING TABLES!

14 Financial Calculator Solution Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th.

15 NI/YR PV PMTFV Heres the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END INPUTS OUTPUT

16 b(2) Whats the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and its the reverse of compounding % PV = ?

17 PV = Solve FV n = PV(1 + i ) n for PV: PV = $100 / ( ) = = $100(0.7513) = $ FV n (1 + i) n n

18 Financial Calculator Solution INPUTS OUTPUT NI/YR PV PMTFV Either PV or FV must be negative. Here PV = Put in $75.13 today, take out $100 after 3 years.

19 EXCEL SOLUTION LOOK AT FUNCTIONS PAGE FOR EXCEL/LOTUS.

20 Spreadsheet Solution Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. = FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -100) =

21 Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, 100) =

22 c. If sales grow at 20% per year, how long before sales double? Solve for n: Time line ? FV n = PV(1 + i) n 2= 1(1.20) n (1.20) n = 2 n ln (1.20)= ln 2 n(0.1823)= n= / = 3.8 years.

23 INPUTS OUTPUT Graphical Illustration: FV 3.8 Years NI/YR PV PMTFV 3.8 Beware: Some Calculators round up.

24 Spreadsheet Solution Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.20, 0, -1, 2) = 3.8 Correction

25 ADDITIONAL QUESTION A FARMER CAN SPEND $60/ACRE TO PLANT PINE TREES ON SOME MARGINAL LAND. THE EXPECTED REAL RATE OF RETURN IS 4%, AND THE EXPECTED INFLATION RATE IS 6%. WHAT IS THE EXPECTED VALUE OF THE TIMBER AFTER 20 YEARS?

26 ADDITIONAL QUESTION Bill Veeck once bought the Chicago White Sox for $10 million and then sold it five years later for $20 million. In short, he doubled his money in five years. What compound rate of return did Veeck earn on his investment?

27 RULE OF 72 A good approximation of the interest rate--or number of years--required to double your money. n * k required to double = 72 In this case, 5 * k required to double = 72 k = 14.4 Correct answer was 14.87, so for ball-park approximation, use Rule of 72.

28 ADDITIONAL QUESTION John Jacob Astor bought an acre of land in Eastside Manhattan in 1790 for $58. If average interest rate is 5%, did he make a good deal?

29 d. Whats the difference between an ordinary annuity and an annuity due?

30 d. Whats the difference between an ordinary annuity and an annuity due? PMT 0123 i% PMT 0123 i% PMT Annuity Due Ordinary Annuity 36

31 HINT ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.

32 e(1). Whats the FV of a 3-year ordinary annuity of $100 at 10%? % FV =

33 e(1). Whats the FV of a 3-year ordinary annuity of $100 at 10%? % FV = 331

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

35 Financial calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Financial Calculator Formula for Annuities Correct but confusing!

36 NI/YRPVPMTFV Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT

37 Spreadsheet Solution Use the FV function: see spreadsheet. = FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) =

38 e(2). Whats the PV of this ordinary annuity? % _____ = PV

39 Whats the PV of this ordinary annuity? % = PV

40 INPUTS OUTPUT N I/YR PVPMTFV Have payments but no lump sum FV, so enter 0 for future value.

41 Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) =

42 e(3). Find the FV and PV if the annuity were an annuity due % 100

43 INPUTS OUTPUT NI/YRPV PMTFV Could, on the 12C, switch from End to Begin; i.e. f Begin. Then enter variables to find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $

44 Another HINT FV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE FV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k) (slide 30) PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k)

45 HINT, illlustrated The PV of this regular annuity was Multiply this by (1 +.10), and you get: , the PV of the annuity due. This avoids the necessity of having to switch from end to begin.

46 PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: = (PV of ordinary annuity) (1+i) = (248.69) ( ) = FV of annuity due: = (FV of ordinary annuity) (1+i) = (331.00) ( ) = 364.1

47 NI/YRPVPMTFV Switch from End to Begin. Then enter variables to find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $ INPUTS OUTPUT

48 Excel Function for Annuities Due Change the formula to: =PV(10%,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(10%,3,-100,0,1)

49 EXCEL SOLUTION

50 (f) What is the PV of this uneven cash flow stream? % ______ = PV

51 (f) What is the PV of this uneven cash flow stream? % = PV

52 Input in CFLO register: CF 0 =0 CF 1 =100 CF 2 =300 CF 3 =300 CF 4 =-50 Enter I = 10%, then press NPV button to get NPV =

53 CALCULATOR SOLUTION 0 g CF g CF j 300 g CF j 2 g N j 50 CHS g CF j 10 i f NPV

54 EXCEL SOLUTION REMEMBER EXCEL/LOTUS READS THE 1ST CASH FLOW AS OCCURING ONE PERIOD HENCE.

55 Spreadsheet Solution Excel Formula in cell A3: =NPV(10%,B2:E2) ABCDE

56 g. What interest rate would cause $100 to grow to $ in 3 years? INPUTS OUTPUT NI/YR PVFVPMT 8.00% $100 (1 + i ) 3 = $

57 EXCEL SOLUTION

58 h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why?

59 h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why? LARGER!If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

60 % % Annually: FV 3 = 100(1.10) 3 = Semiannually: FV 6 = 100(1.05) 6 =

61 Periodic rate = i Per = i 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. Examples: 8% quarterly: i Per = 8/4 = 2%. 8% daily (365): i Per = 8/365 = %.

62 Effective Annual Rate (EAR = EFF%): The annual rate which causes PV to grow to the same FV as under multiperiod compounding.

63 An investment with monthly compounding is different from one with quarterly compounding. Must put on EAR% basis to compare rates of return. Banks say interest paid daily. Same as compounded daily.

64 h(3).How do we find EAR for a nominal rate of 10%, compounded semiannually? Or use a financial calculator (not 12C)

65 EAR = (1+k nom /m) m ENT 2 divide Y x 1 - =.1025 (1 + EAR) = (1+k nom /m) m

66 CALCULATOR WHAT IS EAR IF COMPOUNDING QUARTERLY? COMPOUNDING DAILY? COMPOUNDING CONTINUOUSLY?

67 EAR = EFF% of 10% EAR Annual = 10%. EAR Q =( /4) 4 - 1= 10.38%. EAR M =( /12) = 10.47%. EAR D(360) =( /360) = 10.52%.

68 For multiple years, n (1 + EAR) = (1 + K nom /m) m (1 + EAR) n = (1 + K nom /m) mn To multiply by a $ of dollars, PRIN PRIN * (1 + EAR) n = PRIN * (1 + K nom /m) mn

69 i. 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, EFF% will always be greater than the nominal rate.

70 When is each rate used? i Nom : Written into contracts, quoted by banks and brokers. May be used in calculations or shown on time lines when compounding is annual. OR USE EAR!

71 Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom. Can always use EAR! i Per :

72 EAR = EFF%:Used to compare returns on investments with different compounding patterns. Also used for calculations if dealing with annuities where payments dont match interest compounding periods.

73 FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily? = $100(1.05) 6 = $ FV 3Q = $100(1.025) 12 = $ FV = PV1.+ i m n Nom mn FV = $ S 2x3

74 ALTERNATE SOLUTION USING EAR FOR SEMIANNUAL COMPOUNDING, EAR = 10.25% FOR 3 YEARS: 100*(1.1025) 3 = $ FOR Quarterly COMPOUNDING and 3 years: 100*(1.1038) 3 = $134.49

75 j(3). Whats the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually? % mos. periods 100

76 Payments occur annually, but compounding occurs each 6 months. So we cant use normal annuity valuation techniques.

77 st Method: Compound Each CF % FVA 3 = 100(1.05) (1.05) =

78 Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity I.E. USE EAR EAR = ( 1 + ) - 1 = 10.25%

79 Or, to find EAR with a 17 OR 19b Calculator: NOM%=10 P/YR= 2 EFF%=10.25

80 INPUTS OUTPUT NI/YRPVFV PMT b. The cash flow stream is an annual annuity whose EFF% = 10.25%. c.

81 j(2) Whats the PV of this stream? %

82 Whats the PV of this stream? %

83 Calculator solution 100 PMT 3 n i f NPV =

84 Whats the FV of this stream under quarterly compouning?

85 EAR WORKSHEET

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

87 Step 1: Find the required payments. PMT % INPUTS OUTPUT NI/YRPVFV PMT

88 ALGEBRA PMT/(1+k) + PMT/(1+k) 2 + PMT/(1+k) 3 = 1000 PMT [1/(1+k) + 1/(1+k) 2 + 1/(1+k) 3 ] = 1000, or PMT = 1000/ [1/(1+k) + 1/(1+k) 2 + 1/(1+k) 3 ]

89 Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = 1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = = $

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

91 Interest declines. Tax implications. BEGPRINEND YRBALPMTINT REDUCTION BAL 1$1,000$402$100$302$ TOT1, ,000

92 $ Interest Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

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

94 EXCEL SOLUTION

95 NEW PROBLEM: On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

96 i Per = 10.0% / 365 = % per day. FV=? % -100 Note: % in calculator, decimal in equation. FV = $ = $ = $

97 INPUTS OUTPUT N I/YRPVFV PMT i Per =i Nom /m =10.0/365 = % per day. Enter i in one step. Leave data in calculator.

98 Now suppose you leave your money in the bank for 21 months, which is 1.75 years or = 638 days. How much will be in your account at maturity? Answer:Override N = 273 with N = 638. FV = $

99 i Per = % per day. FV = days -100 FV=$100( /365) 638 =$100( ) 638 =$100(1.1910) =$

100 ALTERNATIVE SOLUTION USING EAR FIND EAR.10 ENTER 365 DIVIDE Y x [= (1 + EAR)].75 Y x [= (1 + EAR).75 ] 100 MULTIPLY = $107.79

101 MORE PRECISELY, instead of.75 exponent: Calculate [1+EAR] as above, then 273 ENTER 365 DIVIDE [=(1+EAR)] Y x [=(1+EAR) (273/365)] 100MULTIPLY = $107.77

102 PROBLEM SUPPOSE THAT YOU WERE TOLD THAT THE EFFECTIVE ANNUAL RATE WERE 10%, WITH DAILY COMPOUNDING. WHAT THE STATED, OR NOMINAL RATE BE IN THIS CASE?

103 ALGEBRA (1 + EAR) = (1 + k nom /m) m (1 + EAR) (1/m) = (1 + k nom /m) (1 + EAR) (1/m) - 1 = k nom /m m*[(1 + EAR) (1/m) - 1] = k nom

104 m*[(1 + EAR) (1/m) - 1] = k nom Using the calculator, EAR = 10%, daily compounding. 1.1 ENTER 365 1/X Y x MULTIPLY = 9.53%

105 n. 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 7.0% nominal rate, with 365 daily compounding, which is a daily rate of % and an EAR of 7.25%. You plan to leave the money in the bank if you dont buy the note. The note is riskless. Should you buy it?

106 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% i Per = % per day. 1, days -850

107 Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with notes FV = $1000. FV Bank =$850( ) 456 =$ in bank. Buy the note: $1000 > $

108 INPUTS OUTPUT NI/YRPVFV PMT Calculator Solution to FV: i Per =i Nom /m =7.0/365 = % per day. Enter i Per in one step.

109 Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV=$1000/[( ) 456 ] =$

110 INPUTS OUTPUT NI/YRPVFV PMT 7/365 = PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

111 Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FV n = PV(1 + i) n 1000 = $850(1 + i) 456 Now we must solve for i. 3. Rate of Return

112 % per day INPUTS OUTPUT NI/YRPV FV PMT Convert % to decimal: Decimal = /100 = EAR = EFF%= ( ) = 13.89%.

113 Using interest conversion: P/YR=365 NOM%= (365)= EFF%=13.89 Since 13.89% > 7.25% opportunity cost, buy the note.

114 ADDITIONAL PROBLEM #2 IT IS NOW JANUARY 1. YOU PLAN TO MAKE 5 DEPOSITS OF $100 EACH, ONE EVERY 6 MONTHS, WITH THE FIRST PAYMENT BEING MADE TODAY. IF THE BANK PAYS A NOMINAL INTEREST RATE OF 12 PERCENT, SEMIANNUAL COMPOUNDING, HOW MUCH WILL BE IN YOUR ACCOUNT AFTER 10 YEARS?

115 ADDITIONAL PROBLEM #3 IT IS NOW JANUARY 1, YOU ARE OFFERED A NOTE UNDER WHICH SOMEONE PROMISES TO MAKE 5 PAYMENTS OF $100 EACH, WITH THE FIRST PAYMENT ON JULY 1, 1997 AND SUBSEQUENT PAYMENTS ON EACH JULY 1 THEREAFTER THROUGH JULY 1, 2001, PLUS A FINAL PAYMENT OF $1000 TO BE MADE ON JANUARY 1, WITH A NOMINAL DISCOUNT RATE OF 10 PERCENT, QUARTERLY COMPOUNDING, WHAT IS THE PV OF THE NOTE?

116 JOHNM PROBLEMS


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