5 Time value of moneyCompinding future value FVDiscounting present value PVInterest rate exchange rate between earlier and later money❖FV = PV(1 + r)t❖PV = FV / (1 + r)t▪r = period interest rate, expressed as a decimal▪t = number of periods▪Future value interest factor = (1 + r)t复利终值系数▪1/(1+r) t present value factor 现值系数For a given interest rate – the longer the time period, the lower the present valueFor a given time period – the higher the interest rate, the smaller the present value❖r = (FV / PV)1/t– 1❖t = ln(FV / PV) / ln(1 + r)6 Discounted Cash Flow Valuation❖Future and Present Values of Multiple Cash FlowsYou think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest、You currently have $7,000 in the account、How much will you have in three years? In four years?Today (year 0 CF): 3 N; 8 I/Y; -7,000 PV; CPT FV = 8,817、98 FV = 7000(1、08)3 = 8,817、98 Year 1 CF: 2 N; 8 I/Y; -4000 PV; CPT FV = 4,665、60 FV = 4,000(1、08)2 = 4,665、60Year 2 CF: 1 N; 8 I/Y; -4000 PV; CPT FV = 4,320 FV = 4,000(1、08) = 4,320Year 3 CF: value = 4,000Total value in 3 years = 8817、98 + 4665、60 + 4320 + 4000 = 21,803、58Value at year 4: 1 N; 8 I/Y; -21803、58 PV; CPT FV = 23,547、87You are offered an investment that will pay you $200 in one 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?Point out that the question could also be phrased as “How much is this investment worth?”Calculator:Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178、57Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = -318、88Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = -427、07Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508、41Total PV = 178、57 + 318、88 + 427、07 + 508、41 = 1,432、93现金流都发生在年末 e/g 后付变先付 相当于原来得数据*（1+r ）Valuing Level Cash Flows: Annuities and Perpetuities Annuity 年金连续相同得现金流 – finite series of equal payments that occur atregular intervals 曠緩颟鎵叶洼颁。

If the first payment occurs at the end of the period, it is called an ordinaryannuity 普通年金If the first payment occurs at the beginning of the period, it is called an annuitydue 先付年金Perpetuity – infinite series of equal payments 永续年金只有现值没有终值 期数无限❖ Perpetuity: PV = C / r ❖ Annuities 优先股有永续年金得特征杠杆：自由资金撬动借有资产 美国杠杆高借贷方便FVDN 先付 FVIFA 普通 Growing Perpetuity 增长年金A growing stream of cash flows with a fixed maturity 增长率一定A growing stream of cash flows that lasts forever 用来计算股票得价格 产品未来所有得价值现在得现值 相当于股票现在得价格不就是交易价格The expected dividend next year is $1、30, and dividends are expected to grow at 5% forever 、 If the discount rate is 10%, what is the value of this promised dividend stream?如果现在卖价小于26即可买入❖ Comparing Rates: The Effect of CompoundingEffective Annual Rate (EAR)This is the actual rate paid (or received) after accounting for compounding that occurs during the t r t tr tFVIFA C r r C FV PVIFA C r r C PV ,,1)1()1(11⨯=⎥⎦⎤⎢⎣⎡-+=⨯=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+-=)1(,r FVIFA C FVD n i N +⨯⨯=)1(,r PVIFA C PVD n i +⨯⨯=t t r g C r g C r C PV )1()1()1()1()1(12++⨯++++⨯++=-Λ⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛++--=t r g g r C PV )1()1(1Λ+++⨯+++⨯++=322)1()1()1()1()1(r g C r g C r C PV g r C PV -=00.26$05.10.30.1$=-=PVyear。

If you want to compare two alternative investments with different compounding periods,比较不同期间得rate不能直接比较Annual Percentage Rate（APR❖By definition APR = period rate times the number of periods per yearConsequently, to get the period rate we rearrange the APR equation:▪Period rate = APR / number of periods per yearYou should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rateContinuous Compounding EAR = e q– 1Example: What is the effective annual rate of 7% compounded continuously?EAR = e、07– 1 = 、0725 or 7、25%Pure Discount LoansThe principal amount is repaid at some future date, without any periodic interest payments、纯折现贷款中间不付息例如Treasury billsInterest-Only Loan 每年支付利息到期一次性还本金加最后一次利息Pay interest each period and repay the entire principal at some point in the futureThis cash flow stream is similar to the cash flows on corporate bonds、鰥堯頭缍誤网奮。

❖Loan Types and Loan Amortization 每年偿还利息加一部分本金❖Make single, fixed payment every period❖5,000=C*{[1-(1/1、095)]/0、09}❖C=1285、457 Interest Rates and Bond Valuation现金流折现得三个重要信息现金流折现率期限评估资产价值对资产未来产生得现金流进行估计把所有现金流折现加总得到零时刻得价值（价值大于价格则买进）❖Bonds and Bond ValuationPar value (face value) – the principal 本金Coupon rate – fixed when the bond issued票面利率发行方许诺支付1mrateQuoted1EARm-⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡+=1-EAR)(1mAPR m1Coupon payment – par value * coupon rateMaturity date 到期日票面利息Yield or Yield to maturity – interest rate required in market on a bond 持有到期收益率Bond Value = PV of coupons + PV of parBond Value = PV of annuity + PV of lump sum 本金现值，bond prices decreaseC：cash flow 左侧年金复利系数右侧本金折现由于市场利率就是变化得，发行时候算得就是发行当时得市场利率，而后可能变化因此到期日要瞧到期日利率YTMIf YTM > coupon rate, then par value > bond price The discount provides yield above coupon ratePrice below par value, called a discount bond平价债券par bond 折现出来得价值就就是面值❖Coupon rate小于market折价债券discount bondConsider a bond with a coupon rate of 10% and annual coupons、The par value is $1,000, and the bond has 5 years to maturity、The market interest rate is 11%、What is the value of the bond?B = PV of annuity + PV of lump sumB = 100[1 – 1/(1、11)5] / 、11 + 1,000 / (1、11)5B = 369、59 + 593、45 = 963、04Coupon rate大于market溢价债券premium bondCoupon and Yield❖If YTM = coupon rate, then par value = bond price❖If YTM < coupon rate, then par value < bond price▪Why? Higher coupon rate causes value above par▪Price above par value, called a premium bondInterest Rate Risk（IRR）the risk that arises for bond owners from fluctuating interest rate利率风险市场得利率变化会导致债券价格得变化All other things being equal, the long the time to maturity, the greater the interest rate risk All other things being equal, the lower the coupon rate, the greater the interest rate risk两个除了期限以外全部相同得债券债券条款：❖Debt ：hort long term borrowing 债务融资▪Not an ownership interest▪Creditors do not have voting rightsInterest is considered a cost of doing business and is tax deductibleCreditors have legal recourse if interest or principal payments are missedExcess debt can lead to financial distress and bankruptcy❖Equity ：onership interest 权益融资▪Ownership interestCommon stockholders vote for the board of directors and other issuesDividends are not considered a cost of doing business and are not tax deductibleDividends are not a liability of the firm, and stockholders have no legal recourseif dividends are not paidAn all equity firm can not go bankrupt merely due to debt since it has no debt利息抵税债权人不参与管理借钱方式有两种银行贷款与发股票The Bond Indenture 债券契约Contract between the company and the bondholders that includes▪The basic terms of the bonds 基本条款▪The total amount of bonds issued 发行总额A description of property used as security, if applicable担保说明有形资产做抵押品▪Sinking fund provisions偿债基金条款▪Call provisions 赎回条款▪Details of protective covenants 保护性条款❖价格受条款影响有抵押价格高、偿债基金高、没有赎回条款高、有保护条款高Bond Classifications❖Registered记名债券Bearer Forms 无记名债券国库券认券不认人❖Security担保性质Collateral – secured by financial securities质押债券质押物通常为公司普通股Mortgage – secured by real property, normally land or buildings 抵押债券通常为不动产▪Debentures – unsecured 信用债券无担保债券以企业信用发行Notes – unsecured debt with original maturity less than 10 years ❖Seniority 优先权比次级债券更优先获得补偿❖Bond RatingsHigh GradeMoody’s Aaa and S&P AAA – capacity to pay is extremely strong 标准普尔Moody’s Aa and S&P AA – capacity to pay is very strongMedium Grade 还款能力强但受环境变化Moody’s A and S&P A – capacity to pay is strong, but more susceptible to changes incircumstancesMoody’s Baa and S&P BBB –capacity to pay is adequate, adverse conditions willhave more impact on the firm’s ability to payBond ratings are concerned only with the possibility of defaultLow GradeMoody’s Ba and BS&P BB and BConsidered possible that the capacity to pay will degenerate、还款能力具有投机性Very Low Grade垃圾债券Moody’s C (and below) and S&P C (and below)income bonds with no interest being paid, or•in default with principal and interest in arrears❖Some Different Types of BondsGovernment BondsTreasury SecuritiesFederal government debt 国债T-bills – pure discount bonds with original maturity of one year or lessT-notes – coupon debt with original maturity between one and ten yearsT-bonds – coupon debt with original maturity greater than ten years Municipal SecuritiesDebt of state and local governmentsVarying degrees of default risk, rated similar to corporate debtInterest received is tax-exempt at the federal level 部分地方政府债可免部分税❖Bond MarketsZero Coupon BondsMake no periodic interest payments (coupon rate = 0%)完全无票息得债券The entire yield-to-maturity comes from the difference between the purchase price and the parvalueCannot sell for more than par value 发行价不能高于面值Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)Treasury Bills and principal-only Treasury strips are good examples of zeroes面值发行折扣债券Floating-Rate BondsCoupon rate floats depending on some index valueExamples – adjustable rate mortgages and inflation-linked TreasuriesThere is less price risk with floating rate bondsThe coupon floats, so it is less likely to differ substantially from the yield-to-maturityCoupons may have a “collar” –the rate cannot go above a specified “ceiling” or below a specified “floorOther Bond TypesCat bonds大灾债券Income bonds 收益债券票息随公司收益波动Convertible bonds 可转换债券可转换为股票Put bonds（call）债券持有者有权要求发行方按照一定价格回购以保护持有者利益Bond MarketsTrading volume is much larger than volume in stocksPrimarily over-the-counter transactions with dealers connected electronicallyExtremely large number of bond issues, but generally low daily volume in single issuesMakes getting up-to-date prices difficult, particularly on small company or municipal issues❖Treasury Quotations Highlighted quote in Figure 7、4Nov 2021 8 136:29 136:30 5 4、36到期日票面利率/买卖价差bidask 相比昨天变化分母为32What is the coupon rate on the bond?When does the bond mature?What is the bid price? What does this mean?What is the ask price? What does this mean?How much did the price change from the previous day?What is the yield based on the ask price?Clean price: quoted price净价/报价Dirty price: price actually paid = quoted price plus accrued interest 含息价Example: Consider a T-bond with a 4% semiannual yield and a clean price of $1,282、50:Number of days since last coupon = 61Number of days in the coupon period = 184Accrued interest = (61/184)(、04*1000) = $13、26Dirty price = $1,282、50 + $13、26 = $1,295、76So, you would actually pay $ 1,295、76 for the bond8 Stock Valuation❖ Common Stock ValuationIf you buy a share of stock, you can receive cash in two ways The company pays dividendsYou sell your shares, either to another investor in the market or back to the company As with bonds, the price of the stock is the present value of these expected cash flows 估计股利Estimating Dividends: Special CasesConstant dividend 高速增长不长久 达到稳定The firm will pay a constant dividend foreverThis is like preferred stockThe price is computed using the perpetuity formulaConstant dividend growthThe firm will increase the dividend by a constant percent every period The price is computed using the growing perpetuity model Supernormal growthDividend growth is not consistent initially, but settles down to constant growth eventually The price is computed using a multistage modelZero GrowthIf dividends are expected at regular intervals forever, then this is a perpetuity and the present value of expected future dividends can be found using the perpetuity formula P 0 = D / R 必要报酬率Dividend Growth ModelDividends are expected to grow at a constant percent per period 、P0 = D1 /(1+R) + D2 /(1+R)2 + D3 /(1+R)3 + …P0 = D0(1+g)/(1+R) + D0(1+g)2/(1+R)2 + D0(1+g)3/(1+R)3 + … 股利增长模型 注意： just paid a dividend 与is expected to pay 可以得到得就是D0预期得就是1Nonconstant Growth Problem Statement 不稳定增长Suppose a firm is expected to increase dividends by 20% in one year and by 15% in two years 、 After that, dividends will increase at a rate of 5% per year indefinitely 、 If the last dividend was $1 and the required return is 20%, what is the price of the stock?Remember that we have to find the PV of all expected future dividends 、 Compute the dividends until growth levels offD 1 = 1(1、2) = $1、20D 2 = 1、20(1、15) = $1、38D 3 = 1、38(1、05) = $1、449Find the expected future priceP 2 = D 3 / (R – g) = 1、449 / (、2 - 、05) = 9、66Find the present value of the expected future cash flows g-R D g -R g)1(D P 100=+=P0 = 1、20 / (1、2) + (1、38 + 9、66) / (1、2)2 = 8、67 股利增长模型求R对投资者必要报酬率对企业获取资金得成本普通股得featureVoting Rights 股东投票权Proxy voting代理投票Classes of stock 我国无股票分级一股一票Other RightsShare proportionally in declared dividendsShare proportionally in remaining assets during liquidationPreemptive right – first shot at new stock issue to maintain proportional ownership if desired 优先认股保持比例防止稀释9 Net Present Value and Other Investment Criteriawhen evaluating capital budgeting decision rules:4个标准▪Does xx account for the time value of money?▪Does xx account for the risk of the cash flows?Does xxprovide an indication about the increase in value?Should we consider xx for our primary decision rule?Net Present Value：difference between the market value of a project and its costHow much value is created from undertaking an investment?▪ 1 estimate the expected future cash flows、2 estimate the required return for projects of this risk level、3 find the present value of the cash flows and subtract the initial investment、Decision rule: If the NPV is positive, accept the project NPV正就接受the project is expected to add value to the firm and will therefore increase the wealth of the owners、Since our goal is to increase owner wealth, NPV is a direct measure of how well this project will meet our goal、NPV=-30,000+6,000*[1-(1/1、15)8)]/0、15+(2000/1、158) =-2,422Payback PeriodHow long does it take to get the initial cost back in a nominal sense?ComputationgPDgPg)1(DRg-RDg-Rg)1(DP11+=++==+=Estimate the cash flowsSubtract the future cash flows from the initial cost until the initial investment has been recovereddecision Rule Accept if the payback period is less than some preset limitYear 1: 165,000 – 63,120 = 101,880 still to recover▪Year 2: 101,880 – 70,800 = 31,080 still to recoverYear 3: 31,080 – 91,080 = -60,000 project pays back in year 3Advantages and Disadvantages of Payback▪Easy to understand▪Adjusts for uncertainty of later cash flows▪Biased toward liquidity▪Risk screening deviceDisadvantage▪Ignores the time value of money▪Requires an arbitrary cutoff point▪Ignores cash flows beyond the cutoff dateBiased against long-term projects, such as research and development, and new projectsNPV(short)=-250+100/1、15+200/1、152=-11、81NPV(long)=-250+100*[1-(1/1、154)]/0、15=35、50❖The Discounted PaybackCompute the present value of each cashflow and then determine how long it takesto pay back on a discounted basis箫誰缨臥维璎镇。