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CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

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## CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

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**CHAPTER 2Discounted Cash Flow AnalysisTime Value of**MoneyFinancial Mathematics • Future value • Present value • Rates of return • Amortization • Annuities, AND • Many Examples**MINICASE 2SIMPLE?**p. 88 Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS.**MINICASE 2**• Why is financial mathematics (time value of money) so important in financial analysis?**a.Time lines show timing of cash flows.**ALWAYS A GOOD IDEA TO DRAW A TIME LINE. 0 1 2 3 i% CF0 CF1 CF2 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.**Time line for a $100 lump sum due at the end of Year 2.**0 1 2 Years i% 100**Time line for an ordinary annuity of $100 for 3 years.**0 1 2 3 i% 100 100 100**Time line for uneven CFs -$50 at t = 0 and $100, $75, and**$50 at the end of Years 1 through 3. 0 1 2 3 i% -50 100 75 50**b(1) What’s the FV of an initial$100 after 3 years if i =**10%? 0 1 2 3 10% 100 FV = ? Finding FVs is compounding.**b(1) What’s the FV of an initial$100 after 3 years if i =**10%? 0 1 2 3 10% 100 FV = ? 110 ? Finding FVs is compounding.**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.**After 3 years**FV3 = PV(1 + i)3 = 100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n.**Four Ways to Find FVs**• Solve the equation with a regular calculator • Use tables • Use a financial calculator • Use a spreadsheet**USING TABLES: See handout**• 3 PERIODS • 10 % • = 1.3310 • times 100 = $133.10 • SAY GOOD-BYE TO USING TABLES!**Financial Calculator Solution**Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th.**Here’s the setup to find FV:**3 10 -100 0 N I/YR PV PMT FV INPUTS OUTPUT 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END**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 1 2 3 10% PV = ? 100**Solve FVn = PV(1 + i )n for PV:**PV = FVn (1 + i)n n PV = $100/() = = $100(0.7513) = $75.13. 3 1.10**Financial Calculator Solution**3 10 0 100 N I/YR PV PMT FV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.**EXCEL SOLUTION**• LOOK AT FUNCTION’S PAGE FOR EXCEL/LOTUS.**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**Spreadsheet Solution**• Use the PV function: see spreadsheet. • = PV(Rate, Nper, Pmt, FV) • = PV(0.10, 3, 0, 100) = -75.13**c. If sales grow at 20% per year, how long before sales**double? Solve for n: Time line ? FVn = PV(1 + i)n 2 = 1(1.20)n (1.20)n = 2 n ln(1.20) = ln 2 n(0.1823) = 0.6931 n = 0.6931/0.1823 = 3.8 years.**20 -1 0 2**N I/YR PV PMT FV 3.8 Beware:Some Calculators round up. INPUTS OUTPUT Graphical Illustration: FV 2 3.8 1 Years 0 1 2 3 4**Spreadsheet Solution**• Use the NPER function: see spreadsheet. • = NPER(Rate, Pmt, PV, FV) • = NPER(0.20, 0, -1, 2) = 3.8 Correction**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?**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?**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.**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?**d. What’s the difference between an ordinary annuity and**an annuity due?**d. What’s the difference between an ordinary annuity and**an annuity due? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT 36**HINT**• ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.**e(1). What’s the FV of a 3-year ordinary annuity of $100**at 10%? 0 1 2 3 10% 100 100 100 FV =**e(1). What’s the FV of a 3-year ordinary annuity of $100**at 10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331**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:**Financial Calculator Formula**for Annuities Financial calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Correct but confusing!**Financial Calculator Solution**INPUTS 3 10 0 -100 331.00 N I/YR PV PMT FV OUTPUT Have payments but no lump sum PV, so enter 0 for present value.**Spreadsheet Solution**• Use the FV function: see spreadsheet. • = FV(Rate, Nper, Pmt, Pv) • = FV(0.10, 3, -100, 0) = 331.00**e(2). What’s the PV of this ordinary annuity?**0 1 2 3 10% 100 100 100 _____ = PV**What’s the PV of this ordinary annuity?**0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.69 = PV**3 10 100 0**-248.69 INPUTS N PV PMT FV I/YR OUTPUT Have payments but no lump sum FV, so enter 0 for future value.**Spreadsheet Solution**• Use the PV function: see spreadsheet. • = PV(Rate, Nper, Pmt, Fv) • = PV(0.10, 3, 100, 0) = -248.69**e(3). Find the FV and PV if theannuity were an annuity due.**0 1 2 3 10% 100 100 100**Could, on the 12C, switch from “End” to “Begin”;**i.e.f Begin. Then enter variables to find PVA3 = $273.55. 3 10 100 0 -273.55 INPUTS N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.**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)**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.**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**Switch from “End” to “Begin”.**Then enter variables to find PVA3 = $273.55. INPUTS 3 10 100 0 -273.55 N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.**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)**(f) What is the PV of this uneven cashflow stream?**0 1 2 3 4 10% 100 300 300 -50 ______ = PV