3 Time value of money.docx

1、3 Time value of moneyTime Value of MoneyChapter 5 and 6Topics:1. Future Value of a Single Sum2. Present Value of a Single Sum 3. Net Present Value 4. Rate of Return 5. Perpetuity 6. Growing Perpetuities7. Present Value of an Ordinary Annuity (and Annuity Due)8. Future Value of an Ordinary Annuity (a

2、nd Annuity Due) 9. Growing annuity10. Periodic Rate 11. Annual Percentage Rate (APR)12. Effective (equivalent) Annual Rate (EFF or EAR)13. Continuous Compounding14. Inflation and Interest Rates 15. Amortizing LoanCash flows are discounted for two reasons Time and risk:1. A dollar today is worth more

3、 than a dollar tomorrow.2. A safe dollar is worth more than a risky one.1. Future Value of a Single SumEXCELFV(Rate, NPER, PMT,PV)_2. Present Value of a Single Sum You will require $700 in 5 years, if you earn 5% interest on your funds, how much will you need to invest today in order to reach your s

4、avings goal? EXCELPV(Rate, NPER, PMT,FV)EXCEL:rate(nper,pmt,pv,fv)_nper(rate,pmt,pv,fv)_3. Net Present Value NPV = PV (cash flow) PV (cash outflow)EXCEL:C0 +npv(rate,value1, value2, .)_4. Rate of Return _5. Perpetuity These are bonds that the government is under no obligation to repay but that offer

5、 a fixed income (C) for each year to perpetuity_6. Growing Perpetuitiesyou can use this formula only when g r. As g approaches r, the stock price becomes infinite._You own an oil pipeline which will generate a $2 million cash return over the coming year. The pipelines operating costs are negligible,

6、 and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by 4% per year. The discount rate is 10%.What is the PV of the pipelines cash flows if its cash flows are assumed to last forever?7. Present Value of an Ord

7、inary Annuity (and Annuity Due)An asset that pays a fixed sum of each year for a specific number of years. Its value is the difference between the values of two perpetuities_EXCELPV(Rate, NPER, PMT,FV, type)_8. Future Value of an Ordinary Annuity (and Annuity Due) EXCELFV(Rate, NPER, PMT,PV, type)_9

8、. Growing annuityYou own an oil pipeline which will generate a $2 million cash return over the coming year. The pipelines operating costs are negligible, and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by

9、4% per year. The discount rate is 10%.a. What is the PV of the pipelines cash flows if its cash flows are assumed to last forever?b. What is the PV of the cash flows if the pipeline is scrapped after 20 years?_10. Periodic Rate where m is number of compounding periods per year. _11. Annual Percentag

10、e Rate (APR)Interest rate that is annualized using simple interest rate.APR = (rPer)(m), Simple compounding _12. Effective (equivalent) Annual Rate (EFF or EAR) The annual rate that causes PV to grow to the same FV as under multi-period compounding.EAR = (1+ rnom/m)m - 1 Excel:Effect(nominal rate, n

11、pery)_13. Continuous Compoundinglim m1 + (1/m)m = e = 2.718lim m1 + (r/m)m = er lim m1 + (r/m)mt = erte is the year-end value to which a principle of $1 will grow if interest at the rate of 100 per annum is compounded continuously.If a bank pays 6 percent interest with continuous compounding. What i

12、s the effective annual rate?_14. Inflation and Interest Rates Current dollar cash flows must be discounted by nominal interest rate; real cash flows must be discounted by the real interest rate.By discounting real cash flows at the real interest rate, you get the same PV that you get when you discou

13、nt the nominal cash flows at the nominal interest rate.15. Amortizing Loan1. You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You pla

14、n to leave the money in the bank if you dont buy the note. The note is riskless. Should you buy it?2. Assume that your father is now 50 years old, that he plans to retire in 10 years, and that he expects to live for 25 years after he retires, that is, until he is 85. He wants a fixed retirement inco

15、me that has the same purchasing power at the time he retires a $ 40,000 has today (he realizes that the real value of his retirement income will decline year by year after he retires). His retirement income will begin the day he retires, 10 years from today, and he will then get 24 additional annual

16、 payments. Inflation is expected to be 5 percent per year from today forward; he currently has $100,000 saved up; and he expects to earn a return on his saving of 8% per year, annual compounding. To the nearest dollar, how much must he save during each of the next 10 years (with deposit being made a

17、t the end of each year) to meet his retirement goal?1. Future Value (Single Sum)r = 0.1nper = t =3Period 0Period 1period 2period 375.13148FV(r,nper,pmt,pv)FV(.1,3,0,100)$100.00 FVIF = (1+r)t 1.331FV = C (FVIF)100.00_2. Present Value (Single Sum)r = 0.1nper = t =3Period 0Period 1period 2period 3100PV

18、(rate,nper,pmt,fv)PV(.1,3,0,100)($75.13)PVIF = 1 /(1+r)t =0.751314801PV = C (PVIF)75.13148009_3. Net Present Valuer = 0.1nper = t =3Period 0Period 1period 2period 3-200100100100C0 +npv(rate,value1, value2, .)=$48.69 _7. Present Value of an Annuityr =0.1t =3Period 0Period 1period 2period 3100100100PV

19、(rate,nper,pmt,fv)PV(.1,3,100,0)($248.69)PVIFA =1/r - (1/r)* (1/(1+r)t =2.4869PV = C PVIFA248.69_Period 0Period 1period 2period 31001001001000PV(.1,3,100,1000)($1,000.00)_8. Future Value of an Annuityr =0.1t =3Period 0Period 1period 2period 3Annuity-100-100-100FV(.1,3,-100,0)$331.00 FVIFA =(1+r)t -1

20、/r =3.31FV = C (FVIFA)331_Period 0Period 1period 2period 3-100-100-100-1000FV(.1,3,-100,-1000) =$1,662.00 _r =0.1g =0.05t =3a =0.954545C = 1005. Perpetuity PV = C/r =10006. Growth PerpetuityPV = C/(r-g)20009. Growing annuityPV = C/(1+r) * (1-at)/(1-a)260.5184072_r = 0.1m =1210. Periodic Rate rper =

21、r/m0.00833333311. Effective Rate (EAR)(1 + r/m)m) - 1 = 0.104712. Annual Percentage Rate (APR)0.1APR = (m)(rper) =_13. Continuous Compoundingr =0.1t =5C =100(Continuous Compounding) FV = C e(rc t)164.8720716(Compounding) FV = C (1+r)t161.051_Other FunctionsAssume r is unknownC0C1C2C3rate(nper,pmt,pv,fv)100100100rate(3,100,-248.69,0)-248.6910%Assume NPER is unknownnper(rate,pmt,pv,fv)C0C1C2C?nper(.1,100,-248.69,0)1001001003.00-248.69Assume NPER is unknownnper(rate,pmt,pv,fv)C0C?nper(.04,0,-400,1000)-400100023.36241894_