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CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future value Present value Rates of return Amortization Annuities, AND Many Examples 2-1 2-2 MINICASE 2 SIMPLE? p. 88 Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS. 2-3 MINICASE 2 Why is financial mathematics (time value of money) so important in financial analysis? 2-4 a.Time lines show timing of cash flows. ALWAYS A GOOD IDEA TO DRAW A TIME LINE. 0 CF0 i% 1 2 CF1 CF2 3 CF3 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. 2-5 Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Years i% 100 2-6 Time line for an ordinary annuity of $100 for 3 years. 0 i% 1 2 3 100 100 100 2-7 Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 0 -50 i% 1 2 3 100 75 50 2-8 b(1) What’s the FV of an initial $100 after 3 years if i = 10%? 0 100 10% 1 2 3 FV = ? Finding FVs is compounding. 2-9 b(1) What’s the FV of an initial $100 after 3 years if i = 10%? 0 100 10% 1 110 2 ? 3 FV = ? Finding FVs is compounding. 2 - 10 After 1 year FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years FV2 = FV1(1 + i) = PV(1 + i)2 = $100(1.10)2 = $121.00. 2 - 11 After 3 years FV3 = PV(1 + i)3 = 100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n. 2 - 12 Four Ways to Find FVs Solve the equation with a regular calculator Use tables Use a financial calculator Use a spreadsheet 2 - 13 USING TABLES: See handout 3 PERIODS 10 % = 1.3310 times 100 = $133.10 SAY GOOD-BYE TO USING TABLES! 2 - 14 Financial Calculator Solution Financial calculators solve this equation: F V n P V 1 i . n There are 4 variables. If 3 are known, the calculator will solve for the 4th. 2 - 15 Here’s the setup to find FV: INPUTS 3 N 10 I/YR -100 PV 0 PMT OUTPUT FV 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END 2 - 16 b(2) What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 PV = ? 10% 1 2 3 100 2 - 17 Solve FVn = PV(1 + i )n for PV: n FVn PV = (1 + i)n 3 ( ) PV = $100/ 1.1 = 0 = $100(0.7513) = $75.13. 2 - 18 Financial Calculator Solution INPUTS OUTPUT 3 N 10 I/YR PV -75.13 0 PMT 100 FV Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. 2 - 19 EXCEL SOLUTION LOOK AT FUNCTION’S PAGE FOR EXCEL/LOTUS. 2 - 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) = 133.10 2 - 21 Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, 100) = -75.13 2 - 22 c. If sales grow at 20% per year, how long before sales double? Solve for n: FVn 2 (1.20)n n ln (1.20) n(0.1823) n Time line ? = PV(1 + i)n = 1(1.20)n =2 = ln 2 = 0.6931 = 0.6931/0.1823 = 3.8 years. 2 - 23 INPUTS OUTPUT N 3.8 20 I/YR -1 PV 0 PMT 2 FV Beware:Some Calculators round up. Graphical Illustration: FV 2 3.8 1 Years 0 1 2 3 4 2 - 24 Spreadsheet Solution Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.20, 0, -1, 2) = 3.8 Correction 2 - 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? 2 - 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? 2 - 27 RULE OF 72 A good approximation of the interest rate--or number of years--required to double your money. n * krequired to double = 72 In this case, 5 * krequired to double = 72 • k = 14.4 Correct answer was 14.87, so for ball-park approximation, use Rule of 72. 2 - 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? 2 - 29 d. What’s the difference between an ordinary annuity and an annuity due? 2 - 30 d. What’s the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 1 2 PMT PMT 1 2 PMT PMT 3 i% PMT Annuity Due 0 3 i% PMT 36 2 - 31 HINT ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT. 2 - 32 e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 100 100 100 10% FV = 2 - 33 e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 100 100 100 10% 110 121 FV = 331 2 - 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: n PMT (1 i) 1 i 3 100 (1 0 . 10 ) 1 0.10 331 . 2 - 35 Financial Calculator Formula for Annuities Financial calculators solve this equation: FV n PV 1 i n n 1 (1 i) PMT 0. i There are 5 variables. If 4 are known, the calculator will solve for the 5th. Correct but confusing! 2 - 36 Financial Calculator Solution INPUTS OUTPUT 3 10 0 -100 N I/YR PV PMT FV 331.00 Have payments but no lump sum PV, so enter 0 for present value. 2 - 37 Spreadsheet Solution Use the FV function: see spreadsheet. = FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) = 331.00 2 - 38 e(2). What’s the PV of this ordinary annuity? 0 10% _____ = PV 1 2 3 100 100 100 2 - 39 What’s the PV of this ordinary annuity? 0 10% 90.91 82.64 75.13 248.69 = PV 1 2 3 100 100 100 2 - 40 INPUTS OUTPUT 3 10 N I/YR PV 100 0 PMT FV -248.69 Have payments but no lump sum FV, so enter 0 for future value. 2 - 41 Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) = -248.69 2 - 42 e(3). Find the FV and PV if the annuity were an annuity due. 0 100 10% 1 2 100 100 3 2 - 43 Could, on the 12C, switch from “End” to “Begin”; i.e. f Begin. Then enter variables to find PVA3 = $273.55. INPUTS OUTPUT 3 10 N I/YR PV 100 0 PMT FV -273.55 Then enter PV = 0 and press FV to find FV = $364.10. 2 - 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) PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k) (slide 30) 2 - 45 HINT, illlustrated The PV of this regular annuity was 248.69. Multiply this by (1 + .10), and you get: 273.55, the PV of the annuity due. This avoids the necessity of having to switch from end to begin. 2 - 46 PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: = (PV of ordinary annuity) (1+i) = (248.69) (1+ 0.10) = 273.56 FV of annuity due: = (FV of ordinary annuity) (1+i) = (331.00) (1+ 0.10) = 364.1 2 - 47 Switch from “End” to “Begin”. Then enter variables to find PVA3 = $273.55. INPUTS OUTPUT 3 10 N I/YR PV 100 0 PMT FV -273.55 Then enter PV = 0 and press FV to find FV = $364.10. 2 - 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) 2 - 49 EXCEL SOLUTION 2 - 50 (f) What is the PV of this uneven cash flow stream? 0 1 2 3 4 100 300 300 -50 10% ______ = PV 2 - 51 (f) What is the PV of this uneven cash flow stream? 0 10% 1 2 3 4 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV 2 - 52 Input in “CFLO” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I = 10%, then press NPV button to get NPV = 530.09. 2 - 53 CALCULATOR SOLUTION 0 100 300 2 50 10 f g CF0 g CFj g CFj g Nj CHS g CFj i NPV 2 - 54 EXCEL SOLUTION REMEMBER EXCEL/LOTUS READS THE 1ST CASH FLOW AS OCCURING ONE PERIOD HENCE. 2 - 55 Spreadsheet Solution 1 A B C D E 0 1 2 3 4 100 300 300 -50 2 3 530.09 Excel Formula in cell A3: =NPV(10%,B2:E2) 2 - 56 g. What interest rate would cause $100 to grow to $125.97 in 3 years? $100 (1 + i )3 = $125.97. INPUTS 3 N OUTPUT -100 I/YR 8.00% PV 0 PMT 125.97 FV 2 - 57 EXCEL SOLUTION 2 - 58 h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why? 2 - 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. 2 - 60 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 0 0 1 1 2 3 2 4 5 3 6 5% 100 134.01 Semiannually: FV6 = 100(1.05)6 = 134.01. 2 - 61 Periodic rate = iPer = iNom/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: iPer = 8/4 = 2%. 8% daily (365): iPer = 8/365 = 0.021918%. 2 - 62 Effective Annual Rate (EAR = EFF%): The annual rate which causes PV to grow to the same FV as under multiperiod compounding. 2 - 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. 2 - 64 h(3).How do we find EAR for a nominal rate of 10%, compounded semiannually? EFF% 1 = + i Nom m m - 1 2 = = = 0 .1 0 1+ - 1 .0 2 2 1 .0 5 - 1 .0 0 .1 0 2 5 = 1 0 .2 5 % . Or use a financial calculator (not 12C) EAR = (1+knom/m)m - 1 .10 ENT 2 divide 1 + 2 Yx 1 = .1025 (1 + EAR) = (1+knom/m)m 2 - 65 2 - 66 CALCULATOR WHAT IS EAR IF COMPOUNDING QUARTERLY? COMPOUNDING DAILY? COMPOUNDING CONTINUOUSLY? 2 - 67 EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 - 1 = 10.38%. EARM = (1 + 0.10/12)12 - 1 = 10.47%. EARD(360) = (1 + 0.10/360)360 - 1 = 10.52%. 2 - 68 For multiple years, n (1 + EAR) = (1 + Knom/m)m (1 + EAR)n = (1 + Knom/m)mn To multiply by a $ of dollars, PRIN PRIN * (1 + EAR)n = PRIN * (1 + Knom/m)mn 2 - 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. 2 - 70 When is each rate used? iNom: 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! 2 - 71 iPer: Used in calculations, shown on time lines. If iNom has annual compounding, then iPer = iNom/1 = iNom. Can always use EAR! 2 - 72 EAR = EFF%: Used to compare returns on investments with different compounding patterns. Also used for calculations if dealing with annuities where payments don’t match interest compounding periods. 2 - 73 FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily? iNom FVn = PV 1 + m FV3S FV3Q mn 0.10 = $100 1 + 2 . 2x3 = $100(1.05)6 = $134.01. = $100(1.025)12 = $134.49. 2 - 74 ALTERNATE SOLUTION USING EAR FOR SEMIANNUAL COMPOUNDING, EAR = 10.25% FOR 3 YEARS: 100*(1.1025)3 = $134.01 FOR Quarterly COMPOUNDING and 3 years: 100*(1.1038)3 = $134.49 2 - 75 j(3). What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually? 0 1 2 3 4 5% 100 100 5 6 6-mos. periods 100 2 - 76 Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques. 2 - 77 1st Method: Compound Each CF 0 5% 1 2 3 4 6 100.00 110.25 121.55 331.80 FVA3 = 100(1.05)4 + 100(1.05)2 + 100 = 331.80. 100 100 5 2 - 78 2nd Method: Treat as an Annuity I.E. USE EAR Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = ( 0.10 1+ 2 2 ) - 1 = 10.25%. 2 - 79 Or, to find EAR with a 17 OR 19b Calculator: NOM% P/YR EFF% = = = 10 2 10.25 2 - 80 b. The cash flow stream is an annual annuity whose EFF% = 10.25%. c. INPUTS OUTPUT 3 10.25 N I/YR 0 -100 PV PMT FV 331.80 2 - 81 j(2) What’s the PV of this stream? 0 5% 1 2 3 100 100 100 2 - 82 What’s the PV of this stream? 0 5% 90.70 82.27 74.62 247.59 1 2 3 100 100 100 2 - 83 Calculator solution 100 PMT 3 n 10.25 i f NPV = 247.59 2 - 84 What’s the FV of this stream under quarterly compouning? 0 1 2 3 100 100 100 2 - 85 EAR WORKSHEET EAR worksheet knom = m= EAR = 0.12 INPUT 8 INPUT 0.126493 EAR= m= knom= 0.1255 INPUT 4 INPUT 0.119992 2 - 86 k. Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments. 2 - 87 Step 1: Find the required payments. 0 10% -1000 INPUTS OUTPUT 1 2 3 PMT PMT PMT 3 10 -1000 N I/YR PV 0 PMT 402.11 FV 2 - 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] 2 - 89 Step 2: Find interest charge for Year 1. INTt = Beg balt (i) INT1 = 1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = 402.11 - 100 = $302.11. 2 - 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. 2 - 91 YR BEG BAL PRIN PMT INT 1 $1,000 $402 $100 2 698 402 70 3 366 402 37 TOT 1,206.34 206.34 REDUCTION $302 332 366 1,000 END BAL $698 366 0 Interest declines. Tax implications. 2 - 92 $ 402.11 Interest 302.11 Principal Payments 0 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. 2 - 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. 2 - 94 EXCEL SOLUTION 2 - 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.) 2 - 96 iPer 0 = 10.0% / 365 = 0.027397% per day. 1 2 273 0.027397% FV=? -100 FV273 = $1001.00027397 = $1001.07765 = $107.77. 273 Note: % in calculator, decimal in equation. 2 - 97 iPer = iNom/m = 10.0/365 = 0.027397% per day. INPUTS 273 N OUTPUT -100 I/YR PV 0 FV PMT 107.77 Enter i in one step. Leave data in calculator. 2 - 98 Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days. How much will be in your account at maturity? Answer: Override N = 273 with N = 638. FV = $119.10. 2 - 99 iPer = 0.027397% per day. 0 -100 365 638 days FV = 119.10 FV = $100(1 + 0.10/365)638 = $100(1.00027397)638 = $100(1.1910) = $119.10. 2 - 100 ALTERNATIVE SOLUTION USING EAR FIND EAR .10 ENTER 365 DIVIDE 1 + 365 Yx [= (1 + EAR)] .75 Yx [= (1 + EAR).75] 100 MULTIPLY = $107.79 2 - 101 MORE PRECISELY, instead of .75 exponent: Calculate [1+EAR] as above, then 273 ENTER 365 Yx DIVIDE [=(1+EAR)] [=(1+EAR)(273/365)] 100 MULTIPLY = $107.77 2 - 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? 2 - 103 ALGEBRA (1 + EAR) = (1 + knom /m)m (1 + EAR)(1/m) = (1 + knom /m) (1 + EAR)(1/m) - 1 = knom /m m*[(1 + EAR)(1/m) - 1] = knom 2 - 104 m*[(1 + EAR)(1/m) - 1] = knom Using the calculator, EAR = 10%, daily compounding. 1.1 365 1 365 = ENTER 1/X Yx MULTIPLY 9.53% 2 - 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 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it? 2 - 106 iPer =0.019178% per day. 0 -850 365 456 days 1,000 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% 2 - 107 1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1000. FVBank = = $850(1.00019178)456 $927.67 in bank. Buy the note: $1000 > $927.67. 2 - 108 Calculator Solution to FV: iPer = iNom/m = 7.0/365 = 0.019178% per day. INPUTS 456 N I/YR -850 0 PV PMT OUTPUT Enter iPer in one step. FV 927.67 2 - 109 2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1000/[(1.00019178)456] = $916.27. 2 - 110 INPUTS 7/365 = 456 .019178 N OUTPUT I/YR 0 PV 1000 PMT FV -916.27 PV of note is greater than its $850 cost, so buy the note. Raises your wealth. 2 - 111 3. Rate of Return Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FVn = PV(1 + i)n 1000 = $850(1 + i)456 Now we must solve for i. 2 - 112 INPUTS 456 N OUTPUT -850 I/YR PV 0.035646% per day 0 1000 PMT FV Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. EAR = EFF% = (1.00035646)365 - 1 = 13.89%. 2 - 113 Using interest conversion: P/YR NOM% EFF% = = = 365 0.035646(365) = 13.01 13.89 Since 13.89% > 7.25% opportunity cost, buy the note. 2 - 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? 2 - 115 ADDITIONAL PROBLEM #3 IT IS NOW JANUARY 1, 1997. 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, 2002. WITH A NOMINAL DISCOUNT RATE OF 10 PERCENT, QUARTERLY COMPOUNDING, WHAT IS THE PV OF THE NOTE? 2 - 116 JOHNM PROBLEMS