CHAPTER 6 Time Value of MoneyTime linesDrawing time lines: $100 lump sum due in 2 years; 3-year $100 ordinary annuityDrawing time lines: Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?Solving for FV: The arithmetic methodSolving for FV: The calculator methodWhat is the present value (PV) of $100 due in 3 years, if I/YR = 10%?Solving for PV: The arithmetic methodSolving for PV: The calculator methodSolving for N: If sales grow at 20% per year, how long before sales double?What is the difference between an ordinary annuity and an annuity due?Solving for FV: 3-year ordinary annuity of $100 at 10%Solving for PV: 3-year ordinary annuity of $100 at 10%Solving for FV: 3-year annuity due of $100 at 10%Solving for PV: 3 year annuity due of $100 at 10%What is the PV of this uneven cash flow stream?Solving for PV: Uneven cash flow streamSolving for I: What interest rate would cause $100 to grow to $125.97 in 3 years?The Power of Compound InterestSolving for FV: Savings problemSolving for FV: Savings problem, if you wait until you are 40 years old to startSolving for PMT: How much must the 40-year old deposit annually to catch the 20-year old?Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant?Classifications of interest ratesSlide 26Why is it important to consider effective rates of return?Can the effective rate ever be equal to the nominal rate?When is each rate used?What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?Method 1: Compound each cash flowMethod 2: Financial calculatorFind the PV of this 3-year ordinary annuity.Loan amortizationStep 1: Find the required annual paymentStep 2: Find the interest paid in Year 1Step 3: Find the principal repaid in Year 1Step 4: Find the ending balance after Year 1Constructing an amortization table: Repeat steps 1 – 4 until end of loanIllustrating an amortized payment: Where does the money go?Partial amortizationCalculating annual loan paymentsDetermining the balloon payment6-1CHAPTER 6Time Value of MoneyFuture valuePresent valueAnnuitiesRates of returnAmortization6-2Time linesShow the timing of cash flows.Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.CF0CF1CF3CF20 1 2 3i%6-3Drawing time lines:$100 lump sum due in 2 years;3-year $100 ordinary annuity100 1001000 1 2 3i%3 year $100 ordinary annuity1000 1 2i%$100 lump sum due in 2 years6-4Drawing time lines:Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50 100 50 750 1 2 3i%-50Uneven cash flow stream6-5What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding.FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.FV = ?0 1 2 310%1006-6Solving for FV:The arithmetic methodAfter 1 year:FV1 = PV ( 1 + i ) = $100 (1.10) = $110.00After 2 years:FV2 = PV ( 1 + i )2 = $100 (1.10)2 =$121.00After 3 years:FV3 = PV ( 1 + i )3 = $100 (1.10)3 =$133.10After n years (general case):FVn = PV ( 1 + i )n6-7Solving for FV:The calculator methodSolves the general FV equation.Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.)INPUTSOUTPUTN I/YR PMTPV FV3 10 0133.10-1006-8PV = ? 100What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding).The PV shows the value of cash flows in terms of today’s purchasing power.0 1 2 310%6-9Solving for PV:The arithmetic methodSolve the general FV equation for PV:PV = FVn / ( 1 + i )nPV = FV3 / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.136-10Solving for PV:The calculator methodSolves the general FV equation for PV.Exactly like solving for FV, except we have different input information and are solving for a different variable.INPUTSOUTPUTN I/YR PMTPV FV3 10 0 100-75.136-11Solving for N:If sales grow at 20% per year, how long before sales double?Solves the general FV equation for N.Same as previous problems, but now solving for N.INPUTSOUTPUTN I/YR PMTPV FV3.820 0 2-16-12What is the difference between an ordinary annuity and an annuity due?Ordinary AnnuityPMT PMTPMT0 1 2 3i%PMT PMT0 1 2 3i%PMTAnnuity Due6-13Solving for FV:3-year ordinary annuity of $100 at 10%$100 payments occur at the end of each period, but there is no PV.INPUTSOUTPUTN I/YR PMTPV FV3 10 -10033106-14Solving for PV:3-year ordinary annuity of $100 at 10%$100 payments still occur at the end of each period, but now there is no FV.INPUTSOUTPUTN I/YR PMTPV FV3 10 100 0-248.696-15Solving for FV:3-year annuity due of $100 at 10%Now, $100 payments occur at the beginning of each period.Set calculator to “BEGIN” mode.INPUTSOUTPUTN I/YR PMTPV FV3 10 -100364.1006-16Solving for PV:3 year annuity due of $100 at 10%Again, $100 payments occur at the beginning of each period.Set calculator to “BEGIN” mode.INPUTSOUTPUTN I/YR PMTPV FV3 10 100 0-273.556-17What is the PV of this uneven cash flow stream?010013002300310%-504 90.91247.93225.39 -34.15530.08 = PV6-18Solving for PV:Uneven cash flow streamInput cash flows in the calculator’s “CFLO” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.)6-19Solving for I:What interest rate would cause $100 to grow to $125.97 in 3 years?Solves the general FV equation for I.INPUTSOUTPUTN I/YR PMTPV FV380125.97-1006-20The Power of Compound InterestA 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095) in an online stock account. The stock account has an expected annual return of 12%.How much money will she have when she is 65 years old?6-21Solving for FV:Savings problemIf she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is