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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-1Chapter FiveFirst Principles of Valuation:The Time Value of Money
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-25.1 Future Value and Compounding5.2 Present Value and Discounting5.3 More on Present and Future Values5.4 Present and Future Values of Multiple Cash Flows5.5 Valuing Equal Cash Flows: Annuities and Perpetuities5.6 Comparing Rates: The Effect of Compounding Periods5.7 Loan Types and Loan Amortisation5.8 Summary and ConclusionsChapter Organisation
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-3Chapter Objectives• Distinguish between simple and compound interest.• Calculate the present value and future value of a single amountfor both one period and multiple periods.• Calculate the present value and future value of multiple cashflows.• Calculate the present value and future value of annuities.• Compare nominal interest rates (NIR) and effective annualinterest rates (EAR).• Distinguish between the different types of loans and calculate thepresent value of each type of loan.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-4Time Value Terminology• Future value (FV) is the amount an investment is worth afterone or more periods.• Present value (PV) is the current value of one or more futurecash flows from an investment.0 1 2 3 4PV FV
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-5Time Value Terminology• The number of time periods between the presentvalue and the future value is represented by ‘t’.• The rate of interest for discounting orcompounding is called ‘r’.• All time value questions involve four values: PV,FV, r and t. Given three of them, it is alwayspossible to calculate the fourth.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-6Interest Rate Terminology• Simple interest refers to interest earned only on theoriginal capital investment amount.• Compound interest refers to interest earned onboth the initial capital investment and on theinterest reinvested from prior periods.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-7Future Value of a Lump SumYou invest $100 in a savings account that earns 10 per centinterest per annum (compounded) for three years.After one year: $100 × (1 + 0.10) = $110After two years: $110 × (1 + 0.10) = $121After three years: $121 × (1 + 0.10) = $133.10
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-8Future Value of a Lump Sum• The accumulated value of this investment at theend of three years can be split into twocomponents:– original principal $100– interest earned $33.10• Using simple interest, the total interest earnedwould only have been $30. The other $3.10 is fromcompounding.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-9Future Value of a Lump Sum• In general, the future value, FVt, of $1 investedtoday at r per cent for t periods is:• The expression (1 + r)tis the future value interestfactor (FVIF). Refer to Table A.1.( )tt r+×= 1$1FV
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-10Example—Future Value of a LumpSum• What will $1000 amount to in five years time if interest is 12per cent per annum, compounded annually?• From the example, now assume interest is 12 per cent perannum, compounded monthly.• Always remember that t is the number of compoundingperiods, not the number of years.( )762.30$11.7623$10000.121$1000FV5=×=+=( )$1816.701.8167$10000.011$1000FV60=×=+=
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-11Interpretation• The difference in values is due to the largernumber of periods in which interest can compound.• Future values also depend critically on theassumed interest rate—the higher the interest rate,the greater the future value.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-12Future Values at Different InterestRatesFuture value of $100 at various interest ratesNumber ofperiods5% 10% 15% 20%1 $105.00 $110.00 $115.00 $120.002 $110.25 $121.00 $132.25 $144.003 $115.76 $133.10 $152.09 $172.804 $121.55 $146.41 $174.90 $207.365 $127.63 $161.05 $201.14 $248.83
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-13Future Value of $1 for DifferentPeriods and Rates
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-14Present Value of a Lump SumYou need $1000 in three years time. If you can earn 10 percent per annum, how much do you need to invest now?Discount one year: $1000 (1 + 0.10) –1= $909.09Discount two years: $909.09 (1 + 0.10) –1= $826.45Discount three years: $826.45 (1 + 0.10) –1= $751.32
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-15Interpretation• In general, the present value of $1 received in t periods oftime, earning r per cent interest is:• The expression (1 + r)–tis the present value interest factor(PVIF). Refer to Table A.2.( )( )ttrr+=+×=−1$11$1PV
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-16Example—Present Value of a LumpSumYour rich grandmother promises to give you $10 000 in 10 yearstime. If interest rates are 12 per cent per annum, how much isthat gift worth today?( )3220$0.3220000$100.121000$10PV10=×=+×=−
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-17Present Values at Different InterestRatesPresent value of $100 at various interest ratesNumber ofperiods5% 10% 15% 20%1 $95.24 $90.91 $86.96 $83.332 $90.70 $82.64 $75.61 $69.443 $86.38 $75.13 $65.75 $57.874 $82.27 $68.30 $57.18 $48.235 $78.35 $62.09 $49.72 $40.19
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-18Present Value of $1 for DifferentPeriods and RatesPresentvalueof $1 ($)Time(years)r = 0%r = 5%r = 10%r = 15%r = 20%1 2 3 4 5 6 7 8 9 101.00.90.80.70.60.50.40.30.20.10
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-19Solving for the Discount Rate• You currently have $100 available for investment for a 21-year period. At what interest rate must you invest thisamount in order for it to be worth $500 at maturity?• Given any three factors in the present value or future valueequation, the fourth factor can be solved.r can be solved in one of three ways:• Use a financial calculator• Take the nthroot of both sides of the equation• Use the future value tables to find a corresponding value. Inthis example, you need to find the r for which the FVIF after21 years is 5 (500/100).
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-20The Rule of 72• The ‘Rule of 72’ is a handy rule of thumb that states:If you earn r per cent per year, your money will double inabout 72/r per cent years.• For example, if you invest at 6 per cent, your money willdouble in about 12 years.• This rule is only an approximate rule.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-21Future Value of Multiple Cash Flows• You deposit $1000 now, $1500 in one year, $2000 in twoyears and $2500 in three years in an account paying 10 percent interest per annum. How much do you have in theaccount at the end of the third year?• You can solve by either:– compounding the accumulated balance forward oneyear at a time– calculating the future value of each cash flow first andthen totalling them.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-22Solutions• Solution 1– End of year 1: ($1000 × 1.10) + $1500 = $2600– End of year 2: ($2600 × 1.10) + $2000 = $4860– End of year 3: ($4860 × 1.10) + $2500 = $ 846• Solution 2$1000 × (1.10)3= $1331$1500 × (1.10)2= $1815$2000 × (1.10)1= $2200$2500 × 1.00 = $2500Total = $7846
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-23SolutionsFuture value calculated by compounding forward one period at a timeTime(years)0 1 2 3 4 5$00$0$ 02000$2000$22002000$4200$46202000$6620$72822000$9282$10 210.202000.00$12 210.20x 1.1 x 1.1 x 1.1 x 1.1 x 1.1Time(years)0 1 2 3 4 5$2000 $2000 $2000 $2000 $2000.02200.02420.02662.02928.2$12 210.20x 1.14x 1.13x 1.12x 1.1Total future valueFuture value calculated by compounding each cash flow separatelyFigures 5.6/5.7 — Calculation of FV for Multiple Cash FlowStream
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-24Present Value of Multiple Cash Flows• You will deposit $1500 in one year’s time, $2000 in two yearstime and $2500 in three years time in an account paying 10per cent interest per annum. What is the present value ofthese cash flows?• You can solve by either:– discounting back one year at a time– calculating the present value of each cash flow first andthen totalling them.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-25Solutions• Solution 1– End of year 2: ($2500 × 1.10–1) + $2000 = $4273– End of year 1: ($4273 × 1.10–1) + $1500 = $5385– Present value: ($5385 × 1.10–1) = $4895• Solution 2$2500 × (1.10) –3= $1878$2000 × (1.10) –2= $1653$1500 × (1.10) –1= $1364Total = $4895
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-26SolutionsPresent valuecalculated bydiscounting eachcash flow separately0 1 2 3 4 5$1000 $1000 $1000 $1000$ 943.40890.00839.62792.09747.26$4212.37x 1/1.065Total present valueTime(years)$1000r = 6%x 1/1.064x 1/1.063x 1/1.062x 1/1.060 1 2 3 4 5$4212.370.00$4212.37$3465.111000.00$4465.11$2673.011000.00$3673.01$1833.401000.00$2833.40$ 943.401000.00$1943.40$ 0.001000.00$1000.00Present valuecalculated bydiscounting back oneperiod at a timeTime(years)Total present value = $4212.37r = 6%Figures 5.8/5.9 — Calculation of PV for Multiple Cash FlowStream
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-27Annuities• An ordinary annuity is a series of equal cash flowsthat occur at the end of each period for some fixednumber of periods.• Examples include consumer loans and homemortgages.• A perpetuity is an annuity in which the cash flowscontinue forever.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-28Present Value of an AnnuityC = equal cash flow• The discounting term is called the present valueinterest factor for annuities (PVIFA). Refer toTable A.3.( ){ } +−×=rrCt11/1PV
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-29• Example 1You will receive $500 at the end of each of thenext five years. The current interest rate is 9per cent per annum. What is the present valueof this series of cash flows?( ){ }944.85$13.8897$5000.091.091/1$500PV5=×= −×=
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-30• Example 2You borrow $7500 to buy a car and agree torepay the loan by way of equal monthlyrepayments over five years. The currentinterest rate is 12 per cent per annum,compounded monthly. What is the amount ofeach monthly repayment?( ){ }$191.3539.1961500$70.011.011/1500$760=÷= −×=CC
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-31( )[ ]rrCt11FV−+×=Future Value of an Annuity• The compounding term is called the future valueinterest factor for annuities (FVIFA). Refer to TableA.4.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-32What is the future value of $200 deposited at theend of every year for 10 years if the interest rate is6 per cent per annum?( )$2636.2013.181$2000.061101.06$200FV=×= −×=Example—Future Value of anAnnuity
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-33Perpetuities• The future value of a perpetuity cannot becalculated as the cash flows are infinite.• The present value of a perpetuity is calculated asfollows:rC=PV
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-34Comparing Rates• The nominal interest rate (NIR) is the interest rateexpressed in terms of the interest payment madeeach period.• The effective annual interest rate (EAR) is theinterest rate expressed as if it was compoundedonce per year.• When interest is compounded more frequently thanannually, the EAR will be greater than the NIR.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-35Calculation of EAR11EARmNIRm−+=m = number of times the interest is compounded
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-36Comparing EARS• Consider the following interest rates quoted by three banks:– Bank A:15%, compounded daily– Bank B:15.5%, compounded quarterly– Bank C:16%, compounded annually
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-38Comparing EARS• Which is the best rate? For a saver, Bank B offersthe best (highest) interest rate. For a borrower,Bank C offers the best (lowest) interest rate.• The highest NIR is not necessarily the best.• Compounding during the year can lead to asignificant difference between the NIR and theEAR.
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Copyright 2004 McGraw-Hill AustraliaPty Ltd5-39Types of Loans• A pure discount loan is a loan where the borrowerreceives money today and repays a single lumpsum in the future.• An interest-only loan requires the borrower to onlypay interest each period and to repay the entireprincipal at some point in the future.• An amortised loan requires the borrower to repayparts of both the principal and interest over time.
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