Tài chính doanh nghiệp - Chapter 6: Time value of money

CHAPTER 6Time Value of MoneyFuture valuePresent valueAnnuitiesRates of returnAmortizationTime linesShow 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.CF0CF1CF3CF20123i%Drawing time lines:$100 lump sum due in 2 years;3-year $100 ordinary annuity1001001000123i%3 year $100 ordinary annuity100012i%$100 lump sum due in 2 yearsDrawing time lines:Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50 100 50 750123i%-50Uneven cash flow streamWhat 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 = ?012310%100Solving for FV:The arithmetic methodAfter 1 year:FV1 = PV ( 1 + i ) = $100 (1.10) = $110.00After 2 years:FV2 = PV ( 1 + i )2 = $100 (1.10)2 =$121.00After 3 years:FV3 = PV ( 1 + i )3 = $100 (1.10)3 =$133.10After n years (general case):FVn = PV ( 1 + i )nSolving for FV:The calculator methodSolves the general FV equation.Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.)INPUTSOUTPUTNI/YRPMTPVFV3100133.10-100PV = ?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.012310%Solving for PV:The arithmetic methodSolve the general FV equation for PV:PV = FVn / ( 1 + i )nPV = FV3 / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.13Solving for PV:The calculator methodSolves the general FV equation for PV.Exactly like solving for FV, except we have different input information and are solving for a different variable.INPUTSOUTPUTNI/YRPMTPVFV3100100-75.13Solving 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.INPUTSOUTPUTNI/YRPMTPVFV3.82002-1What is the difference between an ordinary annuity and an annuity due?Ordinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity DueSolving for FV:3-year ordinary annuity of $100 at 10%$100 payments occur at the end of each period, but there is no PV.INPUTSOUTPUTNI/YRPMTPVFV310-1003310Solving 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.INPUTSOUTPUTNI/YRPMTPVFV3101000-248.69Solving 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.INPUTSOUTPUTNI/YRPMTPVFV310-100364.100Solving 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.INPUTSOUTPUTNI/YRPMTPVFV3101000-273.55What is the PV of this uneven cash flow stream?010013002300310%-504 90.91247.93225.39 -34.15530.08 = PVSolving for PV:Uneven cash flow streamInput cash flows in the calculator’s “CFLO” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.)Solving for I:What interest rate would cause $100 to grow to $125.97 in 3 years?Solves the general FV equation for I.INPUTSOUTPUTNI/YRPMTPVFV380125.97-100The 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?Solving for FV:Savings problemIf she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65.INPUTSOUTPUTNI/YRPMTPVFV4512-10951,487,2620Solving for FV:Savings problem, if you wait until you are 40 years old to startIf a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.Lesson: It pays to start saving early.INPUTSOUTPUTNI/YRPMTPVFV2512-1095146,0010Solving for PMT:How much must the 40-year old deposit annually to catch the 20-year old?To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.INPUTSOUTPUTNI/YRPMTPVFV2512-11,154.421,487,2620Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant?LARGER, as the more frequently compounding occurs, interest is earned on interest more often.Annually: FV3 = $100(1.10)3 = $133.10012310%100133.10Semiannually: FV6 = $100(1.05)6 = $134.0101235%456134.011230100Classifications of interest ratesNominal rate (iNOM) – also called the quoted or state rate. An annual rate that ignores compounding effects.iNOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest.Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly.iPER = iNOM / m, where m is the number of compounding periods per year. m = 4 for quarterly and m = 12 for monthly compounding.Classifications of interest ratesEffective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, taking into account compounding.EFF% for 10% semiannual investment EFF% = ( 1 + iNOM / m )m - 1 = ( 1 + 0.10 / 2 )2 – 1 = 10.25%An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.Why is it important to consider effective rates of return?An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See following values of EFF% rates at various compounding levels. EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY (365) 10.52%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.When is each rate used?iNOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.iPER Used in calculations and shown on time lines. If m = 1, iNOM = iPER = EAR.EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.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?Payments occur annually, but compounding occurs every 6 months.Cannot use normal annuity valuation techniques.01100235%451001006123Method 1:Compound each cash flow110.25121.55331.80FV3 = $100(1.05)4 + $100(1.05)2 + $100FV3 = $331.8001100235%451006123100Method 2:Financial calculatorFind the EAR and treat as an annuity.EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.INPUTSOUTPUTNI/YRPMTPVFV310.25-100331.800Find the PV of this 3-year ordinary annuity.Could solve by discounting each cash flow, or Use the EAR and treat as an annuity to solve for PV.INPUTSOUTPUTNI/YRPMTPVFV310.251000-247.59Loan amortizationAmortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc.Financial calculators and spreadsheets are great for setting up amortization tables.EXAMPLE: Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.Step 1:Find the required annual paymentAll input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan.INPUTSOUTPUTNI/YRPMTPVFV310402.110-1000Step 2:Find the interest paid in Year 1The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate. INTt = Beg balt (i) INT1 = $1,000 (0.10) = $100Step 3:Find the principal repaid in Year 1If a payment of $402.11 was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid. PRIN = PMT – INT = $402.11 - $100 = $302.11Step 4:Find the ending balance after Year 1To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance. END BAL = BEG BAL – PRIN = $1,000 - $302.11 = $697.89Constructing an amortization table:Repeat steps 1 – 4 until end of loanInterest paid declines with each payment as the balance declines. What are the tax implications of this?YearBEG BALPMTINTPRINEND BAL1$1,000$402$100$302$6982698402703323663366402373660TOTAL1,206.34206.341,000-Illustrating an amortized payment:Where does the money go?Constant payments.Declining interest payments.Declining balance.$0123402.11Interest302.11Principal PaymentsPartial amortizationBank agrees to lend a home buyer $220,000 to buy a $250,000 home, requiring a $30,000 down payment.The home buyer only has $7,500 in cash, so the seller agrees to take a note with the following terms:Face value = $22,5007.5% nominal interest ratePayments made at the end of the year, based upon a 20-year amortization schedule.Loan matures at the end of the 10th year.Calculating annual loan paymentsBased upon the loan information, the home buyer must make annual payments of $2,207.07 on the loan.INPUTSOUTPUTNI/YRPMTPVFV207.52207.070-22500Determining the balloon paymentUsing an amortization table (spreadsheet or calculator), it can be found that at the end of the 10th year, the remaining balance on the loan will be $15,149.54.Therefore,Balloon payment = $15,149.54Final payment = $17,356.61