第五章货币的时间价值及现金流贴现分析
货币的时间价值和贴现现金流估价概述(PPT 46页)
复利计息次数
m
每年（m=1） 每半年（m=2） 每季（m=4） 每月（m=12） 每天（m=365） 每小时（m=8760）
CF1
1100.00 1102.50 1103.81 1104.71 1105.16 1105.17
i (1 r )m 1 m
实际年利率
0.10 0.1025 0.10381 0.10471 0.10516 0.10517
解答
1.如果我们今天将 $5,000 存在一个支付 10% 利率的账户 里，它需要经过多长时间能增值到 $10,000？
FV C0(1r)T $1,0 00 $0 5,00 (1 0 .1)T 0 (1.10)T $10,0002 $5,000 ln1(.10)T ln2
T ln2 0.69371.27years ln1.(1)0 0.0953
的1美元利息
复利终值
1.基本符号 PV-现值，未来现金流量在今天的价值 FVt-终值，现金流量在未来的价值 r-每期之利率，报酬率，通常1期是1年 t-期数，通常是年数 CF-现金流量
2.复利终值的一般计算公式 FVt = PV(1+ r)t （1＋r)t为普通复利终值系数，经济意义是指现在 的一元t年后的终值
永续年金的现值 C/ r
增长年金与增长型永续年金
增长年金的现值公式推导
利用等比数列的公式：
增长年金与增长型永续年金 增长 年 C 1 金 [1 ( r g - ) 现 g /1 ( r )t值 ]
（P105）
预付年金的终值
FVAt
C 1 r t
年金终值
年金终值系数： 期初年金（annuity due）
第五章 货币的时间价值
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（一）单利终值与现值
单利的计算相对简单，在讨论货币时间价 值时，通常都采用复利计算方法，但对单 利的学习将有助于我们理解复利。
单利条件下，第n期终值的计算公式为：
单利现值：
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（二）复利终值与现值
复利条件下，第n期终值的计算公式为：
与单利比较，复利条件下的资金具有更大的时 间价值，这是由于利息能够产生利息并带来价 值的缘故。而且，随着时间的延长，这两种计 息方式下产生的终值差额还会进一步扩大。
一、终值和现值
终值（future value，FV）是指现在的一笔资金或 一系列收付款项按给定的利息率计算所得到的未来某 个时点的价值，也即是本金和利息之和。 现值（present value，PV）是指未来的一笔资金或 一系列收付款项按给定的利息率计算所得到的现在的 价值，即由终值倒求现值，一般称之为贴现，所使用 的利率又称为贴现率。
由于每次提取的等额准备金类似年金存款，因而同 样可以获得按复利计算的利息，所以债务实际上等 于年金终值，每年提取的偿债基金等于年金，即偿 债基金的计算实际上是年金终值的逆运算。其计算 公式为：
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3. 普通年金现值
普通年金现值是指一定时期内每期期末等 额收付款项的现值之和。普通年金现值的 计算 公式为:
由于不同时点的资金价值不同，在进行价值大 小的比较时，必须将不同时点的资金折算为同 一时点后才可以。因此，预期未来现金流 （cash flow）的时间表和利率水平对金融资 产的定价是至关重要的。
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二、时间轴
时间： 0 10% 1
2
3
货币时间价值与现金流贴现分析
货币时间价值与现金流贴现分析This manuscript was revised by the office on December 22, 2012第七章 货币时间价值与现金流贴现分析小序：资金融通不仅是资金在不同经济主体之间的重新配置，也是资金的跨时配置，因此，在投融资活动中必然要进行资金价值的跨时比较，因此，需要把现在和未来的货币、资金放在一个时间维度中，这就涉及到货币的时间价值问题，要进行终值和现值的计算，要用到各种分析和计算方法。
本章就主要讲解这方面的内容。
主要内容：本章分两节，讨论：货币时间价值（包括终值与现值）；金融投资的分析方法（净现值法、内含报酬率法）。
课时安排：3第一节 货币时间价值一、货币时间价值问题的提出由于在人们观念上利息和利率广泛、深刻的存在，许多与经济有关的事物都被资本化了，因此，人们在经济活动中，特别是在投融资活动中都要进行货币价值的跨期比较。
在考虑到利率的情况下，同量货币或资金在不同时期的价值是不同的，现在一定量的货币或资金比未来同量货币或资金的价值更高，这就是货币的时间价值。
货币的时间价值要通过计算现金流的现值和终值来反映。
二、终值终值是用复利计算的一笔投资在未来某个时间获得的本利和。
比如1万元资金，1年后将能获得多少本利和，这个本利和就是终值。
FV n ＝PV （1＋i ）n (7-1)如果一年为单位，式中的FV n 为第n 年终值，为初始本金，（1＋i ）n 为复利终值系数。
终值系数决定于利率i 和期限n ，它与这两个变量呈正向关系。
利率和期限相同的投资，终值系数也相同。
按照终值公式，利息为FV n －PV ＝PV[（1＋i ）n －1]现实中，人们习惯用年度百分率来表示利率，但许多借贷的计息间隔期并非一年，有按月的，有按季的，这就形成了不同借贷1年内计息次数的差别。
这种差别，造成实际年利率的差别，从而也就造成了终值的差别。
所以，计息次数也是影响终值的一个变量。
理财中的时间价值：了解现金流量和贴现率
理财中的时间价值：了解现金流量和贴现率在理财中，时间价值是一个至关重要的概念。
时间价值的核心理念是，随着时间的推移，货币的价值会发生变化。
因此，对于理财者来说，了解现金流量和贴现率是至关重要的。
首先，让我们来了解一下现金流量。
现金流量是指在一段时间内或特定时期内，进出账户的现金金额。
在财务管理中，现金流量是指企业或个人所有现金流入和流出的情况。
现金流量可以分为三个主要类别：现金流入、现金流出和净现金流。
现金流入是指从外部到企业或个人的现金流动，例如收入和借贷。
现金流出是指从企业或个人流出的现金流动，例如支出和还贷款。
净现金流是指现金流入与现金流出之间的差值，可用于评估企业或个人的现金状况。
了解现金流量对于理财者至关重要。
通过了解现金流量，理财者可以掌握自己的收入来源和支出情况。
这有助于制定合理的预算和理财计划。
通过分析现金流量，理财者可以确定自己的财务目标，并采取相应的措施来实现这些目标。
理财者可以通过降低支出、增加收入或合理利用现有资源来改善现金流量状况。
此外，理财者还可以根据现金流量情况来决定投资和借贷的时间点和额度。
接下来，让我们了解一下贴现率。
贴现率是指将未来的现金流量折现到当前的费率。
在金融学中，贴现率是用于计算资金的时间价值的重要指标。
贴现率可以根据投资项目的风险、市场情况和利率环境等因素而变化。
较高的贴现率意味着现金的未来价值较低，较低的贴现率意味着现金的未来价值较高。
对于理财者来说，了解贴现率的重要性是不言而喻的。
贴现率可以帮助理财者评估不同投资机会的潜在回报。
通过将未来的现金流量折现为现值，理财者可以比较不同投资项目的价值。
理财者可以使用贴现率来计算净现值、内部回报率和投资回收期等指标，以指导自己的投资决策。
通过考虑贴现率，理财者可以避免错误估计投资的回报率，从而降低风险并增加收益。
理财中的时间价值是非常重要的。
时间价值的核心思想是，相同金额的现金在不同时间点具有不同的价值。
理财者利用现金流量和贴现率的知识来评估投资机会和制定理财计划。
货币的时间价值与现金流贴现分析
2013-2014(2)
Financial Economics
2
本章主要内容
1. 复利 2. 复利的频率 3. 现值与折现 4. 其他折现现金流决策规则 5. 复合现金流 6. 年金 7. 永续年金 8. 贷款的分期偿还 9. 汇率与货币的时间价值 10.通货膨胀与折现现金流分析 11.税收与投资决策
总结出一点：当期限非常长时，非常小的利率差别将导
致很大的终值变化。
2013-2014(2)
Financial Economics
10
§3 按不同的利率进行再投资
1. 假定你正面临着一项投资决策：你拥有10000美元可用于两 年期的投资。通过分析，你决定投资于银行定期存款（ CDs）。 两年期存款单的年利率为7%，1年期的存款单的利率为6%， 你应该选择哪一种呢？ 2. 在做决定之前，你必须先判定下一年的1年期存单的利率可能 是多少，这称为再投资利率，是指在计划投资的期限内能再次 进行投资的资金获得的利息率。假设你判定在投资率将为每年 8%。 3. 这时你可以用计算终值的方法进行你的投资决策。首先，计算 出每一种投资方式的终值，然后选择两年后所获资金最多的方 案。两年期的存单的终值为： FV=10 000×1.072=11449（美元） 4. 连续两年投资于1年期存单的终值为： FV=10 000×1.06×1.08=11 448（美元） 5. 因此，投资两年期存单，你的获利将略多。
PV=现值，即你账户中的起始金额。这里是 1000美元。 i=利息率，通常以每年的百分比为单位表示。这里是 10%（用小数表示则为0.1）。 n=计算利息的年数。 FV=n年年末的终值。
2013-2014(2) Financial Economics
货币时间价值与现金流贴现分析
货币时间价值与现金流贴现分析1. 引言货币时间价值（Time Value of Money，简称TVM）和现金流贴现分析是金融领域中重要的概念和工具。
TVM是指相同金额的货币在不同时间点的价值是不同的，而现金流贴现分析则是一种计算未来现金流的现值的方法。
本文将介绍货币时间价值的概念、现金流贴现分析的原理和应用，并通过实例解析其在投资决策中的重要性。
2. 货币时间价值货币时间价值的核心思想是时间对货币的影响。
在TVM的框架下，我们假设货币具有时间的属性，即货币在时间上具有价值差异。
这是由于时间的推移会导致货币面临三种主要的影响：利息、通货膨胀和风险。
2.1 利息利息是货币时间价值的主要因素之一。
根据时间价值的观点，同样的金额，如果能够立即使用，就比将来收到同等金额的价值更高。
这是因为如果钱可以立即使用，就可以进行投资和消费。
而将来收到的同等金额，由于时间的推移，会错过投资和消费的机会，因此其价值相对较低。
2.2 通货膨胀通货膨胀也是货币时间价值的影响之一。
通货膨胀是指货币购买力的下降，也就是物价的普遍上涨。
当存在通货膨胀时，同样的金额在未来的价值会比现在的价值要低，因为相同金额的货币在未来可以购买的商品和服务更少。
2.3 风险风险是另一个影响货币时间价值的因素。
风险是指投资的不确定性和可能面临的亏损。
由于风险的存在，人们对未来的现金流更加谨慎，即对未来现金流的价值降低。
3. 现金流贴现分析现金流贴现分析是一种计算未来现金流现值的方法。
在该方法中，通过将未来现金流通过贴现率进行折现，得到其现值。
这是基于货币时间价值的概念，即将来的现金流价值会随着时间推移而降低。
现金流贴现分析的核心思想是将未来的现金流通过贴现率进行折现，以衡量其在当前时间点的价值。
贴现率是一个由多个因素决定的参数，例如风险、市场利率、投资回报等。
通过贴现现金流，可以比较不同时期的现金流的价值，并用于投资决策、计算净现值、评估项目优劣等。
第05讲-现金流量与资金时间价值
m→∞
m→∞
z 总结：
r = m(m i + 1 −1) = mi期
i = (1 + r / m ) m − 1 = (1 + i期 ) m − 1
i期
=
m
i
+1
−1 =
r m
名义利率：非有效利率。是指按单利方法计算的年利息与本金 之比。
实际利率：有效利率。 是指按复利方法计算的年利息与本 金之比。
2.利率（或利息率、利润率等）概念
利率：一定时期内（一年、半年、月、季度，即一 个计息期），所得的利息额与借贷金额(本金)之比。
利率＝
期利息 本金
×100 %
， 即， i = R期 P
上式表明，利率是单位本金经过一个计息周期后的 增殖额。 （年利率、半年利率、月利率，……）
如果将一笔资金存人银行，这笔资金就称 为本金。经过一段时间之后，储户可在本金之 外再得到一笔利息，这一过程可表示为：
记为：CFt（Cash flow） 记为：COt（Cash outflow） 记为：CIt（Cash inflow） 记为：NCFt（Net cash flow）
2.现金流量图的绘制 现金流量图是表示项目系统在整个寿命周期内各时间点的
现金流入和现金流出状况的一种图示。
①现金流量图的构成：横轴（代表时间）
1.00
12.683
52
12.00 (已知)
0.2308
12.736
365
12.00 (已知)
0.03288
12.748
连续计息
∞
12.00 (已知)
→0
12.750
从表中可知，复利计息周期越短，年名义利率与年实际利 率差别越大，年实际利率越高。
《金融学》答案第四章 货币的时间价值与现金流贴现分析
CHAPTER 4THE TIME VALUE OF MONEY AND DISCOUNTED CASH FLOW ANALYSISObjectives∙To explain the concepts of compounding and discounting, future value and present value.∙To show how these concepts are applied to making financial decisions.Outline4.1Compounding4.2The Frequency of Compounding4.3Present Value and Discounting4.4Alternative Discounted Cash Flow Decision Rules4.5Multiple Cash Flows4.6Annuities4.7Perpetual Annuities4.8Loan Amortization4.9Exchange Rates and Time Value of Money4.10Inflation and Discounted Cash Flow Analysis4.11Taxes and Investment DecisionsSummary∙Compounding is the process of going from present value (PV) to future value (FV). The future value of $1 earning interest at rate i per period for n periods is (1+i)n.∙Discounting is finding the present value of some future amount. The present value of $1 discounted at rate i per period for n periods is 1/(1+i)n.∙One can make financial decisions by comparing the present values of streams of expected future cash flows resulting from alternative courses of action. The present value of cash inflows less the present value of cash outflows is called net present value (NPV). If a course of action has a positive NPV, it is worth undertaking.∙In any time value of money calculation, the cash flows and the interest rate must be denominated in the same currency.∙Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.How to Do TVM Calculations in MS ExcelAssume you have the following cash flows set up in a spreadsheet:A B1t CF20-1003150426053706NPV7IRRMove the cursor to cell B6 in the spreadsheet. Click the function wizard f x in the tool bar and when a menu appears, select financial and then NPV. Then follow the instructions for inputting the discount rate and cash flows. You can input the column of cash flows by selecting and moving it with your mouse. Ultimately cell B6should contain the following:=NPV(0.1,B3:B5)+B2The first variable in parenthesis is the discount rate. Make sure to input the discount rate as a decimal fraction (i.e., 10% is .1). Note that the NPV function in Excel treats the cash flows as occurring at the end of each period, and therefore the initial cash flow of 100 in cell B2 is added after the closing parenthesis. When you hit the ENTER key, the result should be $47.63.Now move the cursor to cell B7to compute IRR. This time select IRR from the list of financial functions appearing in the menu. Ultimately cell B7 should contain the following:=IRR(B2:B5)When you hit the ENTER key, the result should be 34%.Your spreadsheet should look like this when you have finished:A B1t CF20-1003150426053706NPV47.637IRR34%Solutions to Problems at End of Chapter1.If you invest $1000 today at an interest rate of 10% per year, how much will you have 20 years from now,assuming no withdrawals in the interim?2. a. If you invest $100 every year for the next 20 years, starting one year from today and you earninterest of 10% per year, how much will you have at the end of the 20 years?b.How much must you invest each year if you want to have $50,000 at the end of the 20 years?3.What is the present value of the following cash flows at an interest rate of 10% per year?a.$100 received five years from now.b.$100 received 60 years from now.c.$100 received each year beginning one year from now and ending 10 years from now.d.$100 received each year for 10 years beginning now.e.$100 each year beginning one year from now and continuing forever.e.PV = $100 = $1,000.104.You want to establish a “wasting” fund which will provide you with $1000 per year for four years, at which time the fund will be exhausted. How much must you put in the fund now if you can earn 10% interest per year?SOLUTION:5.You take a one-year installment loan of $1000 at an interest rate of 12% per year (1% per month) to be repaid in 12 equal monthly payments.a.What is the monthly payment?b.What is the total amount of interest paid over the 12-month term of the loan?SOLUTION:b. 12 x $88.85 - $1,000 = $66.206.You are taking out a $100,000 mortgage loan to be repaid over 25 years in 300 monthly payments.a.If the interest rate is 16% per year what is the amount of the monthly payment?b.If you can only afford to pay $1000 per month, how large a loan could you take?c.If you can afford to pay $1500 per month and need to borrow $100,000, how many months would it taketo pay off the mortgage?d.If you can pay $1500 per month, need to borrow $100,000, and want a 25 year mortgage, what is thehighest interest rate you can pay?SOLUTION:a.Note: Do not round off the interest rate when computing the monthly rate or you will not get the same answerreported here. Divide 16 by 12 and then press the i key.b.Note: You must input PMT and PV with opposite signs.c.Note: You must input PMT and PV with opposite signs.7.In 1626 Peter Minuit purchased Manhattan Island from the Native Americans for about $24 worth of trinkets. If the tribe had taken cash instead and invested it to earn 6% per year compounded annually, how much would the Indians have had in 1986, 360 years later?SOLUTION:8.You win a $1 million lottery which pays you $50,000 per year for 20 years, beginning one year from now. How much is your prize really worth assuming an interest rate of 8% per year?SOLUTION:9.Your great-aunt left you $20,000 when she died. You can invest the money to earn 12% per year. If you spend $3,540 per year out of this inheritance, how long will the money last?SOLUTION:10.You borrow $100,000 from a bank for 30 years at an APR of 10.5%. What is the monthly payment? If you must pay two points up front, meaning that you only get $98,000 from the bank, what is the true APR on the mortgage loan?SOLUTION:If you must pay 2 points up front, the bank is in effect lending you only $98,000. Keying in 98000 as PV and computing i, we get:11.Suppose that the mortgage loan described in question 10 is a one-year adjustable rate mortgage (ARM), which means that the 10.5% interest applies for only the first year. If the interest rate goes up to 12% in the second year of the loan, what will your new monthly payment be?SOLUTION:Step 2 is to compute the new monthly payment at an interest rate of 1% per month:12.You just received a gift of $500 from your grandmother and you are thinking about saving this money for graduation which is four years away. You have your choice between Bank A which is paying 7% for one-year deposits and Bank B which is paying 6% on one-year deposits. Each bank compounds interest annually. What is the future value of your savings one year from today if you save your money in Bank A? Bank B? Which is the better decision? What savings decision will most individuals make? What likely reaction will Bank B have? SOLUTION:$500 x (1.07) = $535Formula:$500 x (1.06) = $530a.You will decide to save your money in Bank A because you will have more money at the end of the year. Youmade an extra $5 because of your savings decision. That is an increase in value of 1%. Because interestcompounded only once per year and your money was left in the account for only one year, the increase in value is strictly due to the 1% difference in interest rates.b.Most individuals will make the same decision and eventually Bank B will have to raise its rates. However, it isalso possible that Bank A is paying a high rate just to attract depositors even though this rate is not profitable for the bank. Eventually Bank A will have to lower its rate to Bank B’s rate in order to make money.13.Sue Consultant has just been given a bonus of $2,500 by her employer. She is thinking about using the money to start saving for the future. She can invest to earn an annual rate of interest of 10%.a.According to the Rule of 72, approximately how long will it take for Sue to increase her wealth to $5,000?b.Exactly how long does it actually take?SOLUTION:a.According to the Rule of 72: n = 72/10 = 7.2 yearsIt will take approximately 7.2 years for Sue’s $2,500 to double to $5,000 at 10% interest.b.At 10% interestFormula:$2,500 x (1.10)n = $5,000Hence, (1.10)n = 2.0n log 1.10 = log 2.0n = .693147 = 7.27 Years.095310rry’s bank account has a “floating” interest rate on certain deposits. Every year the interest rate is adjusted. Larry deposited $20,000 three years ago, when interest rates were 7% (annual compounding). Last year the rate was only 6%, and this year the rate fell again to 5%. How much will be in his account at the end of this year?SOLUTION:$20,000 x 1.07 x 1.06 x 1.05 = $23,818.2015.You have your choice between investing in a bank savings account which pays 8% compounded annually (BankAnnual) and one which pays 7.5% compounded daily (BankDaily).a.Based on effective annual rates, which bank would you prefer?b.Suppose BankAnnual is only offering one-year Certificates of Deposit and if you withdraw your moneyearly you lose all interest. How would you evaluate this additional piece of information when making your decision?SOLUTION:a.Effective Annual Rate: BankAnnual = 8%.Effective Annual Rate BankDaily = [1 + .075]365 - 1 = .07788 = 7.788%365Based on effective annual rates, you would prefer BankAnnual (you will earn more money.)b.If BankAnnual’s 8% annual return is conditioned upon leaving the money in for one full year, I would need tobe sure that I did not need my money within the one year period. If I were unsure of when I might need the money, it might be safer to go for BankDaily. The option to withdraw my money whenever I might need it will cost me the potential difference in interest:FV (BankAnnual) = $1,000 x 1.08 = $1,080FV (BankDaily) = $1,000 x 1.07788 = $1,077.88Difference = $2.12.16.What are the effective annual rates of the following:a.12% APR compounded monthly?b.10% APR compounded annually?c.6% APR compounded daily?SOLUTION:Effective Annual Rate (EFF) = [1 + APR] m - 1ma.(1 + .12)12 - 1 = .1268 = 12.68%12b.(1 + .10)- 1 = .10 = 10%1c.(1 + .06)365 - 1 = .0618 = 6.18%36517.Harry promises that an investment in his firm will double in six years. Interest is assumed to be paid quarterly and reinvested. What effective annual yield does this represent?EAR=(1.029302)4-1=12.25%18.Suppose you know that you will need $2,500 two years from now in order to make a down payment on a car.a.BankOne is offering 4% interest (compounded annually) for two-year accounts, and BankTwo is offering4.5% (compounded annually) for two-year accounts. If you know you need $2,500 two years from today,how much will you need to invest in BankOne to reach your goal? Alternatively, how much will you need to invest in BankTwo? Which Bank account do you prefer?b.Now suppose you do not need the money for three years, how much will you need to deposit today inBankOne? BankTwo?SOLUTION:PV = $2,500= $2,311.39(1.04)2PV = $2,500= $2,289.32(1.045)2You would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars today in order to reach your goal of $2,500 two years from today.b.PV = $2,500= $2,222.49(1.04)3PV = $2,500= $2,190.74(1.045)3Again, you would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars today in order to reach your goal of $2,500 three years from today.19.Lucky Lynn has a choice between receiving $1,000 from her great-uncle one year from today or $900 from her great-aunt today. She believes she could invest the $900 at a one-year return of 12%.a.What is the future value of the gift from her great-uncle upon receipt? From her great-aunt?b.Which gift should she choose?c.How does your answer change if you believed she could invest the $900 from her great-aunt at only 10%?At what rate is she indifferent?SOLUTION:a. Future Value of gift from great-uncle is simply equal to what she will receive one year from today ($1000). Sheearns no interest as she doesn’t receive the money until next year.b. Future Value of gift from great-aunt: $900 x (1.12) = $1,008.c. She should choose the gift from her great-aunt because it has future value of $1008 one year from today. Thegift from her great-uncle has a future value of $1,000. This assumes that she will able to earn 12% interest on the $900 deposited at the bank today.d. If she could invest the money at only 10%, the future value of her investment from her great-aunt would only be$990: $900 x (1.10) = $990. Therefore she would choose the $1,000 one year from today. Lucky Lynn would be indifferent at an annual interest rate of 11.11%:$1000 = $900 or (1+i) = 1,000 = 1.1111(1+i)900i = .1111 = 11.11%20.As manager of short-term projects, you are trying to decide whether or not to invest in a short-term project that pays one cash flow of $1,000 one year from today. The total cost of the project is $950. Your alternative investment is to deposit the money in a one-year bank Certificate of Deposit which will pay 4% compounded annually.a.Assuming the cash flow of $1,000 is guaranteed (there is no risk you will not receive it) what would be alogical discount rate to use to determine the present value of the cash flows of the project?b.What is the present value of the project if you discount the cash flow at 4% per year? What is the netpresent value of that investment? Should you invest in the project?c.What would you do if the bank increases its quoted rate on one-year CDs to 5.5%?d.At what bank one-year CD rate would you be indifferent between the two investments?SOLUTION:a.Because alternative investments are earning 4%, a logical choice would be to discount the project’s cash flowsat 4%. This is because 4% can be considered as your opportunity cost for taking the project; hence, it is your cost of funds.b.Present Value of Project Cash Flows:PV = $1,000= $961.54(1.04)The net present value of the project = $961.54 - $950 (cost) = $11.54The net present value is positive so you should go ahead and invest in the project.c.If the bank increased its one-year CD rate to 5.5%, then the present value changes to:PV = $1,000= $947.87(1.055)Now the net present value is negative: $947.87 - $950 = - $2.13. Therefore you would not want to invest in the project.d.You would be indifferent between the two investments when the bank is paying the following one-year interestrate:$1,000 = $950 hence i = 5.26%(1+i)21.Calculate the net present value of the following cash flows: you invest $2,000 today and receive $200 one year from now, $800 two years from now, and $1,000 a year for 10 years starting four years from now. Assume that the interest rate is 8%.SOLUTION:Since there are a number of different cash flows, it is easiest to do this problem using cash flow keys on the calculator:22.Your cousin has asked for your advice on whether or not to buy a bond for $995 which will make one payment of $1,200 five years from today or invest in a local bank account.a.What is the internal rate of return on the bond’s cash flows? What additional information do you need tomake a choice?b.What advice would you give her if you learned the bank is paying 3.5% per year for five years(compounded annually?)c.How would your advice change if the bank were paying 5% annually for five years? If the price of thebond were $900 and the bank pays 5% annually?SOLUTION:a.$995 x (1+i)5 = $1,200.(1+i)5 = $1,200$995Take 5th root of both sides:(1+i) =1.0382i = .0382 = 3.82%In order to make a choice, you need to know what interest rate is being offered by the local bank.b.Upon learning that the bank is paying 3.5%, you would tell her to choose the bond because it is earning a higherrate of return of 3.82% .c.If the bank were paying 5% per year, you would tell her to deposit her money in the bank. She would earn ahigher rate of return.5.92% is higher than the rate the bank is paying (5%); hence, she should choose to buy the bond.23.You and your sister have just inherited $300 and a US savings bond from your great-grandfather who had left them in a safe deposit box. Because you are the oldest, you get to choose whether you want the cash or the bond. The bond has only four years left to maturity at which time it will pay the holder $500.a.If you took the $300 today and invested it at an interest rate 6% per year, how long (in years) would ittake for your $300 to grow to $500? (Hint: you want to solve for n or number of periods. Given these circumstances, which are you going to choose?b.Would your answer change if you could invest the $300 at 10% per year? At 15% per year? What otherDecision Rules could you use to analyze this decision?SOLUTION:a.$300 x (1.06)n = $500(1.06)n = 1.6667n log 1.06 = log 1.6667n = .510845 = 8.77 Years.0582689You would choose the bond because it will increase in value to $500 in 4 years. If you tookthe $300 today, it would take more than 8 years to grow to $500.b.You could also analyze this decision by computing the NPV of the bond investment at the different interest rates:In the calculations of the NPV, $300 can be considered your “cost” for acquiring the bond since you will give up $300 in cash by choosing the bond. Note that the first two interest rates give positive NPVs for the bond, i.e. you should go for the bond, while the last NPV is negative, hence choose the cash instead. These results confirm the previous method’s results.24.Suppose you have three personal loans outstanding to your friend Elizabeth. A payment of $1,000 is due today, a $500 payment is due one year from now and a $250 payment is due two years from now. You would like to consolidate the three loans into one, with 36 equal monthly payments, beginning one month from today. Assume the agreed interest rate is 8% (effective annual rate) per year.a.What is the annual percentage rate you will be paying?b.How large will the new monthly payment be?SOLUTION:a.To find the APR, you must first compute the monthly interest rate that corresponds to an effective annual rate of8% and then multiply it by 12:1.08 = (1+ i)12Take 12th root of both sides:1.006434 = 1+ ii = .006434 or .6434% per monthOr using the financial calculator:b.The method is to first compute the PV of the 3 loans and then compute a 36 month annuity payment with thesame PV. Most financial calculators have keys which allow you to enter several cash flows at once. This approach will give the user the PV of the 3 loans.Note: The APR used to discount the cash flows is the effective rate in this case, because this method is assuming annual compounding.25.As CEO of ToysRFun, you are offered the chance to participate, without initial charge, in a project that produces cash flows of $5,000 at the end of the first period, $4,000 at the end of the next period and a loss of $11,000 at the end of the third and final year.a.What is the net present value if the relevant discount rate (the company’s cost of capital) is 10%?b.Would you accept the offer?c.What is the internal rate of return? Can you explain why you would reject a project which has aninternal rate of return greater than its cost of capital?SOLUTION:At 10% discount rate:Net Present Value = - 0 + $5,000 + $4,000 - $11,000 = - 413.22(1.10)(1.10)2 (1.10)3c.This example is a project with cash flows that begin positive and then turn negative--it is like a loan. The 13.6% IRR is therefore like an interest rate on that loan. The opportunity to take a loan at 13.6% when the cost of capital is only 10% is not worthwhile.26.You must pay a creditor $6,000 one year from now, $5,000 two years from now, $4,000 three years from now, $2,000 four years from now, and a final $1,000 five years from now. You would like to restructure the loan into five equal annual payments due at the end of each year. If the agreed interest rate is 6% compounded annually, what is the payment?SOLUTION:Since there are a number of different cash flows, it is easiest to do the first step of this problem using cash flow keys on the calculator. To find the present value of the current loan payments:27.Find the future value of the following ordinary annuities (payments begin one year from today and all interest rates compound annually):a.$100 per year for 10 years at 9%.b.$500 per year for 8 years at 15%.c.$800 per year for 20 years at 7%.d.$1,000 per year for 5 years at 0%.e.Now find the present values of the annuities in a-d.f.What is the relationship between present values and future values?SOLUTION:Future Value of Annuity:e.f.The relationship between present value and future value is the following:FV = PV x (1+i)n28.Suppose you will need $50,000 ten years from now. You plan to make seven equal annual deposits beginning three years from today in an account that yields 11% compounded annually. How large should the annual deposit be?SOLUTION:You will be making 7 payments beginning 3 years from today. So, we need to find the value of an immediate annuity with 7 payments whose FV is $50,000:29.Suppose an investment offers $100 per year for five years at 5% beginning one year from today.a.What is the present value? How does the present value calculation change if one additional payment isadded today?b.What is the future value of this ordinary annuity? How does the future value change if one additionalpayment is added today?SOLUTION:$100 x [(1.05)5] - 1 = $552.56.05If you were to add one additional payment of $100 today, the future value would increase by:$100 x (1.05)5 = $127.63. Total future value = $552.56 + $127.63 = $680.19.Another way to do it would be to use the BGN mode for 5 payments of $100 at 5%, find the future value of that, and then add $100. The same $680.19 is obtained.30.You are buying a $20,000 car. The dealer offers you two alternatives: (1) pay the full $20,000 purchase price and finance it with a loan at 4.0% APR over 3 years or (2) receive $1,500 cash back and finance the rest at a bank rate of 9.5% APR. Both loans have monthly payments over three years. Which should you choose? SOLUTION:31.You are looking to buy a sports car costing $23,000. One dealer is offering a special reduced financing rate of 2.9% APR on new car purchases for three year loans, with monthly payments. A second dealer is offering a cash rebate. Any customer taking the cash rebate would of course be ineligible for the special loan rate and would have to borrow the balance of the purchase price from the local bank at the 9%annual rate. How large must the cash rebate be on this $23,000 car to entice a customer away from the dealer who is offering the special 2.9% financing?SOLUTION:of the 2.9% financing.32.Show proof that investing $475.48 today at 10% allows you to withdraw $150 at the end of each of the next 4 years and have nothing remaining.SOLUTION:You deposit $475.48 and earn 10% interest after one year. Then you withdraw $150. The table shows what happensAnother way to do it is simply to compute the PV of the $150 annual withdrawals at 10% : it turns out to be exactly $475.48, hence both amounts are equal.33.As a pension manager, you are considering investing in a preferred stock which pays $5,000,000 per year forever beginning one year from now. If your alternative investment choice is yielding 10% per year, what is the present value of this investment? What is the highest price you would be willing to pay for this investment? If you paid this price, what would be the dividend yield on this investment?SOLUTION:Present Value of Investment:PV = $5,000,000 = $50,000,000.10Highest price you would be willing to pay is $50,000,000.Dividend yield = $5,000,000 = 10%.$50,000,00034. A new lottery game offers a choice for the grand prize winner. You can receive either a lump sum of $1,000,000 immediately or a perpetuity of $100,000 per year forever, with the first payment today. (If you die, your estate will still continue to receive payments). If the relevant interest rate is 9.5% compounded annually, what is the difference in value between the two prizes?SOLUTION:The present value of the perpetuity assuming that payments begin at the end of the year is:$100,000/.095 = $1,052,631.58If the payments begin immediately, you need to add the first payment. $100,000 + 1,052,632 = $1,152,632.So the annuity has a PV which is greater than the lump sum by $152,632.35.Find the future value of a $1,000 lump sum investment under the following compounding assumptions:a.7% compounded annually for 10 yearsb.7% compounded semiannually for 10 yearsc.7% compounded monthly for 10 yearsd.7% compounded daily for 10 yearse.7% compounded continuously for 10 yearsa.$1,000 x (1.07)10 = $1,967.15b.$1,000 x (1.035)20 = $1,989.79c.$1,000 x (1.0058)120 = $2,009.66d.$1,000 x (1.0019178)3650 = $2,013.62e.$1,000 x e.07x10 = $2,013.7536.Sammy Jo charged $1,000 worth of merchandise one year ago on her MasterCard which has a stated interest rate of 18% APR compounded monthly. She made 12 regular monthly payments of $50, at the end of each month, and refrained from using the card for the past year. How much does she still owe? SOLUTION:Sammy Jo has taken a $1,000 loan at 1.5% per month and is paying it off in monthly installments of $50. We could work out the amortization schedule to find out how much she still owes after 12 payments, but a shortcut on the financial calculator is to solve for FV as follows:37.Suppose you are considering borrowing $120,000 to finance your dream house. The annual percentage rate is 9% and payments are made monthly,a.If the mortgage has a 30 year amortization schedule, what are the monthly payments?b.What effective annual rate would you be paying?c.How do your answers to parts a and b change if the loan amortizes over 15 years rather than 30?EFF = [1 + .09]1238.Suppose last year you took out the loan described in problem #37a. Now interest rates have declined to 8% per year. Assume there will be no refinancing fees.a.What is the remaining balance of your current mortgage after 12 payments?b.What would be your payment if you refinanced your mortgage at the lower rate for 29 years? SOLUTION:Exchange Rates and the Time Value of Money39.The exchange rate between the pound sterling and the dollar is currently $1.50 per pound, the dollar interest rate is 7% per year, and the pound interest rate is 9% per year. You have $100,000 in a one-year account that allows you to choose between either currency, and it pays the corresponding interest rate.a.If you expect the dollar/pound exchange rate to be $1.40 per pound a year from now and are indifferentto risk, which currency should you choose?b.What is the “break-even” value of the dollar/pound exchange rate one year from now?SOLUTION:a.You could invest $1 today in dollar-denominated bonds and have $1.07 one year from now. Or you couldconvert the dollar today into 2/3 (i.e., 1/1.5) of a pound and invest in pound-denominated bonds to have .726667(i.e., 2/3 x 1.09) pounds one year from now. At an exchange rate of $1.4 per pound, this would yield 0.726667(1.4) = $1.017 (this is lower than $1.07), so you would choose the dollar currency.b.For you to break-even the .726667 pounds would have to be worth $1.07 one year from now, so the break-evenexchange rate is $1.07/.726667 or $1.4725 per pound. So for exchange rates lower than $1.4725 per pound one year from now, the dollar currency will give a better return.。
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( 1 r )n 1
P M T [
]
r
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第五章货币时间价值及现金流贴现分析
（3）普通年金的现值
P V O AP M T(11 1r(1 1 r)2(1 1 r)3(1 1 r)4L(1 1 r)n) P M T[1(1r)n] r
例6：为了能在今后3年每年年末得到100元，以年利率5%计算， 当前需要投入多少资金？
FP(1r)n
例 1： 一 笔 年 利 率 为6%、 期 限 为5年 、 金 额 为 100000元 的 贷 款 ， 请 分 别 用 单 利 和 复 利 计 算 利息和本利和。 解：单利方法：
C=P r n 100000 6% 5 30000元 F P (1 r n) 100000 (1 6% 5) 130000元
三、多重现金流及年金 （一）多重现金流的现值与终值
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第五章货币时间价值及现金流贴现分析
1.多重现金流的终值 计算每一笔现金流的终值，然后加总求和，即多重现
金流终值等于每笔现金流终值之和。
F C 0 ( 1 r ) n C 1 ( 1 r ) n 1 C 2 ( 1 r ) n 2 C 3 ( 1 r ) n 3 L C n 1 ( 1 r ) 1 C n
解：（1）方法一：
先计算名义终值：FV=100(1+5%)45 898.50元
再计算物价水平： inf=(1&#平
898.5 3.7816
237.6元
(2)方法二：
先计算实际利率：i=r-p=5%-3%=2% 或 i=1 5% 1 1.9417% 1 3%
方 案 2的 报 酬 率 已 知 为 5 %
方 案 1的 内 涵 报 酬 率 或 到 期 收 益 率 计 算 方 法 为
F P (1 IR R ) 3
1 0 0 0 9 0 0 (1 IR R ) 3 接受方案2 （ 4） 投 资 回 收 期 法
IR R 3.57%
方 案 1的 投 资 回 收 期 为 3年 。
实际现金流作为贴现率进行贴现计算现值。
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第五章货币时间价值及现金流贴现分析
例8：你计划在第5年年末购买一辆汽车，其当前价格为15万元， 你打算现在存入银行一笔钱，用于购买汽车。假设银行存款 利率为5%。（1）如果没有通货膨胀预期，问：现在应当存 在银行多少钱？（2）如果通货膨胀预期为3%，现在应当存 入银行多少钱？
（ 2） 终 值 法
方 案 1的 终 值 已 知 ， 需 要 求 方 案 2的 终 值 ， 根 据 现 值 相 等 ， 知 方 案 2中 P=900
F = P ( 1 + r )n 9 0 0 (1 5 % ) n 9 0 0 1 .1 5 8 1 0 4 2 .2 元
接受方案2
（ 3） 内 涵 报 酬 率 法 或 到 期 收 益 率 法
第五章货币的时间价值及现金流贴现分析
一、货币的时间价值及现金流贴现分析 二、现金流贴现分析与投资决策准则 三、多重现金流及年金 四、通胀、税收及不确定性对货币时间价值的影响
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第五章货币时间价值及现金流贴现分析
一、货币时间价值及复利计息
（一）货币时间价值的含义 货币时间价值是指当前拥有一定量的货币比未来拥有的等量货
300000.952500000.907200000.863 28710453501726091320元
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第五章货币时间价值及现金流贴现分析
（二）年金的现值与终值 1.年金：储蓄计划、投资项目和贷款偿付所产生的未 来现金流每期都是相等的，把系列均等的现金流或 付款称为年金。
➢ 普通年金：现金流发生在每期期末。 ➢ 即时年金：现金流发生在每期期初。 ➢ 永续年金：现金流发生的次数趋向于无穷大。
币具有更高的价值。货币时间价值问题有两个： ➢ 为什么处于不同时点的同样数量的货币会具有不同的价值？ ➢ 如何计算处于不同时点的货币资金的价值？
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第五章货币时间价值及现金流贴现分析
（二）单利与复利 1.单利
C Prn F P(1 rn)
2.复利：是将上期利息 计入本金一并计算利息 的一种方法。
1 re
(1 r )m m
re : 有效年利率；r：年利率；m：计息次数
当m无穷大时，为连续复利有：
1 re
lim(1
m
r )m m
er
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第五章货币时间价值及现金流贴现分析
例2：假定你想要向银行申请一笔1年期贷款10000元，银行提 供三种产品供你选择：A产品每月支付一次利息，年利率为 12%;B产品每半年支付一次利息，年利率为12.2%;C产品贷 款到期时一次支付利息和本金，年利率12.5%
复利方法： F=P （ 1+r）n 100000 (1 6% )5 133822.56元
C F P 33822.56元
CFPP((1r)n 1)
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第五章货币时间价值及现金流贴现分析
(三)计息次数
在金融实务中，不管计算次数如何，金融工具的利 息通常以年利率表示。由于计息次数的不同，采 用有效年利率进行比较。年利率与有效年利率之 间存在如下关系：
方案2：投资于年利率为5%的银行存款，r=5%
（1）净现值法：
方案1的现值已知，需要求方案2的现值,根据终值相等，知方案2中F=1000。
PV= 1000 (15%)3
10000.863863元
所以投资于该国债方案的净现值为：NPV 863900 37
拒绝该项目。
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第五章货币时间价值及现金流贴现分析
2.现值与贴现
➢ 现值是在复利计息方式下，未来一定金额按照某一利率计 算出来的现在的价值。
➢ 贴现：现值的计算为贴现，用于计算的利率称为贴现率。 现值的计算又称为现金流贴现分析。
➢ 贴现系数： 现在
1 (1 r ) n
未来
F P
现
值
F/(1+i)
贴现
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第五章货币时间价值及现金流贴现分析
再计算实际终值=100（1+2%）45 100 2.4379 243.79
或：100 (11.9417%)45 100 2.375933 237.60元
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第五章货币时间价值及现金流贴现分析
2.通货膨胀与现值 （1）方法一：对名义现金流，用名义利率作为贴现
率计算现值； （2）方法二：把名义现金流转化为实际现金流，用
i n 0( 1 C i r)i
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第五章货币时间价值及现金流贴现分析
例5：假设你1年后需要3万元，2年后需要5万元，第3年末需要 2万元，如果年利率是5%，为了满足上述要求，你现在需要 将多少钱存在银行？
解：
PV
30000 (15%)
(15050% 00)2
(12050% 00)3
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第五章货币时间价值及现金流贴现分析
例4：p120 假设面额为1000元的3年期国债的销售价格为900元，在其他 可供选择的投资方案中，最好的方案是年利率为5%的银行存 款，问购买国债是否是一个好的投资？
解：
方案1：投资于面额为1000元的3年期国债,F=1000, t=3, PV=900
问：三种产品中哪个产品利率最高？
解 ： re
(1
r )m m
1
A : re
(1
r )m m
1
(1
0.12 )12 12
1 12.68%
B : re
(1
r )m m
1
(1
0.122 )2 2
1 12.57%
C
: re
(1
r )m m
1
(1
0.125) 1 12.5%
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（2）即时年金的终值 ：等于普通 年金的终值乘以（1+r）
F V I A P M T ( 1 r ) [ ( 1 r ) n 1 ] F V O A ( 1 r ) r
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第五章货币时间价值及现金流贴现分析
(3)即时年金的现值：等于普通年金的现值乘以（1+r）
P V IA P M T ( 1 r )[1 ( 1 r ) n ] P V O A ( 1 r ) r
解 ： （ A）F=P(1+i)n100000(16% )51000001.33813380（ 0元 ） （ B）P=(1 Fi)n100000(16 1% )51000000.74774700(元 )
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第五章货币时间价值及现金流贴现分析
（二）投资决策准则 1.净现值法则(NPV)：未来现金流的现值大于初始投资额的项目 是可以接受的。净现值等于所有的未来流入现金的现值减去所有 流出现金现值的差额。如果一个项目的NPV为正，就接受；如果 NPV为负，就拒绝。 2.终值法则：如果一个项目的终值大于其他项目的终值，就可以 对该项目进行投资。 3.内涵报酬率法则（IRR）或到期收益率：是指未来流入的现值 等于现金流出现值的贴现率。也就是说：IRR是使NPV等于0的贴 现率。内涵报酬率法则是：投资于那些内涵报酬率大于资金机会 成本的项目。 4.投资回收期法：即选择投资回收期最短的项目。
四、通货膨胀、税收及不确定性对货币时间价值的影响 （一）通货膨胀与现金流贴现分析 1.通货膨胀与终值 （1）方法一：利用名义利率计算名义终值，然后剔除通货膨胀