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1、vBe able to compute the future value and present value of a single cash flow or series of cash flowsvUnderstand NPV of an investmentvKnow APR and EARvUnderstand annuities and perpetuities 4.1 Valuation: The One-Period Case4.2 The Multiperiod Case4.3 Compounding Periods4.4 Simplifications4.5 What Is
2、a Firm Worth?vDon is trying to sell a piece of land in Alaska. Yesterday, he was offered $10,000 for the property. He was about ready to accept the offer when another individual offered him $11,424. However, the second offer was to be paid a year from now. Don has satisfied himself that both buyers
3、are honest and financially solvent, so he has no fear that the offer he selects will fall through. Which offer should Don choose?vMike, Dons financial advisor, points out that if Don takes the first offer, he could invest the $10,000 in the bank at an insured rate of 12 percent.qIf you were to inves
4、t $10,000 at 5-percent interest for one year, your investment would grow to $10,500. $500 would be interest ($10,000 .05)$10,000 is the principal repayment ($10,000 1)$10,500 is the total due. It can be calculated as:$10,500 = $10,000(1.05)qThe total amount due at the end of the investment is called
5、 the Future Value (FV). vIn the one-period case, the formula for FV can be written as:FV = C0(1 + r)Where C0 is cash flow today (time zero), and r is the appropriate interest rate.vIf you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9
6、,523.81 in todays dollars. 05. 1000,10$81.523, 9$Note that $10,000 = $9,523.81(1.05)vThe amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is called the Present Value (PV).vIn the one-period case, the formula for PV can be written as:
7、rCPV11Where C1 is cash flow at date 1, and r is the appropriate interest rate.vThe Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.vSuppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500.
8、 Your interest rate is 5%. Should you buy?81.23$81.523, 9$500, 9$05. 1000,10$500, 9$NPVNPVNPVThe present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased.In the one-period case, the formula for NPV can be writ
9、ten as:NPV = Cost + PVvSuppose you has put $500 in a savings account at the bank. The account earns 7%, compounded annually. vHow much will you have at the end of 3 years?vHow much will you have at the end of 3 years if simple interest is used?500+5007%3=605500(1+7%)3=612.52v The general formula for
10、 the future value of an investment over many periods can be written as:FV = C0(1 + r)TWhere C0 is cash flow at date 0, r is the appropriate interest rate, and T is the number of periods over which the cash is invested.TrTFVIFr,)1(简写为称为复利终值系数vHow much would an investor have to set aside today in orde
11、r to have $20,000 five years from now if the current rate is 15%?012345$20,000PV5)15. 1 (000,20$53.943, 9$vIn the multiperiod case, the formula for the present can be written as:TrTTTTTPVIFrrCrCPV,)1 (1)1 (1)1 (简写为叫做复利现值系数If we deposit $5,000 today in an account paying 10%, how long does it take to
12、grow to $10,000?TrCFV)1 (0T)10. 1 (000, 5$000,10$2000, 5$000,10$)10. 1 (T)2ln()10. 1ln(Tyears 27. 70953. 06931. 0)10. 1ln()2ln(TAssume the total cost of a college education will be $50,000 when a child enters college in 12 years. His/Her parents have $5,000 to invest today. What rate of interest mus
13、t his/her parents earn on their investment to cover the cost of the childs education? TrCFV)1 (012)1 (000, 5$000,50$r10000, 5$000,50$)1 (12r12110)1 (r2115.12115. 1110121rAbout 21.15%At 9 percent interest, how long does it take to double your money? To quadruple it?vConsider an investment that pays $
14、200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows?vIf the issuer offers this investment for $1,500, should you purchase it?01234200400600800178.57318.88427.07508.411,432.93Present Val
15、ue Cost Do Not PurchaseThe present value of the following cash flow stream is $6,453 when discounted at 10 percent annually. What is the value of the missing cash flow?Year 1 Cash flow $1,200Year 2 Cash flow ?Year 3 Cash flow $2,400Year 4 Cash flow $2,600Compounding an investment m times a year for
16、T years provides for future value of wealth:TmmrCFV10qFor example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to6%,663250$93.70$)06. 1 (50$212.150$FVIFFVA reasonable question to ask in the above example is “what is the effective annual rate of interest o
17、n that investment?”The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:93.70$)1 (50$3EAR1236.150$93.70$31EARSo, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.vFind the Eff
18、ective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.vWhat we have is a loan with a monthly interest rate of 1.5%.vThis is equivalent to a loan with an annual interest rate of 19.56%.1956. 01)015. 1 (11218.11212EAR11mmrEAR11mmAPREAR4.4 SimplificationsvAnnuity(年金年金) A stream of cons
19、tant cash flows that lasts for a fixed number of periodsvGrowing annuity(增长年金增长年金) A stream of cash flows that grows at a constant rate for a fixed number of periodsvPerpetuity(永续年金永续年金) A constant stream of cash flows that lasts forevervGrowing perpetuity(永续增长年金永续增长年金) A stream of cash flows that g
20、rows at a constant rate foreverA constant stream of cash flows with a fixed maturity01C2C3CTrCrCrCrCPV)1 ()1 ()1 ()1 (32TCTrTTPVIFArrrrCPV,)1(11,)1(11简写为称为年金现值系数01C2C3CTCTrTTTFVIFArrrrCrCrCrCCFV,12,1)1 (1)1 ()1 (.)1 ()1 (简记为叫做年金终值系数If you can afford a $400 monthly car payment, how much car can you a
21、fford if interest rates are 7% on 36-month loans?01$4002$4003$40059.954,12$1207.)1207.1 (11400$36PV36$400What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? 22.297$09. 197.327$01%,997.327$PVIFPV0 1 2 3 4 5$10
22、0 $100 $100 $100$323.97$297.2297.323$100$)09. 1 (100$)09. 1 (100$)09. 1 (100$)09. 1 (100$)09. 1 (100$4%,94321411PVIFAPVttWhat is the present and future value of a four-year annuity of $100 per year if the first payment occurs immediately? (the discount rate is 9%)A growing stream of cash flows with
23、a fixed maturity01CTTrgCrgCrCPV)1 ()1 ()1 ()1 ()1 (12grrgCPVT1112C(1+g)3C (1+g)2T C(1+g)T-1A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%?01$20,000%3%10
24、10. 103. 1120000$40PV2$20,000(1.03)40 $20,000(1.03)39You are evaluating an income generating property. Net rent is received at the end of each year. The first years rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream
25、over the first 5 years if the discount rate is 12%?0 1 2 3 4 5500, 8$)07. 1 (500, 8$2)07. 1 (500, 8$095, 9$65.731, 9$3)07. 1 (500, 8$87.412,10$4)07. 1 (500, 8$77.141,11$34,706.28A constant stream of cash flows that lasts forever01C2C3C32)1 ()1 ()1 (rCrCrCPVrCPV n What is the value of a British conso
26、l that promises to pay 15 every year for ever? n The interest rate is 10-percent.011521531515010.15PVA growing stream of cash flows that lasts forever01C2C(1+g)3C (1+g)2322)1 ()1 ()1 ()1 ()1 (rgCrgCrCPVgrCPVn The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever.
27、 n If the discount rate is 10%, what is the value of this promised dividend stream?01$1.302$1.30(1.05)3$1.30 (1.05)200.26$05.10.30. 1$PVvConceptually, a firm should be worth the present value of the firms cash flows.vThe tricky part is determining the size, timing, and risk of those cash flows.Quick QuizvWhat is PVIF, FVIF, PVIFA and FVIFA? vWhat is the Net Present Value of an investment?vWhat is an EAR, and how is it computed from APR?vWhat is a perpetuity? An annuity?