1、財務(wù)管理,江萍 2010年秋季,Handout 2,Handout Outline,Discounted Cash Flow Valuation 貼現(xiàn)現(xiàn)金流估值 Net Present Value and Other Investment Criteria 凈現(xiàn)值與其他投資標準 Determining the relevant cash flows 確定相關(guān)現(xiàn)金流,1.Discounted Cash Flow Valuation Outline,Introduction to Valuation: The Time Value of Money Future Value and Compoun

2、ding(連續(xù)復利) Present Value and Discounting More on the future and present values Future and Present Values of Multiple (復雜，多重的)Cash Flows Application(應(yīng)用),I. Introduction to Valuation: The Time Value of Money,Basic Definitions (基本定義)： Present Value earlier money on a time line Future Value later money

3、on a time line Interest rate “exchange rate” between earlier money and later money Discount rate(貼現(xiàn)率) Cost of capital(資本成本) Opportunity cost of capital(資本的機會成本) Required return(必要收益率或內(nèi)部收益率),A. Future Value and Compounding Example 1,Suppose you invest $100 for one year at 5% per year. What is the fut

4、ure value in one year? Interest = $100*5% = $5 Value in one year = principal + interest = $100 + $5 = $105 Future Value (FV) = $100(1 + 5%) = $105 Suppose you leave the money in for another year. How much will you have two years from now? FV = $100(1+5%)(1+5%) = $100(1+5%)2 = $110.25,Future Values:

5、General Formula,FV = PV(1 + r)t FV = future value PV = present value r = period interest rate T = number of periods Future value interest factor = (1 + r)t,Future value of $1,FV of $1=(1+r)t,Future Values Example 2,Suppose you invest the $100 from the previous(之前的) example for 5 years. How much woul

6、d you have? FV = $100(1+5%)5 = $127.63,Effects of Compounding,Simple interest (interest is earned only on the original principal) 單利：只對原始本金支付利息 Compound interest (interest is earned on principal and on interest received) 復利：計息的基礎(chǔ)為原始本金，也包括已經(jīng)獲得的利息,Effects of Compounding,Consider the example 1 FV with

7、simple interest = $100 + 5 + 5= $110 FV with compound interest = $110.25 The extra $0.25 comes from the interest of $5*5% = $0.25 earned on the first interest payment Consider example 2 FV with simple interest = $100 + 5 *5= $125 FV with compound interest =$127.63 The effect of compounding is small

8、for a small number of periods, but increases as the number of periods increases (from $0.25 to $2.63).,Future value, simple interest and compound interest,The effect of compounding increases as the number of periods increases. 隨著期限的增加復利對終值的影響逐漸增加。,Future Values Example 3 The case of Peter Minuet and

9、 the Indians,In 1626, Minuet bought all of Manhattan Island(曼哈頓島) for about $24 in goods and trinkets(珠寶飾物). This sounds cheap, but the Indians may have gotten the better end of the deal. To see why, suppose the Indians had sold the goods and invested the $24 at 10%. How much would it be worth today

10、? The future value interest factor is approximately: 1.1380 5，000，000，000，000，000 The FV of $24 will be $24* 5 quadrillion(千萬億） How much is it? Well, if you had it, you could buy the U.S. All of it. Cash. With money left over(剩下的) to buy Canada, Mexico, and the rest of the world.,B. Present Value an

11、d Discounting,How much do I have to invest today to have some amount in the future? FV = PV(1 + r)t Rearrange to solve for PV = FV / (1 + r)t PV interest factor = 1 / (1 + r)t The present value is always less than the future value when we have positive rates of interest.,Present Values,When we talk

12、about discounting, we mean finding the present value of some future amount. When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.(無特殊情況，我們提到價值即指現(xiàn)值),PV One-Period Example,Suppose you need $10,000 in one year

13、for the down payment on a new car. If you can earn 7% annually(每年), how much do you need to invest today? PV = $10,000 / (1.07)1 = $9,345.79,Present Values Example 2,You want to begin saving for your daughters college education and you estimate that she will need $150,000 in 17 years. If you feel co

14、nfident (有信心)that you can earn 8% per year, how much do you need to invest today? PV = $150,000 / (1.08)17 = $40,540.34,PV Important Relationship I,For a given interest rate the longer the time period, the lower the present value (ceteris paribus: all else equal 其他條件不變) What is the present value of

15、$500 to be received in 5 years? 10 years? The discount rate is 10% 5 years: PV = $500 / (1.1)5 = $310.46 10 years: PV = $500 / (1.1)10 = $192.77,PV Important Relationship II,For a given time period the higher the interest rate, the smaller the present value (ceteris paribus) What is the present valu

16、e of $500 received in 5 years if the interest rate is 10%? 15%? Rate = 10%: PV = $500 / (1.1)5 = $310.46 Rate = 15%; PV = $500 / (1.15)5 = $248.59,Present Value of $1,PV of $1 = 1 / (1 + r)t,C. More on Present and Future Values,The Basic PV Equation - Refresher PV = FV / (1 + r)t There are four part

17、s to this equation PV, FV, r, and t If we know any three, we can solve for (求解)the fourth,Discount Rate,Often, we will want to know what the implied interest rate is in an investment. Rearrange(顛倒，轉(zhuǎn)換) the basic PV equation and solve for r FV = PV(1 + r)t r = (FV / PV)1/t 1,Finding the Number of Peri

18、ods,Start with basic equation and solve for t FV = PV(1 + r)t t = ln(FV / PV) / ln(1 + r) ln is the natural logarithm(自然對數(shù)) and can be found on the calculator.,The rule of 72,The rule of 72 is a quick way to estimate how long it will take to double your money. # years to double = 72 / r where r is a

19、 percent, not a decimal. 72/r年后,投資可以翻倍,其中r為百分數(shù). If you want to buy a car which costs ￥200,000. You currently have ￥100,000. Suppose you are able to earn 24% each year. How long will it be before you have enough money to pay cash for the car?,Example: Spreadsheet Strategies,Another TVM formula that c

20、an be found in a spreadsheet(電子表格) is the payment formula PMT(rate,nper,pv,fv) NPER (rate, pmt,pv,fv) Rate(nper,pmt,pv,fv) The same sign convention(慣例，公約) holds as for the PV and FV formulas Click on the Excel icon(圖標) for an example rate 貸款利率 nper 該項貸款的付款總數(shù) pv 現(xiàn)值 fv 為未來值 type 數(shù)字 0 或 1 pmt貸款的每期支付額,I

21、I. Future and Present Values of Multiple Cash Flows,FV and PV calculation when there are multiple(復合的，不同的)cash flows. 不同于multiple periods，ie，一個cash flow，貼現(xiàn)到N年前的case.,Two Ways to Calculate FV for Multiple CFs,(1) compound the accumulated balance forward one year at a time or 計算出每年的現(xiàn)金流余額再求終值 (2) calcu

22、late the future value of each cash flow first and then add these up. 計算出每個現(xiàn)金流的終值再相加。 Both give the same answer, so you can do it either way.,Multiple Cash Flows FV Example,Suppose you plan to deposit $100 into an account(賬戶) in one year and $300 into the account in three years. How much will be in t

23、he account in five years if the interest rate is 8%? FV = $100(1.08)4 + $300(1.08)2 = $136.05 + $349.92 = $485.97,Example Time Line,$100,0,1,2,3,4,5,$300,$136.05,$349.92,$485.97,Multiple Cash Flows PV,It is similar for PV calculation Example: You are offered an investment that will pay you $200 in o

24、ne year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one? (Or “How much is this investment worth?”),Example Time Line,Example: Spreadsheet Strategies,You can use the PV or F

25、V functions in Excel to find the present value or future value of a set of cash flows Setting the data up(構(gòu)建數(shù)組) is half the battle if it is set up properly, then you can just copy the formulas Click on the Excel icon for an example,III. Application,Valuing Annuities and Perpetuities 年金和永續(xù)年金的估值 Loan

26、Types and Loan Amortization 貸款種類和按揭抵押貸款 Valuation of Bond and Stock 債券和股票估值,A. Annuities,Definition The cash flows from this asset are in the form of a three-year, $500 ordinary annuity Annuity (年金) finite series of equal payments that occur at regular intervals If the first payment occurs at the en

27、d of the period, it is called an ordinary annuity(普通年金) If the first payment occurs at the beginning of the period, it is called an annuity due(先付年金）,Ordinary Annuities PV,Ordinary Annuities:,Annuity PV Example Buying a House,You are ready to buy a house and you have $20,000 for a down payment （首付）

28、and closing costs （買方過戶費用）. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment (每月按揭還款)to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compoun

29、ding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?,Buying a House - Continued,Bank loan Monthly income = $36,000 / 12 = $3,000 Maximum payment = .28($3,000) = $840 PV = $8401 1/1.005360 / .005 = $140,105 Total Price Closin

30、g costs = .04($140,105) = $5,604 Down payment = $20,000 5,604 = $14,396 Total Price = $140,105 + 14,396 = $154,501,Ordinary Annuities FV,Ordinary Annuities:,AnnuitiesFV Example,Suppose you begin saving for your retirement(退休) by depositing(存款) $2,000 per year (at the year end) in a retirement accoun

31、t. If the interest rate is 7.5%, how much will you have in 40 years? FV = $2,000(1.07540 1)/.075 = $454,513.04,Spreadsheet Strategies Annuity PV/FV Example,The present value and future value formulas in a spreadsheet include a place for annuity payments Click on the Excel icon to see an example,Find

32、ing the Payment,Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .666666667% per month). If you take a 4-year loan, what is your monthly payment? $20,000 = C1 1 / 1.006666748 / .0066667 C = $488.26,Finding the Number of Payments,You ran a li

33、ttle short on your new year vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000.,Finding the Number of Payments,Start with the eq

34、uation and remember your logs. $1,000 = $20(1 1/1.015t) / .015 .75 = 1 1 / 1.015t 1 / 1.015t = .25 1 / .25 = 1.015t t = ln(1/.25) / ln(1.015) = 93.111 months = 7.75 year And this is only true if you dont charge anything more on the card!,Spreadsheet Strategies Annuity Payment Example,Another TVM for

35、mula that can be found in a spreadsheet is the payment formula PMT(rate,nper,pv,fv) NPER (rate, pmt,pv,fv) Rate(nper,pmt,pv,fv) The same sign convention holds as for the PV and FV formulas Click on the Excel icon for an example,Perpetuities(永續(xù)年金）,Perpetuity: an annuity in which the cash flows contin

36、ue forever. Perpetuity: PV = C / r Preferred stock (優(yōu)先股) is an important example of a perpetuity. When a corporation sells preferred stock, the buyer is promised a fixed cash dividend every period usually every quarter(季度) forever. The dividend must be paid before any dividend can be paid to regular

37、 stockholders, hence the term preferred.,Perpetuity Example,Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to

38、offer if the preferred stock is going to sell?,Perpetuity Example,Perpetuity formula: PV = C / r Current required return: $40 = $1 / r r = .025 or 2.5% per quarter Dividend for new preferred: $100 = C / .025 C = $2.50 per quarter,B. Loan Types and Loan Amortization,Loan Types (I) Pure Discount Loans

39、(純貼現(xiàn)貸款) Simplest form of loan Borrower receives money today and repays a single lump sum (一次性支付一個總額)at some time in the future Treasury bills (T-bills國庫券) are excellent examples of pure discount loans. The principal (本金)amount is repaid at some future date, without any periodic interest payments.,Pu

40、re Discount Loans Example,If a T-bill promises to repay $10,000 in one year, and the market interest rate is 7 percent, how much will the bill sell for in the market? PV = $10,000 / 1.07 = $9,345.79,Loan Types (II),Interest-Only Loan(附息貸款) The borrowers pay interest each period and repay the entire

41、principal at some point in the future.(每期付息，到期還本) If there is just one period, a pure discount loan and an interest only loan are the same thing.,Interest-Only Loan - Example,Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. Wh

42、at would the stream of cash flows (現(xiàn)金流)be? Years 1 4: Interest payments of .07($10,000) = $700 Year 5: Interest + principal = $10,700,Loan Types (III),Amortized Loan The borrowers repay parts of the loan amount over time. The process of paying off a loan by making regular (定期的，規(guī)則的)principal reductio

43、n(減少) is called amortizing the loan. The most common way of amortizing a loan is to have the borrower make a single, fixed payments every period. Almost all consumer loans (such as car loans) and mortgages work this way.,Amortized Loan with Fixed Payment Example,Consider a 4-year loan with annual pa

44、yments. The interest rate is 8% and the principal amount is $5,000. What is the annual payment? $5,000 = C1 1 / 1.084 / .08 C = $1,509.60,Amortization Table for Example,Each payment covers the interest expense; plus, it reduces principal,The reason that the loan balance does not decline to exactly z

45、ero is because of the rounding of the interest payments. Technically, the last payment would be 1,509.61 so that the loan balance would be zero after the last payment. This is a common issue.,Example: Spreadsheet Strategies,Consider a 4-year loan with annual payments. The interest rate is 8% and the

46、 principal amount is $5,000. What is the annual payment? PMT = -1,509.60 Click on the Excel icon to see the amortization table,C. How to Value Bonds and Stocks,Definitions and Example of a Bond How to Value Bonds Bond Concepts The Present Value of Common Stocks Estimates of Parameters in the Dividen

47、d- Discount Model(股利貼現(xiàn)模型參數(shù)估計) Growth Opportunities The Dividend Growth Model and the NPVGO Model(股利增長模型和增長機會模型) Price-Earnings Ratio(市盈率),1. Definition of a Bond,A bond is a legally binding agreement (具有法律約束力的合約)between a borrower and a lender that specifies the: the Par (face) value(面值) the Coupon

48、rate(息票利率) the Coupon payment(利息支付) the Maturity Date(到期日),Definition of a Bond,Suppose AAA Corporation wants to borrow $1000 (mil) for 30 years by issuing bonds. The interest rate of similar corporation bond is 12%. Thus Beck will pay 0.12*$1000=$120 in interest every year for 30 years. At then end

49、 of 30 years, Beck will repay the $1000.,Definition of a Bond,Par value (face value): the principal amount of a bond that is repaid at the end of the term, $1000. Coupon rate: the annual coupon divided by the face value, 12%. Coupon payment: Stated interest payment made on a bond, 120 Maturity(期限):

50、the number of years until the face value is paid, 30 years. The yield to maturity(YTM到期收益率) : the required market interest rate on the bond.,Yield to Maturity (YTM),Yield to maturity is the rate implied by the current bond price or can be considered as the interest required in the market on a bond.

51、到期收益率是當前的債券價格隱含收益率或者看作債券的市場收益率。 The cost of issuing bond is YTM, not coupon rate. 發(fā)行債券的成本是到期收益率而不是票面利率。,2. How to Value Bonds,Primary Principle: Value of financial securities = PV of expected future cash flows Bond value is, therefore, determined by the present value of the coupon payments and par v

52、alue. Interest rates are inversely related to (與負相關(guān))present (i.e., bond) values.,The Bond Pricing Equation,Pure Discount Bonds,Make no periodic interest payments (coupon rate = 0%) The entire yield to maturity comes from the difference between the purchase price and the par value. Cannot sell for mo

53、re than par value(純折現(xiàn)債券的市場價格不會超過面值). Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs原始發(fā)行折價債券) Treasury Bills are good examples of zeroes(零息債券).,Pure Discount Bonds,Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - todays d

54、ate Face value (F) Discount rate (r),Present value of a pure discount bond at time 0:,Pure Discount Bond: Example,Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.,Level Coupon Bonds(息票債券),Make periodic coupon payments in addition to the maturity value(除了期末外其他各期支付

55、相同利息). The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value. Coupon payments are typically semiannual(息票通常是半年支付一次). Effective annual rate (EAR實際年利率) = (1 + R/m)m 1 （其中m為每年計息的次數(shù)，R為名義年利率）,Level Coupon Bond: Example,Consider a U.S.

56、government bond with a 6 3/8% coupon that expires in December 2010. The Par Value of the bond is $1,000. Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). Since the coupon rate is 6 3/8%, the payment is $31.875. On January 1, 2006 the size and timing of cash

57、flows are:,Level Coupon Bond: Example,The YTM is 5%.,Bond Pricing with a Spreadsheet,There are specific formulas for finding bond prices and yields on a spreadsheet. PRICE(Settlement,Maturity,Rate,Yld,Redemption, Frequency,Basis) YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis) Settlem

58、ent and maturity need to be actual dates The redemption and Pr need to be given as % of par value Click on the Excel icon for an example.,Bond Pricing with a Spreadsheet,PRICE(Settlement, Maturity, Rate, Yld, Redemption, Frequency, Basis) 天數(shù)/Settlement是證券的成交日 Maturity為有價證券的到期日 Rate為有價證券的年息票利率 Yld為有價

59、證券的年收益率 Redemption為面值$100的有價證券的清償價值 Frequency為年付息次數(shù)(如果按年支付，frequency=1；按半年期支付，frequency=2；按季支付，frequency=4) Basis為日計數(shù)基準類型(0或省略為30/360，1為實際天數(shù)/實際天數(shù)，2為實際天數(shù)/360，3為實際365，4為歐洲30/360),Bond Pricing with a Spreadsheet,YIELD(Settlement, Maturity, Rate, Pr, Redemption, Frequency, Basis) Settlement是證券的成交日 Maturity為有價證券的到期日 Rate為有價證券的年息票利率 Pr為面值$100的有價證券的價格 Redemption為面值$100的有價證券的清償價值 Frequency為年付息次數(shù)(如果按年支付，frequency=1；按半年期支付，frequency=2；按季支付，frequency=4) Basis為日計數(shù)基準類型(0或省略為30/360，1為實際天數(shù)/實際天數(shù)，2為實際天數(shù)/360，3為實際天數(shù)/365，4為歐