2-1 Chapter Time Value of Money  Future value  Present value  Rates of return  Amortization 2-2 Time lines show timing of cash flows CF1 CF2 CF3 i% CF0 Tick marks at ends of periods, so Time is today; Time is the end of Period 1; or the beginning of Period 2-3 Time line for a $100 lump sum due at the end of Year i% Year 100 2-4 Time line for an ordinary annuity of $100 for years i% 100 100 100 2-5 Time line for uneven CFs: -$50 at t = and $100, $75, and $50 at the end of Years through 3 100 75 50 i% -50 2-6 What’s the FV of an initial $100 after years if i = 10%? 10% 100 FV = ? Finding FVs (moving to the right on a time line) is called compounding 2-7 After year: FV1 = PV + INT1 = PV + PV (i) = PV(1 + i) = $100(1.10) = $110.00 After years: FV2 = FV1(1+i) = PV(1 + i)(1+i) = PV(1+i)2 = $100(1.10)2 = $121.00 2-8 After years: FV3 = FV2(1+i)=PV(1 + i)2(1+i) = PV(1+i)3 = $100(1.10)3 = $133.10 In general, FVn = PV(1 + i)n 2-9 Three Ways to Find FVs  Solve the equation with a regular calculator  Use a financial calculator  Use a spreadsheet 2-10 Financial calculator: HP10BII  Adjust display brightness: hold down ON and push + or -  Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3)  To temporarily show all digits, hit Orange Shift key, then DISP, then = 2-69 What’s the value at the end of Year of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 5% 100 100 6-mos periods 100 2-70  Payments occur annually, but compounding occurs each months  So we can’t use normal annuity valuation techniques 2-71 1st Method: Compound Each CF 5% 100 100 100.00 110.25 121.55 331.80 FVA3 = $100(1.05)4 + $100(1.05)2 + $100 = $331.80 2-72 2nd Method: Treat as an Annuity Could you find the FV with a financial calculator? Yes, by following these steps: a Find the EAR for the quoted rate: EAR = ( 0.10 1+ ) - = 10.25% 2-73 b Use EAR = 10.25% as the annual rate in your calculator: INPUTS 10.25 -100 N OUTPUT I/YR PV PMT FV 331.80 2-74 What’s the PV of this stream? 90.70 82.27 74.62 247.59 100 5% 100 100 2-75 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? 2-76 iPer = 0.018538% per day -850 365 456 days 1,000 Ways to Solve: Greatest future wealth: FV Greatest wealth today: PV Highest rate of return: Highest EFF% 2-77 Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000 FVBank = $850(1.00018538)456 = $924.97 in bank Buy the note: $1,000 > $924.97 2-78 Calculator Solution to FV: iPer = iNom/m = 6.76649%/365 = 0.018538% per day INPUTS 456 N -850 I/YR PV PMT OUTPUT Enter iPer in one step FV 924.97 2-79 Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1,000/(1.00018538)456 = $918.95 2-80 INPUTS 6.76649/365 = 456 018538 N OUTPUT I/YR PV 1000 PMT FV -918.95 PV of note is greater than its $850 cost, so buy the note Raises your wealth 2-81 Rate of Return Find the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital: FVn = PV(1 + i)n $1,000 = $850(1 + i)456 Now we must solve for i 2-82 INPUTS OUTPUT 456 N -850 I/YR PV 0.035646% per day PMT 1000 FV Convert % to decimal: Decimal = 0.035646/100 = 0.00035646 EAR = EFF% = (1.00035646)365 - = 13.89% 2-83 Using interest conversion: P/YR = 365 NOM% = 0.035646(365) = 13.01 EFF% = 13.89 Since 13.89% > 7.0% opportunity cost, buy the note ... + i)n $2 = $1(1 + 0 .20 )n (1 .2) n = $2/ $1 = nLN(1 .2) = LN (2) n = LN (2) /LN(1 .2) n = 0.693/0.1 82 = 3.8 ? 2- 21 Financial Calculator INPUTS N OUTPUT 3.8 20 I/YR -1 PV PMT FV 2- 22 Spreadsheet Solution.. .2- 2 Time lines show timing of cash flows CF1 CF2 CF3 i% CF0 Tick marks at ends of periods, so Time is today; Time is the end of Period 1; or the beginning of Period 2- 3 Time line for... Pmt, PV, FV)  = NPER(0 .20 , 0, -1, 2) = 3.8 2- 23 Finding the Interest Rate -1 ?% FV = PV(1 + i)n $2 = $1(1 + i)3 (2) (1/3) = (1 + i) 1 .25 99 = (1 + i) i = 0 .25 99 = 25 .99% 2- 24 Financial Calculator