1、Topics CoverednPresent Value&Future ValuenAnnuity&PerpetuitynEffective Annual Interest RatenInflationnCurrenciesPresent ValuePresent ValueValue today of a future cash flow.Discount RateInterest rate used to compute present values of future cash flows.Discount FactorPresent value of a$1 future paymen
2、t.Calculation Recommendations Four primary approaches to solving the problems.nUsing the formulas and a standard calculator.You will need basic algebra skills and a calculator that will raise numbers to a power and take roots.nUsing a financial calculator.A wide variety of financial calculators are
3、available,in a wide range of prices.nUsing Excel on a personal computer.Excel contains built-in functions that will perform most Time Value of Money calculations.nUsing tables from the text.Before the ready accessibility of computing power,this method was used for all TVM problems.Definitions and Ab
4、breviationsnannuity:a series of payments or receipts having three specific characteristics,also called ordinary annuity or annuity in arrears（迟付年金）.nannuity due（即付年金）:an annuity where the payments occur at the beginning of evenly spaced time periods.nCF0,CF1,etc:cash flows,a series of payments or re
5、ceipts or both.nEAR:effective annual interest rate,the annual interest rate that reflects the effects of compounding.nFV:future value,the accumulated value of an investment once all payments are made,including interest or return received.ng:growth rate,we assume this is constant over time.nr or i:in
6、terest rate.nlump sum:a single payment or receipt,in contrast with annuity or perpetuity.nn or t:number of time periods,months,half-years,years,etc.nPMT or C:payment,the constant payment or receipt of an annuity or perpetuity.nperpetuity:a series of payments that never ends,where each payment is the
7、 same amount and the payments occur at the end of evenly spaced time periods.nPV:present value,the value on a given date of a future payment or series of future payments.Time Value of Money FormulasFuture value of a lump-sum:Present value of a lump-sum:Present value of a perpetuity:Present value of
8、an annuity:Future value of an annuity:FVPVrn()1PVFVrn()1rCPVnrrrCPV)1(11rrCFVn1)1(Time Value of Money FormulasPresent value of an annuity due:Future value of an annuity due:Present value of a perpetuity with constant growth:Present value of an annuity with constant growth:Future value of an annuity
9、with constant growth:)1()1(11rrrrCPVn)1(1)1(rrrCFVngrgCgrCPV)1(01nrggrCPV1111nngrgrCFV)1()1()(1Compare with PV of an annuityCompare with FV of an annuityPresent ValuesDiscount Factor=DF=PV of$1Discount Factors can be used to compute the present value of any cash flow.D Frt11()Present ValuesnDiscount
10、 Factors can be used to compute the present value of any cash flow.1111 rCCDFPVPresent ValuesnReplacing“1”with“t”allows the formula to be used for cash flows that exist at any point in time.tttrCCDFPV1Present ValuesExampleYou just bought a new computer for$3,000.The payment terms are 2 years same as
11、 cash.If you can earn 8%on your money,how much money should you set aside today in order to make the payment when due in two years?Present ValuesExampleYou just bought a new computer for$3,000.The payment terms are 2 years same as cash.If you can earn 8%on your money,how much money should you set as
12、ide today in order to make the payment when due in two years?PV 30001 08257202(.)$2,.Present ValuesnPVs can be added together to evaluate multiple cash flows.PVCrCr112211()().Present ValuesnGiven two dollars,one received a year from now and the other two years from now,the value of each is commonly
13、called the Discount Factor.Assume r1=20%and r2=7%.Present ValuesnGiven two dollars,one received a year from now and the other two years from now,the value of each is commonly called the Discount Factor.Assume r1=20%and r2=7%.87.83.21)07.1(00.12)20.1(00.11DFDFPresent ValuesExampleAssume that the cash
14、 flows from the construction and sale of an office building is as follows.Given a 7%required rate of return,create a present value worksheet and show the net present value.000,300000,100000,1502Year 1Year 0Year Present ValuesExample-continuedAssume that the cash flows from the construction and sale
15、of an office building is as follows.Given a 7%required rate of return,create a present value worksheet and show the net present value.400,18$900,261000,300873.2500,93000,100935.1000,150000,1500.10ValuePresentFlowCashFactorDiscountPeriod207.1107.11TotalNPVShortcutsnSometimes there are shortcuts that
16、make it very easy to calculate the present value of an asset that pays off in different periods.These tools allow us to cut through the calculations quickly.Short CutsPerpetuity-Financial concept in which a cash flow is theoretically received forever.PVCr luepresent vaflow cashReturnShort CutsPerpet
17、uity-Financial concept in which a cash flow is theoretically received forever.rCPV1ratediscount flow cash FlowCash of PVShort CutsAnnuity-An asset that pays a fixed sum each year for a specified number of years.trrrC111annuity of PVAnnuity Short CutExampleYou agree to lease a car for 4 years at$300
18、per month.You are not required to pay any money up front or at the end of your agreement.If your opportunity cost of capital is 0.5%per month,what is the cost of the lease?Annuity Short CutExample-continuedYou agree to lease a car for 4 years at$300 per month.You are not required to pay any money up
19、 front or at the end of your agreement.If your opportunity cost of capital is 0.5%per month,what is the cost of the lease?10.774,12$005.1005.1005.1300Cost Lease48CostCompound Interest i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii)one year interest rate1
20、6%6%1.06 6.000%2 3 6 1.032 =1.0609 6.0904 1.5 6 1.0154 =1.06136 6.13612 .5 6 1.00512 =1.06168 6.16852 .1154 6 1.00115452 =1.06180 6.180365 .0164 6 1.000164365=1.06183 6.183Compound InterestCompoundingOn many investments or loans,interest is credited or charged more frequently than once a year.Exampl
21、es:Daily(bank accounts)Monthly(mortgages and leases)Semiannual(bonds)Implication The effective annual rate(EAR)can be much different than the stated annual percentage rate(APR)ExampleMy car loan specifies a finance charge on the unpaid balance,computed daily,at the rate of 6.75%per year.What is the
22、effective annual rate?Suppose loan amount of$10,000 and annual payments.How much would I owe in a year?Daily interest rate=0.0675/365=0.0001849After 1 day:balance=10,000.00 x1.000185=10,001.85 2 days:balance=10,001.85 x1.000185=10,003.70 1 year:balance=10,000 x(1.000185)365=10,698.23 EAR=6.9823%Comp
23、ounding,cont.ExampleMortgagesYou would like to buy a house worth$400,000.If you make a 20%down payment,or$80,000,you can get a 30-year,8%APR mortgage on the remaining$320,000.The mortgage requires you to make a flat monthly payment for the life of the loan.What will the monthly payments be?360 month
24、ly payments Monthly interest rate=0.08/12=0.00667 PVAF(360,0.00667)=136.28 C=320,000/136.28=$2,348.14nrrrCPV)1(11ExampleInflationInflation-Rate at which prices as a whole are increasing.Nominal Interest Rate-Rate at which money invested grows.Real Interest Rate-Rate at which the purchasing power of
25、an investment increases.InflationInflation1 real interest rate=1+nominal interest rate1+inflation rateInflation1 real interest rate=1+nominal interest rate1+inflation rateapproximation formulaReal int.ratenominal int.rate-inflation rateInflation,cont.Inflation,cont.InflationExampleIf the interest ra
26、te on one year govt.bonds is 5.9%and the inflation rate is 3.3%,what is the real interest rate?SavingsBondInflationExampleIf the interest rate on one year govt.bonds is 5.9%and the inflation rate is 3.3%,what is the real interest rate?11+real interest rate=real interest rate=1.025real interest rate=
27、.025 or 2.5%1+.0591+.033SavingsBondInflationExampleIf the interest rate on one year govt.bonds is 5.9%and the inflation rate is 3.3%,what is the real interest rate?11+real interest rate=real interest rate=1.025real interest rate=.025 or 2.5%Approximation=.059-.033=.026 or 2.6%1+.0591+.033SavingsBondCurrenciesCurrencies Cont.ExampleExample
